Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T20:53:40.546Z Has data issue: false hasContentIssue false
  • English
  • Français

CARNAP’S DEFENSE OF IMPREDICATIVE DEFINITIONS

Published online by Cambridge University Press:  05 December 2018

VERA FLOCKE
VERA FLOCKE*
Affiliation:
New York University
*
*DEPARTMENT OF PHILOSOPHY NEW YORK UNIVERSITY 5 WASHINGTON PLACE NEW YORK, NY, 10003, USA E-mail: [email protected]URL: veraflocke.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A definition of a property P is impredicative if it quantifies over a domain to which P belongs. Due to influential arguments by Ramsey and Gödel, impredicative mathematics is often thought to possess special metaphysical commitments. The reason is that an impredicative definition of a property P does not have its intended meaning unless P exists, suggesting that the existence of P cannot depend on its explicit definition. Carnap (1937 [1934], p. 164) argues, however, that accepting impredicative definitions amounts to choosing a “form of language” and is free from metaphysical implications. This article explains this view in its historical context. I discuss the development of Carnap’s thought on the foundations of mathematics from the mid-1920s to the mid-1930s, concluding with an account of Carnap’s (1937 [1934]) non-Platonistic defense of impredicativity. This discussion is also important for understanding Carnap’s influential views on ontology more generally, since Carnap’s (1937 [1934]) view, according to which accepting impredicative definitions amounts to choosing a “form of language”, is an early precursor of the view that Carnap presents in “Empiricism, Semantics and Ontology” (1956 [1950]), according to which referring to abstract entities amounts to accepting a “linguistic framework”.

Keywords

00A3001A6003B15Carnapprinciple of toleranceimpredicative definitionsmathematical ontologymetaontologyphilosophy of mathematics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 2 , June 2019 , pp. 372 - 404
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Awodey, S. & Carus, A. (2001). Carnap, Completeness, and Categoricity: The Gabelbarkeitssatz of 1928. Erkenntnis, 54(2), 145–172.CrossRefGoogle Scholar
Awodey, S. & Carus, A. (2004). How Carnap could have replied to Gödel. In Awodey, S. and Klein, C., editors. Carnap Brought Home: The View from Jena. Chicago: Open Court, pp. 203–224.Google Scholar
Awodey, S. & Carus, A. (2007). Carnap’s Dream: Gödel, Wittgenstein and Logical Syntax. Synthese, 159, 23–46.CrossRefGoogle Scholar
Awodey, S. & Carus, A. (2010). Gödel and Carnap. In Feferman, S., Parsons, C., and Simpson, S. G., editors. Kurt Gödel: Essays for his Centennial. New York: Cambridge University Press, pp. 252–274.CrossRefGoogle Scholar
Bernays, P. (1983 [1935]). On Platonism in Mathematics. In Benacerraf, P. and Putnam, H., editors. Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press, pp. 258–271.Google Scholar
Beth, E. W. (1963). Carnap’s Views on the Advantages of Constructed Systems Over Natural Languages in the Philosophy of Science. In Schilpp, P. A., editor. The Philosophy of Rudolf Carnap. LaSalle, IL: Open Court and Cambridge University Press, pp. 469–502.Google Scholar
Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, 46, 481–512.CrossRefGoogle Scholar
Carnap, R. (1927). Eigentliche und Uneigentliche Begriffe. Symposion: Philosophische Zeitschrift für Forschung und Aussprache, 1, 355–374.Google Scholar
