Source: Notre Dame J. Formal Logic Volume 49, Number 3
(2008), 245-260.
It is argued that the "inner" negation $\mathord{\sim}$ familiar from 3-valued logic can be interpreted as a form of
"conditional" negation: $\mathord{\sim}$ is read '$A$ is false if it has a truth value'. It
is argued that this reading squares well with a particular 3-valued
interpretation of a conditional that in the literature has been seen
as a serious candidate for capturing the truth conditions of the
natural language indicative conditional (e.g., "If Jim went to the
party he had a good time"). It is shown that the logic induced by
the semantics shares many familiar properties with classical
negation, but is orthogonal to both intuitionistic and classical
negation: it differs from both in validating the inference from $A \rightarrow \nega B$ to $\nega(A\rightarrow B)$ to .
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