Fuzzy logics as the logics of chains

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Authors:
Libor Běhounek
Petr Cintula
Title: Fuzzy logics as the logics of chains
Journal: Fuzzy Sets and Systems
Volume 157
Number 5
Pages: 604-610
Year: 2006
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Post-publication comments

An author's comment (August 2008)

Even though we still consider our arguments valid, it seems that the proposed delimitation of the class of fuzzy logics (in the technical sense) has not become widely accepted in the community of mathematical fuzzy logic. Perhaps a more neutral term for Cintula's class of weakly implicative fuzzy logics would be more acceptable to other researchers---e.g., semilinear logics (Nick Galatos has pointed out to us that semi-X is often used in universal algebra for the property that all irreducibles are X, like in semi-simple).

The best behaved among "semilinear weakly implicative logics" seem to be those which are substructural in Ono's sense (i.e., logics of residuated lattices),[1] as they also internalize the deductive role of conjunction and implication. Since the name substructural fuzzy logics has already been taken for a slightly different concept by Metcalfe and Montagna in their JSL paper, a suitable name for this class of logics (also called deductive fuzzy logics in the paper On the difference between traditional and deductive fuzzy logic) could be "semilinear substructural logics", i.e., the logics of semilinear residuated lattices.

These proposals, however, are not intended to suggest replacing the name "mathematical fuzzy logic" for the discipline of logic, and only regard the name of the formally defined class of logics (which is not regarded as coinciding with the agenda of mathematical fuzzy logic by many researchers anyway).

-- LBehounek 20:10, 2 September 2008 (CEST)
(The comment was inspired by a discussion with Petr Cintula, Nick Galatos, Rosťa Horčík, Petr Hájek, and Carles Noguera in Prague, August 2008. However, it only expresses my own opinion immediately after the discussion; the opinions of other participants of the discussion may differ, and my own opinion may easily change in the future.)

References for this page

  1. Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.