Visser frames for sublogics of $\mathbf{IL}$

TL;DR

This paper investigates the modal completeness and finite frame property of various sublogics of interpretability logic, establishing new results about Visser frames and solving a longstanding problem regarding the logic of conservativity.

Contribution

It proves that the logic of conservativity has the finite frame property with respect to Visser frames, addressing Ignatiev's problem.

Findings

Proves finite frame property for the logic of conservativity.

Establishes modal completeness for several sublogics of IL.

Provides new insights into Visser frames and their relation to interpretability logic.

Abstract

We study the modal completeness and the finite frame property of several sublogics of the logic $\mathbf{IL}$ of interpretability with respect to Visser frames, which are also called simplified Veltman frames. Among other things, we prove that the logic $\mathbf{CL}$ of conservativity has the finite frame property with respect to that frames. This is an affirmative solution to Ignatiev's problem.