The Quest for the Basic Fuzzy Logic

  • Petr CintulaEmail author
  • Rostislav Horčík
  • Carles Noguera
Part of the Outstanding Contributions to Logic book series (OCTR, volume 6)


The quest for basic fuzzy logic was initiated by Petr Hájek when he proposed his basic fuzzy logic BL, complete with respect to the semantics given by all continuous t-norms. Later weaker systems, such as MTL, UL or \(\mathrm{psMTL}^r\), complete with respect to broader (but still meaningful for fuzzy logics) semantics, have been introduced and disputed the throne of the basic fuzzy logic. We contribute to the quest with our own proposal of a basic fuzzy logic. Indeed, we put forth a very weak logic called \(\mathrm {SL}^\ell \), introduced and studied in earlier works of the authors, and propose it as a base of a new framework which allows to work in a uniform way with both propositional and first-order fuzzy logics.


Mathematical fuzzy logic Basic fuzzy logic T-Norm Core fuzzy logics Core semilinear logics Non-associative substructural logics Standard completeness 



The authors are deeply indebted to the anonymous referees for their valuable comments and remarks. P. Cintula and R. Horčík were supported by the grant GAP202/10/1826 of the Czech Science Foundation and RVO 67985807. C. Noguera was supported by the grant GA13-14654S of the Czech Science Foundation, by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584) and initially by the MICINN project TASSAT (TIN2010-20967-C04-01).


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Petr Cintula
    • 1
    Email author
  • Rostislav Horčík
    • 1
  • Carles Noguera
    • 2
  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic
  2. 2.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPragueCzech Republic

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