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In [[logic]] and [[mathematics]], 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 ''k''-ary [[boolean function]] defined in such a way 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 0.
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses. 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.
+
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.
  
: <math>\begin{matrix}
+
<ol start="0">
(\ )      & = & 0 & = & \mbox{false} \\
+
 
(x)       & = & \tilde{x} & = & x' \\
+
<li style="padding:8px">
(x, y)   & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
+
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>
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
+
</li>
 +
 
 +
<li style="padding:8px">
 +
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>
 +
</li>
 +
 
 +
<li style="padding:8px">
 +
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"
 +
|
 +
<p>[[Image:Venn Diagram (P,Q).jpg|500px]]</p>
 +
<p><math>\text{Figure 1.}~~\nu(p, q)~\!</math></p>
 +
|}
 +
</li>
 +
 
 +
<li style="padding:8px">
 +
The venn diagram for <math>\nu(p, q, r)~\!</math> is shown in Figure&nbsp;2.
 +
 
 +
{| align="center" cellpadding="8" style="text-align:center"
 +
|
 +
<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>
 +
 
 +
</li></ol>
 +
 
 +
==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 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.&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.&nbsp; The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.
 +
 
 +
{| align="center" cellpadding="8" style="text-align:center"
 +
|
 +
<math>\begin{matrix}
 +
\texttt{()}
 +
& = & \nu_0
 +
& = & 0
 +
& = & \mathrm{false}
 +
\\[6pt]
 +
\texttt{(x)}
 +
& = & \nu_1 (x)
 +
& = & \tilde{x}
 +
& = & x^\prime
 +
\\[6pt]
 +
\texttt{(x, y)}
 +
& = & \nu_2 (x, y)
 +
& = & \tilde{x}y \lor x\tilde{y}
 +
& = & x^\prime y \lor x y^\prime
 +
\\[6pt]
 +
\texttt{(x, y, z)}
 +
& = & \nu_3 (x, y, z)
 +
& = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z}
 +
& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
 
\end{matrix}</math>
 
\end{matrix}</math>
 +
|}
 +
 +
==Formal definition==
 +
 +
To express the general case of <math>\nu_k~\!</math> in terms of familiar operations, it helps to introduce an intermediary concept:
 +
 +
'''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:
 +
 +
{| align="center" cellpadding="8"
 +
| <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.~\!</math>
 +
|}
 +
 +
Then <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!</math> is defined by the following equation:
 +
 +
{| align="center" cellpadding="8"
 +
| <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).~\!</math>
 +
|}
  
It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint.  However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>.
+
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 minimal negation operator ('''mno''') has a legion of aliases:  ''logical boundary operator'', ''[[limen|limen operator]]'', ''threshold operator'', or ''least action operator'', to name but a few. The rationale for these names is visible in the [[venn diagram]]s of the corresponding operations on [[set]]s.
+
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:
  
==Truth tables==
+
{| align="center" cellpadding="4" width="90%"
 +
| valign="top" | <big>&bull;</big>
 +
| The operation <math>x + y~\!</math> is a function equivalent to the exclusive disjunction of <math>x~\!</math> and <math>y,~\!</math> while its fiber of 1 is the relation of inequality between <math>x~\!</math> and <math>y.~\!</math>
 +
|-
 +
| valign="top" | <big>&bull;</big>
 +
| The operation <math>\textstyle\sum_{j=1}^k x_j~\!</math> maps the bit sequence <math>(x_1, \ldots, x_k)~\!</math> to its ''parity''.
 +
|}
  
Table 1 is a [[truth table]] for the sixteen boolean functions of type f&nbsp;:&nbsp;'''B'''<sup>3</sup>&nbsp;&rarr;&nbsp;'''B''', each of which is either a boundary of a point in '''B'''<sup>3</sup> or the complement of such a boundary.
+
The following properties of the minimal negation operators <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!</math> may be noted:
  
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
+
{| align="center" cellpadding="4" width="90%"
|+ Table 1.  Logical Boundaries and Their Complements
+
| valign="top" | <big>&bull;</big>
| width="20%" | L<sub>1</sub>
+
| The function <math>\texttt{(x, y)}~\!</math> is the same as that associated with the operation <math>x + y~\!</math> and the relation <math>x \ne y.~\!</math>
| width="20%" | L<sub>2</sub>
+
|-
| width="20%" | L<sub>3</sub>
+
| valign="top" | <big>&bull;</big>
| width="20%" | L<sub>4</sub>
+
| In contrast, <math>\texttt{(x, y, z)}~\!</math> is not identical to <math>x + y + z.~\!</math>
 
|-
 
|-
| Decimal
+
| valign="top" | <big>&bull;</big>
| Binary
+
| More generally, the function <math>\nu_k (x_1, \dots, x_k)~\!</math> for <math>k > 2~\!</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.~\!</math>
| Sequential
 
| Parenthetical
 
 
|-
 
|-
 +
| valign="top" | <big>&bull;</big>
 +
| 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.
 +
|}
 +
 +
==Truth tables==
 +
 +
Table&nbsp;3 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.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
 +
|+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}~\!</math>
 +
|- style="background:ghostwhite"
 +
| <math>\mathcal{L}_1~\!</math>
 +
| <math>\mathcal{L}_2~\!</math>
 +
| <math>\mathcal{L}_3~\!</math>
 +
| <math>\mathcal{L}_4~\!</math>
 +
|- style="background:ghostwhite"
 
