Publications

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Books

• Epistemic Risk and the Demands of Rationality (2022). Oxford University Press. Webpage Abstract & Reviews
  How much does rationality constrain what we should believe on the basis of our evidence? According to this book, not very much. For most people and most bodies of evidence, there is a wide range of beliefs that rationality permits them to have in response to that evidence. The argument, which takes inspiration from William James’ ideas in ‘The Will to Believe’, proceeds from two premises. The first is a theory about the basis of epistemic rationality. It’s called epistemic utility theory, and it says that what it is epistemically rational for you to believe is what it would be rational for you to choose if you were given the chance to pick your beliefs and, when picking them, you were to care only about their epistemic value. So, to say which beliefs are permitted, we must say how to measure epistemic value, and which decision rule to use when picking your beliefs. The second premise is a claim about attitudes to epistemic risk, and it says that rationality permits many different such attitudes. These attitudes can show up in epistemic utility theory in two ways: in the way you measure epistemic value; and in the decision rule you use to pick beliefs. This book explores the latter. The result is permissivism about epistemic rationality: different attitudes to epistemic risk lead to different choices of prior beliefs; given most bodies of evidence, different priors lead to different posteriors; and even once we fix your attitudes to epistemic risk, if they are at all risk-inclined, there is a range of different priors and therefore different posteriors they permit.

  • Mind, Lara Buchak. Journal
  • Notre Dame Philosophical Reviews, Boris Babic. Webpage
• Choosing for Changing Selves (2020). Oxford University Press. Webpage Abstract & Reviews
  What you value and the extent to which you value it changes over the course of your life. A person might currently greatly value pursuing philosophy, and value spending time in nature much less; but, having watched their parents as they have grown older, and noting that they are very much like their parents, that person might have good reason to think that they will value the pursuit of philosophy much less when they are sixty, and value spending time in nature much more. Given that we make our decisions on the basis of what we believe about the world and what we value in the world, the fact that the latter may change throughout our lives poses a problem for decision-making — in particular, for making decisions whose consequences will start to be felt or continue to be felt later in our lives. To which values should I appeal when making such a decision? My current values? My future values at the time when the decision will have its most significant effect? My past values? Some amalgamation of them all — past, present, and future — perhaps with some of them given more weight than others? (If so, how are the weightings assigned?) Or such an amalgamation only of a few of them? (If so, which ones?) In this book, I aim to provide a comprehensive account of rational decision-making for agents who recognise that their values will change over time and whose decisions will affect those future times. Included in the analysis will be not only agents who recognise that their values will inevitably change in certain ways, but also those who recognise that some of their decisions will lead to consequences that will change their values — thus, in effect, they will choose to change their values.

  • Economics and Philosophy, Arif Ahmed. Journal
  • Mind, Krister Bykvist. Journal
  • Philosophical Review, L. A. Paul. Journal
  • Philosophy of Science, Susanne Burri. Journal
  • Analysis, Nilanjan Das. Journal
  • Philosophical Quarterly, David Strohmaier. Journal
• Who Are Universities For? (with Tom Sperlinger and Josie McLellan) (2018). Bristol University Press. Webpage Abstract
  The university system is no longer fit for purpose. UK higher education was designed for much smaller numbers of students and a very different labour market. Students display worrying levels of mental health issues, exacerbated by unprecedented levels of debt, and the dubious privilege of competing for poorly-paid graduate internships. Meanwhile who goes to university is still too often determined by place of birth, gender, class or ethnicity. Who are universities for? argues for a large-scale shake up of how we organise higher education, how we combine it with work, and how it fits into our lives. It includes radical proposals for reform of the curriculum and how we admit students to higher education, with part-time study (currently in crisis in England) becoming the norm. A short, polemical but also deeply practical book, Who are universities for? offers concrete solutions to the problems facing UK higher education and a way forward for universities to become more inclusive and more responsive to local and global challenges.
• Accuracy and the Laws of Credence (2016). Oxford University Press. Webpage Abstract & Reviews
  In this book, we offer an extended investigation into a particular way of justifying the rational principles that govern our credences (or degrees of belief). The main principles that we justify are the central tenets of Bayesian epistemology, though we meet many other related principles along the way. These are: Probabilism, the claims that credences should obey the laws of probability; the Principal Principle, which says how credences in hypotheses about the objective chances should relate to credences in other propositions; the Principle of Indifference, which says that, in the absence of evidence, we should distribute our credences equally over all possibilities we entertain; and Conditionalization, the Bayesian account of how we should plan to respond when we receive new evidence. Ultimately, then, the book is a study in the foundations of Bayesianism. To justify these principles, we look to decision theory. We treat an agent’s credences as if they were a choice she makes between different options. We give an account of the purely epistemic utility enjoyed by different sets of credences. And we appeal 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 we give is the veritist’s: the sole fundamental source of epistemic utility for credences is their accuracy. Thus, this is an investigation in the version of epistemic utility theory known as accuracy-first epistemology. The book can also be read as an extended reply on behalf of the veritist to the evidentialist’s objection that veritism cannot account for certain evidential principles of credal rationality, such as the Principal Principle, the Principle of Indifference, and Conditionalization.

