Is it coherent to speak of the probability of a probability, and the probability of a probability of a probability, and so on? We show that it is, in the sense that a regress of higher-order probabilities can lead to convergent sequences that determine all these probabilities. By constructing an implementable model which is based on coin-making machines, we demonstrate the consistency of our regress.

Recent work by Peijnenburg, Atkinson, and Herzberg suggests that infinitists who accept a probabilistic construal of justification can overcome significant challenges to their position by attending to mathematical treatments of infinite probabilistic regresses. In this essay, it is argued that care must be taken when assessing the significance of these formal results. Though valuable lessons can be drawn from these mathematical exercises (many of which are not disputed here), the essay argues that it is entirely unclear that the form of (...) infinitism that results meets a basic requirement: namely, providing an account of infinite chains of propositions qua reasons made available to agents. (shrink)

Suppose q is some proposition, and let P(q) = v0 (1) be the proposition that the probability of q is v0.1 How can one know that (1) is true? One cannot know it for sure, for all that may be asserted is a further probabilistic statement like P(P(q) = v0) = v1, (2) which states that the probability that (1) is true is v1. But the claim (2) is also subject to some further statement of an even higher probability: P(P(P(q) (...) = v0) = v1) = v2, (3) and so on. Thus, an infinite regress emerges of probabilities of probabilities, and the question arises as to whether this regress is vicious or harmless. Radical probabilists would like to claim that it is harmless, but Nicholas Rescher (2010), in his scholarly and very stimulating Infinite Regress: The Theory and History of Varieties of Change, argues that it is vicious. He believes that an infinite hierarchy of probabilities makes it impossible to know anything about the probability of the original proposition q: unless some claims are going to be categorically validated and not just adjudged probabilistically, the radically probabilistic epistemology envisioned here is going to be beyond the prospect of implementation. . . . If you can indeed be certain of nothing, then how can you be sure of your probability assessments. If all you ever have is a nonterminatingly regressive claim of the format . . . the probability is .9 that (the probability is .9 that (the probability of q is .9)) then in the face of such a regress, you would know effectively nothing about the condition of q. After all, without a categorically established factual basis of some sort, there is no way of assessing probabilities. But if these requisites themselves are never categorical but only probabilistic, then we are propelled into a vitiating regress of presuppositions. (shrink)

This note employs the recently established consistency theorem for infinite regresses of probabilistic justification (Herzberg in Stud Log 94(3):331–345, 2010) to address some of the better-known objections to epistemological infinitism. In addition, another proof for that consistency theorem is given; the new derivation no longer employs nonstandard analysis, but utilises the Daniell–Kolmogorov theorem.

In an earlier paper we have shown that a proposition can have a well-defined probability value, even if its justification consists of an infinite linear chain. In the present paper we demonstrate that the same holds if the justification takes the form of a closed loop. Moreover, in the limit that the size of the loop tends to infinity, the probability value of the justified proposition is always well-defined, whereas this is not always so for the infinite linear chain. This (...) suggests that infinitism sits more comfortably with a coherentist view of justification than with an approach in which justification is portrayed as a linear process. (shrink)

This Open Access book addresses the age-old problem of infinite regresses in epistemology. How can we ever come to know something if knowing requires having good reasons, and reasons can only be good if they are backed by good reasons in turn? The problem has puzzled philosophers ever since antiquity, giving rise to what is often called Agrippa's Trilemma. The current volume approaches the old problem in a provocative and thoroughly contemporary way. Taking seriously the idea that good reasons are (...) typically probabilistic in character, it develops and defends a new solution that challenges venerable philosophical intuitions and explains why they were mistakenly held. Key to the new solution is the phenomenon of fading foundations, according to which distant reasons are less important than those that are nearby. The phenomenon takes the sting out of Agrippa's Trilemma; moreover, since the theory that describes it is general and abstract, it is readily applicable outside epistemology, notably to debates on infinite regresses in metaphysics. (shrink)

In this article we analyze the claim that a probabilistic interpretation of the infinite epistemic regress problem leads to a unique solution, the so called “completion” of the regress. This claim is implicitly based on the assumption that the standard Kolmogorov axioms of probability theory are suitable for describing epistemic probability. This assumption, however, has been challenged in the literature, by various authors. One of the alternatives that have been suggested to replace the Kolmogorov axioms in case of an epistemic (...) interpretation of probability, are belief functions, introduced by Shafer in 1976. We show that when one uses belief functions to describe the infinite epistemic regress problem, it is no longer the case that the solution is unique. We also argue that this complies with common sense. (shrink)