4-dimensional Space of Algebraic Logic: a Uniied Functorial Framework for Propositional-like and Equational-like Logics

Appetizer. Algebraic logic (AL) is a well established discipline yet some natural questions remain out of its scope. For instance, it is well known that the Horn and universal fragments of FOL are original logics close to equational logic rather than just sublogics of FOL, and, very similarly, Gentzen's axiomatization of FOL seems has much in common with the universal equational logic: how can these phenomena be placed in the AL framework? In general, what is equational-like logic, and how are these logics related to propositional logics? Another block of questions is whether cylindric or polyadic algebraization of FOL make it possible to consider it as a kind of (complex yet) propositional logic, or there are principal diierences? Has it sense to ask about the extent to which a given logic is propositional-like? And if even the questions above can be treated formally, will such an eeort be helpful in logic as such or will remain a purely metalogical achievement? A more technical but principal question is about size problems. Indeed, logics arising from semantics are not bound to be compact, moreover, there are big logics which are axiomatizable by a class rather than a set of inference rules. How can they be managed in AL? On the other hand, categorical logic (CL) has achieved a great success in metalogical studies, and, no doubts, it also manifests the power of the algebraic approach to logic. How can one describe the diierence between these two paradigms? What are their comparative advantages and disadvantages? Whether one of them subsumes the other or they are "orthogonal"? In general, what parameters do determine the essence of one or another style of algebraizing logic, in other words, what are main axes of the space where diierent algebraizions of diierent logics can be placed? What is the structure of this space? Does it possess some non-trivial algebra of operators mapping/building logics? For example, are there standard operators of passing from a logic to its Horn or Gentzen derivative, or from a (propositional) logic to its polyadic version? The present paper (together with its predecessor 8] and the companion 9]) aim at demonstrating that the questions above can be consistently approached within some unexpectedly simple framework constituted by a small set of functors organized into several commutative diagrams. The functorial framework gives a concise language and states meta-studies of algebraic logic on the rm algebraic ground of …