Various philosophers accept moral views that are impartial, additive, and risk-neutral with respect to moral betterness. But, if that risk neutrality is spelt out according to expected value theory alone, such views face a dire reductio ad absurdum. If the expected sum of value in humanity's future is undefined--if, e.g., the probability distribution over possible values of the future resembles the Pasadena game, or a Cauchy distribution--then those views say that no option is ever better than any other. And, as (...) I argue, this holds in practice: our evidence supports such a probability distribution. Indeed, it supports a probability distribution that cannot be evaluated even if we adopt one of the various extensions of expected value theory proposed in the literature. Must we therefore reject all impartial, additive, risk-neutral moral theories? It turns out that we need not. I develop an alternative solution: a new method of extending expected value theory, which allows us to deal with this distribution and to salvage those moral views. I also examine how this solution affects one of the most notable implications of those views--namely, longtermism. (shrink)

What is it for an imprecise credence to be justified? It might be thought that this is not a particularly urgent question for friends of imprecise credences to answer. For one might think that its answer just depends on how a well-trodden issue in epistemology plays out—namely, that of which theory of doxastic justification, be it reliabilism, evidentialism, or some other theory, is correct. I’ll argue, however, that it’s difficult for reliabilists to accommodate imprecise credences, at least if we understand (...) such credences to be determinate first-order attitudes. If I’m right, reliabilists will have to reject imprecise credences, and friends of imprecise credences will have to reject reliabilism. Near the end of the paper, I’ll also consider whether reliabilism can accommodate indeterminate credences. (shrink)

This paper is a response to Paul Bartha’s ‘Making Do Without Expectations’. We provide an assessment of the strengths and limitations of two notable extensions of standard decision theory: relative expectation theory and Paul Bartha’s relative utility theory. These extensions are designed to provide intuitive answers to some well-known problems in decision theory involving gaps in expectations. We argue that both RET and RUT go some way towards providing solutions to the problems in question but neither extension solves all the (...) relevant problems. (shrink)

Does rationality require imprecise credences? Many hold that it does: imprecise evidence requires correspondingly imprecise credences. I argue that this is false. The imprecise view faces the same arbitrariness worries that were meant to motivate it in the first place. It faces these worries because it incorporates a certain idealization. But doing away with this idealization effectively collapses the imprecise view into a particular kind of precise view. On this alternative, our attitudes should reflect a kind of normative uncertainty: uncertainty (...) about what to believe. This view refutes the claim that precise credences are inappropriately informative or committal. Some argue that indeterminate evidential support requires imprecise credences; but I argue that indeterminate evidential support instead places indeterminate requirements on credences, and is compatible with the claim that rational credences may always be precise. (shrink)

Can we extend accuracy-based epistemic utility theory to imprecise credences? There's no obvious way of proceeding: some stipulations will be necessary for either (i) the notion of accuracy or (ii) the epistemic decision rule. With some prima facie plausible stipulations, imprecise credences are always required. With others, they’re always impermissible. Care is needed to reach the familiar evidential view of imprecise credence: that whether precise or imprecise credences are required depends on the character of one's evidence. I propose an epistemic (...) utility theoretic defense of a common view about how evidence places demands on imprecise credence: that your spread of credence should cover the range of chance hypotheses left open by your evidence. I argue that objections to the form of epistemic utility theoretic argument that I use will extend to the standard motivation for epistemically mandatory imprecise credences. (shrink)

Suppose we want to do the most good we can with a particular sum of money, but we cannot be certain of the consequences of different ways of making use of it. This article explores how our attitudes towards risk and ambiguity bear on what we should do. It shows that risk-avoidance and ambiguity-aversion can each provide good reason to divide our money between various charitable organizations rather than to give it all to the most promising one. It also shows (...) how different attitudes towards risk and ambiguity affect whether we should give to an organization which does a small amount of good for certain or to one which does a large amount of good with some small or unknown probability. (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)

We propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: learning observable evidence obtained by (...) repeated sampling from the unknown distribution; and learning higher-order information about the distribution. The first changes only the plausibility map, but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions, as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning. (shrink)

Imprecise probabilities are often modelled with representors, or sets of probability functions. In the recent literature, two ways of interpreting representors have emerged as especially prominent: vagueness interpretations, according to which each probability function in the set represents how the agent's beliefs would be if any vagueness were precisified away; and comparativist interpretations, according to which the set represents those comparative confidence relations that are common to all probability functions therein. I argue that these interpretations have some important limitations. I (...) also propose an alternative—the functional interpretation—according to which representors are best interpreted by reference to the roles they play in the theories that make use of them. (shrink)

