Implicational (semilinear) logics I: a new hierarchy

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 process one considers the Leibniz relation of indiscernible 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. © Springer-Verlag 2010.Petr Cintula was partly supported by grant ICC/08/E018 of the Grant Agency of the Czech Republic (a part of ESF Eurocores-LogICCC project FP006) and partly by Institutional Research Plan AVOZ10300504. Carles Noguera acknowledges partial support from the grant 2006-BP-A-10043 of the Departament d’Educació i Universitats of the Generalitat de Catalunya and the Spanish project MULOG2 (TIN2007-68005-C04) and ESF Eurocores-LogICCC / MICINN project FFI2008-03126-E/FILO).Peer Reviewe