Skip to main content

Kripke’s Thought-Paradox and the 5th Antinomy

  • Chapter
  • First Online:
Unifying the Philosophy of Truth

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 36))

  • 1020 Accesses

Abstract

In ‘A Puzzle about Time and Thought’ Saul Kripke published a new paradox. The paradox is clearly a relative of Russell’s paradox; but it deploys, as well as the notion of set, an intentional notion, thought. This ensures that it raises significantly different issues from Russell’s paradox. Notably, the solution to Russell’s paradox provided by ZF does not apply in any obvious way to this paradox.

In this paper I will first explain Kripke’s paradox and compare it with another paradox which deploys the notion of thought. I will then show that it fits the Inclosure Schema, and so may be expected to have a solution which is the same as other inclosure paradoxes. Next, the paradox is stripped down to a much more acute form. Finally, in the light of this, some thoughts concerning possibilities for resolving the paradox are offered.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Kripke (2011).

  2. 2.

    For a general discussion of intensional paradoxes, see Priest (1991).

  3. 3.

    Dedekind (1888), Theorem 66. ‘Thought’ here means content, not act. Actual thoughts must give out after some finite time.

  4. 4.

    Presumably, there are ordinal-many; but it doesn’t really matter if the sequence peters out before the ordinals are exhausted.

  5. 5.

    Priest (2002), hereafter, BLoT.

  6. 6.

    BLoT, 6.9.

  7. 7.

    I note that there is also a Curried version of the paradox. Let \(k'=\{t\in T:\exists s\subseteq T(\theta_{t}s\wedge(t\in s\rightarrow\bot))\}\). Then reasoning in a natural way, one establises that \(\tau\in k'\rightarrow(\tau\in k'\rightarrow\bot)\), and hence, by contraction, that \(\tau\in k'\rightarrow\bot\). It follows that \(\exists s\subseteq T(\theta_{\tau}s\wedge(\tau\in s\rightarrow\bot))\), that is, \(\tau\in k'\); hence ⊥.

  8. 8.

    BLoT, 9.4, 11.0, 17.2.

  9. 9.

    ε is an indefinite description operator. Since times are not well-ordered, one cannot assume that any non-empty set of times has a first member. But thinkers being finite, if there are any times at which I think of just x, there is, presumably, a first. The indefinite description could therefore be traded in for a definite description.

  10. 10.

    This is argued in BLoT, Chap. 11.

  11. 11.

    See, e.g, BLoT, Chaps. 9 and 10.

  12. 12.

    See Priest (2006), Chap. 1, and Priest (2010a).

  13. 13.

    Or other paradoxes that do not employ the LEM, such as Berry’s and König’s paradoxes. See Priest (2010a), Sect. 6. Note that the 5th Antinomy does not even deploy the least-number principle, as these do.

  14. 14.

    For a more general critique of Prior, see Priest (1991).

  15. 15.

    Thus, in the case of the sorites paradox, the argument forces the conclusion that at least one of the objects in the sorites progression is contradictory. See Priest (2010b). More generally, see BLoT, p. 130, fn. 7.

  16. 16.

    Both possibilities are considered in Priest (2006), Chap. 18.

  17. 17.

    This form of the principle is found in Routley (1977) and Weber (2010). It is known to be very powerful, but non-trivial.

  18. 18.

    It is important to note that in this version of paraconsistent set theory, sets that are extensionally equivalent, in a certain sense, may be distinct. Thus, one may be able to show that there is nothing satisfying either \(\varphi(x)\) or \(\psi(x)\). But this does not suffice to establish that \(\forall x(\varphi(x)\leftrightarrow\psi(x))\), and so that \(\{x:\varphi(x)\}=\{x:\psi(x)\}\). In particular, then, the set I have labelled ‘1’ need not be the same as the von Neumann ordinal 1.

  19. 19.

    The connection between thinking and referring is noted in BLoT, 4.8.

  20. 20.

    Versions of this paper were given at a one day workshop of the Melbourne Logic Group in August 2011, a conference on Kripke’s Philosophical Troubles, at the Graduate Center, CUNY, in September 2011, and to the Logic Group at the University of Indiana, Bloomington, in October 2011. Thanks go to the people in those audiences for their thoughts, and particularly to Colin Caret, Lloyd Humberstone, and, especially, Zach Weber.

References

  • Dedekind, R. (1888). Was sind und was sollen die Zahlen? Reprinted in English translation as Part II of Essays on the theory of numbers, 1955. New York: Dover Publications.

    Google Scholar 

  • Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.

    Article  Google Scholar 

  • Kripke, S. (2011). A puzzle about time and thought (ch. 13 of Philosophical troubles). Oxford: Oxford University Press.

    Google Scholar 

  • Priest, G. (1991). Intensional paradoxes. Notre Dame Journal of Formal Logic, 32, 193–211.

    Article  Google Scholar 

  • Priest, G. (2002). Beyond the limits of thought (2nd (extended) ed.). Oxford: Oxford University Press.

    Book  Google Scholar 

  • Priest, G. (2006). In contradiction (2nd (extended) ed.). Oxford: Oxford University Press.

    Book  Google Scholar 

  • Priest, G. (2010a). Hopes fade for saving truth. Philosophy, 85, 109–140.

    Google Scholar 

  • Priest, G. (2010b). Inclosures, vagueness, and self-reference. Notre Dame Journal of Formal Logic, 51, 69–84.

    Google Scholar 

  • Prior, A. (1961). On a family of paradoxes. Notre Dame Journal of Formal Logic, 2, 16–32.

    Article  Google Scholar 

  • Routley, R. (1977). Ultralogic as universal? Relevance Logic Newsletter, 2, 51–89. Reprinted as an appendix to Exploring Meinong’s jungle and beyond, 1980. Canberra: Australian National University.

    Google Scholar 

  • Weber, Z. (2010). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3, 1–22.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Graham Priest .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Priest, G. (2015). Kripke’s Thought-Paradox and the 5th Antinomy. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_24

Download citation

Publish with us

Policies and ethics