  • Book symposium in Philosophy and Phenomenological Research. Journal
  • Book symposium in Episteme. Journal
  • Notre Dame Philosophical Reviews, Kenny Easwaran. Journal
  • Philosophy of Science, Chad Marxen and Gerard Rothfus. Journal
  • Metascience, Erik Olsson. Journal
  • Journal of Philosophy, R. A. Briggs. Journal
  • Mind, Scott Sturgeon. Journal

  Shorter books

  • Opinion Pooling (with Lee Elkin) (2025). Cambridge University Press. Webpage Abstract
    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. In this Element, we provide an opinionated survey on some ways of pooling opinions: linear pooling, multiplicative pooling (including geometric), and pooling through imprecise probabilities. While we give significant attention to the axiomatic approach in evaluating pooling strategies, we also evaluate them in terms of the epistemic and practical goals they might meet. In doing so, we connect 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.
  • Dutch Book Arguments (2020). Cambridge University Press. Webpage Abstract
    Our beliefs come in degrees. I'm 70% confident it will rain tomorrow, and 0.001% sure my lottery ticket will win. What's more, we think these degrees of belief should abide by certain principles if they are to be rational. For instance, you shouldn't believe that a person's taller than 6ft more strongly than you believe that they're taller than 5ft, since the former entails the latter. In Dutch Book arguments, we try to establish the principles of rationality for degrees of belief by appealing to their role in guiding decisions. In particular, we show that degrees of belief that don't satisfy the principles will always guide action in some way that is bad or undesirable. In this Element, we present Dutch Book arguments for the principles of Probabilism, Conditionalization, and the Reflection Principle, among others, and we formulate and consider the most serious objections to them.

