## Minimal Negation Operators • 3

It will take a few more rounds of stage-setting before I can get to concrete examples of applications but the following should indicate the direction of generalization embodied in minimal negation operators.

To begin, let’s observe two different ways of generalizing the operation of exclusive disjunction (XOR) or symmetric difference.

Let $\mathbb{B}$ = the boolean domain $\{ 0, 1 \}.$

1. XOR or symmetric difference, sometimes indicated by a delta or small triangle $(\vartriangle),$ is a boolean function $\vartriangle : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ identical to the field addition $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$  This is also known as addition mod 2 or GF(2) addition.

Generalizing $p + q$ in that sense would continue the sequence as $p\!+\!q\!+\!r,$  $p\!+\!q\!+\!r\!+\!s,$  $p\!+\!q\!+\!r\!+\!s\!+\!t,$  and so on.  These are known as parity sums, returning $0$ if there are an even number of $1$’s in the sum, returning $1$ if there are an odd number of $1$’s in the sum.

2. The equivalent expressions $\texttt{(} p \texttt{,} q \texttt{)} = \nu(p, q) = p + q = p \vartriangle q$ can also be read with a different connotation, indicating the “next-door-neighbors” or venn diagram cells adjacent to the conjunction $p \land q.$  Generalizing $\texttt{(} p \texttt{,} q \texttt{)}$ in that direction would continue the sequence as $\nu(p, q, r),$  $\nu(p, q, r, s),$  $\nu(p, q, r, s, t),$  and so on.  That sequence of operators differs from the sequence of parity sums once it passes the 2-variable case.

The triple sum can be written in terms of 2-place minimal negations as follows:

$p + q + r ~=~ \texttt{((} p \texttt{,} q \texttt{),} r \texttt{)} ~=~ \texttt{(} p \texttt{,} \texttt{(} q \texttt{,} r \texttt{))}$

It is important to note that these expressions are not equivalent to the 3-place minimal negation $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}.$

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