On the mathematical synthesis of equational logics

Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation, and thereby formalised what is now our notion of (abstract) algebra. Subsequently he set up the model theory of equational presentations (varieties) and analysed their structure from the standpoint of logical inference for algebraic languages. In doing so, he introduced Equational Logic as a sound and complete deductive system for reasoning about equational assertions in Algebraic Theories. Since Birkhoff’s work, our understanding of algebraic structure has deepened; having been both systematised and extended (see e.g. Lawvere [1963], Ehresmann [1968], Burroni [1981], Kelly and Power [1993], Power [1999]). On the other hand, the development of equational logics has remained ad hoc. The aim of the current work is to fill in this gap. Our standpoint is that equational logics should arise from algebraic structure. In this direction, our first purpose is to provide a mathematical theory and methodology for synthesising equational logics from algebraic metatheories (see Part I). Our second purpose is to establish the practicality of the approach. In this respect, we illustrate our methodology by means of two applications: a rational reconstruction of Birkhoff’s Equational Logic and a new equational logic for reasoning about algebraic structure with name-binding operators (see Part II).