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 is a logical connective which says “just one false” of its logical arguments. The first four cases are described below.
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If the list of arguments is empty, as expressed in the form
then it cannot be true exactly one of the arguments is false, so
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If
is the only argument then
says
is false, so
expresses the negation of the proposition
Written in several common notations we have the following equivalent expressions.
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If
and
are the only two arguments then
says exactly one of
is false, so
says the same thing as
Expressing
in terms of ands
ors
and nots
gives the following form.
It is permissible to omit the dot
in contexts where it is understood, giving the following form.
The venn diagram for
is shown in Figure 1.
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The venn diagram for
is shown in Figure 2.
The center cell is the region where all three arguments
hold true, so
holds true in just the three neighboring cells. In other words:
Resources
cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Minimal Negation Operators • Laws of Form
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