The Liar Paradox and the Inclosure Schema

Abstract

In Beyond the Limits of Thought [2002], Graham Priest argues that logical and semantic paradoxes have the same underlying structure (which he calls the Inclosure Schema). He also argues that, in conjunction with the Principle of Uniform Solution (same kind of paradox, same kind of solution), this is sufficient to ‘sink virtually all orthodox solutions to the paradoxes’, because the orthodox solutions to the paradoxes are not uniform. I argue that Priest fails to provide a non-question-begging method to ‘sink virtually all orthodox solutions’, and that the Inclosure Schema cannot be the structure that underlies the Liar paradox. Moreover, Ramsey was right in thinking that logical and semantic paradoxes are paradoxes of different kinds.

Notes

²All unqualified references to (1), (2) and (3) hereafter refer to these three principles.

¹This paper benefited from useful comments from Greg Ray, Graham Priest, William Butchard and participants in the Florida Philosophical Association Conference, Tallahassee 2007 and the Central APA meeting, Chicago, 2008.

³It seems that Ramsey misattributed Grelling's paradox to Weyl.

⁵Chihara 1979 distinguishes the diagnostic problem from the preventative problem of the paradox. The latter is part of the project of replacing the notion of truth with a new notion which does not fall prey to paradoxes.

⁴‘Solution’ here must be understood in a wide enough sense: it should cover not only accounts that indicate a way to block the argument towards contradiction, but also those accounts that take the derivation of a contradiction to be inescapable. I will keep using (following Priest) the word ‘solution’, although Chihara's notion of diagnosis would be more apt in this context.

⁶Priest [136, n. 18] refers to [Anderson et al. 1992 for more details about functional dependency. To his credit, he acknowledges that the problem of being able to tell when a functional dependency is genuine is a ‘tricky and unresolved’ problem.

⁷What Anderson et al. 1992 call syntactic dependency might come closer to what Priest has in mind. A function (formula) M depends syntactically on a variable x ‘just in case M can be seen by syntactic inspection to be semantically strict in the variable x’ [Anderson et al. 1992: 397]. A function (formula) M is semantically strict in the variable x ‘just in case, if the value of x is undefined, so is the value of M’ [ibid. 397].

⁸Moreover, this would reinforce Ramsey's idea that an essential aspect of semantic paradoxes is the reference to language or symbolism.

⁹It might seem that one could get the desired functional dependency by singling out a privileged class of expressions that refer to Ω and its definable subsets. For instance, given a certain definable subset of Ω, s, one could take ‘a’, in the diagonalizer, to be the first expression in lexicographic order that refers to s. The diagonalizer would in this case be functionally dependent, but to prove that this is the desired kind of functional dependency, one would need a principled reason why it is this arbitrarily selected mechanism (rather than any alternative functionally dependent mechanism) that provides the explanation of the paradox. Thanks to Graham Priest for helpful comments on this and some previous points.