Topological semantics of conservativity and interpretability logics

TL;DR

This paper develops a topological semantics framework for conservativity and interpretability logics, proving compactness and strong completeness results for various extensions of the conservativity logic.

Contribution

It introduces a novel topological semantics for these logics and extends Shehtman's ultrabouquet method to establish key completeness theorems.

Findings

Proves topological compactness theorem for conservative logic extensions

Shows strong completeness of several interpretability logic extensions

Extends Shehtman's ultrabouquet construction to new framework

Abstract

We introduce and develop a topological semantics of conservativity logics and interpretability logics. We prove the topological compactness theorem of consistent normal extensions of the conservativity logic $\mathbf{CL}$ by extending Shehtman's ultrabouquet construction method to our framework. As a consequence, we prove that several extensions of $\mathbf{CL}$ such as $\mathbf{IL}$, $\mathbf{ILM}$, $\mathbf{ILP}$ and $\mathbf{ILW}$ are strongly complete with respect to our topological semantics.