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A Cut-free Sequent Calculus for Bi-Intuitionistic Logic: Extended Version
Linda Buisman,Rajeev Goré +1 more
TLDR
This work presents a new cut-free sequent calculus for BiInt, and proves it sound and complete with respect to its Kripke semantics.Abstract:
Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ``cut-free'' sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose.read more
Citations
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Symmetric and dual
TL;DR: Two new first-order paraconsistent logics with De Morgan- type negations and co-implication, called symmetricParaconsistent logic (SPL) and dual paraconsistant logic (DPL), are introduced as Gentzen- type sequent calculi.
Journal ArticleDOI
Foundations for an Abstract Proof Theory in the Context of Horn Rules
TL;DR: In this paper , the authors introduce a logic-independent framework for the study of sequent-style proof systems, which covers a number of proof-theoretic formalisms and concrete proof systems that appear in the literature.
References
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Tableau Methods for Modal and Temporal Logics
TL;DR: This chapter gives a systematic and unified introduction to tableau methods for automating deduction in modal and temporal logics and focuses on the propositional fragments restricted to a two-valued (classical) basis.
Related Papers (5)
Proof Search and Counter-Model Construction for Bi-intuitionistic Propositional Logic with Labelled Sequents
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