Dov Samet was here presenting a new paper of his recently, and he happened to mention the Miller-Popper refutation of learning by induction. Everyone knows Hume’s two refutations: that induction of the form “A has always happened before, hence A will be true in the future/in the larger sample” is invalid because the necessary uniformity of nature assumption requires an inductive step itself, and that statements like “A has always happened before, hence A is likely to happen in the future” suffer a near identical flaw. But what of Bayesians? A Bayesian takes his prior, gathers evidence, and updates inductively. Can induction provide probabilistic support for a theory? Miller-Popper says no, and the proof is easy.
Let h be a theory and e be evidence. Let p(h) be the prior belief of the validity of h, and p(h|e) the conditional belief. Let the Bayesian support for h by e be defined as s(h|e)=p(h|e)-p(h); if this is positive, then e supports h. Note that any proposition h is identical in truth value, for all propositions e, to (h or e)&(h or [not e]). Replace h in the support function with that statement, and you get
s(h|e)
= p([h or e]&[h or not e]|e)-p([h or e]&[h or not e])
= p([h or e]|e)+p(h or not e|e)-p(h or e)-p(h or not e)
= s(h or e|e)+s(h or not e|e)
with the first equality holding by the obvious independence of terms inside the probability operator. So what does this mean? It means that the support of evidence e for theory h is just the sum of two types of support: that given to the proposition “h or e” and that given to the proposition “h or not e”. The first by definition is positive, and the second term by definition is negative. So only the first term can said to be providing positive support for h from the evidence. But the truth of (h or e) follows deductively from the assumed truth of e. Every part of the support for h from e that does not deductively follow from e is the second term, but that term is negative! Induction does not work. Another way to see this is with a slight restatement of the above proof: induction only provides probabilistic support for a theory if p(if e then h|e) is greater than p(if e then h). The above math shows that such a statement can never be true, for any h and any e. (There is a huge literature dealing with whether this logic is flawed or not – Popper and Miller provide the fullest explanation of their theorem in this 1987 article).
So an interesting proof. And this segues to the main paper of this post quite nicely. David Miller, still writing today, is one of the few prominent members of a Popper-style school of thought called critical rationalism. I (and hopefully you!) generally consider Popper-style falsification an essentially defunct school of thought when it comes to philosophy of science. There are many well-known reasons: Quine told us that “falsifying one theory” isn’t really possible given auxiliary assumptions, Lakatos worried about probabalistic evidence, Kuhn pointed out that no one thinks we should throw out a theory after one counterexample since we ought instead just assume there was a mistake in the lab, etc. And as far as guiding our everyday work as social scientists, “learn empirical truth as a disinterested body” is neither realistic (scientists cheat, they lie, they have biases) nor even the most important question in philosophy of science, which instead is about asking interesting questions. Surely it is agreed that even if a philosophy of science which provided an entirely valid way of learning truth about the world, it would still miss an important component, the method for deciding what truths are worth learning with our finite quantity of research effort. There are many other problems with Popper-influenced science, of course.
That’s what makes this Miller paper so interesting. He first notes that Popper is often misunderstood: if you think of falsification from the standpoint of a logician, the question is not “What demarcates science, where science is in some way linked to truth?” but rather “What research programs are valid ways of learning from empirical observation?” And since induction is invalid, justificationist theories (“we have good evidence for X because of Y and Z) are also invalid, whereas falsification arguments (“of the extant theories that swans are multicolored or always white, we can reject the second since we have seen a black swan”) is not ruled out by Hume. This is an interesting perspective on Popper which I hadn’t come across before.
But Miller also lists six areas where he thinks critical rationalism has done a poor job providing answers thus far, treating these problems through the lens of one who is very sympathetic to Popper. It’s worth reading through to see how he suggests dealing with questions like model selection, “approximate” truth, the meaning of progress in a world of falsification, and other worries. Worth a read.
http://www2.warwick.ac.uk/fac/soc/philosophy/people/associates/miller/prague.pdf (December 2009 working paper)
A Fine Theorem 
I’ve read a good deal of Popper but only heard of this argument when Dov mentioned it; I spent some time yesterday reading up on it. Refuting the argument is a cute and perhaps educational intellectual puzzle, like refuting a proof that all triangles are isoceles, but it’s hard to believe people took it so seriously. Is there any reason that deductive support couldn’t be a special limiting case of inductive support, in which case it could very well be that
(deductive support for 1st component)+(inductive anti-support for 2nd component)= inductive support?
Popper-Miller take it as self-evident that this couldn’t be so, without further proof or a formal definition of inductive support. I also like Good’s refutation involving independence…you can’t decompose something into a conjunction and talk about the support for each component without also worrying about how the evidence affects any dependence. (Generically, you would have negative dependence ex ante wbich becomes zero dependence ex post.)
About as convincing as Descartes’ arguments about existence of a higher power. If a two-line argument says there is no induction by your definition, i take this as telling me something about your definition, not about induction.
Did you read the 1987 royal society paper I linked to? They point out that the definition of induction used was pretty common and was not really challenged until they used the definition in the above proof. I think the better counterarguments are the ones about the support function itself: if you were asked to define “Bayesian probablistic support,” you would probably go with a change in likelihood ratio, right? And that changes everything.
That said, I don’t mind 2 line refutations. Hume’s argument about the impossiblity of induction is also basically 2 lines long, and nonetheless completely correct.
Very interesting post. One point I’d like to add though is that all of the criticisms of critical rationalism you mentioned in the third from last paragraph were all answered adequately by Popper himself- some as part of his original theory and some later in response to critics.
For instance, he wrote extensively about the Quine-Duhem thesis decades before Quine ever mentioned it. His response is that falsifications are not conclusive and the decision where to place locate the falsified hypotheses is itself conjectural.
Moreover, in order for a theory to be objectively falsified, the falsifying experiment must be *reproducible*. This is also part of Popper’s original theory and is not a change to accommodate Kuhn. Nor does Popper say that the theory must then be completely abandoned- rather he simply points out that relative to the reproducible experimental result, the theory (taken as a whole together with the auxiliary assumptions) is falsified. This is simply a matter of logic.
Popper also didn’t make any claims about the supposed disinterestedness of particular scientists and instead stressed that it is the social aspect of science that is so necessary in correcting these biases, which are unavoidable anyhow.
In my view nearly all the popular criticisms of Popper’s theory of science are invalid. There’s some additional notes here:
http://www.criticalrationalism.net/2012/10/27/short-rejoinder-to-standard-crits-of-cr-and-popperism/
“I (and hopefully you!) generally consider Popper-style falsification an essentially defunct school of thought when it comes to philosophy of science.”
That is absurd, Popper’s contribution went far beyond the logic of testing and he carefully replied to all the so-called criticisms of his position when he wrote his first book. The literature is full of invalid claims about Popper’s ideas because apparently people just repeat what they are told by critics and don’t check the original sources. For a bigger picture of Popper’s contribution, consider the “turns” that he introduced.
http://www.the-rathouse.com/Pop-Schol/PopperTurns.html
Rafe and Steven – I agree completely with you guys. This is why I am careful to blame “Popper-style” or “Popper-influenced” naive falsificationism, rather than Popper’s own beliefs.
Ah, cool! Personally, I’m not even greatly opposed to naive falsificationism as a first approximation- it’s certainly over-simplified, but the basic logical structure is fundamentally correct I think.
This account by Feynman for instance, while very schematic, is not false: