Minimal negation operator

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A minimal negation operator $$(\nu)~\!$$ is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments. The first four cases are described below.

 If the list of arguments is empty, as expressed in the form $$\nu,~\!$$ then it cannot be true that exactly one of the arguments is false, so $$\nu = \mathrm{false}.~\!$$ 

 If $$p~\!$$ is the only argument then $$\nu(p)~\!$$ says that $$p~\!$$ is false, so $$\nu(p)~\!$$ expresses the logical negation of the proposition $$p.~\!$$ Written in several different notations, we have the following equivalent expressions.

$$\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}~\!$$ 

 If $$p~\!$$ and $$q~\!$$ are the only two arguments then $$\nu(p, q)~\!$$ says that exactly one of $$p, q~\!$$ is false, so $$\nu(p, q)~\!$$ says the same thing as $$p \neq q.~\!$$ Expressing $$\nu(p, q)~\!$$ in terms of ands $$(\cdot),~\!$$ ors $$(\lor),~\!$$ and nots $$(\tilde{~})~\!$$ gives the following form.

$$\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}~\!$$

It is permissible to omit the dot $$(\cdot)~\!$$ in contexts where it is understood, giving the following form.

$$\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}~\!$$

The venn diagram for $$\nu(p, q)~\!$$ is shown in Figure 1.



 The venn diagram for $$\nu(p, q, r)~\!$$ is shown in Figure 2.

The center cell is the region where all three arguments $$p, q, r~\!$$ hold true, so $$\nu(p, q, r)~\!$$ holds true in just the three neighboring cells. In other words:

$$\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}~\!$$



Initial definition
The minimal negation operator $$\nu~\!$$ is a multigrade operator $$(\nu_k)_{k \in \mathbb{N}}~\!$$ where each $$\nu_k~\!$$ is a $$k~\!$$-ary boolean function defined by the rule that $$\nu_k (x_1, \ldots, x_k) = 1~\!$$ if and only if exactly one of the arguments $$x_j~\!$$ is $$0.~\!$$

In contexts where the initial letter $$\nu~\!$$ 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, $$\texttt{(x, y, z)} = \nu (x, y, z).~\!$$

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
To express the general case of $$\nu_k~\!$$ in terms of familiar operations, it helps to introduce an intermediary concept:

Definition. Let the function $$\lnot_j : \mathbb{B}^k \to \mathbb{B}~\!$$ be defined for each integer $$j~\!$$ in the interval $$[1, k]~\!$$ by the following equation:

Then $${\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!$$ is defined by the following equation:

If we take the boolean product $$x_1 \cdot \ldots \cdot x_k~\!$$ or the logical conjunction $$x_1 \land \ldots \land x_k~\!$$ to indicate the point $$x = (x_1, \ldots, x_k)~\!$$ in the space $$\mathbb{B}^k~\!$$ then the minimal negation $$\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!$$ indicates the set of points in $$\mathbb{B}^k~\!$$ that differ from $$x~\!$$ in exactly one coordinate. This makes $$\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!$$ 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 $$(\textstyle\sum)~\!$$ both refer to addition mod 2. Unless otherwise noted, the boolean domain $$\mathbb{B} = \{ 0, 1 \}~\!$$ is interpreted for logic in such a way that $$0 = \mathrm{false}~\!$$ and $$1 = \mathrm{true}.~\!$$ This has the following consequences:

The following properties of the minimal negation operators $${\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!$$ may be noted:

Truth tables
Table 3 is a truth table for the sixteen boolean functions of type $$f : \mathbb{B}^3 \to \mathbb{B}~\!$$ whose fibers of 1 are either the boundaries of points in $$\mathbb{B}^3~\!$$ or the complements of those boundaries.

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 $$\mathbb{B}^k~\!$$ of $$2^k~\!$$ points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of $$\mathbb{B}^k~\!$$ with a unique point of the $$k~\!$$-dimensional hypercube. The venn diagram picture associates each point of $$\mathbb{B}^k~\!$$ with a unique "cell" of the venn diagram on $$k~\!$$ "circles".

In addition, each point of $$\mathbb{B}^k~\!$$ is the unique point in the fiber of truth $$[|s|]~\!$$ of a singular proposition $$s : \mathbb{B}^k \to \mathbb{B},~\!$$ and thus it is the unique point where a singular conjunction of $$k~\!$$ literals is $$1.~\!$$

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

To pass from these limiting examples to the general case, observe that a singular proposition $$s : \mathbb{B}^k \to \mathbb{B}~\!$$ can be given canonical expression as a conjunction of literals, $$s = e_1 e_2 \ldots e_{k-1} e_k~\!$$. Then the proposition $$\nu (e_1, e_2, \ldots, e_{k-1}, e_k)~\!$$ is $$1~\!$$ on the points adjacent to the point where $$s~\!$$ is $$1,~\!$$ and 0 everywhere else on the cube.

For example, consider the case where $$k = 3.~\!$$ Then the minimal negation operation $$\nu (p, q, r)~\!$$ &mdash; written more simply as $$\texttt{(p, q, r)}~\!$$ &mdash; has the following venn diagram:

For a contrasting example, the boolean function expressed by the form $$\texttt{((p),(q),(r))}~\!$$ has the following venn diagram:

Glossary of basic terms

 * Boolean domain
 * A boolean domain $$\mathbb{B}~\!$$ is a generic 2-element set, for example, $$\mathbb{B} = \{ 0, 1 \},~\!$$ whose elements are interpreted as logical values, usually but not invariably with $$0 = \mathrm{false}~\!$$ and $$1 = \mathrm{true}.~\!$$


 * Boolean variable
 * A boolean variable $$x~\!$$ is a variable that takes its value from a boolean domain, as $$x \in \mathbb{B}.~\!$$


 * Proposition
 * In situations where boolean values are interpreted as logical values, a boolean-valued function $$f : X \to \mathbb{B}~\!$$ or a boolean function $$g : \mathbb{B}^k \to \mathbb{B}~\!$$ is frequently called a proposition.


