Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2025-01-02T17:15:31.606Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEGLECTED RESOLUTION OF RUSSELL’S PARADOX OF PROPOSITIONS

Published online by Cambridge University Press:  31 March 2015

GABRIEL UZQUIANO
GABRIEL UZQUIANO*
Affiliation:
University of Southern California and University of St Andrews UNIVERSITY OF ST ANDREWS, ARCHÉ, 17-19 COLLEGE STREET, ST ANDREWS, FIFE KY16 9AL, UK
*
*UNIVERSITY OF SOUTHERN CALIFORNIA, SCHOOL OF PHILOSOPHY, 3709 TROUSDALE PARKWAY, LOS ANGELES, CA 90089, USA E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell’s paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the uses to which modern descendants of Russell’s paradox of propositions have been put in recent literature.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 8 , Issue 2 , June 2015 , pp. 328 - 344
Copyright
Copyright © Association for Symbolic Logic 2015 

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

Adams, R. M. (1974). Theories of actuality. Noûs, 8(3), 211–231.Google Scholar
Adams, R. M. (1981). Actualism and thisness. Synthese, 49(1), 3–41.CrossRefGoogle Scholar
Barwise, J. (1975). Admissible Sets and Structures: An Approach to Definability Theory. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Barwise, J., & Moss, L. (1996). Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. Stanford, CA: Center for the Study of Language and Information Publications.Google Scholar
Beall, J. (2000). A neglected response to the Grim result. Analysis, 60(1), 38–41.CrossRefGoogle Scholar
Bell, J. (1995). Type reducing correspondences and well-orderings: Frege’s and Zermelo’s constructions re-examined. The Journal of Symbolic Logic, 60(1), 209–221.Google Scholar
Bernays, P. (1942). A system of axiomatic set theory: Part IV. General set theory. The Journal of Symbolic Logic, 7(4), 133–145.CrossRefGoogle Scholar
Boolos, G. (1997). Constructing cantorian counterexamples. Journal of Philosophical Logic, 26(3), 237–239.CrossRefGoogle Scholar
Bringsjord, S. (1985). Are there set theoretic possible worlds? Analysis, 45(1), 64.Google Scholar
Cartwright, R. L. (2001). A question about sets. In Byrne, A., Stalnaker, R., and Wedwood, R., editors. Fact and Value. Cambridge, MA: MIT Press, pp. 29–46.Google Scholar
Chihara, C. (1998). The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic. Oxford: Clarendon Press.Google Scholar
Davies, M. (1981). Meaning, Quantification, Necessity: Themes in Philosophical Logic. London: Routledge & Kegan Paul.Google Scholar
Deutsch, H. (2014). Resolution of some paradoxes of propositions. Analysis, 74(1), 26–34.Google Scholar
Forster, T. (2008). The iterative conception of set. The Review of Symbolic Logic, 1(1), 97–110.Google Scholar
Fraenkel, A., Bar-Hillel, Y., & Levy, A. (1973). Foundations of Set Theory. Studies in Logic and the Foundations of Mathematics. Amsterdam, The Netherlands: North Holland.Google Scholar
Friedman, H. (2004). Faithful representation in set theory with atoms. Available from: http://www.cs.nyu.edu/pipermail/fom/2004-January/007845.html.Google Scholar
Grim, P. (1984). There is no set of all truths. Analysis, 44(4), 206–208.Google Scholar
Jech, T. J. (1973). The Axiom of Choice. Amsterdam, The Netherlands: North Holland.Google Scholar
Kanamori, A. (2004). Zermelo and set theory. Bulletin of Symbolic Logic, 10(4) 487–553.Google Scholar
Kelley, J. L. (1955). General Topology. Princeton, NJ: Van Nostrand.Google Scholar
Klement, K. C. (2001). Russell’s paradox in appendix B of the Principles of Mathematics: Was Frege’s response adequate? History and Philosophy of Logic, 22(1), 13–28.Google Scholar
Kripke, S. A. (1980). Naming and Necessity. Cambridge, MA: Harvard University Press.Google Scholar
Lewis, D. K. (1986). On the Plurality of Worlds. Oxford: Blackwell Publishers.Google Scholar
McGee, V., & Rayo, A. (2000). A puzzle about de rebus belief. Analysis, 60(4), 297–299.Google Scholar
Mendelson, E. (1997). Introduction to Mathematical Logic (Fourth Edition). Discrete Mathematics and Its Applications. Boca Raton, FL: Taylor & Francis.Google Scholar
Menzel, C. (2012). Sets and worlds again. Analysis, 72(2), 304–309.Google Scholar
Menzel, C. (2014). Wide sets, ZFCU, and the iterative conception. Journal of Philosophy, 111(2), 57–83.Google Scholar
Plantinga, A. (1976). Actualism and possible worlds. Theoria, 42(1–3), 139–160.Google Scholar
Plantinga, A. (1992). The Nature of Necessity. Oxford: Clarendon Press.Google Scholar
Prior, A. N. (1977). Worlds, Times, and Selves. London: Duckworth.Google Scholar
Russell, B. (1903). The Principles of Mathematics. London: W. W. Norton & Company.Google Scholar
Stalnaker, R. C. (1976). Possible worlds. Noûs, 10(1), 65–75.CrossRefGoogle Scholar
Uzquiano, G. (2003). Plural quantification and classes. Philosophia Mathematica, 11(1), 67–81.CrossRefGoogle Scholar
Zalta, E. (1999, December). Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege’s Grundgesetze in object theory. Journal of philosophical logic, 28(6), 617–658.CrossRefGoogle Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16, 29–47.Google Scholar