TITLE:
A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus
AUTHORS:
Aldo Victorio Figallo, Estela Bianco, Alicia Ziliani
KEYWORDS:
Mathematical Logic, Hilbert-Style Propositional Calculus, Non-Classical Logics, Four-Valued Monteiro Propositional Calculus
JOURNAL NAME:
Open Journal of Philosophy,
Vol.4 No.3,
August
7,
2014
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 theimplication → is one of its basic
binary operations.