A uni ed 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 c a n these phenomena be placed in the AL framework? In general, what is equational-like logic, and how are these logics related to propo-sitional 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 c a n b e treated formally, will such an eeort be helpful in logic as such o r 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