Carnap, R. (1929). Abriss der Logistik. Mit besonderer Berücksichtigung der Relationstheorie und ihrer Anwendungen. Wien: Springer.Google Scholar
Carnap, R. (1930). Die Mathematik als Zweig der Logik. Blätter für deutsche Philosophie, 4, 298–310.Google Scholar
Carnap, R. (1930/1931). Bericht über Untersuchungen zur allgemeinen Axiomatik. Erkenntnis, 1, 303–307.CrossRefGoogle Scholar
Carnap, R. (1931). Die Logizistische Grundlegung der Mathematik. Erkenntnis, 2, 91–105.CrossRefGoogle Scholar
Carnap, R. (1937 [1934]). The Logical Syntax of Language. London: Routledge.Google Scholar
Carnap, R. (1956 [1947]). Meaning and Necessity. A Study in Semantics and Modal Logic (second edition). Chicago: The University of Chicago Press.Google Scholar
Carnap, R. (1956 [1950]). Empiricism, Semantics, and Ontology. In Meaning and Necessity. Chicago: The University of Chicago Press, pp. 205–221.Google Scholar
Carnap, R. (1963a). Carnap’s Intellectual Autobiography. In Schilpp, P. A., editor. The Philosophy of Rudolf Carnap. La Salle, IL: Open Court and Cambridge University Press, pp. 3–84.Google Scholar
Carnap, R. (1963b). E. W. Beth on Constructed Language Systems. In Schilpp, P. A. editor. The Philosophy of Rudolf Carnap. La Salle, IL: Open Court and Cambridge University Press, pp. 927–933.Google Scholar
Carnap, R. (1998 [1928]). Der Logische Aufbau der Welt. Hamburg: Meiner.CrossRefGoogle Scholar
Carnap, R. (2000). Untersuchungen zur Allgemeinen Axiomatik. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
Carnap, R. (2004 [1928]). Scheinprobleme in der Philosophie. In Mormann, T., editor. Scheinprobleme in der Philosophie und Andere Metaphysikkritische Schriften. Hamburg: Felix Meiner, pp. 3–48.Google Scholar
Carnap, R. (2004 [1930]). Die Alte und die Neue Logik. In Mormann, T., editor. Scheinprobleme in der Philosophie und Andere Metaphysikkritische Schriften. Hamburg: Meiner, pp. 63–80.Google Scholar
Carnap, R. (forthcoming). Tagebücher 1908–1935. Hamburg: Meiner.Google Scholar
Cartwright, R. L. (1994). Speaking of Everything. Noûs, 28, 1–20.CrossRefGoogle Scholar
Chihara, C. S. (1973). Ontology and the Vicious Circle Principle. Ithaca, NY: Cornell University Press.Google Scholar
Church, A. (1940). A Formulation of the Simple Theory of Types. The Journal of Symbolic Logic, 5, 56–68.CrossRefGoogle Scholar
Chwistek, L. (2012 [1922]). The Principles of the Pure Type Theory. History and Philosophy of Logic, 33, 329–352.Google Scholar
Coffa, A. (1993). The Semantic Tradition from Kant to Carnap. Cambridge: Cambridge University Press.Google Scholar
Curry, H., Feys, R., & Craig, W. (1958). Combinatory Logic. Amsterdam: North-Holland.Google Scholar
Dedekind, R. (1963 [1872]). Continuity and Irrational Numbers. In Essays on the Theorey of Numbers. New York: Dover, pp. 1–30.Google Scholar
Ebbs, G. (2001). Carnap’s Logical Syntax of Language. In Gaskin, R., editor. Grammar in Early Twentieth-Century Philosophy. London: Routledge, pp. 218–237.Google Scholar
Feferman, S. (2000). The Significance of Hermann Weyl’s “Das Kontinuum”. In Hendricks, V. F., Pedersen, S. A., and Jørgensen, K. F., editors. Proof Theory: History and Philosophical Significance. Dordrecht: Springer, pp. 179–194.CrossRefGoogle Scholar
Feferman, S. (2005). Predicativity. In Shapiro, S., editor. The Oxford Handbook of the Philosophy of Mathematics and Logic. New York: Oxford University Press, pp. 590–624.CrossRefGoogle Scholar
