Archive for Mathematical Logic 49 (4):417-446 (2010)

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Abstract
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
Keywords Abstract algebraic logic  Hierarchy of implicational logics  Implicative logics  Leibniz hierarchy  Linearly ordered logical matrices  Mathematical fuzzy logic  Non-classical logics  Semilinear logics
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DOI 10.1007/s00153-010-0178-7
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References found in this work BETA

An Algebraic Approach to Non-Classical Logics.Helena Rasiowa - 1974 - Amsterdam, Netherlands: Warszawa, Pwn - Polish Scientific Publishers.
A Survey of Abstract Algebraic Logic.J. M. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):13 - 97.
A Propositional Calculus with Denumerable Matrix.Michael Dummett - 1959 - Journal of Symbolic Logic 24 (2):97-106.
Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.

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Citations of this work BETA

Order algebraizable logics.James G. Raftery - 2013 - Annals of Pure and Applied Logic 164 (3):251-283.

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