Studia Logica

, Volume 97, Issue 3, pp 385–413 | Cite as

Tracks of Relations and Equivalences-based Reasoning

  • G. Shtakser
  • L. Leonenko


It is known that the Restricted Predicate Calculus (RPC) can be embedded in an elementary theory, the signature of which consists of exactly two equivalences. Some special models for the mentioned theory were constructed to prove this fact. Besides formal adequacy of these models, a question may be posed concerning their conceptual simplicity, “transparency” of interpretations they assigned to the two stated equivalences. In works known to us these interpretations are rather complex, and can be called “technical”, serving only the purpose of embedding. We propose a conversion method, which transforms an arbitrary model of RPC into some model of the elementary theory TR, which includes three equivalences. RPC is embeddable in TR, and it appears possible to assign some “natural” interpretations to three equivalences using the “Track of Relation” concept (abbreviated to TR).


elementary theories equivalence embedding procedure tracks of relation finite axiomatizability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    von Bertalanffy L.: ‘General System Theory – A Critical Rewiew’. General Systems VII, 1–20 (1962)Google Scholar
  2. 2.
    Church, A., Introduction to Mathematical Logic, Princeton Univ. Press., 1956.Google Scholar
  3. 3.
    Eberle R.: Nominalistic systems. D. Reidel Publ. Co., Dordrecht (1970)Google Scholar
  4. 4.
    Ershov, Yu. L., I. A. Lavrov, A.D. Taimanov, and M. A. Taitslin, ‘Elementary theories’, Russian Math. Surveys 20:4(124): 35–105, 1965.Google Scholar
  5. 5.
    Goodman, N., and W. V. O. Quine, ‘Steps toward a constructive nominalism’, Journal of Symbolic Logic 12: 105–122, 1947; Google Scholar
  6. 6.
    Klement, K. ‘Russell’s Logical Atomism’, Stanford Encyclopedia of Philosophy 2009;
  7. 7.
    Lavrov I.A.: ‘Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain theories’. Algebra and Logic 2(1), 5–18 (1963) (In Russian)Google Scholar
  8. 8.
    Lesniewski, St., ‘O podstawach matematyki’, Przeglad Filozoficzny XXX: 164–206, 1927. (Reprinted in: Lesniewski S. Collected Works - Ed. by S. J. Surina, J. T. Srzednicki, D. I. Barnett and V. F.Rickey, Kluwer, Dordrecht, 1992.)Google Scholar
  9. 9.
    Materna, P., Rehabilitation of Concepts, 2002;
  10. 10.
    Passmore J.: A Hundred Years of Philosophy, 2nd ed. Basic Books, New York (1966)Google Scholar
  11. 11.
    Quine W.V.O.: Philosophy of Logic. Prentis-hall inc., N.Y. (1970)Google Scholar
  12. 12.
    Rickey, F. V., ‘Interpretations of Lesniewski′s Ontology’, Dialectica 39 (3): 181–192, 1985;
  13. 13.
    Russell, B. ‘The Philosophy of Logical Atomism. Lecture III: Atomic and Molecular Propositions’, The Monist Jan 1919: 32–63, 1919;
  14. 14.
    Tichy P.: The Foundations of Frege′s Logic. de Gruyter, Berlin, N.Y. (1988)Google Scholar
  15. 15.
    Uyemov A.I.: Dinge, Eigenschaften und Relationen. Akademie-Verlag, Berlin (1965) (In German)Google Scholar
  16. 16.
    Uyemov, A. I., ‘The language of ternary description as a deviant logic’, Boletim da Sociedade Paranaense de Matematica 15 (1-2): 25–35, 1995; 17 (1-2): 71–81, 1997; 18 (1-2): 173–190, 1998.Google Scholar
  17. 17.
    Uyemov, A. I., ‘The Ternary Description Language as a Formalism for the Parametrical General Systems Theory’, General Systems 28 (4-5): 351–366, 1999; 31 (2): 131– 151, 2002; 32 (6): 583–623, 2003.Google Scholar
  18. 18.
    Candlish, S., and P. Basile. ‘Francis Herbert Bradley’, Stanford Encyclopedia of Philosophy, 2009;

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Odessa National Academy of TelecommunicationsOdessaUkraine
  2. 2.Odessa National Academy of TelecommunicationsOdessaUkraine

Personalised recommendations