An Abstract Algebraic Logic view of some multiple-valued logics

Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by WÓJCKICI and NOWAK) of deÞning logics using a scheme of “preservation of degrees of truth” (as opposed to the more usual one of “preservation of truth”) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a “strong version” deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truth-preserving scheme, that is, the corresponding Þnite-valued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnite-valued logic of Łukasiewicz.Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by WÓJCKICI and NOWAK) of deÞning logics using a scheme of “preservation of degrees of truth” (as opposed to the more usual one of “preservation of truth”) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a “strong version” deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truth-preserving scheme, that is, the corresponding Þnite-valued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnite-valued logic of Łukasiewicz.