Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

I lay out the framework for my theory of sensory imagination in “Imagining as a guide to possibility.” Sensory imagining involves mental imagery , and crucially, in describing the content of imagining, I distinguish between qualitative content and assigned content. Qualitative content derives from the mental image itself; for visual imaginings, it is what is “pictured.” For example, visually imagine the Philadelphia Eagles defeating the Pittsburgh Steelers to win their first Super Bowl. You picture the greenness of the field and (...) the football’s brown oblong shape. Some of what you imagine isn’t explicitly pictured, however. That it is Sunday, that it is the Super Bowl: these facts are assigned. (shrink)

This chapter focuses on the relations between objective probabilities in physical theories at different levels. In general philosophy of probability, it is frequently assumed that a fundamental deterministic theory cannot support probabilistic phenomena at any higher level, or more generally that there cannot be non-trivial probabilities in higher-level theories that are not encoded in probabilities at the lower level. These assumptions face significant challenges from some well-understood physical theories – I focus on statistical mechanics and Bohmian mechanics – where a (...) deterministic description at some lower level gives rise to an effectively probabilistic theory at some higher level; in each case, constraints arising from an objective physical limitation on the acquisition of evidence concerning the lower level plays a crucial role in supporting the higher-level probabilities. (shrink)

The Indifference Principle, its Paradoxes and Kolmogorov's Probability Space.

The Principle of Indifference (POI) is a rule for rationally assigning precise degrees of confidence to possibilities among which we have no reason to discriminate. Despite criticism of the principle stemming from Bertrand's paradox, many have recently come to the defense of POI or adopted some restricted version of that principle, especially in discussions of self-locating belief. I argue that POI in both unrestricted and restricted forms is untenable, and that arguments for the more restricted principles are hostage to problems (...) similar to those that bedevil arguments for traditional POI. (shrink)

The Principle of Indifference is a central element of the ‘classical’ conception of probability, but, for all its strong intuitive appeal, it is widely believed that it faces a devastating objection: the so-called (by Poincare´) ‘Bertrand paradoxes’ (in essence, cases in which the same probability question receives different answers). The puzzle has fascinated many since its discovery, and a series of clever solutions (followed promptly by equally clever rebuttals) have been proposed. However, despite the long-standing interest in this problem, an (...) important assumption, necessary for its generation, has been overlooked. My aim in this paper is to identify this assumption. Since what it claims turns out to be prima facie problematic, I will urge that the burden of proof now shifts to the objectors to PI; they have to provide reasons why this assumption holds. (shrink)

The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing (...) a generalized approach to conditionalization. The technique is illustrated by application to simple examples. (shrink)

A symposium on Michael Strevens' book "Tychomancy", concerning the psychological roots and historical significance of physical intuition about probability in physics, biology, and elsewhere.

We argue that in spite of their apparent dissimilarity, the methodologies employed in the a priori and a posteriori assessment of probabilities can both be justified by appeal to a single principle of inductive reasoning, viz., the principle of symmetry. The difference between these two methodologies consists in the way in which information about the single-trial probabilities in a repeatable chance process is extracted from the constraints imposed by this principle. In the case of a posteriori reasoning, these constraints inform (...) the analysis by fixing an a posteriori determinant of the probabilities, whereas, in the case of a priori reasoning, they imply certain claims which then serve as the basis for subsequent probabilistic deductions. In a given context of inquiry, the particular form which a priori or a posteriori reason may take depends, in large part, on the strength of the underlying symmetry assumed: the stronger the symmetry, the more information can be acquired a priori and the less information about the long-run behavior of the process is needed for an a posteriori assessment of the probabilities. In the context of this framework, frequency-based reasoning emerges as a limiting case of a posteriori reasoning, and reasoning about simple games of chance, as a limiting case of a priori reasoning. Between these two extremes, both a priori and a posteriori reasoning can take a variety of intermediate forms. (shrink)

This paper refutes two important and influential views in one fell stroke. The first is G.E. Moore’s view that assertions of the form ‘Q but I don’t believe that Q’ are inherently “absurd.” The second is Gareth Evans’s view that justification to assert Q entails justification to assert that you believe Q. Both views run aground the possibility of being justified in accepting eliminativism about belief. A corollary is that a principle recently defended by John Williams is also false, namely, (...) that justification to believe Q entails justification to believe that you believe Q. (shrink)

?Those ice cubes melted because by melting total entropy increased and entropy increase has a very high objective chance.? What role does the chance in this explanation play? I argue that it contributes to the explanation by entailing that the melting was almost necessary, and defend the claim that the fact that some event was almost necessary can, in the right circumstances, constitute a causal explanation of that event.

