  A finitary propositional logic can be given an algebraic reading in two
different ways: by translating formulas into equations and logical rules into
quasi-equations, or by translating logical rules directly into equations. The
former type of algebraic interpretation has been extensively studied and
underlies the theory of algebraization. Here we shall develop a systematic
theory of the latter type of algebraic interpretation. More precisely, we
consider a semantic form of this property which we call the equational
definability of compact filters (EDCF). Paralleling the well-studied hierarchy
of variants of the deduction--detachment theorem (DDT), this property also
comes in local, parametrized, and parametrized local variants. Our main results
give a semantic characterization of each of these variants of the EDCF in a
spirit similar to the existing characterizations of the DDT. While the EDCF
hierarchy and the DDT hierarchy coincide for algebraizable logics, part of the
interest of the EDCF stems from the fact it is often enjoyed even by logics
which are not well-behaved in terms of other existing classifications in
algebraic logic.
