## Minimal Negation Operators • 4

Defining minimal negation operators over a more conventional basis is next in order of logic, 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 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 $\texttt{(} x \texttt{,} y \texttt{)}$ 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.