Skip to main content

Physics, inconsistency, and quasi-truth


In this work, the first of a series, we study the nature of informal inconsistency in physics, focusing mainly on the foundations of quantum theory, and appealing to the concept of quasi-truth. We defend a pluralistic view of the philosophy of science, grounded on the existence of inconsistencies and on quasi-truth. Here, we treat only the ‘classical aspects’ of the subject, leaving for a forthcoming paper the ‘non-classical’ part.

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

Fig. 1
Fig. 2
Fig. 3


  1. Below we shall distinguish also between axiomatized and formal theories. The former (which can be also called ‘material (or informal) axiomatics’) can be though of as formal in potentia, so we shall remain speaking in terms of the latter.

  2. Sometimes even ideological ideas enter the scenario. The history of science is full of examples of this kind; just remember Lysenko’s genetic. But it is a task for the scientific practice to discipline these situations. Of course, Einstein’s case cannot be compared with these pathological ideological situations.

  3. In the sense that higher-order logic (theory of types) is, in a certain sense, strictly weaker than ZF set theory.

  4. Suppes himself worked always within informal set theory, so he could presuppose the step-theories as given in advance.

  5. DC is not sufficient to prove that there are nonmeasurable subsets of the real numbers.

  6. Another interesting point that deserves investigation and to which we have dedicated some works is related to the concepts of identity, individuality and their relation to sets, for assuming that some quantum objects are entities without individuality (according to a possible interpretation), their collections cannot be sets such as those in ZF; for details, see French and Krause (2006).

  7. A typical example is the value of the dipolar momentum of the electron, experimentally measured as \(1.00115965219 \pm 0.00000000001\) times the combination of constants given by Dirac; see t’Hooft (2001, p. 72).


  • Arnol’d, V. I. (1978). Mathematical methods of classical mechanics. New York: Springer.

    Book  Google Scholar 

  • Bitbol, M. (1996). Schrödinger’s philosophy of quantum mechanics. Boston studies in the philosophy of science. Dordrecht: Kluwer Academic Press.

    Google Scholar 

  • Bourbaki, N. (1958). Théorie des ensembles. Paris: Hermann.

    Google Scholar 

  • Chernoff, P. R. (2009). Andy Gleason and quantum mechanics. Notices of the American Mathematical Society, November, pp. 1253–1259.

  • da Costa, N. C. A. (1980). Ensaio sobre os Fundamentos da Lógica, S. Paulo, EdUSP-Hucitec (3rd. ed. 2008 by Hucitec).

  • da Costa, N. C. A., & Chuaqui, R. (1988). On Suppes’ set theoretical predicates. Erkenntnis, 29, 95–112.

    Article  Google Scholar 

  • da Costa, N. C. A., & French, S. (2003). Science and partial truth: A unitary approach to models and scientific reasoning. Oxford: Oxford University Press.

    Book  Google Scholar 

  • da Costa, N. C. A., & Krause, D. (2006). The logic of complementarity. In J. van Benthem, G. Heinzmann, M. Rebuschi, & H. Visser (Eds.), The age of alternative logics: Assessing philosophy of logic today. Dordrecht: Springer.

  • da Costa, N. C. A., & Krause, D. (2008). Physics and non-classical logic. In D. Dégremont, L. Keiff, & H. Rückert (Eds.), Dialogues, logics, and other strange things: Essays in honour of Shahid Rahman (pp. 105–122). London: College.

    Google Scholar 

  • da Costa, N. C. A., Krause, D., & Bueno, O. (2006). Paraconsistent logics and paraconsistency. In D. Jacquette, D. M. Gabbay, P. Thagard, & J. Woods (Eds.), Handbook of the philosophy of science (Vol. 5, pp. 655–781)., Philosophy of logic Amsterdam: Elsevier.

    Google Scholar 

  • Dalla Chiara, M. L., & Toraldo di Francia, G. (1981). Le Teorie Fisiche, Un’ Analisi Formale. Torino: Boringhieri.

    Google Scholar 

  • Dalla Chiara, M. L., Giuntini, R., & Greechie, R. (2004). Reasoning in quantum theory: Sharp and unsharp quantum logics (Vol. 22)., Trends in logic Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • d’Espagnat, B. (2006). On physics and reality. Princeton: Princeton & Oxford University Press.

    Google Scholar 

  • d’Espagnat, B. (2009) State of Art and perspectives: quantum physics and the ontological problem.

  • Domenech, G., Holik, F., & Krause, D. (2009). Q-spaces and the foundations of quantum mechanics. Foundations of Physics, 38, 969–994.

    Article  Google Scholar 

  • Domenech, G., Holik, F., Kniznik, L., & Krause, D. (2009). No labeling quantum mechanics of indiscernible particles. International Journal of Theoretical Physics, 49, 3085.

    Article  Google Scholar 

  • French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Heisenberg, W. (1989). Encounters with Einstein and other essays on people, places, and particles. Princeton: Princeton University Press.

    Google Scholar 

  • Holland, P. (1993). The quantum theory of motion: An account of de Broglie–Bohm interpretation of quantum mechanics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Ignatieff, Y. A. (1996). The mathematical world of Walter Noll: A scientific biography. Berlin: Springer.

    Book  Google Scholar 

  • Ludwig, G. (1985). An axiomatic basis for quantum mechanics (Vol. 1). Berlin: Springer.

    Book  Google Scholar 

  • Maitland Wright, J. D. (1973). All operators on a Hilbert space are bounded. Bulletin of the American Mathematical Society, 79(6), 1247–1250.

    Article  Google Scholar 

  • McKinsey, J. C. C., Sugar, A. C., & Suppes, P. (1953). Axiomatic foundations of classical particle mechanics. Journal of Rational Mechanics, 2(2), 253–272.

    Google Scholar 

  • Mikenberg, I., da Costa, N. C. A., & Chuaqui, R. (1986). Pragmatic truth and approximation to truth. The Journal of Symbolic Logic, 51, 201–221.

    Article  Google Scholar 

  • Priest, G. (2006). Doubt truth to be a Liar. Oxford: Clarendon Press.

    Google Scholar 

  • Prugoveĉki, E. (1981). Quantum Mechanics in Hilbert Space (2nd ed.). Dover, NY: Mineola.

  • Reichenbach, H. (1998). Philosophic foundations of quantum mechanics. Dover, NY: Mineola.

    Google Scholar 

  • Suppes, P. (2002). Representation and invariance of scientific structures. Stanford: CSLI Publications.

    Google Scholar 

  • Thalapilli, A. M. (2006). The SU(5) grand unified theory.

  • t’Hooft, G. (2001). Partículas elementales. Barcelona: Crítica.

  • van Fraassen, B. (1980). The scientific image. Oxford: Clarendon Press.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Newton C. A. da Costa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

da Costa, N.C.A., Krause, D. Physics, inconsistency, and quasi-truth. Synthese 191, 3041–3055 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Inconsistencies
  • Quasi-truth
  • Foundations of science