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:

	\(\displaystyle \)	\(\displaystyle \nu_0\)	\(=\)	\(\displaystyle \mathrm F\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \nu_1 \left({p}\right)\)	\(=\)	\(\displaystyle \neg p\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \nu_2 \left({p, q}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \nu_3 \left({p, q, r}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q \land r}\right) \lor \left({p \land \neg q \land r}\right) \lor \left({p \land q \land \neg r}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \nu_4 \left({p, q, r, s}\right)\)	\(=\)	\(\displaystyle \left({\neg p \land q \land r \land s}\right) \lor \left({p \land \neg q \land r \land s}\right) \lor \left({p \land q \land \neg r \land s}\right) \lor \left({p \land q \land r \land \neg s}\right)\)	\(\displaystyle \)

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.