The Consistency of Probabilistic Regresses. A Reply to Jeanne Peijnenburg and David Atkinson

Abstract

In a recent paper, Jeanne Peijnenburg and David Atkinson [Studia Logica, 89(3): 333-341 (2008)] have challenged the foundationalist rejection of infinitism by giving an example of an infinite, yet explicitly solvable regress of probabilistic justification. So far, however, there has been no criterion for the consistency of infinite probabilistic regresses, and in particular, foundationalists might still question the consistency of the solvable regress proposed by Peijnenburg and Atkinson.In this paper, we employ Robinsonian nonstandard analysis to prove that a probabilistic regress is already consistent if it is admissible in the sense that its forward-iteration solution does not lead to obvious contradictions; naturally, the converse also holds true. As a consequence, it turns out that there is a rich class of probabilistic regresses, which generically will fail to be solvable.We therefore propose a weaker version of the Probabilistic Regress Problem which concedes the existence of solvable regresses, but denies their genericity.

A Graded Bayesian Coherence Notion

“…Another complaint against our graded coherence notion might be (iii) that our proposed formalisation could assign a non-minimal degree of coherence even to belief systems S that encode infinite regresses of probabilistic justification, provided P S is sufficiently large. In response, we point to a recent result by Herzberg [36,37] that gives a criterion for the probabilistic consistency simpliciter of certain probabilistic regresses of epistemic justification which in turn can be used to refute influential arguments against regresses of epistemic justification.…”

The Solvability of Probabilistic Regresses. A Reply to Frederik Herzberg

“…In an interesting comment involving a pretty use of nonstandard analysis, Frederik Herzberg [5] has proved that our counterexamples are consistent, in the sense that they have a model. However, he also argues that they are of a very special kind.…”

“…(for the meaning of the symbols β and γ see Herzberg's [5] and our [2]). Following Herzberg, we shall say that Eq.…”

Probabilistic Regresses and the Availability Problem for Infinitism

“…The progressive feature of justification has been given an altogether more precise treatment by Peijnenburg (), Peijnenburg and Atkinson (, , ), and Herzberg (, ). By attending to the formal details of various infinite probabilistic regresses, they have provided several mathematical illustrations of progressive justification.…”

“…Debates over the success of infinitism continue to rage on in the literature. Recent work (e.g., Peijnenburg ; Peijnenburg and Atkinson , , ; 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.…”