Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

CIFOL: Case-Intensional First Order Logic

(I) Toward a Theory of Sorts


This is part I of a two-part essay introducing case-intensional first order logic (CIFOL), an easy-to-use, uniform, powerful, and useful combination of first-order logic with modal logic resulting from philosophical and technical modifications of Bressan’s General interpreted modal calculus (Yale University Press 1972). CIFOL starts with a set of cases; each expression has an extension in each case and an intension, which is the function from the cases to the respective case-relative extensions. Predication is intensional; identity is extensional. Definite descriptions are context-independent terms, and lambda-predicates and -operators can be introduced without constraints. These logical resources allow one to define, within CIFOL, important properties of properties, viz., extensionality (whether the property applies, depends only on an extension in one case) and absoluteness, Bressan’s chief innovation that allows tracing an individual across cases without recourse to any notion of “rigid designation” or “trans-world identity.” Thereby CIFOL abstains from incorporating any metaphysical principles into the quantificational machinery, unlike extant frameworks of quantified modal logic. We claim that this neutrality makes CIFOL a useful tool for discussing both metaphysical and scientific arguments involving modality and quantification, and we illustrate by discussing in diagrammatic detail a number of such arguments involving the extensional identification of individuals via absolute (substance) properties, essential properties, de re vs. de dicto, and the results of possible tests.


  1. 1.

    Bacon, J. (1980). Substance and first-order quantification over individual-concepts. Journal of Symbolic Logic, 45(2), 193–203.

  2. 2.

    Barcan, R. (1947). The identity of individuals in a strict functional calculus of second order. Journal of Symbolic Logic, 12, 12–15.

  3. 3.

    Belnap, N. (2006). Bressan’s type-theoretical combination of quantification and modality. In H. Lagerlund, S. Lindström, R. Sliwinski (Eds.), Modality matters: Twenty-five essays in honour of Krister Segerberg (pp. 31–53). Uppsala: Uppsala Philosophical Studies, Vol. 53, Uppsala University.

  4. 4.

    Belnap, N. (2013). Internalizing case-relative truth in CIFOL. In T. Müller (Ed.), Nuel Belnap’s work on indeterminism and free action. Berlin: Springer (forthcoming).

  5. 5.

    Belnap, N., & Müller, T. (2012). BH-CIFOL: Case-intensional first order logic. (II) Branching histories (forthcoming).

  6. 6.

    Bressan, A. (1972). A general interpreted modal calculus. New Haven, CT: Yale University Press.

  7. 7.

    Bressan, A. (1973). The interpreted type-free modal calculus MC . Rendiconti del Seminario Matematico della Università di Padova, 49, 157–194.

  8. 8.

    Bugno, M., Słota, E., Pieńkowska-Schelling, A., Schelling, C., et al. (2009). Identification of chromosome abnormalities in the horse using a panel of chromosome-specific painting probes generated by microdissection. Acta Veterinaria Hungarica, 57(3), 369.

  9. 9.

    Butterfield, J. (2006). Against pointillisme about mechanics. British Journal for the Philosophy of Science, 57(4), 709–753.

  10. 10.

    Carnap, R. (1947). Meaning and necessity: A study in semantics and modal logic. Chicago, IL: University of Chicago Press. Enlarged edition, 1956.

  11. 11.

    Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.

  12. 12.

    Fitting, M.C. (2004). First-order intensional logic. Annals of Pure and Applied Logic, 127, 171–193.

  13. 13.

    Fitting, M.C. (2011). Intensional logic. In E.N. Zalta (Ed.), The Stanford encyclopedia of philosophy (winter 2011 edn.).

  14. 14.

    Gallin, D. (1975). Intensional and higher-order modal logic: With applications to Montague semantics. Mathematical studies (Vol. 19). Amsterdam: North Holland.

  15. 15.

    Garson, J. (2005). Unifying quantified modal logic. Journal of Philosophical Logic, 34, 621–649.

  16. 16.

    Geach, P.T. (1962). Reference and generality: An examination of some medieval and modern theories. Ithaca, NY: Cornell University Press.

  17. 17.

    Gibbard, A. (1975). Contingent identity. Journal of Philosophical Logic, 4, 187–221.

  18. 18.

    Gupta, A. (1980). The logic of common nouns: An investigation in quantified modal logic. New Haven, CT: Yale University Press.

  19. 19.

    Hughes, G.E., & Cresswell, M.J. (1996). An new introduction to modal logic. London: Routledge.

  20. 20.

    Kishida, K. (2010). Generalized topological semantics for first-order modal logic. Ph.D. thesis, University of Pittsburgh.

  21. 21.

    Kripke, S. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 1–15.

  22. 22.

    Kripke, S. (1963). Semantical considerations in modal logic. Acta Philosophica Fennica, 16, 83–94.

  23. 23.

    Lewis, D.K. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, 65(5), 113–126.

  24. 24.

    Lowe, E.J. (2009). More kinds of being. Oxford: Blackwell.

  25. 25.

    Montague, R. (1973). The proper treatment of quantification in ordinary English. In J. Hintikka, J. Moravcsik, P. Suppes (Eds.), Approaches to natural language: Proceedings of the 1970 Stanford workshop on grammar and semantics (pp. 221–242). Dordrecht: D. Reidel. Reprinted as Chap. 8 of Montague, R. (1974). Formal philosophy: Selected papers of Richard Montague. New Haven, CT: Yale University Press. Edited and with an introduction by R.H. Thomason.

  26. 26.

    Muskens, R. (2007). Higher-order modal logic. In P. Blackburn, J. van Benthem, F. Wolter (Eds.), Handbook of modal logic (pp. 621–654). Amsterdam: Elsevier.

  27. 27.

    Parks, Z. (1972). Classes and change. Journal of Philosophical Logic, 1, 162–169.

  28. 28.

    Quine, W. (1960). Word and object. Cambridge, MA: MIT Press.

  29. 29.

    Sider, T. (2000). The stage view and temporary intrinsics. Analysis, 60, 84–88.

  30. 30.

    Suppes, P. (1957). Introduction to logic. Princeton: D. van Nostrand.

  31. 31.

    Thomason, R.H. (1969). Modal logic and metaphysics. In K. Lambert (Ed.), The logical way of doing things (pp. 119–146). New Haven, CT: Yale University Press.

  32. 32.

    Thomason, R.H. (1970). Indeterminist time and truth-value gaps. Theoria, 36, 264–281.

  33. 33.

    Tichý, P. (1988). The foundations of Frege’s logic. Berlin: De Gruyter.

  34. 34.

    Van Leeuwen, J. (1991). Individuals and sortal concepts. An essay in logical descriptive metaphysics. Ph.D. thesis, Universiteit van Amsterdam.

  35. 35.

    Wiggins, D. (2001). Sameness and substance renewed. Cambridge: Cambridge University Press.

  36. 36.

    Williamson, T. (2013). Barcan formulas in second-order modal logic. In M. Frauchiger (Ed.), Themes from Barcan Marcus, Lauener Library of Analytical Philosophy (Vol. 3, pp. 1–31). Frankfurt: Ontos Verlag.

Download references

Author information

Correspondence to Thomas Müller.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Belnap, N., Müller, T. CIFOL: Case-Intensional First Order Logic. J Philos Logic 43, 393–437 (2014).

Download citation


  • Modal logic
  • Quantification
  • Sortal
  • Tracing
  • Substance