On the logics of algebra.
Graham D. Barbour · ResearchSpace (University of KwaZulu-Natal) · 2008
As the candidates supervisor, I have approved this dissertation for submission Signed: Name: Date: We present and consider a number of logics that arise naturally from universal algebraic considerations, but which are ‘inherently unalgebraizable ’ in the sense of [BP89a], essentially because they have no theo-rems. Of particular interest is the membership logic of a quasivariety, which is determined by its theorems, which are the relative congruence classes of the term algebra together with the empty-set in the case that the quasivariety is non-trivial. The membership logic arises by a more general technique developed in this text, for inducing deductive systems from closed systems on the free algebras of quasivarieties. In order to formalize this technique, we develop a theory of logics over constructs, where constructs are concrete categories. With this theory in place, we are able to view a closed system over an algebra as a logic, and in particular a structural logic, structural with respect to a suitable construct, typically the construct con-sisting of all algebras in a quasivariety and all algebra homomorphisms between these algebras. Of course, in such a case, none of these logics are generally sentential (i.e., structural and finitary deductive systems