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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
<font size="3">&#9758;</font> 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 function]]s, [[logical connective]]s, [[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 [[isomorphism]]s.
'''''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.&nbsp; 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==
==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.
By way of initial orientation, Table&nbsp;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.


<br>
<br>


{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="background:#e6e6ff"
|- style="height:40px; background:ghostwhite"
! style="width:15%" | L<sub>1</sub>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math>
! style="width:15%" | L<sub>2</sub>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math>
! style="width:15%" | L<sub>3</sub>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math>
! style="width:15%" | L<sub>4</sub>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math>
! style="width:15%" | L<sub>5</sub>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math>
! style="width:15%" | L<sub>6</sub>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math>
|- style="background:#e6e6ff"
|- style="background:ghostwhite"
| &nbsp;
| &nbsp;
| align="right" | x :
| align="right" | <math>x\colon\!</math>
| 1 1 0 0  
| <math>1~1~0~0\!</math>
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- style="background:#e6e6ff"
|- style="background:ghostwhite"
| &nbsp;
| &nbsp;
| align="right" | y :
| align="right" | <math>y\colon\!</math>
| 1 0 1 0
| <math>1~0~1~0\!</math>
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|-
|-
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0
| 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>
|-
|-
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x &and; ¬y
| valign="bottom" |
|-
<math>\begin{matrix}
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x &and; y
f_{8}
|-
\\[4pt]
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x
f_{9}
|-
\\[4pt]
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; ¬y
f_{10}
|-
\\[4pt]
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y
f_{11}
|-
\\[4pt]
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y
f_{12}
|-
\\[4pt]
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x &or; ¬y
f_{13}
|-
\\[4pt]
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x &and; y
f_{14}
|-
\\[4pt]
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
f_{15}
|-
\end{matrix}\!</math>
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
| valign="bottom" |
|-
<math>\begin{matrix}
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y
f_{1000}
|-
\\[4pt]
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
f_{1001}
|-
\\[4pt]
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y
f_{1010}
|-
\\[4pt]
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y
f_{1011}
|-
\\[4pt]
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
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>
|}
|}


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These six languages for the sixteen boolean functions are conveniently described in the following order:
These six languages for the sixteen boolean functions are conveniently described in the following order:


* Language '''L<sub>3</sub>''' describes each boolean function ''f'' : '''B'''<sup>2</sup> &#8594; '''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 <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>&nbsp; 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 '''L<sub>2</sub>''' lists the sixteen functions in the form '''f<sub>i</sub>''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L<sub>3</sub>'''.
* 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 '''L<sub>1</sub>''' notates the boolean functions '''f<sub>i</sub>''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L<sub>2</sub>'''.
* 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 '''L<sub>4</sub>''' 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:
* 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:


: <math>\begin{matrix}
{| align="center" cellpadding="8" style="text-align:center"
(\ )     & = & 0 & = & \mbox{false} \\
|
(x)       & = & \tilde{x} & = & x' \\
<math>\begin{matrix}
(x, y)   & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
\texttt{()}
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
& = & 0
\end{matrix}</math>
& = & \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 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>.
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 '''L<sub>5</sub>''' 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}_{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 '''L<sub>6</sub>''' expresses the sixteen functions in one of several notations that are commonly used in formal logic.
* Language <math>\mathcal{L}_{6}\!</math> expresses the sixteen functions in one of several notations that are commonly used in formal logic.


==Translations==
==Translations==
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===Focal nodes===
===Focal nodes===


{{col-begin}}
{{col-break}}
* [[Inquiry Live]]
* [[Inquiry Live]]
{{col-break}}
* [[Logic Live]]
* [[Logic Live]]
{{col-end}}


===Peer nodes===
===Peer nodes===


{{col-begin}}
* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
{{col-break}}
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://mathweb.org/wiki/Zeroth_order_logic Zeroth Order Logic @ MathWeb Wiki]
* [http://netknowledge.org/wiki/Zeroth_order_logic Zeroth Order Logic @ NetKnowledge]
* [http://wiki.oercommons.org/mediawiki/index.php/Zeroth_order_logic Zeroth Order Logic @ OER Commons]
{{col-break}}
* [http://p2pfoundation.net/Zeroth_Order_Logic Zeroth Order Logic @ P2P Foundation]
* [http://semanticweb.org/wiki/Zeroth_order_logic Zeroth Order Logic @ SemanticWeb]
* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
* [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]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]
{{col-end}}


===Logical operators===
===Logical operators===
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===Related articles===
===Related articles===


* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
{{col-begin}}
 
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
* [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://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
* [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://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
* [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://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 
{{col-end}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;]


==Document history==
==Document history==
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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.
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.


{{col-begin}}
* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
{{col-break}}
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/encyclopedia/ZerothOrderLogic.html Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
* [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://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://getwiki.net/-Zeroth-Order_Logic Zeroth Order Logic], [http://getwiki.net/ GetWiki]
{{col-break}}
* [http://wikinfo.org/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/ Wikinfo]
* [http://textop.org/wiki/index.php?title=Zeroth_order_logic Zeroth Order Logic], [http://textop.org/wiki/ Textop Wiki]
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
* [http://altheim.com/cs/zol.html Zeroth Order Logic], [http://altheim.com/cs Altheim.com]
* [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]
{{col-end}}


[[Category:Inquiry]]
[[Category:Inquiry]]
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[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]
[[Category:Normative Sciences]]
[[Category:Semiotics]]

Latest revision as of 03:08, 9 November 2015

☞ This page belongs to resource collections on Logic and 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 and abstract type in a number of different languages for zeroth order logic.


       
       


These six languages for the sixteen boolean functions are conveniently described in the following order:

  • Language describes each boolean function by means of the sequence of four boolean values,   Such a sequence, perhaps in another order, and perhaps with the logical values and instead of the boolean values and respectively, would normally be displayed as a column in a truth table.
  • Language lists the sixteen functions in the form where the index is a bit string formed from the sequence of boolean values in
  • Language notates the boolean functions with an index that is the decimal equivalent of the binary numeral index in
  • Language 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 operators, the first few of which are given in the following variant notations:

It may be noted that is the same function as and The inclusive disjunctions indicated for and for may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. However, the function is not the same thing as the function

  • Language lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
  • Language expresses the sixteen functions in one of several notations that are commonly used in formal logic.

Translations

Syllabus

Focal nodes

Peer nodes

Logical operators

Related topics

Relational concepts

Information, Inquiry

Related articles

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.

Retrieved from "https://subwiki.org/w/index.php?title=Zeroth_order_logic&oldid=735"