| &nbsp;
 
| &nbsp;
| align="right" | p =
+
| align="right" | <math>p\colon~\!</math>
| 1 1 1 1 0 0 0 0
+
| <math>1~1~1~1~0~0~0~0~\!</math>
 
| &nbsp;
 
| &nbsp;
|-
+
|- style="background:ghostwhite"
 
| &nbsp;
 
| &nbsp;
| align="right" | q =
+
| align="right" | <math>q\colon~\!</math>
| 1 1 0 0 1 1 0 0
+
| <math>1~1~0~0~1~1~0~0~\!</math>
 
| &nbsp;
 
| &nbsp;
|-
+
|- style="background:ghostwhite"
 
| &nbsp;
 
| &nbsp;
| align="right" | r =
+
| align="right" | <math>r\colon~\!</math>
| 1 0 1 0 1 0 1 0
+
| <math>1~0~1~0~1~0~1~0~\!</math>
 
| &nbsp;
 
| &nbsp;
|}
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:80%"
 
 
|-
 
|-
| width="20%" | f<sub>104</sub>
+
|
| width="20%" | f<sub>01101000</sub>
+
<math>\begin{matrix}
| width="20%" | 0 1 1 0 1 0 0 0
+
f_{104}
| width="20%" | ( p , q , r )
+
\\[4pt]
 +
f_{148}
 +
\\[4pt]
 +
f_{146}
 +
\\[4pt]
 +
f_{97}
 +
\\[4pt]
 +
f_{134}
 +
\\[4pt]
 +
f_{73}
 +
\\[4pt]
 +
f_{41}
 +
\\[4pt]
 +
f_{22}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{01101000}
 +
\\[4pt]
 +
f_{10010100}
 +
\\[4pt]
 +
f_{10010010}
 +
\\[4pt]
 +
f_{01100001}
 +
\\[4pt]
 +
f_{10000110}
 +
\\[4pt]
 +
f_{01001001}
 +
\\[4pt]
 +
f_{00101001}
 +
\\[4pt]
 +
f_{00010110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0~1~0~0~0
 +
\\[4pt]
 +
1~0~0~1~0~1~0~0
 +
\\[4pt]
 +
1~0~0~1~0~0~1~0
 +
\\[4pt]
 +
0~1~1~0~0~0~0~1
 +
\\[4pt]
 +
1~0~0~0~0~1~1~0
 +
\\[4pt]
 +
0~1~0~0~1~0~0~1
 +
\\[4pt]
 +
0~0~1~0~1~0~0~1
 +
\\[4pt]
 +
0~0~0~1~0~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(~p~,~q~,~r~)}
 +
\\[4pt]
 +
\texttt{(~p~,~q~,(r))}
 +
\\[4pt]
 +
\texttt{(~p~,(q),~r~)}
 +
\\[4pt]
 +
\texttt{(~p~,(q),(r))}
 +
\\[4pt]
 +
\texttt{((p),~q~,~r~)}
 +
\\[4pt]
 +
\texttt{((p),~q~,(r))}
 +
\\[4pt]
 +
\texttt{((p),(q),~r~)}
 +
\\[4pt]
 +
\texttt{((p),(q),(r))}
 +
\end{matrix}</math>
 
|-
 
|-
| f<sub>148</sub> || f<sub>10010100</sub> || 1 0 0 1 0 1 0 0 || ( p , q , (r))
+
|
 +
<math>\begin{matrix}
 +
f_{233}
 +
\\[4pt]
 +
f_{214}
 +
\\[4pt]
 +
f_{182}
 +
\\[4pt]
 +
f_{121}
 +
\\[4pt]
 +
f_{158}
 +
\\[4pt]
 +
f_{109}
 +
\\[4pt]
 +
f_{107}
 +
\\[4pt]
 +
f_{151}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{11101001}
 +
\\[4pt]
 +
f_{11010110}
 +
\\[4pt]
 +
f_{10110110}
 +
\\[4pt]
 +
f_{01111001}
 +
\\[4pt]
 +
f_{10011110}
 +
\\[4pt]
 +
f_{01101101}
 +
\\[4pt]
 +
f_{01101011}
 +
\\[4pt]
 +
f_{10010111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~1~1~0~1~0~0~1
 +
\\[4pt]
 +
1~1~0~1~0~1~1~0
 +
\\[4pt]
 +
1~0~1~1~0~1~1~0
 +
\\[4pt]
 +
0~1~1~1~1~0~0~1
 +
\\[4pt]
 +
1~0~0~1~1~1~1~0
 +
\\[4pt]
 +
0~1~1~0~1~1~0~1
 +
\\[4pt]
 +
0~1~1~0~1~0~1~1
 +
\\[4pt]
 +
1~0~0~1~0~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(((p),(q),(r)))}
 +
\\[4pt]
 +
\texttt{(((p),(q),~r~))}
 +
\\[4pt]
 +
\texttt{(((p),~q~,(r)))}
 +
\\[4pt]
 +
\texttt{(((p),~q~,~r~))}
 +
\\[4pt]
 +
\texttt{((~p~,(q),(r)))}
 +
\\[4pt]
 +
\texttt{((~p~,(q),~r~))}
 +
\\[4pt]
 +
\texttt{((~p~,~q~,(r)))}
 +
\\[4pt]
 +
\texttt{((~p~,~q~,~r~))}
 +
\end{matrix}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
==Charts and graphs==
 +
 
