¿Hacia un replanteamiento del cálculo proposicional clásico?

Abstract

Classical Propositional Calculus deals with propositions that are tautologies, antilogies or contingent, by means of the basic connectives disjunction, conjunction, and negation, as well as the material conditional for translating conditional statements into symbolism. Since a Boolean Algebra is obtained, Propositional Calculus could be reduced to just Boolean Calculus. This conceptual simplification comes since the partial order is equivalent to asserting the material conditional. Moreover, this methodology allows to analyze which parts of the classical calculus could be directly translated into weaker, and more general algebraic structures as, for instance, orthomodular non boolean ortholattices, and De Morgan algebras.
Propositional calculus is deductive; hence, it is developed in the paper from a Tarski’s Consequence Operator. Once the deductive model is newly viewed in this way, it is easy to reach other consequences different from those usually appearing in the Textbooks. In particular, and instead of only the two well-known schemes of Modus Ponens and Modus Tollens (notwithstanding, specially considered), four similar schemes of reasoning are studied.
Among the algebraic structures of fuzzy sets, there are neither Boolean algebras, nor proper ortholattices, but only some De Morgan-Kleene algebras. So, Fuzzy Propositional Calculus should only be done in the particular algebraic structure corresponding at each case, and once it is choosen a conditional function verifying Modus Ponens (fuzzy conditional).