Aldo Victorio Figallo, Estela Bianco, Alicia Ziliani

Abstract: In the XII Latin American Symposium on Mathematical Logic we presented a work introducing a Hilbert-style propositional calculus called four-valued Monteiro propositional calculus. This calculus, denoted by M₄, is introduced in terms of the binary connectives (implication), → (weak implication), ∧ (conjunction) and the unary ones (negation) and ▽ (modal operator). In this paper, it is proved that M₄ belongs to the class of standard systems of implicative extensional propositional calculi as defined by Rasiowa (1974). Furthermore, we show that the definitions of four-valued modal algebra and M₄ -algebra are equivalent and, in addition, obtain the completeness theorem for M₄. We also introduce the notion of modal distributive lattices with implication and show that these algebras are more convenient than four-valued modal algebras for the study of four-valued Monteiro propositional calculus from an algebraic point of view. This follows from the fact that the implication → is one of its basic binary operations.

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Keywords: Mathematical Logic, Hilbert-Style Propositional Calculus, Non-Classical Logics, Four-Valued Monteiro Propositional Calculus

Cite this paper: Figallo, A. , Bianco, E. and Ziliani, A. (2014) A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus. Open Journal of Philosophy, 4, 319-331. doi: 10.4236/ojpp.2014.43036.

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