## Minimal Negation Operators • 1

To accommodate moderate levels of complexity in the application of logical graphs our organon needs a class of organules called “minimal negation operators”.

### Brief Introduction

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

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

$\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}$

3. If $p$ and $q$ are the only two arguments then $\nu(p, q)$ says that 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.

$\text{Figure 1.}~~\nu(p, q)$

4. 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:

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