Flocke, V. (forthcoming). Carnap’s Noncognitivism about Ontology. Noûs, doi: 10.1111/nous.12267.Google Scholar
Gödel, K. (1931). Diskussion zur Grundlegung der Mathematik am Sonntag, dem 7. September 1930. Erkenntnis, 2, 135–151.CrossRefGoogle Scholar
Gödel, K. (1984 [1944]). Russell’s Mathematical Logic. In Benacerraf, P. and Putnam, H. editors. Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge Unversity Press, pp. 447–469.CrossRefGoogle Scholar
Gödel, K. (1995). Collected Works. Vol. III, Unpublished Essays and Lectures (edited by Feferman, S., Dawson, J. W., Goldfarb, W., Parsons, C., and Solovay, R. N.). New York: Oxford Unversity Press.Google Scholar
Gödel, K. (1995 [1953/1959]). Is Mathematics Syntax of Language? In Feferman, S., Dawson, J. W., Goldfarb, W., Parsons, C., and Solovay, R. N., editors. Collected Works. Vol. III, Unpublished Essays and Lectures. New York: Oxford Unversity Press, pp. 334–363.Google Scholar
Gödel, K. (2003). Collected Works. Vol. IV, Correspondence A-G (edited by Feferman, S., Dawson, J. W., Goldfarb, W., Parsons, C., and Solovay, R. N.). Oxford: Clarendon Press.Google Scholar
Goldfarb, W. (1989). Russell’s Reason for Ramification. In Savage, C. W. and Anderson, C. A., editors. Rereading Russell: Essays on Bertrand Russell’s Metaphysics and Epistemology. Minneapolis, MN: Unversity of Minnesota Press, pp. 24–40.Google Scholar
Goldfarb, W. (1996). The Philosophy of Mathematics in Early Positivism. In Giere, R. N. and Richardson, A. W., editors. Minnesota Studies in the Philosophy of Science, Vol. XVI. Minneapolis, MN: University of Minnesota Press, pp. 213–230.Google Scholar
Goldfarb, W. (1997). Semantics in Carnap: A Rejoinder to Alberto Coffa. Philosophical Topics, 25, 51–66.CrossRefGoogle Scholar
Goldfarb, W. (2003). Rudolf Carnap: Introductory Note by Warren Goldfarb. In Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Sieg, W. editors. Kurt Gödel: Collected Works. Vol. IV, Correspondence A-G. Oxford: Clarendon Press, pp. 335–341.Google Scholar
Goldfarb, W. (2005). On Gödel’s Way In: The Influence of Rudolf Carnap. The Bulletin of Symbolic Logic, 11, 185–193.CrossRefGoogle Scholar
Goldfarb, W. & Ricketts, T. (1992). Carnap and the Philosophy of Mathematics. In Bell, D. and Vossenkuhl, W., editors. Science and Subjectivity. The Vienna Circle and Twentienth Century Philosophy. Berlin: Akademie Verlag, pp. 61–78.Google Scholar
Hindley, J. R. (1997). Basic Simple Type Theory. New York: Cambridge University Press.CrossRefGoogle Scholar
Hodes, H. T. (2013). A Report on Some Ramifield-Type Assignment Systems and their Model-Theoretic Semantics. In Griffin, N. and Linky, B., editors. The Palgrave Centenary Companion to Principia Mathematica. Basingstoke: Palgrave Macmillan, pp. 305–336.CrossRefGoogle Scholar
Hodes, H. T. (2015). Why Ramify? Notre Dame Journal of Formal Logic, 56, 379–415.CrossRefGoogle Scholar
Hodges, W. (2014). Tarski’s Truth Definitions. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Fall 2014 Edition). Available at: http://plato.stanford.edu/archives/fall2014/entries/tarski-truth/.Google Scholar
Hylton, P. (2005). Propositions, Functions and Analysis: Selected Essays in Russell’s Philosophy. New York: Oxford University Press.CrossRefGoogle Scholar
Iemhoff, R. (2016). Intuitionism in the Philosophy of Mathematics. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Winter 2016 Edition). Available at: https://plato.stanford.edu/entries/intuitionism/.Google Scholar
