The Logic of Paradox as a substructural logic

Abstract: Graham Priest’s simple three-valued logic LP has many curious properties. It has the same valid formulas as classical logic, but differs from classical logic when it comes to valid sequents. The valid sequents do not uniquely characterise the logic: it is possible to have more than one different LP-“negation”, each of which satisfies all the LP-requirements, without being equivalent. (The situation is not unlike modal operators in your favoured modal logic. A modal logic like S5 does not uniquely determine the meaning of the modal operators. The same goes for LP when characterised in terms of its consequence relation.) One consequence of this fact is that extant proof-first characterisations of LP are unwieldy. In some systems, connectives are given rules featuring negation and rules without; in others, formulas occur positively or negatively signed, and in others, sequents have three positions in which formulas can occur instead of two. This makes relating LP to familiar…