Recent work has formally linked the traditional axiomatisation of incomplete preferences à la Anscombe-Aumann with the theory of desirability developed in the context of imprecise probability, by showing in particular that they are the very same theory. The equivalence has been established under the constraint that the set of possible prizes is finite. In this paper, we relax such a constraint, thus de facto creating one of the most general theories of rationality and decision making available today. We provide the (...) theory with a sound interpretation and with basic notions, and results, for the separation of beliefs and values, and for the case of complete preferences. Moreover, we discuss the role of conglomerability for the presented theory, arguing that it should be a rationality requirement under very broad conditions. (shrink)

Epistemic states of uncertainty play important roles in ethical and political theorizing. Theories that appeal to a “veil of ignorance,” for example, analyze fairness or impartiality in terms of certain states of ignorance. It is important, then, to scrutinize proposed conceptions of ignorance and explore promising alternatives in such contexts. Here, I study Lerner’s probabilistic egalitarian theorem in the setting of imprecise probabilities. Lerner’s theorem assumes that a social planner tasked with distributing income to individuals in a population is “completely (...) ignorant” about which utility functions belong to which individuals. Lerner models this ignorance with a certain uniform probability distribution, and shows that, under certain further assumptions, income should be equally distributed. Much of the criticism of the relevance of Lerner’s result centers on the representation of ignorance involved. Imprecise probabilities provide a general framework for reasoning about various forms of uncertainty including, in particular, ignorance. To what extent can Lerner’s conclusion be maintained in this setting? (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

It is natural to think of precise probabilities as being special cases of imprecise probabilities, the special case being when one’s lower and upper probabilities are equal. I argue, however, that it is better to think of the two models as representing two different aspects of our credences, which are often vague to some degree. I show that by combining the two models into one model, and understanding that model as a model of vague credence, a natural interpretation arises that (...) suggests a hypothesis concerning how we can improve the accuracy of aggregate credences. I present empirical results in support of this hypothesis. I also discuss how this modeling interpretation of imprecise probabilities bears upon a philosophical objection that has been raised against them, the so-called inductive learning problem. (shrink)

The notion of comparative probability defined in Bayesian subjectivist theory stems from an intuitive idea that, for a given pair of events, one event may be considered “more probable” than the other. Yet it is conceivable that there are cases where it is indeterminate as to which event is more probable, due to, e.g., lack of robust statistical information. We take that these cases involve indeterminate comparative probabilities. This paper provides a Savage-style decision-theoretic foundation for indeterminate comparative probabilities.

The basic Bayesian model of credence states, where each individual’s belief state is represented by a single probability measure, has been criticized as psychologically implausible, unable to represent the intuitive distinction between precise and imprecise probabilities, and normatively unjustifiable due to a need to adopt arbitrary, unmotivated priors. These arguments are often used to motivate a model on which imprecise credal states are represented by sets of probability measures. I connect this debate with recent work in Bayesian cognitive science, where (...) probabilistic models are typically provided with explicit hierarchical structure. Hierarchical Bayesian models are immune to many classic arguments against single-measure models. They represent grades of imprecision in probability assignments automatically, have strong psychological motivation, and can be normatively justified even when certain arbitrary decisions are required. In addition, hierarchical models show much more plausible learning behavior than flat representations in terms of sets of measures, which—on standard assumptions about update—rule out simple cases of learning from a starting point of total ignorance. (shrink)

The strategy of updating credences by minimizing the relative entropy has been questioned by many authors, most strongly by means of the Judy Benjamin puzzle. I present a new analysis of Judy Benjamin–like forms of new information and defend the thesis that in general the rational posterior is indeterminate, meaning that a family of posterior credence functions rather than a single one is the rational response when that type of information becomes available. The proposed thesis extends naturally to all cases (...) in which new information is traditionally handled by minimizing the relative entropy. (shrink)

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

It is a consequence of the theory of imprecise credences that there exist situations in which rational agents inevitably become less opinionated toward some propositions as they gather more evidence. The fact that an agent's imprecise credal state can dilate in this way is often treated as a strike against the imprecise approach to inductive inference. Here, we show that dilation is not a mere artifact of this approach by demonstrating that opinion loss is countenanced as rational by a substantially (...) broader class of normative theories than has been previously recognised. Specifically, we show that dilation-like phenomena arise even when one abandons the basic assumption that agents have (precise or imprecise) credences of any kind, and follows directly from bedrock norms for rational comparative confidence judgements of the form `I am at least as confident in p as I am in q'. We then use the comparative confidence framework to develop a novel understanding of what exactly gives rise to dilation-like phenomena. By considering opinion loss in this more general setting, we are able to provide a novel assessment of the prospects for an account of inductive inference that is not saddled with the inevitability of rational opinion loss. (shrink)

