Logica Universalis 4 (1):41-66 (2010)
AbstractIn this paper, we study multiplicative extensions of propositional many-place sequent calculi for finitely-valued logics arising from those introduced in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) through their translation by means of singularity determinants for logics and restriction of the original many-place sequent language. Our generalized approach, first of all, covers, on a uniform formal basis, both the one developed in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) for singular finitely-valued logics (when singularity determinants consist of a variable alone) and conventional Gentzen-style (i.e., two-place sequent) calculi suggested in Pynko (Bull Sect Logic 33(1):23–32, 2004) for finitely-valued logics with equality determinant. In addition, it provides a universal method of constructing Tait-style (i.e., one-place sequent) calculi for finitely-valued logics with singularity determinant (in particular, for Łukasiewicz finitely-valued logics) that fits the well-known Tait calculus (Lecture Notes in Mathematics, Springer, Berlin, 1968) for the classical logic. We properly extend main results of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) and explore calculi under consideration within the framework of Sect. 7 of Pynko (Arch Math Logic 45:267–305, 2006), generalizing the results obtained in Sect. 7.5 of Pynko (Arch Math Logic 45:267–305 2006) for two-place sequent calculi associated with finitely-valued logics with equality determinant according to Pynko (Bull Sect Logic 33(1):23–32, 2004). We also exemplify our universal elaboration by applying it to some denumerable families of well-known finitely-valued logics.
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References found in this work
Definitional Equivalence and Algebraizability of Generalized Logical Systems.Alexej P. Pynko - 1999 - Annals of Pure and Applied Logic 98 (1-3):1-68.
Functional Completeness and Axiomatizability Within Belnap's Four-Valued Logic and its Expansions.Alexej P. Pynko - 1999 - Journal of Applied Non-Classical Logics 9 (1):61-105.
Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle.Karl Schröter - 1955 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 1 (4):241-251.
Methoden zur Axiomatisierung beliebiger Aussagen‐ und Prädikatenkalküle.Karl Schröter - 1955 - Mathematical Logic Quarterly 1 (4):241-251.
Algebraic Completeness Results for R-Mingle and its Extensions.J. Michael Dunn - 1970 - Journal of Symbolic Logic 35 (1):1-13.
Citations of this work
Minimal Sequent Calculi for Łukasiewicz’s Finitely-Valued Logics Pokaż Uproszczony Rekord.Alexej P. Pynko - 2015 - Bulletin of the Section of Logic 44 (3/4).
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