Abstract
The paper starts by arguing that sorites paradoxes are inclosure paradoxes, and therefore require the same sort of solution as other inclosure paradoxes, notably the paradoxes of self-reference. It then puts forward such a solution. Tolerance conditionals are argued to be material conditionals, and sorites arguments fail because of the failure of detachment for such conditionals. Soritical arguments show that a contradiction occurs somewhere down the length of the sorites statements, though they do not locate where. The final part of the paper considers higher-order vagueness, and argues that it is essentially the same as extended paradoxes of self-reference, to be handled in the same way, by constructing a single “soritically closed” language.
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- 1.
For details of the following, see especially Priest (1995, Part 3).
- 2.
In naive set theory, the comprehension schema gives: y ∈ { x : Px} ↔ Py, and contraposition gives y∉{x : Px)} ↔ ¬Py.
- 3.
The Technical Appendix to Part 3 of Priest (1995) constructs models of the Inclosure Schema where some ordinals are consistent and some are not. Sect. 4 of the Appendix gives a model in which inconsistent ordinals need not be consecutive.
- 4.
It is clear from the diagram that {x : Px} ∩ { x : ¬Px} is not empty. But since this set is \(\{x : Px\} \cap \overline{\{x : Px\}}\), it is empty as well. It is difficult to represent this fact in a consistent diagram!
- 5.
In what follows, we will take this to be the logic LP of Priest (1987, Chap. 5); but matters are much the same in virtually every paraconsistent logic.
- 6.
For the sake of definiteness, let this be the conditional of Priest (1987, §19.8).
- 7.
Specifically, no inconsistencies involving only the grounded sentences of the language (in the sense of Kripke) are provable. See Priest (2002, § 8.2).
- 8.
Strictly speaking, {x; x ≥ m}, since every natural number must be in either the extension or the anti-extension of P. But what happens for numbers greater than n in irrelevant for our example.
- 9.
This important observation is due to Colyvan (2007).
References
Colyvan, M. 2007. What is the principle of uniform solution and why should we buy it? Presented at the annual meeting of the Australasian Association of Philosophy, University of New England, Armidale NSW.
Hyde, D. 1997. From heaps and gaps to heaps of gluts. Mind 106: 641–660.
Priest, G. 1987. In Contradiction: A study of the transconsistent. Dordrecht: Martinus Nijhoff; second, extended, edition, Oxford: Oxford University Press, 2006. References are to the second edition.
Priest, G. 1995. Beyond the limits of thought. Cambridge: Cambridge University Press; second, extended, edition, Oxford: Oxford University Press, 2002. References are to the second edition.
Priest, G. 2002. Paraconsistent logic. In Handbook of philosophical logic, 2nd edn., vol. 6, ed. D. Gabbay and F. Guenthner, 287–393. Dordrect: Kluwer.
Priest, G. 2010. Inclosures, vagueness, and self-reference. Notre Dame Journal of Formal Logic 51: 69–84.
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Priest, G. (2013). Vague Inclosures. 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_20
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