A new algebraic counterpart of four - valued Monteiro propositional calculus

Abstract

In the XII Latin American Symposium on Mathematical Logic We presented a Hilbert-style propositional calculus called four-valued Monteiro propositional calculus in terms of the connectives implication ( → ), conjunction ( ∧ ), negation ( ∼ ) and a modal operator ( ∇ )which has tetravalent modal algebras as an algebraic counterpart. In this paper, following the study of this calculus, we prove that it belongs to the class of standard systems of implicative extensional propositional. Furthermore, we prove that the completeness theorem for this calculus holds (H. Rasiowa, An Algebra ic Ap proac h to Non-Classica l Logics, Nor th-Hollan d, Warzawa-Amster dam,1974). Besides, we introduce the notion of modal distributive lattices with implication and we prove that these algebras are more convenient than tetravalent modal algebras in order to study four –valued Monteiro propositional calculus from analgebraic point of view. This last statement follows from the fact that the implication → is one of its basic binar y operations.