Consider the following valid piece of reasoning:[2]

1.Anyone with 100,000,000 cents (a millionaire) is rich.

2. For any n, if a person with n + 1 cents is rich, a person with n cents is rich.

3.

This argument can be formalized and logically proven from the rule of modus ponens (from A and ‘if A then B^, infer B) and universal instantiation (from ‘everything is F infer ‘a is F ’)—both of which are principles accepted by pretty much everyone who has participated in this debate. Anyone who accepts the premises of this argument and closes their beliefs under modus ponens and universal instantiation will therefore accept the conclusion. Yet the premises are seemingly acceptable and the conclusion, seemingly, is not.

When I present this paradox to students for the first time, or when I am explaining what I do to friends and family, I commonly receive the following dismissive response:

There is no paradox here, it all just depends on how you define the word ‘rich’.

I am sure many reading this are familiar with this response. How exactly does this remark solve the paradox? Does it recommend accepting the conclusion, and if not which premise does it recommend denying? After a little thought some will elaborate as follows:

Once you have said what you mean by ‘rich’ it will become clear which premise to deny. If you stipulate ‘rich’ to mean having more than 20,000,000 cents ($200,000) then the second premise fails, but there is nothing puzzling about this—read this way the second premise entails ‘if a person with 20,000,001 cents has more than 20,000,000 cents, then a person with 20,000,000 cents has more than 20,000,000 cents’ and this is quite clearly false.

Of course, there is nothing particularly special about the second premise.

I hope that everybody agrees that this ‘solution’ is not satisfactory. Whether someone is rich or not does not depend on what you have decided to stipulate the word ‘rich’ to mean. There is a quite straightforward empirical test one can perform at home to demonstrate this: log into your bank account, stipulate away, and observe that you do not become one bit richer. Your balance will remain exactly the same—if you were rich before you will remain rich and if you weren’t you will remain that way too. This is a get-rich-quick scheme that will not work.

To think that you have solved the sorites paradox by saying something about the word ‘rich’ is wrong headed—the word ‘rich’ and the way it is used has nothing intrinsically to do with being rich. There is, of course, another sorites paradox not about rich people but about people to whom the word ‘rich’ applies when it is being stipulated that ‘rich’ means such and such. However, deflating this paradox does very little to address the original.

This is the explanation I give to my students, and for what it is worth, it is the explanation I gave before I endorsed the views defended in this book. It is also, I hope you’ll agree, good old common sense.

It should be extremely surprising, then, to discover that almost all contemporary accounts of vagueness commit something like the conflation noted above. Invariably the conflation is less blatant, but it is there nonetheless.

Even with such a preliminary sketch of the view, it is hard not to think that it is ignoring the moral we drew from our discussion of the naive response to the paradox—that one cannot solve the sorites paradox just by saying something about the word ‘rich’. Even if we could convince the entire English speaking population to use the word ‘rich’ differently, doing so would not make you any richer or poorer. The account therefore does very little to explain why we find it hard to imagine that one person could be rich while another person possessing one less cent isn’t. The most it does is explain why it’s hard to imagine that the word ‘rich’ could apply in English to one person without applying to someone with one less cent. This would be fine if we had asked about the variant sorites involving people to whom ‘rich’ applies in English, but once again we have done little to address the paradox we started with which was about rich people. (Indeed, some semantic indecision theorists are even quite explicit about the centrality of the variant sorites. For example McGee and McLaughlin, after proving there must be a sharp cutoff concerning what looks red to someone, write ‘How do we get from the thesis that, for some n the nth tile looks red to you but the n + 1th tile does not to the metatheoretical conclusion that the concept expressed by the phrase “looks red to you” has a sharp boundary? TThis, it seems to us, is the crux of the sorites problem.' (McGee and McLaughlin, [104]).)

To see why such accounts are explanatorily unsatisfactory note that one doesn't need to be familiar with the English language to appreciate the original paradox.

These preliminary points, I think, will leave many people unconvinced. After all, if there's one thing that almost everyone agrees on—whether you're an epistemicist or supervaluationist or something else—it's that vagueness has something to do with the way we use language. And if it isn't linguistic it must be either metaphysical or purely epistemic, and neither of these options seem particularly promising. The primary aim of this book is to outline an account of vagueness that isn't fundamentally a linguistic phenomenon or, indeed, a metaphysical or purely epistemic phenomenon (indeed, this seems to be a false trilemma).

Rather than treat the study of vagueness as a branch of the philosophy of language, the alternative view places the study of vagueness squarely in epistemological terms— here vagueness is characterized by its role within a theory of rational propositional attitudes; specifically belief and desire (see the chapters in part II). I outline the main features of this view in chapter 3.

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