‘Those ice cubes melted because by melting total entropy increased and entropy increase has a very high objective chance.’ What role does the chance in this explanation play? I argue that it contributes to the explanation by entailing that the melting was almost necessary, and defend the claim that the fact that some event was almost necessary can, in the right circumstances, constitute a causal explanation of that event.

We often use symmetries to infer outcomes’ probabilities, as when we infer that each side of a fair coin is equally likely to come up on a given toss. Why are these inferences successful? I argue against answering this with an a priori indifference principle. Reasons to reject that principle are familiar, yet instructive. They point to a new, empirical explanation for the success of our probabilistic predictions. This has implications for indifference reasoning in general. I argue that a priori (...) symmetries need never constrain our probability attributions, even when it comes to our initial credences. (shrink)

Recently in epistemology a number of authors have mounted Bayesian objections to dogmatism. These objections depend on a Bayesian principle of evidential confirmation: Evidence E confirms hypothesis H just in case Pr(H|E) > Pr(H). I argue using Keynes' and Knight's distinction between risk and uncertainty that the Bayesian principle fails to accommodate the intuitive notion of having no reason to believe. Consider as an example an unfamiliar card game: at first, since you're unfamiliar with the game, you assign credences based (...) on the indifference principle. Later you learn how the game works and discover that the odds dictate you assign the very same credences. Examples like this show that if you initially have no reason to believe H, then intuitively E can give you reason to believe H even though Pr(H|E) ≤ Pr(H). I show that without the principle, the objections to dogmatism fail. (shrink)

Epistemic theories of objective chance hold that chances are idealised epistemic probabilities of some sort. After giving a brief history of this approach to objective chance, I argue for a particular version of this view, that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. The main argument for this view is the demonstration that it entails all of the commonly-accepted properties of chance. For example, this analysis entails that chances (...) supervene on the physical facts, that the chance function is an expert probability, that the existence of stable frequencies results from invariant chances in repeated experiments, and that chances are probably close to long-run relative frequencies in repeated experiments. Despite these virtues, epistemic approaches to chance have been neglected in recent decades, on account of their conflict with accepted views about closely related topics such as causation, laws of nature, and epistemic probability. However, existing views on these topics are also very problematic, and I believe that the epistemic view of chance is a key piece of this puzzle that, once in place, allows all the other pieces to fit together into a new and coherent way. I also respond to some criticisms of the epistemic theory that I favour. (shrink)

One currently popular view about the nature of objective probabilities, or objective chances, is that they – or some of them, at least – are primitive features of the physical world, not reducible to anything else nor explicable in terms of frequencies, degrees of belief, or anything else. In this paper I explore the question of what the semantic content of primitive chance claims could be. Every attempt I look at to supply such content either comes up empty-handed, or begs (...) important questions against the skeptic who doubts the meaningfulness of primitive chance claims. In the second half of the paper I show that, by contrast, there are clear, and clearly contentful, ways to understand objective chance claims if we ground them on deterministic physical underpinnings. (shrink)

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference (...) class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains. (shrink)

Frequentism and Bayesianism represent very different approaches to hypothesis testing, and this presents a skeptical challenge for Bayesians. Given that most empirical research uses frequentist methods, why (if at all) should we rely on it? While it is well known that there are conditions under which Bayesian and frequentist methods agree, without some reason to think these conditions are typically met, the Bayesian hasn’t shown why we are usually safe in relying on results reported by significance testers. In this article, (...) I provide arguments that such conditions will usually be met; the Bayesian can maintain her theoretical disagreement with the frequentist while holding that her error is mostly harmless in practice. (shrink)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that it (...) is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

