A hierarchy of implicational (semilinear) logics: the propositional case

In Abstract Algebraic Logic the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this kind of process one considers the Leibniz relation of indiscernible, i.e. logically equivalent, formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field. Moreover, the role of generalized disjunction connectives is considered in a similar abstract fashion and their relation with implications and semilinearity is studied. In particular, the classical law of Proof by Cases is shown to be equivalent to semilinearity of the logic under certain natural conditions.