Definition:Minimal Negation Operator

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 in such a way 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. 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.

$$\begin{matrix} (~)      & = & 0 & = & \mbox{false} \\ (x)      & = & \tilde{x} & = & x' \\ (x, y)   & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}$$

It may also be noted that $$(x, y)\!$$ is the same function as $$x + y\!$$ and $$x \ne y$$, and that the inclusive disjunctions indicated for $$(x, y)\!$$ and for $$(x, y, z)\!$$ may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function $$(x, y, z)\!$$ is not the same thing as the function $$x + y + z\!$$.

The minimal negation operator (mno) has a legion of aliases: logical boundary operator, limen operator, threshold operator, or least action operator, to name but a few. The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.

Truth tables
Table 1 is a truth table for the sixteen boolean functions of type $$f : \mathbb{B}^3 \to \mathbb{B},$$ each of which is either a boundary of a point in $$\mathbb{B}^3$$ or the complement of such a boundary.

The next section discusses two ways of visualizing the operation 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 necessary definitions are relegated to a Glossary at the end of the article.

Charts and graphs
Two common ways of visualizing the space $$\mathbb{B}^k$$ of $$2^k\!$$ points are the hypercube picture and the venn diagram picture. Depending on how literally or figuratively one regards these pictures, each point of $$\mathbb{B}^k$$ 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".

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:
 * $$(x_1)(x_2)\ldots(x_{n-1})(x_n) = 1.$$

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 $$\nu (p, q, r),\!$$ when there is no risk of confusion written more simply as $$(p, q, r),\!$$ has the following venn diagram:

o-o o-o Figure 1. (p, q, r)
 * 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-o  o-o---o        |
 * /      \`````````o`````````/       \      |
 * o            o---o-o---o             o   |
 * |       Q        |`````|        R        |   |
 * o                o`````o                 o   |
 * \                o                 /      |
 * o-o  o-o        |
 * |       Q        |`````|        R        |   |
 * o                o`````o                 o   |
 * \                o                 /      |
 * o-o  o-o        |
 * \                o                 /      |
 * o-o  o-o        |
 * \                o                 /      |
 * o-o  o-o        |
 * o-o  o-o        |

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

o-o o-o Figure 2. ((p),(q),(r))
 * 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-o```o-o---o       |
 * /```````\        o         /```````\      |
 * o`````````````o---o-o---o`````````````o  |
 * |``````` Q ```````|    |``````` R ```````|   |
 * o`````````````````o    o`````````````````o   |
 * \`````````````````o`````````````````/     |
 * o-o  o-o        |
 * |``````` Q ```````|    |``````` R ```````|   |
 * o`````````````````o    o`````````````````o   |
 * \`````````````````o`````````````````/     |
 * o-o  o-o        |
 * \`````````````````o`````````````````/     |
 * o-o  o-o        |
 * \`````````````````o`````````````````/     |
 * o-o  o-o        |
 * o-o  o-o        |

Glossary of basic terms

 * Boolean Domain
 * A boolean domain $$\mathbb{B}$$ is a generic 2-element set, say, $$\mathbb{B} = \{ 0, 1 \},$$ whose elements are interpreted as logical values, usually, but not invariably, with $$0 = \operatorname{false}$$ and $$1 = \operatorname{true}.$$


 * 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 viewed 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
 * A basic proposition is one of the above coordinate projections, $$x_j : \mathbb{B}^k \to \mathbb{B},$$ viewed in the light of a logical interpretation, typically, $$0 = \operatorname{false}$$ and $$1 = \operatorname{true}.$$ The set of $$k\!$$ basic propositions generates the full set of $$2^{2^k}$$ propositions over $$\mathbb{B}^k.$$


 * Literal
 * A literal is one of the $$2k\!$$ propositions $$x_1, \ldots, x_k, (x_1), \ldots, (x_k),$$ in other words, either a posited basic proposition $$x_j\!$$ or a negated basic proposition $$(x_j),\!$$ for some $$j = 1 \mbox{ to } k.\!$$


 * 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 $$f^{-1}(y) \in X.$$


 * Boolean Fiber
 * In the case of a boolean-valued function $$f : X \to \mathbb{B},$$ there are just two fibers:
 * The fiber of 0 under $$f,\!$$ defined as $$f^{-1}(0),\!$$ is the set of points in $$X\!$$ where $$f\!$$ is 0.
 * The fiber of 1 under $$f,\!$$ defined as $$f^{-1}(1),\!$$ is the set of points in $$X\!$$ where $$f\!$$ is 1.


 * Logical Fiber, Fiber of Truth
 * When 1 is interpreted as the logical value 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
 * When the boolean domain $$\mathbb{B} = \{ 0, 1 \}$$ is given a logical interpretation, 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 the set of propositions of type $$\mathbb{B}^k \to \mathbb{B}$$ is a conjunction of $$k\!$$ literals that includes just one conjunct of the pair $$\{ x_j,\ \nu (x_j) \}$$ for each $$j = 1 \mbox{ to } k.\!$$


 * A singular proposition $$s : \mathbb{B}^k \to \mathbb{B}$$ can be expressed as a singular conjunction:


 * $$\begin{array}{lccl}

& s  & = & e_1 e_2 \ldots e_{k-1} e_k \\ \mbox{where} & e_j & = & x_j                       \\ \mbox{or}   & e_j & = & \nu (x_j)                  \\ \mbox{for}  & j   & = & 1 \mbox{ to } k.           \\ \end{array}$$