Definition:Minimal Negation Operator

Definition
Let $\Bbb B$ be a Boolean domain:
 * $\Bbb B = \left\{{\mathrm F, \mathrm T}\right\}$

The minimal negation operator $\nu$ is a multiary operator:
 * $\nu_k: \Bbb B^k \to \Bbb B$

where:
 * $k \in \N$
 * $\nu_k$ is a boolean function defined as:


 * $\nu_k \left({x_1, x_2, \ldots, x_k}\right) = \begin{cases}

\mathrm T & : \exists! x_j \in \left\{{x_1, x_2, \ldots, x_k}\right\}: x_j = \mathrm F \\ \mathrm F & : \text {otherwise} \end{cases}$

That is: $\nu_k \left({x_1, x_2, \ldots, x_k}\right)$ is true iff exactly one of its arguments is false.

Examples
Expressed in disjunctive normal form, the first few instances of $\nu_k$ are as follows:

It can directly be seen that:
 * $\nu_0$ is the false constant, or a contradiction $\bot$
 * $\nu_1$ is the same operator as the logical Not operator $\neg$


 * $\nu_2$ is the same operator as the exclusive or operator $\oplus$

For $k > 2$ there is no immediate correspondence between $\nu_k$ and conventional logical operators.

Notation
The symbol $\nu$ used for this is the Greek letter nu.