Difference between revisions of "Implicational (Semilinear) Logics I: A New Hierarchy"

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{{Paper|author=Petr Cintula|author2=Carles Noguera|title=Implicational (Semilinear) Logics I: A New Hierarchy|journal=Submitted|volume=|number=|pages=|year=|preprint=http://www.carlesnoguera.cat/files/Implicational1.pdf}}
{{Paper|author=Petr Cintula|author2=Carles Noguera|title=Implicational (Semilinear) Logics I: A New Hierarchy|journal=Archive for Mathematical Logic|volume=|number=|pages=|year=|preprint=http://www.carlesnoguera.cat/files/Implicational1.pdf}}
== Abstract ==
== Abstract ==

Revision as of 09:03, 20 January 2010

Petr Cintula
Carles Noguera
Title: Implicational (Semilinear) Logics I: A New Hierarchy
Journal: Archive for Mathematical Logic


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.