Can stable regularities be explained without appealing to governing laws or any other modal notion? In this paper, I consider what I will call a ‘Humean system’—a generic dynamical system without guiding laws—and assess whether it could display stable regularities. First, I present what can be interpreted as an account of the rise of stable regularities, following from Strevens [2003], which has been applied to explain the patterns of complex systems (such as those from meteorology and statistical mechanics). Second, since (...) this account presupposes that the underlying dynamics displays deterministic chaos, I assess whether it can be adapted to cases where the underlying dynamics is not chaotic but truly random—that is, cases where there is no dynamics guiding the time evolution of the system. If this is so, the resulting stable, apparently non-accidental regularities are the fruit of what can be called statistical necessity rather than of a primitive physical necessity. (shrink)

If the laws of nature are as the Humean believes, it is an unexplained cosmic coincidence that the actual Humean mosaic is as extremely regular as it is. This is a strong and well-known objection to the Humean account of laws. Yet, as reasonable as this objection may seem, it is nowadays sometimes dismissed. The reason: its unjustified implicit assignment of equiprobability to each possible Humean mosaic; that is, its assumption of the principle of indifference, which has been attacked on (...) many grounds ever since it was first proposed. In place of equiprobability, recent formal models represent the doxastic state of total ignorance as suspension of judgment. In this paper I revisit the cosmic coincidence objection to Humean laws by assessing which doxastic state we should endorse. By focusing on specific features of our scenario I conclude that suspending judgment results in an unnecessarily weak doxastic state. First, I point out that recent literature in epistemology has provided independent justifications of the principle of indifference. Second, given that the argument is framed within a Humean metaphysics, it turns out that we are warranted to appeal to these justifications and assign a uniform and additive credence distribution among Humean mosaics. This leads us to conclude that, contrary to widespread opinion, we should not dismiss the cosmic coincidence objection to the Humean account of laws. (shrink)

I argue that any broadly dispositional analysis of probability will either fail to give an adequate explication of probability, or else will fail to provide an explication that can be gainfully employed elsewhere (for instance, in empirical science or in the regulation of credence). The diversity and number of arguments suggests that there is little prospect of any successful analysis along these lines.

An adequate account of laws should satisfy at least five desiderata: it should provide a unified account of laws and chances, it should yield plausible relations between laws and chances, it should vindicate numerical chance assignments, it should accommodate dynamical and non-dynamical chances, and it should accommodate a plausible range of nomic possibilities. No extant account of laws satisfies these desiderata. This paper presents a non-Humean account of laws, the Nomic Likelihood Account, that does.

In an influential paper, James Lenman argues that consequentialism can provide no basis for ethical guidance, because we are irredeemably ignorant of most of the consequences of our actions. If our ignorance of distant consequences is great, he says, we can have little reason to recommend one action over another on consequentialist grounds. In this article, I show that for reasons to do with statistical theory, the cluelessness objection is too pessimistic. We have good reason to believe that certain patterns (...) of action will tend to have better consequences, and we have good reason to recommend acting in accordance with strategies based on those advantageous patterns. I close by saying something about the strategies that this argument should lead us to favour. (shrink)

This article surveys several interrelated issues in the metaphysics of chance. First, what is the relationship between the probabilities associated with types of trials (for instance, the chance that a twenty‐eight‐year old develops diabetes before age thirty) and the probabilities associated with individual token trials (for instance, the chance that I develop diabetes before age thirty)? Second, which features of the the world fix the chances: are there objective chances at all, and if so, are there non‐chancy facts on which (...) they supervene? Third, can chance be reconciled with determinism, and if so, how? (shrink)

Traditional Bayesianism requires that an agent’s degrees of belief be represented by a real-valued, probabilistic credence function. However, in many cases it seems that our evidence is not rich enough to warrant such precision. In light of this, some have proposed that we instead represent an agent’s degrees of belief as a set of credence functions. This way, we can respect the evidence by requiring that the set, often called the agent’s credal state, includes all credence functions that are in (...) some sense compatible with the evidence. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In this article I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid it without compromising the initial evidentialist motivation. _1_ Introduction _2_ Precision and Its Problems _3_ Imprecise Bayesianism and Respecting Ambiguous Evidence _4_ Local Belief Inertia _5_ From Local to Global Belief Inertia _6_ Responding to Global Belief Inertia _7_ Conclusion. (shrink)

I describe a realist, ontologically objective interpretation of probability, "far-flung frequency (FFF) mechanistic probability". FFF mechanistic probability is defined in terms of facts about the causal structure of devices and certain sets of frequencies in the actual world. Though defined partly in terms of frequencies, FFF mechanistic probability avoids many drawbacks of well-known frequency theories and helps causally explain stable frequencies, which will usually be close to the values of mechanistic probabilities. I also argue that it's a virtue rather than (...) a failing of FFF mechanistic probability that it does not define single-case chances, and compare some aspects of my interpretation to a recent interpretation proposed by Strevens. (shrink)

The weather report says that the chance of a hurricane arriving later today is 90%. Forewarned is forearmed: expecting a hurricane, before leaving home you pack your hurricane lantern.

