Chance-Credence Principles

(This is for the series Elements of Decision Theory published by Cambridge University Press and edited by Martin Peterson) -/- Our beliefs come in degrees. I believe some things more strongly than I believe others. I believe very strongly that global temperatures will continue to rise during the coming century; I believe slightly less strongly that the European Union will still exist in 2029; and I believe much less strongly that Cardiff is east of Edinburgh. My credence in something is (...) a measure of the strength of my belief in it; it represents my level of confidence in it. These are the states of mind we report when we say things like ‘I’m 20% confident I switch off the gas before I left' or ‘I’m 99.9% confident that it is raining outside'. -/- There are laws that govern these credences. For instance, I shouldn't be more confident that sea levels will rise by over 2 metres in the next 100 years than I am that they'll rise by over 1 metre, since the latter is true if the former is. This book is about a particular way we might try to establish these laws of credence: the Dutch Book arguments (For briefer overviews of these arguments, see Alan Hájek’s entry in the Oxford Handbook of Rational and Social Choice and Susan Vineberg’s entry in the Stanford Encyclopaedia.) -/- We begin, in Chapter 2, with the standard formulation of the various Dutch Book arguments that we'll consider: arguments for Probabilism, Countable Additivity, Regularity, and the Principal Principle. In Chapter 3, we subject this standard formulation to rigorous stress-testing, and make some small adjustments so that it can withstand various objections. What we are left with is still recognisably the orthodox Dutch Book argument. In Chapter 4, we set out the Dutch Strategy argument for Conditionalization. In Chapters 5 and 6, we consider two objections to Dutch Book arguments that cannot be addressed by making small adjustments. Instead, we must completely redesign those arguments, replacing them with ones that share a general approach but few specific details. In Chapter 7, we consider a further objection to which I do not have a response. In Chapter 8, we'll ask what happens to the Dutch Book arguments if we change certain features of the basic framework in which we've been working: first, we ask how Dutch Book arguments fare when we consider credences in self-locating propositions, such as It is Monday; second, we lift the assumption that the background logic is classical and explore Dutch Book arguments for non-classical logics; third, we lift the assumption that an agent's credal state can be represented by a single assignment of numerical values to the propositions she considers. In Chapter 9, we present the mathematical results that underpin these arguments. (shrink)

Direct inferences identify certain probabilistic credences or confirmation-function-likelihoods with values of objective chances or relative frequencies. The best known version of a direct inference principle is David Lewis’s Principal Principle. Certain kinds of statements undermine direct inferences. Lewis calls such statements inadmissible. We show that on any Bayesian account of direct inference several kinds of intuitively innocent statements turn out to be inadmissible. This may pose a significant challenge to Bayesian accounts of direct inference. We suggest some ways in which (...) these challenges may be addressed. (shrink)

Lewis’s Principal Principle is widely recognized as a rationality constraint that our credences should satisfy throughout our epistemic life. In practice, however, our credences often fail to satisfy this principle because of our various epistemic limitations. Facing such violations, we should correct our credences in accordance with this principle. In this paper, I will formulate a way of correcting our credences, which will be called the Adams Correcting Rules and then show that such a rule yields non-commutativity between conditionalizing and (...) correcting. With the help of the notion of ‘accuracy’, then, I attempt to provide a vindication of the Adams Correcting Rule and show how we can respond to the non-commutativity in question. (shrink)

Many have claimed that epistemic rationality sometimes requires us to have imprecise credal states (i.e. credal states representable only by sets of credence functions) rather than precise ones (i.e. credal states representable by single credence functions). Some writers have recently argued that this claim conflicts with accuracy-centered epistemology, i.e., the project of justifying epistemic norms by appealing solely to the overall accuracy of the doxastic states they recommend. But these arguments are far from decisive. In this essay, we prove some (...) new results, which show that there is little hope for reconciling the rationality of credal imprecision with accuracy-centered epistemology. (shrink)

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.

I present an account of deterministic chance which takes, as its jumping-off point, the physico-mathematical approach to theorizing about deterministic chance known as 'the method of arbitrary functions'. This approach promisingly yields deterministic probabilities which align with what we take the chances to be---it tells us that there is approximately a 1/2 probability of a spun roulette wheel stopping on black, and approximately a 1/2 probability of a flipped coin landing heads up---but it requires some probabilistic materials to work with. (...) I contend that the right probabilistic materials are found in reasonable initial credence distributions. I note that, with some normative assumptions, the resulting account entails that deterministic chances obey a variant of Lewis's 'principal principle'. I additionally argue that deterministic chances, so understood, are capable of explaining long-run frequencies. (shrink)

Hawthorne, Landes, Wallmann and Williamson argue that the Principal Principle implies a version of the Principle of Indifference. We show that what the Authors take to be the Principle of Indifference can be obtained without invoking anything which would seem to be related to the Principal Principle. In the Appendix we also discuss several Conditions proposed in the same paper.