 +
This Section focuses on visual representations of minimal negation operators.  A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article.
 +
 
 +
Two ways of visualizing the space <math>\mathbb{B}^k~\!</math> of <math>2^k~\!</math> points are the [[hypercube]] picture and the [[venn diagram]] picture.  The hypercube picture associates each point of <math>\mathbb{B}^k~\!</math> with a unique point of the <math>k~\!</math>-dimensional hypercube.  The venn diagram picture associates each point of <math>\mathbb{B}^k~\!</math> with a unique "cell" of the venn diagram on <math>k~\!</math> "circles".
 +
 
 +
In addition, each point of <math>\mathbb{B}^k~\!</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]~\!</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},~\!</math> and thus it is the unique point where a ''singular conjunction'' of <math>k~\!</math> ''literals'' is <math>1.~\!</math>
 +
 
 +
For example, consider two cases at opposite vertices of the cube:
 +
 
 +
{| align="center" cellpadding="4" width="90%"
 +
| valign="top" | <big>&bull;</big>
 +
| The point <math>(1, 1, \ldots , 1, 1)~\!</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,~\!</math> namely, the point where:
 
|-
 
|-
| f<sub>146</sub> || f<sub>10010010</sub> || 1 0 0 1 0 0 1 0 || ( p , (q), r )
+
| &nbsp;
 +
| align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.~\!</math>
 
|-
 
|-
| f<sub>97</sub> || f<sub>01100001</sub> || 0 1 1 0 0 0 0 1 || ( p , (q), (r))
+
| valign="top" | <big>&bull;</big>
 +
| The point <math>(0, 0, \ldots , 0, 0)~\!</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,~\!</math> namely, the point where:
 
|-
 
|-
| f<sub>134</sub> || f<sub>10000110</sub> || 1 0 0 0 0 1 1 0 || ((p), q , r )
+
| &nbsp;
|-
+
| align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.~\!</math>
| f<sub>73</sub>  || f<sub>01001001</sub> || 0 1 0 0 1 0 0 1 || ((p), q , (r))
 
|-
 
| f<sub>41</sub>  || f<sub>00101001</sub> || 0 0 1 0 1 0 0 1 || ((p), (q), r )
 
|-
 
| f<sub>22</sub> || f<sub>00010110</sub> || 0 0 0 1 0 1 1 0 || ((p), (q), (r))
 
 
|}
 
|}
{|  align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:80%"
+
 
|-
+
To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}~\!</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k~\!</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)~\!</math> is <math>1~\!</math> on the points adjacent to the point where <math>s~\!</math> is <math>1,~\!</math> and 0 everywhere else on the cube.
| width="20%" | f<sub>233</sub>
+
 
| width="20%" | f<sub>11101001</sub>
+
For example, consider the case where <math>k = 3.~\!</math> Then the minimal negation operation <math>\nu (p, q, r)~\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}~\!</math> &mdash; has the following venn diagram:
| width="20%" | 1 1 1 0 1 0 0 1
+
 
| width="20%" | (((p), (q), (r)))
+
{| align="center" cellpadding="8" style="text-align:center"
|-
+
|
| f<sub>214</sub> || f<sub>11010110</sub> || 1 1 0 1 0 1 1 0 || (((p), (q), r ))
+
<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
|-
+
<p><math>\text{Figure 4.}~~\texttt{(p, q, r)}~\!</math></p>
| f<sub>182</sub> || f<sub>10110110</sub> || 1 0 1 1 0 1 1 0 || (((p), q , (r)))
 
|-
 
| f<sub>121</sub> || f<sub>01111001</sub> || 0 1 1 1 1 0 0 1 || (((p), q , r ))
 
|-
 
| f<sub>158</sub> || f<sub>10011110</sub> || 1 0 0 1 1 1 1 0 || (( p , (q), (r)))
 
|-
 
| f<sub>109</sub> || f<sub>01101101</sub> || 0 1 1 0 1 1 0 1 || (( p , (q), r ))
 
|-
 
| f<sub>107</sub> || f<sub>01101011</sub> || 0 1 1 0 1 0 1 1 || (( p , q , (r)))
 
|-
 
| f<sub>151</sub> || f<sub>10010111</sub> || 1 0 0 1 0 1 1 1 || (( p , q , r ))
 
 
|}
 
|}
<br>
 
  
The next section discusses two ways of visualizing the operation of minimal negation operators.  A few bits of terminology will be needed as a language for talking about the pictures, but the formal details are tedious reading, and may be familiar to many readers.  As a result, the full definitions of the terms marked in '''bold''' are relegated to a Glossary at the end of the article.
+
For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}~\!</math> has the following venn diagram:
  
==Charts and graphs==
+
{| align="center" cellpadding="8" style="text-align:center"
 +
|
 +
<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
 +
<p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}~\!</math></p>
 +
|}
  
