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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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==Initial definition==

==Initial definition==

The '''minimal negation operator''' <math>\nu~\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}~\!</math> where each <math>\nu_k~\!</math> is a <math>k~\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1~\!</math> in just those cases where exactly one of the arguments <math>x_j~\!</math> is <math>0.~\!</math>

The '''minimal negation operator''' <math>\nu~\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}~\!</math> where each <math>\nu_k~\!</math> is a <math>k~\!</math>-ary [[boolean function]] defined by the rule that <math>\nu_k (x_1, \ldots , x_k) = 1~\!</math> if and only if exactly one of the arguments <math>x_j~\!</math> is <math>0.~\!</math>

In contexts where the initial letter <math>\nu~\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}~\!</math> = <math>\nu (x, y, z).~\!</math>

In contexts where the initial letter <math>\nu~\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.&nbsp; In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)} = \nu (x, y, z).~\!</math>

The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.

The first four members of this family of operators are shown below.&nbsp; The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.

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To express the general case of <math>\nu_k~\!</math> in terms of familiar operations, it helps to introduce an intermediary concept:

To express the general case of <math>\nu_k~\!</math> in terms of familiar operations, it helps to introduce an intermediary concept:

'''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}~\!</math> be defined for each integer <math>j~\!</math> in the interval <math>[1, k]~\!</math> by the following equation:

'''Definition.'''&nbsp; Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}~\!</math> be defined for each integer <math>j~\!</math> in the interval <math>[1, k]~\!</math> by the following equation:

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If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k~\!</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k~\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,~\!</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}~\!</math> indicates the set of points in <math>\mathbb{B}^k~\!</math> that differ from <math>x~\!</math> in exactly one coordinate. This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}~\!</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few.  The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.

If we take the boolean product <math>x_1 \cdot \ldots \cdot x_k~\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k~\!</math> to indicate the point <math>x = (x_1, \ldots, x_k)~\!</math> in the space <math>\mathbb{B}^k~\!</math> then the minimal negation <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!</math> indicates the set of points in <math>\mathbb{B}^k~\!</math> that differ from <math>x~\!</math> in exactly one coordinate.&nbsp; This makes <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!</math> a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more exactly, a point-omitted distance-one neighborhood.&nbsp; In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.

The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)~\!</math> and the summation symbol <math>(\textstyle\sum)~\!</math> both refer to addition modulo 2. Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted so that <math>0 = \mathrm{false}~\!</math> and <math>1 = \mathrm{true}.~\!</math> This has the following consequences:

The remainder of this discussion proceeds on the algebraic convention that the plus sign <math>(+)~\!</math> and the summation symbol <math>(\textstyle\sum)~\!</math> both refer to addition mod 2.&nbsp; Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted for logic in such a way that <math>0 = \mathrm{false}~\!</math> and <math>1 = \mathrm{true}.~\!</math>&nbsp; This has the following consequences:

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| valign="top" | <big>&bull;</big>

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| The inclusive disjunctions indicated for the <math>\nu_k~\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint.

| The inclusive disjunctions indicated for the <math>\nu_k~\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.

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