David Lewis' "Principal Principle" is a purported principle of rationality connecting credence and objective chance. Almost all of the discussion of the Principal Principle in the philosophical literature assumes classical probability theory, which is unfortunate since the theory of modern physics that, arguably, speaks most clearly of objective chance is the quantum theory, and quantum probabilities are not classical probabilities. Given the generally accepted updating rule for quantum probabilities, there is a straight forward sense in which the Principal Principle is (...) a theorem of quantum probability theory for any credence function satisfying a suitable additivity requirement. No additional principle of rationality is needed to bring credence into line with objective chance. (shrink)

The world is said to contain crystal balls whenever the present carries news of the as-yet-undetermined parts of the future. Many philosophers believe that crystal balls are metaphysically possible. In this essay, I argue that they are not. Whether crystal balls are possible matters, for at least two reasons. The first is epistemological. According to a simple, user-friendly chance norm for credence, which I call the Present Principle, agents are rationally required to conform their credences to their expectations of the (...) present chances, deferring to the present chances as they would to an expert. I would like to defend the Present Principle since its truth would do much to simplify the relation between chance and credence. But the Present Principle is counterexample-free, and hence defensible, only if crystal balls are impossible. The second reason is decision-theoretic. The problem of crystal balls is one of the main objections to causal decision theory. But crystal ball cases can be counterexample to causal decision theory only if crystal balls are possible, and, as I argue, crystal balls are not possible. (shrink)

Rather than entailing that a particular outcome will occur, many scientific theories only entail that an outcome will occur with a certain probability. Because scientific evidence inevitably falls short of conclusive proof, when choosing between different theories it is standard to make reference to how probable the various options are in light of the evidence. A full understanding of probability in science needs to address both the role of probabilities in theories, or chances, as well as the role of probabilistic (...) judgment in theory choice. In this chapter, the author introduces and distinguishes the two sorts of probability from one another and attempt to offer a satisfactory characterization of how the different uses for probability in science are to be understood. A closing section turns to the question of how views about the chance of some outcome should guide our confidence in that outcome. (shrink)

Researchers doing welfare-related science frequently mischaracterize either situations of decision-theoretic mathematical/scientific uncertainty as situations of risk, or situations of risk as those of uncertainty. The paper outlines this epistemic/ethical problem ; surveys its often-deadly, welfare-related consequences in environmental-health sciences; and uses recent research on diesel particulate matter to reveal 7 specific methodological ways that scientists may mischaracterize lethal risks instead as situations of uncertainty, mainly by using methods and assumptions with false-negative biases. The article closes by outlining two normative strategies (...) for curbing misrepresentations of risk and uncertainty, especially in welfare-affecting science. (shrink)

In his, Roger White argued for a principle of indifference. Hart and Titelbaum showed that White’s argument relied on an intuition about conditioning on biconditionals which, while widely shared, is incorrect. In their, Hawthorne, Landes,Wallmann, and Williamson argue for a principle of indifference. Remarkably, their argument relies on the same faulty intuition. We explain their intuition, explain why it’s faulty, and show how it generates their principle of indifference.

A synchronic probability is the probability at a time that an outcome occurs at that very time. Common sense invokes synchronic probabilities with values between 0 and 1, as do scientific theories such as classical statistical mechanics. Recently, philosophers have argued about whether any synchronic probabilities are best interpreted as objective chances. I add to this debate an underappreciated reason we might have to believe in synchronic chance; it might turn out that the best interpretation of our common sense and (...) scientific theories is one in which the macrophysical properties of physical systems are partly determined by synchronic chance distributions over microphysical properties of those systems. Additionally, I argue against the common charge that synchronic probability fails to satisfy various platitudes about chance—most notably Lewis’s Principal Principle. (shrink)

GRW Theory postulates a stochastic mechanism assuring that every so often the wave function of a quantum system is `hit', which leaves it in a localised state. How are we to interpret the probabilities built into this mechanism? GRW theory is a firmly realist proposal and it is therefore clear that these probabilities are objective probabilities (i.e. chances). A discussion of the major theories of chance leads us to the conclusion that GRW probabilities can be understood only as either single (...) case propensities or Humean objective chances. Although single case propensities have some intuitive appeal in the context of GRW theory, on balance it seems that Humean objective chances are preferable on conceptual grounds because single case propensities suffer from various well know problems such as unlimited frequency tolerance and lack of a rationalisation of the principal principle. (shrink)

