A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus

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.