Paradoxes

The fourth in Francis Moorcroft's series looking at some the classic philosophical paradoxes.

No. 4 The Sorites Paradox

Francis Moorcroft

So far we have only considered paradoxes that involve self-reference. 'This sentence is false', 'The set of all sets that does not contain itself’, 'No-one knows this proposition'. It's now time to look at other kinds of paradox.

A man with 10,000 hairs on his head isn't bald and surely subtracting one hair from his head can't make a man who isn't bald into a bald man. So a man with 9,999 hairs on his head isn't bald and subtracting one hair from his head can't make a man who isn't bald into a bald man. So a man with 9,998 hairs on his head isn't bald… So a man with 1 hair on his head isn't bald and subtracting one hair from his head can't make a man who isn't bald into a bald man. So a man with no hairs on his head is not bald.

To give a second example: 1 stone is not a heap of stones and adding one stone to what is not a heap cannot make it into a heap. So 2 stones are not a heap and adding a stone to what is not a heap cannot make it into heap. So 3 stones are not a heap../. So 9,999 stones are not a heap and adding another stone cannot make it into a heap. So 10,000 stones are not a heap.

Paradoxes of this form are known as Sorites and are credited to Eubulides. The title 'Sorites' is actually a pun. In Greek it means 'heap' and the second example above involves a heap; but also it stresses the form of argument that is involved: the argument relies on a step by step addition (or subtraction), which is a heap of premises, and asking the question. when is something a heap - or bald - or not?

To make this a little clearer, it may be worth saying more about what a paradox is, and stating the paradox more formally. A paradox can be defined as an argument which starts from premises which appear to be true and yet, after reasoning that looks valid, ends up with an apparently false conclusion. To put the second example above more formally, we have the premises

1 stone is not a heap of stones
2 stones are not a heap of stones
3 stones are not a heap of stones
:
:
and the conclusion

Therefore, 10 000 stones are not a heap of stones.

So the premises appear to be true, the conclusion seems to follow validly from the premises (by reasoning that adding one stone to what is not a heap cannot form a heap) yet the conclusion is false - a paradox. As the conclusion cannot be accepted then either the premises must be shown to be false or the reasoning shown to be invalid. But which?

One way of resolving these paradoxes is to recognise that they involve vague concepts such as ^‘bald' and 'heap' and it is very difficult in certain situations to decide whether or not these words apply to a particular collection of stones or a particular person - such terms are difficult to apply in borderline cases. So maybe we should either precisely define what we mean by a heap or just accept that using vague concepts leads to incoherence and avoid using them.

The first option would involve giving a numerical value for what we mean by 'a heap', that is stating exactly the value of n such that n stones aren't a heap but that n + 1 stones are a heap. But surely any such value would be arbitrary. The second option is tempting until we realise just how many concepts used in ordinary language are vague: for example, I am tall and my sister isn't, but what about my girlfriend, when her height is midway between the two? When do I have a long day at work, a late night, or a large meal? Certainly language contains some concepts that are not vague - such as 'is 1.95 metres tall' or 'worked for 10 hours and 47 minutes' - but when such a large amount of our concepts are vague we may start to worry that no-one ever understands what anyone else is talking about.

While recognising that there is a great deal of vagueness in language, it should be distinguished from other problematic aspects. The word 'bank' for example, is not vague but ambiguous: It may apply to the side of a river or a financial institution, The word 'game' is neither vague nor ambiguous but applies to a great many different activities by virtue of its generality. Such distinctions may be of use in assessing what counts as vagueness or not.

One final question: when we describe something as 'vague' do we always have a clear idea of whether the concept applies or not? That is, is ‘vague’ a vague concept or not...

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The paradoxes series will be updated in mid-November

Previous articles in the Paradox series

1. The Paradox of the Liar
2. Russell's Paradox
3. The Paradox of Prediction