Fading Foundations: Probability and the Regress Problem

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. 1.1 Reasons for Reasons: Agrippa’s Trilemma We believe many things: that the earth is a spheroid, that Queen Victoria reigned for more that sixty years, that Stockholm is the capital of Finland, that the Russians were the first to land on the moon. Some of these beliefs are true, others are false. A belief might be true by accident. Suppose I have a phobia which makes me believe that there is a poisonous snake under my bed. After many visits to a psychiatrist and intensive therapy I gradually try to convince myself that this belief stems from traumatic and suppressed childhood experiences. One fine day I finally reach the point where I, nervous and trembling, force myself to get into bed before first looking under it. Unbeknownst to me or the psychiatrist, however, a venomous snake has escaped from the zoo and has ensconced itself under my bed. My belief in the proposition ‘There is a poisonous snake under my bed’ is true, but it is accidentally true. I do not have a good reason for this belief, since I am © The Author(s) 2017 D. Atkinson, J. Peijnenburg, Fading Foundations, Synthese Library 383, DOI 10.1007/978-3-319-58295-5_1 1 2 1 The Regress Problem ignorant of the escape and agree with the psychiatrist that reasons based on my phobia are not good reasons. If however a belief is based on good reasons, we say that it is epistemically justified. Had I been aware of the fact that the snake had escaped and in fact had made its way to my bedroom, I would have been in possession of a good reason, and would have been epistemically justified in believing that the animal was lying under my bed. According to a venerable philosophical tradition, a true and justified belief is a candidate for knowledge. One of the things that is needed in order for me to know that there is a snake under my bed is that the good reason I have for it (namely my belief that the reptile had slipped away and is hiding in my room) is itself justified. Without that condition, my reason might be itself a fabrication of my phobic mind, and thus ultimately fall short of being a good reason. What would count as a good reason for believing that a snake has escaped and installed itself in my bedroom? Here is one: an anxious neighbour knocks on my door, agitatedly telling me about the escape. But how do I know that what the neighbour says is true? It seems I need a good reason for that as well. My friendly neighbour shows me a text message on his cellphone, just sent by the police, which contains the alarming news. That seems to be quite a good reason — although, how do I know that the police are well informed? I need a good reason for that as well. I call the head of police, who confirms the news, and says that he was apprised of it by the director of the zoo; I call the director, who tells me that the escape has been reported to her by the curator of the reptile house, and so on. True, my actions are somewhat curious, and they may well signal that a phobia for snakes is not the only mental affliction that plagues me. The point however is not a practical but a principled one. It is that a reason is only a good reason if it is backed up by another good reason, which in turn is backed up by still another other good reason, and so on. We thus arrive at a chain of reasons, where the proposition ‘There is a dangerous snake under my bed’ (the target proposition q) is justified by ‘A neighbour knocks on my door and tells me that a snake has escaped’ (reason A1), which is justified by ‘The police sent my neighbour a text message about the escape’ (reason A2), which is justified by A3, and so on: q ←− A1 ←− A2 ←− A3 ←− A4 . . . (1.1) Such a justificatory chain, as we shall call it, gives rise to the regress problem. It places us in a position where we have to choose between two equally unattractive options: either the chain must be continued, for otherwise we 1.1 Reasons for Reasons: Agrippa’s Trilemma 3 cannot be said to know the proposition q, or the chain must come to a stop, but then it seems we are not justified in claiming that we really can know q, since there is no reason for stopping. Laurence Bonjour called considerations relating to the regress problem “perhaps the most crucial in the entire theory of knowledge”, and Robert Audi observes that no epistemologist quite knows how to handle the problem.1 The roots of the regress problem extend far back into epistemological history, and scholars often refer to the Greek philosopher Agrippa. Little is known about Agrippa, apart from the fact that he probably lived in the first century A.D. and might have been among the group of sceptics discussed by Sextus Empiricus, a philosopher and practising