Leibniz Operator - Hierarchy

Hierarchy

The Leibniz operator and the study of various of its properties that may or may not be satisfied for particular sentential logics have given rise to what is now known as the abstract algebraic hierarchy or Leibniz hierarchy of sentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart. The properties of the Leibniz operator that help classify the logics are monotonicity, injectivity, continuity and commutativity with inverse substitutions. For instance, protoalgebraic logics, forming the widest class in the hierarchy, i.e., the one that lies in the bottom of the hierarchy and contains all other classes, are characterized by the monotonicity of the Leibniz operator on their theories. Other famous classes are formed by the equivalential logics, the weakly algebraizable logics, the algebraizable logics etc.

By now, there is a generalization of the Leibniz operator in the context of Categorical Abstract Algebraic Logic, that makes it possible to apply a wide variety of techniques that were previously applicable in the sentential logic framework to logics formalized as -institutions. The -institution framework is significantly wider in scope than the framework of sentential logics because it allows incorporating multiple signatures and quantifiers in the language and it provides a mechanism for handling logics that are not syntactically-based.