Responding to the Sorites
We began this chapter with a paradox. Any solution worth its salt must at minimum say which premise to reject. I reject the second premise; I deny that for any n, if a person with n + 1 cents is rich, so is a person with n cents.
Let us say that number (of cents) is a cutoff for the property of being rich if the corresponding conditional is false.
This definition generalizes:Definition: An element, x, of a sorites sequence for F is an interior point if and only if, if x is F then X (xs successor in the sequence) is F.
Definition: An element, x, of a sorites sequence for F is a boundary, or cutoff point if and only if it's not an interior point. I.e. it is not the case that if x is F then x' is F.
class=a7 style='text-indent:0cm'>I also think that properties, like being rich, have cutoff points. If one assumes classical logic this fact can be inferred, via contraposition, from the validity of our initial argument. Thus, from the fact that anyone with 100,000,000 cents is rich, and the fact that someone with 0 cents is not rich it follows that for some n, it’s not the case that someone with n cents is rich if someone with n + 1 is. That is, the property of being rich has a cutoff point.This means there is a number, n, such that the person with n + 1 cents is rich, whilst the person with n cents is not: a single cent can make the difference between being rich and not being rich.[3] If that number is 159,927,821 cents, for example, it follows that people with 159,927,821 cents are rich while people 159,927,820 cents are not rich.
Although this might sound wild, it is important to stress at this point that this conclusion was derived in classical logic only from the claim that millionaires are rich and the claim that people who have nothing are not rich. This conclusion is therefore common to anyone who accepts classical reasoning and these two premises.
The existence of cutoff points is not just the domain of epistemic theories of vagueness, it is quite general.Throughout the rest of this book I shall assume that the reader is familiar with, and is at least able to get into the mindset of someone who accepts this result; unless they do so they will get very little out of this book. It will be worth our while, therefore, to briefly say a bit about the alternatives. The alternatives can, broadly speaking, be divided into two kinds: (i) those that reject classical logic, and (ii) those that reject the first premise or accept the conclusion—i.e. those who maintain that either nobody or everybody is rich. Since the dialectic will be pretty symmetrical I clump both those options into the same category. I shall now argue that these alternatives are even worse than the result that a single cent can make the difference between being rich and not rich.
1.1
More on the topic Responding to the Sorites:
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- Bacon Andrew. Vagueness and Thought. Oxford University Press,2018. — 361 p. — (Oxford Philosophical Monographs), 2018
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