Definition:Minimal Negation Operator
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Definition
Let $\Bbb B$ be a Boolean domain:
- $\Bbb B = \set {\F, \T}$
The minimal negation operator $\nu$ is a multiary operator:
- $\nu_k: \Bbb B^k \to \Bbb B$
where:
- $k \in \N$ is a natural number
- $\nu_k$ is a boolean function defined as:
- $\map {\nu_k} {x_1, x_2, \ldots, x_k} = \begin {cases} \T & : \exists! x_j \in \set {x_1, x_2, \ldots, x_k}: x_j = \F \\ \F & : \text {otherwise} \end{cases}$
That is:
- $\map {\nu_k} {x_1, x_2, \ldots, x_k}$ is true if and only if exactly one of its arguments is false.
Notation
The symbol $\nu$ used for this is the Greek letter nu.
Examples
Expressed in disjunctive normal form, the first few instances of $\nu_k$ are as follows:
\(\ds \nu_0\) | \(=\) | \(\ds \F\) | ||||||||||||
\(\ds \map {\nu_1} p\) | \(=\) | \(\ds \neg p\) | ||||||||||||
\(\ds \map {\nu_2} {p, q}\) | \(=\) | \(\ds \paren {\neg p \land q} \lor \paren {p \land \neg q}\) | ||||||||||||
\(\ds \map {\nu_3} {p, q, r}\) | \(=\) | \(\ds \paren {\neg p \land q \land r} \lor \paren {p \land \neg q \land r} \lor \paren {p \land q \land \neg r}\) | ||||||||||||
\(\ds \map {\nu_4} {p, q, r, s}\) | \(=\) | \(\ds \paren {\neg p \land q \land r \land s} \lor \paren {p \land \neg q \land r \land s} \lor \paren {p \land q \land \neg r \land s} \lor \paren {p \land q \land r \land \neg s}\) |
Example: $\nu_0$
$\nu_0$ is the false constant, or the contradiction operator $\bot$:
- $\nu_0 = \bot$
Example: $\nu_1$
$\nu_1$ is the same operator as the logical not operator $\neg$:
- $\map {\nu_1} p = \neg p$
Example: $\nu_2$
$\nu_2$ is the same operator as the exclusive or operator $\oplus$:
- $\map {\nu_2} {p, q} = p \oplus q$
For $k > 2$ there is no immediate correspondence between $\nu_k$ and conventional logical operators.
Also see
- Results about the minimal negation operator can be found here.
Sources
- This article incorporates material from minimal negation operator on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.