, Volume 79, Issue 2, pp 405–429 | Cite as

Submodels in Carnap’s Early Axiomatics Revisited

Original Article


G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent from Carnap’s work at that time, but even contradicts it. As a consequence, Schiemer’s defense of Carnap’s extremal axioms against Hintikka’s criticism fails. We will reconcile Carnap’s treatment of extremal axioms and Hintikka’s observation by taking into account the practice of axiomatics in the early twentieth century. If one realises that, in Carnap’s time, a predicate for the domain of discourse was often introduced in the formal theory, and that Carnap defined such predicates from the basic relations of an axiom system, the apparent disagreement between optimising relations and optimising domains disappears.


Object Language Basic Relation Axiom System Propositional Function Completeness Axiom 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author thanks Georg Schiemer, for sending her his recent papers, for discussions, and for his comments on an earlier version of this paper. She thanks Stefan Roski for many interesting discussions on Carnap and Carnap’s notion of model. She also thanks Arianna Betti, Hein van den Berg, Rik Peels, Jeroen de Ridder, Stefan Roski, Jeroen Smid, and René van Woudenberg for comments on an earlier version of this paper. Finally she thanks both referees for their insightful remarks. Work on this paper was made possible by ERC Starting Grant TRANH 203194. All passages from the Rudolf Carnap Papers are quoted by permission of the University of Pittsburgh. All rights reserved.


  1. Awodey, S., & Reck, E. H. (2003). Completeness and categoricity. Part I: Nineteenth-century axiomatics to twentieth-century metalogic. History and Philosophy of Logic, 23, 1–30.CrossRefGoogle Scholar
  2. Betti, A., & Loeb, I. (2012). On Tarski’s foundations of the geometry of solids. Bulletin of Symbolic Logic, 18(2), 230–260.CrossRefGoogle Scholar
  3. Carnap, R. (1928). Der logische Aufbau der Welt (2nd edn.). Hamburg 1961.Google Scholar
  4. Carnap, R. (1929). Abriss der Logistik. Wien.Google Scholar
  5. Carnap, R. (1934a). Die Antinomien und die Unvollständigkeit der Mathematik. Monatshefte für Mathematik, 41(1), 263–284.Google Scholar
  6. Carnap, R. (1934b). Logische Syntax der Sprache. Wien, translated into English as Carnap (1937).Google Scholar
  7. Carnap, R. (1937). The logical syntax of language. London: Kegan Paul, Trench, Trubner.Google Scholar
  8. Carnap, R. (2000). Untersuchungen zur allgemeinen Axiomatik. Wissenschaftliche Buchgesellschaft Darmstadt.Google Scholar
  9. Carnap, R., & Bachmann, F. (1936). Über Extremalaxiome. Erkenntnis, 166–188, translated as Carnap and Bachmann (1981).Google Scholar
  10. Carnap, R., & Bachmann, F. (1981). On extremal axioms. History and Philosophy of Logic, 2, 67–85, translation of Carnap and Bachmann (1936) by H.G. Bohnert.Google Scholar
  11. Cassirer, E. (1910). Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin: Cassirer.Google Scholar
  12. Coffa, J. (1991). The semantic tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  13. Friedman, M. (1988). Logical truth and analyticity in Carnap. In W. Aspray, & P. Kitcher (Eds.), History and philosophy of modern mathematics (pp. 82–94). Minneapolis: University of Minnesota PressGoogle Scholar
  14. Givant, S. R., & Mackenzie, R. (Eds.). (1986). Alfred Tarski: Collected Papers (Vol. 1). Birkhäuser.Google Scholar
  15. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Mh. Math. Phys., xxxviii, 173–198CrossRefGoogle Scholar
  16. Gómez-Torrente, M. (1996) Tarski on logical concequence. Notre Dame Journal of Formal Logic, 37(1), 125–151CrossRefGoogle Scholar
  17. Hilbert, D. (1903). Grundlagen der Geometrie. B.G. Teubner (2nd edn.) (first edition: 1889).Google Scholar
  18. Hintikka, J. (1991). Carnap, the universality of language and extremality axioms. Erkenntnis, 35(1–3), 325–336.Google Scholar
  19. Mancosu, P. (2006) Tarski on models and logical consequence. In J. Ferreirós, & J. J. Gray (Eds.), The architecture of modern mathematics (pp. 209–237). Oxford: Oxford University PressGoogle Scholar
  20. Mancosu, P. (2010). Fixed- versus variable-domain interpretations of Tarski’s account of logical consequence. Philosophy Compass, 5(9), 745–759.CrossRefGoogle Scholar
  21. Mormann T. (2006). Between Heidelberg and Marburg: The Aufbau’s Neo-Kantian Origins and the AP/CP-Divide. Sapere Aude!, 1, 22–50Google Scholar
  22. Rédei, M. (Ed.). (2005). John von Neumann: Selected letters. Providence, RI: American Mathematical SocietyGoogle Scholar
  23. Schiemer, G. (2012). Carnap on extremal axioms, “completeness of the models,” and categoricity. The Review of Symbolic Logic, 5(4), 613–641.CrossRefGoogle Scholar
  24. Schiemer, G. (2013). Carnap’s early semantics. Erkenntnis, 78(1), 487–522.CrossRefGoogle Scholar
  25. Schiemer, G., & Reck, E. H. (forthcoming) Logic in the 1930s: Type theory and model theory. The Bulletin of Symbolic Logic.Google Scholar
  26. Tarski, A. (1929). Les fondaments de la géométrie des corps. Annales de la Société Polonaise de Mathématiques, 29–34, reprinted in Givant and Mackenzie (1986).Google Scholar
  27. Tarski, A. (1933). Pojęcie prawdy w językach nauk dedukcyjnych. Nakładem / Prace Towarzystwa Naukowego Warszawskiego, wydzial III, 34.Google Scholar
  28. Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405.Google Scholar
  29. Whitehead, A., & Russell, B. (1910, 1912, 1913, 1925). Principia Mathematica. Cambridge: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Faculty of Philosophy and Network InstituteVU University AmsterdamAmsterdamThe Netherlands

Personalised recommendations