This thesis modifies the wording of William Clifford’s 1877 evidence principle (that ‘it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence’) to propose an explicitly moral principle, restricted to descriptive beliefs (about what is or is not the case) and excluding prescriptive beliefs (about what ought or ought not to be the case). It considers potential counter-examples, particularly William James’s 1896 defence of religious belief; and concludes that the modified principle survives unscathed. It then searches (...) for suitable criteria for ‘sufficient’ and ‘insufficient’ evidence, first within a Bayesian framework enriched by pragmatic considerations, and finally by returning to Clifford’s conception of our shared responsibilities to our shared epistemic asset. Combined with elements of Edward Craig’s (1990, 1999) ‘state-of-nature’ theory of knowledge this gets us to a minimum threshold to avoid insufficient evidence and an aspirational criterion of sufficient evidence. (shrink)

Or better: time asymmetry in thermodynamics. Better still: time asymmetry in thermodynamic phenomena. “Time in thermodynamics” misleadingly suggests that thermodynamics will tell us about the fundamental nature of time. But we don’t think that thermodynamics is a fundamental theory. It is a theory of macroscopic behavior, often called a “phenomenological science.” And to the extent that physics can tell us about the fundamental features of the world, including such things as the nature of time, we generally think that only fundamental (...) physics can. On its own, a science like thermodynamics won’t be able to tell us about time per se. But the theory will have much to say about everyday processes that occur in time; and in particular, the apparent asymmetry of those processes. The pressing question of time in the context of thermodynamics is about the asymmetry of things in time, not the asymmetry of time, to paraphrase Price ( , ). I use the title anyway, to underscore what is, to my mind, the centrality of thermodynamics to any discussion of the nature of time and our experience in it. The two issues—the temporal features of processes in time, and the intrinsic structure of time itself—are related. Indeed, it is in part this relation that makes the question of time asymmetry in thermodynamics so interesting. This, plus the fact that thermodynamics describes a surprisingly wide range of our ordinary experience. We’ll return to this. First, we need to get the question of time asymmetry in thermodynamics out on the table. (shrink)

Bayesian confirmation theory—abbreviated to in these notes—is the predominant approach to confirmation in late twentieth century philosophy of science. It has many critics, but no rival theory can claim anything like the same following. The popularity of the Bayesian approach is due to its flexibility, its apparently effortless handling of various technical problems, the existence of various a priori arguments for its validity, and its injection of subjective and contextual elements into the process of confirmation in just the places where (...) critics of earlier approaches had come to think that subjectivity and sensitivity to context were necessary. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

The prospects of a causal interpretation of probability are examined. Various accounts both from the history of scientific method and from recent developments in the tradition of the method of arbitrary functions, in particular by Strevens, Rosenthal, and Abrams, are briefly introduced and assessed. I then present a specific account of causal probability with the following features: First, the link between causal probability and a particular account of induction and causation is established, namely eliminative induction and the related difference-making account (...) of causation in the tradition of Bacon, Herschel, and Mill. Second, it is shown how a causal approach is useful beyond applications of the method of arbitrary functions and is able to deal with various shades of both ontic and epistemic probabilities. Furthermore, I clarify the notion of causal symmetry as a central element of an objective version of the principle of indifference and relate probabilistic independence to causal irrelevance. (shrink)

There is a huge chasm between the notion of lawful determination that figures in fundamental physics, and the notion of causal determination that figures in the "folk physics" of everyday objects. In everyday life, we think of the behavior of an ordinary object as being determined by a small set of simple conditions. But in fundamental physics, no such conditions suffice to determine an ordinary object's behavior. What bridges the chasm is that fundamental physical laws make the folk picture of (...) the world approximately true in certain domains. How? In part, by entailing that many objects are approximately isolated from most of their environments. Dynamical laws yield this result only in conjunction with appropriate statistical assumptions about initial conditions. (shrink)