Search

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.

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.

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.

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.

InA 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.

In August 2015 the Faculty of Philosophy in Groningen hosted JustGroningen, a three-day conference on epistemic justification. The event was funded by the Netherlands Organisation of Scientific Research (NWO) as part of a project on infinite regresses in epistemology in which both of us are involved (Project Number 360-20-280).The present special issue sprang from this conference. It contains some papers that were presented there together with some others. We are very happy to have been able to bring together a collection that highlights so many different aspects of epistemic justification, and uses such a variety of methods to explore them.

A great deal of Anglo-Saxon epistemology is marked by the controversy between foundationalists and anti-foundationalists. The key question is as to how propositions, or beliefs in propositions, are to be justified. Is our body of knowledge sustained by basic beliefs, as foundationalists claim? Or are there no basic beliefs, and is there only mutual support between the elements of the structure, as the anti-foundationalists maintain? The matter is made especially difficult by the fact that no-one knows what precisely epistemic justification is. Foundationalists appear to see justification in terms of transmission: when a proposition justifies another proposition a property is transferred from the one to the other lock, stock and barrel. Anti-foundationalists on the other hand think in terms of emergence: epistemic justification is brought about gradually within a network of propositions as a whole. While transmission can at least be elucidated by a comparison with deductive reasoning, emergence is more difficult to explicate. That is one of the reasons why anti-foundationalist positions are so hard to develop. In this article I show how epistemic justification can emerge in a network of propositions that support one another. I do this by means of a debate that at first sight seems not to have anything to do with the matter, namely the disagreement between the American logician and epistemologist Clarence Irving Lewis and the German-American philosopher of science Hans Reichenbach. This will enable us to establish the phenomenon of fading foundations, which will then yield a mechanism for the emergence of justification.

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.

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.

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.

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.

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.

This introduction presents selected proceedings of a two-day meeting on the regress problem, sponsored by the Netherlands Organization for Scientific Research (NWO) and hosted by Vanderbilt University in October 2013, along with other submitted essays. Three forms of research on the regress problem are distinguished: metatheoretical, developmental, and critical work.

Some series can go on indefinitely, others cannot, and epistemologists want to know in which class to place epistemic chains. Is it sensible or nonsensical to speak of a proposition or belief that is justified by another proposition or belief, ad infinitum? In large part the answer depends on what we mean by “justification.” Epistemologists have failed to find a definition on which everybody agrees, and some have even advised us to stop looking altogether. In spite of this, the present essay submits a few candidate definitions. It argues that, although not giving the final word, these candidates tell us something about the possibility of infinite epistemic chains. And it shows that they can short-circuit a debate about doxastic justification.

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent.May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a probability of a probability of a probability, and so on, ad infinitum? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. We show that an infinite hierarchy of probabilities of probabilities is consistent.The proof consists in a model involving coin-making machines.Weak conditions are given for the convergence of the infinite system.

This paper is the third and final one in a sequence of three. All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically. The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precursor. In the present paper we consider the more complicated case of two-dimensional nets, where each "child" proposition is probabilistically justified by two "parent" propositions. Surprisingly, it turns out that probabilistic justification in two dimensions takes on the form of Mandelbrot's iteration. Like so many patterns in nature, probabilistic reasoning might in the end be fractal in character.

The original article has been corrected. Erroneously, a comma and a space were added in line 164 to 500, 500, and the authors would like readers to know that this should instead read 500,500.

In an earlier paper I argued that there are cases in which an infinite probabilistic chain can be completed (Peijnenburg 2007). According to Jeremy Gwiazda, however, I have merely shown that the chain in question can be computed, not that it can be completed (Gwiazda 2010). Gwiazda thereby discriminates between two terms that I used as synonyms. In the present paper I discuss to what extent computability and completability can be meaningfully distinguished.