Two common ways of visualizing the space '''B'''<sup>''k''</sup> of 2<sup>''k''</sup> points are the [[hypercube]] picture and the [[venn diagram]] picture.  Depending on how literally or figuratively one regards these pictures, each point of '''B'''<sup>k</sup> is either identified with or represented by a point of the ''k''-cube and also by a cell of the venn diagram on ''k'' "circles". 
+
==Glossary of basic terms==
  
In addition, each point of '''B'''<sup>k</sup> is the unique point in the '''[[fiber (mathematics)|fiber]] of truth''' <math>[|s|]</math> of a '''singular proposition''' ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;&rarr;&nbsp;'''B''', and thus it is the unique point where a '''singular conjunction''' of ''k'' '''literals''' is 1.
+
; Boolean domain
 +
: A ''[[boolean domain]]'' <math>\mathbb{B}~\!</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},~\!</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \mathrm{false}~\!</math> and <math>1 = \mathrm{true}.~\!</math>
  
For example, consider two cases at opposite vertices of the cube:
+
; Boolean variable
 +
: A ''[[boolean variable]]'' <math>x~\!</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.~\!</math>
  
* The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to 1, namely, the point where:
+
; Proposition
:: <math>x_1\ x_2\ \ldots\ x_{n-1}\ x_n = 1.</math>
+
: In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}~\!</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}~\!</math> is frequently called a ''[[proposition]]''.
  
* The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to 1, namely, the point where:
+
; Basis element, Coordinate projection
:: <math>(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.</math>
+
: Given a sequence of <math>k~\!</math> boolean variables, <math>x_1, \ldots, x_k,~\!</math> each variable <math>x_j~\!</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k~\!</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.~\!</math>
  
To pass from these limiting examples to the general case, observe that a singular proposition ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is 1 on the points adjacent to the point where ''s'' is 1, and 0 everywhere else on the cube.
+
; Basic proposition
 +
: This means that the set of objects <math>\{ x_j : 1 \le j \le k \}~\!</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}~\!</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}~\!</math> propositions over <math>\mathbb{B}^k.~\!</math>
  
For example, consider the case where ''k'' = 3.  Then the minimal negation operation <math>\nu (p, q, r)\!</math>, when there is no risk of confusion written more simply as <math>(p, q, r)\!</math>, has the following venn diagram:
+
; Literal
 +
: A ''literal'' is one of the <math>2k~\!</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},~\!</math> in other words, either a ''posited'' basic proposition <math>x_j~\!</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},~\!</math> for some <math>j = 1 ~\text{to}~ k.~\!</math>
  
<pre>
+
; Fiber
o-------------------------------------------------o
+
: In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y~\!</math> under a function <math>f : X \to Y~\!</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.~\!</math>
|                                                |
 
|                                                |
 
|                o-------------o                |
 
|                /              \               |
 
|              /                \               |
 
|              /                   \             |
 
|            /                    \             |
 
|            o                      o            |
 
|            |          P          |            |
 
|            |                      |            |
 
|            |                      |            |
 
|        o---o---------o  o---------o---o        |
 
|      /     \`````````\ /`````````/    \      |
 
|      /      \`````````o`````````/      \      |
 
|    /        \```````/ \```````/        \    |
 
|    /          \`````/  \`````/          \    |
 
|  o            o---o-----o---o            o  |
 
|  |                |`````|                |  |
 
|  |                |`````|                |  |
 
|  |        Q        |`````|        R        |  |
 
|  o                o`````o                o  |
 
|    \                \```/                /    |
 
|    \                \`/                /    |
 
|      \                o                /      |
 
|      \              / \              /      |
 
|        o-------------o  o-------------o        |
 
|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
Figure 1(p, q, r)
 
</pre>
 
  
For a contrasting example, the boolean function expressed by the form <math>((p),(q),(r))\!</math> has the following venn diagram:
+
: In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},~\!</math> there are just two fibers:
 +
: The fiber of <math>0~\!</math> under <math>f,~\!</math> defined as <math>f^{-1}(0),~\!</math> is the set of points where the value of <math>f~\!</math> is <math>0.~\!</math>
 +
: The fiber of <math>1~\!</math> under <math>f,~\!</math> defined as <math>f^{-1}(1),~\!</math> is the set of points where the value of <math>f~\!</math> is <math>1.~\!</math>
  
<pre>
+
; Fiber of truth
o-------------------------------------------------o
+
: When <math>1~\!</math> is interpreted as the logical value <math>\mathrm{true},~\!</math> then <math>f^{-1}(1)~\!</math> is called the ''fiber of truth'' in the proposition <math>f.~\!</math>  Frequent mention of this fiber makes it useful to have a shorter way of referring to it.  This leads to the definition of the notation <math>[|f|] = f^{-1}(1)~\!</math> for the fiber of truth in the proposition <math>f.~\!</math>
|                                                |
 
