In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes1. In this paper I show that a number of logics are susceptible to a strengthened version of Curry’s paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in Łukasiewicz’s continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature.
Contractionless logic ω-Inconsistency Łukasiewicz logic Curry’s paradox Naïve truth theory