Minimal negation operator

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A minimal negation operator is a logical connective that says “just one false” of its logical arguments.  The first four cases are described below.

  1. If the list of arguments is empty, as expressed in the form then it cannot be true that exactly one of the arguments is false, so
  2. If is the only argument then says that is false, so expresses the logical negation of the proposition   Written in several different notations, we have the following equivalent expressions.

  3. If and are the only two arguments then says that exactly one of is false, so says the same thing as Expressing in terms of ands ors and nots gives the following form.

    It is permissible to omit the dot in contexts where it is understood, giving the following form.

    The venn diagram for is shown in Figure 1.

    Venn Diagram (P,Q).jpg

  4. The venn diagram for is shown in Figure 2.

    Minimal Negation Operator (P,Q,R).jpg

    The center cell is the region where all three arguments hold true, so holds true in just the three neighboring cells.  In other words:

Initial definition[edit]

The minimal negation operator is a multigrade operator where each is a -ary boolean function defined by the rule that if and only if 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,

The first four members of this family of operators are shown below.  The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.

Formal definition[edit]

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

Definition.  Let the function be defined for each integer in the interval by the following equation:

Then is defined by the following equation:

If we take the boolean product or the logical conjunction to indicate the point in the space then the minimal negation indicates the set of points in that differ from in exactly one coordinate.  This makes a discrete functional analogue of a point-omitted neighborhood in ordinary real 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.

The remainder of this discussion proceeds on the algebraic convention that the plus sign and the summation symbol both refer to addition mod 2.  Unless otherwise noted, the boolean domain is interpreted for logic in such a way that and   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 maps the bit sequence to its parity.

The following properties of the minimal negation operators may be noted:

The function is the same as that associated with the operation and the relation
In contrast, is not identical to
More generally, the function for is not identical to the boolean sum
The inclusive disjunctions indicated for the of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.

Truth tables[edit]

Table 3 is a truth table for the sixteen boolean functions of type whose fibers of 1 are either the boundaries of points in or the complements of those boundaries.


   
   
   


Charts and graphs[edit]

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 of points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of with a unique point of the -dimensional hypercube. The venn diagram picture associates each point of with a unique "cell" of the venn diagram on "circles".

In addition, each point of is the unique point in the fiber of truth of a singular proposition 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 with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to namely, the point where:
 
The point with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to namely, the point where:
 

To pass from these limiting examples to the general case, observe that a singular proposition can be given canonical expression as a conjunction of literals, . Then the proposition 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 — has the following venn diagram:

Minimal Negation Operator (P,Q,R).jpg

For a contrasting example, the boolean function expressed by the form has the following venn diagram:

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

Glossary of basic terms[edit]

Boolean domain
A boolean domain is a generic 2-element set, for example, whose elements are interpreted as logical values, usually but not invariably with and
Boolean variable
A boolean variable is a variable that takes its value from a boolean domain, as
Proposition
In situations where boolean values are interpreted as logical values, a boolean-valued function or a boolean function is frequently called a proposition.
Basis element, Coordinate projection
Given a sequence of boolean variables, each variable may be treated either as a basis element of the space or as a coordinate projection
Basic proposition
This means that the set of objects is a set of boolean functions subject to logical interpretation as a set of basic propositions that collectively generate the complete set of propositions over
Literal
A literal is one of the propositions in other words, either a posited basic proposition or a negated basic proposition for some
Fiber
In mathematics generally, the fiber of a point under a function is defined as the inverse image
In the case of a boolean function 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 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 is a boolean function whose fiber of is a single point of
Singular proposition
In the interpretation where equals 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
Singular conjunction
A singular conjunction in is a conjunction of literals that includes just one conjunct of the pair for each
A singular proposition can be expressed as a singular conjunction:
,

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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.