  Edited volumes

  • The Open Handbook of Formal Epistemology (with Jonathan Weisberg) (2019). PhilPapers Foundation. Webpage Abstract
    In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluctuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we explore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. How should those credences hang together? How should the credences change in response to evidence? And how should those credences guide the person’s actions? This is the approach of the first six chapters of this handbook. In the second half, we consider different representations—the set of propositions a person believes; their ranking of propositions by their plausibility. And in each case we ask again what the norms are that govern a person so represented. Or, we might represent them as having both credences and full beliefs, and then ask how those two representations should interact with one another. This handbook is incomplete, as such ventures often are. Formal epistemology is a much wider topic than we present here. One omission, for instance, is social epistemology, where we consider not only individual believers but also the epistemic aspects of their place in a social world. Michael Caie’s entry on doxastic logic touches on one part of this topic, but there is much more. Relatedly, there is no entry on epistemic logic, nor any on knowledge more generally. There are still more gaps. These omissions should not be taken as ideological choices. This material is missing, not because it is any less valuable or interesting, but because we v failed to secure it in time. Rather than delay publication further, we chose to go ahead with what is already a substantial collection. We anticipate a further volume in the future that will cover more ground. Why an open access handbook on this topic? A number of reasons. The topics covered here are large and complex and need the space allowed by the sort of 50 page treatment that many of the authors give. We also wanted to show that, using free and open software, one can overcome a major hurdle facing open access publishing, even on topics with complex typesetting needs. With the right software, one can produce attractive, clear publications at reasonably low cost. Indeed this handbook was created on a budget of exactly £0 (≈ $0). Our thanks to PhilPapers for serving as publisher, and to the authors: we are enormously grateful for the effort they put into their entries.
  • Continuum Companion to Philosophical Logic (with Leon Horsten) (2011). Bloomsbury. Webpage Abstract
    Logical methods are used in all area of philosophy. By introducing and advancing central to topics in the discipline, The Bloomsbury Companion to Philosophical Logic emphasizes the crucial role logic plays in understanding philosophical problems. Covering stages in the history of logic and of modern logic, this comprehensive Companion looks ahead to new areas of research and explores issues pertaining to classical logic and its rivals, semantics for parts of natural language, and the application of logic in the theory of rationality. Experts in the field provide a mix of technical chapters that offer excellent encyclopaedias of results in the area and chapters of philosophical discussions that survey a range of philosophical positions. To facilitate further study, this volumes also includes a series of research tools such as a detailed index, an up-to-date list of resources and an annotated bibliography. Balancing technical exposition with philosophical discussion, The Bloomsbury Companion to Philosophical Logic not only provides students and lecturers with the basis of a course in philosophical logic, it offers anyone working in this key area of contemporary philosophy a valuable research resource.

  Papers

  In Preparation

  • The epistemology of accurate credences. PDF Abstract
    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).
  • Knowledge is not always more valuable than true belief. PDF Abstract
    Knowledge is not always more valuable than mere true belief. If it were, we’d want to acquire it for those we care about when doing so would require negligible effort; we’d hope those we care about have it when we’re uncertain whether they do, and we’d feel relief were we to learn they do. But we don’t always do or feel these things nor think we should. I argue that the veritist is best able to make sense of these facts.
  • What is purely epistemic normativity, and why? PDF Abstract
    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 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 epistemic version of rule consequentialism. In a coda, I offer an alternative account of the function of knowledge ascriptions that improves on Edward Craig’s.
  • Contractualism, Prioritarianism, and Universalizability. PDF Abstract
    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.
  • Consent and the formation of preferences. PDF Abstract
    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.
  • The Value of Information and the Epistemology of Inquiry. PDF Abstract
    In the analytic tradition, 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.
  • Seeing Like a Dean. PDF Abstract
    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.
  • Believing is said of groups in many ways (and so it should be said of them in none). PDF Abstract
    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.

  Forthcoming

  • On Choosing How to Choose (with Catrin Campbell-Moore and Jason Konek). Theory and Decision. PDF Abstract
    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.
  • Formal Methods. Cambridge Handbook of Analytic Philosophy (ed.) Marcus Rossberg. PDF Abstract
    In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such as Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology.
  • Geometric Pooling: A User’s Guide (with Jonathan Weisberg). The British Journal for the Philosophy of Science. PDF Abstract
    Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.
  • Consequences of Calibration (with J. R. G. Williams). The British Journal for the Philosophy of Science. PDF Abstract
    In this paper, we offer a new set of axioms that characterise the epistemic utility functions that are most often used in arguments in favour of Bayesian norms such as Probabilism and Conditionalization. These are the additive and continuous strictly proper epistemic utility functions. Our characterization is based on a suggestion by Frank P. Ramsey and appeals to the virtue of calibration. We begin by describing Ramsey’s proposal and making our characterization precise; then we answer objections inspired by other treatments of calibration in epistemic utility theory.

  2025

  • Jeffrey Pooling (with Jonathan Weisberg). Philosophers' Imprint. PDF Abstract
    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.
  • Three questions for liberals. Social Research. PDF Abstract
    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.
  • What is the characteristic wrong of testimonial injustice? Philosophical Quarterly. PDF Abstract
    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.