|                                                |
 
|                o-------------o                |
 
|                /```````````````\               |
 
|              /`````````````````\               |
 
|              /```````````````````\             |
 
|            /`````````````````````\            |
 
|            o```````````````````````o            |
 
|            |`````````` P ``````````|            |
 
|            |```````````````````````|            |
 
|            |```````````````````````|            |
 
|        o---o---------o```o---------o---o        |
 
|      /`````\         \`/         /`````\       |
 
|      /```````\        o        /```````\      |
 
|    /`````````\      / \      /`````````\    |
 
|    /```````````\    /  \    /```````````\    |
 
|  o`````````````o---o-----o---o`````````````o  |
 
|  |`````````````````|    |`````````````````|  |
 
|  |`````````````````|    |`````````````````|  |
 
|  |``````` Q ```````|    |``````` R ```````|  |
 
|  o`````````````````o    o`````````````````o  |
 
|    \`````````````````\  /`````````````````/    |
 
|    \`````````````````\ /`````````````````/    |
 
|      \`````````````````o`````````````````/      |
 
|      \```````````````/ \```````````````/      |
 
|        o-------------o  o-------------o        |
 
|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
Figure 2.  ((p),(q),(r))
 
</pre>
 
  
==Glossary of basic terms==
+
; Singular boolean function
 +
: A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}~\!</math> is a boolean function whose fiber of <math>1~\!</math> is a single point of <math>\mathbb{B}^k.~\!</math>
  
* A '''[[boolean domain]]''' '''B''' is a generic 2-element [[set]], say, '''B''' = {0, 1}, whose elements are interpreted as [[logical value]]s, usually but not invariably with 0 = ''false'' and ''1'' = ''true''.
+
; Singular proposition
 +
: In the interpretation where <math>1~\!</math> equals <math>\mathrm{true},~\!</math> a singular boolean function is called a ''singular proposition''.
  
* A '''[[boolean variable]]''' ''x'' is a [[variable]] that takes its value from a boolean domain, as ''x''&nbsp;&isin;&nbsp;'''B'''.  
+
: Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.~\!</math>
  
* In situations where [[boolean value]]s are interpreted as [[logical value]]s, a [[boolean-valued function]] ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;'''B''' or a [[boolean function]] ''g''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' is frequently called a '''[[proposition]]'''.
+
; Singular conjunction
 +
: A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}~\!</math> is a conjunction of <math>k~\!</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}~\!</math> for each <math>j = 1 ~\text{to}~ k.~\!</math>
  
* Given a sequence of ''k'' boolean variables, ''x''<sub>1</sub>,&nbsp;…,&nbsp;''x''<sub>''k''</sub>, each variable ''x''<sub>''j''</sub> may be treated either as a [[basis element]] of the space '''B'''<sup>''k''</sup> or as a [[coordinate projection]] ''x''<sub>''j''</sub>&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B'''.
+
: A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}~\!</math> can be expressed as a singular conjunction:
  
* This means that the ''k'' objects ''x''<sub>''j''</sub> for ''j'' = 1 to ''k'' are just so many boolean functions ''x''<sub>''j''</sub>&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' , subject to logical interpretation as a set of ''basic propositions'' that generate the complete set of <math>2^{2^k}</math> propositions over '''B'''<sup>''k''</sup>.
+
{| align="center" cellspacing"10" width="90%"
 +
| height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k~\!</math>,
 +
|-
 +
|
 +
<math>\begin{array}{llll}
 +
\text{where} & e_j & = & x_j
 +
\\[6pt]
 +
\text{or}    & e_j & = & \nu (x_j),
 +
\\[6pt]
 +
\text{for}  & j  & = & 1 ~\text{to}~ k.
 +
\end{array}</math>
 +
|}
  
* A '''literal''' is one of the 2''k'' propositions ''x''<sub>1</sub>,&nbsp;…,&nbsp;''x''<sub>''k''</sub>, (''x''<sub>1</sub>),&nbsp;…,&nbsp;(''x''<sub>''k''</sub>), in other words, either a ''posited'' basic proposition ''x''<sub>''j''</sub> or a ''negated'' basic proposition (''x''<sub>''j''</sub>), for some ''j'' = 1 to ''k''.
+
==Resources==
  
* In mathematics generally, the '''[[fiber (mathematics)|fiber]]''' of a point ''y'' under a function ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' is defined as the inverse image <math>f^{-1}(y)</math>.
+
* [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs]
 +
** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)]
 +
** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index]
 +
** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons]
 +
** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples]
 +
** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas]
  
* In the case of a boolean function ''f''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''', there are just two fibers:
+
==Syllabus==
** The fiber of 0 under ''f'', defined as <math>f^{-1}(0)</math>, is the set of points where ''f'' is 0.
 
** The fiber of 1 under ''f'', defined as <math>f^{-1}(1)</math>, is the set of points where ''f'' is 1.
 
  
* When 1 is interpreted as the logical value ''true'', then <math>f^{-1}(1)</math> is called the '''fiber&nbsp;of&nbsp;truth''' in the proposition ''f''.  Frequent mention of this fiber makes it useful to have a shorter way of referring to it.  This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition ''f''.
+
===Focal nodes===
  
* A '''singular boolean function''' ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' is a boolean function whose fiber of 1 is a single point of '''B'''<sup>''k''</sup>.
+
* [[Inquiry Live]]
 +
* [[Logic Live]]
  
* In the interpretation where 1 equals ''true'', a singular boolean function is called a '''singular proposition'''.
+
===Peer nodes===
  