physician who allegedly flourished a century later. Sextus’ most famous work, Outlines of Pyrrhonism, contains an explanation and defence of what he takes to be the philosophy of another shadowy figure, namely Pyrrho of Elis (c. 365–270 B.C.), who himself wrote nothing, but became known for his sober life style and his aversion to academic or theoretical reasoning. So-called Pyrrhonian scepticism advocates the attainment of ataraxia, a state of serene calmness in which one is free from moods or other disturbances. An important technique for reaching this state is the practicing of argument strategies known as tropoi or modes, i.e. means to engender suspension of judgement by undermining any claim that conclusive knowledge or justification has been attained. For example, if it were claimed that a particular sound is known to be soft, a Pyrrhonian would point out that to a dog it is loud, and that we cannot judge the loudness or softness independently of the hearer. Typically, a Pyrrhonian will try to thoroughly acquaint himself with the modes, so that reacting in accordance with them becomes as it were a second nature. In this manner he will be able to routinely refrain from assenting to any weighty proposition q or ¬q, and thus avoid getting caught up in one of those rigid intellectual positions that he loathes so much. In Book 1 of Outlines of Pyrrhonism, Sextus discusses five modes which he attributes to “the more recent Sceptics” (to be distinguished from what he calls “the older Sceptics”), and which Diogenes Laertius in the third century would identify with “Agrippa and his school”.2 Of these five modes the 1 Bonjour 1985, p.18; Audi 1998, 183–184. The thought is echoed by Michael Huemer when he writes that regress arguments “concern some of the most fundamental and important issues in all of human inquiry” (Huemer 2016, 16). 2 Sextus Empiricus, Outlines of Pyrrhonism, Book I, 164; see p. 40 in the translation Outlines of scepticism by Julia Annas and Jonathan Barnes. Diogenes Laertius, Lives of eminent philosophers, Volume 2, Book 9, 88. We thank Tamer Nawar and an anonymous referee for guidance in matters of ancient philosophy. 4 1 The Regress Problem three that are of especial interest are the Mode of Infinite Regress, the Mode of Hypothesis, and the Mode of Circularity or Reciprocation. Here is how Sextus explains them: In the mode deriving from infinite regress, we say that what is brought forward as a source of conviction for the matter proposed itself needs another source, which itself needs another, and so on ad infinitum, so that we have no point from which to begin to establish anything, and suspension of judgement follows. . . . We have the mode from hypothesis when the Dogmatists, being thrown back ad infinitum, begin from something which they do not establish but claim to assume simply and without proof in virtue of a concession. The reciprocal mode occurs when what ought to be confirmatory of the object under investigation needs to be made convincing by the object under investigation; then, being unable to take either in order to establish the other, we suspend judgement about both.3 In other words, whenever a ‘dogmatist’ (as Sextus calls any philosopher who is not a Pyrrhonian sceptic) claims that he knows a proposition q, the Pyrrhonian sceptic will ask him what his reason is for q. After the dogmatist has given his answer, for example reason A1, the sceptic will ask further: what is your reason for A1? In the end it will become clear that the dogmatist has only three options open to him, jointly known as ‘Agrippa’s Trilemma’: 1. He goes on giving reasons for reasons for reasons, without end. 2. He stops at a particular reason, claiming that this reason essentially justifies all the others that he has given. 3. He reasons in a circle, where his final reason is identical to his first. In the first case the justificatory chain is infinitely long, in the second case it comes to a halt, and in the third case it forms a loop. The sceptic is quick to point out that none of these options can be accepted as a justification for q. The first option is impossible from a practical point of view, since we are ordinary human beings with a restricted lifespan. Moreover, even if we were to live forever, continuing to give reason after reason, we would never reach the origin of the justification, since by definition the chain does not have an origin. The second option is also unsatisfying. For why do we stop at this particular reason and not at another? If we can answer this question, we have a reason for what we claimed is without a reason, so we actually did not stop the chain. And if we cannot answer the question, then stopping at this particular reason is arbitrary. The third option is likewise unacceptable

