Where were we? Robert Paul refers me to a very long and technical article in the Stanford Encyclopedia that, he says afair, presents at least two versions of inductive probability. Afair RP does not identify these two versions [when there are least three (e.g. enumerative induction, Bayesian logicism, Bayesian subjectivism)]. Nor does RP indicate which of these, if any, he supports ? or why. In other words, Robert Paul acts as though the ball is back in my court. But without indicating where the ball is ? instead I guess I am supposed to find it in a thicket of technicality. While this is one way to respond, it ?leaves a lot to be desired? ? it even leaves unclear whether the ball has, in fact, been returned. There is another less than satisfactory aspect to responding to an argument (I know, I know ? RP says there is no argument; but this does not mean he is right) by referring to a lengthy, technical article on the topic in general. It gives the other side a target that is too big and unwieldy for useful discussion ? it?s as if someone suggests to me that its going to rain because there are dark clouds above and I respond ?I see no argument and so offer none back, but why don?t you wade through this technical general manual on weather forecasting?? While I may take time to deal with some of the various matters discussed in the article (as best I can), might I refer Robert Paul to a much, much shorter and much, much less technical piece by Popper that can be found in the ?Introduction 1982? at xxxvii of ?Realism and the Aim of Science?. I am assuming that whatever his view of my posts on three boxes, Robert Paul will agree that at least Popper?s piece contains an argument ? indeed one, like the Stanford article, that invokes Bayes?s theorem. Perhaps Robert Paul would like to pick apart this short argument, pinpoint its errors etc. Others also might like to do so and, since they may not have access to Popper?s book, here is the argument (reproduced without italics and subscript):- ?It is sometimes objected to my theory that it cannot answer Nelson Goodman?s paradox. That this is not so will be seen from the following considerations which show, by a simple calculation, that the evidence statement e, ?all emeralds observed before the 1st January 2000 are green? does not make the hypothesis h, ?all emeralds are green, at least until February 2000? more probable than the hypothesis ?all emeralds are blue, for ever and ever, with the exception of those observed before the year 2000, which are green?. This is not a paradox, to be formulated and dissolved by linguistic investigations, but it is a demonstrable theorem of the calculus of probability. The theorem can be formulated as follows: The calculus of probability is incompatible with the conjecture that probability is ampliative (and therefore inductive). The idea that probability is ampliative is widely held. It is the idea that evidence e ? say, that all swans in Austria are white ? will somehow increase the probability of a statement that goes beyond e, such as h2, ?all (or most) swans in regions bordering Austria are white?. In other words, the idea is that the evidence makes things *beyond* what it actually asserts at least a little more probable. (This view was strongly defended by Carnap, for example). The view that probability is ampliative was suggested especially by the following theorem (h=hypothesis; e=empirical evidence; b=background knowledge): Let p(h,b) ≠ 0. Further, let e be favourable evidence (that is, e follows from h in the presence of b, so that p(e,b) ≠ 1 and p(e,hb) = 1. Then p(h,eb) > p(h,b). That is to say, the favourable evidence e makes h more probable, even though h says more than e. And this holds for every new e1, e2 ?., which satisfy similar conditions. It therefore seems that increasing favourable evidence goes on supporting h; and so it seems that the support is ampliative. But this is an illusion, as can be shown as follows: Let h1 and h2 be any two hypotheses supported by e in the presence of b, so that p(e,b) ≠ 1 and p(e,h1b) = p(e,h2b) = 1 Let R1,2 (prior) = p(h1b)/p(h2b) be the ratio of the probabilities of h1 and h2 prior to the evidence e, and let R1,2 (posterior) = p(h1,eb)/p(h2,eb) be the ratio of the two probabilities posterior to the evidence e. Then we have, for any h1,h2 and e that satisfy the above conditions: R1,2 (prior) = R1,2 (posterior). This follows almost immediately from p(a,bc) = p(ab,c)/p(b,c), that is, from Bayes?s theorem. What does R1,2 (posterior) = R1,2 (prior) signify? It says that the evidence does not change the ratio of the prior probabilities, whether we have calculated them or freely assumed them, provided the two hypotheses can both explain the evidence e. But this means that if we let h1 = all swans in some region greater than Austria are white; h2 = all swans in the world are non-white except those in Austria which are white; e = all swans in Austria are white, then, assuming any prior probability for h1 and h2 you like: their ratio R1,2 (prior) remains unaffected by the evidence. Thus there is no spill-over, no ampliative support: there is no ampliative probability, neither for swans nor for emeralds. And this is not absurd, but tautological (and it is unaffected by translation).? As I understand it, this argument might be put in less formal terms ? layman?s terms, if you like ? as follows:- where increasing amounts of evidence are consistent with two inconsistent hypotheses, that evidence does not increase their probability relative to each other, and this means it does not increase either of their probabilities full stop ? not in any inductive sense. (Both will, however, remain more probable than hypotheses falsified by the increasing evidence). Is this wrong? Donal ___________________________________________________________ Try the all-new Yahoo! Mail. "The New Version is radically easier to use" ? The Wall Street Journal http://uk.docs.yahoo.com/nowyoucan.html ------------------------------------------------------------------ To change your Lit-Ideas settings (subscribe/unsub, vacation on/off, digest on/off), visit www.andreas.com/faq-lit-ideas.html