Kaufmann, F. (1978 [1930]). The Infinite in Mathematics. Logico-Mathematical Writings. Dordrecht: Reidel.CrossRefGoogle Scholar
Köhler, E. (1991). Gödel und der Wiener Kreis. In Kruntorad, P., editor. Jour Fixe der Venunft. Der Wiener Kreis und die Folgen. Wien: Verlag Hölder-Pichler-Tempsky, pp. 127–158.Google Scholar
Lavers, G. (2015). Carnap, Quine, Quantification and Ontology. In Torza, A., editor. Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics and Language. Cham: Springer, pp. 271–300.CrossRefGoogle Scholar
Lavers, G. (unpublished). Hitting a Moving Target: Gödel, Carnap, and Mathematics as Logical Syntax. Unpublished Manuscript.Google Scholar
Limbeck-Lilienau, C. & Stadler, F. (2015). Der Wiener Kreis. Texte und Bilder zum Logischen Empirismus. Wien: LIT Verlag.Google Scholar
Linnebo, Ø. (2004). Predicative Fragments of Frege Arithmetic. The Bulletin of Symbolic Logic, 10, 153–174.CrossRefGoogle Scholar
Linnebo, Ø. (2017). Predicative and Impredicative Definitions. The Internet Encyclopedia of Philosophy. Available at: https://www.iep.utm.edu/predicat/.Google Scholar
Linnebo, Ø. (unpublished). Generality Explained. Unpublished Manuscript.Google Scholar
Linsky, B. (1999). Russell’s Metaphysical Logic. Stanford, CA: CSLI Publications.Google Scholar
Myhill, J. (1979). A Refutation of an Unjustified Attack on the Axiom of Reducibility. In Russell, B. and Roberts, G. W., editors. Bertrand Russell Memorial Volume. New York: Humanities Press, pp. 81–90.Google Scholar
Quine, W. (1953 [1948]). On What There Is. In From a Logical Point of View. Cambridge, MA: Harvard University Press, pp. 1–19.Google Scholar
Quine, W. (1960 [1954]). Carnap and Logical Truth. Synthese, 12, 350–374.CrossRefGoogle Scholar
Ramsey, F. P. (1931 [1926]). The Foundations of Mathematics. In Braithwaite, R. B., editor. The Foundations of Mathematics and other Logical Essays. London: Routledge, pp. 1–61.Google Scholar
Reck, E. H. (2007). Carnap and Modern Logic. In Friedman, M. and Creath, R., editors. The Cambridge Companion to Carnap. Cambridge: Cambridge University Press, pp. 176–199.CrossRefGoogle Scholar
Ricketts, T. (1994). Carnap’s Principle of Tolerance, Empiricism and Conventionalism. In Clark, P. and Hale, B. editors. Reading Putnam. Oxford: Blackwell, pp. 176–200.Google Scholar
Russell, B. (1908). Mathematical Logic as Based on the Theory of Types. American Journal of Mathematics, 30, 222–262.CrossRefGoogle Scholar
Russell, B. & Whitehead, A. N. (1927 [1910]). Principia Mathematica, Vol I (second edition). Cambridge: Cambridge University Press.Google Scholar
Schiemer, G., Zach, R., & Reck, E. (2017). Carnap’s Early Metatheory: Scope and Limits. Synthese, 194, 33–65.CrossRefGoogle Scholar
Tarski, A. (1956 [1936]). The Concept of Truth in Formalized Languages. In Corcoran, J, editor. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. London: Oxford at the Clarendon Press, pp. 152–278.Google Scholar
Tarski, A. (2002 [1936]). On the Concept of Following Logically. History and Philosophy of Logic, 23, pp. 155–196.CrossRefGoogle Scholar
Verein, E. M. (1929). Wissenschaftliche Weltauffassung. Der Wiener Kreis. Wien: Veröffentlichungen des Vereines Ernst Mach.Google Scholar
Wagner, P. (2009). Carnap’s Logical Syntax of Language. Basingstoke: Palgrave Macmillan.CrossRefGoogle Scholar
Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Verlag von Veit.Google Scholar
Wittgenstein, L. (1921). Logisch-Philosophische Abhandlung. Annalen der Naturphilosophie, 14, 185–262.Google Scholar