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The attempt to justify our beliefs leads to the regress problem. We briefly recount the problem’s history and recall the two traditional solutions, foundationalism and coherentism, before turning to infinitism. According to infinitists, the regress problem is not a genuine difficulty, since infinite chains of reasons are not as troublesome as they may seem. A comparison with causal chains suggests that a proper assessment of infinitistic ideas requires that the concept of justification be made clear.

What is the nature of the justifier and of the justified, and how are they related? The answers to these questions depend on whether one embraces internalism or externalism. As far as the formal side of the justification relation is concerned, however, the difference between internalism and externalism seems irrelevant. Roughly, there are three proposals for the formal relation. One of them conceives the justification relation as probabilistic support; in fact, however, probabilistic support is only a necessary and not a sufficient condition for justification.

During more than twenty years Clarence Irving Lewis and Hans Reichenbach pursued an unresolved debate that is relevant to the question of whether infinite epistemic chains make sense. Lewis, the nay-sayer, held that any probability statement presupposes a certainty, but Reichenbach profoundly disagreed. We present an example of a benign probabilistic regress, thus showing that Reichenbach was right. While in general one lacks a criterion for distinguishing a benign from a vicious regress, in the case of probabilistic regresses the watershed can be precisely delineated. The vast majority (‘the usual class’) is benign, while its complement (‘the exceptional class’) is vicious.

A probabilistic regress, if benign, is characterized by the feature of fading foundations: the effect of the foundational term in a finite chain diminishes as the chain becomes longer, and completely dies away in the limit. This feature implies that in an infinite chain the justification of the target arises exclusively from the joint intermediate links; a foundation or ground is not needed. The phenomenon of fading foundations sheds light on the difference between propositional and doxastic justification, and it helps us settle the question whether justification is transmitted from one link in the chain to another, as foundationalists claim, or whether it emerges from a chain or network as a whole, as is maintained by coherentists and infinitists.

Can finite minds encompass an infinite number of beliefs? There is a difference between being able to complete an infinite series and being able to compute its outcome; and justification is more than mere calculation. Yet the number of propositions or beliefs that are needed in order to reach a desired level of justification for the target can be determined without computing an infinite number of terms: only a finite number of reasons are required for any desired level of accuracy. This suggests a view of epistemic justification as a trade-off between the accuracy of the target and the number of reasons taken into consideration.

There are two conceptual objections to the idea of justification by an infinite regress. First, there is no ground from which the justification can originate. Second, if a regress could justify a proposition, another regress could be found to justify its negation. We show that both objections are pertinent to a regress of entailments, but fail for a probabilistic regress. However, the core notion of such a regress, i.e. probabilistic support, leaves something to be desired: it is not sufficient for justification, so something has to be added. A threshold condition? A closure requirement? Both? Furthermore, the notion is said to have inherent problems, involving symmetry and nontransitivity.

At first sight, a hierarchical regress formed by probability statements about probability statements appears to be different from the probabilistic regress of the previous chapters. After all, the former involves higher and higherorder probabilities, whereas the latter is an epistemic chain in which one proposition or belief probabilistically supports another. Closer examination, however, teaches us that the two regresses are in fact isomorphic. A model based on coin-making machines demonstrates that the hierarchical regress is consistent.

The analysis so far concerned only one-dimensional epistemic chains. In this chapter two extensions are investigated. The first treats loops rather than chains. We show that generally, i.e. in what we have called the usual class, infinite loops yield the same value for the target as do infinite chains; it is only in the exceptional class that the values differ. The second extension involves multi-dimensional networks, where the chains fan out in many different directions. As it turns out, the uniform version of the networks yields the fractal iteration of Mandelbrot. Surprising as it may seem, justificatory systems that mushroom out greatly resemble fractals.