On the Algebraization of Many-Sorted Logics

The theory of abstract algebraic logic aims at drawing a strong bridge between logic and universal algebra, namely by generalizing the well known connection between classical propositional logic and Boolean algebras. Despite of its successfulness, the current scope of application of the theory is rather limited. Namely, logics with a many-sorted language simply fall out from its scope. Herein, we propose a way to extend the existing theory in order to deal also with many-sorted logics, by capitalizing on the theory of many-sorted equational logic. Besides showing that a number of relevant concepts and results extend to this generalized setting, we also analyze in detail the examples of first-order logic and the paraconsistent logic C1 of da Costa.