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Inquiry Live

2020-09-17T13:15:57Z

Jon Awbrey: update

☞ This page serves as a '''focal node''' for a collection of related resources.

==Participants==

Interested parties may add their names on [[Inquiry Live/Participants|this page]].

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Peirce's law

2015-11-18T16:14:14Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form:

{| align="center" cellpadding="10"
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math>
|}

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

==History==

Here is Peirce's own statement and proof of the law:

{| align="center" cellpadding="8" width="90%"
|
A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

<center>
<math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math>
</center>

This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP 3.384).
|}

Peirce goes on to point out an immediate application of the law:

{| align="center" cellpadding="8" width="90%"
|
From the formula just given, we at once get:

<center>
<math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math>
</center>

where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP 3.384).
|}

'''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math>

==Graphical proof==

Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation.

{| align="center" border="0" cellpadding="10" cellspacing="0"
| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1)
|}

'''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law 1.0 Marquee Title.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference Band Collect p.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]]
|-
| [[Image:Equational Inference Band Quit ((q)).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]]
|-
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]]
|-
| [[Image:Equational Inference Band Delete p.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]]
|-
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (2)
|}

The following animation replays the steps of the proof.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law 2.0 Animation.gif]]
|}
| (3)
|}

==Equational form==

A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible:

{| align="center" cellpadding="10"
| <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math>
|}

===Proof 1===

Given what precedes, it remains to show that:

{| align="center" cellpadding="10"
| <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math>
|}

But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math>

===Proof 2===

Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation:

{| align="center" border="0" cellpadding="10" cellspacing="0"
| [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4)
|}

Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner:

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference Rule Collect p.png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]]
|-
| [[Image:Equational Inference Rule Quit ((q)).png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]]
|-
| [[Image:Equational Inference Rule Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (5)
|}

The following animation replays the steps of the proof.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law Strong Form 2.0 Animation.gif]]
|}
| (6)
|}

==Bibliography==

* [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted (CP 3.359–403), (CE 5, 162–190).

* Peirce, Charles Sanders (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).

* Peirce, Charles Sanders (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki]
* [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/PeircesLaw Peirce's Law], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Peirce's_law Peirce's Law], [http://www.wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Peirce's_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Cybernetics]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Systems]]
[[Category:Graph Theory]]
[[Category:History of Logic]]
[[Category:History of Mathematics]]
[[Category:Inquiry]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Logical Graphs]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Semiotics]]
[[Category:Visualization]]

Logic Live

2015-11-17T15:35:19Z

Jon Awbrey:

☞ This page serves as a '''focal node''' for a collection of related resources.

==Participants==

* Interested parties may add their names on [[Logic Live/Participants|this page]].

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki]
* [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]

Inquiry Live

2015-11-17T15:28:52Z

Jon Awbrey: update

☞ This page serves as a '''focal node''' for a collection of related resources.

==Participants==

Interested parties may add their names on [[Inquiry Live/Participants|this page]].

==Rudiments of organization==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]

'''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content — especially as they develop in time across different environments through interaction with diverse populations — but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.

[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]

Truth theory

2015-11-17T14:50:28Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''truth theory''' or a '''theory of truth''' is a conceptual framework that underlies a particular conception of truth, such as those used in art, ethics, logic, mathematics, philosophy, the sciences, or any discussion that either mentions or makes use of a notion of truth. A truth theory can be anything from an informal theory, based on implicit or tacit ideas, to a formal theory, constructed from explicit axioms and definitions and developed by means of definite rules of inference. The scope of a truth theory can be restricted to tightly-controlled and well-bounded universes of discourse or its horizon may extend to the limits of the human imagination.

==Truth in perspective==

Notions of truth are notoriously difficult to disentangle from many of our most basic concepts — meaning, reality, and values in general, to mention just a few.

The subjects of meaning and truth are commonly treated together, the idea being that a thing must be meaningful before it can be true or false. This association is found in ancient times, and has become standard in modern times under the heading of ''semantics'', especially ''formal semantics'' and ''model theory''. Another association of longstanding interest is the relation between truth and ''logical validity'', "because the fundamental notion of logic is validity and this is definable in terms of truth and falsehood" (Kneale and Kneale, 16). Though not the main subjects of this article, meaning and validity are truth's neighbors, and incidental inquiries of them can serve to cast light on truth's character.

Beyond this minor note of accord, hardly universal, suggesting that meaning is necessary to truth, reflectors on the idea of truth just as quickly disperse into schools of thought that barely comprehend each other's thinking. A few of the more notable points of departure are these:

# One of the first partings of the ways occurs at the watershed between literal and symbolic meanings, leading to a corresponding division in truths. People often speak of truth in art, truth in drama, truth in fiction, human truth, moral, religious, and spiritual truth, along with the difference between truth in principle and truth in practice. These topics demand a perspective on meaning, reality, and truth that looks beyond the bounds of literal truth and the branches of philosophy that are limited to it.
# Merely resolving that meaning precedes truth, logically speaking, only brings up a host of new questions, since the meaning of the word ''meaning'' is notoriously hard to pin down. There are just to start at least two different dimensions of meaning that are commonly recognized, namely, ''connotative meaning'' and ''denotative meaning''.

In one classical formulation, truth is defined as the good of [[logic]], where logic is treated as a [[normative science]], that is, an [[inquiry]] into a ''good'' or a ''value'' that seeks knowledge of it and the means to achieve it. In this scheme of ideas, truth is the positive quality of a sign that indicates the right course of action for reaching a value that we value for its own sake. As such, truth takes its place among justice and beauty, whose normative sciences are ethics and aesthetics, respectively. Viewed in this light, it is pointless to discuss truth in isolation from a frame of reference that encompasses the topics of inquiry, knowledge, logic, meaning, practice, and value, all very broadly conceived.

==Historical overview==

In an ancient fragment of text called the ''Dissoi Logoi'', a writer is evidently trying to prove the impossibility of speaking consistently about truth and falsehood. One of the conundrums put forward to confound the reader cites the case of the verbal form, "I am an initiate", which is true when ''A'' says it but false when ''B'' says it. Escape from befuddlement seems easy enough if one observes that it is not the verbal expression, the sentence, to which the predicates of truth and falsity apply but what the sentence expresses, the proposition that it states. (Cf. Kneale and Kneale, 16). This same tension between strings of characters and their meanings remains with us to this day.

In his early work Περι Ερμηνειαs (''Peri Hermeneias'' or ''On Interpretation'') Aristotle strikes a chord that not only sets the key for a number of philosophical movements down through the ages but supplies the initial motif for many themes in the logic of meaning and truth that are still undergoing active development in our time.

<blockquote>
Words spoken (phoné) are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words (graphomena) are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, ''On Interpretation'', 1.16a4).
</blockquote>

Some of the points to be noted in this passage are these:

# Aristotle employs a distinction in Greek that is drawn between natural or physical signs (semeia) and artificial or cultural signs (symbola).
# The passage mentions three principal domains of elements, namely, the ''objects'' (pragmata), the ''signs'' (semeia, symbola), and the psychological elements (pathemata). The last domain extends over the full range of a human being's affective and cognitive experiences, for brevity summed up as ''ideas'' and ''impressions'', where these words are taken in their broadest conceivable senses.
# This means that the phenomena under investigation have to do with the types of [[three-place relation]]s that conceivably exist among three domains of this sort. As a general rule, three-place relations can be very complex, and a commonly-tried strategy for approaching their complexity is to consider the [[two-place relation]]s that are left when the presence of a selected domain is simply ignored.
# There are two types of two-place relation on the face of the overall three-place relation that Aristotle takes the trouble to mention, namely these:Sign <math>\longrightarrow</math> Idea. Words spoken are signs or symbols of pathemata.Idea <math>\longrightarrow</math> Object. Pathemata are icons (homoiomata) of pragmata.
# More incidentally, but still bearing heavily on many later discussions, Aristotle holds that the relation between writing and speech is analogous to the relation between speech and the realm of experiences, feelings, and thoughts.Writing <math>\longrightarrow</math> Speech. Written words are symbols of spoken words. Speech <math>\longrightarrow</math> Ideation. Spoken words are symbols of impressions.

==Elements of theory==

It is customary in philosophy to refer to a distinctive treatment of a particular subject matter, frequently summed up in a succinctly stated thesis, as a ''theory'', whether or not it qualifies as a theory by strict empirical or logical standards. When there is any risk of confusion, an informal thesis of this kind may be referred to as an ''account'', a ''perspective'', a ''treatment'', or so on, reserving the term ''theory'' for the type of [[formal system]] that serves in logic and science.

Theories of truth can be classified according to the following features:

* Primary subjects. What kinds of things are potentially meaningful enough to be asserted or not, believed or not, or considered true or false?
* Relevant objects. What kinds of things, in addition to primary subjects, are pertinent to deciding whether to assert them or not, believe them or not, or consider them true or false?
* Value predicates. What kinds of things are legitimate to say about primary subjects, either in themselves, or in relation to relevant objects?

In some discussions of meaning and truth that consider forms of expression well beyond the limits of literally-interpreted linguistic forms, potentially meaningful elements are called ''[[representation]]s'', or ''[[sign (semiotics)|signs]]'' for short, taking these words in the broadest conceivable senses.

Most treatments of truth draw an important distinction at this point, though the language in which they draw it may vary. On the one hand there is a type of incomplete sign that is nevertheless said to be true or false of various objects. For example, in logic there are ''terms'' such as "man" or "woman" that are true of some things and false of others, and there are ''predicates'' such as "__is a man" or "__is a woman" that are true or false in the same way. On the other hand there is a type of complete sign that expresses what grammarians traditionally call a ''complete thought''. Here one speaks of ''sentences'' and ''propositions''. Some considerations of truth admit both types of signs, ''terms'' and ''sentences'', while others admit only the bearers of complete thoughts into the arena of judgment. In a number of recent discussions that focus on linguistic analysis, the vehicles of complete thoughts are described as ''truthbearers'', with no intention of prejudging whether they bear truth or falsehood. The things that can be said about any of these representations, signs, or truthbearers are expressed in what most truth theorists describe as ''truth predicates''.

Most inquiries into the character of truth begin with a notion of an informative, meaningful, or significant element, the truth of whose information, meaning, or significance may be put into question and needs to be evaluated. Depending on the context, this element might be called an ''artefact'', ''expression'', ''image'', ''impression'', ''lyric'', ''mark'', ''performance'', ''picture'', ''sentence'', ''sign'', ''string'', ''symbol'', ''text'', ''thought'', ''token'', ''utterance'', ''word'', ''work'', and so on. For the sake of brevity, it is convenient to use the term ''sign'' for any one of these elements. Whatever the case, one has the task of judging whether the bearers of information, meaning, or significance are indeed ''truth-bearers''. This judgment is typically expressed in the form of a specific ''truth predicate'', whose positive application to a sign asserts that the sign is true.

Considered within the broadest horizon, there is little reason to imagine that the process of judging a ''work'', that leads to a predication of false or true, is necessarily amenable to formalization, and it may always remain what is commonly called a ''judgment call''. But there are indeed many well-circumscribed domains where it is useful to consider disciplined forms of evaluation, and the observation of these limits allows for the institution of what is called a ''[[method]]'' of judging truth and falsity.

One of the first questions that can be asked in this setting is about the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says "that's true", is there anything useful at all that can be said in general terms about the relationship between these two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it just an insubstantial echo of the original sign?

Theories of truth may be described according to several dimensions of description that affect the character of the predicate "true". The truth predicates that are used in different theories may be classified by the number of things that have to be mentioned in order to assess the truth of a sign, counting the sign itself as the first thing. In formal logic, this number is called the ''[[arity]]'' of the predicate. The kinds of truth predicates may then be subdivided according to any number of more specific characters that various theorists recognize as important.

# A ''monadic'' truth predicate is one that applies to its main subject ? typically a concrete representation or its abstract content ? independently of reference to anything else. In this case one can say that a truth bearer is true in and of itself.
# A ''dyadic'' truth predicate is one that applies to its main subject only in reference to something else, a second subject. Most commonly, the auxiliary subject is either an ''object'', an ''interpreter'', or a ''language'' to which the representation bears some [[relation (mathematics)|relation]].
# A ''triadic'' truth predicate is one that applies to its main subject only in reference to a second and a third subject. For example, in a pragmatic theory of truth, one has to specify both the object of the sign, and either its interpreter or another sign called the ''interpretant'' before one can say that the sign is true ''of'' its object ''to'' its interpreting agent or sign.

Several qualifications must be kept in mind with respect to any such radically simple scheme of classification, as real practice seldom presents any pure types, and there are settings in which it is useful to speak of a theory of truth that is "almost" ''k''-adic, or that "would be" ''k''-adic if certain details can be abstracted away and neglected in a particular context of discussion. That said, given the generic division of truth predicates according to their arity, further species can be differentiated within each genus according to a number of more refined features.

The truth predicate of interest in a typical [[correspondence theory of truth]] tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is ''true of'' an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an ''[[analogy]]'', an ''[[icon]]'', or a ''[[morphism]]'', whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure.

===Signs===

In some branches of philosophy and fields of science the domain of potentially meaningful entities may include almost any kind of informative or significant element. The generic terms ''sign'' or ''representation'' suffice for these, with the qualification that the terms are used equivocally up and down a full spectrum from the more abstract ''types'' to the more concrete ''tokens'' that are associated with each other. More specifically, the linguistic turn in analytic philosophy begins with a focus on the syntactic character of the ''sentence'', from which is abstracted its meaningful content, referred to as the corresponding ''proposition''. A proposition is the content expressed by a sentence, held in a belief, or affirmed in an assertion or judgment.

''Truthbearer'' is used by a number of writers to refer to any entity that can be judged true or false. The term ''truthbearer'' may be applied to propositions, sentences, statements, ideas, beliefs, and judgments. Some writers exclude one or more of these categories, or argue that some of them are true (or false) only in a derivative sense. Other writers may add additional entities to the list.

Truthbearers typically have two possible values, true or false. Fictional forms of expression are usually regarded as false if interpreted literally, but may be said to bear a species of truth if interpreted suitably. Still other truthbearers may be judged true or false to a greater or lesser degree.

===Higher order signs===

As ''predicate terms'', most discussions of truth allow for a number of phrases that are used to say in what ways signs or sentences or their abstract senses are regarded as true, either by themselves or in relation to other things. Theorists who admit the term call these phrases ''[[truth predicate]]s''. A truth predicate that is used to ascribe truth to something, in and of itself, in effect treating truth as an [[intrinsic property (philosophy)|intrinsic property]] of the thing, is called a ''one-place'' or ''monadic'' truth predicate. Other forms of truth predicates may be used to say that something is true in relation to specified numbers and types of other things. These are called ''many-place'' or ''polyadic'' truth predicates.

In ordinary parlance, the things that one says about a subject are expressed in predicates. If one says that a sentence is true, then one is predicating truth of that sentence. Is this the same thing as asserting the sentence? This question serves as useful touchstone for sorting out some of the theories of truth.

===Propositional attitudes===

<blockquote>
What sort of name shall we give to verbs like 'believe' and 'wish' and so forth? I should be inclined to call them 'propositional verbs'. This is merely a suggested name for convenience, because they are verbs which have the ''form'' of relating an object to a proposition. As I have been explaining, that is not what they really do, but it is convenient to call them propositional verbs. Of course you might call them 'attitudes', but I should not like that because it is a psychological term, and although all the instances in our experience are psychological, there is no reason to suppose that all the verbs I am talking of are psychological. There is never any reason to suppose that sort of thing. (Russell 1918, 227).
</blockquote>

What a proposition is, is one thing. How we feel about it, or how we regard it, is another. We can accept it, assert it, believe it, command it, contest it, declare it, deny it, doubt it, enjoin it, exclaim it, expect it, imagine it, intend it, know it, observe it, prove it, question it, suggest it, or wish it were so. Different attitudes toward propositions are called ''propositional attitudes'', and they are also discussed under the headings of ''intentionality'' and ''linguistic modality''. The formal properties of verbs like ''assert'', ''believe'', ''command'', ''consider'', ''deny'', ''doubt'', ''hunt'', ''imagine'', ''judge'', ''know'', ''want'', ''wish'', and a host of others, are studied under these headings by linguists and logicians alike.

Many problematic situations in real life arise from the circumstance that many different propositions in many different modalities are in the air at once. In order to compare propositions of different colors and flavors, as it were, we have no basis for comparison but to examine the underlying propositions themselves. Thus we are brought back to matters of language and logic. Despite the name, propositional attitudes are not regarded as psychological attitudes proper, since the formal disciplines of linguistics and logic are concerned with nothing more concrete than what can be said in general about their formal properties and their patterns of interaction.

The variety of attitudes that a proposer can bear toward a single proposition is a critical factor in evaluating its truth. One topic of central concern is the relation between the modalities of assertion and belief, especially when viewed in the light of the proposer's intentions. For example, we frequently find ourselves faced with the question of whether a person's assertions conform to his or her beliefs. Discrepancies here can occur for many reasons, but when the departure of assertion from belief is intentional, we usually call that a ''lie''.

Other comparisons of multiple modalities that frequently arise are the relationships between belief and knowledge and the discrepancies that occur among observations, expectations, and intentions. Deviations of observations from expectations are commonly perceived as ''surprises'', phenomena that call for ''explanations'' to reduce the shock of amazement. Deviations of observations from intentions are commonly experienced as ''problems'', situations that call for plans of action to reduce the drive of dissatisfaction. Either type of discrepancy forms an impulse to ''[[inquiry]]'' (Awbrey and Awbrey 1995).

===Reflection and quotation===

The study of propositional attitudes is no sooner begun than it leads to the all-important philosophical distinction between (1) using a meaning-bearer to bear its meaning in an active manner and (2) mentioning a meaning-bearer in a form that keeps its meaning in a more inert or inhibited state. The reasons for doing the latter are various, but involve the need to reflect on a potential meaning, to compare and contrast it with others, to criticize and evaluate both its logical implications and its practical consequences, all before deciding whether to put its meaning into action or not.

The word “quote” derives from the Latin verb ''quotare'', which refers to the practice of numbering references and referring to pieces of text by marking their numbers. There is a certain aesthetic distance involved in this practice, and it leads, if only for moments at a time, to viewing each piece of text as a string of characters that bears its own litter of meanings, but meanings to be reflected on and critically compared with others, both in and out of their litter. It is hardly an accident, then, that matters of Gödel numbers, quotation, and reflection are bound up with each other in mathematical logic and computation theory.

==Varieties of truth theory==

===Nominal truth theories===

A ''nominal truth'' theory is defined by the axiom that the concept ''truth'' is a mere name. In traditional systems of logic, a concept is always a symbol, specifically, a mental symbol, and so the word ''mere'' in the nominal axiom says that ''truth'' is nothing more than a symbol. One of the aims of nominal philosophies, generally speaking, is to clear away the conceptual clutter of excess metaphysical ideas through the searching examination of their verbal formulations. Thus the question arises whether ''truth'' is one of the essentials or one of the excesses of rational thought. One method of critical analysis that is commonly brought to bear at this juncture is based on the nominal corollary that if one can do without the word in every linguistic context, then one can do without the concept, which is after all nothing but the word.

===Real truth theories===

===Formal truth theories===

There is a generally acknowledged distinction between merely contemplating or entertaining a proposition, and actually asserting or believing it. This does not mean that there is general agreement as to the precise nature of the distinction. Although there are many ways of talking about the distinction, words alone do not guarantee clarity, and they often lead to the problem of having to decide which descriptions say the same thing and which say something different.

For example, formal logic provides symbolic operators for indicating the assertion of a sentence, or the assertion of the proposition that comes from interpreting the sentence relative to a particular context of discussion. Another way of saying something about a sentence or the corresponding proposition is by means of various semantic predicates, including truth predicates as a special case. This raises the question of how these operators and predicates are related to one another. As noted before, one of the first questions of this sort is whether asserting a proposition amounts to the same thing as predicating truth of that proposition.

===Semantic relations===

A ''denotation relation'', or a ''name relation'', is a [[relation (mathematics)|relation]] between symbols (formulas, words, phrases) and the things that they are interpreted as denoting or naming in a particular context of discussion (Church 1962). The things denoted, which may be quite literally anything that can be talked about or thought about, are called the ''objects'' of denotation.

Different theories of meaning vary in their use of denotation relations and the properties that they require of them. The following are two criteria that serve to distinguish particular theories of denotation:
# How many things can a symbol denote? For instance, can a symbol denote more than one thing, or must a symbol always denote at most one thing?
# Is denoting the same sort of relation as ''being true of'', and thus a state of affairs that can be described by a particular type of truth predicate, or is denoting a very different sort of relation than that?

==Truth and the conduct of life==

<blockquote>
Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation ([[arete (excellence)|αρετηs]] κυβερνητικηs), do you perceive what must happen to him and his fellow sailors? (Plato, ''Alcibiades'', 135A).
</blockquote>

==References==

* Aristotle, “On Interpretation”, Harold P. Cooke (trans.), pp. 111–179 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938.

* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), “Interpretation as Action : The Risk of Inquiry”, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40–52. [https://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry Online].

* Blackburn, S. (1996), ''The Oxford Dictionary of Philosophy'', Oxford University Press, Oxford, UK, 1994. Paperback edition with new Chronology, 1996.

* Blackburn, S., and Simmons, K. (eds., 1999), ''Truth'', Oxford University Press, Oxford, UK.

* Church, A. (1962a), “Name Relation, or Meaning Relation”, p. 204 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ.

* Church, A. (1962b), “Truth, Semantical”, p. 322 in Dagobert D. Runes (ed.), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ.

* Kneale, W., and Kneale, M. (1962), ''The Development of Logic'', Oxford University Press, London, UK, 1962. Reprinted with corrections, 1975.

* Plato, “Alcibiades 1”, W.R.M. Lamb (trans.), pp. 93–223 in ''Plato, Volume 12'', Loeb Classical Library, William Heinemann, London, UK, 1927.

* Russell, B. (1918), “The Philosophy of Logical Atomism”, ''The Monist'', 1918. Reprinted, pp. 177–281 in ''Logic and Knowledge: Essays 1901–1950'', Robert Charles Marsh (ed.), Unwin Hyman, London, UK, 1956. Reprinted, pp. 35–155 in ''The Philosophy of Logical Atomism'', David Pears (ed.), Open Court, La Salle, IL, 1985.

==Further reading==

* Beaney, M. (ed., 1997), ''The Frege Reader'', Blackwell Publishers, Oxford, UK.

* Dewey, J. (1900–1901), ''Lectures on Ethics 1900?1901'', Donald F. Koch (ed.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1991.

* Dewey, John (1932), ''Theory of the Moral Life'', Part 2 of John Dewey and James H. Tufts, ''Ethics'', Henry Holt and Company, New York, NY, 1908. 2nd edition, Holt, Rinehart, and Winston, 1932. Reprinted, Arnold Isenberg (ed.), Victor Kestenbaum (Preface), Irvington Publishers, New York, NY, 1980.

* Dummett, M. (1991), ''Frege and Other Philosophers'', Oxford University Press, Oxford, UK.

* Dummett, M. (1993), ''Origins of Analytical Philosophy'', Harvard University Press, Cambridge, MA.

* Foucault, M. (1997), ''Essential Works of Foucault, 1954–1984, Volume 1, Ethics : Subjectivity and Truth'', Paul Rabinow (ed.), Robert Hurley et al. (trans.), The New Press, New York, NY.

* Gadamer, H.-G. (1986), ''The Idea of the Good in Platonic–Aristotelian Philosophy'', P. Christopher Smith (trans.), Yale University Press, New Haven, CT. 1st published, ''Die Idee des Guten zwischen Plato und Aristoteles'', J.C.B. Mohr, Heidelberg, Germany, 1978.

* Grover, Dorothy (1992), ''A Prosentential Theory of Truth'', Princeton University Press, Princeton, NJ.

* Habermas, J. (1979), ''Communication and the Evolution of Society'', Thomas McCarthy (trans.), Beacon Press, Boston, MA.

* Habermas, J. (1990), ''Moral Consciousness and Communicative Action'', Christian Lenhardt and Shierry Weber Nicholsen (trans.), Thomas McCarthy (intro.), MIT Press, Cambridge, MA.

* Habermas, J. (2003), ''Truth and Justification'', Barbara Fultner (trans.), MIT Press, Cambridge, MA.

* Kirkham, R.L. (1992), ''Theories of Truth'', MIT Press, Cambridge, MA.

* Kripke, S.A. (1975), “An Outline of a Theory of Truth”, ''Journal of Philosophy'' 72 (1975), 690?716.

* Kripke, S.A. (1980), ''Naming and Necessity'', Harvard University Press, Cambridge, MA.

* Lewis, C.I. (1946), ''An Analysis of Knowledge and Valuation'', The Paul Carus Lectures, Series 8, Open Court, La Salle, IL.

* Linsky, L. (ed., 1971), ''Reference and Modality'', Oxford University Press, London, UK.

* Martin, R.L. (ed., 1984), ''Recent Essays on Truth and the Liar Paradox'', Oxford University Press, Oxford, UK.

* Moody, E.A. (1953), ''Truth and Consequence in Mediaeval Logic'', North-Holland, Amsterdam, Netherlands, 1953. Reprinted, Greenwood Press, Westport, CT, 1976.

* Nietzsche, Friedrich (1873/1968). “Uber Wahrheit und Lüge im aussermoralischen Sinn”, (“On Truth and Lying in an Extra-moral Sense”), in Jürgen Habermas (ed.), ''Erkenntnistheoretische Schriften'', Suhrkamp, Frankfurt, Germany.

* Putnam, Hilary (1981), ''Reason, Truth, and History'', Cambridge University Press, Cambridge, UK.

* Quine, W.V. (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.

* Quine, W.V. (1992), ''Pursuit of Truth'', Harvard University Press, Cambridge, MA, 1990. Revised edition, Harvard University Press, Cambridge, MA, 1992.

* Quine, W.V., and Ullian, J.S. (1978), ''The Web of Belief'', Random House, New York, NY, 1970. 2nd edition, Random House, New York, NY, 1978.

* Rawls, J. (2000), ''Lectures on the History of Moral Philosophy'', Barbara Herman (ed.), Harvard University Press, Cambridge, MA.

* Rescher, N. (1973), ''The Coherence Theory of Truth'', Oxford University Press, Oxford, UK.

* Rorty, R. (1991), ''Objectivity, Relativism, and Truth : Philosophical Papers, Volume 1'', Cambridge University Press, Cambridge, UK.

* Russell, B. (1913), ''Theory of Knowledge (The 1913 Manuscript)'', Elizabeth Ramsden Eames (ed.), Kenneth Blackwell (collab.), George Allen & Unwin, 1984. Reprinted, Routledge, London, UK, 1992.

* Russell, B. (1940), ''An Inquiry into Meaning and Truth'', 'The William James Lectures for 1940 Delivered at Harvard University', George Allen & Unwin, 1950. Reprinted, Thomas Baldwin (intro.), Routledge, London, UK, 1992.

* Salmon, N., and [[Scott Soames|Soames, Scott]] (eds., 1988), ''Propositions and Attitudes'', Oxford University Press, Oxford, UK.

* Smart, N. (1969), ''The Religious Experience of Mankind'', Charles Scribner's Sons, New York, NY.

* Tarski, A. (1944), “The Semantic Conception of Truth and the Foundations of Semantics”, ''Philosophy and Phenomenological Research'' 4 (3), 341–376.

* Wallace, A.F.C.]] (1966), ''Religion, An Anthropological View'', Random House, New York, NY.

* Williams, B. (2002), ''Truth and Truthfulness: An Essay in Genealogy'', Princeton University Press, Princeton, NJ.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Truth_theory Truth Theory @ InterSciWiki]
* [http://mywikibiz.com/Truth_theory Truth Theory @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Truth_theory Truth Theory @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Truth_theory Truth Theory @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_theory Truth Theory @ Wikiversity Beta]

===Logical operators===

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* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
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* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

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* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
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* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
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* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
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===Relational concepts===

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* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
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* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
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* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

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* [[Inquiry]]
* [[Dynamics of inquiry]]
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* [[Semeiotic]]
* [[Logic of information]]
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* [[Descriptive science]]
* [[Normative science]]
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* [[Pragmatic maxim]]
* [[Truth theory]]
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===Related articles===

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* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

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* [http://semanticweb.org/wiki/Truth_theory Truth Theory], [http://semanticweb.org/ Semantic Web]
* [http://ref.subwiki.org/wiki/Truth_theory Truth Theory], [http://ref.subwiki.org/ Subject Wikis]
* [http://wikinfo.org/w/index.php/Truth_theory Truth Theory], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Truth_theory Truth Theory], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_theory Truth Theory], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Truth_Theory Truth Theory], [http://en.wikipedia.org/ Wikipedia]

This article contains material from an earlier version of the former Wikipedia article, [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Truth_Theory Truth Theory], no longer extant. The Wikipedia article was deleted and its edit history destroyed by Wikipedia administrators, in violation of the GNU Free Documentation License. A record of the Wikipedia AFD (Article For Deletion) proceedings can be found at the following locations:

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Pragmatic maxim

2015-11-17T04:35:25Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

The '''pragmatic maxim''', also known as the ''maxim of pragmatism'' or the ''maxim of pragmaticism'', is a maxim of logic formulated by [[Charles Sanders Peirce]]. Serving as a normative recommendation or a regulative principle in the [[normative science]] of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

==Seven ways of looking at a pragmatic maxim==

Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus.

* The first excerpt appears in the form of a dictionary entry, intended as a definition of ''pragmatism''.

<blockquote>
Pragmatism. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: “Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.” (Peirce, CP 5.2, 1878/1902).
</blockquote>

* The second excerpt presents another version of the pragmatic maxim, a recommendation about a way of clarifying meaning that can be taken to stake out the general philosophy of pragmatism.

<blockquote>
Pragmaticism was originally enounced in the form of a maxim, as follows: Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object. (Peirce, CP 5.438, 1878/1905).
</blockquote>

* The third excerpt puts a gloss on the meaning of a ''practical bearing'' and provides an alternative statement of the maxim.

<blockquote>
Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a “practical consideration”. Hence is justified the maxim, belief in which constitutes pragmatism; namely:

''In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception.'' (Peirce, CP 5.9, 1905).
</blockquote>

* The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim in an alternative way. In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a whole philosophy.

<blockquote>
On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this:

Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood. (Peirce, CP 5.18, 1903).
</blockquote>

* The fifth excerpt is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic maxim.

<blockquote>
The doctrine appears to assume that the end of man is action — a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought. (Peirce, CP 5.3, 1902).
</blockquote>

* A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection.

<blockquote>
The study of philosophy consists, therefore, in reflexion, and ''pragmatism'' is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. … It will be seen that ''pragmatism'' is not a ''Weltanschauung'' but is a method of reflexion having for its purpose to render ideas clear. (Peirce, CP 5.13 note 1, 1902).
</blockquote>

* The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a ''post mortem'' analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers.

<blockquote>
This employment five times over of derivates of ''concipere'' must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot. (Peirce, CP 5.402 note 3, 1906).
</blockquote>

==References==

* Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. (Cited as CP ''n''.''m'' for volume ''n'', paragraph ''m'').

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim → ThoughtMesh]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Pragmatic_maxim Pragmatic Maxim @ InterSciWiki]
* [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Pragmatic_maxim Pragmatic Maxim], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Pragmatic_maxim Pragmatic Maxim], [http://mywikibiz.com/ MyWikiBiz]
* [http://web.archive.org/web/20050427095007/http://suo.ieee.org/ontology/msg05407.html Pragmatic Maxim], [http://web.archive.org/web/20131214093401/http://suo.ieee.org/ontology/thrd1.html Ontology List]
* [http://semanticweb.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://semanticweb.org/ Semantic Web]
* [http://vectors.usc.edu/thoughtmesh/publish/141.php Pragmatic Maxim], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php/Pragmatic_maxim Pragmatic Maxim], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Pragmatic_maxim Pragmatic Maxim], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Pragmatic_maxim&oldid=45528828 Pragmatic Maxim], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Critical Thinking]]
[[Category:Cybernetics]]
[[Category:Education]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Inquiry]]
[[Category:Intelligence Amplification]]
[[Category:Learning Organizations]]
[[Category:Knowledge Representation]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Systems Science]]

Normative science

2015-11-16T22:44:42Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''normative science''' is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes.

The three '''normative sciences''', according to traditional conceptions in philosophy, are ''aesthetics'', ''ethics'', and ''logic''.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science @ InterSciWiki]
* [http://mywikibiz.com/Normative_science Normative Science @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Normative_science Normative Science @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Normative_science Normative Science], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Normative_science Normative Science], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Normative_science Normative Science], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Normative_science Normative Science], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Normative_science Normative Science], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Normative_science&oldid=51993011 Normative Science], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Normative Sciences]]
[[Category:Philosophy]]
[[Category:Philosophy of Science]]
[[Category:Science]]

Descriptive science

2015-11-16T21:56:04Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''descriptive science''', also called a '''special science''', is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of phenomena.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science @ InterSciWiki]
* [http://mywikibiz.com/Descriptive_science Descriptive Science @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Descriptive_science Descriptive Science @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Descriptive_science Descriptive Science], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Descriptive_science Descriptive Science], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Descriptive_science Descriptive Science], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Descriptive_science&oldid=51990248 Descriptive Science], [http://en.wikipedia.org/ Wikipedia]

[[Category:Descriptive Sciences]]
[[Category:Inquiry]]
[[Category:Philosophy]]
[[Category:Philosophy of Science]]
[[Category:Science]]

Triadic relation

2015-11-16T21:25:38Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations.

Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms.

==Examples from mathematics==

For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner.

The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way.

The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math>

The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math>

The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math>

In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math>

The relation <math>L_0\!</math> is defined as follows:

{| align="center" cellpadding="6" width="90%"
| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math>
|}

The relation <math>L_0\!</math> is the set of four triples enumerated here:

{| align="center" cellpadding="6" width="90%"
| <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>
|}

The relation <math>L_1\!</math> is defined as follows:

{| align="center" cellpadding="6" width="90%"
| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math>
|}

The relation <math>L_1\!</math> is the set of four triples enumerated here:

{| align="center" cellpadding="6" width="90%"
| <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>
|}

The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|}

 

==Examples from semiotics==

The study of signs — the full variety of significant forms of expression — in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''.

The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''.

For example, consider the aspects of sign use that concern two people — let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> — in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively.

Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math>

The set-up so far is summarized as follows:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccc}
L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\
\\
O & = & \{ \mathrm{A}, \mathrm{B} \} \\
\\
S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
\\
I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
\\
\end{array}</math>
|}

The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\{ &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
\\
&
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) &
\}.
\end{array}</math>
|}

The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>

The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\{ &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
\\
&
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) &
\}.
\end{array}</math>
|}

The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>

The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
|<math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}

 

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki]
* [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Triadic_relation Triadic Relation], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
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[[Category:Charles Sanders Peirce]]
[[Category:Cognitive Sciences]]
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[[Category:Syntax]]

Relation theory

2015-11-16T21:05:04Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

This article treats relations from the perspective of combinatorics, in other words, as a subject matter in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory, or the theory of relations, is distinguished from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

==Preliminaries==

Two definitions of the relation concept are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another.

