Jeanne Peijnenburg

It has been argued that an infinite regress of entailments cannot justify a proposition, _q_. For if it could, then it can be shown that any proposition can be justified in that manner, incuding ¬_q_. This chapter shows that this _reductio ad absurdum_ weakens in the face of a so-called probabilistic regress, in which justification is only probabilistic. The reason is that, whereas a regress of entailments offers no entry point for the world, in the probabilistic regress the empirical thrust is carried by an infinite number of conditional probabilities. This chapter explains to what extent these conditional probabilities differ from the basic beliefs that lie at the heart of epistemic foundationalism.

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.

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.

The Berkson effect shows that two independent diseases, A and B, become negatively correlated if they are confined within the walls of a hospital. We explain that, simply by adding a third disease, C, the negative correlation may flip into a positive one, and we identify the point where this happens. That leads to a necessary and sufficient condition for a positive as well as a negative correlation between A and B. We further explain that a flip from negative to positive is impossible if C is independent of A, of B, and of the disjunction of A and B: with these three independences in place, the Berkson effect remains in force. However, if only two of the three independences hold, the effect is not guaranteed.

The present volume collects papers on ten female thinkers who directly or indirectly contributed to the development of analytic philosophy but who did not always receive the attention they deserve. In this introduction, we briefly recall the standard account of analytic philosophy as we know it from the textbooks, provide an overview of the research that has been done on the role of women in analytic philosophy in the past few years, and offer a quantitative analysis of 3,274 publications in the major analytic journals between 1896 and 1960. We conclude our introduction with summaries of the ten chapters in this volume.

As is well known, implication is transitive but probabilistic support is not. Eells and Sober, followed by Shogenji, showed that screening off is a sufficient constraint for the transitivity of probabilistic support. Moreover, this screening off condition can be weakened without sacrificing transitivity, as was demonstrated by Suppes and later by Roche. In this paper we introduce an even weaker sufficient condition for the transitivity of probabilistic support, in fact one that can be made as weak as one wishes. We explain that this condition has an interesting property: it shows that transitivity is retained even though the Simpson paradox reigns. We further show that by adding a certain restriction the condition can be turned into one that is both sufficient and necessary for transitivity.

Heisenberg’s uncertainty principle is a milestone of twentieth-century physics. We sketch the history that led to the formulation of the principle, and we recall the objections of Grete Hermann and Niels Bohr. Then we explain that there are in fact two uncertainty principles. One was published by Heisenberg in the Zeitschrift für Physik of March 1927 and subsequently targeted by Bohr and Hermann. The other one was introduced by Earle Kennard in the same journal a couple of months later. While Kennard’s principle remains untarnished, the principle of Heisenberg has recently been criticized in a way that is very different from the objections by Bohr and Hermann: there are reasons to believe that Heisenberg’s formula is not valid. Experimental results seem to support this claim.

In The Structure of Science, Ernest Nagel finds fault with Werner Heisenberg’s explication of the uncertainty principle. Nagel’s complaint is that this principle does not follow from the impossibility of measuring with precision both the position and the momentum of a particle, as Heisenberg intimates, rather it is the other way around. Recent developments in theoretical physics have shown that Nagel’s argument is more substantial than he could have envisaged. In particular it has become clear that there are in fact two uncertainty principles; as a result, there are four pairs of quantities to examine, whereas Heisenberg considers only one. These findings throw new light on Nagel’s criticism. They enable us to see that his intuition was surprisingly apposite, but also make clear where his argument misses the mark.

Eells and Sober proved in 1983 that screening off is a sufficient condition for the transitivity of probabilistic causality, and in 2003 Shogenji noted that the same goes for probabilistic support. We start this paper by conjecturing that Hans Reichenbach may have been aware of this fact. Then we consider the work of Suppes and Roche, who demonstrated in 1986 and 2012 respectively that screening off can be generalized, while still being sufficient for transitivity. We point out an interesting difference between Reichenbach’s screening off and the generalized version, which we illustrate with an example about haemophilia among the descendants of Queen Victoria. Finally, we embark on a further generalization: we develop a still weaker condition, one that can be made as weak as one wishes.

How certain is Heisenberg’s uncertainty principle? Heisenberg’s uncertainty principle is at the heart of the orthodox or Copenhagen interpretation of quantum mechanics. We first sketch the history that led up to the formulation of the principle. Then we recall that there are in fact two uncertainty principles, both dating from 1927, one by Werner Heisenberg and one by Earle Kennard. Finally, we explain that recent work in physics gives reason to believe that the principle of Heisenberg is invalid, while that of Kennard still stands.

