The raven paradox

4 July, 2017 at 19:03 | Posted in Theory of Science & Methodology | 2 Comments

 

Besides illustrating that it is simply not a good description of how we make inferences in science to assume that non-black armchairs confirm the hypothesis that all ravens are black, Hempel’s paradox — at least in my reading of it — makes a good argument for a causal account of confirmation of empirical generalizations. Contrary to positivist theories of confirmation, the paradox shows that to have a good explanation in sciences, we have to make references to causes. Observed uniformity does not per se confirm generalizations. We also have to be able to show that uniformity does not appear by chance, but is the result of causal forces at work (such as e.g. genes in the case of ravens.

Assume you’re a Bayesian turkey (chicken) and hold a nonzero probability belief in the hypothesis H that “people are nice vegetarians that do not eat turkeys and that every day I see the sun rise confirms my belief.” For every day you survive, you update your belief according to Bayes’ Rule

P(H|e) = [P(e|H)P(H)]/P(e),

where evidence e stands for “not being eaten” and P(e|H) = 1. Given that there do exist other hypotheses than H, P(e) is less than 1 and a fortiori P(H|e) is greater than P(H). Every day you survive increases your probability belief that you will not be eaten. This is totally rational according to the Bayesian definition of rationality. Unfortunately — as Bertrand Russell famously noticed — for every day that goes by, the traditional Christmas dinner also gets closer and closer …

Studying only surface relations won’t do. Not knowing the nature of the causal structures and relations that give rise to what we observe, explanations serve us as badly as the one used by the turkey. Not knowing why things are the way they are, we run the same risk as the Russellian turkey.

No causality, no confirmation/explanation.

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2 Comments

  1. Wikipedia has a somewhat tortuous discussion of the raven paradox. As a mathematician it seems to me that Hempel is using terms like ‘confirmation’ very loosely. There are various precise interpretations one can try, but then – as far as I can see – the paradox disappears.

    I am not sure how you are making sense of the video, but there are clearly two kinds of ‘causality’ that are relevant. One is ‘in the world’, the other is in how you are sampling the world. The latter is clearly very relevant. Are you suggesting that the former is also?

    As a thought experiment, suppose that an urn contains many balls with two labels: one the name of an object (e.g. ‘raven’, ‘armchair’) the other a colour (e.g. ‘black’). An oracle can select a ball at random whose label matches some criteria (e.g. ‘is a raven’). You ask for balls according to relevant criteria and inspect their labels. The result is consistent with the notion that ‘all ravens are black’. I see no paradox, and nor do I see a role for the first type of causality. Am I missing something?

    • I think Peter Lipton has the best argument around this paradox in his “Inference to the best explanation” (2nd ed, chapter 6). My own view is highly influenced by Lipton’s.


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