The same sort of ambiguity arose in the development of the function concept and it may save some effort to follow the pattern of resolution that worked itself out there.

When we speak of a function <math>f : X \to Y\!</math> we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> So far so good.

Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far.

When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math>

Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math>

In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in connection with many different types.

Following the pattern of the functional case, let the notation <math>{}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}\!</math> bring to mind a mathematical object that is specified by three pieces of data, the set <math>X,\!</math> the set <math>Y,\!</math> and a particular subset of their cartesian product <math>{X \times Y}.\!</math> As before we have two choices, either let <math>L = (X, Y, \mathrm{graph}(L))\!</math> or let <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(L)\!</math> and choose another name for the triple.

==Definition==

It is convenient to begin with the definition of a ''<math>k\!</math>-place relation'', where <math>k\!</math> is a positive integer.

'''Definition.''' A ''<math>k\!</math>-place relation'' <math>L \subseteq X_1 \times \ldots \times X_k\!</math> over the nonempty sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>(X_1, \ldots, X_k, L)\!</math> where <math>L\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k.\!</math>

==Remarks==

Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets <math>X_1, \ldots, X_k\!</math> are called the ''domains'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> with <math>{X_j}\!</math> being the <math>j^\text{th}\!</math> domain. If all of the <math>{X_j}\!</math> are the same set <math>X\!</math> then <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is more simply described as a <math>k\!</math>-place relation over <math>X.\!</math> The set <math>L\!</math> is called the ''graph'' of the relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> on analogy with the graph of a function. If the sequence of sets <math>X_1, \ldots, X_k\!</math> is constant throughout a given discussion or is otherwise determinate in context, then the relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is determined by its graph <math>L,\!</math> making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective ''<math>k\!</math>-place'' are ''<math>k\!</math>-adic'' and ''<math>k\!</math>-ary'', all of which leads to the integer <math>k\!</math> being called the ''dimension'', ''adicity'', or ''arity'' of the relation <math>L.\!</math>

==Local incidence properties==

A ''local incidence property'' (LIP) of a relation <math>L\!</math> is a property that depends in turn on the properties of special subsets of <math>L\!</math> that are known as its ''local flags''. The local flags of a relation are defined in the following way:

Let <math>L\!</math> be a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>

Select a relational domain <math>{X_j}\!</math> and one of its elements <math>x.\!</math> Then <math>L_{x \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is referred to as the ''flag'' of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math> or the ''<math>x \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object that has the following definition:

{| align="center" cellpadding="8" style="text-align:center"
| <math>L_{x \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j = x \}.\!</math>
|}

Any property <math>C\!</math> of the local flag <math>L_{x \operatorname{at} j} \subseteq L\!</math> is said to be a ''local incidence property'' of <math>L\!</math> with respect to the ''locus'' <math>x \operatorname{at} j.\!</math>

A <math>k\!</math>-adic relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is said to be ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> has the property <math>C,\!</math> where <math>x\!</math> is taken to vary over the ''theme'' of the fixed domain <math>X_j.\!</math>

Expressed in symbols, <math>L\!</math> is ''<math>C\!</math>-regular'' at <math>j\!</math> if and only if <math>C(L_{x \operatorname{at} j})\!</math> is true for all <math>x\!</math> in <math>X_j.\!</math>

==Regional incidence properties==

The definition of a local flag can be broadened from a point <math>x\!</math> in <math>{X_j}\!</math> to a subset <math>M\!</math> of <math>X_j,\!</math> arriving at the definition of a ''regional flag'' in the following way:

Suppose that <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Then <math>L_{M \operatorname{at} j}\!</math> is a subset of <math>L\!</math> that is said to be the ''flag'' of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math> or the ''<math>M \operatorname{at} j\!</math>-flag'' of <math>L,\!</math> an object which has the following definition:

{| align="center" cellpadding="8" style="text-align:center"
| <math>L_{M \operatorname{at} j} ~=~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~:~ x_j \in M \}.\!</math>
|}

==Numerical incidence properties==

A ''numerical incidence property'' (NIP) of a relation is a local incidence property that depends on the cardinalities of its local flags.

For example, <math>L\!</math> is said to be ''<math>c\!</math>-regular at <math>j\!</math>'' if and only if the cardinality of the local flag <math>L_{x \operatorname{at} j}\!</math> is <math>c\!</math> for all <math>x\!</math> in <math>{X_j},\!</math> or, to write it in symbols, if and only if <math>|L_{x \operatorname{at} j}| = c\!</math> for all <math>{x \in X_j}.\!</math>

In a similar fashion, one can define the NIPs, ''<math>(<\!c)\!</math>-regular at <math>{j},\!</math>'' ''<math>(>\!c)\!</math>-regular at <math>{j},\!</math>'' and so on. For ease of reference, a few of these definitions are recorded here:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
L & \text{is} & c\textit{-regular} & \text{at}\ j
& \text{if and only if} &
|L_{x \operatorname{at} j}| & = & c & \text{for all}\ x \in X_j.
\\[4pt]
L & \text{is} & (<\!c)\textit{-regular} & \text{at}\ j
& \text{if and only if} &
|L_{x \operatorname{at} j}| & < & c & \text{for all}\ x \in X_j.
\\[4pt]
L & \text{is} & (>\!c)\textit{-regular} & \text{at}\ j
& \text{if and only if} &
|L_{x \operatorname{at} j}| & > & c & \text{for all}\ x \in X_j.
\end{matrix}</math>
|}

==Species of 2-adic relations==

Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq S \times T\!</math> be an arbitrary 2-adic relation. The following properties of <math>L\!</math> can be defined:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
L & \text{is} & \textit{total} & \text{at}~ S
& \text{if and only if} &
L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S.
\\[4pt]
L & \text{is} & \textit{total} & \text{at}~ T
& \text{if and only if} &
L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T.
\\[4pt]
L & \text{is} & \textit{tubular} & \text{at}~ S
& \text{if and only if} &
L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S.
\\[4pt]
L & \text{is} & \textit{tubular} & \text{at}~ T
& \text{if and only if} &
L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T.
\end{matrix}</math>
|}

If <math>L \subseteq S \times T\!</math> is tubular at <math>S\!</math> then <math>L\!</math> is called a ''partial function'' or a ''prefunction'' from <math>S\!</math> to <math>T.\!</math> This is sometimes indicated by giving <math>L\!</math> an alternate name, say, <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>L = p : S \rightharpoonup T.\!</math>

Just by way of formalizing the definition:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
L & = & p : S \rightharpoonup T
& \text{if and only if} &
L & \text{is} & \text{tubular} & \text{at}~ S.
\end{matrix}</math>
|}

If <math>L\!</math> is a prefunction <math>p : S \rightharpoonup T\!</math> that happens to be total at <math>S,\!</math> then <math>L\!</math> is called a ''function'' from <math>S\!</math> to <math>T,\!</math> indicated by writing <math>L = f : S \to T.\!</math> To say that a relation <math>L \subseteq S \times T\!</math> is ''totally tubular'' at <math>S\!</math> is to say that it is <math>1\!</math>-regular at <math>S.\!</math> Thus, we may formalize the following definition:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
L & = & f : S \to T
& \text{if and only if} &
L & \text{is} & 1\text{-regular} & \text{at}~ S.
\end{matrix}</math>
|}

In the case of a function <math>f : S \to T,\!</math> one has the following additional definitions:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
f & \text{is} & \textit{surjective}
& \text{if and only if} &
f & \text{is} & \text{total} & \text{at}~ T.
\\[4pt]
f & \text{is} & \textit{injective}
& \text{if and only if} &
f & \text{is} & \text{tubular} & \text{at}~ T.
\\[4pt]
f & \text{is} & \textit{bijective}
& \text{if and only if} &
f & \text{is} & 1\text{-regular} & \text{at}~ T.
\end{matrix}</math>
|}

==Variations==

Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide variety of terminology that the reader may run across in connection with the subject.

One dimension of variation is reflected in the names that are given to <math>k\!</math>-place relations, for <math>k = 0, 1, 2, 3, \ldots,\!</math> with some writers using the Greek forms, ''medadic'', ''monadic'', ''dyadic'', ''triadic'', ''<math>k\!</math>-adic'', and other writers using the Latin forms, ''nullary'', ''unary'', ''binary'', ''ternary'', ''<math>k\!</math>-ary''.

The number of relational domains may be referred to as the ''adicity'', ''arity'', or ''dimension'' of the relation. Accordingly, one finds a relation on a finite number of domains described as a ''polyadic'' relation or a ''finitary'' relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to <math>k,\!</math> then the relation may be described as a ''<math>k\!</math>-adic'' relation, a ''<math>k\!</math>-ary'' relation, or a ''<math>k\!</math>-dimensional'' relation, respectively.

A more conceptual than nominal variation depends on whether one uses terms like ''predicate'', ''relation'', and even ''term'' to refer to the formal object proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical differences over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else one is derivative in relationship to the other. Historically speaking, just about every combination of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated.

==Examples==

: ''See the articles on [[relation (mathematics)|relations]], [[relation composition]], [[relation reduction]], [[sign relation]]s, and [[triadic relation]]s for concrete examples of relations.''

Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as ''binary operations'', and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations.

==References==

* Peirce, C.S., “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'', 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429.

* Ulam, S.M. and Bednarek, A.R., “On the Theory of Relational Structures and Schemata for Parallel Computation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990.

==Bibliography==

* Barr, Michael, and Wells, Charles (1990), ''Category Theory for Computing Science'', Prentice Hall, Hemel Hempstead, UK.
* Bourbaki, Nicolas (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany.
* Carnap, Rudolf (1958), ''Introduction to Symbolic Logic with Applications'', Dover Publications, New York, NY.
* Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands.
* van Dalen, Dirk (1980), ''Logic and Structure'', 2nd edition, Springer-Verlag, Berlin, Germany.
* Devlin, Keith J. (1993), ''The Joy of Sets : Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY.
* Halmos, Paul Richard (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ.
* van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879–1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
* Kelley, John L. (1955), ''General Topology'', Van Nostrand Reinhold, New York, NY.
* Kneale, William; and Kneale, Martha (1962/1975), ''The Development of Logic'', Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
* Lawvere, Francis William; and Rosebrugh, Robert (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK.
* Lawvere, Francis William; and Schanuel, Stephen H. (1997/2000), ''Conceptual Mathematics : A First Introduction to Categories'', Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000.
* Manin, Yu. I. (1977), ''A Course in Mathematical Logic'', Neal Koblitz (trans.), Springer-Verlag, New York, NY.
* Mathematical Society of Japan (1993), ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA.
* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. (Introduction to Tarskian relation theory and relational programming.)
* Mitchell, John C. (1996), ''Foundations for Programming Languages'', MIT Press, Cambridge, MA.
* Peirce, Charles Sanders (1870), ``Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", ''Memoirs of the American Academy of Arts and Sciences'' 9 (1870), 317–378. Reprinted (CP 3.45–149), (CE 2, 359–429).
* Peirce, Charles Sanders (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).
* Peirce, Charles Sanders (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).
* Poizat, Bruno (2000), ''A Course in Model Theory : An Introduction to Contemporary Mathematical Logic'', Moses Klein (trans.), Springer-Verlag, New York, NY.
* Quine, Willard Van Orman (1940/1981), ''Mathematical Logic'', 1940. Revised edition, Harvard University Press, Cambridge, MA, 1951. New preface, 1981.
* Royce, Josiah (1961), ''The Principles of Logic'', Philosophical Library, New York, NY.
* Runes, Dagobert D. (ed., 1962), ''Dictionary of Philosophy'', Littlefield, Adams, and Company, Totowa, NJ.
* Shoenfield, Joseph R. (1967), ''Mathematical Logic'', Addison-Wesley, Reading, MA.
* Styazhkin, N.I. (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA.
* Suppes, Patrick (1957/1999), ''Introduction to Logic'', 1st published 1957. Reprinted, Dover Publications, New York, NY, 1999.
* Suppes, Patrick (1960/1972), ''Axiomatic Set Theory'', 1st published 1960. Reprinted, Dover Publications, New York, NY, 1972.
* Tarski, Alfred (1956/1983), ''Logic, Semantics, Metamathematics : Papers from 1923 to 1938'', J.H. Woodger (trans.), Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.
* Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), “On the Theory of Relational Structures and Schemata for Parallel Computation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990.
* Ulam, Stanislaw Marcin (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.
* Ullman, Jeffrey D. (1980), ''Principles of Database Systems'', Computer Science Press, Rockville, MD.
* Venetus, Paulus (1472/1984), ''Logica Parva : Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory @ InterSciWiki]
* [http://mywikibiz.com/Relation_theory Relation Theory @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_theory Relation Theory @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_theory Relation Theory], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_theory Relation Theory], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/RelationTheory Relation Theory], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Relation_theory Relation Theory], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Relation_theory Relation Theory], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_theory Relation Theory], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_theory Relation Theory], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Theory_of_relations&oldid=45042729 Relation Theory], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Formal Sciences]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Set Theory]]

Relative term

2015-11-16T20:22:04Z

Jon Awbrey: + link

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''relative term''' is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate''1 of the relative term in question.  A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, “lover of __”, or “giver of __ to __”.

1. Pronounced with the accent on the first syllable.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Relative_term Relative Term], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Relation Theory]]
[[Category:Science]]
[[Category:Semiotics]]

Relative term

2015-11-16T19:58:03Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''relative term''' is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate''1 of the relative term in question.  A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, “lover of __”, or “giver of __ to __”.

1. Pronounced with the accent on the first syllable.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Relation Theory]]
[[Category:Science]]
[[Category:Semiotics]]

Logic of information

2015-11-16T03:26:34Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

The '''logic of information''', or the ''logical theory of information'', considers the information content of logical signs — everything from bits to books and beyond — along the lines initially developed by [[Charles Sanders Peirce]].  In this line of development the concept of information serves to integrate the aspects of logical signs that are separately covered by the concepts of denotation and connotation, or, in roughly equivalent terms, by the concepts of extension and comprehension.

Peirce began to develop these ideas in his lectures “On the Logic of Science” at Harvard University (1865) and the Lowell Institute (1866).  Here is one of the starting points:

{| align="center" cellpadding="8" width="90%"
|
Let us now return to the information.  The information of a term is the measure of its superfluous comprehension.  That is to say that the proper office of the comprehension is to determine the extension of the term.  For instance, you and I are men because we possess those attributes — having two legs, being rational, &tc. — which make up the comprehension of ''man''.  Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead.

Thus, let us commence with the term ''colour'';  add to the comprehension of this term, that of ''red''.  ''Red colour'' has considerably less extension than ''colour'';  add to this the comprehension of ''dark'';  ''dark red colour'' has still less [extension].  Add to this the comprehension of ''non-blue'' — ''non-blue dark red colour'' has the same extension as ''dark red colour'', so that the ''non-blue'' here performs a work of supererogation;  it tells us that no ''dark red colour'' is blue, but does none of the proper business of connotation, that of diminishing the extension at all.

Thus information measures the superfluous comprehension.  And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension.  I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of ''information''.  (C.S. Peirce, “The Logic of Science, or, Induction and Hypothesis” (1866), CE 1, 467).
|}

==References==

* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* De Tienne, André (2006), "Peirce's Logic of Information", Seminario del Grupo de Estudios Peirceanos, Universidad de Navarra, 28 Sep 2006. [http://www.unav.es/gep/SeminariodeTienne.html Online].

* Peirce, C.S. (1867), "Upon Logical Comprehension and Extension", [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information → ThoughtMesh]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_information Logic of Information @ InterSciWiki]
* [http://mywikibiz.com/Logic_of_information Logic of Information @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Logic_of_information Logic of Information @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Logic_of_information Logic of Information @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_information Logic of Information], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Logic_of_information Logic of Information], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Logic_of_information Logic of Information], [http://semanticweb.org/ Semantic Web]
* [http://vectors.usc.edu/thoughtmesh/publish/145.php Logic of Information], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php/Logic_of_information Logic of Information], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Logic_of_information Logic of Information], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logic_of_information Logic of Information], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Logic_of_information&oldid=67770000 Logic of Information], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Cybernetics]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Information Theory]]
[[Category:Inquiry]]
[[Category:Intelligence Amplification]]
[[Category:Knowledge Representation]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
[[Category:Science]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Systems Science]]

Dynamics of inquiry

2015-11-16T00:28:19Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

<blockquote>
Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. (Peirce, "On Time and Thought", CE 3, 68–69.)
</blockquote>

All through the 1860s the young Charles Peirce was busy establishing a conceptual base-camp and a technical supply line for the intellectual adventures of a lifetime. Taking the long view of this activity and trying to choose the best titles for the story, it all seems to have something to do with the dynamics of inquiry. This broad subject area has a part that is given by nature and a part that is ruled by nurture. On first approach, it is possible to see a question of articulation and a question of explanation:

:* What is needed to articulate the workings of the active form of representation that is known as ''conscious experience''?

:* What is needed to account for the workings of the reflective discipline of inquiry that is known as ''science''?

The pursuit of answers to these questions finds them to be so entangled with each other that it's ultimately impossible to comprehend them apart from each other, but for the sake of exposition it's convenient to organize our study of Peirce's assault on the ''summa'' by following first the trails of thought that led him to develop a ''[[theory of signs]]'', one that has come to be known as '[[semiotic]]', and tracking next the ways of thinking that led him to develop a ''[[theory of inquiry]]'', one that would be up to the task of saying 'how science works'.

Opportune points of departure for exploring the dynamics of representation, such as led to Peirce's theories of [[inference]] and [[information]], inquiry and signs, are those that he took for his own springboards. Perhaps the most significant influences radiate from points on parallel lines of inquiry in [[Aristotle]]'s work, points where the intellectual forerunner focused on many of the same issues and even came to strikingly similar conclusions, at least about the best ways to begin. Staying within the bounds of what will give us a more solid basis for understanding Peirce, it serves to consider the following ''loci'' in Aristotle:

:* The basic terminology of [[psychology]], in ''[[On the Soul]]''.

:* The founding description of [[sign relations]], in ''[[On Interpretation]]'';

:* The differentiation of the genus of reasoning into three species of [[inference]] that are commonly translated into English as ''[[Abductive reasoning|abduction]]'', ''[[Deductive reasoning|deduction]]'', and ''[[Inductive reasoning|induction]]'', in the ''[[Prior Analytics]]''.

In addition to the three elements of inference, that Peirce would assay to be [[irreducible]], [[Aristotle]] analyzed several types of [[compound inference]], most importantly the type known as 'reasoning by [[analogy]]' or 'reasoning from [[example]]', employing for the latter description the Greek word 'paradeigma', from which we get our word '[[paradigm]]'.

Inquiry is a form of reasoning process, in effect, a particular way of conducting thought, and thus it can be said to institute a specialized manner, style, or turn of thinking. Philosophers of the school that is commonly called 'pragmatic' hold that all
thought takes place in signs, where 'sign' is the word they use for the broadest conceivable variety of characters, expressions, formulas, messages, signals, texts, and so on up the line, that might be imagined. Even intellectual concepts and mental ideas are held to be a special class of signs, corresponding to internal states of the thinking agent that both issue in and result from the interpretation of external signs.

The subsumption of inquiry within reasoning in general and the inclusion of thinking within the class of sign processes allows us to approach the subject of inquiry from two different perspectives:

:* The ''[[syllogistic]]'' approach treats inquiry as a species of logical process, and is limited to those of its aspects that can be related to the most basic laws of inference.

:* The ''[[sign-theoretic]]'' approach views inquiry as a genus of ''[[semiosis]]'', an activity taking place within the more general setting of [[sign relation]]s and [[sign process]]es.

The distinction between signs denoting and objects denoted is critical to the discussion of Peirce's theory of signs. Wherever needed in the rest of this article, therefore, in order to mark this distinction a little more emphatically than usual, double quotation marks placed around a given sign, for example, a string of zero or more characters, will be used to create a new sign that denotes the given sign as its object.

===Semeiotic : Peirce's theory of signs===

Peirce referred to his general study of signs, based on the concept of a [[triadic relation|triadic]] [[sign relation]], as ''[[semeiotic]]'' or ''[[semiotic]]'', either of which terms are currently used in both singular of plural forms. Peirce began writing on semeiotic in the 1860s, around the time that he devised his system of three categories. He eventually defined ''[[semiosis]]'' as an "action, or influence, which is, or involves, a cooperation of ''three'' subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs". (Houser 1998: 411, written 1907). This triadic relation grounds the semeiotic.

In order to understand what a ''sign'' is we need to understand what a ''[[sign relation]]'' is, for signhood is a way of being in relation, not a way of being in itself. In order to understand what a sign relation is we need to understand what a ''[[triadic relation]]'' is, for the role of a sign is constituted as one among three, where roles in general are distinct even when the things that fill them are not. In order to understand what a triadic relation is we need to understand what a ''[[relation (mathematics)|relation]]'' is, and here there are traditionally two ways of understanding what a relation is, both of which are necessary if not sufficient to complete understanding, namely, the way of ''[[extension (semantics)|extension]]'' and the way of ''[[intension]]''. To these traditional approximations, Peirce adds a third way, the way of ''[[semiotic information theory|information]]'', that integrates the other two approaches in a unified whole.

====Sign relations====

: ''Main article'' : [[Sign relation]]

With that hasty map of relations and relatives sketched above, we may now trek into the terrain of ''sign relations'', the main subject matter of Peirce's ''semeiotic'', or theory of signs.

====Types of signs====

Peirce proposes several typologies and definitions of the signs. More than 76 definitions of what a sign is have been collected throughout Peirce's work. Some canonical typologies can nonetheless be observed, one crucial one being the distinction between "icons", "indices" and "symbols" (CP 2.228, CP 2.229 and CP 5.473). This typology emphasizes the different ways in which the ''representamen'' (or its ''ground'') addresses or refers to its ''object'', through a particular mobilisation of an ''interpretant'' (but Peirce proposes also other typologies based on other criteria).

* An '''icon''' is a sign that denotes its objects by virtue of a quality that it shares with them. The sign is perceived as resembling or imitating the object it refers to (e.g. fork on a sign by the road indicating a rest stop). In other words, an icon thus "resembles" to its object. It shares a character or an aspect with it, which allows for it to be interpreted as a sign even if the object does not exist. It signifies essentially on the basis of its "ground".

* An '''index''' is a sign that denotes its objects by virtue of an existential connection that it has with them. For an index to signify, the relation to the object is crucial. The ''representamen'' is directly connected in some way (physically or casually) to the object it denotes (e.g. smoke coming from a building is an index of fire). Hence, an index refers to the object because it is really affected or modified by it, and thus may stand as a trace of the existence of the object.

* A '''symbol''' is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so. The ''representamen'' does not resemble the object signified but is fundamentally conventional, so that the signifying relationship must be learned and agreed upon (e.g. the word "cat"). A symbol thus denotes, primarily, by virtue of its ''interpretant''. Its action (''semeiosis'') is ruled by a convention, a more or less systematic set of associations that guarantees its interpretation, independently of any resemblance or any material relation with its object.

Note that these definitions are specific to Peirce's theory of signs and are not exactly equivalent to general uses of the notion of "[[icon]]", "[[symbol]]" or "[[index]]".

===Theory of inquiry===

: ''Main article'' : [[Inquiry]]

: Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy:
<center> Do not block the way of inquiry.</center>
: Although it is better to be methodical in our investigations, and to consider the economics of research, yet there is no positive sin against logic in ''trying'' any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted. (Peirce, "F.R.L." (c. 1899), CP 1.135–136.)

Peirce extracted the pragmatic model or [[theory]] of [[inquiry]] from its raw materials in classical logic and refined it in parallel with the early development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, processes that are currently known as ''[[abductive]]'', ''[[deductive]]'', and ''[[inductive]]'' [[inference]].

In the roughest terms, [[abductive reasoning|abduction]] is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a [[phenomenon]] of interest or a [[problem]] of concern, while [[deductive reasoning|deduction]] is used to clarify, to derive, and to explicate the relevant consequences of the selected [[hypothesis]], and [[inductive reasoning|induction]] is used to test the sum of the predictions against the sum of the data.

These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the [[knowledge]] or [[skills]], in other words, an [[augmentation]] in the [[competence]] or [[performance]], of the agent or community engaged in the inquiry.

In the pragmatic way of thinking every thing has a purpose, and the purpose of any thing is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''knowledge'' or ''certainty''. It needs to be appreciated that the three kinds of inference, insofar as they contribute to the end of inquiry, describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others.

For instance, the purpose of [[abductive reasoning|abduction]] is to generate guesses of a kind that [[deductive reasoning|deduction]] can explicate and that [[inductive reasoning|induction]] can evaluate. This places a mild but meaningful [[constraint]] on the production of hypotheses, since it is not just any wild guess at [[explanation]] that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of [[inference]] realizes its purpose only in accord with its proper role in the whole [[cycle of inquiry]]. No matter how much it may be necessary to study these processes in abstraction from each other, the [[integrity]] of inquiry places strong limitations on the effective modularity of its principal components.

If we then think to inquire, "What sort of [[constraint]], exactly, does pragmatic thinking place on our guesses?", we have asked the question that is generally recognized as the problem of "giving a rule to abduction". Peirce's way of answering it is given in terms of the so-called ''[[pragmatic maxim]]'', and this in turn gives us a clue as to the central role of abductive reasoning in Peirce's pragmatic philosophy.

===Logic of information===

: ''Main article'' : [[Logic of information]]

<blockquote>
Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes — having two legs, being rational, &tc. — which make up the comprehension of ''man''. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead. (C.S. Peirce, "The Logic of Science, or, Induction and Hypothesis" (1866), CE 1, 467.)
</blockquote>

==Source materials==

[[C.S. Peirce]], “On Time and Thought”, MS 215, 8 March 1873.

<blockquote>
Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. But is it pre-supposed in the conception of a logical mind, that the temporal succession in its ideas is continuous, and not by discrete steps? A continuum such as we suppose time and space to be, is defined as something any part of which itself has parts of the same kind. So that the point of time or the point of space is nothing but the ideal limit towards which we approach, but which we can never reach in dividing time or space; and consequently nothing is true of a point which is not true of a space or a time. A discrete quantum, on the other hand, has ultimate parts which differ from any other part of the quantum in their absolute separation from one another. If the succession of images in the mind is by discrete steps, time for that mind will be made up of indivisible instants. Any one idea will be absolutely distinguished from every other idea by its being present only in the passing moment. And the same idea can not exist in two different moments, however similar the ideas felt in the two different moments may, for the sake of argument, be allowed to be. Now an idea exists only so far as the mind thinks it; and only when it is present to the mind. An idea therefore has no characters or qualities but what the mind thinks of it at the time when it is present to the mind. It follows from this that if the succession of time were by separate steps, no idea could resemble another; for these ideas if they are distinct, are present to the mind at different times. Therefore at no time when one is present to the mind, is the other present. Consequently the mind never compares them nor thinks them to be alike; and consequently they are not alike; since they are only what they are thought to be at the time when they are present. It may be objected that though the mind does not directly think them to be alike; yet it may think together reproductions of them, and thus think them to be alike. This would be a valid objection were it not necessary, in the first place, in order that one idea should be the representative of another, that it should resemble that idea, which it could only do by means of some representation of it again, and so on to infinity; the link which is to bind the first two together which are to be pronounced alike, never being found. In short the resemblance of ideas implies that some two ideas are to be thought together which are present to the mind at different times. And this never can be, if instants are separated from one another by absolute steps. This conception is therefore to be abandoned, and it must be acknowledged to be already presupposed in the conception of a logical mind that the flow of time should be continuous. Let us consider then how we are to conceive what is present to the mind. We are accustomed to say that nothing is present but a fleeting instant, a point of time. But this is a wrong view of the matter because a point differs in no respect from a space of time, except that it is the ideal limit which, in the division of time, we never reach. It can not therefore be that it differs from an interval of time in this respect that what is present is only in a fleeting instant, and does not occupy a whole interval of time, unless what is present be an ideal something which can never be reached, and not something real. The true conception is, that ideas which succeed one another during an interval of time, become present to the mind through the successive presence of the ideas which occupy the parts of that time. So that the ideas which are present in each of these parts are more immediately present, or rather less mediately present than those of the whole time. And this division may be carried to any extent. But you never reach an idea which is quite immediately present to the mind, and is not made present by the ideas which occupy the parts of the time that it occupies. Accordingly, it takes time for ideas to be present to the mind. They are present during a time. And they are present by means of the presence of the ideas which are in the parts of that time. Nothing is therefore present to the mind in an instant, but only during a time. The events of a day are less mediately present to the mind than the events of a year; the events of a second less mediately present than the events of a day. (C.S. Peirce, CE 3, pp. 68–70).
</blockquote>

Charles Sanders Peirce, MS 215, 1873, [“On Time and Thought”], pp. 68–71 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 3, 1872–1878'', Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry @ InterSciWiki]
* [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Dynamics_of_inquiry Dynamics of Inquiry], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://semanticweb.org/ SemanticWeb]
* [http://wikinfo.org/w/index.php/Dynamics_of_inquiry Dynamics of Inquiry], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Dynamics_of_inquiry Dynamics of Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Charles_Sanders_Peirce&oldid=111891138#Dynamics_of_inquiry Dynamics of Inquiry], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Computer Science]]
[[Category:Critical Thinking]]
[[Category:Cybernetics]]
[[Category:Education]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Information Theory]]
[[Category:Inquiry]]
[[Category:Inquiry Driven Systems]]
[[Category:Intelligence Amplification]]
[[Category:Learning Organizations]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
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[[Category:Semantics]]
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[[Category:Systems Science]]

Inquiry

2015-11-15T22:36:11Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Inquiry''' is any proceeding or process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.

==Classical sources==

===Deduction===

{| align="center" cellpadding="4" width="90%" 
|
When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By 'middle term' I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by 'extremes' (a) that which is contained in another, and (b) that in which another is contained. For if ''A'' is predicated of all ''B'', and ''B'' of all ''C'', ''A'' must necessarily be predicated of all ''C''. … I call this kind of figure the First. (Aristotle, ''Prior Analytics'', 1.4).
|}

===Induction===

{| align="center" cellpadding="4" width="90%" 
|
Induction, or inductive reasoning, consists in establishing a relation between one extreme term and the middle term by means of the other extreme; for example, if ''B'' is the middle term of ''A'' and ''C'', in proving by means of ''C'' that ''A'' applies to ''B''; for this is how we effect inductions. (Aristotle, ''Prior Analytics'', 2.23).
|}

===Abduction===

The ''locus classicus'' for the study of [[abductive reasoning]] is found in [[Aristotle]]'s ''[[Prior Analytics]]'', Book 2, Chapt. 25. It begins this way:

{| align="center" cellpadding="4" width="90%" 
|
We have Reduction (απαγωγη, [[abductive reasoning|abduction]]):

<ol>

<li>When it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet is nevertheless more probable or not less probable than the conclusion;</li>

<li>Or if there are not many intermediate terms between the last and the middle;</li>

</ol>

For in all such cases the effect is to bring us nearer to knowledge.
|}

By way of explanation, [[Aristotle]] supplies two very instructive examples, one for each of the two varieties of abductive inference steps that he has just described in the abstract:

{| align="center" cellpadding="4" width="90%" 
|
<ol>

<li>For example, let ''A'' stand for "that which can be taught", ''B'' for "knowledge", and ''C'' for "morality". Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if ''BC'' is not less probable or is more probable than ''AC'', we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that ''AC'' is true.</li>

<li>Or again we have reduction if there are not many intermediate terms between ''B'' and ''C''; for in this case too we are brought nearer to knowledge. For example, suppose that ''D'' is "to square", ''E'' "rectilinear figure", and ''F'' "circle". Assuming that between ''E'' and ''F'' there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of [[lunule]]s — we should approximate to knowledge.</li>

</ol>

([[Aristotle]], "[[Prior Analytics]]", 2.25, with minor alterations)
|}

Aristotle's latter variety of abductive reasoning, though it will take some explaining in the sequel, is well worth our contemplation, since it hints already at streams of inquiry that course well beyond the syllogistic source from which they spring, and into regions that Peirce will explore more broadly and deeply.

==Inquiry in the pragmatic paradigm==

In the pragmatic philosophies of [[Charles Sanders Peirce]], [[William James]], [[John Dewey]], and others, inquiry is closely associated with the [[normative science]] of [[logic]]. In its inception, the pragmatic model or theory of inquiry was extracted by Peirce from its raw materials in classical logic, with a little bit of help from [[Kant]], and refined in parallel with the early development of symbolic logic by [[Boole]], [[De Morgan]], and Peirce himself to address problems about the nature and conduct of scientific reasoning. Borrowing a brace of concepts from [[Aristotle]], Peirce examined three fundamental modes of reasoning that play a role in inquiry, commonly known as [[abductive reasoning|abductive]], [[deductive reasoning|deductive]], and [[inductive reasoning|inductive]] [[inference]].

In rough terms, ''[[abductive reasoning|abduction]]'' is what we use to generate a likely [[hypothesis]] or an initial [[diagnosis]] in response to a phenomenon of interest or a problem of concern, while ''[[deductive reasoning|deduction]]'' is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and ''[[inductive reasoning|induction]]'' is used to test the sum of the predictions against the sum of the data. It needs to be observed that the classical and pragmatic treatments of the types of reasoning, dividing the generic territory of inference as they do into three special parts, arrive at a different characterization of the environs of reason than do those accounts that count only two.

These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in knowledge or in skills.

In the pragmatic way of thinking everything has a purpose, and the purpose of each thing is the first thing we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call ''[[knowledge]]'' or ''[[certainty]]''. As they contribute to the end of inquiry, we should appreciate that the three kinds of inference describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective [[modularity]] of its principal components.

===Art and science of inquiry===

For our present purposes, the first feature to note in distinguishing the three principal modes of reasoning from each other is whether each of them is exact or approximate in character. In this light, deduction is the only one of the three types of reasoning that can be made exact, in essence, always deriving true conclusions from true premisses, while abduction and induction are unavoidably approximate in their modes of operation, involving elements of fallible judgment in practice and inescapable error in their application.

The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena and problems that will no doubt continue to exceed the capacities of any finite resource, human or machine, to master. Situated in this larger reality, approximations can be judged appropriate only in relation to their context of use and can be judged fitting only with regard to a purpose in view.

A parallel distinction that is often made in this connection is to call deduction a ''[[demonstrative]]'' form of inference, while abduction and induction are classed as ''[[non-demonstrative]]'' forms of reasoning. Strictly speaking, the latter two modes of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved as useful heuristic strategies in the repertoire of the agent. But [[non-demonstrative]] ways of thinking are inherently subject to error, and must be constantly checked out and corrected as needed in practice.

In classical terminology, forms of judgment that require attention to the context and the purpose of the judgment are said to involve an element of 'art', in a sense that is judged to distinguish them from 'science', and in their renderings as expressive judgments to implicate arbiters in styles of [[rhetoric]], as contrasted with [[logic]].

In a figurative sense, this means that only deductive logic can be reduced to an exact theoretical science, while the practice of any empirical science will always remain to some degree an art.

