## Combining Derivations and Refutations for Cut-free Completeness in Bi-Intuitionistic Logic (2008)

Citations: | 7 - 0 self |

### BibTeX

@MISC{Goré08combiningderivations,

author = {Rajeev Goré and Linda Postniece},

title = {Combining Derivations and Refutations for Cut-free Completeness in Bi-Intuitionistic Logic},

year = {2008}

}

### OpenURL

### Abstract

Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cut-free ” sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1

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Citation Context ...gh they clearly have some proof/refutation search content. If we were interested only in decision procedures, we could obtain a decision procedure for BiInt by embedding it into the tense logic Kt.S4 =-=[34]-=-, and using tableaux for description logics with inverse roles [19]. However, an embedding into Kt.S4 provides no proof-theoretic insights into BiInt itself. Moreover, the restart technique of Horrock... |

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Citation Context ...istic logic (e.g. [31, 6]), the case for bi-intuitionistic logic is less satisfactory. Although Rauszer presented a sequent calculus for bi-intuitionistic logic and “proved” it cut-free [26], Uustalu =-=[32]-=- has recently shown that it fails cut-elimination. Rauszer’s sequent calculus is incomplete without cut because it does not handle the interaction between intuitionistic implication → and dual intuiti... |