Skip to main content

On Generalization of Definitional Equivalence to Non-Disjoint Languages


For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages.

This is a preview of subscription content, access via your institution.


  1. Andréka, H., Madarász, J.X., Németi, I. (2005). Mutual definability does not imply definitional equivalence, a simple example. Mathematical Logic Quarterly, 51(6), 591–597.

    Article  Google Scholar 

  2. Andréka, H., Madarász, J.X., Németi, I. (2008). Defining new universes in many-sorted logic. Research report, Alfréd Rényi Institute of Mathematics, Hungar. Acad. Sci., Budapest.

  3. Andréka, H., Madarász, J.X., Németi, I. (2002). with contributions from: Andai, A., Sági, G., Sain, I., Töke, C.: On the logical structure of relativity theories. Research report, Alfréd Rényi Institute of Mathematics, Hungar. Acad. Sci., Budapest.

  4. Andréka, H., & Németi, I. (2014). Definability theory course notes.

  5. Andréka, H., Németi, I., Sain, I. (2001). Algebraic logic. In Handbook of philosophical logic Volume II (pp. 133–248): Springer.

  6. Barrett, T.W., & Halvorson, H. (2016). Glymour and Quine on theoretical equivalence. Journal of Philosophical Logic, 45(5), 467–483.

    Article  Google Scholar 

  7. Barrett, T.W., & Halvorson, H. (2016). Morita equivalence. The Review of Symbolic Logic, 9(3), 556–582.

    Article  Google Scholar 

  8. de Bouvère, K.L. (1965). Logical synonymy. Indagationes Mathematicae, 27, 622–629.

    Article  Google Scholar 

  9. de Bouvère, K.L. (1965). Synonymous theories. In The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley (pp. 402–406). North Holland.

  10. Chang, H. (2012). Is water h 2 o? evidence, realism and pluralism. Dordrecht: Springer.

    Book  Google Scholar 

  11. Corcoran, J. (1980). On definitional equivalence and related topics. History and Philosophy of Logic, 1(1-2), 231–234.

    Article  Google Scholar 

  12. Friedman, H.A., & Visser, A. (2014). When bi-interpretability implies synonymy.

  13. Friend, M., Khaled, M., Lefever, K., Székely, G. (2018). Distances between formal theories. arXiv:1807.01501.

  14. Fujimoto, K. (2010). Relative truth definability of axiomatic truth theories. Bulletin of Symbolic Logic, 16(3), 305–344.

    Article  Google Scholar 

  15. Glymour, C. (1970). Theoretical realism and theoretical equivalence. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 1970, 275–288.

    Google Scholar 

  16. Glymour, C. (1977). Symposium on space and time: The epistemology of geometry. Noûs, 11(3), 227–251.

    Article  Google Scholar 

  17. Glymour, C. (1980). Theory and evidence. Princeton.

  18. Heath, T.L. (1956). The Thirteen Books of Euclid’s Elements ([Facsimile. Original publication: Cambridge University Press, 1908] 2nd ed.) Dover Publications.

  19. Henkin, L., Monk, J., Tarski, A. (1971). Cylindric algebras Part I. North-Holland.

  20. Henkin, L., Monk, J., Tarski, A. (1985). Cylindric algebras part II. North-Holland.

  21. Hodges, W. (1993). Model theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  22. Hodges, W. (1997). A shorter model theory. Cambridge: Cambridge University Press.

    Google Scholar 

  23. Kuhn, T. (1957). The copernican revolution: planetary astronomy in the development of western thought. Cambridge: Harvard University Press.

    Google Scholar 

  24. Lefever, K. (2017). Using logical interpretation and definitional equivalence to compare classical kinematics and special relativity theory. Ph.D. thesis, Vrije Universiteit Brussel.

  25. Lefever, K., & Székely, G. (2018). Comparing classical and relativistic kinematics in first-order-logic. Logique et Analyse, 61(241), 57–117.

    Google Scholar 

  26. Madarász, J.X. (2002). Logic and relativity (in the light of definability theory). Ph.D. thesis, Eötvös Loránd Univ., Budapest.

  27. Montague, R. (1956). Contributions to the axiomatic foundations of set theory. Ph.D. thesis, Berkeley.

  28. Pinter, C.C. (1978). Properties preserved under definitional equivalence and interpretations. Zeitschr. f. math Logik und Grundlagen d. nlath., 24, 481–488.

    Article  Google Scholar 

  29. Playfair, J. (1846). Elements of geometry. W. E. Dean.

  30. Quine, W.V. (1946). Concatenation as a basis for arithmetic. The Journal of Symbolic Logic, 11(4), 105–114.

    Article  Google Scholar 

  31. Tarski, A., Mostowski, A., Robinson, R. (1953). Undecidable theories. New York: Elsevier.

    Google Scholar 

  32. Visser, A. (2006). Categories of theories and interpretations. In Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22, 2003, volume 26 of Lecture Notes in Logic (pp. 284–341). Wellesley, Mass: ASL, A.K. Peters, Ltd.

  33. Visser, A. (2015). Extension & interpretability. Logic Group preprint series 329. 1874/319941.

Download references


The writing of the current paper was induced by questions by Marcoen Cabbolet and Sonja Smets during the public defence of [24]. We are also grateful to Hajnal Andréka, Michèle Friend, Mohamed Khaled, Amedé Lefever, István Németi and Jean Paul Van Bendegem for enjoyable discussions and feedback while writing this paper, as well as to the two anonymous referees who made valuable remarks which helped significantly improving our paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Koen Lefever.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lefever, K., Székely, G. On Generalization of Definitional Equivalence to Non-Disjoint Languages. J Philos Logic 48, 709–729 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • First-order logic
  • Definability theory
  • Definitional equivalence
  • Logical translation
  • Logical interpretation