Generic Modal Cut Elimination Applied to Conditional Logics

TL;DR

This paper introduces a general criterion for cut elimination in sequent calculi for propositional modal and conditional logics, simplifying proofs and extending applicability beyond normal modal logic.

Contribution

It provides a unified framework for cut elimination applicable to a wide range of modal and conditional logics, including new internalized calculi with simpler proofs.

Findings

Successfully applied to various conditional logics

Achieved simpler, more efficient proof systems

Resolved open problems in cut elimination for certain logics

Abstract

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature.