A Note on Freedom from Detachment in the Logic of Paradox

Abstract

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\phi$ and $\left(\phi \odot \psi \right)$ also satisfies $\psi$, except trivially. We derive this as a corollary of a more general result concerning variable sharing.