Various arguments have been put forward to show that Zeno-like paradoxes are still with us. A particularly interesting one involves a cube composed of colored slabs that geometrically decrease in thickness. We first point out that this argument has already been nullified by Paul Benacerraf. Then we show that nevertheless a further problem remains, one that withstands Benacerraf's critique. We explain that the new problem is isomorphic to two other Zeno-like predicaments: a problem described by Alper and Bridger in 1998 and a modified version of the problem that Benardete introduced in 1964. Finally, we present a solution to the three isomorphic problems.

We have earlier shown by construction that a proposition can have a well-defined nonzero probability, even if it is justified by an infinite probabilistic regress. We thought this to be an adequate rebuttal of foundationalist claims that probabilistic regresses must lead either to an indeterminate, or to a determinate but zero probability. In a comment, Frederik Herzberg has argued that our counterexamples are of a special kind, being what he calls 'solvable'. In the present reaction we investigate what Herzberg means by solvability. We discuss the advantages and disadvantages of making solvability a sine qua non, and we ventilate our misgivings about Herzberg's suggestion that the notion of solvability might help the foundationalist.We further show that the canonical series arising from an infinite chain of conditional probabilities always converges, and also that the sum is equal to the required unconditional probability if a certain infinite product of conditional probabilities vanishes.

Tom Stoneham put forward an argument purporting to show that coherentists are, under certain conditions, committed to the conjunction fallacy. Stoneham considers this argument a reductio ad absurdum of any coherence theory of justification. I argue that Stoneham neglects the distinction between degrees of confirmation and degrees of probability. Once the distinction is in place, it becomes clear that no conjunction fallacy has been committed.

Could some evidence confirm a conjunction of two hypotheses more than it confirms either of the hypotheses separately? We show that it might, moreover under conditions that are the same for ten different measures of confirmation. Further, we demonstrate that it is even possible for the conjunction of two disconfirmed hypotheses to be confirmed by the same evidence.

We discuss two objections that foundationalists have raised against infinite chains of probabilistic justification. We demonstrate that neither of the objections can be maintained.

Reichenbach’s use of ‘posits’ to defend his frequentistic theory of probability has been criticized on the grounds that it makes unfalsifiable predictions. The justice of this criticism has blinded many to Reichenbach’s second use of a posit, one that can fruitfully be applied to current debates within epistemology. We show first that Reichenbach’s alternative type of posit creates a difficulty for epistemic foundationalists, and then that its use is equivalent to a particular kind of Jeffrey conditionalization. We conclude that, under particular circumstances, Reichenbach’s approach and that of the Bayesians amount to the same thing, thereby presenting us with a new instance in which chance and credence coincide.

Richard Jeffrey's radical probabilism ('probability all the way down') is augmented by the claim that probability cannot be turned into certainty, except by data that logically exclude all alternatives. Once we start being uncertain, no amount of updating will free us from the treadmill of uncertainty. This claim is cast first in objectivist and then in subjectivist terms.

Like many discussions on the pros and cons of epistemic foundationalism, the debate between C.I. Lewis and H. Reichenbach dealt with three concerns: the existence of basic beliefs, their nature, and the way in which beliefs are related. In this paper we concentrate on the third matter, especially on Lewis’s assertion that a probability relation must depend on something that is certain, and Reichenbach’s claim that certainty is never needed. We note that Lewis’s assertion is prima facie ambiguous, but argue that this ambiguity is only apparent if probability theory is viewed within a modal logic. Although there are empirical situations where Reichenbach is right, and others where Lewis’s reasoning seems to be more appropriate, it will become clear that Reichenbach’s stance is the generic one. This follows simply from the fact that, if P(E|G) > 0 and P(E|not-G) > 0, then P(E) > 0. We conclude that this constitutes a threat to epistemic foundationalism.

Galileo claimed inconsistency in the Aristotelian dogma concerning failing bodies and stated that all bodies must fall at the same rate. However, there is an empirical situation where the speeds of falling bodies are proportional to their weights; and even in vacuo all bodies do not fall at the same rate under terrestrial conditions. The reason for the deficiency of Galileo's reasoning is analyzed, and various physical scenarios are described in which Aristotle's claim is closer to the truth than is Galileo's. The purpose is not to reinstate Aristotelian physics at the expense of Galileo and Newton, but rather to provide evidence in support of the verdict that empirical knowledge does not come from prior philosophy. (C) 2003 Elsevier Ltd. All rights reserved.