* Singular boolean functions and singular propositions serve as functional or logical representatives of the points in '''B'''<sup>''k''</sup>.
+
* [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki]
 +
* [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz]
 +
* [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis]
 +
* [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta]
  
* A '''singular conjunction''' in '''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' is a conjunction of ''k'' literals that includes just one conjunct of the pair <math>\{ x_j,\ \nu (x_j) \}</math> for each ''j'' = 1 to ''k''.
+
===Logical operators===
  
* A singular proposition ''s''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;→&nbsp;'''B''' can be expressed as a singular conjunction:
+
{{col-begin}}
<br>
+
{{col-break}}
:{|
+
* [[Exclusive disjunction]]
|
+
* [[Logical conjunction]]
| <math>s\ \ =\ e_1 e_2 \ldots e_{k-1} e_k</math>,
+
* [[Logical disjunction]]
|-
+
* [[Logical equality]]
| where
+
{{col-break}}
| <math>e_j\ =\ x_j\!</math>
+
* [[Logical implication]]
|-
+
* [[Logical NAND]]
| or
+
* [[Logical NNOR]]
| <math>e_j\ =\ \nu (x_j)\!</math>,
+
* [[Logical negation|Negation]]
|-
+
{{col-end}}
| for
 
| <math>j\ \ =\ 1\ \mbox{to}\ k</math>.
 
|}
 
  
==See also==
+
===Related topics===
  
{|
+
{{col-begin}}
| valign=top |
+
{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
* [[Anamnesis]]
+
* [[Boolean domain]]
* [[BACD]]
 
* [[George Boole|Boole, George]]
 
* [[Boolean algebra]]
 
 
* [[Boolean function]]
 
* [[Boolean function]]
* [[Boolean logic]]
 
 
* [[Boolean-valued function]]
 
* [[Boolean-valued function]]
| valign=top |
+
* [[Differential logic]]
* [[Continuous predicate]]
+
{{col-break}}
* [[Differentiable manifold]]
 
* [[Differential topology]]
 
* [[Dynamical system]]
 
* [[Exclusive disjunction]]
 
* [[Gottfried Leibniz|Leibniz, G.W.]]
 
* [[Logical connective]]
 
 
* [[Logical graph]]
 
* [[Logical graph]]
| valign=top |
+
* [[Minimal negation operator]]
* [[Meno]]
 
 
* [[Multigrade operator]]
 
* [[Multigrade operator]]
 
* [[Parametric operator]]
 
* [[Parametric operator]]
* [[Charles Sanders Peirce|Peirce, C.S.]]
+
* [[Peirce's law]]
 +
{{col-break}}
 +
* [[Propositional calculus]]
 
* [[Sole sufficient operator]]
 
* [[Sole sufficient operator]]
 
* [[Truth table]]
 
* [[Truth table]]
* [[Universal algebra]]
+
* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
|}
+
{{col-end}}
 +
 
 +
===Relational concepts===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [[Continuous predicate]]
 +
* [[Hypostatic abstraction]]
 +
* [[Logic of relatives]]
 +
* [[Logical matrix]]
 +
{{col-break}}
 +
* [[Relation (mathematics)|Relation]]
 +
* [[Relation composition]]
 +
* [[Relation construction]]
 +
* [[Relation reduction]]
 +
{{col-break}}
 +
* [[Relation theory]]
 +
* [[Relative term]]
 +
* [[Sign relation]]
 +
* [[Triadic relation]]
 +
{{col-end}}
 +
 
 +
===Information, Inquiry===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [[Inquiry]]
 +
* [[Dynamics of inquiry]]
 +
{{col-break}}
 +
* [[Semeiotic]]
 +
* [[Logic of information]]
 +
{{col-break}}
 +
* [[Descriptive science]]
 +
* [[Normative science]]
 +
{{col-break}}
 +
* [[Pragmatic maxim]]
 +
* [[Truth theory]]
 +
{{col-end}}
 +
 
 +
===Related articles===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 +
{{col-end}}
  
==External links==
+
==Document history==
  
* [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs]
+
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
** [http://atlas.wolfram.com/01/01/ ECARs (Elementary Cellular Automata Rules)]
 
** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index]
 
** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Rule Icons]
 
** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples]
 
** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Boolean Forms]
 
  
{{aficionados}}<sharethis />
+
* [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator], [http://intersci.ss.uci.edu/ InterSciWiki]
 +
* [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz]
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* [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath]
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* [http://wikinfo.org/w/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/w/ Wikinfo]
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* [http://en.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikiversity.org/ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia]
  
 
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Latest revision as of 13:54, 3 September 2017

This page belongs to resource collections on Logic and Inquiry.

A minimal negation operator \((\nu)~\!\) is a logical connective that says “just one false” of its logical arguments.  The first four cases are described below.

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

    \(\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}~\!\)

  3. If \(p~\!\) and \(q~\!\) are the only two arguments then \(\nu(p, q)~\!\) says that exactly one of \(p, q~\!\) is false, so \(\nu(p, q)~\!\) says the same thing as \(p \neq q.~\!\) Expressing \(\nu(p, q)~\!\) in terms of ands \((\cdot),~\!\) ors \((\lor),~\!\) and nots \((\tilde{~})~\!\) gives the following form.