  2024

  • Epistemic utility Arguments for Epistemic Norms. Stanford Encyclopedia of Philosophy (ed.) E. Zalta. Webpage
  • Should longtermists recommend hastening extinction rather than delaying it? The Monist. PDF Abstract
    Longtermism is the view that the most urgent global priorities, and those to which we should devote the largest portion of our current resources, are those that focus on ensuring a long future for humanity, and perhaps sentient or intelligent life more generally, and improving the quality of those lives in that long future. The central argument for this conclusion is that, given a fixed amount of a resource that we are able to devote to global priorities, the longtermist's favoured interventions have greater expected goodness 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 granting the longtermist's axiology and their consequentialist ethics, we are not morally required to choose whatever option maximises expected utility, and may not be permitted to do so. Instead, if their axiology and 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. And such decision theories do not recommend longtermist interventions. Indeed, sometimes, they recommend hastening human extinction. Many, though not all, will take this as a reductio of the longtermist's axiology or consequentialist ethics. I remain agnostic on the conclusion we should draw.
  • Updating on the Evidence of Others (with Jonathan Weisberg). Philosophical Studies. PDF Abstract
    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.

  2023

  • Credences. Routledge Encyclopedia of Philosophy.
  • How should your beliefs change when your awareness grows? Episteme. PDF Abstract
    Epistemologists who study partial beliefs, or credences, have a well-developed account of how you should change your credences 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 each of 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.
  • Nudging for Changing Selves. Synthese. PDF Abstract
    When is it legitimate for a government to 'nudge' its citizens, in the sense described by Richard Thaler and Cass Sunstein? In their original work on the topic, Thaler and Sunstein developed the 'as judged by themselves' (or AJBT) test to answer this question. In a recent paper, L. A. Paul and Sunstein raised a concern about this test: it often seems to give the wrong answer in cases in which we are nudged to make a decision that leads to what Paul calls a personally transformative experience, that is, one that results in our values changing. In those cases, the nudgee will judge the nudge to be legitimate after it has taken place, but they might not have judged it legitimate before.

  2022

  • Autonomy for Changing Selves. Routledge Handbook of Autonomy (ed.) Ben Colburn. PDF Abstract
    Our values change. What we value, want, desire, prefer, and how much; for nearly everyone, these will be different at different times in their life. These changes can be gradual or abrupt; they can be long-lasting or short-lived; and they can be induced by forces outside yourself or they can come from within or they can have no specific catalyst at all. Such preference change raises a number of questions for our theorising about rational choice, and these have been discussed at length. In §2 and §3, I’ll outline two of these questions along with some of the putative solutions that have been proposed. But preference change also raises questions for our theorising about autonomy, and these have hardly been considered at all. In §4, I’ll outline three problems for personal autonomy; and in §5, I’ll outline one problem for political autonomy. In §6, I conclude.
  • Competing reasons, incomplete preferences, and framing effects. Behavioral and Brain Sciences. Journal Abstract
    (Commentary on José Luis Bermúdez ‘Rational Framing Effects: A Multidisciplinary Case’) The quasi-cyclical preferences that Bermúdez ascribes to Agamemnon and others in analogous situations do not best represent them. I offer two alternative accounts. One works best if the preference ordering is taken to be the agent’s personal betterness ordering of acts; the other works best if it is taken to provide a summary of the agent’s dispositions to act.
  • Aggregating agents with opinions about different propositions. Synthese. PDF Abstract
    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.
  • Accuracy-first epistemology without Additivity. Philosophy of Science. PDF Abstract
    Accuracy arguments for the core tenets of Bayesian epistemology differ mainly in the conditions they place on the legitimate ways of measuring the inaccuracy of our credences. The best existing arguments rely on three conditions: Continuity, Additivity, and Strict Propriety. In this paper, I show how to strengthen the arguments based on these conditions by showing that the central mathematical theorem on which each depends goes through without assuming Additivity.

