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
= 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)