In a recent paper in this journal, James Hawthorne, Jürgen Landes, Christian Wallmann, and Jon Williamson argue that the principal principle entails the principle of indifference. In this paper, I argue that it does not. Lewis’s version of the principal principle notoriously depends on a notion of admissibility, which Lewis uses to restrict its application. HLWW base their argument on certain intuitions concerning when one proposition is admissible for another: Conditions 1 and 2. There are two ways of reading their (...) argument, depending on how you understand the status of these conditions. Reading 1: The correct account of admissibility is determined independently of these two principles, and yet these two principles follow from that correct account. Reading 2: The correct account of admissibility is determined in part by these two principles, so that the principles follow from that account but only because the correct account is constrained so that it must satisfy them. HLWWshow that, given an account of admissibility on which Conditions 1 and 2 hold, the principal principle entails the principle of indifference. I argue that, on either reading of the argument, it fails. First, I argue that there is a plausible account of admissibility on which Conditions 1 and 2 are false. That defeats reading 1. Next, I argue that the intuitions that lead us to assent to Condition 2 also lead us to assent to other very closely related principles that are inconsistent with Condition 2. This, I claim, casts doubt on the reliability of those intuitions, and thus removes our justification for Condition 2. This defeats the second reading of the HLWW argument. Thus, the argument fails. (shrink)

According to David Lewis’s Principal Principle, our beliefs about the objective chances of outcomes determine our rational credences in those outcomes. Lewis influentially argues that any adequate metaphysics of objective chance must explain why the Principal Principle holds. Since no theory of chance is widely agreed to have met this burden, I suggest we change tack. On the view I develop, a central aspect of the Principal Principle holds not because of what objective chances are but rather because of the (...) explanatory role that objective chances play for the rational agents who believe in them. (shrink)

In a recent paper in the British Journal for the Philosophy of Science, James Hawthorne, Jürgen Landes, Christian Wallmann, and Jon Williamson (henceforth HLWW) argue that the Principal Principle entails the Principle of Indifference. In this paper, I argue that it does not. Lewis’ version of the Principal Principle notoriously depends on a notion of admissibility, which Lewis uses to restrict its application. HLWW do not give a precise account either, but they do appeal to two principles concerning admissibility, which (...) they call Condition 1 and Condition 2. There are two ways of reading their argument, depending on how you understand the status of Conditions 1 and 2. Reading 1: The correct account of admissibility is determined independently of these two principles, and yet these two principles follow from that correct account. Reading 2: The correct account of admissibility is determined in part by these two principles, so that the principles follow from that account but only because the correct account is constrained so that it must satisfy them. HLWW then show that, given an account of admissibility on which Conditions 1 and 2 hold, the Principal Principle entails the Principle of Indifference. I will argue that, on either reading of the argument, it fails. I will argue that there is a plausible account of admissibility on which Conditions 1 and 2 are false. That defeats the first reading of the argument. I will then argue that the intuitions that lead us to assent to Condition 2 also lead us to assent to other very closely related principles that are inconsistent with Condition 2. This, I claim, casts doubt on the reliability of those intuitions, and thus removes our justification for Condition 2. This defeats the second reading of the HLWW argument. Thus, the argument fails. (shrink)

David Lewis' "Principal Principle" is a purported principle of rationality connecting credence and objective chance. Almost all of the discussion of the Principal Principle in the philosophical literature assumes classical probability theory, which is unfortunate since the theory of modern physics that, arguably, speaks most clearly of objective chance is the quantum theory, and quantum probabilities are not classical probabilities. This paper develops an account of how chance works in quantum theory that reveals a connection between credence and quantum chance (...) quite unlike what is envisioned in the philosophical literature: as a theorem of quantum probability, updating a completely additive chance function on a knowledge of chance brings credence into line with chance. The account also suggests a way of construing the Humean supervenience of chance that has the virtue of dissolving some puzzles about the "undermining" of chances. A number of interpretative moves in quantum theory are needed to generate the account of quantum chance on offer here, and they can all be disputed. But engaging in these disputes is part and parcel of naturalized metaphysics, and as such it can be more productive than engaging in the battle of intuitions among analytical metaphysicians about how chance ought to work this and other possible worlds. -/- . (shrink)

