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Kripke’s Thought-Paradox and the 5th Antinomy

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Unifying the Philosophy of Truth

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

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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.

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  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.


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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.

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