Skip to main content
SpringerLink
Log in

Axiomatic Extensions of IMT3 Logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. BLOK, W., and D. PIGOZZI, ‘Algebraizable Logics’ Mem. Amer. Math. Soc. 396, vol. 77, Amer. Math. Soc., Providence, 1989.

  2. BURRIS, S., and H. P. SANKAPPANAVAR, A Course in Universal Algebra, Graduate text in Math. vol. 78, Springer-Verlag, New York, 1981.

    Google Scholar 

  3. CIABATTONI, A., F. ESTEVA, and L. GODO, ‘T-norm based logic with n-contraction’, Newral Network World 5 (2002), 441–452.

    Google Scholar 

  4. ESTEVA, F., and L. GODO, ‘Monoidal t-norm based logic: towards a logic for left-continuous t-norms’, Fuzzy Sets and Systems 124 (2001), 271–288.

    Article  Google Scholar 

  5. ESTEVA, F., J. GISPERT, L. GODO, and F. MONTAGNA, ‘On the Standard and Rational completness of some Axiomatic extensions of Monoidal t-norm Logic’, Studia Logica 71 (2002), 199–226.

    Article  Google Scholar 

  6. GISPERT, J., ‘Axiomatic extensions of the Nilpoten Minimum Logic’, Reports on Mathematical Logic 37 (2003), 113–123.

  7. HÖHLE, U., ‘Commutative residuated 1-monoids’ in U. Hohle and E.P. Klement (eds). Non-classical logics and their applications to fuzzy subsets. A handbook of the Mathematical foundations of Fuzzy set theory. Kluwer, Dordrecht 1995, pp. 53–106.

    Google Scholar 

  8. KOWALSKY, T., and H. ONO ‘Residuated lattices: An algebraic glimpse at logics without contractions’, Manuscript (preliminary report).

  9. NOGUERA, C., F. ESTEVA, and J. GISPERT ‘Perfect and bipartite IMTL-algebras and disconnected rotations on basic semihoops’ to appear in Archive for Mathematical Logic

Download references

Author information

Authors and Affiliations

  1. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, Pati de Ciències 2 pis, 08007, Barcelona, Spain

    Joan Gispert & Antoni Torrens

Authors
  1. Joan Gispert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoni Torrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joan Gispert.

Additional information

Both authors are partially supported by Grants MTM2004-031012001-3329 and TIN 2004-07933-C03-02 of Spain and by Grant 2001SGR-0017 of D.G.R. of Generalitat de Catalunya

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gispert, J., Torrens, A. Axiomatic Extensions of IMT3 Logic. Stud Logica 81, 311–324 (2005). https://doi.org/10.1007/s11225-005-4647-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-005-4647-7

Keywords

Search

Navigation