Generalisation of proof simulation procedures for Frege systems by M.L. Bonet and S.R. Buss

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics.
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DOI 10.1080/11663081.2018.1525208
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Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
Introduction to Metamathematics.Stephen Cole Kleene - 1968 - Journal of Symbolic Logic 33 (2):290-291.
Natural 3-Valued Logics—Characterization and Proof Theory.Arnon Avron - 1991 - Journal of Symbolic Logic 56 (1):276-294.

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