===Zeroth order inquiry===

Many aspects of inquiry can be recognized and usefully studied in very basic logical settings, even simpler than the level of [[syllogism]], for example, in the realm of reasoning that is variously known as ''[[boolean algebra]]'', ''[[propositional logic|propositional calculus]]'', ''[[sentential calculus]]'', or ''[[zeroth-order logic]]''. By way of approaching the learning curve on the gentlest availing slope, we may well begin at the level of ''[[zeroth-order inquiry]]'', in effect, taking the syllogistic approach to inquiry only so far as the propositional or sentential aspects of the associated reasoning processes are concerned. One of the bonuses of doing this in the context of Peirce's logical work is that it provides us with doubly instructive exercises in the use of his [[logical graph]]s, taken at the level of his so-called '[[alpha graph]]s'.

In the case of propositional calculus or sentential logic, deduction comes down to applications of the [[transitive law]] for conditional implications and the approximate forms of inference hang on the properties that derive from these. In describing the various types of inference I will employ a few old terms of art from classical logic that are still of use in treating these kinds of simple problems in reasoning.

: '''Deduction''' takes a Case, the [[minor premiss]] <math>X \Rightarrow Y</math>
: and combines it with a Rule,the [[major premiss]] <math>Y \Rightarrow Z</math>
: to arrive at a Fact, the demonstrative [[conclusion]] <math>X \Rightarrow Z.</math>

: '''Induction''' takes a Case of the form <math>X \Rightarrow Y</math>
: and matches it with a Fact of the form <math>X \Rightarrow Z</math>
: to infer a Rule of the form <math>Y \Rightarrow Z.</math>

: '''Abduction''' takes a Fact of the form <math>X \Rightarrow Z</math>
: and matches it with a Rule of the form <math>Y \Rightarrow Z</math>
: to infer a Case of the form <math>X \Rightarrow Y.</math>

For ease of reference, Figure 1 and the Legend beneath it summarize the classical terminology for the three types of inference and the relationships among them.

{| align="center" cellpadding="8" style="text-align:center"
|
<pre>
o-------------------------------------------------o
| |
| Z |
| o |
| |\ |
| | \ |
| | \ |
| | \ |
| | \ |
| | \ R U L E |
| | \ |
| | \ |
| F | \ |
| | \ |
| A | \ |
| | o Y |
| C | / |
| | / |
| T | / |
| | / |
| | / |
| | / C A S E |
| | / |
| | / |
| | / |
| | / |
| |/ |
| o |
| X |
| |
| Deduction takes a Case of the form X => Y, |
| matches it with a Rule of the form Y => Z, |
| then adverts to a Fact of the form X => Z. |
| |
| Induction takes a Case of the form X => Y, |
| matches it with a Fact of the form X => Z, |
| then adverts to a Rule of the form Y => Z. |
| |
| Abduction takes a Fact of the form X => Z, |
| matches it with a Rule of the form Y => Z, |
| then adverts to a Case of the form X => Y. |
| |
| Even more succinctly: |
| |
| Abduction Deduction Induction |
| |
| Premiss: Fact Rule Case |
| Premiss: Rule Case Fact |
| Outcome: Case Fact Rule |
| |
o-------------------------------------------------o
Figure 1. Elementary Structure and Terminology
</pre>
|}

In its original usage a statement of Fact has to do with a deed done or a record made, that is, a type of event that is openly observable and not riddled with speculation as to its very occurrence. In contrast, a statement of Case may refer to a hidden or a hypothetical cause, that is, a type of event that is not immediately observable to all concerned. Obviously, the distinction is a rough one and the question of which mode applies can depend on the points of view that different observers adopt over time. Finally, a statement of a Rule is called that because it states a regularity or a regulation that governs a whole class of situations, and not because of its syntactic form. So far in this discussion, all three types of constraint are expressed in the form of conditional propositions, but this is not a fixed requirement. In practice, these modes of statement are distinguished by the roles that they play within an argument, not by their style of expression. When the time comes to branch out from the syllogistic framework, we will find that propositional constraints can be discovered and represented in arbitrary syntactic forms.

===Kinds of inference===

The three kinds of inference that Peirce would come to refer to as ''abductive'', ''deductive'', and ''inductive'' inference he gives his earliest systematic treatment in two series of lectures on the logic of science: the [[Harvard University]] Lectures of 1865 and the [[Lowell Institute]] Lectures of 1866. There he sums up the characters of the three kinds of reasoning in the following terms:

{| align="center" cellpadding="4" width="90%" 
|
We have then three different kinds of inference:

   Deduction or inference ''[[a priori and a posteriori (philosophy)|à priori]]'',

   Induction or inference ''[[à particularis]]'', and

   Hypothesis or inference ''[[à posteriori]]''.

(Peirce, "On the Logic of Science" (1865), CE 1, 267).
|}

Early in the first series of lectures Peirce gives a very revealing illustration of how he then thinks of the natures, operations, and relationships of this trio of inference types:

{| align="center" cellpadding="4" width="90%" 
|
 If I reason that certain conduct is wise
 because it has a character which belongs
 ''only'' to wise things, I reason ''à priori''.

 If I think it is wise because it once turned out
 to be wise, that is, if I infer that it is wise on
 this occasion because it was wise on that occasion,
 I reason inductively [''à particularis''].

 But if I think it is wise because a wise man does it,
 I then make the pure hypothesis that he does it
 because he is wise, and I reason ''à posteriori''.

(Peirce, "On the Logic of Science" (1865), CE 1, 180).
|}

We may begin the analysis of Peirce's example by making the following assignments of letters to the qualitative attributes mentioned in it:

:* A = 'Wisdom',
:* B = 'a certain character',
:* C = 'a certain conduct',
:* D = 'done by a wise man',
:* E = 'a certain occasion'.

Recognizing that a little more concreteness will serve as an aid to the understanding, let's augment the Spartan features of Peirce's illustration in the following way:

:* B = 'Benevolence', a certain character,
:* C = 'Contributes to Charity', a certain conduct,
:* E = 'Earlier today', a certain occasion.

The converging operation of all three reasonings is shown in Figure 2.

{| align="center" cellpadding="8" style="text-align:center"
|
<pre>
o---------------------------------------------------------------------o
| |
| D ("done by a wise man") |
| o |
| \* |
| \ * |
| \ * |
| \ * |
| \ * |
| \ * |
| \ * A ("a wise act") |
| \ o |
| \ /| * |
| \ / | * |
| \ / | * |
| . | o B ("benevolence", a certain character) |
| / \ | * |
| / \ | * |
| / \| * |
| / o |
| / * C ("contributes to charity", a certain conduct) |
| / * |
| / * |
| / * |
| / * |
| / * |
| /* |
| o |
| E ("earlier today", a certain occasion) |
| |
o---------------------------------------------------------------------o
Figure 2. A Thrice Wise Act
</pre>
|}

One of the styles of syntax that Aristotle uses for syllogistic propositions suggests the composite symbols that geometers have long used for labeling line intervals in a geometric figure, and it comports quite nicely with the Figure that we have just drawn. Specifically, the proposition that predicates X of the subject Y is represented by the digram 'XY' and associated with the line interval XY that descends from the point X to the point Y in the corresponding lattice diagram. In this wise we make the following observations:

The common proposition that concludes each argument is AC. Introducing the symbol "⇒" to denote the relation of logical implication, the proposition AC can be written as C ⇒ A, and read as "C implies A". Adopting the parenthetical form of Peirce's alpha graphs, in their ''existential interpretation'', AC can be written as (C (A)), and most easily comprehended as "not C without A". In the context of the present example, all of these forms are equally good ways of expressing the same concrete proposition, namely, "contributing to charity is wise".

:* Deduction could have obtained the Fact AC from the Rule AB, "benevolence is wisdom", along with the Case BC, "contributing to charity is benevolent".

:* Induction could have gathered the Rule AC, after a manner of saying that "contributing to charity is exemplary of wisdom", from the Fact AE, "the act of earlier today is wise", along with the Case CE, "the act of earlier today was an instance of contributing to charity".

:* Abduction could have guessed the Case AC, in a style of expression stating that "contributing to charity is explained by wisdom", from the Fact DC, "contributing to charity is done by this wise man", and the Rule DA, "everything that is wise is done by this wise man". Thus, a wise man, who happens to do all of the wise things that there are to do, may nevertheless contribute to charity for no good reason, and even be known to be charitable to a fault. But all of this notwithstanding, on seeing the wise man contribute to charity we may find it natural to conjecture, in effect, to consider it as a possibility worth examining further, that charity is indeed a mark of his wisdom, and not just the accidental trait or the immaterial peculiarity of his character — in essence, that wisdom is the ''cause'' of his contribution or the ''reason'' for his charity.

As a general rule, and despite many obvious exceptions, an English word that ends in ''-ion'' denotes equivocally either a process or its result. In our present application, this means that each of the words ''abduction'', ''deduction'', ''induction'' can be used to denote either the process of inference or the product of that inference, that is, the proposition to which the inference in question leads.

One of the morals of Peirce's illustration can now be drawn. It demonstrates in a very graphic fashion that the three kinds of inference are three kinds of process and not three kinds of proposition, not if one takes the word ''kind'' in its literal sense as denoting a ''genus'' of being, essence, or substance. Said another way, it means that being an abductive Case, a deductive Fact, or an inductive Rule is a category of relation, indeed, one that involves at the very least a triadic relation among propositions, and not a category of essence or substance, that is, not a property that inheres in the proposition alone.

This category distinction between the absolute, essential, or monadic predicates and the more properly relative predicates constitutes a very important theme in Peirce's architectonic. There is of course a parallel application of it in the theory of sign relations, or semiotics, where the distinctions among the sign relational roles of Object, Sign, and Interpretant are distinct ways of relating to other things, modes of relation that may vary from moment to moment in the extended trajectory of a sign process, and not distinctions that mark some fixed and eternal essence of the thing in itself.

In the normal course of inquiry, the elementary types of inference proceed in the order: Abduction, Deduction, Induction. However, the same building blocks can be assembled in other ways to yield different types of complex inferences. Of particular importance, reasoning by analogy can be analyzed as a combination of induction and deduction, in other words, as the abstraction and the application of a rule. Because a complicated pattern of analogical inference will be used in our example of a complete inquiry, it will help to prepare the ground if we first stop to consider an example of analogy in its simplest form.

====Abduction====

: ''Main article : [[Abductive reasoning]]''

Much of Peirce's work deals with the scientific and logical questions of [[knowledge]] and [[truth]], questions grounded in his experience as a working logician and experimental scientist, one who was a member of the international community of scientists and thinkers of his day. He made important contributions to [[deductive logic]] (see below), but was primarily interested in the logic of science and specifically in what he called [[abduction (logic)|abduction]] or "hypothesis", as opposed to [[deductive reasoning|deduction]] and [[inductive reasoning|induction]]. Abduction is the process whereby a hypothesis is generated, so that surprising facts may be explained. "There is a more familiar name for it than abduction", Peirce wrote, "for it is neither more nor less than guessing". Indeed, Peirce considered abduction to be at the heart not only of scientific research but of native human intelligence as well.

In his "Illustrations of the Logic of Science" (CE 3, 325-326), Peirce gives the following example of how abduction nests with deductive and inductive reasoning. Peirce begins by positing the following three statements:

:*''Rule'': "All the beans from this bag are white."

:*''Case'': "These beans are from this bag."

:*''Result'': "These beans are white."

Now let any two of these statements be Givens (their order not mattering), and let the remaining statement be the Conclusion. The result is an ''argument'', of which three kinds are possible:

{| align="center" cellpadding="4"
|-
!   !! Deduction !! Induction !! Abduction
|-
|- style="border-top:1px solid #999;"
|-
| ''Premiss'' || Rule || Case || Rule
|-
| ''Premiss'' || Case || Fact || Fact
|-
| ''Conclusion'' || Fact || Rule || Case
|}

====Deduction====

: ''Main article : [[Deductive reasoning]]''

====Induction====

: ''Main article : [[Inductive reasoning]]''

====Analogy====

The classic description of analogy in the syllogistic frame comes from Aristotle, who called this form of inference by the name ''paradeigma'', that is, reasoning by way of example or through the parallel comparison of cases.

{| align="center" cellpadding="4" width="90%" 
|
We have an Example [παραδειγμα, analogy] when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third. (Aristotle, "Prior Analytics", 2.24).
|}

Aristotle illustrates this pattern of argument with the following sample of reasoning. The setting is a discussion, taking place in Athens, on the issue of going to war with Thebes. It is apparently accepted that a war between Thebes and Phocis is or was a bad thing, perhaps from the objectivity lent by non-involvement or perhaps as a lesson of history.

{| align="center" cellpadding="4" width="90%" 
|
For example, let ''A'' be 'bad', ''B'' 'to make war on neighbors', ''C'' 'Athens against Thebes', and ''D'' 'Thebes against Phocis'. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, for example, that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is war against neighbors, it is evident that war against Thebes is bad. (Aristotle, "Prior Analytics", 2.24, with minor alterations).
|}

Aristotle's sample of argument from analogy may be analyzed in the following way:

First, a Rule is induced from the consideration of a similar Case and a relevant Fact:

:* Case: D ⇒ B, Thebes vs Phocis is war against neighbors.
:* Fact: D ⇒ A, Thebes vs Phocis is bad.
:* Rule: B ⇒ A, War against neighbors is bad.

Next, the Fact to be proved is deduced from the application of the previously induced Rule to the present Case:

:* Case: C ⇒ B, Athens vs Thebes is war against neighbors.
:* Rule: B ⇒ A, War against neighbors is bad.
:* Fact: C ⇒ A, Athens vs Thebes is bad.

In practice, of course, it would probably take a mass of comparable cases to establish a rule. As far as the logical structure goes, however, this quantitative confirmation only amounts to 'gilding the lily'. Perfectly valid rules can be guessed on the first try, abstracted from a single experience or adopted vicariously with no personal experience. Numerical factors only modify the degree of confidence and the strength of habit that govern the application of previously learned rules.

Figure 3 gives a graphical illustration of Aristotle's example of 'Example', that is, the form of reasoning that proceeds by Analogy or according to a Paradigm.

{| align="center" cellpadding="8" style="text-align:center"
|
<pre>
o-----------------------------------------------------------o
| |
| A |
| o |
| /*\ |
| / * \ |
| / * \ |
| / * \ |
| / * \ |
| / * \ |
| / R u l e \ |
| / * \ |
| / * \ |
| / * \ |
| / * \ |
| F a c t o F a c t |
| / * B * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / C a s e C a s e \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| o o |
| C D |
| |
| A = Atrocious, Adverse to All, A bad thing |
| B = Belligerent Battle Between Brethren |
| C = Contest of Athens against Thebes |
| D = Debacle of Thebes against Phocis |
| |
| A is a major term |
| B is a middle term |
| C is a minor term |
| D is a minor term, similar to C |
| |
o-----------------------------------------------------------o
Figure 3. Aristotle's "War Against Neighbors" Example
</pre>
|}

In this analysis of reasoning by Analogy, it is a complex or a mixed form of inference that can be seen as taking place in two steps:

:* The first step is an Induction that abstracts a Rule from a Case and a Fact.

:: Case: D ⇒ B, Thebes vs Phocis is a battle between neighbors.
:: Fact: D ⇒ A, Thebes vs Phocis is adverse to all.
:: Rule: B ⇒ A, A battle between neighbors is adverse to all.

:* The final step is a Deduction that applies this Rule to a Case to arrive at a Fact.

:: Case: C ⇒ B, Athens vs Thebes is a battle between neighbors.
:: Rule: B ⇒ A, A battle between neighbors is adverse to all.
:: Fact: C ⇒ A, Athens vs Thebes is adverse to all.

As we see, Aristotle analyzed analogical reasoning into a phase of inductive reasoning followed by a phase of deductive reasoning. Peirce would pick up the story at this juncture and eventually parse analogy in a couple of different ways, both of them involving all three types of inference: abductive, deductive, and inductive.

==Example of inquiry==

Examples of inquiry, that illustrate the full cycle of its abductive, deductive, and inductive phases, and yet are both concrete and simple enough to be suitable for a first (or zeroth) exposition, are somewhat rare in Peirce's writings, and so let us draw one from the work of fellow pragmatician John Dewey, analyzing it according to the model of zeroth-order inquiry that we developed above.

{| align="center" cellpadding="4" width="90%" 
|
A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something ''suggested''. The pedestrian ''feels'' the cold; he ''thinks of'' clouds and a coming shower. (John Dewey, ''How We Think'', pp. 6–7).
|}

===Once over quickly===

Let's first give Dewey's elegant example of inquiry in everyday life the quick once over, hitting just the high points of its analysis into Peirce's three kinds of reasoning.

====Abductive phase====

In Dewey's 'Rainy Day' or 'Sign of Rain' story, we find our peripatetic hero presented with a surprising Fact:

:* Fact: C ⇒ A, In the Current situation the Air is cool.

Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule:

:* Rule: B ⇒ A, Just Before it rains, the Air is cool.

This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, C ⇒ A, in its consequential feature A.

All of this suggests that the present Case may be one in which it is just about to rain:

:* Case: C ⇒ B, The Current situation is just Before it rains.

The whole mental performance, however automatic and semi-conscious it may be, that leads up from a problematic Fact and a previously settled knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling an [[abductive inference]].

====Deductive phase====

The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth. For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation. Thus, he now reflects on the Case just assumed:

:* Case: C ⇒ B, The Current situation is just Before it rains.

He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter B, we may safely suppose that our reasoner has already detached the consequence of the abduced Case, C ⇒ B, and has begun to expand on its further implications. So let us imagine that our up-looker has a more deliberate purpose in mind, and that his search for additional data is driven by the new-found, determinate Rule:

:* Rule: B ⇒ D, Just Before it rains, Dark clouds appear.

Contemplating the assumed Case in combination with this new Rule leads him by an immediate deduction to predict an additional Fact:

:* Fact: C ⇒ D, In the Current situation Dark clouds appear.

The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of [[deductive inference]].

====Inductive phase====

Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis. The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this process of inquiry necessary.

===Looking more closely===

====Seeding hypotheses====

Figure 4 gives a graphical illustration of Dewey's example of inquiry, isolating for the purposes of the present analysis the first two steps in the more extended proceedings that go to make up the whole inquiry.

{| align="center" cellpadding="8" style="text-align:center"
|
<pre>
o-----------------------------------------------------------o
| |
| A D |
| o o |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ R u l e R u l e / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * B * / |
| F a c t o F a c t |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \ C a s e / |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \*/ |
| o |
| C |
| |
| A = the Air is cool |
| B = just Before it rains |
| C = the Current situation |
| D = a Dark cloud appears |
| |
| A is a major term |
| B is a middle term |
| C is a minor term |
| D is a major term, associated with A |
| |
o-----------------------------------------------------------o
Figure 4. Dewey's "Rainy Day" Inquiry
</pre>
|}

In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps:

:* The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule.

:: Fact: C ⇒ A, In the Current situation the Air is cool.
:: Rule: B ⇒ A, Just Before it rains, the Air is cool.
:: Case: C ⇒ B, The Current situation is just Before it rains.

:* The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact.

:: Case: C ⇒ B, The Current situation is just Before it rains.
:: Rule: B ⇒ D, Just Before it rains, a Dark cloud will appear.
:: Fact: C ⇒ D, In the Current situation, a Dark cloud will appear.

This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start.

One other thing needs to be noticed here, the formal [[duality]] between this expansion phase of inquiry and the argument from [[analogy]]. This can be seen most clearly in the propositional [[lattice]] diagrams shown in Figures 3 and 4, where analogy exhibits a rough "A" shape and the first two steps of inquiry exhibit a rough "V" shape, respectively. Since we find ourselves repeatedly referring to this expansion phase of inquiry as a unit, let's give it a name that suggests its duality with [[analogical reasoning|analogy]] — '[[catalogical reasoning|catalogy]]' will do for the moment. This usage is apt enough if one thinks of a catalogue entry for an item as a text that lists its salient features. Notice that [[analogical reasoning|analogy]] has to do with the examples of a given quality, while [[catalogical reasoning|catalogy]] has to do with the qualities of a given example. Peirce noted similar forms of duality in many of his early writings, leading to the consummate treatment in his 1867 paper [http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm "On a New List of Categories"] (CP 1.545-559, CE 2, 49-59).

====Weeding hypotheses====

In order to comprehend the bearing of [[inductive reasoning]] on the closing phases of inquiry there are a couple of observations that we need to make:

:* First, we need to recognize that smaller inquiries are typically woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by a single agent or by a complex community.

:* Further, we need to consider the different ways in which the particular instances of inquiry can be related to ongoing inquiries at larger scales. Three modes of inductive interaction between the micro-inquiries and the macro-inquiries that are salient here can be described under the headings of the 'Learning', the 'Transfer', and the 'Testing' of rules.

====Analogy of experience====

Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from the masses of experience where they are learned to the moments of experience where they are applied. Inductive reasoning is involved in the learning and the transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application.

:* Learning. The principal way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules that goes into the knowledge base, or ever gets used along the way.

:* Transfer. The continuing way that induction contributes to an ongoing inquiry is through the exploit of analogy, a two-step combination of induction and deduction that serves to transfer rules from one context to another.

:* Testing. Finally, every inquiry that makes use of a knowledge base constitutes a 'field test' of its accumulated contents. If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there is a prima facie reason to reconsider and possibly to amend some of its rules.

Let's now consider how these principles of learning, transfer, and testing apply to John Dewey's 'Sign of Rain' example.

=====Learning=====

Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference.

For example, a rule like:

:* Rule: B ⇒ A, Just Before it rains, the Air is cool,

is usually induced from a consideration of many past events, in a manner that can be rationally reconstructed as follows:

:* Case: C ⇒ B, In Certain events, it is just Before it rains,
:* Fact: C ⇒ A, In Certain events, the Air is cool,
: ------------------------------------------------------------------------------------------
:* Rule: B ⇒ A, Just Before it rains, the Air is cool.

However, the very same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a presumptive theory.

=====Transfer=====

What is it that gives a distinctively inductive character to the acquisition of a knowledge base? It is evidently the 'analogy of experience' that underlies its useful application. Whenever we find ourselves prefacing an argument with the phrase 'If past experience is any guide …' then we can be sure that this principle has come into play. We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application. What we mean in practice is this: 'If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience'. This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules.

Here are the details of how this notion of transfer works out in the case of the 'Sign of Rain' example:

Let K(pres) be a portion of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules, as follows:

:* K(pres) = (B ⇒ A) and (B ⇒ D).

K(pres) is the present knowledge base, expressed in the form of a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements in terms of their [[logical model]]s, that is, in terms of the primitive circumstances or the elements of experience over which they hold true.

:* Let E(past) be the chosen set of experiences, or the circumstances that we have in mind when we refer to 'past experience'.

:* Let E(poss) be the collective set of experiences, or the projective total of possible circumstances.

:* Let E(pres) be the present experience, or the circumstances that are present to the reasoner at the current moment.

If we think of the knowledge base K(pres) as referring to the 'regime of experience' over which it is valid, then all of these sets of models can be compared by the simple relations of [[set inclusion]] or [[logical implication]].

Figure 5 schematizes this way of viewing the 'analogy of experience'.

{| align="center" cellpadding="8" style="text-align:center"
|
<pre>
o-----------------------------------------------------------o
| |
| K(pres) |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / Rule \ |
| / | \ |
| / | \ |
| / | \ |
| / E(poss) \ |
| Fact / o \ Fact |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * Case Case * \ |
| / * * \ |
| / * * \ |
| /* *\ |
| o<<<---------------<<<---------------<<<o |
| E(past) Analogy Morphism E(pres) |
| More Known Less Known |
| |
o-----------------------------------------------------------o
Figure 5. Analogy of Experience
</pre>
|}

In these terms, the ''analogy of experience'' proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence:

Inductive Phase:

:* Given Case: E(past) ⇒ E(poss), Chosen events fairly sample Collective events.
:* Given Fact: E(past) ⇒ K(pres), Chosen events support the Knowledge regime.
: -----------------------------------------------------------------------------------------------------------------------------
:* Induce Rule: E(poss) ⇒ K(pres), Collective events support the Knowledge regime.

Deductive Phase:

:* Given Case: E(pres) ⇒ E(poss), Current events fairly sample Collective events.
:* Given Rule: E(poss) ⇒ K(pres), Collective events support the Knowledge regime.
: --------------------------------------------------------------------------------------------------------------------------------
:* Deduce Fact: E(pres) ⇒ K(pres), Current events support the Knowledge regime.

=====Testing=====

If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the utility or the validity of his knowledge base. But we must leave our foulweather friend for now and defer the logical analysis of this testing phase to another occasion.

==References==

* [[Dana Angluin|Angluin, Dana]] (1989), "Learning with Hints", pp. 167–181 in David Haussler and Leonard Pitt (eds.), ''Proceedings of the 1988 Workshop on Computational Learning Theory'', MIT, 3–5 August 1988, Morgan Kaufmann, San Mateo, CA, 1989.

* [[Aristotle]], "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181–531 in ''Aristotle, Volume 1'', [[Loeb Classical Library]], [[Heinemann (book publisher)|William Heinemann]], London, UK, 1938.

* Awbrey, S.M., and Awbrey, J.L. (May 2001), "Conceptual Barriers to Creating Integrative Universities", ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269–284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].

* Awbrey, S.M., and Awbrey, J.L. (September 18, 1999), "Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century", ''Second International Conference of the Journal ''Organization'' '', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].

* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), "Interpretation as Action : The Risk of Inquiry", ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40–52. [http://www.chss.montclair.edu/inquiry/fall95/awbrey.html Online].

* Awbrey, J.L., and Awbrey, S.M. (June 1992), "Interpretation as Action : The Risk of Inquiry", ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.

* Awbrey, S.M., and Awbrey, J.L. (May 1991), "An Architecture for Inquiry : Building Computer Platforms for Discovery", ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874–875. [http://home.m04.itscom.net/hhomey/tmp-a.html Online].

* Awbrey, J.L., and Awbrey, S.M. (January 1991), "Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry", Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.

* Awbrey, J.L., and Awbrey, S.M. (August 1990), "Exploring Research Data Interactively. Theme One : A Program of Inquiry", ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9–15.

* [[Cornelius F. Delaney|Delaney, C.F.]] (1993), ''Science, Knowledge, and Mind: A Study in the Philosophy of C.S. Peirce'', University of Notre Dame Press, Notre Dame, IN.

* [[John Dewey|Dewey, John]] (1910), ''How We Think'', [[D.C. Heath]], Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.

* Dewey, John (1938), ''Logic: The Theory of Inquiry'', Henry Holt and Company, New York, NY, 1938. Reprinted as pp. 1–527 in ''John Dewey, The Later Works, 1925–1953, Volume 12 : 1938'', Jo Ann Boydston (ed.), Kathleen Poulos (text. ed.), [[Ernest Nagel]] (intro.), Southern Illinois University Press, Carbondale and Edwardsville, IL, 1986.

* [[Susan Haack|Haack, Susan]] (1993), ''Evidence and Inquiry : Towards Reconstruction in Epistemology'', Blackwell Publishers, Oxford, UK.

* [[Norwood Russell Hanson|Hanson, Norwood Russell]] (1958), ''Patterns of Discovery, An Inquiry into the Conceptual Foundations of Science'', Cambridge University Press, Cambridge, UK.

* [[Vincent F. Hendricks|Hendricks, Vincent F.]] (2005), ''Thought 2 Talk : A Crash Course in Reflection and Expression'', Automatic Press, New York, NY.

* [[Cheryl J. Misak|Misak, Cheryl J.]] (1991), ''Truth and the End of Inquiry, A Peircean Account of Truth'', Oxford University Press, Oxford, UK.

* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Sanders Peirce|Peirce, C.S.]], (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph.

* [[Robert C. Stalnaker|Stalnaker, Robert C.]] (1984), ''Inquiry'', MIT Press, Cambridge, MA.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry Inquiry @ InterSciWiki]
* [http://mywikibiz.com/Inquiry Inquiry @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Inquiry Inquiry @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Inquiry Inquiry @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Inquiry Inquiry @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry Inquiry], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Inquiry Inquiry], [http://mywikibiz.com/ MyWikiBiz]
* [http://forum.wolframscience.com/showthread.php?threadid=595 Inquiry], [http://forum.wolframscience.com/ NKS Forum]
* [http://semanticweb.org/wiki/Inquiry Inquiry], [http://semanticweb.org/ SemanticWeb]
* [http://wikinfo.org/w/index.php/Inquiry Inquiry], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Inquiry Inquiry], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Inquiry Inquiry], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Inquiry&oldid=71880922 Inquiry], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Computer Science]]
[[Category:Critical Thinking]]
[[Category:Cybernetics]]
[[Category:Education]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Information Theory]]
[[Category:Inquiry]]
[[Category:Inquiry Driven Systems]]
[[Category:Intelligence Amplification]]
[[Category:Learning Organizations]]
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[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
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[[Category:Science]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Systems Science]]

Triadic relation

2015-11-15T22:04:05Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a '''ternary relation'''. One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations.

Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms.

==Examples from mathematics==

For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner.

The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence. This space is constructed as a 3-fold [[cartesian power]] in the following way.

The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math>

The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math>

The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math>

In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math>

The relation <math>L_0\!</math> is defined as follows:

{| align="center" cellpadding="6" width="90%"
| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math>
|}

The relation <math>L_0\!</math> is the set of four triples enumerated here:

{| align="center" cellpadding="6" width="90%"
| <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>
|}

The relation <math>L_1\!</math> is defined as follows:

{| align="center" cellpadding="6" width="90%"
| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math>
|}

The relation <math>L_1\!</math> is the set of four triples enumerated here:

{| align="center" cellpadding="6" width="90%"
| <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>
|}

The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|}

 

==Examples from semiotics==

The study of signs — the full variety of significant forms of expression — in relation to the things that signs are significant ''of'', and in relation to the beings that signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''. As just described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''.

The term ''[[semiosis]]'' refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''.

For example, consider the aspects of sign use that concern two people — let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> — in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math> For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively.

Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math> In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math>

The set-up so far is summarized as follows:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccc}
L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\
\\
O & = & \{ \mathrm{A}, \mathrm{B} \} \\
\\
S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
\\
I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
\\
\end{array}</math>
|}

The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\{ &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
\\
&
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) &
\}.
\end{array}</math>
|}

The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>

The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\{ &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
\\
&
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) &
\}.
\end{array}</math>
|}

The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs. For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>

The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
|<math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}

 

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki]
* [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath]
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Cognitive Sciences]]
[[Category:Computer Science]]
[[Category:Formal Sciences]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Information Theory]]
[[Category:Inquiry]]
[[Category:Intelligence Amplification]]
[[Category:Knowledge Representation]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
[[Category:Relation Theory]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Syntax]]

Sign relation

2015-11-15T21:35:04Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''sign relation''' is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce.

==Anthesis==

{| align="center" cellpadding="6" width="90%"
|
Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun. (C.S. Peirce, “Syllabus” (''c''. 1902), ''Collected Papers'', CP 2.274).
|}

In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or ''[[semiosis]]'', Peirce uses the technical term ''representamen'' for his concept of a sign, but the shorter word is precise enough, so long as one recognizes that its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

==Definition==

One of Peirce's clearest and most complete definitions of a sign is one that he gives in the context of providing a definition for ''logic'', and so it is informative to view it in that setting.

{| align="center" cellpadding="6" width="90%"
|
Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, NEM 4, 20–21).
|}

In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ''ontological'' property of a thing, or whether it is a relational, interpretive, and mutable role that a thing can be said to have only within a particular context of relationships.

Peirce's definition of a ''sign'' defines it in relation to its ''object'' and its ''interpretant sign'', and thus it defines signhood in ''[[logic of relatives|relative terms]]'', by means of a predicate with three places. In this definition, signhood is a role in a [[triadic relation]], a role that a thing bears or plays in a given context of relationships — it is not as an ''absolute'', ''non-relative'' property of a thing-in-itself, one that it possesses independently of all relationships to other things.

Some of the terms that Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

:* '''Correspondence.''' From the way that Peirce uses this term throughout his work, it is clear that he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself. In particular, his use of this term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations that are bandied about in contemporary controversies about “correspondence theories of truth”.

:* '''Determination.''' Peirce's concept of determination is broader in several directions than the sense of the word that refers to strictly deterministic causal-temporal processes. First, and especially in this context, he is invoking a more general concept of determination, what is called a ''formal'' or ''informational'' determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms. Second, he characteristically allows for what is called ''determination in measure'', that is, an order of determinism that admits a full spectrum of more and less determined relationships.

:* '''Non-psychological.''' Peirce's “non-psychological conception of logic” must be distinguished from any variety of ''anti-psychologism''. He was quite interested in matters of psychology and had much of import to say about them. But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a ''[[normative science]]'' where psychology is a ''[[descriptive science]]'', and so they have very different aims, methods, and rationales.

==Signs and inquiry==
: ''Main article : [[Inquiry]]''

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]]. In fact, the correspondence between the two studies exhibits so many congruences and parallels that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (John Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

==Examples of sign relations==

Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: “Ann”, “Bob”, “I”, “you”.

:* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math>

:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math>

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[three-place relation]] called the ''sign relation'' of that interpreter.

Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math>

Broadly speaking, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations typically contemplated in formal settings are usually constrained to having <math>I \subseteq S\!</math>. In this case interpretants are just a special variety of signs and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''. In the forthcoming examples <math>S\!</math> and <math>I\!</math> are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math>

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:

{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{ccl}
O & = & \text{Object Domain}
\\[6pt]
S & = & \text{Sign Domain}
\\[6pt]
I & = & \text{Interpretant Domain}
\end{array}</math>
|}

Introducing a few abbreviations for use in considering the present Example, we have the following data:

{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{cclcl}
O
& = &
\{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \}
\\[6pt]
S
& = &
\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}
& = &
\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}
\\[6pt]
I
& = &
\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}
& = &
\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}
\end{array}</math>
|}

In the present example, <math>S = I = \text{Syntactic Domain}\!</math>.

The next two Tables give the sign relations associated with the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.\!</math>

 

{| align="center" style="width:92%"
| width="50%" |
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 1a.} ~~ {L_\mathrm{A}} = \text{Sign Relation of Interpreter A}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
| width="50%" |
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 1b.} ~~ {L_\mathrm{B}} = \text{Sign Relation of Interpreter B}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
|<math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|}

 

These Tables codify a rudimentary level of interpretive practice for the agents <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <math>(o, s, i)\!</math> called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

==Dyadic aspects of sign relations==

For an arbitrary triadic relation <math>L \subseteq O \times S \times I,\!</math> whether it is a sign relation or not, there are six dyadic relations that can be obtained by ''projecting'' <math>L\!</math> on one of the planes of the <math>OSI\!</math>-space <math>O \times S \times I.\!</math> The six dyadic projections of a triadic relation <math>L\!</math> are defined and notated as follows:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Dyadic Projections of Triadic Relations}\!</math>
|
<math>\begin{matrix}
L_{OS} & = & \mathrm{proj}_{OS}(L) & = & \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}
\\[6pt]
L_{SO} & = & \mathrm{proj}_{SO}(L) & = & \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}
\\[6pt]
L_{IS} & = & \mathrm{proj}_{IS}(L) & = & \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}
\\[6pt]
L_{SI} & = & \mathrm{proj}_{SI}(L) & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}
\\[6pt]
L_{OI} & = & \mathrm{proj}_{OI}(L) & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}
\\[6pt]
L_{IO} & = & \mathrm{proj}_{IO}(L) & = & \{ (i, o) \in I \times O ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}
\end{matrix}</math>
|}

 

By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language.

