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.

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

    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:



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



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  
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  
A literal is one of the   propositions   in other words, either a posited basic proposition   or a negated basic proposition   for some  
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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