Darren Bradley has recently appealed to observation selection effects to argue that conditionalization presents no special problem for Everettian quantum mechanics, and to defend the ‘halfer’ answer to the puzzle of Sleeping Beauty. I assess Bradley’s arguments and conclude that while he is right about confirmation in Everettian quantum mechanics, he is wrong about Sleeping Beauty. This result is doubly good news for Everettians: they can endorse Bayesian confirmation theory without qualification, but they are not thereby compelled to adopt the (...) unpopular ‘halfer’ answer in Sleeping Beauty. These considerations suggest that objective chance is playing an important and under-appreciated role in Sleeping Beauty. (shrink)

I argue—from a Humean perspective—for the falsity of what I call the “Admissibility of Historical Information Thesis”. According to the AHIT, propositions that describe past events are always admissible with respect to propositions that describe future events. I first demonstrate that this thesis has some counter-intuitive implications and argue that a Humean can explain the intuitive attractiveness of the AHIT by arguing that it results from a wrong understanding of the concept of chance. I then demonstrate how a Humean “best (...) system” analysis of chance predicts the existence of inadmissible historical information and discuss the implications of this conclusion to the debate between Humeans and non-Humeans. (shrink)

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. (shrink)

David Lewis proposed the Principal Principle and a “reformulation” which later on he called ‘OP’. Reacting to his belief that these principles run into trouble, Lewis concluded that they should be replaced with the New Principle. This conclusion left Lewis uneasy, because he thought that an inverse form of NP is “quite messy”, whereas an inverse form of OP, namely the simple and intuitive PP, is “the key to our concept of chance”. I argue that, even if OP should be (...) discarded, FP need not be. Moreover, far from being messy, an inverse form of NP is a simple and intuitive Conditional Principle. Finally, both PP and CP are special cases of a General Principle : it follows that so are PP and NP, which are thus compatible rather than competing. (shrink)

The existence of "undermining futures" appears to show that a contradiction can be deduced from the conjunction of Humean supervenience about chance and the Principal Principle. A number of strategies for rescuing HS from this problem have been proposed recently. In this paper, a novel way of defending HS from the threat is presented, and it is argued that this defense has advantages not shared by others. In particular, it requires no revisionism about chance, and it is equally available to (...) defenders of HS who hold HS to be necessary and those who hold it to be contingent. (shrink)

Dorr et al. present a case that poses a challenge for a number of plausible principles about knowledge and objective chance. Implicit in their discussion is an interesting new argument against KK, the principle that anyone who knows p is in a position to know that they know p. We bring out this argument, and investigate possible responses for defenders of KK, establishing new connections between KK and various knowledge-chance principles.

In a deterministically evolving world, the usefulness of nontrivial probabilities can seem mysterious. I use the ‘Mentaculus’ machinery developed by David Albert and Barry Loewer to show how all probabilities in such a world can be derived from a single, initial chance event. I go on to argue that this is the only genuine chance event. Perhaps surprisingly, we have good evidence of its existence and nature. I argue that the existence of this chance event justifies our epistemic reliance on (...) nontrivial probabilities. (shrink)

I follow Hájek (Synthese 137:273–323, 2003c) by taking objective probability to be a function of two propositional arguments—that is, I take conditional probability as primitive. Writing the objective probability of q given r as P(q, r), I argue that r may be chosen to provide less than a complete and exact description of the world’s history or of its state at any time. It follows that nontrivial objective probabilities are possible in deterministic worlds and about the past. A very simple (...) chance–credence relation is also then natural, namely that reasonable credence equals objective probability. In other words, we should set our actual credence in a proposition equal to the proposition’s objective probability conditional on available background information. One advantage of that approach is that the background information is not subject to an admissibility requirement, as it is in standard formulations of the Principal Principle. Another advantage is that the “undermining” usually thought to follow from Humean supervenience can be avoided. Taking objective probability to be a two-argument function is not merely a technical matter, but provides us with vital flexibility in addressing significant philosophical issues. (shrink)

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...

This is a discussion of Richard Pettigrew's book "Accuracy and the Laws of Credence". I target Pettigrew's application of the accuracy framework to derive chance-credence principles. My principal contention is that Pettigrew's preferred version of the argument might in one sense be circular and, moreover, that Pettigrew's premises have content that go beyond that of standard chance-credence principles.

