Nested Sequents for the Logic of Conditional Belief

Abstract

The logic of conditional belief, called Conditional Doxastic Logic (\(\mathsf {CDL}\)), was proposed by Board, Baltag and Smets to model revisable belief and knowledge in a multi-agent setting. We present a proof system for \(\mathsf {CDL}\) in the form of a nested sequent calculus. To the best of our knowledge, ours is the first internal and standard calculus for this logic. We take as primitive a multi-agent version of the “comparative plausibility operator”, as in Lewis’ counterfactual logic. The calculus is analytic and provides a decision procedure for \(\mathsf {CDL}\). As a by-product we also obtain a nested sequent calculus for multi-agent modal logic \(\mathsf {S5}_i\).