{| align="center" cellpadding="6" width="90%"
|
The dyadic relation that results from the projection of <math>L\!</math> on the <math>OS\!</math>-plane <math>O \times S\!</math> is written briefly as <math>L_{OS}\!</math> or written more fully as <math>\mathrm{proj}_{OS}(L),\!</math> and it is defined as the set of all ordered pairs <math>(o, s)\!</math> in the cartesian product <math>O \times S\!</math> for which there exists an ordered triple <math>(o, s, i)\!</math> in <math>L\!</math> for some interpretant <math>i\!</math> in the interpretant domain <math>I.\!</math>
|}

In the case where <math>L\!</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L\!</math> can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies.

===Denotation===

One aspect of a sign's complete meaning is concerned with the reference that a sign has to its objects, which objects are collectively known as the ''denotation'' of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane that is spanned by its object domain and its sign domain.

The dyadic relation that makes up the ''denotative'', ''referential'', or ''semantic'' aspect or component of a sign relation <math>L\!</math> is notated as <math>\mathrm{Den}(L).\!</math> Information about the denotative aspect of meaning is obtained from <math>L\!</math> by taking its projection on the object-sign plane, in other words, on the 2-dimensional space that is generated by the object domain <math>O\!</math> and the sign domain <math>S.\!</math> This component of a sign relation <math>L\!</math> can be written in any of the forms, <math>\mathrm{proj}_{OS} L,\!</math>   <math>L_{OS},\!</math>   <math>\mathrm{proj}_{12} L,\!</math>   <math>L_{12},\!</math> and it is defined as follows:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{Den}(L)
& = &
\mathrm{proj}_{OS} L
& = &
\{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.
\end{matrix}</math>
|}

Looking to the denotative aspects of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> various rows of the Tables specify, for example, that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B},\!</math> whereas <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to denote <math>\mathrm{B}\!</math> and <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to denote <math>\mathrm{A}.\!</math> All of these denotative references are summed up in the projections on the <math>OS\!</math>-plane, as shown in the following Tables:

 

{| align="center" style="width:90%"
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 3a.} ~~ \mathrm{Den}(L_\mathrm{A}) = \mathrm{proj}_{OS}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 3b.} ~~ \mathrm{Den}(L_\mathrm{B}) = \mathrm{proj}_{OS}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|}

 

===Connotation===

Another aspect of meaning concerns the connection that a sign has to its interpretants within a given sign relation. As before, this type of connection can be vacuous, singular, or plural in its collection of terminal points, and it can be formalized as the dyadic relation that is obtained as a planar projection of the triadic sign relation in question.

The connection that a sign makes to an interpretant is here referred to as its ''connotation''. In the full theory of sign relations, this aspect of meaning includes the links that a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language.

Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. For a given sign relation <math>L,\!</math> the dyadic relation that constitutes the ''connotative aspect'' or ''connotative component'' of <math>L\!</math> is notated as <math>\mathrm{Con}(L).\!</math>

The connotative aspect of a sign relation <math>L\!</math> is given by its projection on the plane of signs and interpretants, and is therefore defined as follows:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{Con}(L)
& = &
\mathrm{proj}_{SI} L
& = &
\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.
\end{matrix}</math>
|}

All of these connotative references are summed up in the projections on the <math>SI\!</math>-plane, as shown in the following Tables:

 

{| align="center" style="width:90%"
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 4a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 4b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|}

 

===Ennotation===

The aspect of a sign's meaning that arises from the dyadic relation of its objects to its interpretants has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role, but here we have to consider a dyadic relation between objects and interpretants that is mediated by the sign from an off-stage position, as it were. As a relation between objects and interpretants that is mediated by a sign, this aspect of meaning may be referred to as the ''ennotation'' of a sign, and the dyadic relation that constitutes the ''ennotative aspect'' of a sign relation <math>L\!</math> may be notated as <math>\mathrm{Enn}(L).\!</math>

The ennotational component of meaning for a sign relation <math>L\!</math> is captured by its projection on the plane of the object and interpretant domains, and it is thus defined as follows:

{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{Enn}(L)
& = &
\mathrm{proj}_{OI} L
& = &
\{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}.
\end{matrix}</math>
|}

As it happens, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are fully symmetric with respect to exchanging signs and interpretants, so all the data of <math>\mathrm{proj}_{OS} L_\mathrm{A}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{A}\!</math> and all the data of <math>\mathrm{proj}_{OS} L_\mathrm{B}\!</math> is echoed unchanged in <math>\mathrm{proj}_{OI} L_\mathrm{B}.\!</math>

 

{| align="center" style="width:90%"
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 5a.} ~~ \mathrm{Enn}(L_\mathrm{A}) = \mathrm{proj}_{OI}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 5b.} ~~ \mathrm{Enn}(L_\mathrm{B}) = \mathrm{proj}_{OI}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|}

 

==Semiotic equivalence relations==

A ''semiotic equivalence relation'' (SER) is a special type of equivalence relation that arises in the analysis of sign relations. As a general rule, any equivalence relation is closely associated with a family of equivalence classes that partition the underlying set of elements, frequently called the ''domain'' or ''space'' of the relation. In the case of a SER, the equivalence classes are called ''semiotic equivalence classes'' (SECs) and the partition is called a ''semiotic partition'' (SEP).

The sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> have many interesting properties that are not possessed by sign relations in general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> on the <math>SI\!</math>-plane, notated as <math>\mathrm{proj}_{SI} L_\mathrm{A}\!</math> and <math>\mathrm{proj}_{SI} L_\mathrm{B},\!</math> respectively. The 2-adic relations on <math>S \times I\!</math> induced by these projections are also referred to as the ''connotative components'' of the corresponding sign relations, notated as <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B}),\!</math> respectively. Tables 6a and 6b show the corresponding connotative components.

 

{| align="center" style="width:90%"
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 6a.} ~~ \mathrm{Con}(L_\mathrm{A}) = \mathrm{proj}_{SI}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>S\!</math>
| width="50%" | <math>I\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}</math>
|}
| width="50%" |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 6b.} ~~ \mathrm{Con}(L_\mathrm{B}) = \mathrm{proj}_{SI}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>S\!</math>
| width="50%" | <math>I\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
\\[4pt]
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}</math>
|}
|}

 

One nice property possessed by the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is that their connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> form a pair of equivalence relations on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such a structure as a ''semiotic equivalence relation'' (SER) since it equates signs that mean the same thing to some interpreter. Each of the SERs, <math>\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S\!</math> partitions the whole collection of signs into ''semiotic equivalence classes'' (SECs). This makes for a strong form of representation in that the structure of the interpreters' common object domain <math>\{ \mathrm{A}, \mathrm{B} \}\!</math> is reflected or reconstructed, part for part, in the structure of each of their ''semiotic partitions'' (SEPs) of the syntactic domain <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> But it needs to be observed that the semiotic partitions for interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.

Information about the contrasting patterns of semiotic equivalence induced by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> is summarized in Tables 7a and 7b. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> are orthogonal to each other.

 

{| align="center" style="width:92%"
| width="50%" |
{| align="center" cellpadding="20" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 7a.} ~~ \text{Semiotic Partition for Interpreter A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="20" cellspacing="1" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 7b.} ~~ \text{Semiotic Partition for Interpreter B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| style="border-left:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|}
|}

 

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

As a general consideration, if <math>E\!</math> is an equivalence relation on a set <math>X,\!</math> then every element <math>x\!</math> of <math>X\!</math> belongs to a unique equivalence class under <math>E\!</math> called ''the equivalence class of <math>x\!</math> under <math>E\!</math>''. Convention provides the ''square bracket notation'' for denoting this equivalence class, either in the subscripted form <math>[x]_E\!</math> or in the simpler form <math>[x]\!</math> when the subscript <math>E\!</math> is understood. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'' or an ''equivalence'' and may be expressed in any of the following ways:

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{array}{clc}
(x, y) & \in & E
\\[4pt]
x & \in & [y]_E
\\[4pt]
y & \in & [x]_E
\\[4pt]
[x]_E & = & [y]_E
\\[4pt]
x & =_E & y
\end{array}</math>
|}

Thus we have the following definitions:

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{array}{ccc}
[x]_E & = & \{ y \in X : (x, y) \in E \}
\\[6pt]
x =_E y & \Leftrightarrow & (x, y) \in E
\end{array}</math>
|}

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If <math>L\!</math> is a sign relation whose connotative component or syntactic projection <math>L_{SI}\!</math> is an equivalence relation on <math>S = I,\!</math> let <math>[s]_L\!</math> be the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math> That is to say, <math>[s]_L = [s]_{L_{SI}}.\!</math> A statement that the signs <math>x\!</math> and <math>y\!</math> are equivalent under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and may be written in either of the following equivalent forms:

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{array}{clc}
[x]_L & = & [y]_L
\\[6pt]
x & =_L & y
\end{array}</math>
|}

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i)\!</math> in a sign relation <math>L,\!</math> it is permissible to let <math>[o]_L\!</math> be defined as <math>[s]_L.\!</math> These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

The semiotic equivalence relation for interpreter <math>\mathrm{A}\!</math> yields the following semiotic equations:

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{matrix}
[ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}}
& = &
[ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}}
\\[6pt]
[ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}}
& = &
[ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}}
\end{matrix}</math>
|}

or

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
& =_{L_\mathrm{A}} &
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
& =_{L_\mathrm{A}} &
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\end{matrix}</math>
|}

Thus it induces the semiotic partition:

{| align="center" cellpadding="12" style="text-align:center; width:100%"
| <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.\!</math>
|}

The semiotic equivalence relation for interpreter <math>\mathrm{B}\!</math> yields the following semiotic equations:

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{matrix}
[ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}}
& = &
[ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}}
\\[6pt]
[ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}}
& = &
[ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}}
\end{matrix}</math>
|}

or

{| align="center" style="text-align:center; width:100%"
|
<math>\begin{matrix}
{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}
& =_{L_\mathrm{B}} &
{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}
\\[6pt]
{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}
& =_{L_\mathrm{B}} &
{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}
\end{matrix}</math>
|}

Thus it induces the semiotic partition:

{| align="center" cellpadding="12" style="text-align:center; width:100%"
| <math>\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.\!</math>
|}

==Graphical representations==

The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties.

By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math>

The denotative components <math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I =\!</math> <math>\{ \mathrm{A}, \mathrm{B}, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> The arcs are given as follows:

{| align="center" cellspacing="6" width="90%"
|
<math>\mathrm{Den}(L_\mathrm{A})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math>
|-
|
<math>\mathrm{Den}(L_\mathrm{B})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}\!</math> to <math>\mathrm{A}\!</math> and an arc from each point of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}\!</math> to <math>\mathrm{B}.\!</math>
|}

<math>\mathrm{Den}(L_\mathrm{A})\!</math> and <math>\mathrm{Den}(L_\mathrm{B})\!</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> evaluate the signs in <math>S\!</math> according to their own frames of reference.

The connotative components <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =\!</math> <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> Since <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> are semiotic equivalence relations, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows:

{| align="center" cellspacing="6" width="90%"
|
<math>\mathrm{Con}(L_\mathrm{A})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math>
|-
|
<math>\mathrm{Con}(L_\mathrm{B})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}.\!</math>
|}

Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively.

==Six ways of looking at a sign relation==

In the context of 3-adic relations in general, Peirce provides the following illustration of the six ''converses'' of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation:

: So in a triadic fact, say, for example 
{| align="center" cellspacing="8" style="width:72%"
| align="center" | ''A'' gives ''B'' to ''C''
|}
: we make no distinction in the ordinary logic of relations between the ''subject nominative'', the ''direct object'', and the ''indirect object''. We say that the proposition has three ''logical subjects''. We regard it as a mere affair of English grammar that there are six ways of expressing this: 
{| align="center" cellspacing="8" style="width:72%"
| style="width:36%" | ''A'' gives ''B'' to ''C''
| style="width:36%" | ''A'' benefits ''C'' with ''B''
|-
| ''B'' enriches ''C'' at expense of ''A''
| ''C'' receives ''B'' from ''A''
|-
| ''C'' thanks ''A'' for ''B''
| ''B'' leaves ''A'' for ''C''
|}
: These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, “The Categories Defended”, MS 308 (1903), EP 2, 170–171).

===IOS===

(Text in Progress)

===ISO===

(Text in Progress)

===OIS===

{| align="center" cellpadding="6" width="90%"
|
Words spoken are symbols or signs (σύμβολα) of affections or impressions (παθήματα) of the soul (ψυχή); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (σημεια), are the same for the whole of mankind, as are also the objects (πράγματα) of which those affections are representations or likenesses, images, copies (ομοιώματα). (Aristotle, ''De Interpretatione'', 1.16a4).
|}

===OSI===

(Text in Progress)

===SIO===

{| align="center" cellpadding="6" width="90%"
|
Logic will here be defined as ''formal semiotic''. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign determined or created by it, into the same sort of correspondence with something, ''C'', its ''object'', as that in which itself stands to ''C''. It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized. (C.S. Peirce, “Application to the Carnegie Institution”, L75 (1902), NEM 4, 20–21).
|}

===SOI===

{| align="center" cellpadding="6" width="90%"
|
A ''Sign'' is anything which is related to a Second thing, its ''Object'', in respect to a Quality, in such a way as to bring a Third thing, its ''Interpretant'', into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ''ad infinitum''. (CP 2.92, quoted in Fisch 1986, p. 274)
|}

==References==

==Bibliography==

===Primary sources===

* [[Charles Sanders Peirce (Bibliography)]]

===Secondary sources===

* Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40–52. [http://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [http://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].

* Deledalle, Gérard (2000), ''C.S. Peirce's Philosophy of Signs'', Indiana University Press, Bloomington, IN.

* Eisele, Carolyn (1979), in ''Studies in the Scientific and Mathematical Philosophy of C.S. Peirce'', Richard Milton Martin (ed.), Mouton, The Hague.

* Esposito, Joseph (1980), ''Evolutionary Metaphysics : The Development of Peirce's Theory of Categories'', Ohio University Press (?).

* Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Indiana University Press, Bloomington, IN.

* Houser, N., Roberts, D.D., and Van Evra, J. (eds.)(1997), ''Studies in the Logic of C.S. Peirce'', Indiana University Press, Bloomington, IN.

* Liszka, J.J. (1996), ''A General Introduction to the Semeiotic of C.S. Peirce'', Indiana University Press, Bloomington, IN.

* Misak, C. (ed.)(2004), ''Cambridge Companion to C.S. Peirce'', Cambridge University Press.

* Moore, E., and Robin, R. (1964), ''Studies in the Philosophy of C.S. Peirce, Second Series'', University of Massachusetts Press, Amherst, MA.

* Murphey, M. (1961), ''The Development of Peirce's Thought''. Reprinted, Hackett, Indianapolis, IN, 1993.

* Percy, Walker (2000), pp. 271–291 in ''Signposts in a Strange Land'', P. Samway (ed.), Saint Martin's Press.

==Resources==

* [http://www.helsinki.fi/science/commens/dictionary.html The Commens Dictionary of Peirce's Terms]
** [http://www.helsinki.fi/science/commens/terms/semeiotic.html Semeiotic, Semiotic, Semeotic, Semeiotics]
* [http://forum.wolframscience.com/archive/ A New Kind Of Science • Forum Archive]
** [http://forum.wolframscience.com/archive/topic/647.html Excerpts from Peirce on Sign Relations]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation @ InterSciWiki]
* [http://mywikibiz.com/Sign_relation Sign Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Sign_relation Sign Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Sign_relation Sign Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Sign_relation Sign Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/SignRelation Sign Relation], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Sign_relation Sign Relation], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Sign_relation Sign Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Sign_relation Sign Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
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[[Category:Computer Science]]
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[[Category:Information Theory]]
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[[Category:Knowledge Representation]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Pragmatism]]
[[Category:Relation Theory]]
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[[Category:Semiotics]]
[[Category:Syntax]]

Relative term

2015-11-15T15:27:02Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''relative term''', also called a '''rhema''' or a '''rheme''', is a logical term that requires reference to any number of other objects, called the ''correlates'' of the term, in order to denote a definite object, called the ''relate'' (pronounced with the accent on the first syllable) of the relative term in question.  A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks, for example, ''lover of __'', or ''giver of __ to __''.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term @ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relative_term Relative Term @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relative_term Relative Term], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relative_term Relative Term], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relative_term Relative Term], [http://semanticweb.org/ Semantic Web]
* [http://en.wikiversity.org/wiki/Relative_term Relative Term], [http://em.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relative_term Relative Term], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relative_term&oldid=35330741 Relative Term], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Relation Theory]]
[[Category:Science]]
[[Category:Semiotics]]

Relation reduction

2015-11-15T03:36:00Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In logic and mathematics, '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by a set of other relations, called the ''relation dataset''. The relation under examination is called the ''reductandum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a set of other relations, called the ''reduciens'' or the ''relational base'', each of which is properly simpler in a specified way than the relation under examination.

A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens.

A relation that is not uniquely determined by a particular relation dataset is said to be ''irreducible'' in just that respect. A relation that is not uniquely determined by any relation dataset in a particular class of relation datasets is said to be ''irreducible'' in respect of that class.

==Discussion==

The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of “getting new relations from old” in order to say precisely what is meant by the claim that the relation <math>L\!</math> is reducible to the set of relations <math>\{ L_j : j \in J \}.\!</math> This amounts to claiming one can be given a set of ''properly simpler'' relations <math>L_j\!</math> for values <math>j\!</math> in a given index set <math>J\!</math> and that this collection of data would suffice to fix the original relation <math>L\!</math> that one is seeking to analyze, determine, specify, or synthesize.

In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely:

# Reduction under composition.
# Reduction under projections.

As it happens, there is an interesting relationship between these two notions of reducibility, the implications of which may be taken up partly in parallel with the discussion of the basic concepts.

==Projective reducibility of relations==

It is convenient to begin with the ''projective reduction'' of relations, partly because this type of reduction is simpler and more intuitive (in the visual sense), but also because a number of conceptual tools that are needed in any case arise quite naturally in the projective setting.

The work of intuiting how projections operate on multidimensional relations is often facilitated by keeping in mind the following sort of geometric image:

* Picture a <math>k\!</math>-adic relation <math>L\!</math> as a body that resides in a <math>k\!</math>-dimensional space <math>X.\!</math> If the domains of the relation <math>L\!</math> are <math>X_1, \ldots, X_k,\!</math> then the ''extension'' of the relation <math>L\!</math> is a subset of the cartesian product <math>X = X_1 \times \ldots \times X_k.\!</math>

In this setting the interval <math>K = [1, k] = \{ 1, \ldots, k \}\!</math> is called the ''index set'' of the ''indexed family'' of sets <math>X_1, \ldots, X_k.\!</math>

For any subset <math>F\!</math> of the index set <math>K,\!</math> there is the corresponding subfamily of sets, <math>\{ X_j : j \in F \},\!</math> and there is the corresponding cartesian product over this subfamily, notated and defined as <math>\textstyle X_F = \prod_{j \in F} X_j.\!</math>

For any point <math>x\!</math> in <math>X,\!</math> the ''projection'' of <math>x\!</math> on the subspace <math>X_F\!</math> is notated as <math>\mathrm{proj}_F (x).\!</math>

More generally, for any relation <math>L \subseteq X,\!</math> the projection of <math>L\!</math> on the subspace <math>X_F\!</math> is written as <math>\mathrm{proj}_F (L)\!</math> or still more simply as <math>\mathrm{proj}_F L.\!</math>

The question of ''projective reduction'' for <math>k\!</math>-adic relations can be stated with moderate generality in the following way:

* Given a set of <math>k\!</math>-place relations in the same space <math>X\!</math> and a set of projections from <math>X\!</math> to the associated subspaces, do the projections afford sufficient data to tell the different relations apart?

==Projective reducibility of triadic relations==
: ''Main article : [[Triadic relation]]''

By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject.

===Examples of projectively irreducible relations===

The 3-adic relations <math>L_0\!</math> and <math>L_1\!</math> are shown in the next two Tables:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
|-
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|}

 

A ''2-adic projection'' of a 3-adic relation <math>L\!</math> is the 2-adic relation that results from deleting one column of the table for <math>L\!</math> and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored.

In the case of the above two relations, <math>{L_0, L_1 ~\subseteq~ X \times Y \times Z ~\cong~ \mathbb{B}^3},\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

 

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XY} L_0\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_0\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_0\!</math>
|- style="height:40px; background:ghostwhite"
| <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|}
|}

 

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XY} L_1\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Y\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XZ} L_1\!</math>
|- style="height:40px; background:ghostwhite"
| <math>X\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{YZ} L_1\!</math>
|- style="height:40px; background:ghostwhite"
| <math>Y\!</math> || <math>Z\!</math>
|-
| <math>0\!</math> || <math>1\!</math>
|-
| <math>1\!</math> || <math>0\!</math>
|-
| <math>0\!</math> || <math>0\!</math>
|-
| <math>1\!</math> || <math>1\!</math>
|}
|}

 

It is clear on inspection that the following three equations hold:

{| align="center" cellpadding="8" style="text-align:center; width:90%"
| <math>\mathrm{proj}_{XY}(L_0) ~=~ \mathrm{proj}_{XY}(L_1)~\!</math>
| <math>\mathrm{proj}_{XZ}(L_0) ~=~ \mathrm{proj}_{XZ}(L_1)~\!</math>
| <math>\mathrm{proj}_{YZ}(L_0) ~=~ \mathrm{proj}_{YZ}(L_1)~\!</math>
|}

These equations say that <math>L_0\!</math> and <math>L_1\!</math> cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, each relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations.

===Examples of projectively reducible relations===

The 3-adic relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are shown in the next two Tables:

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}

 

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:33%" | <math>\text{Object}\!</math>
| style="width:33%" | <math>\text{Sign}\!</math>
| style="width:33%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
|<math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}

 

In the case of the two sign relations, <math>L_\mathrm{A}, L_\mathrm{B} ~\subseteq~ X \times Y \times Z ~\cong~ O \times S \times I,\!</math> the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

 

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{A})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|}
|}

 

{| align="center" style="width:90%"
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XY}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{XZ}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{A}\!</math> || <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>\mathrm{B}\!</math> || <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\mathrm{proj}_{YZ}(L_\mathrm{B})\!</math>
|- style="height:40px; background:ghostwhite"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
|-
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
|}
|}

 

It is clear on inspection that the following three inequalities hold:

{| align="center" cellpadding="8" style="text-align:left; width:90%"
| <math>\mathrm{proj}_{XY}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XY}(L_\mathrm{B})\!</math>
| <math>\mathrm{proj}_{XZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{XZ}(L_\mathrm{B})\!</math>
| <math>\mathrm{proj}_{YZ}(L_\mathrm{A}) ~\ne~ \mathrm{proj}_{YZ}(L_\mathrm{B})\!</math>
|}

These inequalities say that <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from ''every'' other 3-adic relation on the basis of the 2-adic projection data alone.

In other words, to show that a 3-adic relation <math>L\!</math> on <math>O \times S \times I\!</math> is ''reducible'' or ''reconstructible'' in the 2-adic projective sense, it is necessary to show that no distinct <math>L'\!</math> on <math>O \times S \times I\!</math> exists such that <math>L\!</math> and <math>L'\!</math> have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on <math>O \times S \times I\!</math> than looking merely at one or two relations at a time.

'''Fact.''' As it happens, each of the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below.

Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles.

* If <math>L\!</math> is a relation over a set of domains that includes the domains <math>U\!</math> and <math>V,\!</math> then the abbreviated notation <math>L_{UV}\!</math> can be used for the projection <math>\mathrm{proj}_{UV}(L).\!</math>

* The operation of reversing a projection asks what elements of a bigger space project onto given elements of a smaller space. The set of elements that project onto <math>x\!</math> under a given projection <math>f\!</math> is called the ''fiber'' of <math>x\!</math> under <math>f\!</math> and is written <math>f^{-1}(x)\!</math> or <math>f^{-1}x.\!</math>

* If <math>X\!</math> is a finite set, the ''cardinality'' of <math>X,\!</math> written <math>\mathrm{card}(X)\!</math> or <math>|X|,\!</math> means the number of elements in <math>X.\!</math>

'''Proof.''' Let <math>L\!</math> be either one of the relations <math>L_\mathrm{A}\!</math> or <math>L_\mathrm{B}.\!</math> Consider any coordinate position <math>(s, i)\!</math> in the <math>SI\!</math>-plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>o\!</math> in <math>O\!</math> is <math>(o, s, i)\!</math> in the fiber <math>\mathrm{proj}_{SI}^{-1}(s, i)?\!</math> Now, the circumstance that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s\!</math> in <math>S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i\!</math> in <math>I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> All together, this proves that both <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are ''projectively 2-adically reducible''.

===Summary===

The ''projective analysis'' of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result:

* Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction Relation Reduction @ InterSciWiki]
* [http://mywikibiz.com/Relation_reduction Relation Reduction @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction Relation Reduction], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_reduction Relation Reduction], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relation_reduction Relation Reduction], [http://semanticweb.org/ Semantic Web]
* [http://ref.subwiki.org/wiki/Relation_reduction Relation Reduction], [http://ref.subwiki.org/ Subject Wikis]
* [http://wikinfo.org/w/index.php/Relation_reduction Relation Reduction], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_reduction Relation Reduction], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_reduction Relation Reduction], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_reduction&oldid=39828834 Relation Reduction], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Formal Sciences]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Semiotics]]
[[Category:Set Theory]]

Relation construction

2015-11-14T19:24:03Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In logic and mathematics, '''relation construction''' and '''relational constructibility''' have to do with the ways that one [[relation (mathematics)|relation]] is determined by an indexed family or a sequence of other relations, called the ''relation dataset''.  The relation in the focus of consideration is called the ''faciendum''.  The relation dataset typically consists of a specified relation over sets of relations, called the ''constructor'', the ''factor'', or the ''method of construction'', plus a specified set of other relations, called the ''faciens'', the ''ingredients'', or the ''makings''.

[[Relation composition]] and [[relation reduction]] are special cases of relation constructions.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_construction Relation Construction @ InterSciWiki]
* [http://mywikibiz.com/Relation_construction Relation Construction @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_construction Relation Construction @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_construction Relation Construction @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_construction Relation Construction @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_construction Relation Construction], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_construction Relation Construction], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/RelationConstruction Relation Construction], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Relation_construction Relation Construction], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Relation_construction Relation Construction], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_construction Relation Construction], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_construction Relation Construction], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_construction&oldid=39070184 Relation Construction], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Formal Sciences]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Set Theory]]

Relation composition

2015-11-14T18:18:41Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of function composition, or the composition of functions.  The following treatment of relation composition takes the “strongly typed” approach to relations that is outlined in the article on [[relation theory]].

==Preliminaries==

There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general.

The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations.

As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation.

{| align="center" cellpadding="4" width="90%"
|
The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.
|-
|
The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.
|}

These two factors together generate the following four styles of syntax:

{| align="center" cellpadding="4" width="90%"
| LALA = left application, left association.
|-
| LARA = left application, right association.
|-
| RALA = right application, left association.
|-
| RARA = right application, right association.
|}

==Definition==

A notion of relational composition is to be defined that generalizes the usual notion of functional composition:

{| align="center" cellpadding="4" width="90%"
|
Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math>

results in a ''composite function'' formulated as <math>fg : X \to Z.</math>
|-
|
Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math>

results in a ''composite function'' formulated as <math>gf : X \to Z.</math>
|}

Note on notation. The ordinary symbol for functional composition is the ''composition sign'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' “<math>\cdot</math>”, as <math>f \cdot g.</math>

Generalizing the paradigm along parallel lines, the ''composition'' of a pair of dyadic relations is formulated in the following two ways:

{| align="center" cellpadding="4" width="90%"
|
Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math>

results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math>
|-
|
Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math>

results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math>
|}

==Geometric construction==

There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations.

This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called ''tacit extensions''.

The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely:

:* The use of [[logical conjunction]], as denoted by the symbol <math>\land,\!</math> in expressions of the form <math>F(x, y, z) = G(x, y) \land H(y, z),\!</math> to define a triadic relation <math>F\!</math> in terms of a pair of dyadic relations <math>G\!</math> and <math>H.\!</math>

:* The concepts of dyadic ''projection'' and ''projective determination'', that are invoked in the “weak” notion of ''projective reducibility''.

The relational composition <math>G \circ H\!</math> of a pair of dyadic relations <math>G\!</math> and <math>H\!</math> will be constructed in three stages, first, by taking the tacit extensions of <math>G\!</math> and <math>H\!</math> to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the ''prima facie'' dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition <math>G \circ H\!</math> of the relations <math>G\!</math> and <math>H.\!</math>

The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to isomorphism'' as the conventional saying goes, that is, any objects that have the “same form” are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, <math>G, H \subseteq X \times Y,\!</math> as shown in Figure 1.

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o o |
| |\ |\ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | * \ | * \ |
| X * Y X * Y |
| \ * | \ * | |
| \ G | \ H | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \| \| |
| o o |
| |
o-------------------------------------------------o
Figure 1. Dyadic Relations G, H c X x Y
</pre>
|}

The dyadic relations <math>G\!</math> and <math>H\!</math> cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen:

:* The first type of case occurs when <math>X = Y.\!</math> In this case, both of the compositions <math>G \circ H\!</math> and <math>H \circ G\!</math> are defined.

:* The second type of case occurs when <math>X\!</math> and <math>Y\!</math> are distinct, but when it nevertheless makes sense to speak of a dyadic relation <math>\hat{H}\!</math> that is isomorphic to <math>H,\!</math> but living in the plane <math>YZ,\!</math> that is, in the space of the cartesian product <math>Y \times Z,\!</math> for some set <math>Z.\!</math>

Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o o |
| |\ /| |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | * \ / * | |
| X * Y Y * Z |
| \ * | | * / |
| \ G | | Ĥ / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 2. Dyadic Relations G c X x Y and Ĥ c Y x Z
</pre>
|}

With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition <math>P \circ Q\!</math> of a pair of dyadic relations <math>P, Q
\subseteq X \times X.\!</math>

: '''Definition.''' <math>P \circ Q = \mathrm{proj}_{13} (P \times X ~\cap~ X \times Q).\!</math>

To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form <math>L \subseteq X \times X.\!</math> So, if one has started out with a dyadic relation of the shape <math>L \subseteq U \times V,\!</math> one merely lets <math>X = U \cup V,\!</math> trading in the initial <math>L\!</math> for a new <math>L \subseteq X \times X\!</math> as need be.

The projection <math>\mathrm{proj}_{13}\!</math> is just the projection of the cartesian cube <math>X \times X \times X\!</math> on the space of shape <math>X \times X\!</math> that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places.

Finally, the notation of the cartesian product sign “<math>\times\!</math>” is extended to signify two other products with respect to a dyadic relation <math>L \subseteq X \times X\!</math> and a subset <math>W \subseteq X,\!</math> as follows:

: '''Definition.''' <math>L \times W ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in L ~\mathrm{and}~ z \in W \}.\!</math>

: '''Definition.''' <math>W \times L ~=~ \{ (x, y, z) \in X^3 ~:~ x \in W ~\mathrm{and}~ (y, z) \in L \}.\!</math>

Applying these definitions to the case <math>P, Q \subseteq X \times X,\!</math> the two dyadic relations whose relational composition <math>P \circ Q \subseteq X \times X\!</math> is about to be defined, one finds:

: <math>P \times X ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in P ~\mathrm{and}~ z \in X \},\!</math>

: <math>X \times Q ~=~ \{ (x, y, z) \in X^3 ~:~ x \in X ~\mathrm{and}~ (y, z) \in Q \}.\!</math>

These are just the appropriate special cases of the tacit extensions already defined.

: <math>P \times X ~=~ \mathrm{te}_{12}^3 (P),~\!</math>

: <math>X \times Q ~=~ \mathrm{te}_{23}^1 (Q).~\!</math>

In summary, then, the expression:

: <math>\mathrm{proj}_{13} (P \times X ~\cap~ X \times Q)\!</math>

is equivalent to the expression:

: <math>\mathrm{proj}_{13} (\mathrm{te}_{12}^3 (P) ~\cap~ \mathrm{te}_{23}^1 (Q))\!</math>

and this form is generalized — although, relative to one's school of thought, perhaps inessentially so — by the form that was given above as follows:

: '''Definition.''' <math>P \circ Q ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (P) ~\cap~ \mathrm{te}_{YZ}^X (Q)).\!</math>

Figure 3 presents a geometric picture of what is involved in formulating a definition of the triadic relation <math>F \subseteq X \times Y \times Z\!</math> by way of a conjunction between the dyadic relation <math>G \subseteq X \times Y\!</math> and the dyadic relation <math>H \subseteq Y \times Z,\!</math> as done for example by means of an expression of the following form:

:* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 3. Projections of F onto G and H
</pre>
|}

To interpret the Figure, visualize the triadic relation <math>F \subseteq X \times Y \times Z\!</math> as a body in <math>XYZ\!</math>-space, while <math>G\!</math> is a figure in <math>XY\!</math>-space and <math>H\!</math> is a figure in <math>YZ\!</math>-space.

The dyadic '''projections''' that accompany a triadic relation over <math>X, Y, Z\!</math> are defined as follows:

:* <math>\mathrm{proj}_{XY} (L) ~=~ \{ (x, y) \in X \times Y : (x, y, z) \in L ~\text{for some}~ z \in Z) \},\!</math>

:* <math>\mathrm{proj}_{XZ} (L) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in L ~\text{for some}~ y \in Y) \},\!</math>

:* <math>\mathrm{proj}_{YZ} (L) ~=~ \{ (y, z) \in Y \times Z : (x, y, z) \in L ~\text{for some}~ x \in X) \}.\!</math>

For many purposes it suffices to indicate the dyadic projections of a triadic relation <math>L\!</math> by means of the briefer equivalents listed next:

:* <math>L_{XY} ~=~ \mathrm{proj}_{XY}(L),\!</math>

:* <math>L_{XZ} ~=~ \mathrm{proj}_{XZ}(L),\!</math>

:* <math>L_{YZ} ~=~ \mathrm{proj}_{YZ}(L).\!</math>

In light of these definitions, <math>\mathrm{proj}_{XY}\!</math> is a mapping from the set <math>\mathcal{L}_{XYZ}\!</math> of triadic relations over the domains <math>X, Y, Z\!</math> to the set <math>\mathcal{L}_{XY}\!</math> of dyadic relations over the domains <math>X, Y,\!</math> with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions.

