View source for Minimal negation operator

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A '''minimal negation operator''' <math>(\nu)~\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.&nbsp; The first four cases are described below.

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If the list of arguments is empty, as expressed in the form <math>\nu(),~\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\nu() = \mathrm{false}.~\!</math>
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If <math>p~\!</math> is the only argument then <math>\nu(p)~\!</math> says that <math>p~\!</math> is false, so <math>\nu(p)~\!</math> expresses the logical negation of the proposition <math>p.~\!</math>&nbsp; Written in several different notations, we have the following equivalent expressions.

<p style="padding:8px; text-align:center"><math>\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}~\!</math></p>
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If <math>p~\!</math> and <math>q~\!</math> are the only two arguments then <math>\nu(p, q)~\!</math> says that exactly one of <math>p, q~\!</math> is false, so <math>\nu(p, q)~\!</math> says the same thing as <math>p \neq q.~\!</math>  Expressing <math>\nu(p, q)~\!</math> in terms of ands <math>(\cdot),~\!</math> ors <math>(\lor),~\!</math> and nots <math>(\tilde{~})~\!</math> gives the following form.

<p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}~\!</math></p>

It is permissible to omit the dot <math>(\cdot)~\!</math> in contexts where it is understood, giving the following form.

<p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}~\!</math></p>

The venn diagram for <math>\nu(p, q)~\!</math> is shown in Figure&nbsp;1.

{| align="center" cellpadding="8" style="text-align:center"
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<p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p>
<p><math>\text{Figure 1.}~~\nu(p, q)~\!</math></p>
|}
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The venn diagram for <math>\nu(p, q, r)~\!</math> is shown in Figure&nbsp;2.

{| align="center" cellpadding="8" style="text-align:center"
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<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
<p><math>\text{Figure 2.}~~\nu(p, q, r)~\!</math></p>
|}

The center cell is the region where all three arguments <math>p, q, r~\!</math> hold true, so <math>\nu(p, q, r)~\!</math> holds true in just the three neighboring cells.&nbsp; In other words:

<p style="padding:8px; text-align:center"><math>\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}~\!</math></p>

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