Minimal negation operator

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A minimal negation operator Failed to parse (Missing texvc executable; please see math/README to configure.):

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 Failed to parse (Missing texvc executable; please see math/README to configure.):

then it cannot be true that exactly one of the arguments is false, so Failed to parse (Missing texvc executable; please see math/README to configure.): 


If is the only argument, then Failed to parse (Missing texvc executable; please see math/README to configure.):

says that  is false, so Failed to parse (Missing texvc executable; please see math/README to configure.): 
expresses the logical negation of the proposition   Wrtten in several different notations, Failed to parse (Missing texvc executable; please see math/README to configure.): 


If and are the only two arguments, then Failed to parse (Missing texvc executable; please see math/README to configure.):

says that exactly one of  is false, so Failed to parse (Missing texvc executable; please see math/README to configure.): 
says the same thing as   Expressing Failed to parse (Missing texvc executable; please see math/README to configure.): 
in terms of ands Failed to parse (Missing texvc executable; please see math/README to configure.): 
ors Failed to parse (Missing texvc executable; please see math/README to configure.): 
and nots Failed to parse (Missing texvc executable; please see math/README to configure.): 
gives the following form.
Failed to parse (Missing texvc executable; please see math/README to configure.):

As usual, one drops the dots Failed to parse (Missing texvc executable; please see math/README to configure.):

in contexts where they are understood, giving the following form.
Failed to parse (Missing texvc executable; please see math/README to configure.):

The venn diagram for Failed to parse (Missing texvc executable; please see math/README to configure.):

is shown in Figure 1.

Venn Diagram (P,Q).jpg

Failed to parse (Missing texvc executable; please see math/README to configure.):

The venn diagram for Failed to parse (Missing texvc executable; please see math/README to configure.):

is shown in Figure 2.

Venn Diagram (P,Q,R).jpg

Failed to parse (Missing texvc executable; please see math/README to configure.):

The center cell is the region where all three arguments hold true, so Failed to parse (Missing texvc executable; please see math/README to configure.):

holds true in just the three neighboring cells.  In other words, Failed to parse (Missing texvc executable; please see math/README to configure.): 


Initial definition

The minimal negation operator is a multigrade operator Failed to parse (Missing texvc executable; please see math/README to configure.):

where each  is a -ary boolean function defined in such a way that Failed to parse (Missing texvc executable; please see math/README to configure.): 
in just those cases where exactly one of the arguments  is 

In contexts where the initial letter 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, Failed to parse (Missing texvc executable; please see math/README to configure.):

= 

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.

Failed to parse (Missing texvc executable; please see math/README to configure.):

Formal definition

To express the general case of in terms of familiar operations, it helps to introduce an intermediary concept:

Definition. Let the function Failed to parse (Missing texvc executable; please see math/README to configure.):

be defined for each integer  in the interval  by the following equation:
Failed to parse (Missing texvc executable; please see math/README to configure.):

Then Failed to parse (Missing texvc executable; please see math/README to configure.):

is defined by the following equation:
Failed to parse (Missing texvc executable; please see math/README to configure.):

If we think of the point Failed to parse (Missing texvc executable; please see math/README to configure.):

as indicated by the boolean product Failed to parse (Missing texvc executable; please see math/README to configure.): 
or the logical conjunction Failed to parse (Missing texvc executable; please see math/README to configure.): 
then the minimal negation Failed to parse (Missing texvc executable; please see math/README to configure.): 
indicates the set of points in Failed to parse (Missing texvc executable; please see math/README to configure.): 
that differ from  in exactly one coordinate.  This makes Failed to parse (Missing texvc executable; please see math/README to configure.): 
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 and the summation symbol Failed to parse (Missing texvc executable; please see math/README to configure.):

both refer to addition modulo 2.  Unless otherwise noted, the boolean domain Failed to parse (Missing texvc executable; please see math/README to configure.): 
is interpreted so that Failed to parse (Missing texvc executable; please see math/README to configure.): 
and Failed to parse (Missing texvc executable; please see math/README to configure.): 
 This has the following consequences:
The operation is a function equivalent to the exclusive disjunction of and while its fiber of 1 is the relation of inequality between and
The operation Failed to parse (Missing texvc executable; please see math/README to configure.):
maps the bit sequence  to its parity.

The following properties of the minimal negation operators Failed to parse (Missing texvc executable; please see math/README to configure.):

may be noted:
The function Failed to parse (Missing texvc executable; please see math/README to configure.):
is the same as that associated with the operation  and the relation Failed to parse (Missing texvc executable; please see math/README to configure.): 
In contrast, Failed to parse (Missing texvc executable; please see math/README to configure.):
is not identical to 
More generally, the function Failed to parse (Missing texvc executable; please see math/README to configure.):
for  is not identical to the boolean sum Failed to parse (Missing texvc executable; please see math/README to configure.): 
The inclusive disjunctions indicated for the 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 Failed to parse (Missing texvc executable; please see math/README to configure.):

whose fibers of 1 are either the boundaries of points in Failed to parse (Missing texvc executable; please see math/README to configure.): 
or the complements of those boundaries.


Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.):
  Failed to parse (Missing texvc executable; please see math/README to configure.):  
  Failed to parse (Missing texvc executable; please see math/README to configure.):  
  Failed to parse (Missing texvc executable; please see math/README to configure.):  


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

of  points are the hypercube picture and the venn diagram picture.  The hypercube picture associates each point of Failed to parse (Missing texvc executable; please see math/README to configure.): 
with a unique point of the -dimensional hypercube.  The venn diagram picture associates each point of Failed to parse (Missing texvc executable; please see math/README to configure.): 
with a unique "cell" of the venn diagram on  "circles".

In addition, each point of Failed to parse (Missing texvc executable; please see math/README to configure.):

is the unique point in the fiber of truth  of a singular proposition Failed to parse (Missing texvc executable; please see math/README to configure.): 
and thus it is the unique point where a singular conjunction of  literals is 

For example, consider two cases at opposite vertices of the cube:

The point Failed to parse (Missing texvc executable; please see math/README to configure.):
with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to  namely, the point where:
  Failed to parse (Missing texvc executable; please see math/README to configure.):
The point Failed to parse (Missing texvc executable; please see math/README to configure.):
with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to  namely, the point where:
  Failed to parse (Missing texvc executable; please see math/README to configure.):

To pass from these limiting examples to the general case, observe that a singular proposition Failed to parse (Missing texvc executable; please see math/README to configure.):

can be given canonical expression as a conjunction of literals, Failed to parse (Missing texvc executable; please see math/README to configure.): 

. Then the proposition Failed to parse (Missing texvc executable; please see math/README to configure.):

is  on the points adjacent to the point where  is  and 0 everywhere else on the cube.

For example, consider the case where Then the minimal negation operation — written more simply as Failed to parse (Missing texvc executable; please see math/README to configure.):

— has the following venn diagram:

Venn Diagram (P,Q,R).jpg

Failed to parse (Missing texvc executable; please see math/README to configure.):

For a contrasting example, the boolean function expressed by the form Failed to parse (Missing texvc executable; please see math/README to configure.):

has the following venn diagram:

Venn Diagram ((P),(Q),(R)).jpg

Failed to parse (Missing texvc executable; please see math/README to configure.):

Glossary of basic terms

Boolean domain
A boolean domain is a generic 2-element set, for example, Failed to parse (Missing texvc executable; please see math/README to configure.):
whose elements are interpreted as logical values, usually but not invariably with Failed to parse (Missing texvc executable; please see math/README to configure.): 
and Failed to parse (Missing texvc executable; please see math/README to configure.): 


Boolean variable
A boolean variable is a variable that takes its value from a boolean domain, as Failed to parse (Missing texvc executable; please see math/README to configure.):


Proposition
In situations where boolean values are interpreted as logical values, a boolean-valued function Failed to parse (Missing texvc executable; please see math/README to configure.):
or a boolean function Failed to parse (Missing texvc executable; please see math/README to configure.): 
is frequently called a proposition.
Basis element, Coordinate projection
Given a sequence of boolean variables, Failed to parse (Missing texvc executable; please see math/README to configure.):
each variable  may be treated either as a basis element of the space Failed to parse (Missing texvc executable; please see math/README to configure.): 
or as a coordinate projection Failed to parse (Missing texvc executable; please see math/README to configure.): 


Basic proposition
This means that the set of objects Failed to parse (Missing texvc executable; please see math/README to configure.):
is a set of boolean functions Failed to parse (Missing texvc executable; please see math/README to configure.): 
subject to logical interpretation as a set of basic propositions that collectively generate the complete set of Failed to parse (Missing texvc executable; please see math/README to configure.): 
propositions over Failed to parse (Missing texvc executable; please see math/README to configure.): 


Literal
A literal is one of the propositions Failed to parse (Missing texvc executable; please see math/README to configure.):
in other words, either a posited basic proposition  or a negated basic proposition Failed to parse (Missing texvc executable; please see math/README to configure.): 
for some Failed to parse (Missing texvc executable; please see math/README to configure.): 


Fiber
In mathematics generally, the fiber of a point Failed to parse (Missing texvc executable; please see math/README to configure.):
under a function  is defined as the inverse image Failed to parse (Missing texvc executable; please see math/README to configure.): 


In the case of a boolean function Failed to parse (Missing texvc executable; please see math/README to configure.):
there are just two fibers:
The fiber of under defined as is the set of points where the value of is
The fiber of under defined as is the set of points where the value of is
Fiber of truth
When is interpreted as the logical value Failed to parse (Missing texvc executable; please see math/README to configure.):
then  is called the fiber of truth in the proposition   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  for the fiber of truth in the proposition 
Singular boolean function
A singular boolean function Failed to parse (Missing texvc executable; please see math/README to configure.):
is a boolean function whose fiber of  is a single point of Failed to parse (Missing texvc executable; please see math/README to configure.): 


Singular proposition
In the interpretation where equals Failed to parse (Missing texvc executable; please see math/README to configure.):
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 Failed to parse (Missing texvc executable; please see math/README to configure.):


Singular conjunction
A singular conjunction in Failed to parse (Missing texvc executable; please see math/README to configure.):
is a conjunction of  literals that includes just one conjunct of the pair Failed to parse (Missing texvc executable; please see math/README to configure.): 
for each Failed to parse (Missing texvc executable; please see math/README to configure.): 


A singular proposition Failed to parse (Missing texvc executable; please see math/README to configure.):
can be expressed as a singular conjunction:
Failed to parse (Missing texvc executable; please see math/README to configure.):

,

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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.