At its origins, analytic philosophy is an interest in language, science, logic, analysis, and a systematic rather than a historical approach to philosophical problems. Early analytic philosophers were famous for making clear conceptual distinctions and for couching them in comprehensible and lucid sentences. It is argued that this situation is changing, that analytic philosophy is turning into its mirror image and is thereby becoming more like the kind of philosophy that it used to oppose.

It is argued that the recent revival of theakrasia problem in the philosophy of mind is adirect, albeit unforeseen result of the debate onaction explanation in the philosophy of science. Asolution of the problem is put forward that takesaccount of the intimate links between the problem ofakrasia and this debate. This solution is basedon the idea that beliefs and desires have degrees ofstrength, and it suggests a way of giving a precisemeaning to that idea. Finally, it is pointed out thatthe solution captures certain intuitions of bothSocrates and Aristotle.

It is argued that probability should be defined implicitly by the distributions of possible measurement values characteristic of a theory. These distributions are tested by, but not defined in terms of, relative frequencies of occurrences of events of a specified kind. The adoption of an a priori probability in an empirical investigation constitutes part of the formulation of a theory. In particular, an assumption of equiprobability in a given situation is merely one hypothesis inter alia, which can be tested, like any other assumption. Probability in relation to some theories - for example quantum mechanics - need not satisfy the Kolmogorov axioms. To illustrate how two theories about the same system can generate quite different probability concepts, and not just different probabilistic predictions, a team game for three players is described. If only classical methods are allowed, a 75% success rate at best can be achieved. Nevertheless, a quantum strategy exists that gives a 100% probability of winning.

We have never entirely agreed with Daniel Cohnitz on the status and r?le of thought experi ments. Several years ago, enjoying a splendid lunch together in the city of Ghent, we cheer fully agreed to disagree on the matter; and now that Cohnitz has published his considered opinion of our views, we are glad that we have the opportunity to write a rejoinder and to explicate some of our disagreements. We choose not to deal here with all the issues that Cohnitz raises, but rather to restrict ourselves to three specific points. The first point concerns the way in which Cohnitz represents our views. He summa rizes them in five theses, (PA1)-(PA5), and while we have no dispute with the wording of (PA1)-(PA4), we consider (PA5) to be somewhat misleading. Putting into our mouths the declamation that "In philosophy we cannot turn to crucial experiments (whereas in science we can)" wrongly suggests that we take crucial experiments to constitute the pivotal difference between philosophy and science. At no place do we claim that disputes over thought experi ments in physics are always settled by carrying out a crucial experiment. Cohnitz writes: "As Peijnenburg and Atkinson seem to admit, this [the settling by a crucial experiment] is not the case for all physical thought experiments." Seem to admit? Nay, we roundly asseverate the fact! After that sentence, Cohnitz continues as follows

A major objection to epistemic infinitism is that it seems to make justification impossible. For if there is an infinite chain of reasons, each receiving its justification from its neighbour, then there is no justification to inherit in the first place. Some have argued that the objection arises from misunderstanding the character of justification. Justification is not something that one reason inherits from another; rather it gradually emerges from the chain as a whole. Nowhere however is it made clear what exactly is meant by emergence. The aim of this paper is to fill that lacuna: we describe a detailed procedure for the emergence of justification that enables us to see exactly how justification surfaces from a chain of reasons.

The notion of probabilistic support is beset by well-known problems. In this paper we add a new one to the list: the problem of transitivity. Tomoji Shogenji has shown that positive probabilistic support, or confirmation, is transitive under the condition of screening off. However, under that same condition negative probabilistic support, or disconfirmation, is intransitive. Since there are many situations in which disconfirmation is transitive, this illustrates, but now in a different way, that the screening-off condition is too restrictive. We therefore weaken this condition to what we call ‘partial’ screening off. We show that the domain defined by partial screening off comprises two mutually exclusive subdomains. In one subdomain disconfirmation is indeed transitive, but confirmation is then intransitive. In the other, confirmation is transitive, but here disconfirmation is once more intransitive.

