Minimal Negation Operator/Examples

Examples of the Minimal Negation Operator

Let $\nu_k$ denote the minimal negation operator with $k$ arguments.

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.