Minimal Sequent Calculi for Łukasiewicz’s Finitely-Valued Logics

Bulletin of the Section of Logic 44 (3/4):149-153 (2015)
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Abstract

The primary objective of this paper, which is an addendum to the author’s [8], is to apply the general study of the latter to Łukasiewicz’s n-valued logics [4]. The paper provides an analytical expression of a 2(n−1)-place sequent calculus (in the sense of [10, 9]) with the cut-elimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi in the sense of a complexity of systems of premises of introduction rules. This together with a quite effective procedure of construction of an equality determinant (in the sense of [5]) for the logics involved to be extracted from the constructive proof of Proposition 6.10 of [6] yields an equally effective procedure of construction of both Gentzen-style [2] (i.e., 2-place) and Tait-style [11] (i.e., 1-place) minimal sequent calculi following the method of translations described in Subsection 4.2 of [7].

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Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle.Karl Schröter - 1955 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 1 (4):241-251.

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