Abstract
In many ontological debates there is a familiar challenge. Consider a debate over X s. The “small” or anti-X side tries to show that they can paraphrase the pro-X or “big” side’s claims without any loss of expressive power. Typically though, when the big side adds whatever resources the small side used in their paraphrase, the symmetry breaks down. The big side plus small’s resources is a more expressively powerful and thus more theoretically fruitful theory. In this paper, I show that there is a very general solution to this problem, for the small side. Assuming the resources of set theory, small can successfully paraphrase big. This result depends on a theorem about models of set theory with urelements. After proving this theorem, I discuss some of its philosophical ramifications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett, K. (2009). Composition, Colocation, and Metaontology. In Chalmers, M., & Wasserman (Eds.) Metametaphysics (pp. 38–76). Oxford: Oxford University Press.
Bernays, P. (1954). A system of axiomatic set theory VII. Journal of Symbolic Logic, 19, 81–96.
Boolos, G. (1984). To be is to be the value of a variable (or some value of some variables). Journal of Philosophy, 81, 430–449.
Boolos, G. (1985). Nominalist platonism. Philosophical Review, 94, 327–344.
Bunt, H.C. (1985). Mass terms and model-theoretic semantics. Cambridge: Cambridge University Press.
Chihara, C.S. (1990). Constructibility and mathematical existence. Oxford: Clarendon Press.
Eklund, M. (2008). The picture of reality as an amorphous lump. In Contemporary Debates in Metaphysics: Blackwell Publishing.
Eklund, M. (2009). Carnap and Ontological Pluralism. In Metametaphysics: New essays on the foundations of ontology. Oxford: Oxford University Press.
Gödel, K. (1931). Ü Formal unentscheidbare sätze der Principia Mathematica und verwandter Systeme I. Montashefte für Mathematik und Physik, 38, 173–198.
Gödel, K. (1939). Consistency-Proof For the generalized Continuum-Hypothesis. Proceedings of the National Academy of Sciences 25.
Hawthorne, J. (2006). Plenitude, Convention, and Ontology. In Metaphysical Essays. Oxford: Oxford University Press.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon Press.
Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathmatik Volume II. Berlin: Springer.
Hirsch, E., & Warren, J. (2019). Quantifier variance and the demand for a semantics. Philosophy and Phenomenological Research, 98(3), 592–605.
Jech, T. (1973). The axiom of choice. Amsterdam: North-Holland.
Jech, T. (2003). Set theory: The third millennium edition, revised and expanded. New York: Springer.
Kunen, K. (1980). Set theory: An introduction to independence proofs. Amsterdam: Elsevier.
Lewis, D. (1991). Parts of classes. Oxford: Basil Blackwell.
Löwe, B. (2006). Set theory with and without urelements and categories of interpretations. Notre Dame Journal of Formal Logic, 47(1), 83–91.
McGee, V. (1997). How we learn mathematical language. Philosophical Review, 106(1), 35–68.
McKay, T.J. (2006). Plural predication. Oxford: Clarendon Press.
Oliver, A., & Smiley, T. (2016). Plural logic: 2nd edn, Revised and expanded. Oxford: Oxford University Press.
Rayo, A. (2006). Beyond plurals. In Rayo, & Uzquiano (Eds.) Absolute Generality (pp. 220–254). Oxford: Oxford University Press.
Rieger, L. (1957). A Contribution to gödel’s Axiomatic Set Theory. Czechoslovak Mathematical Journal, 7, 323–357.
Russell, J.S. (2016). Indefinite divisibility. Inquiry, 59(3), 239–263.
Tarski, A., Andrzej, M., & Raphael, R. (1953). Undecidable theories. Amsterdam: North Holland.
Uzquiano, G. (2004). Plurals and simples. The Monist, 87, 429–451.
Uzquiano, G. (2006). Unrestricted unrestricted quantification: The cardinal problem of absolute generality. In Rayo, & Uzquiano (Eds.) Absolute generality. Oxford: Clarendon Press.
van Inwagen, P. (1990). Material beings. Ithaca: Cornell University Press.
Warren, J. (2015). Quantifier variance and the collapse argument. Philosophical Quarterly, 65(259), 241–253.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Warren, J. Ontology, Set Theory, and the Paraphrase Challenge. J Philos Logic 50, 1231–1248 (2021). https://doi.org/10.1007/s10992-021-09597-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-021-09597-6