Skip to main content
SpringerLink
Log in

Formal Axiomatic Theories Based on a Three-Valued Logic

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Formal axiomatic theories based on the three-valued logic of Lukasiewicz are considered. Main notions related to these theories, in particular, those of Luk-model, Luk-consistent theory, and Luk-complete theory are introduced. Logical calculuses that describe such theories are defined; counterparts of the classical compactness and completeness theorems are proved. Theories of arithmetic based on Lukasiewicz’s logic and on its constructive (intuitionistic) variant are investigated; the theorem on effective Luk-incompleteness is proved for a large class of arithmetic systems. This theorem is a three-valued counterpart of the famous Godel theorem on incompleteness of formal theories. Three-valued counterparts of Presburger’s arithmetic system are defined and proved to be Luk-complete but incomplete in the classical sense. Bibliography: 29 titles.

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. L. Bolc and P. Borowik, Many-Valued Logics 1: Theoretical Foundations, Springer-Verlag (1992).

  2. H. Bolibekyan, “The compactness theorem in many-valued logics,” Preprint of IIAP of ANAS and YSU, No. 02-001 (2002).

  3. K. Godel, “Die Vollstandigkeit der Axiome des logischen Funktionenkalkuls,” Monatsh. Math. Phys., 37, 349–360 (1930).

    Article  Google Scholar 

  4. K. Godel, “Uber formal unentscheidbare Satze der Principia mathematica und verwandter Systeme,” Monatsh. Math. Phys., 38, 173–198 (1931).

    Article  Google Scholar 

  5. R. Haenle and G. Escalada-Imaz, “Deduction in many-valued logics: a survey,” Marthware Soft Computing, IV, No.2, 69–97 (1997).

    Google Scholar 

  6. D. Hilbert and P. Bernays, Grundlagen der Mathematik, 1, Springer-Verlag, Berlin (1934).

    Google Scholar 

  7. S. C. Kleene, Introduction to Metamathematics, Amsterdam-Groningen (1952).

  8. C. I. Lewis and C. H. Langford, Symbolic Logic, New York-London (1932).

  9. J. Lukasiewicz, “O logice trojwartosciowej,” Ruch Filozof. (Lwow), 5, 168–171 (1920).

    Google Scholar 

  10. A. I. Mal’cev, “Untersuchungen aus der Gebiete der mathematischen Logik,” Mat. Sborn., 1, No.43, 323–326 (1936).

    Google Scholar 

  11. M. Presburger, “Uber die Vollstaendigkeit eines gewissen System der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt,” in: Sprawozdanie z Kongresu Matematikow Krajow Slowianskich (Comptes rendu du I congres des mathematiciens des pays slaves), Warszawa (1930), pp. 92–101.

  12. N. Resher, Many-Valued Logic, McGraw-Hill, New York (1969).

    Google Scholar 

  13. J. B. Rosser, “Extensions of some theorems of Godel and Church,” J. Symb. Logic, 1, 87–91 (1936).

    Google Scholar 

  14. J. B. Rosser and A. R. Turquette, Many-Valued Logics. Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam (1952).

    Google Scholar 

  15. J. Rousseau, “Sequents in many-valued logics. I,” Fund. Mat., 67, No.1, 23–33 (1967).

    Google Scholar 

  16. J. Rousseau, “Sequents in many-valued logics. II,” Fund. Mat., 70, No.1, 125–131 (1970).

    Google Scholar 

  17. I. D. Zaslavsky, “On the predicate and arithmetic calculuses of the symmetric constructive logic,” Sov. Math. Dokl., 14, 751–754 (1973).

    Google Scholar 

  18. I. D. Zaslavsky, Symmetric Constructive Logic [in Russian], Acad. Sci. Arm. SSR (1978).

  19. I. D. Zaslavsky, “On the symmetric constructive truth of arithmetic formulas,” in: 7th All-Union Conference on Mathematical Logic, Abstracts of Reports, Novosibirsk (1984), p. 67.

  20. I. D. Zaslavsky, “Symmetric constructive logic and some of its applications,” Doctoral Thesis, Yerevan (1987).

  21. I. D. Zaslavsky, “On the symmetric constructive arithmetic,” in: 9th All-Union Conference on Mathematical Logic, Abstracts of Reports, Leningrad (1988), p. 59.

  22. I. D. Zaslavsky, “On biconstructive arithmetic formulas,” in: 10th All-Union Conference on Mathematical Logic, Abstracts of Reports, Alma-Ata (1990), p. 67.

  23. I. D. Zaslavsky, “On iterations of determining algorithms of the symmetric constructive logic,” in: 11th Interrepublican Conference on Mathematical Logic, Abstracts of Reports, Kazan (1992), p. 62.

  24. I. D. Zaslavsky, “On the theory of constructive reducibility,” in: Proceedings of the International Conference on Computer Science and Information Technologies (CSIT’97), Yerevan (1997), pp. 10–11.

  25. I. D. Zaslavsky, “On a logically but not functionally complete calculus in three-valued logic,” in: The Tbilisi Symposium on Logic, Language and Computation, Selected Papers, CSLI Publications, Stanford, California (1998), pp. 309–313.

    Google Scholar 

  26. I. D. Zaslavsky, “On three-valued arithmetic,” in: Proceedings of the International Conference on Computer Science and Information Technologies (CSIT’01), Yerevan (2001), pp. 133–136.

  27. I. D. Zaslavsky, “Dual realizability in symmetric logic,” Ann. Pure Appl. Logic, 113, 389–397 (2002).

    Article  Google Scholar 

  28. I. D. Zaslavsky “On three-valued logical models and their applications to arithmetical theories,” Denis Richard 60th Birthday Conference, Clermont-Ferrand, France (2002) [submitted].

    Google Scholar 

  29. I. D. Zaslavsky “On the completeness properties of the systems of symmetric constructive and three-valued logic,” in: International Conference “21st Days of Weak Arithmetics,” Abstracts, St. Petersburg (2002), pp. 17–18.

Download references

Author information

Authors and Affiliations

  1. Institute for Informatics and Automation Problems, National Academy of Sciences of Armenia and Yerevan State University, Yerevan, Armenia

    I. D. Zaslavsky

Authors
  1. I. D. Zaslavsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 304, 2002, pp. 19–74.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zaslavsky, I.D. Formal Axiomatic Theories Based on a Three-Valued Logic. J Math Sci 130, 4578–4597 (2005). https://doi.org/10.1007/s10958-005-0353-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-005-0353-2

Keywords

Search

Navigation