Richard Pettigrew

Since its inception, analytic philosophy has appealed to mathematical tools to represent the phenomena that concern it, to formulate claims about them with a certain sort of precision, and to turbocharge its arguments by calling upon non-trivial mathematical theorems to deduce surprising claims from seemingly weak premises. A certain tool I've found useful in my own work is the mathematical notion of a divergence, and in particular a family of divergences known as the Bregman divergences. I learned about them first from a paper that uses them to score probabilistic beliefs by how far they lie from the truth. But, as I've used them more and more, I've became convinced they are a useful tool outside epistemology as well. The purpose of these notes is to describe these mathematical objects and their properties, and explain how they have been used in rather different parts of philosophy. I illustrate mainly with examples from probabilistic epistemology and welfare ethics, but my hope is that this introduction to the mathematics of these divergences will inspire others to use them for different purposes.

In _Seeing Like a State_, James C. Scott argues that governments act badly when they see the society they govern in distorted and incomplete ways. I argue that this is also a common pitfall of university management. In particular, I scrutinise a particular way of thinking and speaking and deliberating that arises when the part of the organisation over which the manager has control is too large to allow them to think and speak always of the goals, viewpoints, and values of individual members. In response, managers often begin to think and speak and deliberate in terms of groups—for instance, the philosophy department, the HR division, the student body—and ascribes beliefs, goals, and values to them as group agents. But such attitude ascriptions to group agents are badly ambiguous in ways that aren’t resolved by context. This, I argue, leads to bad decision-making.

Pettigrew focuses on trade-off objections to epistemic consequentialism. Such objections are similar to familiar objections from ethics where an intuitively wrong action (e.g., killing a healthy patient) leads to a net gain in value (e.g., saving five other patients). The objection to the epistemic consequentialist concerns cases where adopting an intuitively wrong belief leads to a net gain in epistemic value. Pettigrew defends the epistemic consequentialist against such objections by accepting that the unintuitive verdicts of consequentialism are unintuitive, but offering an error theory for why these intuitions do not show the view to be false.

We often ask for the opinion of a group of individuals. How strongly does the scientific community believe that the rate at which sea levels are rising has increased over the last 200 years? How likely does the UK Treasury think it is that there will be a recession if the country leaves the European Union? What are these group credences that such questions request? And how do they relate to the individual credences assigned by the members of the particular group in question? According to the credal judgement aggregation principle, linear pooling, the credence function of a group should be a weighted average or linear pool of the credence functions of the individuals in the group. In this chapter, I give an argument for linear pooling based on considerations of accuracy. And I respond to two standard objections to the aggregation principle.

I give an overview of some of the central ideas, arguments, and results in accuracy-first epistemology (also known as epistemic utility theory or epistemic decision theory), as well as a lengthy technical appendix in which I present some of the central proofs. To begin, I describe the three central principles of Bayesian epistemology--Probabilism, Conditionalization, and the Principal Principle--and ask why they are requirements of good reasoning. I give a brief overview of the pragmatist's answer, due to Frank Ramsey (1931) and Bruno de Finetti (1937), but my topic is the more recent, purely epistemic answer, due to James M. Joyce (1999) and others. I'll describe that answer and then explain how it can furnish us with a foundation for norms of reasoning under uncertainty. It grounds norms that govern both ideal and non-ideal agents; it provides a natural account of the goal of inquiry, its value, and the norms that govern how we choose what to learn; it sheds light on questions from social epistemology about the opinions of collective agents and the norms that govern epistemic communities; and it interacts naturally with the pragmatist's approach to credal epistemology. My aim is to give a sense of the power of this framework and the breadth of its application, as well as the interest of the results already established and arguments already formulated.

What is the distinction between what we ought to believe _simpliciter_ and what we ought to believe _epistemically speaking_, and why do we draw that distinction? I motivate this question through a series of examples, consider various existing answers, and find them wanting. Then I propose and explore an alternative based on a version of Susan Wolf's rule consequentialism transposed to the epistemic realm: the norms that determine what we ought to do epistemically speaking are those such that broad adherence to them across your epistemic community would give the best results from an epistemic point of view. I argue that norms that exhort us to believe only what we ought to believe epistemically speaking follow from this Wolfian version of epistemic rule consequentialism. In a coda, I offer an alternative account of the function of knowledge ascriptions that improves on Edward Craig's.

Consider Phoebe and Daphne. Phoebe has credences in 1 million propositions. Daphne, on the other hand, has credences in all of these propositions, but she's also got credences in 999 million other propositions. Phoebe's credences are all very accurate. Each of Daphne's credences, in contrast, are not very accurate at all; each is a little more accurate than it is inaccurate, but not by much. Whose doxastic state is better, Phoebe's or Daphne's? It is clear that this question is analogous to a question that has exercised ethicists over the past thirty years. How do we weigh a population consisting of some number of exceptionally happy and satisfied individuals against another population consisting of a much greater number of people whose lives are only just worth living? This is the question that occasions population ethics. In this paper, I go in search of the correct population ethics for credal states.