A principle endorsed by many theories of objective chance, and practically forced on us by the standard interpretation of the Kolmogorov semantics for chance, is the principle that when a proposition P has a chance, any proposition Q that is necessarily equivalent to P will have the same chance as P. Call this principle SUB (for the substitution of necessary equivalents into chance ascriptions). I will present some problems for a theory of chance, and will argue that the best way (...) to resolve these problems is to reject SUB, and similar principles e.g. for the chances of outcomes or the chances of events. Objective chance, it turns out, carves things more finely than necessary equivalence does. (shrink)

The Doomsday argument purports to show that the risk of the human species going extinct soon has been systematically underestimated. This argument has something in common with controversial forms of reasoning in other areas, including: game theoretic problems with imperfect recall, the methodology of cosmology, the epistomology of indexical belief, and the debate over so-called fine-tuning arguments for the design hypothesis. The common denominator is a certain premiss: the Self-Sampling Assumption. We present two strands of argument in favor of this (...) assumption. Through a series of throught experiments we then investigate some bizarre _prima facie_ consequences - backward causation, psychic powers, and an apparent conflict with the Principal Principle. (shrink)

Weak and strong consistency of thePrincipal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract Principal Principle is both weakly and strongly consistent. The Abstract Principal Principle is strengthened by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined. It is shown that the Stable Abstract Principal Principle is weakly consistent. Strong consistency of the Stable Abstract Principal principle remains an open question.

Rédei and Gyenis suggest that Lewis’s Principal Principle is meaningful only if it satisfies certain consistency conditions: starting from any assignment of subjective probabilities to some algebra of events, we should always be able to extend our algebra with events of the form “the value of the objective probability of event E is p” and assign subjective probabilities to such events in a consistent manner. We show that this extension is indeed possible in most cases. However, we also argue that (...) this requirement is not necessary: the Principal Principle concerns subjective believes about objective chance, hence events concerning those probabilities are meant to be in the algebra initially, as Lewis’s text suggests clearly. (shrink)

One argument for the thirder position on the Sleeping Beauty problem rests on direct inference from objective probabilities. In this paper, I consider a particularly clear version of this argument by John Pollock and his colleagues (The Oscar Seminar 2008). I argue that such a direct inference is defeated by the fact that Beauty has an equally good reason to conclude on the basis of direct inference that the probability of heads is 1/2. Hence, neither thirders nor halfers can find (...) direct support in an appeal to objective probabilities. (shrink)

It is a commonplace in epistemology that credences should equal known chances. It is less clear, however, that conditional credences should do so, too. Following Ramsey, this paper proposes a counterfactual interpretation of conditional probability which provides a justification for this equality without relying on the Principal Principle. As a result, we obtain a refined view of Bayesian inference where both learning and supposing have a place.

_Philosophy of Probability: Contemporary Readings_ is the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others, the book looks in depth at the following key topics: subjective probability and credence probability updating: conditionalization and reflection Bayesian confirmation theory classical, logical, and evidential probability frequentism physical probability: propensities and objective chances. The book (...) features a useful primer on the mathematics of probability, and each section includes an introduction by the editor, as well as a guide to further reading. A broad-ranging and highly accessible exploration of the subject, _Philosophy of Probability_ is ideal for any student of formal epistemology, philosophy of science, metaphysics, or philosophy of mathematics. (shrink)

In this dissertation I aim to clarify the concept of probability. There are three kinds of interpretation of probability: the objective interpretation, the logical interpretation, and the subjective interpretation. The objective interpretation understands probability as a natural property that exists independently of our minds. I will clarify and defend one version of objective probability---the propensity interpretation of probability. I reject the logical interpretation of probability that treats probability as a logical relation between evidence and hypothesis, and maintain that probability is (...) an empirical relation. The subjective interpretation of probability, which is quite influential in recent literature, understands probability as the subject's degree of confidence that is only concerned with the states of one's mind. There are two arguments to support the claim that partial beliefs satisfy the probability measure, the Dutch Book Argument and the Representation Theorem . I argue that the subjective probability derived from both arguments is in fact the subjective estimate of objective chance, rather than subjective uncertainty of the mind assumed by subjective interpretation. The clarification of the concept of probability, I think, is of crucial importance to the studies in the philosophy of science, epistemology, and statistics. (shrink)

Richard Pettigrew offers an extended investigation into a particular way of justifying the rational principles that govern our credences. The main principles that he justifies are the central tenets of Bayesian epistemology, though many other related principles are discussed along the way. Pettigrew looks to decision theory in order to ground his argument. He treats an agent's credences as if they were a choice she makes between different options, gives an account of the purely epistemic utility enjoyed by different sets (...) of credences, and then appeals to the principles of decision theory to show that, when epistemic utility is measured in this way, the credences that violate the principles listed above are ruled out as irrational. The account of epistemic utility set out here is the veritist's: the sole fundamental source of epistemic utility for credences is their accuracy. Thus, Pettigrew conducts an investigation in the version of epistemic utility theory known as accuracy-first epistemology. (shrink)