On axiomatization of Łukasiewicz's four-valued modal logic

Marcin Tkaczyk

DOI: http://dx.doi.org/10.12775/LLP.2011.012

Abstract


Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £.


Keywords


Łukasiewicz’s modal logic; axiomatization; modal logic

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References


Font, J.M., and P. Hájek, “On Łukasiewicz’s four-valued modal logic”, Studia Logica 70, 2 (2002): 157–182.

Kripke, S., “Semantical analysis of modal logic II”, pages 206–220 in: The Theory of Models. Proc. of 1963 International Symposium at Berkeley, J.W.L. Henkin and A. Tarski (eds.), Nord-Holland, Amsterdem, 1965.

Lemmon, E.J., “Algebraic semantics for modal logics”, The Journal of Symbolic Logic 31 (1966), part I: p. 46–65; part II: 191–218.

Łukasiewicz, J., “On variable functors of propositional arguments”, Proceedings of the Royal Irish Academy 54 (1951), A 2: 25–35.

Łukasiewicz, J., “A system of modal logic”, pages 352–390 in: J. Łukasiewicz, Selected Works, L. Borkowski (ed.), transl. from Polish by O. Wojtasiewicz, North-Holland, Amsterdam 1970.

Smiley, T.J., “On Łukasiewicz’s Ł-modal System”, Notre Dame Journal of Formal Logic 2 (1961): 149–153.








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