    \(\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}~\!\)

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

    \(\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}~\!\)

    The venn diagram for \(\nu(p, q)~\!\) is shown in Figure 1.

    Venn Diagram (P,Q).jpg

    \(\text{Figure 1.}~~\nu(p, q)~\!\)

  4. The venn diagram for \(\nu(p, q, r)~\!\) is shown in Figure 2.

    Venn Diagram (P,Q,R).jpg

    \(\text{Figure 2.}~~\nu(p, q, r)~\!\)

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

    \(\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}~\!\)

Initial definition

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

In contexts where the initial letter \(\nu~\!\) 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, \(\texttt{(x, y, z)} = \nu (x, y, z).~\!\)

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

\(\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}\)

Formal definition

To express the general case of \(\nu_k~\!\) in terms of familiar operations, it helps to introduce an intermediary concept:

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

\(\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.~\!\)

Then \({\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!\) is defined by the following equation:

\(\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).~\!\)

If we take the boolean product \(x_1 \cdot \ldots \cdot x_k~\!\) or the logical conjunction \(x_1 \land \ldots \land x_k~\!\) to indicate the point \(x = (x_1, \ldots, x_k)~\!\) in the space \(\mathbb{B}^k~\!\) then the minimal negation \(\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!\) indicates the set of points in \(\mathbb{B}^k~\!\) that differ from \(x~\!\) in exactly one coordinate.  This makes \(\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!\) a discrete functional analogue of a point-omitted neighborhood in ordinary real 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.

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

The operation \(x + y~\!\) is a function equivalent to the exclusive disjunction of \(x~\!\) and \(y,~\!\) while its fiber of 1 is the relation of inequality between \(x~\!\) and \(y.~\!\)
The operation \(\textstyle\sum_{j=1}^k x_j~\!\) maps the bit sequence \((x_1, \ldots, x_k)~\!\) to its parity.

The following properties of the minimal negation operators \({\nu_k : \mathbb{B}^k \to \mathbb{B}}~\!\) may be noted:

The function \(\texttt{(x, y)}~\!\) is the same as that associated with the operation \(x + y~\!\) and the relation \(x \ne y.~\!\)
In contrast, \(\texttt{(x, y, z)}~\!\) is not identical to \(x + y + z.~\!\)
More generally, the function \(\nu_k (x_1, \dots, x_k)~\!\) for \(k > 2~\!\) is not identical to the boolean sum \(\textstyle\sum_{j=1}^k x_j.~\!\)
The inclusive disjunctions indicated for the \(\nu_k~\!\) of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.

Truth tables

Table 3 is a truth table for the sixteen boolean functions of type \(f : \mathbb{B}^3 \to \mathbb{B}~\!\) whose fibers of 1 are either the boundaries of points in \(\mathbb{B}^3~\!\) or the complements of those boundaries.


\(\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}~\!\)
\(\mathcal{L}_1~\!\) \(\mathcal{L}_2~\!\) \(\mathcal{L}_3~\!\) \(\mathcal{L}_4~\!\)
  \(p\colon~\!\) \(1~1~1~1~0~0~0~0~\!\)  
  \(q\colon~\!\) \(1~1~0~0~1~1~0~0~\!\)  
  \(r\colon~\!\) \(1~0~1~0~1~0~1~0~\!\)  

\(\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}\)

\(\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}\)

\(\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}\)

\(\begin{matrix} \texttt{(~p~,~q~,~r~)} \\[4pt] \texttt{(~p~,~q~,(r))} \\[4pt] \texttt{(~p~,(q),~r~)} \\[4pt] \texttt{(~p~,(q),(r))} \\[4pt] \texttt{((p),~q~,~r~)} \\[4pt] \texttt{((p),~q~,(r))} \\[4pt] \texttt{((p),(q),~r~)} \\[4pt] \texttt{((p),(q),(r))} \end{matrix}\)

\(\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}\)

\(\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}\)

\(\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}\)

\(\begin{matrix} \texttt{(((p),(q),(r)))} \\[4pt] \texttt{(((p),(q),~r~))} \\[4pt] \texttt{(((p),~q~,(r)))} \\[4pt] \texttt{(((p),~q~,~r~))} \\[4pt] \texttt{((~p~,(q),(r)))} \\[4pt] \texttt{((~p~,(q),~r~))} \\[4pt] \texttt{((~p~,~q~,(r)))} \\[4pt] \texttt{((~p~,~q~,~r~))} \end{matrix}\)


Charts and graphs

This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in italics are relegated to a Glossary at the end of the article.

Two ways of visualizing the space \(\mathbb{B}^k~\!\) of \(2^k~\!\) points are the hypercube picture and the venn diagram picture. The hypercube picture associates each point of \(\mathbb{B}^k~\!\) with a unique point of the \(k~\!\)-dimensional hypercube. The venn diagram picture associates each point of \(\mathbb{B}^k~\!\) with a unique "cell" of the venn diagram on \(k~\!\) "circles".

