An $\omega$-Rule for the Logic of Provability and Its Models

Abstract

In this paper, we discuss semantical properties of the logic $\textbf{GL}$ of provability. The logic $\textbf{GL}$ is a normal modal logic which is axiomatized by the the Löb formula $\Box (\Box p\supset p)\supset \Box p$, but it is known that $\textbf{GL}$ can also be axiomatized by an axiom $\Box p\supset \Box \Box p$ and an $\omega$-rule $(\Diamond ^{*})$ which takes countably many premises $\phi \supset \Diamond ^{n}\top$ $(n\in \omega )$ and returns a conclusion $\phi \supset \bot$. We show that the class of transitive Kripke frames which validates $(\Diamond ^{*})$ and the class of transitive Kripke frames which strongly validates $(\Diamond ^{*})$ are equal, and that the following three classes of transitive Kripke frames, the class which validates $(\Diamond ^{*})$, the class which weakly validates $(\Diamond ^{*})$, and the class which is defined by the Löb formula, are mutually different, while all of them characterize $\textbf{GL}$. This gives an example of a proof system P and a class C of Kripke frames such that P is sound and complete with respect to C but the soundness cannot be proved by simple induction on the height of the derivations in P. We also show Kripke completeness of the proof system with $(\Diamond ^{*})$ in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations $\Box x\le \Box \Box x$ and $\bigwedge _{n\in \omega }\Diamond ^{n}1=0$ is not a variety.