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

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{{Paper|author=Petr Cintula|author2=Carles Noguera|title=A hierarchy of implicational (semilinear) logics: the propositional case|journal=Submitted|volume=|number=|pages=|year=|preprint=http://www.carlesnoguera.cat/files/PropHierarchy-revised.pdf}}
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{{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 kind of process one
+
of the Lindenbaum-Tarski process. In this process one
considers the Leibniz relation of indiscernible, i.e. logically
+
considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification
equivalent, formulae. Such approach has resulted in a classification
 
 
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. Moreover, the role of generalized disjunction
+
research in the field.
connectives is considered in a similar abstract fashion and their
 
relation with implications and semilinearity is studied. In
 
particular, the classical law of Proof by Cases is shown to be
 
equivalent to semilinearity of the logic under certain natural
 
conditions.
 

Latest revision as of 10:52, 13 April 2010

Authors:
Petr Cintula
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





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.