Minimal negation operator

In logic and mathematics, 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 in such a way 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. 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.

\(\begin{matrix} (~) & = & 0 & = & \operatorname{false} \'"`UNIQ-MathJax1-QINU`"' * The point \((0, 0, \ldots , 0, 0)\) with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to \(1,\!\) namely, the point where:

\[(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.\]

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 \(\nu (p, q, r)\!\), when there is no risk of confusion written more simply as \((p, q, r)\!\), has the following venn diagram:

Minimal Negation Operator 1.jpg

\(\text{Figure 1.}\quad (p, q, r)\!\)

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

Minimal Negation Operator 2.jpg

\(\text{Figure 2.}\quad ((p),(q),(r))\!\)

Glossary of basic terms

  • A boolean domain \(\mathbb{B}\) is a generic 2-element set, say, \(\mathbb{B} = \{ 0, 1 \},\) whose elements are interpreted as logical values, usually but not invariably with \(0 = \operatorname{false}\) and \(1 = \operatorname{true}.\)
  • Given a sequence of k boolean variables, x1, …, xk, each variable xj may be treated either as a basis element of the space Bk or as a coordinate projection xj : Bk → B.
  • This means that the k objects xj for j = 1 to k are just so many boolean functions xj : Bk → B , subject to logical interpretation as a set of basic propositions that generate the complete set of \(2^{2^k}\) propositions over Bk.
  • A literal is one of the 2k propositions x1, …, xk, (x1), …, (xk), in other words, either a posited basic proposition xj or a negated basic proposition (xj), for some j = 1 to k.
  • In mathematics generally, the fiber of a point y under a function f : X → Y is defined as the inverse image \(f^{-1}(y)\).
  • In the case of a boolean function f : Bk → B, there are just two fibers:
    • The fiber of 0 under f, defined as \(f^{-1}(0)\), is the set of points where f is 0.
    • The fiber of 1 under f, defined as \(f^{-1}(1)\), is the set of points where f is 1.
  • When 1 is interpreted as the logical value 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.
  • A singular boolean function s : Bk → B is a boolean function whose fiber of 1 is a single point of Bk.
  • In the interpretation where 1 equals 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 Bk.
  • A singular conjunction in Bk → B is a conjunction of k literals that includes just one conjunct of the pair \(\{ x_j,\ \nu (x_j) \}\) for each j = 1 to k.
  • A singular proposition s : Bk → B can be expressed as a singular conjunction:
\(s ~=~ e_1 e_2 \ldots e_{k-1} e_k\),

\(\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}\)

See also

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External links

Aficionados



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