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Significance of the paradox of the heap


To: Lishan C.
From: Geoffrey Klempner
Subject: Significance of the paradox of the heap
Date: 25th August 2009 14:14

Dear Lishan,

Thank you for your email of 17 August with your first essay for the Pathways Philosophy of Language program, in response to the question, 'How would you explain to a non-philosopher the philosophical significance of the paradox of the heap?'

As you have probably discovered, it is no easy matter to explain to one's non-philosophical friends why a particular philosophical problem is regarded as significant or important. None more so than with the paradox of the heap. The usual response is sheer exasperation: 'Of course some men are bald, some men are not bald and some are in between!'

What the non-philosopher misses is that here we have an example of an argument with impeccable credentials leading to a contradictory conclusion. You can't do mathematics without the principle of mathematical induction. Yet here it leads to what seems an impossible result.

Your statement of the argument is not quite correct, from a mathematical standpoint. The argument goes:

1. A man with 0 hairs is bald.
2. IF a man with n hairs is bald THEN a man with n+1 hairs is bald.
3. Conclusion: A man with 100000 hairs is bald.

What the non-philosopher probably doesn't realize is that you can run exactly the same argument for 'bald' and 'partially bald' (='in between'). Make as many distinctions as you like, if they are ultimately vague distinctions then the paradox applies, as before.

You say that this argument 'is caused by an informal fallacy', but what exactly is the fallacy? If the premisses 1. and 2. are true, then 3. follows. To question the validity of the argument you would have to question the validity of the principle of mathematical induction.

But IS 2. true? Obviously, we have little difficulty in imagining that if you add one hair to a bald man, he remains bald. Yet, if one were to do this in real life (imagine painstakingly transplanting hairs on to the head of a bald man, one follicle at a time) we WOULD say at some point, 'Hey, you're no longer bald!'

The weird thing is that there is no fixed cut-off point. One observer would say this at, say 900 hairs, another at 1100 hairs.

There are two possible reactions to this. The first is to state that you can't use mathematical induction because 'bald' is a vague concept. The principle of mathematical induction is only valid for precisely defined concepts. That saves mathematics. However, it still leaves us with the problem of explaining what claim is made when we make a statement which employs a vague concept. Worse, it leaves us with the conclusion that the principles of reasoning and logic don't apply universally, but only to neatly defined areas of discourse.

The second alternative is to accept that vague statements have a precise content -- precise truth conditions -- only we are ignorant of what exactly these truth conditions are. In that case, the reason why mathematical induction fails is very simple: there IS a point where adding one hair makes a bald man not-bald. However, competent speakers do not know where this point is (and cannot know -- that's what gives vague concepts their utility). This solution, which seems at first extremely counter-intuitive was proposed by a British philosopher Timothy Williamson a few years ago. It is a very elegant solution which saves logic and demolishes the paradox.

My problem with this is that no explanation has been given of how it is *possible* for vague statements to have truth conditions in this way. What kinds of *facts* about our use of language do these truth-conditions capture? How is it that speakers are largely able to agree (within, say, plus or minus 10%) on how a concept like 'bald' is applied without knowing what these truth conditions are?

In the program, it is suggested that the facts captured by these truth conditions might be statistical facts (like the observation about the 900 or 1100 hairs). Question enough people, and a reliable pattern will emerge, enabling us to define precisely the cut-off point for baldness. My objection is that there is a false assumption here that everyone has equal authority to make these kinds of judgement. But do they? If not, then the attempt to distil truth conditions from the statistics breaks down.

I realize that at this point we have left our non-philosophical friend far behind. What I would say to them is that what the paradox shows is that there is a necessary gap between the world as described by science and the world of human perception. There are truths of ordinary experience which cannot be scientifically or mathematically defined, yet they are part of reality nonetheless.

All the best,