The main question of this paper is: how do we manage to know what our own degrees of belief are? Section 1 briefly reviews and criticizes the traditional functionalist view, a view notably associated with David Lewis and sometimes called the theory-theory. I use this criticism to motivate the approach I want to promote. Section 2, the bulk of the paper, examines and begins to develop the view that we have a special kind of introspective access to our degrees of (...) belief. I give an initial assessment of the view by examining its compatibility with leading theories of introspection. And I identify a challenge for the view, and explain why I'm optimistic that the view can overcome it. (shrink)

It is important to have an adequate model of uncertainty, since decisions must be made before the uncertainty can be resolved. For instance, flood defenses must be designed before we know the future distribution of flood events. It is standardly assumed that probability theory offers the best model of uncertain information. I think there are reasons to be sceptical of this claim. I criticise some arguments for the claim that probability theory is the only adequate model of uncertainty. In particular (...) I critique Dutch book arguments, representation theorems, and accuracy based arguments. Then I put forward my preferred model: imprecise probabilities. These are sets of probability measures. I offer several motivations for this model of uncertain belief, and suggest a number of interpretations of the framework. I also defend the model against some criticisms, including the so-called problem of dilation. I apply this framework to decision problems in the abstract. I discuss some decision rules from the literature including Levi’s E-admissibility and the more permissive rule favoured by Walley, among others. I then point towards some applications to climate decisions. My conclusions are largely negative: decision making under such severe uncertainty is inevitably difficult. I finish with a case study of scientific uncertainty. Climate modellers attempt to offer probabilistic forecasts of future climate change. There is reason to be sceptical that the model probabilities offered really do reflect the chances of future climate change, at least at regional scales and long lead times. Indeed, scientific uncertainty is multi-dimensional, and difficult to quantify. I argue that probability theory is not an adequate representation of the kinds of severe uncertainty that arise in some areas in science. I claim that this requires that we look for a better framework for modelling uncertainty. (shrink)

There is a growing interest in the foundations as well as the application of imprecise probability in contemporary epistemology. This dissertation is concerned with the application. In particular, the research presented concerns ways in which imprecise probability, i.e. sets of probability measures, may helpfully address certain philosophical problems pertaining to rational belief. The issues I consider are disagreement among epistemic peers, complete ignorance, and inductive reasoning with imprecise priors. For each of these topics, it is assumed that belief can be (...) modeled with imprecise probability, and thus there is a non-classical solution to be given to each problem. I argue that this is the case for peer disagreement and complete ignorance. However, I discovered that the approach has its shortcomings, too, specifically in regard to inductive reasoning with imprecise priors. Nevertheless, the dissertation ultimately illustrates that imprecise probability as a model of rational belief has a lot of promise, but one should be aware of its limitations also. (shrink)

Much recent philosophical attention has been devoted to variants on the Best System Analysis of laws and chance. In particular, philosophers have been interested in the prospects of such Best System Analyses for yielding *high-level* laws and chances. Nevertheless, a foundational worry about BSAs lurks: there do not appear to be uniquely appropriate measures of the degree to which a system exhibits theoretical virtues, such as simplicity and strength. Nor does there appear to be a uniquely correct exchange rate at (...) which the theoretical virtues of simplicity, strength, and likelihood trade off against one another in the determination of a best system. Moreover, it may be that there is no *robustly* best system: no system that comes out best under *any* reasonable measures of the theoretical virtues and exchange rate between them. This worry has been noted by several philosophers, with some arguing that there is indeed plausibly a set of tied-for-best systems for our world. Some have even argued that this entails that there are no Best System laws or chances in our world. I argue that, while it *is* plausible that there is a set of tied-for-best systems for our world, it doesn't follow from this that there are no Best System chances. Rather, it follows that the Best System chances for our world are *unsharp*. (shrink)

Many argue that absolutist moral theories -- those that prohibit particular kinds of actions or trade-offs under all circumstances -- cannot adequately account for the permissibility of risky actions. In this dissertation, I defend various versions of absolutism against this critique, using overlooked resources from formal decision theory. Against the prevailing view, I argue that almost all absolutist moral theories can give systematic and plausible verdicts about what to do in risky cases. In doing so, I show that critics have (...) overlooked: (1) the fact that absolutist theories -- and moral theories, more generally -- underdetermine their formal decision-theoretic representations; (2) that decision theories themselves can be generalised to better accommodate distinctively absolutist commitments. Overall, this dissertation demonstrates that we can navigate a risky world without compromising our moral commitments. (shrink)

"Chance" crops up all over philosophy, and in many other areas. It is often assumed -- without argument -- that chances are probabilities. I explore the extent to which this assumption is really sanctioned by what we understand by the concept of chance.