Minimal Negation Operators • 1

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.

  1. 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}.
  2. 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}

  3. 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.

    ν(p, q)

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

  4. The venn diagram for \nu(p, q, r) is shown in Figure 2.

    ν(p, q, r)

    \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}


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