In addition, each point of \(\mathbb{B}^k~\!\) is the unique point in the fiber of truth \([|s|]~\!\) of a singular proposition \(s : \mathbb{B}^k \to \mathbb{B},~\!\) and thus it is the unique point where a singular conjunction of \(k~\!\) literals is \(1.~\!\)

For example, consider two cases at opposite vertices of the cube:

The point \((1, 1, \ldots , 1, 1)~\!\) with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to \(1,~\!\) namely, the point where:
  \(x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.~\!\)
The point \((0, 0, \ldots , 0, 0)~\!\) with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to \(1,~\!\) namely, the point where:
  \(\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.~\!\)

To pass from these limiting examples to the general case, observe that a singular proposition \(s : \mathbb{B}^k \to \mathbb{B}~\!\) can be given canonical expression as a conjunction of literals, \(s = e_1 e_2 \ldots e_{k-1} e_k~\!\). Then the proposition \(\nu (e_1, e_2, \ldots, e_{k-1}, e_k)~\!\) is \(1~\!\) on the points adjacent to the point where \(s~\!\) is \(1,~\!\) and 0 everywhere else on the cube.

For example, consider the case where \(k = 3.~\!\) Then the minimal negation operation \(\nu (p, q, r)~\!\) — written more simply as \(\texttt{(p, q, r)}~\!\) — has the following venn diagram:

Venn Diagram (P,Q,R).jpg

\(\text{Figure 4.}~~\texttt{(p, q, r)}~\!\)

For a contrasting example, the boolean function expressed by the form \(\texttt{((p),(q),(r))}~\!\) has the following venn diagram:

Venn Diagram ((P),(Q),(R)).jpg

\(\text{Figure 5.}~~\texttt{((p),(q),(r))}~\!\)

Glossary of basic terms

Boolean domain
A boolean domain \(\mathbb{B}~\!\) is a generic 2-element set, for example, \(\mathbb{B} = \{ 0, 1 \},~\!\) whose elements are interpreted as logical values, usually but not invariably with \(0 = \mathrm{false}~\!\) and \(1 = \mathrm{true}.~\!\)
Boolean variable
A boolean variable \(x~\!\) is a variable that takes its value from a boolean domain, as \(x \in \mathbb{B}.~\!\)
Proposition
In situations where boolean values are interpreted as logical values, a boolean-valued function \(f : X \to \mathbb{B}~\!\) or a boolean function \(g : \mathbb{B}^k \to \mathbb{B}~\!\) is frequently called a proposition.
Basis element, Coordinate projection
Given a sequence of \(k~\!\) boolean variables, \(x_1, \ldots, x_k,~\!\) each variable \(x_j~\!\) may be treated either as a basis element of the space \(\mathbb{B}^k~\!\) or as a coordinate projection \(x_j : \mathbb{B}^k \to \mathbb{B}.~\!\)
Basic proposition
This means that the set of objects \(\{ x_j : 1 \le j \le k \}~\!\) is a set of boolean functions \(\{ x_j : \mathbb{B}^k \to \mathbb{B} \}~\!\) subject to logical interpretation as a set of basic propositions that collectively generate the complete set of \(2^{2^k}~\!\) propositions over \(\mathbb{B}^k.~\!\)
Literal
A literal is one of the \(2k~\!\) propositions \(x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},~\!\) in other words, either a posited basic proposition \(x_j~\!\) or a negated basic proposition \(\texttt{(} x_j \texttt{)},~\!\) for some \(j = 1 ~\text{to}~ k.~\!\)
Fiber
In mathematics generally, the fiber of a point \(y \in Y~\!\) under a function \(f : X \to Y~\!\) is defined as the inverse image \(f^{-1}(y) \subseteq X.~\!\)
In the case of a boolean function \(f : \mathbb{B}^k \to \mathbb{B},~\!\) there are just two fibers:
The fiber of \(0~\!\) under \(f,~\!\) defined as \(f^{-1}(0),~\!\) is the set of points where the value of \(f~\!\) is \(0.~\!\)
The fiber of \(1~\!\) under \(f,~\!\) defined as \(f^{-1}(1),~\!\) is the set of points where the value of \(f~\!\) is \(1.~\!\)
Fiber of truth
When \(1~\!\) is interpreted as the logical value \(\mathrm{true},~\!\) then \(f^{-1}(1)~\!\) is called the fiber of truth in the proposition \(f.~\!\) Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation \([|f|] = f^{-1}(1)~\!\) for the fiber of truth in the proposition \(f.~\!\)
Singular boolean function
A singular boolean function \(s : \mathbb{B}^k \to \mathbb{B}~\!\) is a boolean function whose fiber of \(1~\!\) is a single point of \(\mathbb{B}^k.~\!\)
Singular proposition
In the interpretation where \(1~\!\) equals \(\mathrm{true},~\!\) a singular boolean function is called a singular proposition.
Singular boolean functions and singular propositions serve as functional or logical representatives of the points in \(\mathbb{B}^k.~\!\)
Singular conjunction
A singular conjunction in \(\mathbb{B}^k \to \mathbb{B}~\!\) is a conjunction of \(k~\!\) literals that includes just one conjunct of the pair \(\{ x_j, ~\nu(x_j) \}~\!\) for each \(j = 1 ~\text{to}~ k.~\!\)
A singular proposition \(s : \mathbb{B}^k \to \mathbb{B}~\!\) can be expressed as a singular conjunction:
\(s ~=~ e_1 e_2 \ldots e_{k-1} e_k~\!\),

\(\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}\)

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