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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

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 as follows:

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></li>

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> Written in several different notations, <math>\nu(p) = \mathrm{not}(p) = \lnot p = \tilde{p} = p^\prime.~\!</math></li>

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.

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.

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

<math>\nu(p, q) = \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}.~\!</math></p>

As usual, one drops the dot <math>(\cdot)~\!</math> in contexts where it's understood, giving the following form.

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

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

<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.

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

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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. In other words:

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:

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<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>

<math>\nu(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.~\!</math></p>

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