Sections 1 through 3 present all of the main ideas behind the probabilistic logic of evidential support. For most readers these three sections will suffice to provide an adequate understanding of the subject. Those readers who want to know more about how the logic applies when the implications of hypotheses about evidence claims (called likelihoods) are vague or imprecise may, after reading sections 1-3, skip to section 6. Sections 4 and 5 are for the more advanced reader who wants a (...) detailed understanding of some telling results about how this logic may bring about convergence to the truth. (shrink)

Eliminative induction is a method for finding the truth by using evidence to eliminate false competitors. It is often characterized as "induction by means of deduction"; the accumulating evidence eliminates false hypotheses by logically contradicting them, while the true hypothesis logically entails the evidence, or at least remains logically consistent with it. If enough evidence is available to eliminate all but the most implausible competitors of a hypothesis, then (and only then) will the hypothesis become highly confirmed. I will argue (...) that, with regard to the evaluation of hypotheses, Bayesian inductive inference is essentially a probabilistic form of induction by elimination. Bayesian induction is an extension of eliminativism to cases where, rather than contradict the evidence, false hypotheses imply that the evidence is very unlikely, much less likely than the evidence would be if some competing hypothesis were true. This is not, I think, how Bayesian induction is usually understood. The recent book by Howson and Urbach, for example, provides an excellent, comprehensive explanation and defense of the Bayesian approach; but this book scarcely remarks on Bayesian induction's eliminative nature. Nevertheless, the very essence of Bayesian induction is the refutation of false competitors of a true hypothesis, or so I will argue. (shrink)

A person required to risk money on a remote digit of π would, in order to comply fully with the theory [of personal probability] have to compute that digit, though this would really be wasteful if the cost of computation were more than the prize involved. For the postulates of the theory imply that you should behave in accordance with the logical implications of all that you know. Is it possible to improve the theory in this respect, making allowance within (...) it for the cost of thinking, or would that entail paradox? (shrink)

The objectivity of Bayesian induction relies on the ability of evidence to produce a convergence to agreement among agents who initially disagree about the plausibilities of hypotheses. I will describe three sorts of Bayesian convergence. The first reduces the objectivity of inductions about simple "occurrent events" to the objectivity of posterior probabilities for theoretical hypotheses. The second reveals that evidence will generally induce converge to agreement among agents on the posterior probabilities of theories only if the convergence is 0 or (...) 1. The third establishes conditions under which evidence will very probably compel posterior probabilities of theories to converge to 0 or 1. (shrink)