How should your opinion change in response to the opinion of an epistemic peer? We show that the pooling rule known as "upco" is the unique answer satisfying some natural desiderata. If your revised opinion will influence your opinions on other matters by Jeffrey conditionalization, then upco is the only standard pooling rule that ensures the order in which peers are consulted makes no difference. Popular proposals like linear pooling, geometric pooling, and harmonic pooling cannot boast the same. In fact, no alternative to upco can if it possesses four minimal properties which these proposals share.

Disagreement is a common feature of a social world. For various reasons, however, we sometimes need to resolve a disagreement into a single set of opinions. This can be achieved by pooling the opinions of individuals that make up the group. This Element provides an opinionated survey on some ways of pooling opinions: linear pooling, multiplicative pooling (including geometric), and pooling through imprecise probabilities. While this Element gives significant attention to the axiomatic approach in evaluating pooling strategies, it also evaluates them in terms of the epistemic and practical goals they might meet. In doing so, this Element connects opinion pooling to some philosophical problems in social epistemology and the philosophy of action, illuminating different perspectives one might take when figuring out how to pool opinions for a given purpose. This title is also available as Open Access on Cambridge Core.

One often learns the opinions of others without getting to hear the evidence behind them. How should you revise your own opinions in such cases? Dietrich (2010) shows that, for opinions about objective chance, the method known as upco effectively adds your interlocutor’s evidence to your own. We provide a simple way of viewing upco that makes properties like Dietrich’s easy to appreciate, and we do three things with it. First, we unify Dietrich’s motivation for upco with another motivation due to Easwaran et al. (2016). Second, we show that laypeople can sometimes use upco to resolve expert disagreements. And third, we use it to cricitize the social argument for the uniqueness thesis.

I offer a new style of argument that supports either utilitarianism or instances of the expected equally distributed equivalent version of prioritarianism (EEDE). The central idea is that the value the welfarist assigns to a state of the world should not lie unnecessarily far from the utilities of the individuals who exist at that state. I describe various ways we might measure distance from a candidate value to the individuals' utilities, characterize a family of such measures, and explain which versions of welfarism follow when we use them. For some, it's utilitarianism; for others, it's different instances of EEDE.

In this paper, I aim to identify the wrong that is done by the hearer to the testifier in all cases of testimonial injustice. I introduce the concept of testimonial injustice, as well as the existing accounts of this characteristic wrong, and I argue that the latter don't work. Then I present my favoured account, which adapts Rachel Fraser's account of the wrong of aesthetic injustice. I argue that this allows us to see that certain putative stock examples of testimonial injustices in fact do not fall under the concept.

In this paper, I ask three questions of the liberal. In each, I fill in philosophical detail around a certain sort of complaint raised in current public debates about their position. In the first, I probe the limits of the liberal's tolerance for civil disobedience; in the second, I ask how the liberal can adjudicate the most divisive moral disputes of the age; and, in the third, I suggest the liberal faces a problem when there is substantial disagreement about the boundaries of the rational and the reasonable.

Under ideal conditions, explicit consent and related actions usually change the moral facts in a distinctive way: they make something permissible that was previously impermissible. But they don't do this if the consent is coerced. And it seems they also don't do it if the preferences on which the consent is based were formed in particular ways: if they were formed via certain mechanisms under pressure from unjust social forces, for instance. In this paper, I describe a range of mechanisms by which our preferences might be formed, determine which of those can give rise to preferences that ground effective consent and which cannot, and explain why each mechanism falls on its side of the division.

In the analytic tradition outside the philosophy of science, epistemology has typically begun at the point at which we have our evidence; it has then asked which beliefs or credences are justified or warranted by that evidence, which are rational and which count as knowledge for someone with that evidence. And yet we are not mere passive recipients of our evidence; we often actively collect it, and collect it in one way rather than another, or act in ways we know will bring us certain pieces of evidence and not others, even when acquiring that evidence is not the primary aim of that act. In this essay, I present and apply a theory that says when we should inquire and how. It is a probabilistic theory: I use probabilities to represent the opinions of the inquirer. And it is teleological: whether we should inquire in this way or that or not at all is determined by the value of the available options, where the value of an inquiry is determined by the value of the option you choose from the point of view of the opinions you have after the inquiry, the purely epistemic value of those opinions, the costs the inquiry incurs, and any other benefits it might bring. The theory says you should choose whether, when, and how to inquire just as you choose whether, when, and how to do anything else, namely, by maximizing value in expectation. This theory has its origins in Janina Hosiasson’s work in Bayesian philosophy of science in the 1930s and David Blackwell’s work in economics in the 1950s. In the first part of the essay, I present this theory in a way that assumes no prior familiarity with Bayesianism or the epistemology of probabilistic opinion more generally. In the second part, I put the theory to work. First, I extend it to cover not only inquiries in which you gather empirical evidence, but also those in which you undertake logical or conceptual reasoning. This allows us to ask, for instance, when we should spend our time gathering new evidence and when we should spend it drawing out the consequences of the evidence we already have. Next, I ask when the normative status of the inquiries that have led to our current opinions affect the normative status of those opinions. And finally I show how the teleological approach might be extended to furnish norms for other cognitive acts and processes, such as forgetting or coarse-graining a credal state so that it becomes a belief.