The set <math>\mathcal{L}_{XYZ},~\!</math> whose members are just the triadic relations over <math>X, Y, Z,\!</math> can be recognized as the set of all subsets of the cartesian product <math>X \times Y \times Z,\!</math> also known as the ''power set'' of <math>X \times Y \times Z,\!</math> and notated here as <math>\mathrm{Pow} (X \times Y \times Z).\!</math>

:* <math>\mathcal{L}_{XYZ} ~=~ \{ L : L \subseteq X \times Y \times Z \} ~=~ \mathrm{Pow} (X \times Y \times Z).\!</math>

Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from <math>\{ X, Y, Z \}.\!</math>

:* <math>\mathcal{L}_{XY} ~=~ \{L : L \subseteq X \times Y \} ~=~ \mathrm{Pow} (X \times Y),~\!</math>

:* <math>\mathcal{L}_{XZ} ~=~ \{L : L \subseteq X \times Z \} ~=~ \mathrm{Pow} (X \times Z),~\!</math>

:* <math>\mathcal{L}_{YZ} ~=~ \{L : L \subseteq Y \times Z \} ~=~ \mathrm{Pow} (Y \times Z).~\!</math>

In mathematics, the inverse relation corresponding to a projection map is usually called an ''extension''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term ''tacit extension''.

Given three sets, <math>X, Y, Z,\!</math> and three dyadic relations,

:* <math>U \subseteq X \times Y,~\!</math>

:* <math>V \subseteq X \times Z,~\!</math>

:* <math>W \subseteq Y \times Z,~\!</math>

the ''tacit extensions'', <math>\mathrm{te}_{XY}^Z, \mathrm{te}_{XZ}^Y, \mathrm{te}_{YZ}^X,~\!</math> of <math>U, V, W,\!</math> respectively, are defined as follows:

:* <math>\mathrm{te}_{XY}^Z (U) ~=~ \{ (x, y, z) : (x, y) \in U \},\!</math>

:* <math>\mathrm{te}_{XZ}^Y (V) ~=~ \{ (x, y, z) : (x, z) \in V \},\!</math>

:* <math>\mathrm{te}_{YZ}^X (W) ~=~ \{ (x, y, z) : (y, z) \in W \}.\!</math>

So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, <math>\mathrm{te}(U), \mathrm{te}(V), \mathrm{te}(W).\!</math>

The definition and illustration of relational composition presently under way makes use of the tacit extension of <math>G \subseteq X \times Y\!</math> to <math>\mathrm{te}(G) \subseteq X \times Y
\times Z\!</math> and the tacit extension of <math>H \subseteq Y \times Z\!</math> to <math>\mathrm{te}(H) \subseteq X \times Y \times Z,\!</math> only.

Geometric illustrations of <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H)\!</math> are afforded by Figures 4 and 5, respectively.

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | * \ |
| o o ** o |
| |\ / \*** /| |
| | \ / *** / | |
| | \ / ***\ / | |
| | \ *** / | |
| | / \*** / \ | |
| | / *** / \ | |
| |/ ***\ / \| |
| o X /** Y Z o |
| |\ \//* | / /| |
| | \ /// | / / | |
| | \ ///\ | / / | |
| | \ /// \ | / / | |
| | \/// \ | / / | |
| | /\/ \ | / / | |
| | *//\ \|/ / * | |
| X */ Y o Y * Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 4. Tacit Extension of G to X x Y x Z
</pre>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / * | \ |
| o ** o o |
| |\ ***/ \ /| |
| | \ *** \ / | |
| | \ /*** \ / | |
| | \ *** / | |
| | / \ ***/ \ | |
| | / \ *** \ | |
| |/ \ /*** \| |
| o X Y **\ Z o |
| |\ \ | *\\/ /| |
| | \ \ | \\\ / | |
| | \ \ | /\\\ / | |
| | \ \ | / \\\ / | |
| | \ \ | / \\\/ | |
| | \ \ | / \/\ | |
| | * \ \|/ /\\* | |
| X * Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 5. Tacit Extension of H to X x Y x Z
</pre>
|}

A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following:

:* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math>

The conjunction that is indicated by “<math>\land\!</math>” corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H).\!</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 6. F as the Intersection of te(G) and te(H)
</pre>
|}

==Algebraic construction==

The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ 7\/// | \\\/7 /| |
| | \ 6// | \\6 / | |
| | \ //5\ | /5\\ / | |
| | \ /// 4\ | /4 \\\ / | |
| | \/// 3\ | /3 \\\/ | |
| | G/\/ 2\ | /2 \/\H | |
| | *//\ 1\|/1 /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\@5 5@// |/5 |
| 4@ \@4 4@/ @4 |
| 3\ @3 3@ /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 7. F as the Intersection of te(G) and te(H)
</pre>
|}

Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on.

For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}

As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use.

Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions:

{| align="center" cellpadding="8" width="90%"
|
<math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math>
|}

Here is the big picture, with all the pieces in place:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / G o H \ |
| X * Z |
| 7\ /|\ /7 |
| 6\ / | \ /6 |
| 5\ / | \ /5 |
| 4@ | @4 |
| 3\ | /3 |
| 2\ | /2 |
| 1\|/1 |
| | |
| | |
| | |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o | o |
| |\ /|\ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | G/\/ \ | / \/\H | |
| | *//\ \|/ /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\@5 5@// |/5 |
| 4@ \@4 4@/ @4 |
| 3\ @3 3@ /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 8. G o H = proj_XZ (te(G) |^| te(H))
</pre>
|}

All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4)
\\[6pt]
& = & 4:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G)
\\[4pt]
& = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z)
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(G)
& = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\
& & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\
& & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\
& & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\
& & 4:3:7 & + & 4:4:7 & + & 4:5:7
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H)
\\[4pt]
& = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4)
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(H)
& = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\
& & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\
& & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\
& & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\
& & 7:3:4 & + & 7:4:4 & + & 7:5:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccl}
\mathrm{te}(G) \cap \mathrm{te}(H)
& = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4
\\[4pt]
G \circ H
& = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H))
\\[4pt]
& = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4)
\\[4pt]
& = & 4:4
\end{array}</math>
|}

==Matrix representation==

We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as ''[[logical matrix|logical matrices]]'', and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in linear algebra.

First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H.\!</math>

Here is the setup that we had before:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X & = & \{ 1, 2, 3, 4, 5, 6, 7 \}\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G & = & 4:3 & + & 4:4 & + & 4:5 & \subseteq & X \times X
\\
H & = & 3:4 & + & 4:4 & + & 5:4 & \subseteq & X \times X
\end{matrix}</math>
|}

Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z,\!</math> the composition of <math>P ~\text{on}~ Q\!</math> is written as <math>P \circ Q,\!</math> or more simply as <math>PQ,\!</math> and obtained as follows:

To compute <math>PQ,\!</math> in general, where <math>P\!</math> and <math>Q\!</math> are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes <math>a:b\!</math> and <math>c:d.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
(a:b)(c:d) & = & (a:d) & \text{if}~ b = c
\\
(a:b)(c:d) & = & 0 & \text{otherwise}
\end{matrix}</math>
|}

To find the relational composition <math>G \circ H,\!</math> one may begin by writing it as a quasi-algebraic product:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4)
\end{matrix}</math>
|}

Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H
& = & (4:3)(3:4) & + & (4:3)(4:4) & + & (4:3)(5:4) & +
\\
& & (4:4)(3:4) & + & (4:4)(4:4) & + & (4:4)(5:4) & +
\\
& & (4:5)(3:4) & + & (4:5)(4:4) & + & (4:5)(5:4)
\end{matrix}</math>
|}

Applying the rule that determines the product of elementary relations produces the following array:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H
& = & 4:4 & + & 0 & + & 0 & +
\\
& & 0 & + & 4:4 & + & 0 & +
\\
& & 0 & + & 0 & + & 4:4
\end{matrix}</math>
|}

Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}G \circ H & = & 4:4\end{matrix}</math>
|}

With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations <math>G\!</math> and <math>H\!</math> together to obtain their relational composite <math>G \circ H.\!</math>

Given the space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> whose cardinality <math>|X|\!</math> is <math>7,\!</math> there are <math>|X \times X| = |X| \cdot |X|\!</math> <math>=\!</math> <math>7 \cdot 7 = 49\!</math> elementary relations of the form <math>i:j,\!</math> where <math>i\!</math> and <math>j\!</math> range over the space <math>X.\!</math> Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
1\!:\!1 & 1\!:\!2 & 1\!:\!3 & 1\!:\!4 & 1\!:\!5 & 1\!:\!6 & 1\!:\!7
\\
2\!:\!1 & 2\!:\!2 & 2\!:\!3 & 2\!:\!4 & 2\!:\!5 & 2\!:\!6 & 2\!:\!7
\\
3\!:\!1 & 3\!:\!2 & 3\!:\!3 & 3\!:\!4 & 3\!:\!5 & 3\!:\!6 & 3\!:\!7
\\
4\!:\!1 & 4\!:\!2 & 4\!:\!3 & 4\!:\!4 & 4\!:\!5 & 4\!:\!6 & 4\!:\!7
\\
5\!:\!1 & 5\!:\!2 & 5\!:\!3 & 5\!:\!4 & 5\!:\!5 & 5\!:\!6 & 5\!:\!7
\\
6\!:\!1 & 6\!:\!2 & 6\!:\!3 & 6\!:\!4 & 6\!:\!5 & 6\!:\!6 & 6\!:\!7
\\
7\!:\!1 & 7\!:\!2 & 7\!:\!3 & 7\!:\!4 & 7\!:\!5 & 7\!:\!6 & 7\!:\!7
\end{matrix}</math>
|}

The relations <math>G\!</math> and <math>H\!</math> may then be regarded as logical sums of the following forms:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G & = & \displaystyle\sum_{ij} G_{ij} (i\!:\!j)
\\[20pt]
H & = & \displaystyle\sum_{ij} H_{ij} (i\!:\!j)
\end{matrix}\!</math>
|}

The notation <math>\textstyle\sum_{ij}\!</math> indicates a logical sum over the collection of elementary relations <math>i\!:\!j\!</math> while the factors <math>G_{ij}\!</math> and <math>H_{ij}\!</math> are values in the ''[[boolean domain]]'' <math>\mathbb{B} = \{ 0, 1 \}~\!</math> that are called the ''coefficients'' of the relations <math>G\!</math> and <math>H,\!</math> respectively, with regard to the corresponding elementary relations <math>i\!:\!j.\!</math>

In general, for a dyadic relation <math>L,\!</math> the coefficient <math>L_{ij}\!</math> of the elementary relation <math>i\!:\!j\!</math> in the relation <math>L\!</math> will be <math>0\!</math> or <math>1,\!</math> respectively, as <math>i\!:\!j\!</math> is excluded from or included in <math>L.\!</math>

With these conventions in place, the expansions of <math>G\!</math> and <math>H\!</math> may be written out as follows:

{| align="center" cellpadding="4" width="90%"
|
<math>\begin{matrix}G & = & 4:3 & + & 4:4 & + & 4:5 & =\end{matrix}</math>
|-
|
<math>\begin{smallmatrix}
0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & +
\\
0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & +
\\
0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & 0 \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & +
\\
0 \cdot (4:1) & + & 0 \cdot (4:2) & + & \mathbf{1} \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & \mathbf{1} \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & +
\\
0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & 0 \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & +
\\
0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & +
\\
0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7)
\end{smallmatrix}</math>
|}

{| align="center" cellpadding="4" width="90%"
|
<math>\begin{matrix}H & = & 3:4 & + & 4:4 & + & 5:4 & =\end{matrix}</math>
|-
|
<math>\begin{smallmatrix}
0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & +
\\
0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & +
\\
0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & \mathbf{1} \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & +
\\
0 \cdot (4:1) & + & 0 \cdot (4:2) & + & 0 \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & 0 \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & +
\\
0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & \mathbf{1} \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & +
\\
0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & +
\\
0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7)
\end{smallmatrix}</math>
|}

Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations <math>G\!</math> and <math>H.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>G ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 1 & 1 & 1 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

These are the logical matrix representations of the dyadic relations <math>G\!</math> and <math>H.\!</math>

If the dyadic relations <math>G\!</math> and <math>H\!</math> are viewed as logical sums then their relational composition <math>G \circ H\!</math> can be regarded as a product of sums, a fact that can be indicated as follows:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ (\sum_{ij} G_{ij} (i\!:\!j))(\sum_{ij} H_{ij} (i\!:\!j)).\!</math>
|}

The composite relation <math>G \circ H\!</math> is itself a dyadic relation over the same space <math>X,\!</math> in other words, <math>G \circ H \subseteq X \times X,\!</math> and this means that <math>G \circ H\!</math> must be amenable to being written as a logical sum of the following form:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ \sum_{ij} (G \circ H)_{ij} (i\!:\!j).\!</math>
|}

In this formula, <math>(G \circ H)_{ij}\!</math> is the coefficient of <math>G \circ H\!</math> with respect to the elementary relation <math>i\!:\!j.\!</math>

One of the best ways to reason out what <math>G \circ H\!</math> should be is to ask oneself what its coefficient <math>(G \circ H)_{ij}\!</math> should be for each of the elementary relations <math>i\!:\!j\!</math> in turn.

So let us pose the question:

{| align="center" cellpadding="8" width="90%"
| <math>(G \circ H)_{ij} ~=~ ?\!</math>
|}

In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ (\sum_{ik} G_{ik} (i\!:\!k))(\sum_{kj} H_{kj} (k\!:\!j)).\!</math>
|}

A moment's thought will tell us that <math>(G \circ H)_{ij} = 1\!</math> if and only if there is an element <math>k\!</math> in <math>X\!</math> such that <math>G_{ik} = 1\!</math> and <math>H_{kj} = 1.\!</math>

Consequently, we have the result:

{| align="center" cellpadding="8" width="90%"
| <math>(G \circ H)_{ij} ~=~ \sum_{k} G_{ik} H_{kj}.\!</math>
|}

This follows from the properties of boolean arithmetic, specifically, from the fact that the product <math>G_{ik} H_{kj}\!</math> is <math>1\!</math> if and only if both <math>G_{ik}\!</math> and <math>H_{kj}\!</math> are <math>1\!</math> and from the fact that <math>\textstyle\sum_{k} F_{k}\!</math> is equal to <math>1\!</math> just in case some <math>F_{k}\!</math> is <math>1.\!</math>

All that remains in order to obtain a computational formula for the relational composite <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H\!</math> is to collect the coefficients <math>(G \circ H)_{ij}\!</math> as <math>i\!</math> and <math>j\!</math> range over <math>X.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}G \circ H
& = & \displaystyle \sum_{ij} (G \circ H)_{ij} (i\!:\!j)
& = & \displaystyle \sum_{ij} (\sum_{k} G_{ik} H_{kj}) (i\!:\!j).
\end{matrix}</math>
|}

This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction.

By way of disentangling this formula, one may notice that the form <math>\textstyle \sum_{k} G_{ik} H_{kj}\!</math> is what is usually called a ''scalar product''. In this case it is the scalar product of the <math>i^\text{th}\!</math> row of <math>G\!</math> with the <math>j^\text{th}\!</math> column of <math>H.\!</math>

To make this statement more concrete, let us go back to the examples of <math>G\!</math> and <math>H\!</math> we came in with:

{| align="center" cellpadding="8" width="90%"
|
<math>G ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 1 & 1 & 1 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

The formula for computing <math>G \circ H\!</math> says the following:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccl}
(G \circ H)_{ij}
& = & \text{the}~ {ij}^\text{th} ~\text{entry in the matrix representation for}~ G \circ H
\\[2pt]
& = & \text{the entry in the}~ {i}^\text{th} ~\text{row and the}~ {j}^\text{th} ~\text{column of}~ G \circ H
\\[2pt]
& = & \text{the scalar product of the}~ {i}^\text{th} ~\text{row of}~ G ~\text{with the}~ {j}^\text{th} ~\text{column of}~ H
\\[2pt]
& = & \sum_{k} G_{ik} H_{kj}
\end{array}</math>
|}

As it happens, it is possible to make exceedingly light work of this example, since there is only one row of <math>G\!</math> and one column of <math>H\!</math> that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of <math>G\!</math> with the fourth column of <math>H\!</math> produces the sole non-zero entry for the matrix of <math>G \circ H.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>G \circ H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

==Graph-theoretic picture==

There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''bipartite graphs'', or ''bigraphs'' for short.

Here is what <math>G\!</math> and <math>H\!</math> look like in the bigraph picture:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ |
| / | \ G |
| / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 9. G = 4:3 + 4:4 + 4:5
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| \ | / |
| \ | / H |
| \|/ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 10. H = 3:4 + 4:4 + 5:4
</pre>
|}

These graphs may be read to say:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G ~\text{puts}~ 4 ~\text{in relation to}~ 3, 4, 5.
\\[2pt]
H ~\text{puts}~ 3, 4, 5 ~\text{in relation to}~ 4.
\end{matrix}</math>
|}

To form the composite relation <math>G \circ H,\!</math> one simply follows the bigraph for <math>G\!</math> by the bigraph for <math>H,\!</math> here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for <math>G \circ H.\!</math>

Here's how it looks in pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ |
| / | \ G |
| / | \ |
| o o o o o o o X |
| \ | / |
| \ | / H |
| \|/ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 11. G Followed By H
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | |
| | G o H |
| | |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 12. G Composed With H
</pre>
|}

Once again we find that <math>G \circ H = 4:4.\!</math>

We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind.

To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula.

Keeping to the same space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> define the dyadic relations <math>M, N \subseteq X \times X\!</math> as follows:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{*{19}{c}}
M & = &
2\!:\!1 & + & 2\!:\!2 & + & 2\!:\!3 & + & 4\!:\!3 & + & 4\!:\!4 & + & 4\!:\!5 & + & 6\!:\!5 & + & 6\!:\!6 & + & 6\!:\!7
\\[2pt]
N & = &
1\!:\!1 & + & 2\!:\!1 & + & 3\!:\!3 & + & 4\!:\!3 & ~ & + & ~ & 4\!:\!5 & + & 5\!:\!5 & + & 6\!:\!7 & + & 7\!:\!7
\end{array}</math>
|}

Here are the bigraph pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 13. Dyadic Relation M
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 14. Dyadic Relation N
</pre>
|}

To form the composite relation <math>M \circ N,\!</math> one simply follows the bigraph for <math>M\!</math> by the bigraph for <math>N,\!</math> arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for <math>M \circ N.\!</math>

Here's how it looks in pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 15. M Followed By N
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| / \ / \ / \ |
| / \ / \ / \ M o N |
| / \ / \ / \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 16. M Composed With N
</pre>
|}

Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation.

The coefficient of the composition <math>M \circ N\!</math> between <math>i\!</math> and <math>j\!</math> in <math>X\!</math> is given as follows:

{| align="center" cellpadding="8" width="90%"
| <math>(M \circ N)_{ij} ~=~ \sum_{k} M_{ik} N_{kj}\!</math>
|}

Graphically interpreted, this is a ''sum over paths''. Starting at the node <math>i,\!</math> <math>M_{ik}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>M\!</math> from node <math>i\!</math> to node <math>k\!</math> and <math>N_{kj}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>N\!</math> from node <math>k\!</math> to node <math>j.\!</math> So the <math>\textstyle\sum_{k}\!</math> ranges over all possible intermediaries <math>k,\!</math> ascending from <math>0\!</math> to <math>1\!</math> just as soon as there happens to be a path of length two between nodes <math>i\!</math> and <math>j.\!</math>

It is instructive at this point to compute the other possible composition that can be formed from <math>M\!</math> and <math>N,\!</math> namely, the composition <math>N \circ M,\!</math> that takes <math>M\!</math> and <math>N\!</math> in the opposite order. Here is the graphic computation:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 17. N Followed By M
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| |
| N o M |
| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 18. N Composed With M
</pre>
|}

In sum, <math>N \circ M = 0.\!</math> This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''non-commutative'' algebraic operation.

==References==

* Ulam, S.M., and Bednarek, A.R., “On the Theory of Relational Structures and Schemata for Parallel Computation” (1977), pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990.

==Bibliography==

* Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.

* Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994.

* Ulam, S.M., ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition @ InterSciWiki]
* [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web]
* [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis]
* [http://wikinfo.org/w/index.php/Relation_composition Relation Composition], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_composition Relation Composition], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia]

[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Formal Sciences]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Set Theory]]

Relation (mathematics)

2015-11-14T17:10:44Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below.

* The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign “<math>=\!</math>” in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign “<math>{<}\!</math>” in a statement like <math>5 < 12.\!</math>  Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''.

* The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.  These are called ''finite-place'' or ''finitary'' relations.  A finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', ''<math>k\!</math>-adic'', or ''<math>k\!</math>-dimensional'' relation.  The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively.

==Informal introduction==

The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.  For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  The facts of a concrete situation could be organized in the form of a Table like the one below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
<math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Person}~ X\!</math>
| <math>\text{Person}~ Y\!</math>
| <math>\text{Person}~ Z\!</math>
|-
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
| <math>\text{Denise}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
|-
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
|}

 

Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>  The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion:

{| align="center" cellpadding="10"
| <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math>
|}

The data of the Table are equivalent to the following set of ordered triples:

{| align="center" cellpadding="10"
|
<math>\begin{smallmatrix}
S
& =
& \{
& \text{(Alice, Bob, Denise)},
& \text{(Charles, Alice, Bob)},
& \text{(Charles, Charles, Alice)},
& \text{(Denise, Denise, Denise)}
& \}
\end{smallmatrix}\!</math>
|}

By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.  The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.  The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package.

The Table for relation <math>S\!</math> is an extremely simple example of a relational database.  The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.  Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation.

For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth.

==Example 1. Divisibility==

A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>  This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math>

To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>  In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.  The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math>

For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math>

==Formal definitions==

There are two definitions of <math>k\!</math>-place relations that are commonly encountered in mathematics.  In order of simplicity, the first of these definitions is as follows:

'''Definition 1.'''   A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a subset of their cartesian product, written <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>  Under this definition, then, a <math>k\!</math>-ary relation is simply a set of <math>k\!</math>-tuples.

The second definition makes use of an idiom that is common in mathematics, saying that “such and such is an <math>n\!</math>-tuple” to mean that the mathematical object being defined is determined by the specification of <math>n\!</math> component mathematical objects.  In the case of a relation <math>L\!</math> over <math>k\!</math> sets, there are <math>k + 1\!</math> things to specify, namely, the <math>k\!</math> sets plus a subset of their cartesian product.  In the idiom, this is expressed by saying that <math>L\!</math> is a <math>(k+1)\!</math>-tuple.

'''Definition 2.'''   A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>L = (X_1, \ldots, X_k, \mathrm{graph}(L)),\!</math> where <math>\mathrm{graph}(L)\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k~\!</math> called the ''graph'' of <math>L.\!</math>

Elements of a relation are sometimes denoted by using boldface characters, for example, the constant element <math>\mathbf{a} = (a_1, \ldots, a_k)\!</math> or the variable element <math>\mathbf{x} = (x_1, \ldots, x_k).\!</math>

A statement of the form “<math>\mathbf{a}\!</math> is in the relation <math>L\!</math>” is taken to mean that <math>\mathbf{a}\!</math> is in <math>L\!</math> under the first definition and that <math>\mathbf{a}\!</math> is in <math>\mathrm{graph}(L)\!</math> under the second definition.

The following considerations apply under either definition:

:* The sets <math>X_j~\!</math> for <math>j = 1 ~\text{to}~ k\!</math> are called the ''domains'' of the relation.  In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains.

:* If all the domains <math>X_j~\!</math> are the same set <math>X,\!</math> then <math>L\!</math> is more simply referred to as a <math>k\!</math>-ary relation over <math>X.\!</math>

:* If any domain <math>X_j~\!</math> is empty then the cartesian product is empty and the only relation over such a sequence of domains is the empty relation <math>L = \varnothing.\!</math>  Most applications of the relation concept will set aside this trivial case and assume that all domains are nonempty.

If <math>L\!</math> is a relation over the domains <math>X_1, \ldots, X_k,\!</math> it is conventional to consider a sequence of terms called ''variables'', <math>x_1, \ldots, x_k,\!</math> that are said to ''range over'' the respective domains.

A ''[[boolean domain]]'' <math>\mathbb{B}\!</math> is a generic 2-element set, say, <math>\mathbb{B} = \{ 0, 1 \},\!</math> whose elements are interpreted as logical values, typically <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math>

The ''characteristic function'' of the relation <math>L,\!</math> written <math>f_L\!</math> or <math>\chi(L),\!</math> is the [[boolean-valued function]] <math>f_L : X_1 \times \ldots \times X_k \to \mathbb{B},\!</math> defined in such a way that <math>f_L (\mathbf{x}) = 1\!</math> just in case the <math>k\!</math>-tuple <math>\mathbf{x} = (x_1, \ldots, x_k)\!</math> is in the relation <math>L.\!</math>  The characteristic function of a relation may also be called its ''indicator function'', especially in probabilistic and statistical contexts.

It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like <math>f_L\!</math> as a <math>k\!</math>-place ''predicate''.  From the more abstract viewpoints of formal logic and model theory, the relation <math>L\!</math> is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible interpretations of a corresponding <math>k\!</math>-place ''predicate symbol'', as that term is used in ''predicate calculus''.

Due to the convergence of many traditions of study, there are wide variations in the language used to describe relations.  The ''extensional'' approach presented in this article treats a relation as the set-theoretic ''extension'' of a relational concept or term.  An alternative, ''intensional approach'' reserves the term ''relation'' to the corresponding logical entity, either the ''logical comprehension'', which is the totality of ''intensions'' or abstract ''properties'' that all the elements of the extensional relation have in common, or else the symbols that are taken to denote those elements and intensions.

==Example 2. Coplanarity==

For lines <math>\ell\!</math> in three-dimensional space, there is a triadic relation picking out the triples of lines that are coplanar.  This does not reduce to the dyadic relation of coplanarity between pairs of lines.

In other words, writing <math>P(\ell, m, n)\!</math> when the lines <math>\ell, m, n\!</math> lie in a plane, and <math>Q(\ell, m)\!</math> for the binary relation, it is not true that <math>Q(\ell, m),\!</math> <math>Q(m, n),\!</math> and <math>Q(n, \ell)\!</math> together imply <math>P(\ell, m, n),\!</math> although the converse is certainly true:  any two of three coplanar lines are necessarily coplanar.  There are two geometrical reasons for this.

In one case, for example taking the <math>x\!</math>-axis, <math>y\!</math>-axis, and <math>z\!</math>-axis, the three lines are concurrent, that is, they intersect at a single point.  In another case, <math>\ell, m, n\!</math> can be three edges of an infinite triangular prism.

What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

==Remarks==

Relations are classified by the number of sets in the cartesian product, in other words, the number of places or terms in the relational expression:

{| align="center" cellspacing="6" width="90%"
| width="18%" | <math>L(a)\!</math>
| Monadic or unary relation, in other words, a property or set
|-
| <math>L(a, b) ~\text{or}~ a L b\!</math>
| Dyadic or binary relation
|-
| <math>L(a, b, c)\!</math>
| Triadic or ternary relation
|-
| <math>L(a, b, c, d)\!</math>
| Tetradic or quaternary relation
|-
| <math>L(a, b, c, d, e)\!</math>
| Pentadic or quinary relation
|}

Relations with more than five terms are usually referred to as <math>k\!</math>-adic or <math>k\!</math>-ary, for example, a 6-adic, 6-ary, or hexadic relation.

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]] (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429.

* Ulam, S.M., and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for Parallel Computation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA.

==Bibliography==

* Bourbaki, N. (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany.

* Halmos, P.R. (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ.

* Lawvere, F.W., and Rosebrugh, R. (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK.

* Maddux, R.D. (2006), ''Relation Algebras'', vol. 150 in Studies in Logic and the Foundations of Mathematics, Elsevier Science.

* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY.

* Minsky, M.L., and Papert, S.A. (1969/1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988.

* [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867–1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN.

* Royce, J. (1961), ''The Principles of Logic'', Philosophical Library, New York, NY.

* Tarski, A. (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st edition, Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.

* Ulam, S.M. (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.

* Venetus, P. (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Semiotics]]
[[Category:Set Theory]]

Relation (mathematics)

2015-11-13T18:14:24Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below.

* The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign “<math>=\!</math>” in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign “<math>{<}\!</math>” in a statement like <math>5 < 12.\!</math>  Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''.

* The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.  These are called ''finite-place'' or ''finitary'' relations.  A finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', a ''<math>k\!</math>-adic'', or a ''<math>k\!</math>-dimensional'' relation.  The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively.

==Informal introduction==

The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.  For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  The facts of a concrete situation could be organized in the form of a Table like the one below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
<math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Person}~ X\!</math>
| <math>\text{Person}~ Y\!</math>
| <math>\text{Person}~ Z\!</math>
|-
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
| <math>\text{Denise}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
|-
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
|}

 

Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>  The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion:

{| align="center" cellpadding="10"
| <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math>
|}

The data of the Table are equivalent to the following set of ordered triples:

{| align="center" cellpadding="10"
|
<math>\begin{smallmatrix}
S
& =
& \{
& \text{(Alice, Bob, Denise)},
& \text{(Charles, Alice, Bob)},
& \text{(Charles, Charles, Alice)},
& \text{(Denise, Denise, Denise)}
& \}
\end{smallmatrix}\!</math>
|}

By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.  The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.  The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package.

The Table for relation <math>S\!</math> is an extremely simple example of a relational database.  The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.  Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation.

For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth.

==Example 1. Divisibility==

A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>  This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math>

To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>  In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.  The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math>

For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math>

==Formal definitions==

There are two definitions of <math>k\!</math>-place relations that are commonly encountered in mathematics.  In order of simplicity, the first of these definitions is as follows:

'''Definition 1.'''   A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a subset of their cartesian product, written <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>  Under this definition, then, a <math>k\!</math>-ary relation is simply a set of <math>k\!</math>-tuples.

The second definition makes use of an idiom that is common in mathematics, saying that “such and such is an <math>n\!</math>-tuple” to mean that the mathematical object being defined is determined by the specification of <math>n\!</math> component mathematical objects.  In the case of a relation <math>L\!</math> over <math>k\!</math> sets, there are <math>k + 1\!</math> things to specify, namely, the <math>k\!</math> sets plus a subset of their cartesian product.  In the idiom, this is expressed by saying that <math>L\!</math> is a <math>(k+1)\!</math>-tuple.

'''Definition 2.'''   A '''relation''' <math>L\!</math> over the sets <math>X_1, \ldots, X_k\!</math> is a <math>(k+1)\!</math>-tuple <math>L = (X_1, \ldots, X_k, \mathrm{graph}(L)),\!</math> where <math>\mathrm{graph}(L)\!</math> is a subset of the cartesian product <math>X_1 \times \ldots \times X_k~\!</math> called the ''graph'' of <math>L.\!</math>

Elements of a relation are sometimes denoted by using boldface characters, for example, the constant element <math>\mathbf{a} = (a_1, \ldots, a_k)\!</math> or the variable element <math>\mathbf{x} = (x_1, \ldots, x_k).\!</math>

A statement of the form “<math>\mathbf{a}\!</math> is in the relation <math>L\!</math>” is taken to mean that <math>\mathbf{a}\!</math> is in <math>L\!</math> under the first definition and that <math>\mathbf{a}\!</math> is in <math>\mathrm{graph}(L)\!</math> under the second definition.

The following considerations apply under either definition:

:* The sets ''X''''j'' for ''j'' = 1 to ''k'' are called the ''[[domain]]s'' of the relation. In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains.

:* If all of the domains ''X''''j'' are the same set ''X'', then ''L'' is more simply referred to as a ''k''-ary relation over ''X''.

:* If any of the domains ''X''''j'' is empty, then the cartesian product is empty, and the only relation over such a sequence of domains is the empty relation ''L'' = <math>\varnothing</math>. As a result, naturally occurring applications of the relation concept typically involve a stipulation that all of the domains be nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that falls under it will be called a 'relation' for the duration of that discussion. If it becomes necessary to distinguish the two alternatives, the latter type of object can be referred to as an ''embedded'' or ''included'' relation.

If ''L'' is a relation over the domains ''X''1, …, ''X''''k'', it is conventional to consider a sequence of terms called ''variables'', ''x''1, …, ''x''''k'', that are said to ''range over'' the respective domains.

A ''[[boolean domain]]'' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true.

The ''[[characteristic function]]'' of the relation ''L'', written ''f''''L'' or χ(''L''), is the [[boolean-valued function]] ''f''''L'' : ''X''1 × … × ''X''''k'' → '''B''', defined in such a way that ''f''''L''(<math>\mathbf{x}</math>) = 1 just in case the ''k''-tuple <math>\mathbf{x}</math> is in the relation ''L''. The characteristic function of a relation may also be called its ''[[indicator function]]'', especially in probability and statistics.

It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like ''f''''L'' as a ''k''-place ''[[predicate]]''. From the more abstract viewpoints of [[formal logic]] and [[model theory]], the relation ''L'' is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible [[interpretation]]s of a corresponding ''k''-place ''predicate symbol'', as that term is used in ''[[first-order logic|predicate calculus]]''.

Due to the convergence of many different styles of study on the same areas of interest, the reader will find much variation in usage here. The variation presented in this article treats a relation as the [[set theory|set-theoretic]] ''[[extension (semantics)|extension]]'' of a relational concept or term. Another variation reserves the term 'relation' to the corresponding logical entity, either the ''[[comprehension (logic)|logical comprehension]]'', which is the totality of ''[[intension]]s'' or abstract ''[[property (philosophy)|properties]]'' that all of the elements of the relation in extension have in common, or else the symbols that are taken to denote these elements and intensions. Further, but hardly finally, some writers of the latter persuasion introduce terms with more concrete connotations, like 'relational structure', for the set-theoretic extension of a given relational concept.

==Example 2. Coplanarity==

For lines ''L'' in three-dimensional space, there is a ternary relation picking out the triples of lines that are [[coplanar]]. This ''does not'' reduce to the binary [[symmetric relation]] of coplanarity of pairs of lines.

In other words, writing ''P''(''L'', ''M'', ''N'') when the lines ''L'', ''M'', and ''N'' lie in a plane, and ''Q''(''L'', ''M'') for the binary relation, it is not true that ''Q''(''L'', ''M''), ''Q''(''M'', ''N'') and ''Q''(''N'', ''L'') together imply ''P''(''L'', ''M'', ''N''); although the converse is certainly true (any pair out of three coplanar lines is coplanar, ''a fortiori''). There are two geometrical reasons for this.

In one case, for example taking the ''x''-axis, ''y''-axis and ''z''-axis, the three lines are concurrent, i.e. intersect at a single point. In another case, ''L'', ''M'', and ''N'' can be three edges of an infinite [[triangular prism]].

What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

==Remarks==

Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression:
:* Unary relation or [[property (philosophy)|property]]: ''L''(''u'')
:* Binary relation: ''L''(''u'', ''v'') or ''u'' ''L'' ''v''
:* Ternary relation: ''L''(''u'', ''v'', ''w'')
:* Quaternary relation: ''L''(''u'', ''v'', ''w'', ''x'')

Relations with more than four terms are usually referred to as ''k''-ary, for example, "a 5-ary relation".

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]] (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429.

* Ulam, S.M., and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for Parallel Computation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA.

==Bibliography==

* Bourbaki, N. (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany.

* Halmos, P.R. (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ.

* Lawvere, F.W., and Rosebrugh, R. (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK.

* Maddux, R.D. (2006), ''Relation Algebras'', vol. 150 in Studies in Logic and the Foundations of Mathematics, Elsevier Science.

* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY.

* Minsky, M.L., and Papert, S.A. (1969/1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988.

* [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867–1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN.

* Royce, J. (1961), ''The Principles of Logic'', Philosophical Library, New York, NY.

* Tarski, A. (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st edition, Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.

* Ulam, S.M. (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.

* Venetus, P. (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Semiotics]]
[[Category:Set Theory]]

Relation (mathematics)

2015-11-12T21:14:57Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

In mathematics, a '''finitary relation''' is defined by one of the formal definitions given below.

* The basic idea is to generalize the concept of a two-place relation, such as the relation of ''equality'' denoted by the sign “<math>=\!</math>” in a statement like <math>5 + 7 = 12\!</math> or the relation of ''order'' denoted by the sign “<math>{<}\!</math>” in a statement like <math>5 < 12.\!</math>  Relations that involve two ''places'' or ''roles'' are called ''binary relations'' by some and ''dyadic relations'' by others, the latter being historically prior but also useful when necessary to avoid confusion with ''binary (base 2) numerals''.

* The concept of a two-place relation is generalized by considering relations with increasing but still finite numbers of places or roles.  These are called ''finite-place'' or ''finitary'' relations.  A finitary relation involving <math>k\!</math> places is variously called a ''<math>k\!</math>-ary'', a ''<math>k\!</math>-adic'', or a ''<math>k\!</math>-dimensional'' relation.  The number <math>k\!</math> is then called the ''arity'', the ''adicity'', or the ''dimension'' of the relation, respectively.

==Informal introduction==

The definition of ''relation'' given in the next section formally captures a concept that is actually quite familiar from everyday life.  For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  The facts of a concrete situation could be organized in the form of a Table like the one below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
<math>\text{Relation}~ S ~:~ X ~\text{suspects that}~ Y ~\text{likes}~ Z\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Person}~ X\!</math>
| <math>\text{Person}~ Y\!</math>
| <math>\text{Person}~ Z\!</math>
|-
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
| <math>\text{Denise}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
| <math>\text{Bob}\!</math>
|-
| <math>\text{Charles}\!</math>
| <math>\text{Charles}\!</math>
| <math>\text{Alice}\!</math>
|-
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
| <math>\text{Denise}\!</math>
|}

 

Each row of the Table records a fact or makes an assertion of the form <math>X ~\text{suspects that}~ Y ~\text{likes}~ Z.\!</math>  For instance, the first row says, in effect, <math>\text{Alice suspects that Bob likes Denise.}\!</math>  The Table represents a relation <math>S\!</math> over the set <math>P\!</math> of people under discussion:

{| align="center" cellpadding="10"
| <math>P ~=~ \{ \text{Alice}, \text{Bob}, \text{Charles}, \text{Denise} \}\!</math>
|}

The data of the Table are equivalent to the following set of ordered triples:

{| align="center" cellpadding="10"
|
<math>\begin{smallmatrix}
S
& =
& \{
& \text{(Alice, Bob, Denise)},
& \text{(Charles, Alice, Bob)},
& \text{(Charles, Charles, Alice)},
& \text{(Denise, Denise, Denise)}
& \}
\end{smallmatrix}\!</math>
|}

By a slight overuse of notation, it is usual to write <math>S (\text{Alice}, \text{Bob}, \text{Denise})\!</math> to say the same thing as the first row of the Table.  The relation <math>S\!</math> is a ''triadic'' or ''ternary'' relation, since there are ''three'' items involved in each row.  The relation itself is a mathematical object, defined in terms of concepts from set theory, that carries all the information from the Table in one neat package.

The Table for relation <math>S\!</math> is an extremely simple example of a relational database.  The theoretical aspects of databases are the specialty of one branch of computer science, while their practical impacts have become all too familiar in our everyday lives.  Computer scientists, logicians, and mathematicians, however, tend to see different things when they look at these concrete examples and samples of the more general concept of a relation.

For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematics is nothing if not concerned with infinity, at the very least, potential infinity. This difference in perspective brings up a number of ideas that are usefully introduced at this point, if by no means covered in depth.

==Example: Divisibility==

A more typical example of a two-place relation in mathematics is the relation of ''divisibility'' between two positive integers <math>n\!</math> and <math>m\!</math> that is expressed in statements like <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} n ~\text{goes into}~ m {}^{\prime\prime}.\!</math>  This is a relation that comes up so often that a special symbol <math>{}^{\backprime\backprime} | {}^{\prime\prime}\!</math> is reserved to express it, allowing one to write <math>{}^{\backprime\backprime} n|m {}^{\prime\prime}\!</math> for <math>{}^{\backprime\backprime} n ~\text{divides}~ m {}^{\prime\prime}.\!</math>

To express the binary relation of divisibility in terms of sets, we have the set <math>P\!</math> of positive integers, <math>P = \{ 1, 2, 3, \ldots \},\!</math> and we have the binary relation <math>D\!</math> on <math>P\!</math> such that the ordered pair <math>(n, m)\!</math> is in the relation <math>D\!</math> just in case <math>n|m.\!</math>  In other turns of phrase that are frequently used, one says that the number <math>n\!</math> is related by <math>D\!</math> to the number <math>m\!</math> just in case <math>n\!</math> is a factor of <math>m,\!</math> that is, just in case <math>n\!</math> divides <math>m\!</math> with no remainder.  The relation <math>D,\!</math> regarded as a set of ordered pairs, consists of all pairs of numbers <math>(n, m)\!</math> such that <math>n\!</math> divides <math>m.\!</math>

For example, <math>2\!</math> is a factor of <math>4,\!</math> and <math>6\!</math> is a factor of <math>72,\!</math> which two facts can be written either as <math>2|4\!</math> and <math>6|72\!</math> or as <math>D(2, 4)\!</math> and <math>D(6, 72).\!</math>

==Formal definitions==

There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows:

'''Definition 1.''' A '''relation''' ''L'' over the [[set]]s ''X''1, …, ''X''''k'' is a [[subset]] of their [[cartesian product]], written ''L'' &sube; ''X''1 × … × ''X''''k''. Under this definition, then, a ''k''-ary relation is simply a set of ''k''-[[tuple]]s.

The second definition makes use of an idiom that is common in mathematics, stipulating that "such and such is an ''n''-tuple" in order to ensure that such and such a mathematical object is determined by the specification of ''n'' component mathematical objects. In the case of a relation ''L'' over ''k'' sets, there are ''k'' + 1 things to specify, namely, the ''k'' sets plus a subset of their cartesian product. In the idiom, this is expressed by saying that ''L'' is a (''k''+1)-tuple.

'''Definition 2.''' A '''relation''' ''L'' over the sets ''X''1, …, ''X''''k'' is a (''k''+1)-tuple ''L'' = (''X''1, …, ''X''''k'', ''G''(''L'')), where ''G''(''L'') is a subset of the cartesian product ''X''1 × … × ''X''''k''. ''G''(''L'') is called the ''[[graph]]'' of ''L''.

Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element <math>\mathbf{a}</math> = (a1, …, a''k'') or the variable element <math>\mathbf{x}</math> = (''x''1, …, ''x''''k'').

A statement of the form "<math>\mathbf{a}</math> is in the relation ''L'' " is taken to mean that <math>\mathbf{a}</math> is in ''L'' under the first definition and that <math>\mathbf{a}</math> is in ''G''(''L'') under the second definition.

The following considerations apply under either definition:

:* The sets ''X''''j'' for ''j'' = 1 to ''k'' are called the ''[[domain]]s'' of the relation. In the case of the first definition, the relation itself does not uniquely determine a given sequence of domains.

:* If all of the domains ''X''''j'' are the same set ''X'', then ''L'' is more simply referred to as a ''k''-ary relation over ''X''.

:* If any of the domains ''X''''j'' is empty, then the cartesian product is empty, and the only relation over such a sequence of domains is the empty relation ''L'' = <math>\varnothing</math>. As a result, naturally occurring applications of the relation concept typically involve a stipulation that all of the domains be nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that falls under it will be called a 'relation' for the duration of that discussion. If it becomes necessary to distinguish the two alternatives, the latter type of object can be referred to as an ''embedded'' or ''included'' relation.

If ''L'' is a relation over the domains ''X''1, …, ''X''''k'', it is conventional to consider a sequence of terms called ''variables'', ''x''1, …, ''x''''k'', that are said to ''range over'' the respective domains.

A ''[[boolean domain]]'' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true.

The ''[[characteristic function]]'' of the relation ''L'', written ''f''''L'' or χ(''L''), is the [[boolean-valued function]] ''f''''L'' : ''X''1 × … × ''X''''k'' → '''B''', defined in such a way that ''f''''L''(<math>\mathbf{x}</math>) = 1 just in case the ''k''-tuple <math>\mathbf{x}</math> is in the relation ''L''. The characteristic function of a relation may also be called its ''[[indicator function]]'', especially in probability and statistics.

It is conventional in applied mathematics, computer science, and statistics to refer to a boolean-valued function like ''f''''L'' as a ''k''-place ''[[predicate]]''. From the more abstract viewpoints of [[formal logic]] and [[model theory]], the relation ''L'' is seen as constituting a ''logical model'' or a ''relational structure'' that serves as one of many possible [[interpretation]]s of a corresponding ''k''-place ''predicate symbol'', as that term is used in ''[[first-order logic|predicate calculus]]''.

Due to the convergence of many different styles of study on the same areas of interest, the reader will find much variation in usage here. The variation presented in this article treats a relation as the [[set theory|set-theoretic]] ''[[extension (semantics)|extension]]'' of a relational concept or term. Another variation reserves the term 'relation' to the corresponding logical entity, either the ''[[comprehension (logic)|logical comprehension]]'', which is the totality of ''[[intension]]s'' or abstract ''[[property (philosophy)|properties]]'' that all of the elements of the relation in extension have in common, or else the symbols that are taken to denote these elements and intensions. Further, but hardly finally, some writers of the latter persuasion introduce terms with more concrete connotations, like 'relational structure', for the set-theoretic extension of a given relational concept.

==Example: coplanarity==

For lines ''L'' in three-dimensional space, there is a ternary relation picking out the triples of lines that are [[coplanar]]. This ''does not'' reduce to the binary [[symmetric relation]] of coplanarity of pairs of lines.

In other words, writing ''P''(''L'', ''M'', ''N'') when the lines ''L'', ''M'', and ''N'' lie in a plane, and ''Q''(''L'', ''M'') for the binary relation, it is not true that ''Q''(''L'', ''M''), ''Q''(''M'', ''N'') and ''Q''(''N'', ''L'') together imply ''P''(''L'', ''M'', ''N''); although the converse is certainly true (any pair out of three coplanar lines is coplanar, ''a fortiori''). There are two geometrical reasons for this.

In one case, for example taking the ''x''-axis, ''y''-axis and ''z''-axis, the three lines are concurrent, i.e. intersect at a single point. In another case, ''L'', ''M'', and ''N'' can be three edges of an infinite [[triangular prism]].

What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

==Remarks==

Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression:
:* Unary relation or [[property (philosophy)|property]]: ''L''(''u'')
:* Binary relation: ''L''(''u'', ''v'') or ''u'' ''L'' ''v''
:* Ternary relation: ''L''(''u'', ''v'', ''w'')
:* Quaternary relation: ''L''(''u'', ''v'', ''w'', ''x'')

Relations with more than four terms are usually referred to as ''k''-ary, for example, "a 5-ary relation".

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]] (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429.

* Ulam, S.M., and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for Parallel Computation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA.

==Bibliography==

* [[Nicolas Bourbaki|Bourbaki, N.]] (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany.

* [[Paul Richard Halmos|Halmos, P.R.]] (1960), ''Naive Set Theory'', D. Van Nostrand Company, Princeton, NJ.

* [[Francis William Lawvere|Lawvere, F.W.]], and [[Robert Rosebrugh|Rosebrugh, R.]] (2003), ''Sets for Mathematics'', Cambridge University Press, Cambridge, UK.

* Maddux, R.D. (2006), ''Relation Algebras'', vol. 150 in 'Studies in Logic and the Foundations of Mathematics', Elsevier Science.

* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY.

* [[Marvin L. Minsky|Minsky, M.L.]], and [[Seymour A. Papert|Papert, S.A.]] (1969/1988), ''[[Perceptron]]s, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Expanded edition, 1988.

* [[Charles Sanders Peirce|Peirce, C.S.]] (1984), ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867-1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN.

* [[Josiah Royce|Royce, J.]] (1961), ''The Principles of Logic'', Philosophical Library, New York, NY.

* [[Alfred Tarski|Tarski, A.]] (1956/1983), ''Logic, Semantics, Metamathematics, Papers from 1923 to 1938'', J.H. Woodger (trans.), 1st edition, Oxford University Press, 1956. 2nd edition, J. Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.

* [[Stanisław Marcin Ulam|Ulam, S.M.]] (1990), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.

* [[Paulus Venetus|Venetus, P.]] (1984), ''Logica Parva, Translation of the 1472 Edition with Introduction and Notes'', Alan R. Perreiah (trans.), Philosophia Verlag, Munich, Germany.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation @ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_(mathematics) Relation @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_(mathematics) Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_(mathematics) Relation], [http://mywikibiz.com/ MyWikiBiz]
* [http://wikinfo.org/w/index.php/Relation_(mathematics) Relation], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_(mathematics) Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_(mathematics)&oldid=73324659 Relation], [http://en.wikipedia.org/ Wikipedia]

[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Semiotics]]
[[Category:Set Theory]]

Logical matrix

2015-11-11T16:16:49Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''logical matrix''', in the finite dimensional case, is a <math>k\!</math>-dimensional array with entries from the [[boolean domain]] <math>\mathbb{B} = \{ 0,1 \}.</math> Such a matrix affords a matrix representation of a <math>k\!</math>-adic [[relation (mathematics)|relation]].

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Logical_matrix Logical Matrix @ InterSciWiki]
* [http://mywikibiz.com/Logical_matrix Logical Matrix @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Logical_matrix Logical Matrix @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Logical_matrix Logical Matrix], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Logical_matrix Logical Matrix], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/LogicalMatrix Logical Matrix], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Logical_matrix Logical Matrix], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logical_matrix Logical Matrix], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Logical_matrix&oldid=43606082 Logical Matrix], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Discrete Mathematics]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Semiotics]]

The Simplest Mathematics

2015-11-11T15:36:55Z

Jon Awbrey:

'''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter 3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January–February 1902 but was not published until the appearance of his ''Collected Papers, Volume 4'' in 1933. Peirce introduces the subject of the paper as “certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration” (CP 4.227).

==Related topic==

* [[Kaina Stoicheia]]

==References==

* [[Charles Sanders Peirce (Bibliography)]].

* Peirce, C.S. (1902), “The Simplest Mathematics”, MS dated January–February 1902, intended as Chapter 3 of the ''Minute Logic'', CP 4.227–323 in ''Collected Papers''.

* Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP volume.paragraph.

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

The Simplest Mathematics

2015-11-11T15:22:01Z

Jon Awbrey: update

'''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter 3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January–February 1902 but was not published until the appearance of his ''Collected Papers, Volume 4'' in 1933. Peirce introduces the subject of the paper as “certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration” (CP 4.227).

==References==

* [[Charles Sanders Peirce (Bibliography)]].

* Peirce, C.S. (1902), “The Simplest Mathematics”, MS dated January–February 1902, intended as Chapter 3 of the ''Minute Logic'', CP 4.227–323 in ''Collected Papers''.

* Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP volume.paragraph.

==See also==

* [[Kaina Stoicheia]]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Logic of relatives

2015-11-10T22:54:22Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

The '''logic of relatives''', more precisely, the '''logic of relative terms''', is the study of [[relation (mathematics)|relation]]s as represented in symbolic forms known as ''rhemes'', ''rhemata'', or ''relative terms''. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter.

The consideration of ''[[relative term]]s'' has its roots in antiquity, but it entered a radically new phase of development with the work of [[Charles Sanders Peirce]], beginning with his 1870 paper “[[Logic of Relatives (1870)|Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic]]”.

==See also==

* [[Logic of Relatives (1870)]]
* [[Logic of Relatives (1883)]]

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 1870. Reprinted, ''Collected Papers'' CP 3.45–149, ''Chronological Edition'' CE 2, 359–429.

* [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Awbrey, J.L., “Peirce's 1870 Logic of Relatives”]

==Bibliography==

* Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938.

* Aristotle, “On Interpretation”, Harold P. Cooke (trans.), pp. 111–179 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938.

* Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938.

* Boole, George, ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.

* Maddux, Roger D., ''Relation Algebras'', vol. 150 in ‘Studies in Logic and the Foundations of Mathematics’, Elsevier Science, 2006.

* Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP volume.paragraph.

* Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition, Volume 2, 1867–1871'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1984. Cited as CE 2.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_relatives Logic of Relatives @ InterSciWiki]
* [http://mywikibiz.com/Logic_of_relatives Logic of Relatives @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Logic_of_relatives Logic of Relatives @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Logic_of_relatives Logic of Relatives], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Logic_of_relatives Logic of Relatives], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Logic_of_Relatives Logic of Relatives], [http://semanticweb.org/ Semantic Web]
* [http://wikinfo.org/w/index.php/Logic_of_relatives Logic of Relatives], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Logic_of_relatives Logic of Relatives], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Logic_of_relatives&oldid=43501411 Logic of Relatives], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Science]]
[[Category:Semiotics]]

The Simplest Mathematics

2015-11-10T19:28:07Z

Jon Awbrey: add article

'''''The Simplest Mathematics''''' is the title of a paper by [[Charles Sanders Peirce]], intended as Chapter 3 of his unfinished magnum opus ''The Minute Logic''. The paper is dated January–February 1902 but was not published until the appearance of his ''Collected Papers, Volume 4'' in 1933. Peirce introduces the subject of the paper as “certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration” (CP 4.227).

==References==

* Peirce, Benjamin (1870), “Linear Associative Algebra”, § 1. See ''American Journal of Mathematics'' 4 (1881).

* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* Peirce, C.S. (1902), “The Simplest Mathematics”, MS dated January–February 1902, intended as Chapter 3 of the ''Minute Logic'', CP 4.227–323 in ''Collected Papers''.

* Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP volume.paragraph.

==See also==

* [[Foundations of mathematics]]
* [[Kaina Stoicheia]]
* [[Philosophy of mathematics]]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Prescisive abstraction

2015-11-10T19:12:07Z

Jon Awbrey:

'''Prescisive abstraction''' or '''prescision''', variously spelled as '''precisive abstraction''' or '''prescission''', is a formal operation that marks, selects, or singles out one feature of a concrete experience to the disregard of others.

The above definition is adapted from the one given by [[Charles Sanders Peirce]] (CP 4.235, “[[The Simplest Mathematics]]” (1902), in ''Collected Papers'', CP 4.227–393).

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.

==See also==

* [[Charles Sanders Peirce]]
* [[Hypostatic abstraction]]
* [[Hypostatic object]]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Hypostatic abstraction

2015-11-10T18:58:54Z

Jon Awbrey:

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>  The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>  Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.  The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''.

The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP 4.235, “[[The Simplest Mathematics]]” (1902), in ''Collected Papers'', CP 4.227–323).

The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process.

For example, a typical case of hypostatic abstraction occurs in the transformation from “honey is sweet” to “honey possesses sweetness”, which transformation can be viewed in the following variety of ways:

 
[[Image:Hypostatic Abstraction Figure 1.png|center]] 
[[Image:Hypostatic Abstraction Figure 2.png|center]] 
[[Image:Hypostatic Abstraction Figure 3.png|center]] 
[[Image:Hypostatic Abstraction Figure 4.png|center]] 

The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective “sweet” from the main predicate “is sweet”, thus arriving at a new, increased-arity predicate “possesses”, and as a by-product of the reaction, as it were, precipitating out the substantive “sweetness” as a new second subject of the new predicate, “possesses”.

==See also==

* [[Hypostatic object]]
* [[Prescisive abstraction]]

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction → ThoughtMesh]

* [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction'']

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath]
* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Hypostatic object

2015-11-10T18:45:04Z

Jon Awbrey: add article

A '''hypostatic object''', also known in certain senses as an '''abstract object''' or a '''formal object''', is an object of discussion or thought that results as the normal product of a process of ''[[hypostatic abstraction]]''.

==See also==

* [[Charles Sanders Peirce]]
* [[Hypostatic abstraction]]
* [[Prescisive abstraction]]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Hypostatic abstraction

2015-11-10T16:10:09Z

Jon Awbrey: + ==See also==

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>  The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>  Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.  The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''.

The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP 4.235, “[[The Simplest Mathematics]]” (1902), in ''Collected Papers'', CP 4.227–323).

The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process.

For example, a typical case of hypostatic abstraction occurs in the transformation from “honey is sweet” to “honey possesses sweetness”, which transformation can be viewed in the following variety of ways:

 
[[Image:Hypostatic Abstraction Figure 1.png|center]] 
[[Image:Hypostatic Abstraction Figure 2.png|center]] 
[[Image:Hypostatic Abstraction Figure 3.png|center]] 
[[Image:Hypostatic Abstraction Figure 4.png|center]] 

The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective “sweet” from the main predicate “is sweet”, thus arriving at a new, increased-arity predicate “possesses”, and as a by-product of the reaction, as it were, precipitating out the substantive “sweetness” as a new second subject of the new predicate, “possesses”.

==See also==

* [[Hypostatic object]]
* [[Prescisive abstraction]]
* [[The Simplest Mathematics]]

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction → ThoughtMesh]

* [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction'']

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath]
* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Prescisive abstraction

2015-11-10T15:14:47Z

Jon Awbrey: add article

'''Prescisive abstraction''' or '''prescision''', variously spelled as '''precisive abstraction''' or '''prescission''', is a formal operation that marks, selects, or singles out one feature of a concrete experience to the disregard of others.

The above definition is adapted from the one given by [[Charles Sanders Peirce]] (CP 4.235, “[[The Simplest Mathematics]]” (1902), in ''Collected Papers'', CP 4.227–393).

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.

==See also==

* [[Hypostatic abstraction]]
* [[Hypostatic object]]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Continuous predicate

2015-11-10T04:10:56Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''continuous predicate''', as described by Charles Sanders Peirce, is a special type of [[relation (mathematics)|relational predicate]] that arises as the limit of an iterated process of [[hypostatic abstraction]].

Here is one of Peirce's definitive discussions of the concept:

<div style="margin-left:5em; margin-right:20em">

When we have analyzed a proposition so as to throw into the subject everything that can be removed from the predicate, all that it remains for the predicate to represent is the form of connection between the different subjects as expressed in the propositional ''form''.  What I mean by “everything that can be removed from the predicate” is best explained by giving an example of something not so removable.

But first take something removable.  “Cain kills Abel.”  Here the predicate appears as “— kills —.”  But we can remove killing from the predicate and make the latter “— stands in the relation — to —.”  Suppose we attempt to remove more from the predicate and put the last into the form “— exercises the function of relate of the relation — to —” and then putting “the function of relate to the relation” into another subject leave as predicate “— exercises — in respect to — to —.”  But this “exercises” expresses “exercises the function”.  Nay more, it expresses “exercises the function of relate”, so that we find that though we may put this into a separate subject, it continues in the predicate just the same.

Stating this in another form, to say that “A is in the relation R to B” is to say that A is in a certain relation to R.  Let us separate this out thus:  “A is in the relation R1 (where R1 is the relation of a relate to the relation of which it is the relate) to R to B”.  But A is here said to be in a certain relation to the relation R1.  So that we can express the same fact by saying, “A is in the relation R1 to the relation R1 to the relation R to B”, and so on ''ad infinitum''.

A predicate which can thus be analyzed into parts all homogeneous with the whole I call a ''continuous predicate''.  It is very important in logical analysis, because a continuous predicate obviously cannot be a ''compound'' except of continuous predicates, and thus when we have carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate elements.

(C.S. Peirce, “Letters to Lady Welby” (14 December 1908), ''Selected Writings'', pp. 396–397).

</div>

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], “Letters to Lady Welby”, pp. 380–432 in ''Charles S. Peirce : Selected Writings (Values in a Universe of Chance)'', Philip P. Wiener (ed.), Dover Publications, New York, NY, 1966.

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate → ThoughtMesh]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate @ InterSciWiki]
* [http://mywikibiz.com/Continuous_predicate Continuous Predicate @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Continuous_predicate Continuous Predicate @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Continuous_predicate Continuous Predicate], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Continuous_predicate Continuous Predicate], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ContinuousPredicate Continuous Predicate], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Continuous_predicate Continuous Predicate], [http://semanticweb.org/ SemanticWeb]
* [http://vectors.usc.edu/thoughtmesh/publish/147.php Continuous Predicate], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php?title=Continuous_predicate Continuous Predicate], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Continuous_predicate Continuous Predicate], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Continuous_predicate&oldid=96870273 Continuous Predicate], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Hypostatic abstraction

2015-11-09T22:30:46Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Hypostatic abstraction''' is a formal operation that takes an element of information, as expressed in a proposition <math>X ~\text{is}~ Y,\!</math> and conceives its information to consist in the relation between that subject and another subject, as expressed in the proposition <math>X ~\text{has}~ Y\!\text{-ness}.\!</math>  The existence of the abstract subject <math>Y\!\text{-ness}\!</math> consists solely in the truth of those propositions that contain the concrete predicate <math>Y.\!</math>  Hypostatic abstraction is known by many names, for example, ''hypostasis'', ''objectification'', ''reification'', and ''subjectal abstraction''.  The object of discussion or thought thus introduced is termed a ''[[hypostatic object]]''.

The above definition is adapted from the one given by introduced [[Charles Sanders Peirce]] (CP 4.235, “[[The Simplest Mathematics]]” (1902), in ''Collected Papers'', CP 4.227–323).

The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the ''arity'', also called the ''adicity'', of the main predicate in the process.

For example, a typical case of hypostatic abstraction occurs in the transformation from “honey is sweet” to “honey possesses sweetness”, which transformation can be viewed in the following variety of ways:

 
[[Image:Hypostatic Abstraction Figure 1.png|center]] 
[[Image:Hypostatic Abstraction Figure 2.png|center]] 
[[Image:Hypostatic Abstraction Figure 3.png|center]] 
[[Image:Hypostatic Abstraction Figure 4.png|center]] 

The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective “sweet” from the main predicate “is sweet”, thus arriving at a new, increased-arity predicate “possesses”, and as a by-product of the reaction, as it were, precipitating out the substantive “sweetness” as a new second subject of the new predicate, “possesses”.

==References==

* [[Charles Sanders Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.

==Resources==

* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction → ThoughtMesh]

* [http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm J. Jay Zeman, ''Peirce on Abstraction'']

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction @ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Hypostatic_abstraction Hypostatic Abstraction], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Hypostatic_abstraction Hypostatic Abstraction], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/HypostaticAbstraction Hypostatic Abstraction], [http://planetmath.org/ PlanetMath]
* [http://vectors.usc.edu/thoughtmesh/publish/146.php Hypostatic Abstraction], [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]
* [http://wikinfo.org/w/index.php?title=Hypostatic_abstraction Hypostatic Abstraction], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Hypostatic_abstraction Hypostatic Abstraction], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Hypostatic_abstraction&oldid=69736615 Hypostatic Abstraction], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Inquiry]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Ontology]]
[[Category:Philosophy]]
[[Category:Pragmatism]]
[[Category:Semiotics]]

Continuous predicate

2015-11-09T15:14:01Z

Jon Awbrey: update

Zeroth order logic

2015-11-09T03:08:28Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.  The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

==Propositional forms on two variables==

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}\!</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math> in a number of different languages for zeroth order logic.

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
|  
|  
|  
|-
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(~)}
\\[4pt]
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
\texttt{(} x \texttt{)}
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{,} ~ y \texttt{)}
\\[4pt]
\texttt{(} x ~ y \texttt{)}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~ y
\\[4pt]
\texttt{((} x \texttt{,} ~ y \texttt{))}
\\[4pt]
y
\\[4pt]
\texttt{(} x ~ \texttt{(} y \texttt{))}
\\[4pt]
x
\\[4pt]
\texttt{((} x \texttt{)} ~ y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\texttt{((~))}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}\!</math>
|}

 

These six languages for the sixteen boolean functions are conveniently described in the following order:

* Language <math>\mathcal{L}_{3}\!</math> describes each boolean function <math>f : \mathbb{B}^2 \to \mathbb{B}\!</math> by means of the sequence of four boolean values, <math>f(1,1),\!</math> <math>f(1,0),\!</math> <math>f(0,1),\!</math> <math>f(0,0).\!</math>  Such a sequence, perhaps in another order, and perhaps with the logical values <math>\mathrm{F}\!</math> and <math>\mathrm{T}\!</math> instead of the boolean values <math>0\!</math> and <math>1,\!</math> respectively, would normally be displayed as a column in a [[truth table]].

* Language <math>\mathcal{L}_{2}\!</math> lists the sixteen functions in the form <math>f_i,\!</math> where the index <math>i\!</math> is a bit string formed from the sequence of boolean values in <math>\mathcal{L}_{3}.\!</math>

* Language <math>\mathcal{L}_{1}\!</math> notates the boolean functions <math>f_i\!</math> with an index <math>i\!</math> that is the decimal equivalent of the binary numeral index in <math>\mathcal{L}_{2}.\!</math>

* Language <math>\mathcal{L}_{4}\!</math> expresses the sixteen functions in terms of [[logical conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations:

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
\texttt{()}
& = & 0
& = & \mathrm{false}
\\[6pt]
\texttt{(} x \texttt{)}
& = & \tilde{x}
& = & x^\prime
\\[6pt]
\texttt{(} x \texttt{,} y \texttt{)}
& = & \tilde{x}y \lor x\tilde{y}
& = & x^\prime y \lor x y^\prime
\\[6pt]
\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}
& = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z}
& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
\end{matrix}\!</math>
|}

It may be noted that <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y.</math> The inclusive disjunctions indicated for <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math> and for <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math> may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. However, the function <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math> is not the same thing as the function <math>x + y + z.\!</math>

* Language <math>\mathcal{L}_{5}\!</math> lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.

* Language <math>\mathcal{L}_{6}\!</math> expresses the sixteen functions in one of several notations that are commonly used in formal logic.

==Translations==

* [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 中文 : 零阶逻辑]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
* [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Computer Science]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]
[[Category:Semiotics]]

Zeroth order logic

2015-11-08T15:50:18Z

Jon Awbrey: update with TeX table

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.  The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

==Propositional forms on two variables==

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic.

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
|  
|  
|  
|-
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(~)}
\\[4pt]
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
\texttt{(} x \texttt{)}
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{,} ~ y \texttt{)}
\\[4pt]
\texttt{(} x ~ y \texttt{)}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~ y
\\[4pt]
\texttt{((} x \texttt{,} ~ y \texttt{))}
\\[4pt]
y
\\[4pt]
\texttt{(} x ~ \texttt{(} y \texttt{))}
\\[4pt]
x
\\[4pt]
\texttt{((} x \texttt{)} ~ y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\texttt{((~))}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}\!</math>
|}

 

These six languages for the sixteen boolean functions are conveniently described in the following order:

* Language '''L3''' describes each boolean function ''f'' : '''B'''2 → '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]].

* Language '''L2''' lists the sixteen functions in the form '''fi''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L3'''.

* Language '''L1''' notates the boolean functions '''fi''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L2'''.

* Language '''L4''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations:

: <math>\begin{matrix}
(\ ) & = & 0 & = & \mbox{false} \\
(x) & = & \tilde{x} & = & x' \\
(x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
\end{matrix}</math>

It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>.

* Language '''L5''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.

* Language '''L6''' expresses the sixteen functions in one of several notations that are commonly used in formal logic.

==Translations==

* [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 中文 : 零阶逻辑]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
* [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Computer Science]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]
[[Category:Semiotics]]

Zeroth order logic

2015-11-08T03:44:42Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.  The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

==Propositional forms on two variables==

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic.

 

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:#e6e6ff"
! style="width:15%" | L1
! style="width:15%" | L2
! style="width:15%" | L3
! style="width:15%" | L4
! style="width:15%" | L5
! style="width:15%" | L6
|- style="background:#e6e6ff"
|  
| align="right" | x :
| 1 1 0 0
|  
|  
|  
|- style="background:#e6e6ff"
|  
| align="right" | y :
| 1 0 1 0
|  
|  
|  
|-
| f0 || f0000 || 0 0 0 0 || ( ) || false || 0
|-
| f1 || f0001 || 0 0 0 1 || (x)(y) || neither x nor y || ¬x &and; ¬y
|-
| f2 || f0010 || 0 0 1 0 || (x) y || y and not x || ¬x &and; y
|-
| f3 || f0011 || 0 0 1 1 || (x) || not x || ¬x
|-
| f4 || f0100 || 0 1 0 0 || x (y) || x and not y || x &and; ¬y
|-
| f5 || f0101 || 0 1 0 1 || (y) || not y || ¬y
|-
| f6 || f0110 || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
|-
| f7 || f0111 || 0 1 1 1 || (x y) || not both x and y || ¬x &or; ¬y
|-
| f8 || f1000 || 1 0 0 0 || x y || x and y || x &and; y
|-
| f9 || f1001 || 1 0 0 1 || ((x, y)) || x equal to y || x = y
|-
| f10 || f1010 || 1 0 1 0 || y || y || y
|-
| f11 || f1011 || 1 0 1 1 || (x (y)) || not x without y || x → y
|-
| f12 || f1100 || 1 1 0 0 || x || x || x
|-
| f13 || f1101 || 1 1 0 1 || ((x) y) || not y without x || x ← y
|-
| f14 || f1110 || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
|-
| f15 || f1111 || 1 1 1 1 || (( )) || true || 1
|}

 

These six languages for the sixteen boolean functions are conveniently described in the following order:

* Language '''L3''' describes each boolean function ''f'' : '''B'''2 → '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]].

* Language '''L2''' lists the sixteen functions in the form '''fi''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L3'''.

* Language '''L1''' notates the boolean functions '''fi''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L2'''.

* Language '''L4''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations:

: <math>\begin{matrix}
(\ ) & = & 0 & = & \mbox{false} \\
(x) & = & \tilde{x} & = & x' \\
(x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
\end{matrix}</math>

It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>.

* Language '''L5''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.

* Language '''L6''' expresses the sixteen functions in one of several notations that are commonly used in formal logic.

==Translations==

* [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 中文 : 零阶逻辑]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
* [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Computer Science]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]

Universe of discourse

2015-11-07T19:14:55Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

The term '''''universe of discourse''''' is generally attributed to Augustus De Morgan (1846).  George Boole (1854) defines it in the following manner:

{| align="center" cellpadding="4" width="90%"
|
In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. … Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. (Boole 1854/1958, p. 42).
|}

==References==

* Boole, George (1854/1958), ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan Publishers, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.

* De Morgan, Augustus (1846), ''Cambridge Philosophical Transactions'', ''viii'', p. 380.

