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 is a logical connective that 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 that exactly one of the arguments is false, so
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If
is the only argument then
says that
is false, so
expresses the logical negation of the proposition
Written in several different notations, we have the following equivalent expressions.
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If
and
are the only two arguments then
says that 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
- Minimal Negation Operator • OEIS Wiki • Wikiversity
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