Drawing on a passage from Ramsey's Truth and Probability, we formulate a simple, plausible constraint on evaluating the accuracy of credences: the Calibration Test. We show that any additive, continuous accuracy measure that passes the Calibration Test will be strictly proper. Strictly proper accuracy measures are known to support the touchstone results of accuracy-first epistemology, for example vindications of probabilism and conditionalization. We show that our use of Calibration is an improvement on previous such appeals by showing how it answers or sidesteps problems that have been raised for previous work in this area.

We often learn the opinions of others without hearing the evidence on which they're based. The orthodox Bayesian response is to treat the reported opinion as evidence itself and update on it by conditionalizing. But sometimes this isn't feasible. In these situations, a simpler way of combining one's existing opinion with opinions reported by others would be useful, especially if it yields the same results as conditionalization. We will show that one method---upco, also known as multiplicative pooling---is specially suited to this role when the opinions you wish to pool concern hypotheses about chances. The result has interesting consequences: it addresses the problem of disagreement between experts; and it sheds light on the social argument for the uniqueness thesis.

Decision theories give guidance about what to do when you face a particular decision. But they also give higher-level advice—depending on how likely you think it is that you’ll face various decision problems, decision theories give advice about the best strategy for picking what to do. For some ways of being uncertain over possible decisions, decision theories that accommodate risk undermine themselves. They simultaneously provide specific advice about what to pick whilst also deeming that very picking strategy to be impermissible. Savage-style expected utility theory does not have this flaw; it always deems its own picking strategies to be the rational ones. Popular decision rules for imprecise probabilities never straightforwardly undermine themselves but they often require adopting strategies that pick in accordance with expected utility theory, thereby coordinating how to pick across different potential decision problems. Some strategies that always pick an option that the decision theory does not reject are deemed impermissible. In particular, randomising amongst non-rejected options is often impermissible. This raises questions about how to use the guidance of an imprecise decision theory when it leaves different options non-rejected.

In the first half of this paper, I argue that group belief ascriptions are highly ambiguous. What's more, in many cases, neither the available contextual factors nor known pragmatic considerations are sufficient to allow the audience to identify which of the many possible meanings is intended. In the second half, I argue that this ambiguity often has bad consequences when a group belief ascription is heard and taken as testimony. And indeed it has these consequences even when the ascription is true on the speaker's intended interpretation, when the speaker does not intend to mislead and indeed intends to cooperatively inform, and when the audience incorporates the evidence from the testimony as they should. I conclude by arguing that these consequences should lead us to stop using such ascriptions.

Longtermism is the view that the most urgent global priorities, and those to which we should devote the largest portion of our resources, are those that focus on (i) ensuring a long future for humanity, and perhaps sentient or intelligent life more generally, and (ii) improving the quality of the lives that inhabit that long future. While it is by no means the only one, the argument most commonly given for this conclusion is that these interventions have greater expected goodness per unit of resource devoted to them than each of the other available interventions, including those that focus on the health and well-being of the current population. In this paper, I argue that, even if we grant the consequentialist ethics upon which this argument depends, and even if we grant one of the axiologies that are typically paired with that ethics to give the argument, we are not morally required to choose an option that maximises expected utility; indeed, we might not even be permitted to do so. Instead, I will argue, if the argument's consequentialism is correct, we should choose using a decision theory that is sensitive to risk, and allows us to give greater weight to worse-case outcomes than expected utility theory does. And, I will show, such decision theories do not always recommend longtermist interventions. Indeed, sometimes, they recommend exactly the opposite: sometimes, they recommend hastening human extinction. Many, though not all, will take this as a reductio of the consequentialism or the axiology of the argument. I remain agnostic on the conclusion we should draw.

Epistemologists who study credences have a well-developed account of how you should change them when you learn new evidence; that is, when your body of evidence grows. What's more, they boast a diverse range of epistemic and pragmatic arguments that support that account. But they do not have a satisfactory account of when and how you should change your credences when you become aware of possibilities and propositions you have not entertained before; that is, when your awareness grows. In this paper, I consider the arguments for the credal epistemologist's account of how to respond to evidence, and I ask whether they can help us generate an account of how to respond to awareness growth. The results are surprising: the arguments that all support the same norms for responding to evidence growth support a number of different norms when they are applied to awareness growth. Some of these norms seem too weak, others too strong. I ask what we should conclude from this, and argue that our credal response to awareness growth is considerably less rigorously constrained than our credal response to new evidence.

There are many reasons we might want to take the opinions of various individuals and pool them to give the opinions of the group they constitute. If all the individuals in the group have probabilistic opinions about the same propositions, there is a host of pooling functions we might deploy, such as linear or geometric pooling. However, there are also cases where different members of the group assign probabilities to different sets of propositions, which might overlap a lot, a little, or not at all. There are far fewer proposals for how to proceed in these cases, and those there are have undesirable features. I begin by considering four proposals and arguing that they don't work. Then I'll describe my own proposal, which is intended to cover the situation in which we want to pool the individual opinions in order to ascribe an opinion to the group considered as an agent in its own right.