In this master's thesis we study a lower part of the so called Leibniz hierarchy in abstract algebraic logic. Abstract algebraic logic concerns itself with taxonomic study of logics. The main classification of logics is the one into the Leibniz hierarchy according to the properties that the Leibniz operator has on the lattice of theories of a given logic. The Leibniz operator is a function that maps a theory of a logic to an indiscernability relation modulo the theory.
We study here two of the main classes in the Leibniz hierarchy — protoalgebraic and equivalential logics — and some of their subclasses. We state and prove the most important characterizations for these classes. We also provide new characterizations for the class of finitely protoalgebraic logics — a class that has previously enjoyed only limited attention.
We recall first some basic facts from universal algebra and lattice theory that are used in the remainder of the thesis. Then we present the abstract definition of logic we work with and give abstract semantics for any logic via logical matrices. We define logics determined by a class of matrices and show how to uniformly associate with a given logic a class of matrices so that the original logic and the logic determined by the class coincide — thus providing an abstract completeness theorem for any logic.
Remainder of the thesis is dedicated to the study of protoalgebraic and equivalential logics. We provide three main families of characterizations for the various classes of logics. The first characterizations are completely syntactic via the existence of sets of formuli satisfying certain properties. The second family of characterizations is via the properties that the Leibniz operator has on the lattice of theories of a given logic. The third and final family of characterizations is via the closure properties of a canonical class of matrices — the class of reduced models — that we associate to any logic.