==Resources==

* [http://www.helsinki.fi/science/commens/dictionary.html Commens Dictionary of Peirce's Terms]
** [http://www.helsinki.fi/science/commens/terms/logicaluniv.html Logical Universe]
** [http://www.helsinki.fi/science/commens/terms/universedisc.html Universe of Discourse]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Universe_of_discourse Universe of Discourse @ InterSciWiki]
* [http://mywikibiz.com/Universe_of_discourse Universe of Discourse @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Universe_of_discourse Universe of Discourse @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Universe_of_discourse Universe of Discourse], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Universe_of_discourse Universe of Discourse], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/UniverseOfDiscourse Universe of Discourse], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Universe_of_discourse Universe of Discourse], [http://semanticweb.org/ SemanticWeb]
* [http://en.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Universe_of_discourse Universe of Discourse], [http://beta.wikiversity.org/ Wikiversity Beta]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Computer Science]]
[[Category:Formal Grammars]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Model Theory]]
[[Category:Normative Sciences]]
[[Category:Pragmatics]]
[[Category:Proof Theory]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Syntax]]

Truth table

2015-11-07T18:31:02Z

Jon Awbrey:

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''truth table''' is a tabular array that illustrates the computation of a ''logical function'', that is, a function of the form <math>f : \mathbb{A}^k \to \mathbb{A},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{A}</math> is the domain of logical values <math>\{ \operatorname{false}, \operatorname{true} \}.</math> The names of the logical values, or ''truth values'', are commonly abbreviated in accord with the equations <math>\operatorname{F} = \operatorname{false}</math> and <math>\operatorname{T} = \operatorname{true}.</math>

In many applications it is usual to represent a truth function by a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math> In most applications <math>\operatorname{false}</math> is represented by <math>0\!</math> and <math>\operatorname{true}</math> is represented by <math>1\!</math> but the opposite representation is also possible, depending on the overall representation of truth functions as boolean functions. The remainder of this article assumes the usual representation, taking the equations <math>\operatorname{F} = 0</math> and <math>\operatorname{T} = 1</math> for granted.

==Logical negation==

'''[[Logical negation]]''' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true.

The truth table of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical Negation}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:50%" | <math>p\!</math>
| style="width:50%" | <math>\lnot p\!</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}

 

The negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" width="45%"
|+ style="height:30px" | <math>\text{Variant Notations}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" align="center" | <math>\text{Notation}\!</math>
| width="50%" | <math>\text{Vocalization}\!</math>
|-
| align="center" | <math>\bar{p}\!</math>
| <math>p\!</math> bar
|-
| align="center" | <math>\tilde{p}\!</math>
| <math>p\!</math> tilde
|-
| align="center" | <math>p'\!</math>
| <math>p\!</math> prime <math>p\!</math> complement
|-
| align="center" | <math>!p\!</math>
| bang <math>p\!</math>
|}

 

==Logical conjunction==

'''[[Logical conjunction]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are true.

The truth table of <math>p ~\operatorname{AND}~ q,</math> also written <math>p \land q\!</math> or <math>p \cdot q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical Conjunction}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \land q</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}

 

==Logical disjunction==

'''[[Logical disjunction]]''', also called '''logical alternation''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are false.

The truth table of <math>p ~\operatorname{OR}~ q,</math> also written <math>p \lor q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical Disjunction}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \lor q</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}

 

==Logical equality==

'''[[Logical equality]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both operands are false or both operands are true.

The truth table of <math>p ~\operatorname{EQ}~ q,</math> also written <math>p = q,\!</math> <math>p \Leftrightarrow q,\!</math> or <math>p \equiv q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical Equality}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p = q\!</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}

 

==Exclusive disjunction==

'''[[Exclusive disjunction]]''', also known as '''logical inequality''' or '''symmetric difference''', is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' just in case exactly one of its operands is true.

The truth table of <math>p ~\operatorname{XOR}~ q,</math> also written <math>p + q\!</math> or <math>p \ne q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Exclusive Disjunction}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p ~\operatorname{XOR}~ q</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}

 

The following equivalents may then be deduced:

{| align="center" cellspacing="10" width="90%"
|
<math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q)
\\[6pt]
& = & (p \lor q) & \land & (\lnot p \lor \lnot q)
\\[6pt]
& = & (p \lor q) & \land & \lnot (p \land q)
\end{matrix}</math>
|}

==Logical implication==

The '''[[logical implication]]''' relation and the '''material conditional''' function are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.

The truth table associated with the material conditional <math>\text{if}~ p ~\text{then}~ q,\!</math> symbolized <math>p \rightarrow q,\!</math> and the logical implication <math>p ~\text{implies}~ q,\!</math> symbolized <math>p \Rightarrow q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical Implication}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \Rightarrow q\!</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|}

 

==Logical NAND==

The '''[[logical NAND]]''' is an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false.

The truth table of <math>p ~\operatorname{NAND}~ q,</math> also written <math>p \stackrel{\circ}{\curlywedge} q\!</math> or <math>p \barwedge q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical NAND}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \stackrel{\circ}{\curlywedge} q\!</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}

 

==Logical NNOR==

The '''[[logical NNOR]]''' (“Neither Nor”) is an operation on two logical values, typically the values of two propositions, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true.

The truth table of <math>p ~\operatorname{NNOR}~ q,</math> also written <math>p \curlywedge q,\!</math> appears below:

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ style="height:30px" | <math>\text{Logical NNOR}\!</math>
|- style="height:40px; background:#f0f0ff"
| style="width:33%" | <math>p\!</math>
| style="width:33%" | <math>q\!</math>
| style="width:33%" | <math>p \curlywedge q\!</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
|-
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
|-
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
|}

 

==Translations==

* [http://zh.wikipedia.org/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8 中文 : 真值表]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table @ InterSciWiki]
* [http://mywikibiz.com/Truth_table Truth Table @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Truth_table Truth Table @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Truth_table Truth Table], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Truth_table Truth Table], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Truth_table Truth Table], [http://semanticweb.org/ SemanticWeb]
* [http://wikinfo.org/w/index.php/Truth_table Truth Table], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Truth_table Truth Table], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Truth_table Truth Table], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Truth_table&oldid=77110085 Truth Table], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Discrete Mathematics]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Semiotics]]

Sole sufficient operator

2015-11-07T18:10:08Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''sole sufficient operator''' is an operator that is sufficient by itself to generate every operator in a specified class of operators.  In the context of [[logic]], it is a logical operator that suffices to generate every [[boolean-valued function]], <math>f : X \to \mathbb{B},\!</math> where <math>X\!</math> is an arbitrary set and where <math>\mathbb{B}\!</math> is a generic two-element set, typically <math>\mathbb{B} = \{ 0, 1 \} = \{ \mathrm{false}, \mathrm{true} \},\!</math> in particular, to generate every finitary [[boolean function]], <math>f : \mathbb{B}^k \to \mathbb{B}.\!</math>

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Sole_sufficient_operator Sole Sufficient Operator @ InterSciWiki]
* [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Sole_sufficient_operator Sole Sufficient Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Sole_sufficient_operator Sole Sufficient Operator], [http://mywikibiz.com/ MyWikiBiz]
* [http://http://planetmath.org/SoleSufficientOperator Sole Sufficient Operator], [http://planetmath.org/ PlanetMath]
* [http://semanticweb.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://semanticweb.org/ SemanticWeb]
* [http://wikinfo.org/w/index.php/Sole_sufficient_operator Sole Sufficient Operator], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Sole_sufficient_operator Sole Sufficient Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Sole_sufficient_operator&oldid=156136346 Sole Sufficient Operator], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Semiotics]]

Propositional calculus

2015-11-07T16:02:26Z

Jon Awbrey: add article

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''propositional calculus''' (or a '''sentential calculus''') is a formal system that represents the materials and the principles of ''propositional logic'' (or ''sentential logic''). Propositional logic is a domain of formal subject matter that is, up to somorphism, constituted by the structural relationships of mathematical objects called ''propositions''.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions (''well-formed formulas'' or ''wffs''), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions.

When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of ''simplifying'' expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical ''axioms''.

The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems.

The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as ''atomic formulas'', ''placeholders'', ''proposition letters'', or ''variables'', and (2) a set of operator symbols, variously interpreted as ''logical operators'' or ''logical connectives''. A ''well-formed formula'' (''wff'') is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols.

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available.

==Abstraction and application==

Although it is possible to construct an abstract formal calculus that has no immediate practical use and next to nothing in the way of obvious applications, the very name ''calculus'' indicates that this species of formal system owes its origin to the utility of its prototypical members in practical calculation. Generally speaking, any mathematical calculus is designed with the intention of representing a given domain of formal objects, and typically with the aim of facilitating the computations and inferences that need to be carried out in this representation. Thus some idea of the intended denotation, the formal objects that the formulas of the calculus are intended to denote, is given in advance of developing the calculus itself.

Viewed over the course of its historical development, a formal calculus for any given subject matter normally arises through a process of gradual abstraction, stepwise refinement, and trial-and-error synthesis from the array of informal notational systems that inform prior use, each of which covers the object domain only in part or from a particular angle.

==Generic description of a propositional calculus==

A '''propositional calculus''' is a formal system <math>\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, whose formulas are constructed in the following manner:

* The ''alpha set'' <math>\Alpha\!</math> is a finite set of elements called ''proposition symbols'' or ''propositional variables''. Syntactically speaking, these are the most basic elements of the formal language <math>\mathcal{L},</math> otherwise referred to as ''atomic formulas'' or ''terminal elements''. In the examples to follow, the elements of <math>\Alpha\!</math> are typically the letters ''p'', ''q'', ''r'', and so on.

* The ''omega set'' <math>\Omega\!</math> is a finite set of elements called ''operator symbols'' or ''logical connectives''. The set <math>\Omega\!</math> is partitioned into disjoint subsets as follows:

::: <math>\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m.\!</math>

: In this partition, <math>\Omega_j\!</math> is the set of operator symbols of ''arity'' <math>j.\!</math>

: In the more familiar propositional calculi, <math>\Omega\!</math> is typically partitioned as follows:

::: <math>\Omega_1 = \{ \lnot \},\!</math>

::: <math>\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \}.\!</math>

: A frequently adopted option treats the constant logical values as operators of arity zero, thus:

::: <math>\Omega_0 = \{ 0, 1 \}.\!</math>

: Some writers use the tilde (~) instead of (¬) and some use the ampersand (&) instead of (∧). Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, {0, 1}, and {<math>\bot</math>, <math>\top</math>} all being seen in various contexts.

* Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas.

The ''language'' of <math>\mathcal{L}</math>, also known as its set of ''formulas'', ''well-formed formulas'' or ''wffs'', is inductively or recursively defined by the following rules:

# Base. Any element of the alpha set <math>\Alpha\!</math> is a formula of <math>\mathcal{L}</math>.
# Step (a). If ''p'' is a formula, then ¬''p'' is a formula.
# Step (b). If ''p'' and ''q'' are formulas, then (''p'' ∧ ''q''), (''p'' ∨ ''q''), (''p'' → ''q''), and (''p'' ↔ ''q'') are formulas.
# Close. Nothing else is a formula of <math>\mathcal{L}</math>.

Repeated applications of these rules permits the construction of complex formulas. For example:

# By rule 1, ''p'' is a formula.
# By rule 2, ¬''p'' is a formula.
# By rule 1, ''q'' is a formula.
# By rule 3, (¬''p'' ∨ ''q'') is a formula.

* The ''zeta set'' <math>\Zeta\!</math> is a finite set of ''transformation rules'' that are called ''inference rules'' when they acquire logical applications.

* The ''iota set'' <math>\Iota\!</math> is a finite set of ''initial points'' that are called ''axioms'' when they receive logical interpretations.

==Example 1. Simple axiom system==

Let <math>\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows:

* The alpha set <math>\Alpha \!</math> is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

::: <math>\Alpha = \{p, q, r, s, t, u \} \,.</math>

Of the three connectives for conjunction, disjunction, and implication (∧, ∨, and →), one can be taken as primitive and the other two can be defined in terms of it and negation (¬). Indeed, all of the logical connectives can be defined in terms of a [[sole sufficient operator]]. The biconditional (↔) can of course be defined in terms of conjunction and implication, with a ↔ b defined as (a → b) ∧ (b → a).

Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set <math>\Omega = \Omega_1 \cup \Omega_2</math> partition as follows:

::: <math>\Omega_1 = \{ \lnot \} \,,</math>

::: <math>\Omega_2 = \{ \Rightarrow \} \,.</math>

An axiom system discovered by [[Jan Łukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows:

::* <math>p \Rightarrow (q \Rightarrow p)</math>

::* <math>(p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))</math>

::* <math>(\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)</math>

The inference rule is ''[[modus ponens]]'':

::* From ''p'', (''p'' ⇒ ''q''), infer ''q''.

Then we have the following definitions:

::* ''p'' &or; ''q'' is defined as ¬''p'' ⇒ ''q''.

::* ''p'' &and; ''q'' is defined as ¬(''p'' ⇒ ¬''q'').

==Example 2. Natural deduction system==

Let <math>\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>, where <math>\Alpha,\ \Omega,\ \Zeta,\ \Iota</math> are defined as follows:

* The alpha set <math>\Alpha \!</math> is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

::: <math>\Alpha = \{p, q, r, s, t, u \} \,.</math>

* The omega set <math>\Omega = \Omega_1 \cup \Omega_2</math> partitions as follows:

::: <math>\Omega_1 = \{ \lnot \} \,,</math>

::: <math>\Omega_2 = \{ \land, \lor, \rightarrow, \leftrightarrow \} \,.</math>
In the following example of a propositional calculus,
the transformation rules are intended to be interpreted as the inference rules of a so-called ''[[natural deduction system]]''. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its [[theorem]]s from an empty axiom set.

* The set of initial points is empty, that is, <math>\Iota = \varnothing \,.</math>

* The set of transformation rules, <math>\Zeta ,\!</math>, is described as follows:

==Graphical calculi==

: ''Main article'' : [[Logical graph]]

It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic.

For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Indeed, many species of graphs arise as ''parse graphs'' in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are wffs or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called ''parsing'' and the inverse mapping from parse graphs to strings is achieved by an operation that is called ''traversing'' the graph.

==References==

* Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

* Chang, C.C., and Keisler, H.J. (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands.

* Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.

* Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.

* Lambek, J., and Scott, P.J. (1986), ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK.

* Mendelson, Elliot (1964), ''Introduction to Mathematical Logic'', D. Van Nostrand Company.

==Resources==

* Klement, Kevin C. (2006), “Propositional Logic”, in James Fieser and Bradley Dowden (eds.), ''Internet Encyclopedia of Philosophy''. [http://www.iep.utm.edu/p/prop-log.htm Online].

* Magnus, P.D. ''[http://www.fecundity.com/logic/ Forall x : An Introduction to Formal Logic]''.

* [http://www.visualstatistics.net/Scaling/Propositional%20Calculus/Elements%20of%20Propositional%20Calculus.htm Elements of Propositional Calculus].

* [http://www.ltn.lv/~podnieks/mlog/ml2.htm Introduction to Mathematical Logic].

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus @ InterSciWiki]
* [http://mywikibiz.com/Propositional_calculus Propositional Calculus @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Propositional_calculus Propositional Calculus @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_calculus Propositional Calculus], [http://intersci.ss.uci.edu/wiki/index.php/Main_Page InterSciWiki]
* [http://mywikibiz.com/Propositional_calculus Propositional Calculus], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/PropositionalCalculus Propositional Calculus], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Propositional_calculus Propositional Calculus], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Propositional_calculus Propositional Calculus], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Propositional_calculus&oldid=77110794 Propositional Calculus], [http://en.wikipedia.org/ Wikipedia]

[[Category:Charles Sanders Peirce]]
[[Category:Computer Science]]
[[Category:Formal Grammars]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Model Theory]]
[[Category:Normative Sciences]]
[[Category:Pragmatics]]
[[Category:Proof Theory]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Syntax]]

Peirce's law

2015-11-07T15:34:53Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form:

{| align="center" cellpadding="10"
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math>
|}

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

==History==

Here is Peirce's own statement and proof of the law:

{| align="center" cellpadding="8" width="90%"
|
A ''fifth icon'' is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

<center>
<math>\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.</math>
</center>

This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent <math>x\!</math> being false while its antecedent <math>(x \,-\!\!\!< y) \,-\!\!\!< x</math> is true. If this is true, either its consequent, <math>x,\!</math> is true, when the whole formula would be true, or its antecedent <math>x \,-\!\!\!< y</math> is false. But in the last case the antecedent of <math>x \,-\!\!\!< y,</math> that is <math>x,\!</math> must be true. (Peirce, CP 3.384).
|}

Peirce goes on to point out an immediate application of the law:

{| align="center" cellpadding="8" width="90%"
|
From the formula just given, we at once get:

<center>
<math>\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,</math>
</center>

where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math> (Peirce, CP 3.384).
|}

'''Note.''' Peirce uses the ''sign of illation'' “<math>-\!\!\!<</math>” for implication. In one place he explains “<math>-\!\!\!<</math>” as a variant of the sign “<math>\le</math>” for ''less than or equal to''; in another place he suggests that <math>A \,-\!\!\!< B</math> is an iconic way of representing a state of affairs where <math>A,\!</math> in every way that it can be, is <math>B.\!</math>

==Graphical proof==

Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation.

{| align="center" border="0" cellpadding="10" cellspacing="0"
| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1)
|}

'''Proof.''' Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law 1.0 Marquee Title.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference Band Collect p.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]]
|-
| [[Image:Equational Inference Band Quit ((q)).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]]
|-
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]]
|-
| [[Image:Equational Inference Band Delete p.png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]]
|-
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (2)
|}

The following animation replays the steps of the proof.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law 2.0 Animation.gif]]
|}
| (3)
|}

==Equational form==

A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible:

{| align="center" cellpadding="10"
| <math>((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p</math>
|}

===Proof 1===

Given what precedes, it remains to show that:

{| align="center" cellpadding="10"
| <math>p \Rightarrow ((p \Rightarrow q) \Rightarrow p)</math>
|}

But this is immediate, since <math>p \Rightarrow (r \Rightarrow p)</math> for any proposition <math>r.\!</math>

===Proof 2===

Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation:

{| align="center" border="0" cellpadding="10" cellspacing="0"
| [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (4)
|}

Using the axioms and theorems listed in the article on [[logical graphs]], the equational form of Peirce's law may be proved in the following manner:

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference Rule Collect p.png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]]
|-
| [[Image:Equational Inference Rule Quit ((q)).png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]]
|-
| [[Image:Equational Inference Rule Cancel (( )).png|500px]]
|-
| [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (5)
|}

The following animation replays the steps of the proof.

{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Peirce's Law Strong Form 2.0 Animation.gif]]
|}
| (6)
|}

==Bibliography==

* [[Charles Sanders Peirce|Peirce, Charles Sanders]] (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted (CP 3.359–403), (CE 5, 162–190).

* Peirce, Charles Sanders (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).

* Peirce, Charles Sanders (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki]
* [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/PeircesLaw Peirce's Law], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Peirce's_law Peirce's Law], [http://www.wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Peirce%27s_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Artificial Intelligence]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Cybernetics]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Systems]]
[[Category:Graph Theory]]
[[Category:History of Logic]]
[[Category:History of Mathematics]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Logical Graphs]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Semiotics]]
[[Category:Visualization]]

Parametric operator

2015-11-07T04:18:37Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''parametric operator''' <math>\Omega\!</math> with '''parameter''' <math>\alpha\!</math> in the '''parameter set''' <math>\Alpha\!</math> is an indexed family of operators <math>(\Omega_\alpha)_\Alpha = \{ \Omega_\alpha : \alpha \in \Alpha \}\!</math> with index <math>\alpha\!</math> in the index set <math>\Alpha.\!</math>

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Parametric_operator Parametric Operator @ InterSciWiki]
* [http://mywikibiz.com/Parametric_operator Parametric Operator @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Parametric_operator Parametric Operator @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Parametric_operator Parametric Operator @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Parametric_operator Parametric Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Parametric_operator Parametric Operator], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ParametricOperator Parametric Operator], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Parametric_operator Parametric Operator], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Parametric_operator Parametric Operator], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Parametric_operator Parametric Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Parametric_operator&oldid=40451935 Parametric Operator], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Automata Theory]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Neural Networks]]
[[Category:Philosophy]]
[[Category:Semiotics]]

Multigrade operator

2015-11-07T04:12:40Z

Jon Awbrey: /* Document history */

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''multigrade operator''' <math>\Omega\!</math> is a ''[[parametric operator]]'' with ''parameter'' <math>k\!</math> in the set <math>\mathbb{N}\!</math> of non-negative integers.

The application of a multigrade operator <math>\Omega\!</math> to a finite sequence of operands <math>(x_1, \ldots, x_k)\!</math> is typically denoted with the parameter <math>k\!</math> left tacit, as the appropriate application is implicit in the number of operands listed. Thus <math>\Omega (x_1, \ldots, x_k)\!</math> may be taken for <math>\Omega_k (x_1, \ldots, x_k).\!</math>

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator @ InterSciWiki]
* [http://mywikibiz.com/Multigrade_operator Multigrade Operator @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Multigrade_operator Multigrade Operator @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Multigrade_operator Multigrade Operator], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/MultigradeOperator Multigrade Operator], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Multigrade_operator Multigrade Operator], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Multigrade_operator Multigrade Operator], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Automata Theory]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Neural Networks]]
[[Category:Philosophy]]
[[Category:Semiotics]]

Multigrade operator

2015-11-07T03:32:40Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''multigrade operator''' <math>\Omega\!</math> is a ''[[parametric operator]]'' with ''parameter'' <math>k\!</math> in the set <math>\mathbb{N}\!</math> of non-negative integers.

The application of a multigrade operator <math>\Omega\!</math> to a finite sequence of operands <math>(x_1, \ldots, x_k)\!</math> is typically denoted with the parameter <math>k\!</math> left tacit, as the appropriate application is implicit in the number of operands listed. Thus <math>\Omega (x_1, \ldots, x_k)\!</math> may be taken for <math>\Omega_k (x_1, \ldots, x_k).\!</math>

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator @ InterSciWiki]
* [http://mywikibiz.com/Multigrade_operator Multigrade Operator @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Multigrade_operator Multigrade Operator @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Multigrade_operator Multigrade Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Multigrade_operator Multigrade Operator], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/MultigradeOperator Multigrade Operator], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Multigrade_operator Multigrade Operator], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Multigrade_operator Multigrade Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Multigrade_operator&oldid=40451309 Multigrade Operator], [http://en.wikipedia.org/ Wikipedia]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Automata Theory]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Neural Networks]]
[[Category:Philosophy]]
[[Category:Semiotics]]

Relation composition

2015-11-07T02:56:46Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

'''Relation composition''', or the composition of [[relation (mathematics)|relations]], is the generalization of function composition, or the composition of functions.  The following treatment of relation composition takes the “strongly typed” approach to relations that is outlined in the article on [[relation theory]].

==Preliminaries==

There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general.

The first order of business is to define the operation on [[relation (mathematics)|relations]] that is variously known as the ''composition of relations'', ''relational composition'', or ''relative multiplication''. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations.

As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation.

{| align="center" cellpadding="4" width="90%"
|
The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.
|-
|
The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.
|}

These two factors together generate the following four styles of syntax:

{| align="center" cellpadding="4" width="90%"
| LALA = left application, left association.
|-
| LARA = left application, right association.
|-
| RALA = right application, left association.
|-
| RARA = right application, right association.
|}

==Definition==

A notion of relational composition is to be defined that generalizes the usual notion of functional composition:

{| align="center" cellpadding="4" width="90%"
|
Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math>

results in a ''composite function'' formulated as <math>fg : X \to Z.</math>
|-
|
Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math>

results in a ''composite function'' formulated as <math>gf : X \to Z.</math>
|}

Note on notation. The ordinary symbol for functional composition is the ''composition sign'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a ''center dot'' “<math>\cdot</math>”, as <math>f \cdot g.</math>

Generalizing the paradigm along parallel lines, the ''composition'' of a pair of dyadic relations is formulated in the following two ways:

{| align="center" cellpadding="4" width="90%"
|
Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math>

results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math>
|-
|
Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math>

results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math>
|}

==Geometric construction==

There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations.

This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called ''tacit extensions''.

The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely:

:* The use of [[logical conjunction]], as denoted by the symbol <math>\land,\!</math> in expressions of the form <math>F(x, y, z) = G(x, y) \land H(y, z),\!</math> to define a triadic relation <math>F\!</math> in terms of a pair of dyadic relations <math>G\!</math> and <math>H.\!</math>

:* The concepts of dyadic ''projection'' and ''projective determination'', that are invoked in the “weak” notion of ''projective reducibility''.

The relational composition <math>G \circ H\!</math> of a pair of dyadic relations <math>G\!</math> and <math>H\!</math> will be constructed in three stages, first, by taking the tacit extensions of <math>G\!</math> and <math>H\!</math> to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the ''prima facie'' dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition <math>G \circ H\!</math> of the relations <math>G\!</math> and <math>H.\!</math>

The construction of a relational composition in a specifically mathematical setting normally begins with [[relation (mathematics)|mathematical relations]] at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only ''up to isomorphism'' as the conventional saying goes, that is, any objects that have the “same form” are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, <math>G, H \subseteq X \times Y,\!</math> as shown in Figure 1.

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o o |
| |\ |\ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | * \ | * \ |
| X * Y X * Y |
| \ * | \ * | |
| \ G | \ H | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \| \| |
| o o |
| |
o-------------------------------------------------o
Figure 1. Dyadic Relations G, H c X x Y
</pre>
|}

The dyadic relations <math>G\!</math> and <math>H\!</math> cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen:

:* The first type of case occurs when <math>X = Y.\!</math> In this case, both of the compositions <math>G \circ H\!</math> and <math>H \circ G\!</math> are defined.

:* The second type of case occurs when <math>X\!</math> and <math>Y\!</math> are distinct, but when it nevertheless makes sense to speak of a dyadic relation <math>\hat{H}\!</math> that is isomorphic to <math>H,\!</math> but living in the plane <math>YZ,\!</math> that is, in the space of the cartesian product <math>Y \times Z,\!</math> for some set <math>Z.\!</math>

Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o o |
| |\ /| |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | * \ / * | |
| X * Y Y * Z |
| \ * | | * / |
| \ G | | Ĥ / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 2. Dyadic Relations G c X x Y and Ĥ c Y x Z
</pre>
|}

With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition <math>P \circ Q\!</math> of a pair of dyadic relations <math>P, Q
\subseteq X \times X.\!</math>

: '''Definition.''' <math>P \circ Q = \mathrm{proj}_{13} (P \times X ~\cap~ X \times Q).\!</math>

To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form <math>L \subseteq X \times X.\!</math> So, if one has started out with a dyadic relation of the shape <math>L \subseteq U \times V,\!</math> one merely lets <math>X = U \cup V,\!</math> trading in the initial <math>L\!</math> for a new <math>L \subseteq X \times X\!</math> as need be.

The projection <math>\mathrm{proj}_{13}\!</math> is just the projection of the cartesian cube <math>X \times X \times X\!</math> on the space of shape <math>X \times X\!</math> that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places.

Finally, the notation of the cartesian product sign “<math>\times\!</math>” is extended to signify two other products with respect to a dyadic relation <math>L \subseteq X \times X\!</math> and a subset <math>W \subseteq X,\!</math> as follows:

: '''Definition.''' <math>L \times W ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in L ~\mathrm{and}~ z \in W \}.\!</math>

: '''Definition.''' <math>W \times L ~=~ \{ (x, y, z) \in X^3 ~:~ x \in W ~\mathrm{and}~ (y, z) \in L \}.\!</math>

Applying these definitions to the case <math>P, Q \subseteq X \times X,\!</math> the two dyadic relations whose relational composition <math>P \circ Q \subseteq X \times X\!</math> is about to be defined, one finds:

: <math>P \times X ~=~ \{ (x, y, z) \in X^3 ~:~ (x, y) \in P ~\mathrm{and}~ z \in X \},\!</math>

: <math>X \times Q ~=~ \{ (x, y, z) \in X^3 ~:~ x \in X ~\mathrm{and}~ (y, z) \in Q \}.\!</math>

These are just the appropriate special cases of the tacit extensions already defined.

: <math>P \times X ~=~ \mathrm{te}_{12}^3 (P),~\!</math>

: <math>X \times Q ~=~ \mathrm{te}_{23}^1 (Q).~\!</math>

In summary, then, the expression:

: <math>\mathrm{proj}_{13} (P \times X ~\cap~ X \times Q)\!</math>

is equivalent to the expression:

: <math>\mathrm{proj}_{13} (\mathrm{te}_{12}^3 (P) ~\cap~ \mathrm{te}_{23}^1 (Q))\!</math>

and this form is generalized — although, relative to one's school of thought, perhaps inessentially so — by the form that was given above as follows:

: '''Definition.''' <math>P \circ Q ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (P) ~\cap~ \mathrm{te}_{YZ}^X (Q)).\!</math>

Figure 3 presents a geometric picture of what is involved in formulating a definition of the triadic relation <math>F \subseteq X \times Y \times Z\!</math> by way of a conjunction between the dyadic relation <math>G \subseteq X \times Y\!</math> and the dyadic relation <math>H \subseteq Y \times Z,\!</math> as done for example by means of an expression of the following form:

:* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 3. Projections of F onto G and H
</pre>
|}

To interpret the Figure, visualize the triadic relation <math>F \subseteq X \times Y \times Z\!</math> as a body in <math>XYZ\!</math>-space, while <math>G\!</math> is a figure in <math>XY\!</math>-space and <math>H\!</math> is a figure in <math>YZ\!</math>-space.

The dyadic '''projections''' that accompany a triadic relation over <math>X, Y, Z\!</math> are defined as follows:

:* <math>\mathrm{proj}_{XY} (L) ~=~ \{ (x, y) \in X \times Y : (x, y, z) \in L ~\text{for some}~ z \in Z) \},\!</math>

:* <math>\mathrm{proj}_{XZ} (L) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in L ~\text{for some}~ y \in Y) \},\!</math>

:* <math>\mathrm{proj}_{YZ} (L) ~=~ \{ (y, z) \in Y \times Z : (x, y, z) \in L ~\text{for some}~ x \in X) \}.\!</math>

For many purposes it suffices to indicate the dyadic projections of a triadic relation <math>L\!</math> by means of the briefer equivalents listed next:

:* <math>L_{XY} ~=~ \mathrm{proj}_{XY}(L),\!</math>

:* <math>L_{XZ} ~=~ \mathrm{proj}_{XZ}(L),\!</math>

:* <math>L_{YZ} ~=~ \mathrm{proj}_{YZ}(L).\!</math>

In light of these definitions, <math>\mathrm{proj}_{XY}\!</math> is a mapping from the set <math>\mathcal{L}_{XYZ}\!</math> of triadic relations over the domains <math>X, Y, Z\!</math> to the set <math>\mathcal{L}_{XY}\!</math> of dyadic relations over the domains <math>X, Y,\!</math> with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions.

The set <math>\mathcal{L}_{XYZ},~\!</math> whose members are just the triadic relations over <math>X, Y, Z,\!</math> can be recognized as the set of all subsets of the cartesian product <math>X \times Y \times Z,\!</math> also known as the ''power set'' of <math>X \times Y \times Z,\!</math> and notated here as <math>\mathrm{Pow} (X \times Y \times Z).\!</math>

:* <math>\mathcal{L}_{XYZ} ~=~ \{ L : L \subseteq X \times Y \times Z \} ~=~ \mathrm{Pow} (X \times Y \times Z).\!</math>

Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from <math>\{ X, Y, Z \}.\!</math>

:* <math>\mathcal{L}_{XY} ~=~ \{L : L \subseteq X \times Y \} ~=~ \mathrm{Pow} (X \times Y),~\!</math>

:* <math>\mathcal{L}_{XZ} ~=~ \{L : L \subseteq X \times Z \} ~=~ \mathrm{Pow} (X \times Z),~\!</math>

:* <math>\mathcal{L}_{YZ} ~=~ \{L : L \subseteq Y \times Z \} ~=~ \mathrm{Pow} (Y \times Z).~\!</math>

In mathematics, the inverse relation corresponding to a projection map is usually called an ''extension''. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term ''tacit extension''.

Given three sets, <math>X, Y, Z,\!</math> and three dyadic relations,

:* <math>U \subseteq X \times Y,~\!</math>

:* <math>V \subseteq X \times Z,~\!</math>

:* <math>W \subseteq Y \times Z,~\!</math>

the ''tacit extensions'', <math>\mathrm{te}_{XY}^Z, \mathrm{te}_{XZ}^Y, \mathrm{te}_{YZ}^X,~\!</math> of <math>U, V, W,\!</math> respectively, are defined as follows:

:* <math>\mathrm{te}_{XY}^Z (U) ~=~ \{ (x, y, z) : (x, y) \in U \},\!</math>

:* <math>\mathrm{te}_{XZ}^Y (V) ~=~ \{ (x, y, z) : (x, z) \in V \},\!</math>

:* <math>\mathrm{te}_{YZ}^X (W) ~=~ \{ (x, y, z) : (y, z) \in W \}.\!</math>

So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, <math>\mathrm{te}(U), \mathrm{te}(V), \mathrm{te}(W).\!</math>

The definition and illustration of relational composition presently under way makes use of the tacit extension of <math>G \subseteq X \times Y\!</math> to <math>\mathrm{te}(G) \subseteq X \times Y
\times Z\!</math> and the tacit extension of <math>H \subseteq Y \times Z\!</math> to <math>\mathrm{te}(H) \subseteq X \times Y \times Z,\!</math> only.

Geometric illustrations of <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H)\!</math> are afforded by Figures 4 and 5, respectively.

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | * \ |
| o o ** o |
| |\ / \*** /| |
| | \ / *** / | |
| | \ / ***\ / | |
| | \ *** / | |
| | / \*** / \ | |
| | / *** / \ | |
| |/ ***\ / \| |
| o X /** Y Z o |
| |\ \//* | / /| |
| | \ /// | / / | |
| | \ ///\ | / / | |
| | \ /// \ | / / | |
| | \/// \ | / / | |
| | /\/ \ | / / | |
| | *//\ \|/ / * | |
| X */ Y o Y * Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 4. Tacit Extension of G to X x Y x Z
</pre>
|}

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / * | \ |
| o ** o o |
| |\ ***/ \ /| |
| | \ *** \ / | |
| | \ /*** \ / | |
| | \ *** / | |
| | / \ ***/ \ | |
| | / \ *** \ | |
| |/ \ /*** \| |
| o X Y **\ Z o |
| |\ \ | *\\/ /| |
| | \ \ | \\\ / | |
| | \ \ | /\\\ / | |
| | \ \ | / \\\ / | |
| | \ \ | / \\\/ | |
| | \ \ | / \/\ | |
| | * \ \|/ /\\* | |
| X * Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 5. Tacit Extension of H to X x Y x Z
</pre>
|}

A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following:

:* <math>F(x, y, z) ~=~ G(x, y) \land H(y, z).\!</math>

The conjunction that is indicated by “<math>\land\!</math>” corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions <math>\mathrm{te}(G)\!</math> and <math>\mathrm{te}(H).\!</math>

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 6. F as the Intersection of te(G) and te(H)
</pre>
|}

==Algebraic construction==

The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ 7\/// | \\\/7 /| |
| | \ 6// | \\6 / | |
| | \ //5\ | /5\\ / | |
| | \ /// 4\ | /4 \\\ / | |
| | \/// 3\ | /3 \\\/ | |
| | G/\/ 2\ | /2 \/\H | |
| | *//\ 1\|/1 /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\@5 5@// |/5 |
| 4@ \@4 4@/ @4 |
| 3\ @3 3@ /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 7. F as the Intersection of te(G) and te(H)
</pre>
|}

Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair <math>(x, y)\!</math> is written <math>x\!:\!y,\!</math> the ordered triple <math>(x, y, z)\!</math> is written <math>x\!:\!y\!:\!z,\!</math> and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like <math>a\!:\!b ~+~ b\!:\!c ~+~ c\!:\!d\!</math> and so on.

