Skip to main content
SpringerLink
Log in

Product Ł ukasiewicz Logic

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract.

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.

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.

Similar content being viewed by others

References

  1. Baaz, M.: Infinite-valued Gödel logic with 0-1 projector and relativisations. In Gödel’96: Logical foundations of mathematics, computer science and physics. Ed. P. Hájek, Lecture notes in logic 6, 23–33 (1996)

    Google Scholar 

  2. Cignoli, R., D’Ottaviano, IML., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, (1999)

  3. Cintula, P.: The ŁΠ and Ł\(\Pi\frac12\) propositional and predicate logics. Fuzzy Sets and Systems 124/3, 21–34 (2001)

    Google Scholar 

  4. Cintula, P.: Advances in the ŁΠ and Ł\(\Pi\frac12\) logics. Arch. Math. Logic 42, 449–468 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Di Nola, A., Dvurečenskij, A.: Product MV-algebras. Multiple-Valued Logic 6, 193–215 No.1–2 (2001)

    MATH  Google Scholar 

  6. Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Logic 39, 103–124 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Esteva, F., Godo, L., Montagna, F.: The ŁΠ and Ł\(\Pi\frac12\) logics: two complete fuzzy systems joining Łukasiewicz and product logics. Arch. Math. Logic 40, 39–67 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Evans, K., Konikoff, M., Madden, J.J., Mathis, R., Whipple, G.: Totally Ordered Commutative Monoids. Semigroup Forum 62, 249–278 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963.

  10. Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation. Research Studies Press, Baldock, 2001.

  11. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.

  12. Hájek, P.: Fuzzy logic and arithmetical hierarchy III. Studia Logica 68, 129–142 (2001)

    Article  MathSciNet  Google Scholar 

  13. Hájek, P., Godo, L., Esteva, F.: A complete many-valued logic with product conjunction. Arch. Math. Logic 35, 191–208 (1996)

    Article  MathSciNet  Google Scholar 

  14. Henriksen, M., Isbell, J.R.: Lattice-ordered Rings and Function Rings. Pacific J. Math. 12, 533–565 (1962)

    MATH  Google Scholar 

  15. Isbell, J.R.: Notes on Ordered Rings. Algebra Univ 1, 393–399 (1972)

    MATH  Google Scholar 

  16. Ł ukasiewicz, J.: O logice trojwartosciowej (On three-valued logic). Ruch filozoficzny 5:170-171, 1920.

    Google Scholar 

  17. Montagna, F.: An algebraic approach to propositional fuzzy logic. Journal of Logic Language and Information 9, 91–124 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Montagna, F.: Functorial Representation Theorems for MV Δ algebras with additional operators. Journal of Algebra 238 (1), 99–125 (2001)

    Article  MATH  Google Scholar 

  19. Montagna, F.: Reducts of MV-algebras with product and product residuation. (draft)

  20. Montagna, F., Panti, G.: Adding structure to MV-algebras. Journal of Pure and Applied Algebra 164 (3), 365–387 (2001)

    Article  MATH  Google Scholar 

  21. Novák, N., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic . Kluwer, Norwell, 1999.

  22. Pavelka, J.: On Fuzzy Logic I,II,III. Zeitschrift für math. Logic und Grundlagen der Math. 25:,119–134 447 (464), 45–52 (1979)

  23. Takeuti, G., Titani, S.: Fuzzy Logic and Fuzzy Set Theory. Arch. Math. Logic 32, 1–32 (1992)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic

    Rostislav Horčík

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, 182 07 Prague 8 and Dept. of Math., Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic

    Petr Cintula

Authors
  1. Rostislav Horčík
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr Cintula
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rostislav Horčík.

Additional information

The work of the first author was supported by the Grant Agency of the Czech Republic under project GACR 201/02/1540, by the Grant Agency of the Czech Technical University in Prague under project CTU 0208613, and by Net CEEPUS SK-042.

The work of the second author was supported by grant IAA1030004 of the Grant Agency of the Academy of Sciences of the Czech Republic.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horčík, R., Cintula, P. Product Ł ukasiewicz Logic. Arch. Math. Logic 43, 477–503 (2004). https://doi.org/10.1007/s00153-004-0214-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-004-0214-6

Keywords

Search

Navigation