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In recent work, Callender and Cohen (2009) and Hoefer (2007) have proposed variants of the account of chance proposed by Lewis (1994). One of the ways in which these accounts diverge from Lewis’s is that they allow special sciences and the macroscopic realm to have chances that are autonomous from those of physics and the microscopic realm. A worry for these proposals is that autonomous chances may place incompatible constraints on rational belief. This chapter examines this worry, and attempts to determine (i) what kinds of conflicts would be problematic, and (ii) whether these proposals lead to problematic conflicts. By working through a pair of cases, it is argued that these proposals do give rise to problematic conflicts. Some morals are considered.

Any credible interpretation of objective chance should make sense of the connection between objective chance and rational degree of belief. Ideally, an account that identifies chance with some objective quantity X should be accompanied by a story that explains, from independently plausible assumptions, why X guides rational credence in the way captured by the Principal Principle. In this chapter, this story is provided for various Humean accounts of chance, including frequentist and Best Systems accounts. Along the way, a generalization of the Principal Principle that allows for dyadic and ‘indefinite’ chances is suggested.

This chapter offers a theory of objective chance in the Everettian context, often seen as the main challenge facing EQM. By supplementing diverging EQM with quantum modal realist bridge principles connecting the physics of quantum mechanics with the metaphysics of modality, we obtain a package deal: Indexicalism. Indexicalist objective chance is an essentially self-locating phenomenon: chances are chances of self-location within the multiverse. I provide three arguments for Indexicalism: it establishes the right qualitative connections between chance and possibility, it establishes the right quantitative connection between chance and prediction, and it establishes the right epistemological story about how quantum mechanics is confirmed by empirical evidence. The resulting theory of chance is naturalistic and reductive; fundamental reality is deterministic, but chance arises at the non-fundamental level of Everett-worldbound perspectives. The theory provides unique resources for motivating an Everettian version of Lewis’s Principal Principle, helping to clarify at last the persistently mysterious connection between chance and rational credence.

This chapter begins Part II of the book, which treats the accuracy argument for various chance-credence principles. This chapter considers David Lewis’ Principal Principle. It introduces the framework in which that principle is stated and discusses some of its features.

This chapter introduces a key notion for the book’s project: counterfactual chance.The desired evaluation constraint is then developed in close analogy to D. Lewis’s Principal Principle. The concept of admissibility is critically discussed and the resulting principle is compared to the operation of imaging.

The word “probability” has long been used in (at least) two distinct senses. One sense has to do with a rational agent’s degree of belief, commonly called credence in the philosophical literature. The other sort of probability is thought to be characteristic of a physical system, such as a roulette wheel; these are “in the world” rather than in our heads. This concept is called chance. It would be a mistake to think of these as rivals for the title of the single correct interpretation of probability. Rather, they are both useful concepts, with different roles to play. This chapter is an introduction to these concepts and their relations. It includes a discussion of the proper formulation and justification of a principle that links the two concepts, the Principal Principle. It is argued that neither of these concepts is dispensable. This raises the question of whether there is a notion of probability that can play the role of objective chance and is compatible with deterministic laws of physics.

The most familiar philosophical conception of objective chance renders determinism incompatible with non-trivial chances. This conception—associated in particular with the work of David Lewis—is not a good fit with use of the word ‘chance’ and cognates in ordinary discourse. A generalized framework for chance reconciles determinism with non-trivial chances, and provides for a more charitable interpretation of ordinary chance-talk. Variation in an admissible ‘evidence base’ generates a spectrum of different chance functions: coarse-grainings of the evidence base generate a partial ordering of chance functions, with finer trumping coarser if known. A contextual mechanism is proposed, according to which users of chance-talk refer to different chance functions in different contexts. Admissible evidence is identified with available evidence: evidence which could be obtained. Consequently, attributions of objective chances inherit the relatively familiar context-sensitivity associated with the modal ‘could’. This context-dependency undermines certain arguments for the incompatibility of chance with determinism.

This chapter introduces a first attempt at an accuracy-based argument for chance-credence principles. It adapts the accuracy-based argument for Probabilism by changing the notion of vindication from that proposed by the veritist to an alternative that has been endorsed by Alan Hájek. The chapter argues that, ultimately, it fails.

This chapter gives two distinct justifications of the Principal Principle (PP) for Humean objective chances (HOCs). The first justification is “consequentialist” in nature: it shows that in practical decision-making, an agent who has to make bets on repeated chancy events of type A, and who knows the chance of A but has no better information (the scenario of PP), will do better setting her credence equal to the chance of A than she can do with any other, significantly different, betting strategy. The second justification shows that an epistemic agent meeting the conditions for application of PP is irrational—logically incoherent, in fact—if she sets her credence to a level substantially different from the chance. This argument is an adaptation of one originally offered by Colin Howson and Peter Urbach (1993) to justify the PP for von Mises–style hypothetical frequentism. It is shown that the argument works better in support of HOC than it did in support of frequentism.

How does Bayesian inference handle the highly idealized nature of many (statistical) models in science? The standard interpretation of probability as degree of belief in the truth of a model does not seem to apply in such cases since all candidate models are most probably wrong. Similarly, it is not clear how chance-credence coordination works for the probabilities generated by a statistical model. We solve these problems by developing a suppositional account of degree of belief where probabilities in scientific modeling are decoupled from our actual (unconditional) degrees of belief. This explains the normative pull of chance-credence coordination in Bayesian inference, uncovers the essentially counterfactual nature of reasoning with Bayesian models, and squares well with our intuitive judgment that statistical models provide “objective” probabilities.

The main purpose of this introduction is to provide background on debates about chance (mostly from the philosophy of probability literature) and about the direction of time (mostly from the philosophy of physics literature). It is designed to get those with a general background in metaphysics up to speed in these more specialized areas. The contributions to the volume are also summarized, and some connections between them are drawn out.

The problem of undermining is an apparent contradiction that arises when Humean chances and the Principal Principle come together. This chapter gives a full discussion of, and resolution of, the undermining/contradiction problem for Humean chance. First, the problem is laid out and earlier attempts to resolve it are shown to be insufficient. It is then argued that the correct way to overcome the undermining problem is via a revised form of the Lewis-Hall response, which involves making a small amendment to the Principal Principle (PP) itself. The amendment is seen to be simpler and better justified than the one Lewis and Hall suggested. The modified version of the PP can be seen to be essentially identical, in all practical scenarios, to the original PP.

In his contribution to this volume, Christopher Meacham identifies a possibly serious defect of the pragmatic Humean approach to laws and chance, compared to the more fundamentalist/reductionist approaches of Lewis and Loewer: a potential inconsistency in the credences that rational agents are advised to have, when two or more different objective chances are ascribed to the same event by distinct parts of the pragmatist’s chance theory. This chapter discusses Meacham’s objections, some of which apply to Hoefer’s approach to objective chance, and some of which apply to Callender and Cohen’s Better Best System account of laws. In the course of defending Hoefer’s account of chance from the inconsistency objection, the chapter also discusses some contentious issues regarding how we should understand the Principal Principle, and in particular whether it should include an ‘admissibility clause’.

This chapter introduces a second attempt at an accuracy-based argument for chance-credence principles. In this attempt, the veritist account of vindication, on which the accuracy-based argument for Probabilism depends, is retained. What is changed instead is the decision-theoretic principle: it is changed from a pure dominance principle to what is called a chance dominance principle. The chapter shows how the argument thus adapted can be used to establish various chance-credence principles and answers an objection that the justification it provides is circular.