Studia Logica 65 (1):53-89 (2000)
AbstractIn this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed.
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References found in this work
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Proof Theory of Finite-valued Logics.Richard Zach - 1993 - Dissertation, Technische Universität Wien
Gentzen-type systems, resolution and tableaux.Arnon Avron - 1993 - Journal of Automated Reasoning 10:265-281.