Abstract
In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n ⩾ 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued Łukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability for the classes of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued Łukasiewicz logics.
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References
Avron, A., 1991, ‘Natural 3-Valued Logics-Characterization and Proof Theory’, The Journal of Symbolic Logic, 56.
Beavers, M. G., 1993, ‘Extensions of the ℵ0-Valued Łukasiewicz Propositional Logic’, Notre Dame Journal of Formal Logic, 34, 251–261.
Boicescu, V., A. Filipoiu, G. Georgescu, S. Rudeanu, 1991, ‘Łukasiewicz-Moisil Algebras (monograph)’, Annals of Discrete Mathematics 49, North-Holland, Amsterdam.
Chang, C. C., 1958, ‘Algebraic Analysis of Many Valued Logics’, Trans. Amer. Math. Soc. 88, 467–490.
Chang, C. C., 1959, ‘A New Proof of the Completeness of the Łukasiewicz Axioms’, Trans. Amer. Math. Soc. 93, 74–80.
Girard, J. Y., 1987, ‘Linear Logic’, Theoretical Computer Science 50, 1–102.
Grigolia, R., 1977, ‘Algebraic Analysis of Łukasiewicz-Tarski's n-Valued Logical Systems’, in: Wójcicki, Malinowski (eds.), Selected Papers on Łukasiewicz Sentential Calculi, Ossolineum, 81–92.
Grishin, V. N., 1976, ‘On the Algebraic Semantics of a Logic without Contraction’, in D.A. Bochvar, V.N. Grishin (eds.), Studies in Set Theory and Nonclassical Logics, Collection of Papers, Nauka, Moskva, 247–264.
Łukasiewicz, J., A. Tarski, 1930, ‘Untersuchungen über den Aussagenkalkül’, in Alfred Tarski: Collected Papers 1, Basel (Birkhauser), 1986, 323–346.
Rosser, J. B., and A. R. Turquette, 1945, ‘Axiom Schemes For M-Valued Propositional Calculi’, The Journal of Symbolic Logic 10, 61–82.
Rosser, J. B., and A. R. Turquette, 1950, ‘A note on the Deductive Completeness of M-Valued Propositional Calculi, The Journal of Symbolic Logic 14, 219–225.
Rosser, J. B., and A. R. Turquette, 1952, ‘Many-valued Logics’, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam.
Tokarz, M., 1974, ‘A Method of Axiomatization of Łukasiewicz Logics’, in Studia Logica 33, Ossolineum, 333–338.
Troelstra, A. S., 1992, ‘Lectures on Linear Logic’, CSLI Lecture Notes, 29, Stanford.
Tuziak, R., 1988, ‘An Axiomatization of the Finite-Valued Łukasiewicz Calculus’, in Studia Logica 47, Ossolineum, 49–55.
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Prijatelj, A. Bounded contraction and Gentzen-style formulation of Łukasiewicz logics. Stud Logica 57, 437–456 (1996). https://doi.org/10.1007/BF00370844
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DOI: https://doi.org/10.1007/BF00370844