Wednesday, May 11, 2011

All elephants are small elephants

To: James D.
From: Geoffrey Klempner
Subject: All elephants are small elephants
Date: 15 May 2002 11:40

Dear Jim,

Thank you for your e-mail of 5 May, with your final essay for the Philosophy of Language, in response to the question:

'A one ounce elephant is a small elephant. Adding one ounce to a small elephant cannot make it not-small. Therefore, all elephants are small elephants.' - Give a logical analysis of this argument.

Well done for completing the program - I hope you enjoyed it. I will be sending a summary tutor's report and Pathways Certificate in due course.

The main problem here is seeing the problem.

It is a universally recognized principle of inference that if accept that the first member of a given series has property F, and if you also accept the conditional principle, that *if* that if any given member of the series has F *then* the next item in the series also has F, then you must accept that all items in the series have F. This is called the 'principle of mathematical induction', and forms the core of the logical analysis of Wang's paradox of the one ounce elephant.

Yet the conclusion, 'all elephants are small elephants' is false. So we have an apparently valid argument with a false conclusion - a paradox.

One extreme reaction would be to say, 'Who cares about logic?!' If we allow this reaction, then we would be saying in effect, that even if an argument leads from true premisses to a false conclusion, we are still fully entitled to believe that the argument is valid. But if leading from true premisses to a false conclusion is not sufficient to show that an argument is invalid, then nothing is! Every argument, even the most absurd, is valid. Few would embrace that conclusion.

But are the premisses true? No-one would deny that a one ounce elephant is a small elephant (as you correctly point out, this means 'small for an elephant' - some writers on Wang's paradox have focused on the fact that 'small' is an 'attributive adjective', requiring completion, 'small for an x', not realizing that many other vague predicates are not attributive adjectives).

What about the other premise? 'If an n ounce elephant is a small elephant, then an n+1 ounce elephant is a small elephant'. Here, there is some room to manoeuvre. As I remark in unit 13, the Oxford philosopher Timothy Williamson has argued that if we accept that statements like, 'Patsy is a small elephant', 'Betsy is not a small elephant' can express truths - in other words, if we accept that these statements have *truth conditions* - then it logically follows that there is, in fact, a precise cut-off point, even though we are necessarily ignorant of where this point is, where adding an ounce to an elephant turns a 'small' elephant into an elephant that is 'not small'.

Logically, this solution is impeccable. The solution also allows us to see how vague predicates can be useful (because we are ignorant of their precise truth conditions). The only problem is grasping how it could possibly be the case that for every size of elephant there is, in reality, a precise answer to the question whether it is 'small' or 'not small'.

My objection is that if we look at how a words 'small', 'elephant', or the phrase 'small elephant' are learned, there is no way we could explain how there can exist precise truth conditions for every statement about a 'small elephant'.

You say, 'in the case we are discussing we can as a society roughly agree as to whether or not an elephant is small or not', and later,' one must be pragmatic and objective in this situation and make an elephant part of the world and not an abstract thought.'

I think this is on the way to an acceptable response. Rough agreement cannot, as I have argued, deliver precise truth conditions. Yet elephants are part of the world, and not an abstract concept that we have invented. How do these two thoughts fit together?

Clearly we are dealing with something which we have, in a sense, 'invented', namely the concept 'small elephant'. Whereas one may suppose that the world in itself is something precise, the linguistic tools we have made for probing different aspects of the world serve a practical purpose.

In using language for a practical purpose, we are not always looking for precision. What we are looking for, however, is *agreement*. Ideally, we aim for complete agreement in the way we use vague terms, an agreement that matches imprecision for imprecision, uncertainty for uncertainty. Ideally, therefore, there ought to exist a cut-off point where adding an ounce to a small elephant makes it not small. This is not the same as saying that the cut-off point actually exists. The cut-off point is an ideal which can never be realized in practice because, as I remark at the end of unit 13, 'the smaller the differences between the objects of our judgements, the greater the effect of the differences between us.'

All the best,