Abstract
In this paper I consider attempts to unify the liar and sorites paradoxes. I argue that while they both may be said to exhibit indeterminacy and be alike in this respect, attempts to model the indeterminacy by way of a paracomplete logic result in the two paradoxes diverging in their logical structure in the face of extended paradoxes. If, on the other hand, a paraconsistent logic is invoked then the paradoxes and associated extended paradoxes may be seen to be of a kind in having their source in the indeterminacy of the relevant predicates involved. Paraconsistency then offers the prospect of a unified treatment of these vexing puzzles.
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Notes
- 1.
Caution is required here. The proposal is paracomplete in the sense that not every sentence is determined to be true or determined to be false; some sentences (e.g. vague ones, liar sentences) remain undetermined. The paracomplete logic advocated is the popular supervaluationism adapted from Van Fraassen (1966).
- 2.
See Field (2008, Part I, §3).
- 3.
We are, of course assuming a semantic conception of vagueness—what Burgess (1998) calls an “indeterminist conception”.
- 4.
If we further suppose that for any m, n such that m < n, if a man with m hairs is not bald then a man with n hairs is not bald, then it follows that there must be some uniquen for which a man with n hairs on his head is bald while a man with n + 1 hairs on his head is not.
- 5.
Where ‘ ⊨ ’ represents the generalised multiple-conclusion consequence relation.
- 6.
This presumption of uniqueness thus amounts to a presumption of consistency and cannot be assumed in a paraconsistent setting, on which more later.
- 7.
This is somewhat simplistic as regards both higher-order vagueness and higher-order sorites arguments. When discussing second-order vagueness one should, I think, recognise not only indeterminacy in respect of determinate baldness but also indeterminacy in respect of determinate non-baldness and indeterminacy in respect in borderline baldness, i.e. indeterminate baldness. Correlatively, one should recognise the existence of second-order sorites arguments using ‘determinately not bald’ and ‘borderline bald’, i.e. ‘indeterminately bald’. And so on for higher orders. The simple discussion offered above, however, is sufficient. It is easy to see how the newly recognised categories of higher-order vagueness might be employed in an attempt to dispel puzzlement engendered by each of the newly recognised higher-order arguments.
- 8.
Given that λ1 says of itself that it is not true, λ1 ↔ ∼ True(λ1). So (7) then simplifies to the claim that: ∼ DTrue( ∼ λ1) and ∼ DTrue(λ1).
- 9.
We are assuming that ‘True( ∼ p)’ is equivalent to ‘False(p)’, that the biconditional in (T) is contraposible and De Morgan’s laws.
- 10.
McGee, in effect, contests the inference from ‘λ2 is true’ to ‘λ2 is determinately true’, thus avoiding contradiction (McGee 1990, p. 7), but many including Field endorse the inference (Field 2003, p. 298). Suffice to say there is a debate to be had here, but space precludes its presentation. That is a story for another day.
- 11.
This presumption of determinateness thus amounts to a presumption of completeness.
- 12.
We shall leave discussion of non-adjunctive paraconsistent approaches for another day.
References
Beall, J.C., and M. Colyvan. 2001. Heaps of gluts and hyde-ing the sorites. Mind 110: 401–408.
Beall, J.C. 2002. Deflationism and gaps: Untying ‘not’s in the debate. Analysis 62: 299–305.
Burgess, J.A. 1998. In defence of an indeterminist theory of vagueness. Monist 81: 233–252.
Field, H. 2003. Semantic paradoxes and the paradoxes of vagueness. In Liars and heaps, ed. J.C. Beall and M. Glanzberg, 262–311. Oxford: Oxford University Press.
Field, H. 2008. Solving the paradoxes, escaping revenge. In Revenge of the liar, ed. J.C. Beall, 78–144. Oxford: Oxford University Press.
Fine, K. 1975. Vagueness, truth and logic. Synthese 30: 265–300.
Keefe, R. 2000. Theories of vagueness. Cambridge: Cambridge University Press.
McGee, V. 1990. Truth, vagueness and paradox. Indianapolis: Hackett.
Rolf, B. 1980. A theory of vagueness. Journal of Philosophical Logic 9: 315–325.
Tappenden, J. 1993. The liar and sorites paradoxes: Toward a nified treatment. Journal of Philosophy 90: 551–77.
Van Fraassen, B.C. 1966. Singular terms, truth-value gaps, and free logic. Journal of Philosophy 63: 481–85.
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Hyde, D. (2013). Are the Sorites and Liar Paradox of a Kind?. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_19
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