, Volume 191, Issue 12, pp 2815–2833 | Cite as

Uniting model theory and the universalist tradition of logic: Carnap’s early axiomatics

  • Iris Loeb


We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature.


Model theory Universalism Carnap Type theory 



The author would like to thank Arianna Betti and Stefan Roski for valuable feedback on earlier versions of this paper. She is grateful for the opportunity to present her work at the Lunch Lecture series of the Theoretical Philosophy Group at Utrecht University, and would like to thank the audience for their helpful comments. The author was supported through ERC Starting Grant TRANH 203194 until September 2013. All passages from the Rudolf Carnap Papers are quoted by permission of the University of Pittsburgh. All rights reserved.


  1. Anellis, I. H. (2012). Jean van Heijenoort’s conception of modern logic, in historical perspective. Logica Universalis. Special Issue: Perspectives on the History and Philosophy of Modern Logic: Van Heijenoort Centenary, 6, 339–409.Google Scholar
  2. Betti, A. (2008). Polish axiomatics and its truth: On Tarski’s Leśniewskian background and the Ajdukiewicz connection. In D. Patterson (Ed.), New essays on Tarski and Philosophy (pp. 44–71). Oxford: Oxford University Press.Google Scholar
  3. Carnap, R. (1927). Eigentliche und uneigentliche Begriffe. Symposion: Philosophische Zeitschrift für Forschung und Aussprache (pp. 355–374).Google Scholar
  4. Carnap, R. (1928). Der logische Aufbau der Welt (2nd ed.). Hamburg: Meiner, 1961.Google Scholar
  5. Carnap, R. (1929). Abriß der Logistik. Wien: Springer.CrossRefGoogle Scholar
  6. Carnap, R. (1934). 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. Darmstadt: 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. Coffa, J. (1991). The semantic tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. Corcoran, J. (1981). From categoricity to completeness. History and Philosophy of Logic, 2, 113–119.CrossRefGoogle Scholar
  13. de Rouilhan, P. (1998). Tarski et l’université de la logique: Remarques sur le post-scriptum au \(\langle \langle \) wahrheitsbegriff\(\rangle \rangle \). Le formalisme en question: Le tournant des anneés 30, F. Nef and D. Vernant, Eds., Paris: Vrin, Problèmes et controverses, 85–102.Google Scholar
  14. de Rouilhan, P. (2012). In defense of logical universalism: Taking issue with Jean van Heijenoort. Logica Universalis. Special Issue: Perspectives on the History and Philosophy of Modern Logic: Van Heijenoort Centenary, 6, 553–586.Google Scholar
  15. Etchemendy, J. (1988). Tarski on truth and logical consequence. The Journal of Symbolic Logic, 53(1), 51–79.CrossRefGoogle Scholar
  16. Frege, G. (1906). Über die Grundlagen der Geometrie II. Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, 377–403.Google Scholar
  17. Friedman, M. (1992). Epistemology in the Aufbau. Synthese, 93(1–2), 15–57.CrossRefGoogle Scholar
  18. Gödel, K. (1931). Über formal unentscheidbare Sätze der principia mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, xxxviii, 173–198.CrossRefGoogle Scholar
  19. Goldfarb, W. D. (1982). Logicism and logical truth. The Journal of Philosophy, 79(11), 692–695.CrossRefGoogle Scholar
  20. Hintikka, J. (1988). On the development of the model-theoretic viewpoint in logical theory. Synthese, 77, 1–36.CrossRefGoogle Scholar
  21. Hintikka, J. (1991). Carnap, the universality of language and extremality axioms. Erkenntnis, 35(1–3), 325–336.Google Scholar
  22. Hintikka, J. (1992). Carnap’s work in the foundations of logic and mathematics in historical perspective. Synthese, 93, 167–189, reprinted in [Hintikka(1997)], pp. 191–213.Google Scholar
  23. Hintikka, J. (1997). Lingua universalis vs calculus ratiocinator. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  24. Huntington, E. (1913). A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion. Mathematische Annalen, 73(4), 522–559.CrossRefGoogle Scholar
  25. Jané, I. (2006). What is Tarski’s common notion of concequence? The Bulletin of Symbolic Logic, 12(1), 1–42.CrossRefGoogle Scholar
  26. Kattsoff, L. O. (1936). Postulational methods. II. Philosophy of Science, 3(1), 67–89.CrossRefGoogle Scholar
  27. Korhonen, A. (2012). Logic as a science and logic as a theory: Remarks on Frege, Russell and the Logocentric predicament. Logica Universalis. Special Issue: Perspectives on the History and Philosophy of Modern Logic: Van Heijenoort Centenary, 6, 597–613.Google Scholar
  28. Loeb, I. (2013). Submodels in Carnap’s early axiomatics revised. Erkenntnis. doi: 10.1007/s10670-013-9501-0.
  29. Loeb, I. (2014). Towards transfinite type theory: Rereading Tarski’s Wahrheitsbegriff. Synthese. doi: 10.1007/s11229-014-0399-0.
  30. 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 Press.Google Scholar
  31. Mancosu, P. (2010). Fixed- versus variable-domain interpretations of Tarski’s account of logical consequence. Philosophy Compass, 5(9), 745–759.CrossRefGoogle Scholar
  32. Mares, E. (2011). Propositional function. The Stanford encyclopedia of philosophy (fall 2011 ed). In E. N. Zalta (Ed.). Stanford, CA.Google Scholar
  33. Peckhaus, V. (2004). Calculus ratiocinator versus characteristica universalis? The two traditions in logic, revisited. History and Philosophy of Logic, 25(1), 3–14.CrossRefGoogle Scholar
  34. Reck, E. H. (2007). Carnap and modern logic. The Cambridge companion to Carnap. Cambridge: Cambridge University Press.Google Scholar
  35. Russell, B. (1903). The principles of mathematics. New York: Norton and Norton.Google Scholar
  36. Sandu, G., & Aho, T. (2009). Logic and semantics in the twentieth century. In L. Haaparanta (Ed.), The history of modern logic (pp. 562–612). New York: Oxford University Press.Google Scholar
  37. Schiemer, G. (2013). Carnap’s early semantics. Erkenntnis, 78(3), 487–522.CrossRefGoogle Scholar
  38. Sluga, H. (1987). Frege against the Booleans. Notre Dame Journal of Formal Logic, 28(1), 80–98.CrossRefGoogle Scholar
  39. Tappenden, J. (1997). Metatheory and mathematical practice in Frege. Philosophical Topics, 25, 213–264.CrossRefGoogle Scholar
  40. Tarski, A. (1933). Pojȩcie prawdy w jȩzykach nauk dedukcyjnych. Nakładem / Prace Towarzystwa Naukowego Warszawskiego, wydzial III, 34.Google Scholar
  41. Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405.Google Scholar
  42. Van Heijenoort, J. (1967). Logic as language and logic as calculus. Synthese, 17(1), 324–330.CrossRefGoogle Scholar
  43. Van Heijenoort, J. (1977). Set-theoretic semantics. Studies in logic and the foundations of mathematics. In Logic Colloquium 76 (Vol. 87, pp. 183–190).Google Scholar
  44. Van Heijenoort, J. (1987). Système et métasystème chez Russell. In Logic Colloquium ’85 (pp. 111–122). Amsterdam: North Holland.Google Scholar
  45. Whitehead, A., & Russell, B. (1910). Principia Mathematica (Vol. I). Cambridge: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesVU University AmsterdamAmsterdamThe Netherlands

Personalised recommendations