On the Equivalence of Logical Theories

Abstract

Definitional equivalence captures a sense in which theories are intertranslatable. Here I show that the same holds for a natural generalization of definitional equivalence to many-sorted theories: Morita equivalence. I show that Morita equivalence, a syntactic notion of equivalence of theories developed by Halvorson and Barrett, coincides with a generalized version of another syntactic notion created by Ahlbrandt and Ziegler (and more recently developed by Visser). This generalizes the earlier result of Halvorson and Barrett which showed that definitional equivalence and intertranslatability coincide in the case of single-sorted first-order theories. Definitional equivalence corresponds to being isomorphic in a category; Morita equivalence corresponds to being naturally isomorphic.