The Logic of Conditional Negation

Abstract

It is argued that the "inner" negation $\sim$ familiar from 3-valued logic can be interpreted as a form of "conditional" negation: $\sim A$ 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 $A\to \sim B$ to $\sim \left(A\to B\right)$.