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

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## Abstract

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.

## Keywords

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

### Acknowledgments

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.

### Compliance with ethical standards

### 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.

## References

- Agliano P, Montagna F (2003) Varieties of BL-algebras. I. General properties. J Pure Appl Algebra 181(2–3):105–129CrossRefzbMATHMathSciNetGoogle Scholar
- Blok WJ, Pigozzi D (1989) Algebraizable logics, vol 396. Memoirs of the American Mathematical Society AMS, ProvidencezbMATHGoogle Scholar
- Bou F, Esteva F, Godo L, Rodríguez R (2011) On the minimum many-valued modal logic over a finite residuated lattice. J Logic Comput 21(5):739–790CrossRefzbMATHMathSciNetGoogle Scholar
- Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4(2):106–112CrossRefGoogle Scholar
- Cintula P (2004) From fuzzy logic to fuzzy mathematics. Ph.D. thesis, Czech Technical University in PragueGoogle Scholar
- Cintula P (2016) A note on axiomatizations of Pavelka-style complete fuzzy logics. Fuzzy Sets Syst 292:160–174CrossRefMathSciNetGoogle Scholar
- Cintula P, Noguera C (2010) Implicational (semilinear) logics I: a new hierarchy. Arch Math Logic 49(4):417–446CrossRefzbMATHMathSciNetGoogle Scholar
- Cintula P, Noguera C (2013) The proof by cases property and its variants in structural consequence relations. Studia Logica 101(4):713–747CrossRefzbMATHMathSciNetGoogle Scholar
- Cintula P, Hájek P, Noguera C (eds) (2011) Handbook of mathematical fuzzy logic (in 2 volumes), volume 37 and 38 of studies in logic, mathematical logic and foundation. College Publications, LondonGoogle Scholar
- Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288CrossRefzbMATHMathSciNetGoogle Scholar
- Esteva F, Godo L, Montagna F (2004) Equational characterization of the subvarieties of BL generated by t-norm algebras. Studia Logica 76(2):161–200Google Scholar
- Givant S, Halmos P (2009) Introduction to boolean algebras. Undergraduate texts in mathematics. Springer, BerlinGoogle Scholar
- Hájek P (1998) Metamathematics of fuzzy logic, volume 4 of trends in logic. Studia Logica Library. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
- Haniková Z (2014) Varieties generated by standard BL-algebras. Order 31(1):15–33CrossRefzbMATHMathSciNetGoogle Scholar
- Jenei S, Montagna F (2002) A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica 70:183–192CrossRefzbMATHMathSciNetGoogle Scholar
- Montagna F (2007) Notes on strong completeness in Łukasiewicz, product and
*BL*logics and in their first-order extensions. In: Algebraic and proof-theoretic aspects of non-classical logics: papers in honor of Daniele Mundici on the occasion of his 60th birthday, LNCS, vol 4460. Springer, pp 247–274Google Scholar - Pavelka J (1979a) On fuzzy logic. I. Many-valued rules of inference. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25(1):45–52CrossRefzbMATHMathSciNetGoogle Scholar
- Pavelka J (1979b) On fuzzy logic. II. Enriched residuated lattices and semantics of propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 25(2):119–134CrossRefzbMATHMathSciNetGoogle Scholar
- Pavelka J (1979c) On fuzzy logic. III. Semantical completeness of some many-valued propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25(5):447–464CrossRefzbMATHMathSciNetGoogle Scholar
- Takeuti G, Titani S (1984) Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J Symb Logic 49(3):851–866CrossRefzbMATHMathSciNetGoogle Scholar
- Vidal A, Esteva F, Godo L (2016) On modal extensions of product fuzzy logic. J Logic Comput. doi: 10.1093/logcom/exv046 (in press)
- Vidal A, Godo L, Esteva F (2015) On strongly standard complete fuzzy logics: \({MTL}^{Q}_*\) and its expansions. In: Alonso JM, Bustince H, Reformat M (eds) Proceedings of the IFSA-EUSFLAT-15, Gijón, Spain. Atlantis Press, pp 828–835Google Scholar