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<font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
<font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
A '''minimal negation operator''' (Mno) is a logical connective that says “just one false” of its logical arguments.
If the list of arguments is empty, as expressed in the form Mno(), then it cannot be true that exactly one of the arguments is false, so Mno() = False.
If p is the only argument, then Mno(p) says that p is false, so Mno(p) = Not(p).
If p and q are the only two arguments, then Mno(p, q) says that exactly one of p, q is false, so Mno(p, q) says the same thing as p ≠ q.
The venn diagram for Mno(p, q, r) is shown in Figure 1.
{| 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 1.}~~\texttt{(p, q, r)}</math></p>
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The center cell is the region where all three arguments p, q, r hold true, so Mno(p, q, r) holds true in just the three neighboring cells. In other words, Mno(p, q, r) = ¬p q r ∨ p ¬q r ∨ p q ¬r.
==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 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>
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==Definition==
==Formal definition==
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:
==Truth tables==
==Truth tables==
Table 1 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries.
Table 2 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries.
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
|+ <math>\text{Table 2.}~~\text{Logical Boundaries and Their Complements}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| <math>\mathcal{L}_1</math>
| <math>\mathcal{L}_1</math>
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<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
<p><math>\text{Figure 2.}~~\texttt{(p, q, r)}</math></p>
<p><math>\text{Figure 3.}~~\texttt{(p, q, r)}</math></p>
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<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
<p><math>\text{Figure 3.}~~\texttt{((p),(q),(r))}</math></p>
<p><math>\text{Figure 4.}~~\texttt{((p),(q),(r))}</math></p>
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