To: Craig S.
From: Geoffrey Klempner
Subject: Carl Hempel's paradox of the ravens
Date: 9th February 2009 11:36

Dear Craig,

Thank you for your emails of 27 and 28 December with your essay in response to the University of London Logic question, 'What is the paradox of the ravens? What is the most effective way of dealing with it?'

This is in many ways a model answer to the question, which on a first reading does not give me much to comment on.

The question asked for the 'most effective way of dealing' with the paradox, rather than a more general reflection on the nature of induction and the light which the paradox casts on our belief that induction is a source of knowledge about the world. You have stuck to this rubric, offering observations which defuse the puzzle to a large extent. Your conclusion: the 'paradox' is not something to be too puzzled about (as Hempel indeed said).

One issue which might raise an eyebrow is your reference to Bayes Theorem. It could be argued that this is using a sledgehammer to crack a nut. An 'effective' method of dealing with a problem is judged on two grounds: its results and also its cost. If the only way to deal with the Ravens Paradox is to appeal to Bayesian probability -- given that at least some applications of this concept of probability are contested -- then that would be a more significant result than (arguably) the original paradox warranted.

This is not a reason to not mention Bayes: rather, it is worth making the point that some solutions have a higher cost than others. It is relevant that the discovery that different ways of judging probability lead to differing probability judgements is itself a very contentious area of debate (see A.J. Ayer's excellent book 'Probability and Evidence').

Another example of an 'expensive' way of dealing with the Ravens paradox would be to side with Popper on the question of the general value of induction as the paradigm of scientific method. (Of course, Popper still has the problem of explaining why some methods of 'corroboration' are better than others.)

You say at the beginning of your essay that 'a hypothesis is confirmed (supported) if its consequences are observed to be true' then claim that this is 'equivalent' to a generalization being confirmed by observing its instances. I wonder about this. Inference to the best explanation (what Pierce called 'abduction') differs in significant ways from the model of confirming/ deriving a generalization from observation of instances. The notion of 'best' explanation explicitly appeals to our sense of what is a good or bad 'theory' whereas the model of induction seems on the surface purely mechanical. Of course, this is not the case, as you show with your examples of background knowledge affecting judgements based on the observation of instances. However, you don't have to be a Popperian to hold that inference to the best explanation is the primary notion.

Bertrand Russell offered the definitive 'refutation' of knowledge by induction: the chicken who spends each day happily pecking around the farmyard -- until the day comes when it is taken to slaughter. This shows in the most vivid way the uselessness of induction in the absence of background knowledge.

Another point that could be made is that generalizations like the colour of ravens are not that significant, given what we know about the variations of colour in members of the same species. It is not a 'deep' property, as the example of albinos shows. No-one has ever built a theory on the blackness of ravens. Why not pick a more realistic example? As soon as you start casting around for suitable examples you realize that the generalizations that interest us are relatively rarely those which you arrive at simply through observing instances.

The reflections here would not be sufficient to defuse Goodman's Paradox, which is an interesting point in itself. Why is Goodman more of a challenge than Hempel? That would be a possible exam question.

All the best,

Geoffrey