Ordering protoalgebraic logics
Josep Maria Font · Journal of Logic and Computation · 2014
This article is a first step in the study of the order structure of the set of all protoalgebraic logics over a fixed (but arbitrary) language. In particular, it is shown herein that the set is a join-complete semilattice, that it has no minimum, and that is not a meet-semilattice. One of the key points in this study is the discovery of a large family of rather weak protoalgebraic logics, from which an infinity of denumerable sequences of protoalgebraic logics of strictly decreasing strength and with no lower protoalgebraic bound is constructed. Other properties of these logics are also studied, such as their classification in the Frege hierarchy and in the Leibniz hierarchy, and several common metalogical properties (conjunction, disjunction, deduction theorems, etc.; in fact, it turns out that they do not possess any of these properties). These logics provide examples of a kind of protoalgebraic logics, few of which have been provided in the literature to date. This article ends by discussing the issue of how to extend the results to the class of all protoalgebraic logics over different languages, ordered under the expansion relation. This article is an example of the use of combinatorial techniques within the field of abstract algebraic logic.