The Fixed Point Property in Modal Logic

Abstract

This paper deals with the modal logics associated with (possibly nonstandard) provability predicates of Peano Arithmetic. One of our goals is to present some modal systems having the fixed point property and not extending the Gödel-Löb system GL. We prove that, for every $ n\geq 2, K+\square(\square^{n-1}p\rightarrow p)\rightarrow\square p$ has the explicit fixed point property. Our main result states that every complete modal logic L having the Craig's interpolation property and such that $ L\vdash\Delta(\nabla(p)\rightarrow p)\rightarrow\Delta(p)$, where $ \nabla(p)$ and $ \Delta(p)$ are suitable modal formulas, has the explicit fixed point property.