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Paradox of the heap for non-philosophers


To: Ian C.
From: Geoffrey Klempner
Subject: Paradox of the heap for non-philosophers
Date: 18th March 2010 12:20

Dear Ian,

Thank you for your email of 7 March, with your first essay for the 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?"

In your essay, you make the case for the utility of vague terms in ordinary English. As you point out, the vague terms that we use, like 'heap', are sufficient for the purposes for which we use them, and when greater precision is needed, we use more precise, or specialized vocabulary.

In making this point, it is important to distinguish between cases where it really doesn't matter what word you used, and cases where a person understands what you said, in the same sense as you intended to say it. If we are driving a noisy truck and I shout to you, 'Be careful there's a --- of sand just around the next corner', I succeed in getting the information across even though you didn't actually hear the word '---'. You surmised, correctly, given the context, that whatever a --- of sand is, it is something to avoid. That's not generally a good model for the way vague predicates work.

You could also have gone on to consider possible definitions, e.g. of a heap, and why they would be of no use. Imagine a council bye-law prohibiting the depositing of heaps of sand, which includes a definition of a heap of sand as one which has such-and-such minimum circumference and height. This definition of a 'heap', if applied to everyday usage, would make a statement like, 'I noticed a heap of sand in your drive' dependent for its truth on certain precise measurements being taken. Moreover, we would need separate definitions for 'heap of books', 'heap of clothing' etc.

A non-philosopher would have no difficulty in agreeing to all of this.

However, the conclusion a reader might come to is that there doesn't seem to be a paradox here, let alone any observation of any philosophical significance. That's two things you need to explain.

Why is there a paradox?

I take it that you admit the validity of logic, and the principle of mathematical induction. Severe consequences follow for human knowledge if we don't accept this.

However, to take your example, it follows from the statement,

1. One grain of sand cannot make a heap


2. If you add one grain of sand to a collection of m grains of sand which cannot make a heap, then the resulting collection of m+1 grains cannot be a heap


3. For all n, a collection of n grains of sand cannot make a heap.

This is a paradox. A paradox is an apparently sound argument which leads to an impossible or contradictory conclusion. There are paradoxes whose philosophical significance would be quite difficult to explain to a non-philosopher, e.g. Russell's paradox. Is this one of those paradoxes?

It is uncontroversial (as you have argued) that many of the terms we use in ordinary language are vague. However, I think that it is more controversial that when we are using ordinary language, we have to abandon logic. Of course, not everyone has an appreciation of what logic is, and why we need it. But I take it that you don't need to be a philosopher in order to appreciate the importance of logic.

This is a non-unfamiliar situation in philosophy. We encounter a problem or a paradox whose solution requires that we give something up, that we reject an assumption which we previously held, not realizing that the assumption in question was problematic.

We need logic. And yet we have to recognize that we can't always use logic. The laws of logic as such have no exceptions. But the application of those laws to the everyday world, paradoxically, does have exceptions. Ordinary language isn't built for logic. It's built for negotiating our way around the world (in order to avoid heaps, or find things in heaps etc.)

The philosophical significance of this observation is that we have to revise what seems at first a very plausible view of the nature of linguistic usage, namely, that words are learned, and used, according to rules. 'Logic' and 'rules' are correlative notions. You either obey the rule or disobey it. An inference is either logically valid or logically invalid. There is no third alternative.

But if words are *not* used according to rules in this sense, then how on earth do we manage to talk coherently? How do words work? What is meaning?

From the fact that we successfully communicate, and collaborate in our activities, that whatever it is that keeps the use of language 'on track' is doing so pretty effectively. As a philosopher, that is something to investigate. A plausible theory or model of the meaning of 'meaning' doesn't work. Hence the challenge for the philosophy of language.

All the best,