Difference between revisions of "Implicational (Semilinear) Logics I: A New Hierarchy"
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− | {{Paper|author=Petr Cintula|author2=Carles Noguera|title= | + | {{Paper|author=Petr Cintula|author2=Carles Noguera|title=Implicational (Semilinear) Logics I: A New Hierarchy|journal=Archive for Mathematical Logic|volume=49|number=4|pages=417-446|year=2010|preprint=http://www.carlesnoguera.cat/files/Implicational1.pdf}} |
== Abstract == | == Abstract == | ||
− | In Abstract Algebraic Logic the general study of propositional | + | In Abstract Algebraic Logic, the general study of propositional |
non-classical logics has been traditionally based on the abstraction | non-classical logics has been traditionally based on the abstraction | ||
− | of the Lindenbaum-Tarski process. In this | + | of the Lindenbaum-Tarski process. In this process one |
− | considers the Leibniz relation of indiscernible | + | considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification |
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of logics partly based on generalizations of equivalence | of logics partly based on generalizations of equivalence | ||
connectives: the ''Leibniz hierarchy''. This paper performs an | connectives: the ''Leibniz hierarchy''. This paper performs an | ||
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implicational semilinear logics that encompasses almost all the | implicational semilinear logics that encompasses almost all the | ||
known examples of fuzzy logics and suggests new directions for | known examples of fuzzy logics and suggests new directions for | ||
− | research in the field | + | research in the field. |
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Latest revision as of 10:52, 13 April 2010
Authors: |
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Title: | Implicational (Semilinear) Logics I: A New Hierarchy | ||
Journal: | Archive for Mathematical Logic | ||
Volume | 49 | ||
Number | 4 | ||
Pages: | 417-446 | ||
Year: | 2010 | ||
Preprint |
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.