Effectively Constructible Fixed Points in Sacchetti’s Modal Logics of Provability

Abstract

We give a purely syntactical proof of the fixed point theorem for Sacchetti’s modal logics $\mathbf{K}+\square {(\square }^{n}p\to p)\to \square p$ ($n\ge 2$) of provability. From our proof, an effective procedure for constructing fixed points in these logics is obtained. We also show the existence of simple fixed points for particular modal formulas.