This note makes a contribution to the issue raised in a paper by Popper and Miller (1983) in which it was claimed that probabilistic support is purely deductive. Developing R. C. Jeffrey's remarks, a new general approach to the crucial concept of "going beyond" is here proposed. By means of it a quantitative measure of the inductive component of a probabilistic inference is reached. This proposal leads to vindicating the view that typical predictive probabilistic inferences by enumeration and analogy are (...) purely inductive. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:David Hume and the Probability of Miracles Barry Gower 1. Introduction Oflate there have been published several discussions ofDavid Hume's famous essay "Of Miracles" which attempt to make precise the reasoning it contains. This, it turns out, requires the use of certain mathematical rules and theorems of the probability calculus which were unknown to Hume or, indeed, to anyone else when the essay was first published. It is suggested, (...) in particular, that the claims he made can best be understood in the light of a theorem of the probability calculus which we now name after a celebrated eighteenth century probabilist, Thomas Bayes. However, the so-called "Bayesian" interpretation of the argument against miracles misrepresents Hume's reasoning in that the manner in which he expressed his thinking does not fit such an interpretation. Moreover, this error needs to be rectified, not just because it involves a mistaken view about the past but, more importantly, because it obscures the legacy of a different mode of thinking evident in Hume's writing about probabilistic inference which deserves to be recovered. Our thinkingisimpoverishedifcertain presumptions aboutprobability become so entrenched that we have great difficulty in seeing them as anything other than obvious. By imputing them to Hume we overlook their contestability and so lose an opportunity to recapture an alternative way of understanding probabilistic reasoning. As is well known, the argument against miracles, though not published until 1748, had been worked out some twelve or thirteen years earlier during Hume's stay at the Jesuit college of La Fleche, where he wrote most of A Treatise of Human Nature. A chapter entitled "Reasonings concerning Miracles" had been written but, at the last moment, it was suppressed for fear that such an explicit attack on religion would give offence and thus deflect attention from the book's main philosophical aim which was to put an end to controversy. On the other hand, Thomas Bayes's famous memoir on probability was not published until 1764, andit is unlikely that Hume knew ofits existence until Richard Price mentionedit, though without identifyingits author, in an elaborate mathematical footnote to a dissertation containing remarks on the probability of miracles, published in 1767. Had Hume taken the trouble to follow up this reference, he would almost certainly Volume XVI Number 1 17 BARRY GOWER have been either baffled by its mathematics or discouraged by its lack of practical advice. That, at least, seems to have been the reaction of Hume's contemporaries, for apart from Price none of them seems to have paid any attention to the memoir. Even today, when anyone with the slightest acquaintance with probability can recite a version of Bayes's theorem, it is hard to follow the reasoning of the memoir and to appreciate the significance of the ideas it contains. As an historian of statistics has recently said, "Bayes's contemporaries were stymied by the work, and modern readers unwilling to devote many hours to it are likely to gain only a superficial understanding ofit." One paradoxical feature of attributing Bayesianism to Hume concerns the role of Richard Price. His dissertation, which had, it seems, been written some years before it was published — perhaps in the early 1750's — is sharply critical of Hume's argument against miracles. Yet it was Price himself who was responsible for the posthumous publication of Bayes's memoir, and his introduction and commentary clearly express approval of its methods and conclusions. We are, then, faced with the suggestion that Hume, not having heard of Bayes, was applying Bayesian ideas about "inverse" inference to miraculous events, whereas Price, who was not only familiar with but also endorsed Bayes's ideas, preferred a quite different approach to reasoning about miracles and about the credibility of testimony generally. Can Hume really be described as a "better intuitive Bayesian" than Price? Before embracing such aparadoxical conclusion we should consider whether it is really justified by what Hume said about the probability ofmiracles. First, though, some attention needs to be paid to a way ofthinking about probable arguments which does not conform to convention, and which implies a non-Bayesian interpretation of the question about miracles. 2. Jakob Bernoulli and the Combination... (shrink)

It is widely supposed that David Hume invented and espoused the "regularity" theory of causation, holding that causal relations are nothing but a matter of one type of thing being regularly followed by another. It is also widely supposed that he was not only right about this, but that it was one of his greatest contributions to philosophy. Strawson here argues that the regularity theory of causation is indefensible, and that Hume never adopted it in any case. Strawson maintains that (...) Hume did not claim that causation in the natural world is just a matter of regular succession, that such a dogmatic metaphysical claim about the nature of reality would have been utterly contrary to his fundamental philosophical principles, and that he rightly took it for granted that there was more to causation than regularity of succession, claiming only that regularity of succession was all that we could ever know of causation. (shrink)