Abstract
In this work we establish some syntactical and semantical links between Łukasiewicz Logics and Linear Logic. First we introduce a new sequent calculus of infinite-valued Łukasiewicz Logic by adding a new rule of inference to those of Affine Linear Logic. The only axioms of this calculus have the form A ⊢ A. Then we compare the (provability) semantics of both logics, respectively given by MV-algebras and phase spaces. We prove that every MV-algebra can be embedded into a phase space, and every complete MV-algebra is isomorphic to some phase space. In fact, completeness is necessary and sufficient for the existence of the isomorphism. Our proof is constructive.
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References
A. Avron. Natural 3-valued Logics. Characterization and Proof Theory. J. of Symbolic Logic, vol. 56. pp. 276–294. 1991.
A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Nonclassical Logics. In: Logic: from Foundations to Applications. W. Hodges, M. Hyland, C. Steinhorn and J. Truss eds., European Logic Colloquium. Oxford Science Pubblications. Clarendon Press. Oxford. 1996.
C.C. Chang. Algebraic analysis of many valued logics. Trans. Amer. Math. Soc., vol. 88. pp. 467–490. 1958.
C. C. Chang. Proof of an axiom of Lukasiewicz. Trans. Am. Math. Soc., vol. 87. pp. 55–56. 1958.
C.C. Chang. A new proof of the Completeness of the Lukasiewicz's axioms. Trans. Amer. Math. Soc., vol. 93. pp. 74–80. 1959.
R. Cignoli, D. Mundici and LM. D'Ottaviano. MV-algebras: the Mathematics of Many-Valued Logic. In preparation.
J. Y. Girard. Linear Logic. Theoretical Comp. Science, vol. 50. pp. 1–102. 1987.
G. Gentzen.Untersuchungen über das logische Schliessen I, II. Mathematische Zeitschrift, vol. 39. pp. 176–210, pp. 405–431. 1934-35.
V.N. Grishin. On the Algebraic Semantics of a Logic without Contraction. In: Studies in Set Theory and Nonclassical Logics, D.A. Bochvar, V.N. Grishin eds., Nauka, Moskva, pp. 247–264. 1976.
U. Höhle. Commutative, residuated l-monoids. In: Nonclassical logics and their applications to fuzzy subsets. U. Höhle and P. Klement eds., Kluwer. Dordrecht. pp. 53–106. 1995.
J. Lukasiewicz. O Logice Trówartosciowej. Ruch Filozoficzny., vol. 5 pp. 170–171. 1920. English Translation: On three-valued logic. In: J. Łukasiewicz Selected Works. North-Holland, Amsterdam. pp. 87–88. 1970.
J. Lukasiewicz and A. Tarski. Untersuchungen über den Aussagenkalkiil. Compt. Rendus de la Soc. des Sciences et des Lett. de Vars.. cl. iii vol. 23. pp. 1–21. 1930. English translation: Investigations into the sentential calculus. Chap. 4. In: Logic, Semantics Metamathematics. Oxford: Clarendon Press. 1956. Reprinted Hackett. Indianapolis. 1983.
C. A. Meredith. The dependency of an axiom of Lukasiewicz. Trans. Amer. Math. Soc., vol. 87. p. 57. 1958.
A. Prijately. Bounded Contraction and Gentzen style Formulation of Lukasiewicz Logics. Studia Logica, vol. 57. pp. 437–456. 1996.
G. Takeuti. Proof Theory. Studies in Logic and the Foundations of Math., vol. 81. North Holland. 1975.
A.S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge University Press. 1996.
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Ciabattoni, A., Luchi, D. (1997). Two connections between Linear Logic and Łukasiewicz Logics. In: Gottlob, G., Leitsch, A., Mundici, D. (eds) Computational Logic and Proof Theory. KGC 1997. Lecture Notes in Computer Science, vol 1289. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63385-5_38
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DOI: https://doi.org/10.1007/3-540-63385-5_38
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