Foundations of Science

, Volume 16, Issue 4, pp 363–382 | Cite as

Axiomatization and Models of Scientific Theories

  • Décio Krause
  • Jonas R. B. Arenhart
  • Fernando T. F. Moraes
Article

Abstract

In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal, for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science.

Keywords

Structures Models Set-theoretical predicates Formal languages 

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References

  1. Andréka, H., Madarász, J. X., Németi, I. & Székely, G. (2010). A logic road from special relativity to general relativity, http://arxiv.org/abs/1005.0960.
  2. Bourbaki, N. (1968). Theory of sets. Reading: Hermann & Addicon Wesley.Google Scholar
  3. Brignole D., da Costa N. C. A. (1971) On supernormal Ehresmann-Dedecker universes. Mathematische Zeitschrift 122: 342–350CrossRefGoogle Scholar
  4. da Costa N. C. A., Chuaqui R. (1988) On Suppes’ Set theoretical predicates. Erkenntnis 29: 95–112CrossRefGoogle Scholar
  5. da Costa, N. C. A., & Krause, D. (forthcoming). Physics, inconsistency, and quasi-truth.Google Scholar
  6. da Costa, N. C. A., Krause, D., & Bueno, O. (2007). Paraconsistent logics and paraconsistency. In D. Jacquette (Ed.), Philosophy of logic, Vol. 5 of D. M. Gabbay, P. Thagard & J. Woods (Eds.), Handbook of the philosophy of science (pp. 655–781).Google Scholar
  7. da Costa N. C. A., Rodrigues A. A. M. (2007) Definability and invariance. Studia Logica 82: 1–30CrossRefGoogle Scholar
  8. French, S. & Krause, D. (2010). Remarks on the theory of quasi-sets, Studia Logica, Online.Google Scholar
  9. Hodges W. (2001) Elementary predicate logic. In: Gabbay D. M., Guenthner F. (eds) Handbook of philosophical logic, Vol. 1, 2nd ed. Springer, BerlinGoogle Scholar
  10. Kaye R. (1991) Models of Peano arithmetic. Clarendon Press, OxfordGoogle Scholar
  11. Kunen K. (2009) The foundations of mathematics. College Publications, LondonGoogle Scholar
  12. Ludwig G. (1985) An axiomatic basis for quantum mechanics, Vol. I. Springer, BerlinGoogle Scholar
  13. Maitland Wright J. D. (1973) All operators on a Hilbert space are bounded. Bulletin of the American Mathematical Society 79(6): 1247–1250CrossRefGoogle Scholar
  14. McKinsey J. C. C., Sugar A. C., Suppes P. (1953) Axiomatic foundations of classical particle mechanics. Journal of Rational Mechanics and Analysis 2(2): 253–272Google Scholar
  15. Moore G. H. (1980) Beyond first-order logic: The historical interplay between mathematical logic and axiomatic set theory. History and Philosophy of Logic 1: 95–137CrossRefGoogle Scholar
  16. Muller, F. A. (2009). Reflections on the revolution at Stanford. Synthese. doi:10.1007/s11229-009-9669-7.
  17. Shapiro S. (1991) Foundations without foundationalism, a case for second-order logic. Clarendon Press, OxfordGoogle Scholar
  18. Suppes P. (1957) Introduction to logic. van Nostrand, New YorkGoogle Scholar
  19. Suppe, F. (ed.) (1977) The structure of scientific theories, 2nd ed. Chicago University Press, Urbana, ChigagoGoogle Scholar
  20. Suppes T. (2002) Representation and invariance of scientific structures. Center for the Study of Language and Information, StanfordGoogle Scholar
  21. Suppes, P. (2009). Future developments of scientific structures closer to experiments: Response to F. A. Muller. Synthese. doi:10.1007/s11229-009-9670-1.

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Décio Krause
    • 1
  • Jonas R. B. Arenhart
    • 1
  • Fernando T. F. Moraes
    • 1
  1. 1.Research Group in Logic and Foundations, Department of PhilosophyFederal University of Santa CatarinaFlorianópolisBrazil

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