Algebraization of logics and beyond
Ricardo Gonçalves · 2004
this paper, he gave the precise connection between Boolean algebra and classical propositional logic, using the idea of looking at the set of formulas as an algebra with operators induced by the logical connectives. He observed that logical equivalence was a congruence on the formula algebra, and a quotient algebra could be built. This is the so-called Lindenbaum-Tarski method. It turns out that the quotient algebra is a Boolean algebra, and the theorems coincide exactly with the formulas equivalent to J. Using this idea, a number of other logics were algebraized, namely the intuitionistic propositional logic of Heyting, the multiple-valued logics of Post and Lukasiewicz and the modal logics S 4 and S 5 of Lewis. In contrast to Boolean, cylindric, polyadic and Wajsberg algebras which were known before the Lindenbaum -Tarski method was first applied to generate them from the appropriate logics, Heyting algebras were first identified precisely by applying the LindenbaumTarski method to intuitionistic propositional logic. Although the focus of algebraic logic was on finding an algebraic counterpart for particular classes of logics, there was also interest, when this counterpart was found, in investigating the relationship between the metalogical properties of the logic and the algebraic properties of the algebraic counterpart. These results are usually called bridge theorems and allow us to use powerful methods of modern algebra in the investigation of metalogical properties of algebraizable logics. The investigation of particular classes of logics gave place to a systematic investigation of broad classes of logics in a more abstract context. The focus has turned to the process of algebraization itself rather than being centered on the algebraization of particular classes...