Implicational (Semilinear) Logics I: A New Hierarchy

0342136 - ÚI 2011 RIV DE eng J - Journal Article
Cintula, Petr - Noguera, C.
Implicational (Semilinear) Logics I: A New Hierarchy.
Archive for Mathematical Logic. Roč. 49, č. 4 (2010), s. 417-446. ISSN 0933-5846. E-ISSN 1432-0665
R&D Projects: GA ČR GEICC/08/E018
Institutional research plan: CEZ:AV0Z10300504
Keywords : abstract algebraic logic * hierarchy of implicational logics * implicative logics * Leibniz hierarchy * linearly ordered logical matrices * mathematical fuzzy logic * non-classical logics * semilinear logics
Subject RIV: BA - General Mathematics
Impact factor: 0.414, year: 2010

In abstract algebraic logic, the general study of propositional logics is based on the abstraction of the Lindenbaum-Tarski process, one considers the Leibniz relation of indiscernible formulae. It leads to the Leibniz hierarchy; a classification of logics based on generalized equivalences. We perform an analogous abstract study of non-classical logics based on generalized implications. It yields the hierarchy of implicational logics which expands Leibniz hierarchy. The notion of implicational semilinear logic is then naturally introduced as a property of the implication, namely a logic is an implicational semilinear logic iff it has an implication and is complete w.r.t. the matrices where this implication induces a linear order, a property which is satisfied by majority of fuzzy logics. This hierarchy is then restricted to the semilinear case obtaining a classification that encompasses almost all the known examples of fuzzy logics and suggests new directions for research.
Permanent Link: http://hdl.handle.net/11104/0184955