Nested sequent calculi for normal conditional logics

Abstract

Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)-modal logic to obtain analytic and modular proof systems for these logics. In this work, we extend the realm of nested sequents by providing nested sequent calculi for the basic conditional logic CK and some of its significant extensions. We provide also a calculus for Kraus Lehman Magidor cumulative logic C. The calculi are internal (a sequent can be directly translated into a formula), cut-free and analytic. Moreover, they can be used to design (sometimes optimal) decision procedures for the respective logics, and to obtain complexity upper bounds. Our calculi are an argument in favour of nested sequent calculi for modal logics and alike, showing their versatility and power.

Nested sequent calculi and theorem proving for normal conditional logics: The theorem prover NESCOND

“…Nested sequent calculi NS are sound and complete with respect to the semantics for the respective logics, a proof can be found in Section 3 in [2].…”

“…In detail, we have the following results, whose proofs can be found in [2]. For systems with (CEM ), we need a more complicated mechanism.…”

“…In [2] it is shown that: are sound, complete and terminating. The terminating calculi can be used to provide a PSPACE decision procedure for their respective logics [2].…”

“…In this work we first describe nested sequent calculi NS for propositional conditional logics, recently introduced in [1,2]. Nested sequent calculi [9,12,21,24] are a natural generalization of ordinary sequent calculi where sequents are allowed to occur within sequents.…”

Standard Sequent Calculi for Lewis’ Logics of Counterfactuals

“…Calculi for some weaker conditional logics are given, e.g., in [1,18] and more recently in [19,15]. Regarding Lewis' counterfactual logics, external labelled calculi have been proposed in [9] and in [16], both based on a relational reformulation of the sphere semantics.…”