rereading
this is a highly recommended introduction to an important topic in the philosophy of logic, philosophy of language and metaphysics. at the heart of the issue is the paradox of the heap (or sorites paradox, from the greek word 'soros' for heap). the greek logician eubulides is credited with discovering the sorites paradox along with the liar paradox - perhaps the deepest two paradoxes there are. eubulides also discussed the sorites paradox in roughly the following form, employing the predicate 'is bald' instead of 'is a heap':
suppose we have someone with a full head of hair in front of us. say he has a million hairs distributed evenly on the top of his head. it seems clearly correct to say that he is not bald. now we pull a single hair from his scalp. it still seems correct to say that he is not bald. indeed, when we consider it, it seems absurd to think that pulling a single hair from someone's head can make that person bald. so we continue pulling hairs from his scalp one by one, each time thinking that pulling one hair can make no difference to whether he is bald or not. but at the end of this process, we will have pulled every hair on his head. so at some point it has become incorrect to say that the guy is not bald and indeed correct to say that he is bald. yet where could have made the leap from the non-bald man to the bald man? at no time did it seem that there was such a leap, but there had to be a leap somewhere, unless we accept that the head no hairs is not bald. there had to be some n number of hairs such that the person with n hairs is not bald, but the person with n-1 hairs is bald. there is thus a magic hair that divides the bald and the non-bald people.
the paradox appears to prove that the following three propositions are inconsistent:
(1) Someone with a million hairs is not bald.
(2) If someone with N number of hairs is not bald, then someone with N-1 hairs is not bald.
(3) Someone with no hairs is bald.
the problem, of course, is that these all seem true but cannot all be true. one sort of reply is to make certain revisions to what is called 'classical logic' in which (1)-(3) is really inconsistent and to show that (1)-(3) are actually consistent in this new logic. another line of reply, which was first advanced by stoic logicisns is to deny (2). but it is natural to think that (2) just characterizes 'bald' as a vague predicate and to deny it is to actually deny that 'bald' is vague.
chrysippus and other stoic logicians following him flatly denied (2) and also rejected that (2) characterizes 'bald' as vague. instead they took the vagueness of 'bald' (or 'heap') to consist in this: for some n, someone with n hairs is not bald, but someone with n-1 hairs is bald, but one cannot know which n this is. so for them vagueness is an epistemic phenomenon and has to do with certain limits in our knowledge. this sort of idea was thought to be a dead end until more recently. logicians working on the problem took for granted that something along the first line of response to the paradox has to be correct and proposed different detailed solutions along those lines. however, each such proposal seems to give rise to paradoxes similar to the original one (eg. one may want to distinguish a notion of determinate true from truth, and base the definition of validity of an argument on the notion of determinate truth rather than truth. on this way of thinking, we can say that (1) and (3) are determinately true, but (2) isn't determinately true. that would still be consistent with (2) being true - so it is consistent to take (1)-(3) to be true. however, we quickly get into trouble when we actually spell this out in detail and we can generate analogous paradoxes using the notion of being determinately bald rather than being bald. - i'm really handwaving here, as it will take a whole lot of time to actually set this out and i can't do that now.)
in his book williamson provides a short history of the problem of vagueness from ancient greece to the middle of the 20th century, and then he scrutinizes the influential theories that have been around since mid-70s to this day. his own account can be seen as a development of the stoic's epistemic solution to the paradox. his most important contribution is to render the epistemic solution more palatable by explaining why on general epistemological principles we should expect there to be limits on our knowledge exemplified by these cases.
williamson is among the best contemporary philosophers out there. he has written many papers on metaphysics, epistemology and logic. i disagree with almost every major thesis he defends in these papers and in this book. but he is very good at explaining exactly where a philosophical problem lies and laying out the land for possible solutions.
all this may sound too academic for many people here, but i suggest taking a look at this book if you can. it doesn't really require prior knowledge of formal logic and semantics, though some familiarity would be of some use. it is a much easier read than much of the stuff mentioned on this board.