From Aristotle onwards, epistemic justification has been conceived as a form of 5 inference. If a proposition En is epistemically justified by a proposition

From 1929 onwards, C. I. Lewis defended the foundationalist claim that judgements of the form 'x is probable' only make sense if one assumes there to be a ground y that is certain (where x and y may be beliefs, propositions, or events). Without this assumption, Lewis argues, the probability of x could not be anything other than zero. Hans Reichenbach repeatedly contested Lewis's idea, calling it "a remnant of rationalism". The last move in this debate was a challenge by Lewis, defying Reichenbach to produce a regress of probability values that yields a number other than zero. Reichenbach never took up the challenge, but we will meet it on his behalf, as it were. By presenting a series converging to a limit, we demonstrate that x can have a definite and computable probability, even if its justification consists of an infinite number of steps. Next we show the invalidity of a recent riposte of foundationalists that this limit of the series can be the ground of justification. Finally we discuss the question where justification can come from if not from a ground.

Infinite regresses have had a mixed reception in philosophy. On the one hand, they are often quickly brushed aside as being unreasonable and even preposterous. If a position or a theory is shown to engender an infinite regress, that pretty well means the end of it: an argumentum ad infinitum is as good as a reductio ad absurdum. On the other hand, the alleged absurdity of infinite regresses has been a source of inspiration and the ground for far-reaching conclusions. It led to the requirement that a regress should come to a stop and thence to the verdict that there exist such things as a Prime Mover, a First Cause, a Highest Good, or a Causa Sui. Infinite regresses have been discussed in practically all branches of philosophy, not only in metaphysics and epistemology, but also in ethics, philosophy of mind, logic, and argumentation theory. In recent years, they have spurred a lively debate in traditional epistemology, where ‘infinitism’ (the idea that an infinite chain of epistemic justification is not prima facie absurd) is on its way to becoming a mature alternative to the traditional positions, foundationalism and coherentism. The papers in the present volume are about infinite regress in decision theory, philosophy of science, and formal epistemology. We are fortunate to have brought together a number of renowned philosophers and promising young scholars: John Barker is associate professor of Philosophy at the University of Illinois at Springfield, and chair of the Philosophy Department. His Ph.D. dissertation, defended in 1998 at Princeton University, is entitled ‘The Inconsistency Theory of Truth’. His areas of interest are philosophical logic and philosophy of language (especially paradoxes and inconsistencies) with side-interests in philosophy of mind and philosophy of science.

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.

Today it is generally assumed that epistemic justification comes in degrees. The consequences, however, have not been adequately appreciated. In this paper we show that the assumption invalidates some venerable attacks on infinitism: once we accept that epistemic justification is gradual, an infinitist stance makes perfect sense. It is only without the assumption that infinitism runs into difficulties.

A distinction is made between imagination in the narrow sense and in the broad sense. Narrow imagination is characterised as the ability to "see" pictures in the mind's eye or to "hear" melodies in the head. Broad imagination is taken to be the faculty of creating, either in the strict sense of making something ex nihilo or in the looser sense of seeing patterns in some data. The article focuses on a particular sort of broad imagination, the kind that has to do with creating, not a work of art, a scientific theory or a political vision but one's own life. We shape our lives through our actions, and these actions not only influence our future-a commonplace-but also determine our past, which is a new and more controversial perspective.

Kuipers’ model of action explanation is compared, first with that of Anscombe, and then with models in the post-Anscombian tradition. Whereas Kuipers and Anscombe differ on the question of the first-person view, the difference with post-Anscombian writers concerns the so-called intentional statement. Kuipers criticizes the models of both Hempel and von Wright for their lack of an intentional statement. Kuipers’ own model seems immune to this criticism, since it contains no less than two intentional statements, a “specific” and an “unspecific” one. I argue that, contrary to appearances, it is not so immune. The call for intentional statements is in fact a call for intentions that are irreducible to beliefs and desires. Kuipers’ intentional statements, however, are about intentions that can be so reduced.