Minimal Negation Operators

Note. I’m including a more detailed definition of minimal negation operators in terms of conventional logical operations largely because readers of particular tastes have asked for it in the past. But it can easily be skipped until one has a felt need for it. Skimmed lightly, though, it can serve to illustrate a major theme in logic and mathematics, namely, the Relativity of Complexity or the Relativity of Primitivity to the basis we have chosen for constructing our conceptual superstructures.

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Defining minimal negation operators over a more conventional basis is next in order of exposition, if not necessarily in order of every reader’s reading. For what it’s worth and against the day when it may be needed, here is a definition of minimal negations in terms of $\land,$ $\lor,$ and $\lnot.$

Formal Definition

To express the general form 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.

$\begin{matrix} \lnot_j (x_1, \ldots, x_j, \ldots, x_k) & = & x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k. \end{matrix}$

Then ${\nu_k : \mathbb{B}^k \to \mathbb{B}}$ is defined by the following equation.

$\begin{matrix} \nu_k (x_1, \ldots, x_k) & = & \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k). \end{matrix}$

We may 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 $\mathbb{B}^k,$ in which case the minimal negation $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ indicates the set of points in $\mathbb{B}^k$ which 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 precisely, a point-omitted distance-one neighborhood. Viewed in that light the minimal negation operator can be recognized as a differential construction, an observation opening a very wide field.

The remainder of this discussion proceeds on the algebraic convention making 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 $\texttt{(} x \texttt{,} y \texttt{)}$ is the same as that associated with the operation $x + y$ and the relation $x \ne y.$
In contrast, $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ 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.

Resources

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