For example, translating the relations <math>F \subseteq X \times Y \times Z, ~ G \subseteq X \times Y, ~ H \subseteq Y \times Z\!</math> into this notation produces the following summary of the data:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}

As often happens with abstract notations for functions and relations, the ''type information'', in this case, the fact that <math>G\!</math> and <math>H\!</math> live in different spaces, is left implicit in the context of use.

Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions:

{| align="center" cellpadding="8" width="90%"
|
<math>G \circ H ~=~ \mathrm{proj}_{XZ} (\mathrm{te}_{XY}^Z (G) ~\cap~ \mathrm{te}_{YZ}^X (H)).\!</math>
|}

Here is the big picture, with all the pieces in place:

{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / G o H \ |
| X * Z |
| 7\ /|\ /7 |
| 6\ / | \ /6 |
| 5\ / | \ /5 |
| 4@ | @4 |
| 3\ | /3 |
| 2\ | /2 |
| 1\|/1 |
| | |
| | |
| | |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o | o |
| |\ /|\ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | G/\/ \ | / \/\H | |
| | *//\ \|/ /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\@5 5@// |/5 |
| 4@ \@4 4@/ @4 |
| 3\ @3 3@ /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 8. G o H = proj_XZ (te(G) |^| te(H))
</pre>
|}

All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4
\\
G & = & 4:3 & + & 4:4 & + & 4:5
\\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4)
\\[6pt]
& = & 4:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(G) & = & \mathrm{te}_{XY}^Z (G)
\\[4pt]
& = & \displaystyle\sum_{z=1}^7 (4\!:\!3\!:\!z ~+~ 4\!:\!4\!:\!z ~+~ 4\!:\!5\!:\!z)
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(G)
& = & 4:3:1 & + & 4:4:1 & + & 4:5:1 & + \\
& & 4:3:2 & + & 4:4:2 & + & 4:5:2 & + \\
& & 4:3:3 & + & 4:4:3 & + & 4:5:3 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 4:3:5 & + & 4:4:5 & + & 4:5:5 & + \\
& & 4:3:6 & + & 4:4:6 & + & 4:5:6 & + \\
& & 4:3:7 & + & 4:4:7 & + & 4:5:7
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(H) & = & \mathrm{te}_{YZ}^X (H)
\\[4pt]
& = & \displaystyle\sum_{x=1}^7 (x\!:\!3\!:\!4 ~+~ x\!:\!4\!:\!4 ~+~ x\!:\!5\!:\!4)
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
\mathrm{te}(H)
& = & 1:3:4 & + & 1:4:4 & + & 1:5:4 & + \\
& & 2:3:4 & + & 2:4:4 & + & 2:5:4 & + \\
& & 3:3:4 & + & 3:4:4 & + & 3:5:4 & + \\
& & 4:3:4 & + & 4:4:4 & + & 4:5:4 & + \\
& & 5:3:4 & + & 5:4:4 & + & 5:5:4 & + \\
& & 6:3:4 & + & 6:4:4 & + & 6:5:4 & + \\
& & 7:3:4 & + & 7:4:4 & + & 7:5:4
\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccl}
\mathrm{te}(G) \cap \mathrm{te}(H)
& = & 4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4
\\[4pt]
G \circ H
& = & \mathrm{proj}_{XZ} (\mathrm{te}(G) \cap \mathrm{te}(H))
\\[4pt]
& = & \mathrm{proj}_{XZ} (4\!:\!3:\!4 ~+~ 4\!:\!4\!:\!4 ~+~ 4\!:\!5\!:\!4)
\\[4pt]
& = & 4:4
\end{array}</math>
|}

==Matrix representation==

We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as ''[[logical matrix|logical matrices]]'', and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in linear algebra.

First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H.\!</math>

Here is the setup that we had before:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}X & = & \{ 1, 2, 3, 4, 5, 6, 7 \}\end{matrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G & = & 4:3 & + & 4:4 & + & 4:5 & \subseteq & X \times X
\\
H & = & 3:4 & + & 4:4 & + & 5:4 & \subseteq & X \times X
\end{matrix}</math>
|}

Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z,\!</math> the composition of <math>P ~\text{on}~ Q\!</math> is written as <math>P \circ Q,\!</math> or more simply as <math>PQ,\!</math> and obtained as follows:

To compute <math>PQ,\!</math> in general, where <math>P\!</math> and <math>Q\!</math> are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes <math>a:b\!</math> and <math>c:d.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
(a:b)(c:d) & = & (a:d) & \text{if}~ b = c
\\
(a:b)(c:d) & = & 0 & \text{otherwise}
\end{matrix}</math>
|}

To find the relational composition <math>G \circ H,\!</math> one may begin by writing it as a quasi-algebraic product:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H & = & (4\!:\!3 ~+~ 4\!:\!4 ~+~ 4\!:\!5)(3\!:\!4 ~+~ 4\!:\!4 ~+~ 5\!:\!4)
\end{matrix}</math>
|}

Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H
& = & (4:3)(3:4) & + & (4:3)(4:4) & + & (4:3)(5:4) & +
\\
& & (4:4)(3:4) & + & (4:4)(4:4) & + & (4:4)(5:4) & +
\\
& & (4:5)(3:4) & + & (4:5)(4:4) & + & (4:5)(5:4)
\end{matrix}</math>
|}

Applying the rule that determines the product of elementary relations produces the following array:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G \circ H
& = & 4:4 & + & 0 & + & 0 & +
\\
& & 0 & + & 4:4 & + & 0 & +
\\
& & 0 & + & 0 & + & 4:4
\end{matrix}</math>
|}

Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result:

{| align="center" cellpadding="8" width="90%"
| <math>\begin{matrix}G \circ H & = & 4:4\end{matrix}</math>
|}

With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations <math>G\!</math> and <math>H\!</math> together to obtain their relational composite <math>G \circ H.\!</math>

Given the space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> whose cardinality <math>|X|\!</math> is <math>7,\!</math> there are <math>|X \times X| = |X| \cdot |X|\!</math> <math>=\!</math> <math>7 \cdot 7 = 49\!</math> elementary relations of the form <math>i:j,\!</math> where <math>i\!</math> and <math>j\!</math> range over the space <math>X.\!</math> Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
1\!:\!1 & 1\!:\!2 & 1\!:\!3 & 1\!:\!4 & 1\!:\!5 & 1\!:\!6 & 1\!:\!7
\\
2\!:\!1 & 2\!:\!2 & 2\!:\!3 & 2\!:\!4 & 2\!:\!5 & 2\!:\!6 & 2\!:\!7
\\
3\!:\!1 & 3\!:\!2 & 3\!:\!3 & 3\!:\!4 & 3\!:\!5 & 3\!:\!6 & 3\!:\!7
\\
4\!:\!1 & 4\!:\!2 & 4\!:\!3 & 4\!:\!4 & 4\!:\!5 & 4\!:\!6 & 4\!:\!7
\\
5\!:\!1 & 5\!:\!2 & 5\!:\!3 & 5\!:\!4 & 5\!:\!5 & 5\!:\!6 & 5\!:\!7
\\
6\!:\!1 & 6\!:\!2 & 6\!:\!3 & 6\!:\!4 & 6\!:\!5 & 6\!:\!6 & 6\!:\!7
\\
7\!:\!1 & 7\!:\!2 & 7\!:\!3 & 7\!:\!4 & 7\!:\!5 & 7\!:\!6 & 7\!:\!7
\end{matrix}</math>
|}

The relations <math>G\!</math> and <math>H\!</math> may then be regarded as logical sums of the following forms:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G & = & \displaystyle\sum_{ij} G_{ij} (i\!:\!j)
\\[20pt]
H & = & \displaystyle\sum_{ij} H_{ij} (i\!:\!j)
\end{matrix}\!</math>
|}

The notation <math>\textstyle\sum_{ij}\!</math> indicates a logical sum over the collection of elementary relations <math>i\!:\!j\!</math> while the factors <math>G_{ij}\!</math> and <math>H_{ij}\!</math> are values in the ''[[boolean domain]]'' <math>\mathbb{B} = \{ 0, 1 \}~\!</math> that are called the ''coefficients'' of the relations <math>G\!</math> and <math>H,\!</math> respectively, with regard to the corresponding elementary relations <math>i\!:\!j.\!</math>

In general, for a dyadic relation <math>L,\!</math> the coefficient <math>L_{ij}\!</math> of the elementary relation <math>i\!:\!j\!</math> in the relation <math>L\!</math> will be <math>0\!</math> or <math>1,\!</math> respectively, as <math>i\!:\!j\!</math> is excluded from or included in <math>L.\!</math>

With these conventions in place, the expansions of <math>G\!</math> and <math>H\!</math> may be written out as follows:

{| align="center" cellpadding="4" width="90%"
|
<math>\begin{matrix}G & = & 4:3 & + & 4:4 & + & 4:5 & =\end{matrix}</math>
|-
|
<math>\begin{smallmatrix}
0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & +
\\
0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & +
\\
0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & 0 \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & +
\\
0 \cdot (4:1) & + & 0 \cdot (4:2) & + & \mathbf{1} \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & \mathbf{1} \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & +
\\
0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & 0 \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & +
\\
0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & +
\\
0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7)
\end{smallmatrix}</math>
|}

{| align="center" cellpadding="4" width="90%"
|
<math>\begin{matrix}H & = & 3:4 & + & 4:4 & + & 5:4 & =\end{matrix}</math>
|-
|
<math>\begin{smallmatrix}
0 \cdot (1:1) & + & 0 \cdot (1:2) & + & 0 \cdot (1:3) & + & 0 \cdot (1:4) & + & 0 \cdot (1:5) & + & 0 \cdot (1:6) & + & 0 \cdot (1:7) & +
\\
0 \cdot (2:1) & + & 0 \cdot (2:2) & + & 0 \cdot (2:3) & + & 0 \cdot (2:4) & + & 0 \cdot (2:5) & + & 0 \cdot (2:6) & + & 0 \cdot (2:7) & +
\\
0 \cdot (3:1) & + & 0 \cdot (3:2) & + & 0 \cdot (3:3) & + & \mathbf{1} \cdot (3:4) & + & 0 \cdot (3:5) & + & 0 \cdot (3:6) & + & 0 \cdot (3:7) & +
\\
0 \cdot (4:1) & + & 0 \cdot (4:2) & + & 0 \cdot (4:3) & + & \mathbf{1} \cdot (4:4) & + & 0 \cdot (4:5) & + & 0 \cdot (4:6) & + & 0 \cdot (4:7) & +
\\
0 \cdot (5:1) & + & 0 \cdot (5:2) & + & 0 \cdot (5:3) & + & \mathbf{1} \cdot (5:4) & + & 0 \cdot (5:5) & + & 0 \cdot (5:6) & + & 0 \cdot (5:7) & +
\\
0 \cdot (6:1) & + & 0 \cdot (6:2) & + & 0 \cdot (6:3) & + & 0 \cdot (6:4) & + & 0 \cdot (6:5) & + & 0 \cdot (6:6) & + & 0 \cdot (6:7) & +
\\
0 \cdot (7:1) & + & 0 \cdot (7:2) & + & 0 \cdot (7:3) & + & 0 \cdot (7:4) & + & 0 \cdot (7:5) & + & 0 \cdot (7:6) & + & 0 \cdot (7:7)
\end{smallmatrix}</math>
|}

Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations <math>G\!</math> and <math>H.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>G ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 1 & 1 & 1 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

These are the logical matrix representations of the dyadic relations <math>G\!</math> and <math>H.\!</math>

If the dyadic relations <math>G\!</math> and <math>H\!</math> are viewed as logical sums then their relational composition <math>G \circ H\!</math> can be regarded as a product of sums, a fact that can be indicated as follows:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ (\sum_{ij} G_{ij} (i\!:\!j))(\sum_{ij} H_{ij} (i\!:\!j)).\!</math>
|}

The composite relation <math>G \circ H\!</math> is itself a dyadic relation over the same space <math>X,\!</math> in other words, <math>G \circ H \subseteq X \times X,\!</math> and this means that <math>G \circ H\!</math> must be amenable to being written as a logical sum of the following form:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ \sum_{ij} (G \circ H)_{ij} (i\!:\!j).\!</math>
|}

In this formula, <math>(G \circ H)_{ij}\!</math> is the coefficient of <math>G \circ H\!</math> with respect to the elementary relation <math>i\!:\!j.\!</math>

One of the best ways to reason out what <math>G \circ H\!</math> should be is to ask oneself what its coefficient <math>(G \circ H)_{ij}\!</math> should be for each of the elementary relations <math>i\!:\!j\!</math> in turn.

So let us pose the question:

{| align="center" cellpadding="8" width="90%"
| <math>(G \circ H)_{ij} ~=~ ?\!</math>
|}

In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form:

{| align="center" cellpadding="8" width="90%"
| <math>G \circ H ~=~ (\sum_{ik} G_{ik} (i\!:\!k))(\sum_{kj} H_{kj} (k\!:\!j)).\!</math>
|}

A moment's thought will tell us that <math>(G \circ H)_{ij} = 1\!</math> if and only if there is an element <math>k\!</math> in <math>X\!</math> such that <math>G_{ik} = 1\!</math> and <math>H_{kj} = 1.\!</math>

Consequently, we have the result:

{| align="center" cellpadding="8" width="90%"
| <math>(G \circ H)_{ij} ~=~ \sum_{k} G_{ik} H_{kj}.\!</math>
|}

This follows from the properties of boolean arithmetic, specifically, from the fact that the product <math>G_{ik} H_{kj}\!</math> is <math>1\!</math> if and only if both <math>G_{ik}\!</math> and <math>H_{kj}\!</math> are <math>1\!</math> and from the fact that <math>\textstyle\sum_{k} F_{k}\!</math> is equal to <math>1\!</math> just in case some <math>F_{k}\!</math> is <math>1.\!</math>

All that remains in order to obtain a computational formula for the relational composite <math>G \circ H\!</math> of the dyadic relations <math>G\!</math> and <math>H\!</math> is to collect the coefficients <math>(G \circ H)_{ij}\!</math> as <math>i\!</math> and <math>j\!</math> range over <math>X.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}G \circ H
& = & \displaystyle \sum_{ij} (G \circ H)_{ij} (i\!:\!j)
& = & \displaystyle \sum_{ij} (\sum_{k} G_{ik} H_{kj}) (i\!:\!j).
\end{matrix}</math>
|}

This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction.

By way of disentangling this formula, one may notice that the form <math>\textstyle \sum_{k} G_{ik} H_{kj}\!</math> is what is usually called a ''scalar product''. In this case it is the scalar product of the <math>i^\text{th}\!</math> row of <math>G\!</math> with the <math>j^\text{th}\!</math> column of <math>H.\!</math>

To make this statement more concrete, let us go back to the examples of <math>G\!</math> and <math>H\!</math> we came in with:

{| align="center" cellpadding="8" width="90%"
|
<math>G ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 1 & 1 & 1 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

{| align="center" cellpadding="8" width="90%"
|
<math>H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

The formula for computing <math>G \circ H\!</math> says the following:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccl}
(G \circ H)_{ij}
& = & \text{the}~ {ij}^\text{th} ~\text{entry in the matrix representation for}~ G \circ H
\\[2pt]
& = & \text{the entry in the}~ {i}^\text{th} ~\text{row and the}~ {j}^\text{th} ~\text{column of}~ G \circ H
\\[2pt]
& = & \text{the scalar product of the}~ {i}^\text{th} ~\text{row of}~ G ~\text{with the}~ {j}^\text{th} ~\text{column of}~ H
\\[2pt]
& = & \sum_{k} G_{ik} H_{kj}
\end{array}</math>
|}

As it happens, it is possible to make exceedingly light work of this example, since there is only one row of <math>G\!</math> and one column of <math>H\!</math> that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of <math>G\!</math> with the fourth column of <math>H\!</math> produces the sole non-zero entry for the matrix of <math>G \circ H.\!</math>

{| align="center" cellpadding="8" width="90%"
|
<math>G \circ H ~=~ \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}</math>
|}

==Graph-theoretic picture==

There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of ''bipartite graphs'', or ''bigraphs'' for short.

Here is what <math>G\!</math> and <math>H\!</math> look like in the bigraph picture:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ |
| / | \ G |
| / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 9. G = 4:3 + 4:4 + 4:5
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| \ | / |
| \ | / H |
| \|/ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 10. H = 3:4 + 4:4 + 5:4
</pre>
|}

These graphs may be read to say:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
G ~\text{puts}~ 4 ~\text{in relation to}~ 3, 4, 5.
\\[2pt]
H ~\text{puts}~ 3, 4, 5 ~\text{in relation to}~ 4.
\end{matrix}</math>
|}

To form the composite relation <math>G \circ H,\!</math> one simply follows the bigraph for <math>G\!</math> by the bigraph for <math>H,\!</math> here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for <math>G \circ H.\!</math>

Here's how it looks in pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ |
| / | \ G |
| / | \ |
| o o o o o o o X |
| \ | / |
| \ | / H |
| \|/ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 11. G Followed By H
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | |
| | G o H |
| | |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 12. G Composed With H
</pre>
|}

Once again we find that <math>G \circ H = 4:4.\!</math>

We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind.

To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula.

Keeping to the same space <math>X = \{ 1, 2, 3, 4, 5, 6, 7 \},\!</math> define the dyadic relations <math>M, N \subseteq X \times X\!</math> as follows:

{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{*{19}{c}}
M & = &
2\!:\!1 & + & 2\!:\!2 & + & 2\!:\!3 & + & 4\!:\!3 & + & 4\!:\!4 & + & 4\!:\!5 & + & 6\!:\!5 & + & 6\!:\!6 & + & 6\!:\!7
\\[2pt]
N & = &
1\!:\!1 & + & 2\!:\!1 & + & 3\!:\!3 & + & 4\!:\!3 & ~ & + & ~ & 4\!:\!5 & + & 5\!:\!5 & + & 6\!:\!7 & + & 7\!:\!7
\end{array}</math>
|}

Here are the bigraph pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 13. Dyadic Relation M
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 14. Dyadic Relation N
</pre>
|}

To form the composite relation <math>M \circ N,\!</math> one simply follows the bigraph for <math>M\!</math> by the bigraph for <math>N,\!</math> arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for <math>M \circ N.\!</math>

Here's how it looks in pictures:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 15. M Followed By N
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| / \ / \ / \ |
| / \ / \ / \ M o N |
| / \ / \ / \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 16. M Composed With N
</pre>
|}

Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation.

The coefficient of the composition <math>M \circ N\!</math> between <math>i\!</math> and <math>j\!</math> in <math>X\!</math> is given as follows:

{| align="center" cellpadding="8" width="90%"
| <math>(M \circ N)_{ij} ~=~ \sum_{k} M_{ik} N_{kj}\!</math>
|}

Graphically interpreted, this is a ''sum over paths''. Starting at the node <math>i,\!</math> <math>M_{ik}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>M\!</math> from node <math>i\!</math> to node <math>k\!</math> and <math>N_{kj}\!</math> being <math>1\!</math> indicates that there is an edge in the bigraph of <math>N\!</math> from node <math>k\!</math> to node <math>j.\!</math> So the <math>\textstyle\sum_{k}\!</math> ranges over all possible intermediaries <math>k,\!</math> ascending from <math>0\!</math> to <math>1\!</math> just as soon as there happens to be a path of length two between nodes <math>i\!</math> and <math>j.\!</math>

It is instructive at this point to compute the other possible composition that can be formed from <math>M\!</math> and <math>N,\!</math> namely, the composition <math>N \circ M,\!</math> that takes <math>M\!</math> and <math>N\!</math> in the opposite order. Here is the graphic computation:

{| align="center" border="0" cellpadding="10"
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| | / | / \ | \ | |
| | / | / \ | \ | N |
| |/ |/ \| \| |
| o o o o o o o X |
| /|\ /|\ /|\ |
| / | \ / | \ / | \ M |
| / | \ / | \ / | \ |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 17. N Followed By M
</pre>
|-
|
<pre>
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o X |
| |
| N o M |
| |
| o o o o o o o X |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 18. N Composed With M
</pre>
|}

In sum, <math>N \circ M = 0.\!</math> This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a ''non-commutative'' algebraic operation.

==References==

* Ulam, S.M., and Bednarek, A.R., “On the Theory of Relational Structures and Schemata for Parallel Computation” (1977), pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', University of California Press, Berkeley, CA, 1990.

==Bibliography==

* Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.

* Mili, A., Desharnais, J., Mili, F., with Frappier, M., ''Computer Program Construction'', Oxford University Press, New York, NY, 1994.

* Ulam, S.M., ''Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990.

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition @ InterSciWiki]
* [http://mywikibiz.com/Relation_composition Relation Composition @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Relation_composition Relation Composition @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Relation_composition Relation Composition], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Relation_composition Relation Composition], [http://mywikibiz.com/ MyWikiBiz]
* [http://semanticweb.org/wiki/Relation_composition Relation Composition], [http://semanticweb.org/ Semantic Web]
* [http://ref.subwiki.org/wiki/Relation_composition Relation Composition], [http://ref.subwiki.org/ Subject Wikis]
* [http://en.wikiversity.org/wiki/Relation_composition Relation Composition], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Relation_composition Relation Composition], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://en.wikipedia.org/w/index.php?title=Relation_composition&oldid=43467878 Relation Composition], [http://en.wikipedia.org/ Wikipedia]

[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Database Theory]]
[[Category:Discrete Mathematics]]
[[Category:Formal Sciences]]
[[Category:Inquiry]]
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[[Category:Mathematics]]
[[Category:Relation Theory]]
[[Category:Set Theory]]

Minimal negation operator

2015-11-06T19:22:24Z

Jon Awbrey: update

☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says “just one false” of its logical arguments.

If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.\!</math>

If <math>p\!</math> is the only argument, then <math>\texttt{Mno}(p)\!</math> says that <math>p\!</math> is false, so <math>\texttt{Mno}(p)\!</math> expresses the logical negation of the proposition <math>p.\!</math> Wrtten in several different notations, <math>\texttt{Mno}(p) = \texttt{Not}(p) = \lnot p = \tilde{p} = p^\prime.\!</math>

If <math>p\!</math> and <math>q\!</math> are the only two arguments, then <math>\texttt{Mno}(p, q)\!</math> says that exactly one of <math>p, q\!</math> is false, so <math>\texttt{Mno}(p, q)\!</math> says the same thing as <math>p \neq q.\!</math> Expressing <math>\texttt{Mno}(p, q)\!</math> in terms of ands <math>(\cdot),\!</math> ors <math>(\lor),\!</math> and nots <math>(\tilde{~})\!</math> gives the following form.

{| align="center" cellpadding="8"
| <math>\texttt{Mno}(p, q) = \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}.\!</math>
|}

As usual, one drops the dots <math>(\cdot)~\!</math> in contexts where they are understood, giving the following form.

{| align="center" cellpadding="8"
| <math>\texttt{Mno}(p, q) = \tilde{p}q \lor p\tilde{q}.\!</math>
|}

The venn diagram for <math>\texttt{Mno}(p, q)\!</math> is shown in Figure 1.

{| align="center" cellpadding="8" style="text-align:center"
|
[[Image:Venn Diagram (P,Q).jpg|500px]]
<math>\text{Figure 1.}~~\texttt{Mno}(p, q)\!</math>
|}

The venn diagram for <math>\texttt{Mno}(p, q, r)\!</math> is shown in Figure 2.

{| align="center" cellpadding="8" style="text-align:center"
|
[[Image:Venn Diagram (P,Q,R).jpg|500px]]
<math>\text{Figure 2.}~~\texttt{Mno}(p, q, r)\!</math>
|}

The center cell is the region where all three arguments <math>p, q, r\!</math> hold true, so <math>\texttt{Mno}(p, q, r)\!</math> holds true in just the three neighboring cells. In other words, <math>\texttt{Mno}(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.\!</math>

==Initial definition==

The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}\!</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1\!</math> in just those cases where exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math>

In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}\!</math> = <math>\nu (x, y, z).\!</math>

The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.

{| align="center" cellpadding="8" style="text-align:center"
|
<math>\begin{matrix}
\texttt{()}
& = & \nu_0
& = & 0
& = & \mathrm{false}
\\[6pt]
\texttt{(x)}
& = & \nu_1 (x)
& = & \tilde{x}
& = & x^\prime
\\[6pt]
\texttt{(x, y)}
& = & \nu_2 (x, y)
& = & \tilde{x}y \lor x\tilde{y}
& = & x^\prime y \lor x y^\prime
\\[6pt]
\texttt{(x, y, z)}
& = & \nu_3 (x, y, z)
& = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z}
& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
\end{matrix}</math>
|}

==Formal definition==

To express the general case of <math>\nu_k\!</math> in terms of familiar operations, it helps to introduce an intermediary concept:

'''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}\!</math> be defined for each integer <math>j\!</math> in the interval <math>[1, k]\!</math> by the following equation:

{| align="center" cellpadding="8"
| <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.\!</math>
|}

Then <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}\!</math> is defined by the following equation:

{| align="center" cellpadding="8"
| <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).\!</math>
|}

If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k\!</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,\!</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}\!</math> indicates the set of points in <math>\mathbb{B}^k\!</math> that differ from <math>x\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}\!</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.

The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)\!</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted so that <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math> This has the following consequences:

{| align="center" cellpadding="4" width="90%"
| valign="top" | <big>•</big>
| The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math>
|-
| valign="top" | <big>•</big>
| The operation <math>\textstyle\sum_{j=1}^k x_j\!</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''.
|}

The following properties of the minimal negation operators <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}\!</math> may be noted:

{| align="center" cellpadding="4" width="90%"
| valign="top" | <big>•</big>
| The function <math>\texttt{(x, y)}\!</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.\!</math>
|-
| valign="top" | <big>•</big>
| In contrast, <math>\texttt{(x, y, z)}\!</math> is not identical to <math>x + y + z.\!</math>
|-
| valign="top" | <big>•</big>
| More generally, the function <math>\nu_k (x_1, \dots, x_k)\!</math> for <math>k > 2\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.\!</math>
|-
| valign="top" | <big>•</big>
| The inclusive disjunctions indicated for the <math>\nu_k\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint.
|}

==Truth tables==

Table 3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3\!</math> or the complements of those boundaries.

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}\!</math>
|- style="background:ghostwhite"
| <math>\mathcal{L}_1\!</math>
| <math>\mathcal{L}_2\!</math>
| <math>\mathcal{L}_3\!</math>
| <math>\mathcal{L}_4\!</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>p\colon\!</math>
| <math>1~1~1~1~0~0~0~0\!</math>
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>q\colon\!</math>
| <math>1~1~0~0~1~1~0~0\!</math>
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>r\colon\!</math>
| <math>1~0~1~0~1~0~1~0\!</math>
|  
|-
|
<math>\begin{matrix}
f_{104}
\\[4pt]
f_{148}
\\[4pt]
f_{146}
\\[4pt]
f_{97}
\\[4pt]
f_{134}
\\[4pt]
f_{73}
\\[4pt]
f_{41}
\\[4pt]
f_{22}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{01101000}
\\[4pt]
f_{10010100}
\\[4pt]
f_{10010010}
\\[4pt]
f_{01100001}
\\[4pt]
f_{10000110}
\\[4pt]
f_{01001001}
\\[4pt]
f_{00101001}
\\[4pt]
f_{00010110}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0~1~0~0~0
\\[4pt]
1~0~0~1~0~1~0~0
\\[4pt]
1~0~0~1~0~0~1~0
\\[4pt]
0~1~1~0~0~0~0~1
\\[4pt]
1~0~0~0~0~1~1~0
\\[4pt]
0~1~0~0~1~0~0~1
\\[4pt]
0~0~1~0~1~0~0~1
\\[4pt]
0~0~0~1~0~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(~p~,~q~,~r~)}
\\[4pt]
\texttt{(~p~,~q~,(r))}
\\[4pt]
\texttt{(~p~,(q),~r~)}
\\[4pt]
\texttt{(~p~,(q),(r))}
\\[4pt]
\texttt{((p),~q~,~r~)}
\\[4pt]
\texttt{((p),~q~,(r))}
\\[4pt]
\texttt{((p),(q),~r~)}
\\[4pt]
\texttt{((p),(q),(r))}
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{233}
\\[4pt]
f_{214}
\\[4pt]
f_{182}
\\[4pt]
f_{121}
\\[4pt]
f_{158}
\\[4pt]
f_{109}
\\[4pt]
f_{107}
\\[4pt]
f_{151}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{11101001}
\\[4pt]
f_{11010110}
\\[4pt]
f_{10110110}
\\[4pt]
f_{01111001}
\\[4pt]
f_{10011110}
\\[4pt]
f_{01101101}
\\[4pt]
f_{01101011}
\\[4pt]
f_{10010111}
\end{matrix}</math>
|
<math>\begin{matrix}
1~1~1~0~1~0~0~1
\\[4pt]
1~1~0~1~0~1~1~0
\\[4pt]
1~0~1~1~0~1~1~0
\\[4pt]
0~1~1~1~1~0~0~1
\\[4pt]
1~0~0~1~1~1~1~0
\\[4pt]
0~1~1~0~1~1~0~1
\\[4pt]
0~1~1~0~1~0~1~1
\\[4pt]
1~0~0~1~0~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(((p),(q),(r)))}
\\[4pt]
\texttt{(((p),(q),~r~))}
\\[4pt]
\texttt{(((p),~q~,(r)))}
\\[4pt]
\texttt{(((p),~q~,~r~))}
\\[4pt]
\texttt{((~p~,(q),(r)))}
\\[4pt]
\texttt{((~p~,(q),~r~))}
\\[4pt]
\texttt{((~p~,~q~,(r)))}
\\[4pt]
\texttt{((~p~,~q~,~r~))}
\end{matrix}</math>
|}

 

==Charts and graphs==

This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article.

Two ways of visualizing the space <math>\mathbb{B}^k\!</math> of <math>2^k\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture. The hypercube picture associates each point of <math>\mathbb{B}^k\!</math> with a unique point of the <math>k\!</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k\!</math> with a unique "cell" of the venn diagram on <math>k\!</math> "circles".

In addition, each point of <math>\mathbb{B}^k\!</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},\!</math> and thus it is the unique point where a ''singular conjunction'' of <math>k\!</math> ''literals'' is <math>1.\!</math>

For example, consider two cases at opposite vertices of the cube:

{| align="center" cellpadding="4" width="90%"
| valign="top" | <big>•</big>
| The point <math>(1, 1, \ldots , 1, 1)\!</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,\!</math> namely, the point where:
|-
|  
| align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.\!</math>
|-
| valign="top" | <big>•</big>
| The point <math>(0, 0, \ldots , 0, 0)\!</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,\!</math> namely, the point where:
|-
|  
| align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.\!</math>
|}

To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k\!</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)\!</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube.

For example, consider the case where <math>k = 3.\!</math> Then the minimal negation operation <math>\nu (p, q, r)\!</math> — written more simply as <math>\texttt{(p, q, r)}\!</math> — has the following venn diagram:

{| align="center" cellpadding="8" style="text-align:center"
|
[[Image:Venn Diagram (P,Q,R).jpg|500px]]
<math>\text{Figure 4.}~~\texttt{(p, q, r)}\!</math>
|}

For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}\!</math> has the following venn diagram:

{| align="center" cellpadding="8" style="text-align:center"
|
[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]
<math>\text{Figure 5.}~~\texttt{((p),(q),(r))}\!</math>
|}

==Glossary of basic terms==

; Boolean domain
: A ''[[boolean domain]]'' <math>\mathbb{B}\!</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},\!</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \mathrm{false}\!</math> and <math>1 = \mathrm{true}.\!</math>

; Boolean variable
: A ''[[boolean variable]]'' <math>x\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.\!</math>

; Proposition
: In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}\!</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}\!</math> is frequently called a ''[[proposition]]''.

; Basis element, Coordinate projection
: Given a sequence of <math>k\!</math> boolean variables, <math>x_1, \ldots, x_k,\!</math> each variable <math>x_j\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k\!</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.\!</math>

; Basic proposition
: This means that the set of objects <math>\{ x_j : 1 \le j \le k \}\!</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}\!</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}\!</math> propositions over <math>\mathbb{B}^k.\!</math>

; Literal
: A ''literal'' is one of the <math>2k\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},\!</math> in other words, either a ''posited'' basic proposition <math>x_j\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},\!</math> for some <math>j = 1 ~\text{to}~ k.\!</math>

; Fiber
: In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y\!</math> under a function <math>f : X \to Y\!</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.\!</math>

: In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},\!</math> there are just two fibers:
: The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math>
: The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math>

; Fiber of truth
: When <math>1\!</math> is interpreted as the logical value <math>\mathrm{true},\!</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math>

; Singular boolean function
: A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> is a boolean function whose fiber of <math>1\!</math> is a single point of <math>\mathbb{B}^k.\!</math>

; Singular proposition
: In the interpretation where <math>1\!</math> equals <math>\mathrm{true},\!</math> a singular boolean function is called a ''singular proposition''.

: Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.\!</math>

; Singular conjunction
: A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}\!</math> is a conjunction of <math>k\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}\!</math> for each <math>j = 1 ~\text{to}~ k.\!</math>

: A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}\!</math> can be expressed as a singular conjunction:

{| align="center" cellspacing"10" width="90%"
| height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k\!</math>,
|-
|
<math>\begin{array}{llll}
\text{where} & e_j & = & x_j
\\[6pt]
\text{or} & e_j & = & \nu (x_j),
\\[6pt]
\text{for} & j & = & 1 ~\text{to}~ k.
\end{array}</math>
|}

==Resources==

* [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs]
** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)]
** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index]
** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons]
** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples]
** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki]
* [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
{{col-end}}

===Relational concepts===

{{col-begin}}
{{col-break}}
* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
{{col-end}}

===Information, Inquiry===

{{col-begin}}
{{col-break}}
* [[Inquiry]]
* [[Dynamics of inquiry]]
{{col-break}}
* [[Semeiotic]]
* [[Logic of information]]
{{col-break}}
* [[Descriptive science]]
* [[Normative science]]
{{col-break}}
* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

{{col-begin}}
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
{{col-break}}
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
{{col-end}}

==Document history==

Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.

* [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikiversity.org/ Wikiversity]
* [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia]

[[Category:Automata Theory]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Inquiry]]
[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Logical Graphs]]
[[Category:Mathematics]]
[[Category:Neural Networks]]
[[Category:Philosophy]]
[[Category:Propositional Calculus]]
[[Category:Semiotics]]