Minimal Sequent Calculi for Łukasiewicz’s Finitely-Valued Logics
Pynko, Alexej P.
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The primary objective of this paper, which is an addendum to the author’s , is to apply the general study of the latter to Łukasiewicz’s n-valued logics . 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 ) for the logics involved to be extracted from the constructive proof of Proposition 6.10 of  yields an equally eﬀective procedure of construction of both Gentzen-style  (i.e., 2-place) and Tait-style  (i.e., 1-place) minimal sequent calculi following the method of translations described in Subsection 4.2 of .