Logic Filters

Abstract

Being a two valued logics, classical logic associates with each proposition one of the two values: true or false. In contrast to the classical logics a non-classical many-valued logics was introduced. In many valued logics any proposition could have many values for the truth, from total false to complete truth. Mathematicians Adolf Lindenbaum (1904-1941) and Alfred Tarski (1901-1983) investigated an approach for establishing correspondence between logic and an algebraic structure. First logic for which such a correspondence has been established was a Boolean algebra that models the classical logic. Similarly, BL-algebras are algebraic structures associated with basic fuzzy logic. MV-algebras in turn rise as Lindenbaum algebras from the Łukasiewicz logic, G-algebras rise from Gödel logic and eventually, product algebras rise from the product logic. The notion of the construction of the Lindenbaum-Tarski algebra is related to the factoring the algebra with the congruence relation. Based on this idea we can derive a quotient algebra for the particular algebraic structure. In our thesis we study quotient algebras generated by the logic filters and deductive systems. Additionally, we study different types of filters, their properties and conditions under which they correspond to each other. We describe filters, implicative filters, deductive systems and their properties. Next we observe quotient algebras generated by the filters and deductive systems.