As is well known, implication is transitive but probabilistic support is not. Eells and Sober, followed by Shogenji, showed that screening off is a sufficient constraint for the transitivity of probabilistic support. Moreover, this screening off condition can be weakened without sacrificing transitivity, as was demonstrated by Suppes and later by Roche. In this paper we introduce an even weaker sufficient condition for the transitivity of probabilistic support, in fact one that can be made as weak as one wishes. We explain that this condition has an interesting property: it shows that transitivity is retained even though the Simpson paradox reigns. We further show that by adding a certain restriction the condition can be turned into one that is both sufficient and necessary for transitivity.

This book contains a selection of papers from the workshop *Women in the History of Analytic Philosophy* held in October 2019 in Tilburg, the Netherlands. It is the first volume devoted to the role of women in early analytic philosophy. It discusses the ideas of ten female philosophers and covers a period of over a hundred years, beginning with the contribution to the Significs Movement by Victoria, Lady Welby in the second half of the nineteenth century, and ending with Ruth Barcan Marcus’s celebrated version of quantified modal logic after the Second World War. The book makes clear that women contributed substantially to the development of analytic philosophy in all areas of philosophy, from logic, epistemology, and philosophy of science, to ethics, metaphysics, and philosophy of language. It illustrates that although women's voices were no different from men's as regards their scope and versatility, they had a much harder time being heard. The book is aimed at historians of philosophy and scholars in gender studies.

We use the method of fixed points to describe a form of probabilistic truth approximation which we illustrate by means of three examples. We then contrast this form of probabilistic truth approximation with another, more familiar kind, where no fixed points are used. In probabilistic truth approximation with fixed points the events are dependent on one another, but in the second kind they are independent. The first form exhibits a phenomenon that we call ‘fading origins’, the second one is subject to a phenomenon known as ‘the washing out of the prior’. We explain that the two phenomena may seem very similar, but are in fact quite different.

In a recent paper Ronald Meester and Timber Kerkvliet argue by example that infinite epistemic regresses have different solutions depending on whether they are analyzed with probability functions or with belief functions. Meester and Kerkvliet give two examples, each of which aims to show that an analysis based on belief functions yields a different numerical outcome for the agent’s degree of rational belief than one based on probability functions. In the present paper we however show that the outcomes are the same. The only way in which probability functions and belief functions can yield different solutions for the agent’s degree of belief is if they are applied to different examples, i.e. to different situations in which the agent finds himself.

One of the most persistent complaints about Peter Klein’s infinitism involves the finite mind objection: given that we are finite, how can we ever handle an infinite series of reasons? Klein’s answer has been that we need not actually produce an infinite series; it is enough that such a series be available to us. In this chapter, a different reply is presented through the reconstruction of epistemic justification as a trade-off. In acting as responsible agents, we are striking a balance between the number of reasons that we can handle and the level of precision that we want our beliefs to have. If we are unable or unwilling to manage a large number of reasons, then we have to pay the price in terms of justificatory inexactitude and thereby of accepting relatively untrustworthy beliefs. As well as being intuitively attractive, this idea of a trade-off is warranted by the mathematics of epistemic justification, understood as involving probabilistic relations.

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.

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.

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.

In A Treatise of Human Nature, David Hume presents an argument according to which all knowledge reduces to probability, and all probability reduces to nothing. Many have criticized this argument, while others find nothing wrong with it. In this paper we explain that the argument is invalid as it stands, but for different reasons than have been hitherto acknowledged. Once the argument is repaired, it becomes clear that there is indeed something that reduces to nothing, but it is something other than what, according to many, Hume had in mind. Thus two views emerge of what exactly it is that reduces. We surmise that Hume failed to distinguish the two, because he lacked the formal means to differentiate between a rendering of his argument that is in accordance with the probability calculus, and one that is not.

Even if two testimonies in a criminal trial are independent, they are not necessarily more trustworthy than one. But if they are independent in the sense that they are screened off from one another by the crime, then two testimonies raise the probability of guilt above the level that one testimony alone could achieve. In fact this screening-off condition can be weakened without changing the conclusion. It is however only a sufficient, not a necessary condition for concluding that two witnesses are better than one. We will discuss two different conditions, each of them necessary as well as sufficient, and we conclude that one of them is slightly better than the other.