The logic of conditional negation
This paper presents a form of conditional negation in the context of familiar 3-valued logic, whereby \(\sim\! A\) is read `\(A\) is false if it has a truth-value'. This is supposed to coordinate well with a 3-valued conditional that captures many aspects of the ordinary-language indicative if-then. The logic of this negation is similar to classical negation, but different from that and from intuitionistic negation in that the inference from \(A\to\,\sim\! B\) to \(\sim\!(A\to B)\) is valid. This logic is presented both semantically, by standard matrices, and proof-theoretically, and the latter shown to be sound and complete with respect to the former. Key to these results is to take validity not in the sense of preserving truth, but as not introducing falsity, so that an inference is valid just in case the conclusion is false only if at least one premise is false. The paper includes valuable informal discussion to motivate the various concepts.
scientific article; zbMATH DE number 5353772