Minimal Negation Operators

To accommodate moderate levels of complexity in the application of logical graphs to practical problems our Organon requires a class of organules called “minimal negation operators”. I outlined the history of their early development from Peirce’s alpha graphs for propositional calculus in a previous series of posts. The next order of business is to sketch their properties in a systematic fashion and to illustrate their uses. As it turns out, taking minimal negations as primitive operators enables efficient expressions for many natural constructs and affords a bridge between boolean domains of two values and domains with finite numbers of values, for example, finite sets of individuals.

Brief Introduction

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

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

$\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}$
If $p$ and $q$ are the only two arguments then $\nu(p, q)$ says 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.

$\text{Figure 1.} ~~ \nu(p, q)$
The venn diagram for $\nu(p, q, r)$ is shown in Figure 2.

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

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:

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

Resources

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