An Interpretation of Łukasiewicz’s 4-Valued Modal Logic
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A simple, bivalent semantics is defined for Łukasiewicz’s 4-valued modal logic Łm4. It is shown that according to this semantics, the essential presupposition underlying Łm4 is the following: A is a theorem iff A is true conforming to both the reductionist (rt) and possibilist (pt) theses defined as follows: rt: the value (in a bivalent sense) of modal formulas is equivalent to the value of their respective argument (that is, ‘ A is necessary’ is true (false) iff A is true (false), etc.); pt: everything is possible. This presupposition highlights and explains all oddities arising in Łm4.
KeywordsMany-valued logics Modal logics 4-valued logics Łukasiewicz’s 4-valued modal logic Bivalent semantics
Work supported by research project FFI2011-28494, financed by the Spanish Ministry of Economy and Competitiveness. -G. Robles is supported by Program Ramón y Cajal of the Spanish Ministry of Economy and Competitiveness. -We sincerely thank the referees of the JPL for their comments and suggestions on a previous draft of this paper.
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