Difference between revisions of "On the Hierachy of t-norm Based Residuated Fuzzy Logics"

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(New page: {{Book chapter| author=Francesc Esteva|author2=Lluís Godo|author3=Àngel García-Cerdaña| chapter=On the Hierachy of t-norm Based Residuated Fuzzy Logics| title=Beyond Two: Theory and Ap...)
 
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editor=Melvin Fitting|
 
editor=Melvin Fitting|
 
editor2=Eva Orlowska|
 
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city=Heidelberg-Berlin-New York|
 
year=2003}}
 
year=2003}}
  
 
== Abstract ==
 
== Abstract ==
 
In this paper we overview recent results, both logical and algebraic, about [0,1]-valued logical systems having a t-norm and its residuum as truth functions for conjunction and implication. We describe their axiomatic systems and algebraic varieties and show they can be suitably placed in a hierarchy of logics depending on their characteristic axioms. We stress that the most general variety generated by residuated structures in [0,1], which are defined by left-continuous t-norms, is not the variety of residuated lattices but the variety of pre-linear residuated lattices, also known as MTL-algebras. Finally, we also relate t-norm based logics to substructural logics to substructural logics, in particular to Ono’s hierarchy of extensions of the Full Lambek Calculus.
 
In this paper we overview recent results, both logical and algebraic, about [0,1]-valued logical systems having a t-norm and its residuum as truth functions for conjunction and implication. We describe their axiomatic systems and algebraic varieties and show they can be suitably placed in a hierarchy of logics depending on their characteristic axioms. We stress that the most general variety generated by residuated structures in [0,1], which are defined by left-continuous t-norms, is not the variety of residuated lattices but the variety of pre-linear residuated lattices, also known as MTL-algebras. Finally, we also relate t-norm based logics to substructural logics to substructural logics, in particular to Ono’s hierarchy of extensions of the Full Lambek Calculus.

Latest revision as of 08:14, 27 April 2016

Authors:
Francesc Esteva
Lluís Godo
Àngel García-Cerdaña
Title of the chapter: On the Hierachy of t-norm Based Residuated Fuzzy Logics
Title of the book: Beyond Two: Theory and Applications of Multiple-Valued Logic
Editor(s):


Melvin Fitting
Eva Orlowska
Pages: 251-272
Publisher: Physica-Verlag
City: Heidelberg-Berlin-New York
Year: 2003




Abstract

In this paper we overview recent results, both logical and algebraic, about [0,1]-valued logical systems having a t-norm and its residuum as truth functions for conjunction and implication. We describe their axiomatic systems and algebraic varieties and show they can be suitably placed in a hierarchy of logics depending on their characteristic axioms. We stress that the most general variety generated by residuated structures in [0,1], which are defined by left-continuous t-norms, is not the variety of residuated lattices but the variety of pre-linear residuated lattices, also known as MTL-algebras. Finally, we also relate t-norm based logics to substructural logics to substructural logics, in particular to Ono’s hierarchy of extensions of the Full Lambek Calculus.