 * Basis element, Coordinate projection
 * Given a sequence of $$k~\!$$ boolean variables, $$x_1, \ldots, x_k,~\!$$ each variable $$x_j~\!$$ may be treated either as a basis element of the space $$\mathbb{B}^k~\!$$ or as a coordinate projection $$x_j : \mathbb{B}^k \to \mathbb{B}.~\!$$


 * Basic proposition
 * This means that the set of objects $$\{ x_j : 1 \le j \le k \}~\!$$ is a set of boolean functions $$\{ x_j : \mathbb{B}^k \to \mathbb{B} \}~\!$$ subject to logical interpretation as a set of basic propositions that collectively generate the complete set of $$2^{2^k}~\!$$ propositions over $$\mathbb{B}^k.~\!$$


 * Literal
 * A literal is one of the $$2k~\!$$ propositions $$x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},~\!$$ in other words, either a posited basic proposition $$x_j~\!$$ or a negated basic proposition $$\texttt{(} x_j \texttt{)},~\!$$ for some $$j = 1 ~\text{to}~ k.~\!$$


 * Fiber
 * In mathematics generally, the fiber of a point $$y \in Y~\!$$ under a function $$f : X \to Y~\!$$ is defined as the inverse image $$f^{-1}(y) \subseteq X.~\!$$


 * In the case of a boolean function $$f : \mathbb{B}^k \to \mathbb{B},~\!$$ there are just two fibers:
 * The fiber of $$0~\!$$ under $$f,~\!$$ defined as $$f^{-1}(0),~\!$$ is the set of points where the value of $$f~\!$$ is $$0.~\!$$
 * The fiber of $$1~\!$$ under $$f,~\!$$ defined as $$f^{-1}(1),~\!$$ is the set of points where the value of $$f~\!$$ is $$1.~\!$$


 * Fiber of truth
 * When $$1~\!$$ is interpreted as the logical value $$\mathrm{true},~\!$$ then $$f^{-1}(1)~\!$$ is called the fiber of truth in the proposition $$f.~\!$$ 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 $$[|f|] = f^{-1}(1)~\!$$ for the fiber of truth in the proposition $$f.~\!$$


 * Singular boolean function
 * A singular boolean function $$s : \mathbb{B}^k \to \mathbb{B}~\!$$ is a boolean function whose fiber of $$1~\!$$ is a single point of $$\mathbb{B}^k.~\!$$


 * Singular proposition
 * In the interpretation where $$1~\!$$ equals $$\mathrm{true},~\!$$ 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 $$\mathbb{B}^k.~\!$$


 * Singular conjunction
 * A singular conjunction in $$\mathbb{B}^k \to \mathbb{B}~\!$$ is a conjunction of $$k~\!$$ literals that includes just one conjunct of the pair $$\{ x_j, ~\nu(x_j) \}~\!$$ for each $$j = 1 ~\text{to}~ k.~\!$$


 * A singular proposition $$s : \mathbb{B}^k \to \mathbb{B}~\!$$ can be expressed as a singular conjunction:

Resources

 * Wolfram Atlas of Simple Programs
 * Elementary Cellular Automata Rules (ECARs)
 * ECAR Index
 * ECAR Icons
 * ECAR Examples
 * ECAR Formulas

Focal nodes

 * Inquiry Live
 * Logic Live

Peer nodes

 * Minimal Negation Operator @ InterSciWiki
 * Minimal Negation Operator @ Subject Wikis
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Logical operators

 * Exclusive disjunction
 * Logical conjunction
 * Logical disjunction
 * Logical equality


 * Logical implication
 * Logical NAND
 * Logical NNOR
 * Negation

Related topics

 * Ampheck
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Differential logic


 * Logical graph
 * Minimal negation operator
 * Multigrade operator
 * Parametric operator
 * Peirce's law


 * Propositional calculus
 * Sole sufficient operator
 * Truth table
 * Universe of discourse
 * Zeroth order logic

Relational concepts

 * Continuous predicate
 * Hypostatic abstraction
 * Logic of relatives
 * Logical matrix


 * Relation
 * Relation composition
 * Relation construction
 * Relation reduction


 * Relation theory
 * Relative term
 * Sign relation
 * Triadic relation

Information, Inquiry

 * Inquiry
 * Dynamics of inquiry


 * Semeiotic
 * Logic of information


 * Descriptive science
 * Normative science


 * Pragmatic maxim
 * Truth theory

Related articles

 * Cactus Language
 * Futures Of Logical Graphs
 * Propositional Equation Reasoning Systems


 * Differential Logic : Introduction
 * Differential Propositional Calculus
 * Differential Logic and Dynamic Systems


 * Prospects for Inquiry Driven Systems
 * Introduction to Inquiry Driven Systems
 * Inquiry Driven Systems : Inquiry Into Inquiry

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.


 * Minimal Negation Operator, InterSciWiki
 * Minimal Negation Operator, PlanetMath
 * Minimal Negation Operator, Wikinfo
 * Minimal Negation Operator, Wikiversity
 * Minimal Negation Operator, Wikiversity Beta
 * Minimal Negation Operator, Wikipedia