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.
 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

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.

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.

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  edit source]
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  edit source]
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 pointomitted neighborhood in ordinary real analysis, more exactly, a pointomitted distanceone 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  edit source]
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.



Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \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}} 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \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}} 


Charts and graphs[edit  edit source]
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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \mathbb{B}^k~\!} of points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of with a unique point of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle k~\!} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \mathbb{B}^k~\!} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle (1, 1, \ldots , 1, 1)~\!} with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to namely, the point where: 
•  The point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle (0, 0, \ldots , 0, 0)~\!} with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle 1,~\!} namely, the point where: 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n1} \texttt{)(} x_n \texttt{)} ~=~ 1.~\!} 
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \nu (e_1, e_2, \ldots, e_{k1}, e_k)~\!} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle 1~\!} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \texttt{((p),(q),(r))}~\!} has the following venn diagram:

Glossary of basic terms[edit  edit source]
 Boolean domain
 A boolean domain is a generic 2element 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 booleanvalued 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:
, 

Resources[edit  edit source]
Syllabus[edit  edit source]
Focal nodes[edit  edit source]
Peer nodes[edit  edit source]
 Minimal Negation Operator @ InterSciWiki
 Minimal Negation Operator @ Subject Wikis
 Minimal Negation Operator @ Wikiversity
 Minimal Negation Operator @ Wikiversity Beta
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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.
 Automata Theory
 Boolean functions
 Charles Sanders Peirce
 Combinatorics
 Computer science
 Differential Logic
 Equational Reasoning
 Inquiry
 Linguistics
 Logic
 Logical Graphs
 Minimal Negation Operators
 Neural Networks
 Philosophy
 Propositional Calculus
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