Second-Order Quantification

  • Guido Imaguire
Part of the Synthese Library book series (SYLI, volume 397)


In this chapter, I will deal with the problem of second-order quantification. I distinguish two senses of ‘second-order quantification’: (i) metaphysically second-order quantification, which occurs whenever we quantify into the position of individual variables of a domain that includes properties or sets and (ii) logically second-order quantification, i.e. cases in which we quantify into the position of variables for first-order predicates. Both kinds of second-order quantification may be used to establish the existence of universals and thus should be discussed. In the first part of Chap. 6, I deal with metaphysically second-order quantification and argue that any metaphysically second-order sentence that seems to refer to properties, e.g., ‘humility is a virtue’ and ‘red resembles orange more than blue’, has an ontologically fundamental first-order paraphrase. I develop and apply what I call the ‘method of grounded paraphrase’. In the second part, I discuss logically second-order quantification. I explain and defend the plausibility of two arguments of the ostrich against the line of reasoning that derives the existence of properties from logically second-order quantification. The first argument aims to cast doubts on the very intelligibility of quantification into the predicate position, the second aims to show that predicates can play an important semantic role even when they do not ‘stand for’ extra-linguistic entities like properties. In the end, I will maintain that even when one rejects these last two arguments, the priority nominalist may not be concerned about logically second-order predication, because all second-order truths are grounded in first-order truths.


  1. Audi, P., and R. Garcia. forthcoming. Do Realists Need Paraphrase Just as Badly as Nominalists Do?Google Scholar
  2. Azzouni, J. 1994. Metaphysical Myths, mathematical practice: The ontology and epistemology of the exact sciences. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  3. ———. 2010. Talking about Nothing: Numbers, Hallucinations and Fictions. Oxford: Oxford University Press.CrossRefGoogle Scholar
  4. ———. 2012. Simple Metaphysics and “Ontological Dependence”. In Metaphysical Grounding: Understanding the Structure of Reality, ed. F. Correia and B. Schneider, 234–253. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  5. Barwise, J. 1977. Handbook of Mathematical Logic. Amsterdam: North-Holland.Google Scholar
  6. Boolos, G. 1975. On Second-Order Logic. Journal of Philosophy 72: 509–527.CrossRefGoogle Scholar
  7. ———. 1984. To be is to be a value of a variable (or to be some values of some variables). The Journal of Philosophy 81 (8): 430–449.CrossRefGoogle Scholar
  8. Bueno, O. 2010. A Defense of Second-Order Logic. Axiomathes 20: 365–383.CrossRefGoogle Scholar
  9. Dudman, V.H. 1993. Bedeutung for Predicates. In Sense and Reference in Frege’s Philosophy, ed. H. Sluga. New York: Garland.Google Scholar
  10. Dummett, M. 1973. Frege: Philosophy of Language. London: Duckworth.Google Scholar
  11. Fine, K. 2012. The Pure Logic of Ground. Review of Symbolic Logic 5 (I): 1–25.CrossRefGoogle Scholar
  12. Frege, G. 1962. Ausführungen über Sinn und Bedeutung. In Nachgelassene Schriften, ed. G. Frege, vol. I. Hamburg: Felix Meiner Verlag.Google Scholar
  13. Goodman, N. 1955. Fact, Fiction, & Forecast. Cambridge, MA: Harvard University Press.Google Scholar
  14. Hirsch, E. 1997. Complex Kinds. Philosophical Papers 26 (1): 47–70.CrossRefGoogle Scholar
  15. van Inwagen, P. 2009. Being, Existence, and Ontological Commitment. In Metametaphysics, ed. D. Chalmers, D. Manley, and R. Wasserman, 472–526. Oxford: Oxford University Press.Google Scholar
  16. Krämer, S. 2014. On What There Is To Things To Be: Ontological Commitment and Second-Order Quantification. Frankfurt: Vittorio Klostermann.Google Scholar
  17. Lewis, D. 1997. New Work for a Theory of Universals. Reprint in Properties, ed. Mellor, D.H., and Oliver, A., 118–227. Oxford: Oxford University Press. Originally in Australasian Journal of Philosophy 61/4: 343–77.Google Scholar
  18. Loux, M. 1979. Substance and Attribute: A Study in Ontology. London: Reidel Publishing Company.Google Scholar
  19. McGee, V. 1997. How We Learn Mathematical Language. Philosophical Review 106: 35–68.CrossRefGoogle Scholar
  20. Melia, J. 1995. The Significance of Non-Standard Models. Analysis 55: 127–134.CrossRefGoogle Scholar
  21. Parisi, A. forthcoming. Sellars, Second-Order Quantification, and Ontological Commitment.Google Scholar
  22. Prior, A. 1971. Objects of Thought. Oxford: Oxford University Press.CrossRefGoogle Scholar
  23. Quine, W.V.O. 1948. On What There Is. In Review of Metaphysics, 2. Reprinted in W. V. O. Quine, From a Logical Point of View, 1–19. Cambridge, MA: Harvard University Press.Google Scholar
  24. ———. 1961. Logic and the Reification of Universals. In From a Logical Point of View, ed. W.V.O. Quine, 2nd ed., 102–129. Cambridge, MA: Harvard University Press.Google Scholar
  25. ———. 1970. Philosophy of Logic. 2nd ed. Englewood Cliffs: Prentice Hall. Harvard University Press. 1986.Google Scholar
  26. Rayo, A., and S. Yablo. 2001. Nominalism Through De-Nominalization. Nous 1: 74–92.CrossRefGoogle Scholar
  27. Rodriguez-Pereyra, G. 2002. Resemblance Nominalism. A Solution to the Problem of Universals. Oxford: Clarendon Press.Google Scholar
  28. ———. 2015. Resemblance Nominalism and Abstract Nouns. Analysis 75 (2): 223–231.CrossRefGoogle Scholar
  29. Schaffer, J. 2012. Grounding, Transitivity, and Contrastivity. In Metaphysical Grounding: Understanding the Structure of Reality, ed. F. Correia and B. Schneider. Cambridge: Cambridge University Press.Google Scholar
  30. Sellars, W. 1960. Grammar and Existence: A Preface to Ontology. Mind 69 (276): 499–533.CrossRefGoogle Scholar
  31. ———. 1979. Naturalism and Ontology. Reseda: Ridgeview Publishing Company.Google Scholar
  32. Shapiro, S. 1985. Second-Order Languages and Mathematical Practice. Journal of Symbolic Logic 50: 714–742.CrossRefGoogle Scholar
  33. ———. 1990. Second-order Logic, Foundations, and Rules. Journal of Philosophy 87: 234–261.CrossRefGoogle Scholar
  34. ———. 1991. Foundations Without Foundationalism: A Case for Second-order Logic. Oxford: Clarendon Press.Google Scholar
  35. Summerford, J. 2003. Neither Universals Nor Nominalism. Kinds and the Problem of Universals. Metaphysical 3 (5): 101–126.Google Scholar

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Authors and Affiliations

  • Guido Imaguire
    • 1
  1. 1.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil

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