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MyWikiBiz, Author Your Legacy — Tuesday April 15, 2025
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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"
+|
+<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
+<p><math>\text{Figure 1.}~~\texttt{(p, q, r)}</math></p>
+|}
+
+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:
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==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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|}
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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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|}
