Nested Sequents for the Logic of Conditional Belief

Abstract

The logic of conditional belief, called Conditional Doxastic Logic (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 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 CDL. As a by-product we also obtain a nested sequent calculus for multi-agent modal logic S5i.

Counterfactual Assumptions and Counterfactual Implications

“…However, the internal systems mentioned above make use of structural operators and specific rules which directly imitate model-theoretic structures involved in the semantics. (Translations of internal into labelled systems and back are considered in Girlando, 2019;Girlando, Negri, and Olivetti, 2018. ) From a foundational point of view-or seeing proof-theoretic semantics as an "alternative to truth-condition semantics" (Schroeder-Heister, 2018, p. 1)-neither an internalization of model-theoretic truth conditions nor an imitation of model-theoretic structures seems to be appealing.…”

Modal Intuitionistic Logics as Dialgebraic Logics

“…The crux is that we want to determine the action of s on a A uniquely. Additional axioms of the logic, for example inspired by [29][30][31][32], may require a more involved definition. 9.7.…”