Soft Computing

, Volume 21, Issue 1, pp 125–147 | Cite as

On strong standard completeness in some MTL\(_\Delta \) expansions

  • Amanda VidalEmail author
  • Félix Bou
  • Francesc Esteva
  • Lluís Godo


In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard \(\mathsf {BL}\)-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm \(*\), find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra Open image in new window This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system \(\mathsf {L}_*^\infty \), expanding the language with \(\Delta \) and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti–Titani density rule. Then we show that \(\mathsf {L}_*^\infty \) is indeed strongly complete with respect to the standard algebra Open image in new window . Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions.


Mathematical fuzzy logic Left-continuous t-norms Monoidal t-norm logic Infinitary rules Standard completeness 



The authors are thankful to an anonymous reviewer for his/her comments that have helped to improve the final layout of this paper. Vidal has been supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L and by the institutional support RVO:67985807. Esteva and Godo have been funded by the FEDER/MINECO Spanish Project TIN2015-71799-C2-1-P and by the Grant 2014SGR-118 from the Catalan Government. Bou thanks the Grant 2014SGR-788 from the Catalan Government.

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Conflicts of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Computer ScienceCzech Academy of SciencesPragueCzech Republic
  2. 2.Artificial Intelligence Research Institute (IIIA-CSIC)BellaterraSpain

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