I am happy to report that A&P succeed in spades. Fading Foundations is an excellent book. It is elegantly written (with most technical details relegated to appendices), and jam-packed with careful argumentation and fascinating results.

I. CHAPTER-BY-CHAPTER OVERVIEW
There are eight chapters in Fading Foundations along with four appendices. I shall ignore the appendices in what follows, and focus on the chapters.
In Ch. 1 and Ch. 2, A&P tend to preliminary matters. In Ch. 1, they introduce the epistemic regress problem (with some historical background), explain coherentism, foundationalism, and infinitism, and discuss the issue of vicious versus benign infinite regresses. In Ch. 2, they examine several distinct ways of understanding the expression 'A j justifies A i '. They do not settle on any particular way of understanding it. But they argue against each of the following theses: They further argue for the following alternative thesis: They stress that the right side of this conditional is a purely formal condition in that it is neutral between different ways of understanding A j (as a belief, as a proposition, as a fact, as an event, as a perceptual experience, as a neural state, etc.) and between different ways of understanding probability.
Ch. 3 and Ch. 4 together constitute the core of Fading Foundations. A&P show that: (*) There are infinite chains A 1 , A 2 , …, A n , … such that (i) A i is probabilistically supported by A i+1 for all i and (ii) A 1 has a high unconditional probability.
Take some infinite chain A 1 , A 2 , …, A n , …, and suppose, for example, that: They show that given this supposition, it follows that: If β is held fixed at, say, 0.04, then Pr(A 1 ) tends to 1 as α tends to 1. Hence (*). This result is ingenious (and, as A&P note, bears on an old debate between C. I. Lewis and Hans Reichenbach). But it might seem puzzling. How can an unconditional probability be fully determined by an infinite number of conditional probabilities?
It will help here to turn briefly from infinite chains to finite chains. First, take a 3-link chain A 1 , A 2 , A 3 such that: If Pr(A 3 ) = 0.9, then Pr(A 1 ) has a high value (roughly 0.890). If Pr(A 3 ) = 0.1, then Pr(A 1 ) has a low value (roughly 0.168). Now consider a 100-link chain A 1 , A 2 , …, A 100 such that: If Pr(A 100 ) = 0.9, then Pr(A 1 ) has a high value (roughly 0.801). If Pr(A 100 ) = 0.1, then, unlike in the prior case, Pr(A 1 ) still has a high value (roughly 0.796).
In the first case, where n = 3, Pr(A n ) has a significant impact on Pr(A 1 ). High values for Pr(A n ) lead to high values for Pr(A 1 ). Low values for Pr(A n ) lead to low values for Pr(A 1 ). In the second case, in contrast, where n = 100, Pr(A n ) has very little impact on Pr(A 1 ). Both high values for Pr(A n ) and low values for Pr(A n ) lead to high values for Pr(A 1 ).
This pattern continues: the impact of Pr(A n ) on Pr(A 1 ) approaches zero as n approaches infinity. Its impact, as it were, fades away to nothing (thus the expression 'fading foundations'), so that Pr(A 1 ) is fully determined by Pr(A 1 | A 2 ), Pr(A 1 | ~A 2 ), Pr(A 2 | A 3 ), Pr(A 2 | ~A 3 ), ….
The remainder of Fading Foundations is organized as follows. In Ch. 5, A&P address the 'Finite Minds Objection'. This objection (or at least one version thereof) is based on the claim, roughly, that finite minds cannot handle infinite chains. A&P argue, drawing on certain parts of Ch. 3 and Ch. 4, that oftentimes a finite chain with a relatively small number of links is enough to get Pr(A 1 ) within a suitable range of values so that an infinite chain is unnecessary. In Ch. 6, A&P address several 'conceptual' (as opposed to 'pragmatic') objections to infinite chains and probabilistic regresses. In Ch. 7, they turn from probabilistic regresses to regresses of higher-order probabilities, and argue that the latter are formally equivalent to the former. In Ch. 8, they consider loops and multidimensional networks.
There is a lot more to Fading Foundations than what is noted above. I cannot do it justice in a brief chapter-by-chapter overview.

II. A CRUCIAL CLARIFICATION
Return to (*). A&P consider a worry about the conditional probabilities at issue. They write: First, how do we know that the conditional probabilities in our chain are 'good' ones, i.e. make contact with the world? What is the difference between our reasonings and those occurring in fiction, in the machinations of a liar, or in the hallucinations of a heroin addict? Or, applied to our example about bacteria, how can we distinguish the regress concerning Barbara and her ancestors from a fairy tale with the same structure in which, instead of the heritable trait T, there is an inheritable magical power, M, to turn a prince into a frog? (p. 96) They then try to answer it: The distinction is not far to seek. It lies in the mundane fact that in the former, but not in the latter, the conditional probabilities arise from observation and experiment. (p.

96)
This is fine as far as it goes. But how exactly does perceptual justification work?
There is a large literature on this question. Different theorists give different answers. Some hold that perceptual beliefs can be justified by perceptual experiences. Others deny this and claim instead that perceptual justification, as with all justification, is ultimately a matter of coherence. Others appeal to reliability. And so on.
A&P, though, offer no worked out view on this front. They simply assume that there is a perception-based story to be told in terms of how the various conditional probabilities at issue in (*) are justified, and then show that the probabilities in question can lead in surprising ways to further justified probabilities. They write: We realize perfectly well that this answer will not convince the confirmed sceptic …. We do not have the temerity to aim at refuting the claim that all our perceptions might be illusory, or at outlawing evil demon scenarios, old and new. We simply assume that there is a real world, and that empirical facts can justify certain propositions, or more generally can sanction the probabilities that certain propositions are true. (p. 96) They never show, or even try to show, how to get justification in the first place. They show, rather, how to use justification already on hand to get-in some quite surprising ways-more justification. 1 This should be borne in mind when considering passages like this: In the end we somehow try to get it all, sketching the contours of an infinitist version of coherentism, which also acknowledges the foundationalist lesson that we should somehow make contact with the world. (p. 11) This should be read as '… sketching the contours of an infinitist version of coherentism on how to use justification already on hand to get more justification …'. This is a clarification, not a criticism, of Fading Foundations. It is a first-rate piece of epistemology. I highly recommend it. 2 WILLIAM ROCHE Texas Christian University, USA 1 For further discussion of this point, see W. Roche, 'Foundationalism with Infinite Regresses of Probabilistic Support', Synthese, forthcoming. 2 Thanks to Jeanne Peijnenburg and David Atkinson for helpful feedback on a prior version of this review.