Monday, November 28, 2011

Hempel's paradox of the ravens

To: Francis M.
From: Geoffrey Klempner
Subject: Hempel's paradox of the ravens
Date: 14 February 2006 11:18

Dear Frank,

Thank you for your email of 5 February, with your essay for the University of London Methodology paper, in response to the question, 'What is the paradox of the Ravens? What is the most effective way of dealing with it?'

I probably don't have to tell you that this is the best essay you have done for me. An excellent piece of work. Well done.

What made the difference this time is that you have really thrown yourself into searching out sources and following up your own questions.

I'm embarrassed this time in finding so little to comment on. If I were looking for ways to make things difficult for you, I would challenge your claim that a single instance does not confirm a generalization, to any degree. I agree with your argument. But just suppose (as is not unlikely) that in the exam you found the question, 'A single instance does not confirm a generalization to any degree.' Discuss.

If you think about it, there are many intuitively plausible counterexamples to this claim. You would have to explain why the general claim can still be maintained, in the face of these counterexamples.

If you stick your hand in a fire, it will get burned. Kiwi fruits are delicious. If you press the blue button, the computer instantly shuts down. And so on.

Generally the colours of things we find in nature do not in general support very interesting generalizations. There is no logical reason why ravens should be black rather than white. No doubt one could concoct a more or less plausible explanation in evolutionary terms, but this is only ad hoc.

However, there is a loose generalization which one can make about birds in general, that on the whole, a given species is one colour rather than a range of colours. I see a bright green bird and someone tells me that it's a tata bird from New Guinea. It would not be rash of me to form the expectation that all tata birds are green. The background assumption supporting this inductive inference is that significantly more species of bird are unicoloured (all members of the species have the same colouring) than non-unicoloured.

As I said, I think your argument is a good one, so the task would be to explain why these alleged counterexamples do not undermine the general point.

When something poses a danger or a threat, like fire, we are much more likely to err on the side of caution. Even if you've never tasted a Kiwi before, you know that fruits have characteristic tastes. Computers are generally designed for ease and predictability of use, so if pressing the blue button does action A on one occasion, its a pretty good bet that it will do the same on another occasion.

I don't have to tell you this, I'm sure you could have worked it out for yourself. Your general point still stands, that considerations of vagueness create a 'threshold' condition in virtue of which a vaguely defined number of instances is required before the process of confirmation can get off the ground.

There is only one point that I put a question mark against. This is where you discuss McMillan's example of the white handkerchief 'confirming' that all ravens are black. There is some controversy whether the first order predicate calculus rendering of 'All ravens are black' (For all x, if x is a raven then x is black') is faithful to how speakers normally understand this statement.

But let's suppose we agree that the first order predicate calculus rendering is not correct. Then 'All ravens are black' becomes,

A. For some x, x is a raven and for all y, if y is a raven then y is black.

While, 'All non-black things are non-ravens' becomes,

B. For some x, x is not black and for all y, if y is not black then y is not a raven.

In other words, the two formulations are no longer logically equivalent. What would be equivalent to A is,

C. For some x, x is a raven and for all y, if y is non-black then y is not a raven.

Thus Macmillan's extra condition, that for a generalization about Ps to be confirmed, Ps must exist, is sufficient to rule out C, even though C is equivalent to A. Being a generalization about non-black things, in order to be confirmed C requires the existence of a non-black thing, but does not require the existence of a raven.

All the best,

Geoffrey