  2021

  • On the pragmatic and epistemic virtues of inference to the best explanation. Synthese. PDF Abstract
    In a series of papers over the past twenty years, and in a new book, Igor Douven (sometimes in collaboration with Sylvia Wenmackers) has argued that Bayesians are too quick to reject versions of inference to the best explanation that cannot be accommodated within their framework. In this paper, I survey Douven’s worries and attempt to answer them using a series of pragmatic and purely epistemic arguments that I take to show that Bayes’ Rule really is the only rational way to respond to your evidence.
  • Radical epistemology, structural explanations, and epistemic weaponry. Philosophical Studies. PDF Abstract
    When is a belief justified? There are three families of arguments we typically use to support different accounts of justification: (i) arguments from our intuitive responses to vignettes that involve the concept; (ii) arguments from the theoretical role we would like the concept to play in epistemology; and (iii) arguments from the practical, moral, and political uses to which we wish to put the concept. I focus particularly on the third sort (iii), and specifically on an argument of this sort offered by Clayton Littlejohn (2012) and Amia Srinivasan (2018) in favour of externalism. I counter Srinivasan’s argument in two ways: (a) first, I show that the internalist’s concept of justification might figure just as easily in the sorts of structural explanation Srinivasan thinks our political goals require us to give; and (b) I argue that the internalist’s concept is needed for a particular political task, namely, to help us build more effective defences against what I call epistemic weapons. I conclude that we should adopt an Alstonian pluralism about the concept of justification.
  • Bayesian updating when what you learn might be false. Erkenntnis. PDF Abstract
    Michael Rescorla (2020) has recently pointed out that the standard arguments for Bayesian Conditionalization assume that, whenever I become certain of something, it is true. Most people would reject this assumption. In response, Rescorla offers an improved Dutch Book argument for Bayesian Conditionalization that does not make this assumption. My purpose in this paper is two-fold. First, I want to illuminate Rescorla’s new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those covered by Conditionalization, and then showing how Rescorla’s version follows as a special case of that. Second, I want to show how to generalise R. A. Briggs and Richard Pettigrew’s Accuracy Dominance argument to avoid the assumption that Rescorla has identified (Briggs & Pettigrew, 2018). In both cases, these arguments proceed by first establishing a very general reflection principle.

  2020

  • Logical ignorance and logical learning. Synthese. PDF Abstract
    According to certain normative theories in epistemology, rationality requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omniscience requirement in Bayesian epistemology. Building on a key insight by Ian Hacking (1967), I develop a version of Bayesianism that permits logical ignorance. This includes an account of the synchronic norms that govern a logically ignorant individual at any given time, as well as an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations.
  • A Note on Deterministic Updating and van Fraassen’s symmetry argument for Conditionalization. Philosophical Studies. PDF Abstract
    In a recent paper, I argue that the pragmatic and epistemic arguments for Bayesian updating are based on an unwarranted assumption, which I call Deterministic Updating, and which says that your updating plan should be deterministic. In that paper, I did not consider whether the symmetry arguments due to Hughes and van Fraassen make the same assumption (Hughes & van Fraassen 1984; van Fraassen 1987). In this note, I show that they do.
  • Transformative experience and the knowledge norms for action: Moss on Paul’s challenge to decision theory. Becoming Someone New: Essays on Experience, Choice, and Change (eds.) Enoch Lambert and John Schwenkler. PDF Abstract
    L. A. Paul (2014, 2015) argues that the possibility of epistemically transformative experiences poses serious and novel problems for the orthodox theory of rational choice, namely, expected utility theory — I call her argument the Utility Ignorance Objection. In a pair of earlier papers, I responded to Paul’s challenge (Pettigrew 2015, 2016), and a number of other philosophers have responded in similar ways (Dougherty, et al. 2015, Harman 2015) — I call our argument the Fine-Graining Response. Paul has her own reply to this response, which we might call the Authenticity Reply. But Sarah Moss has recently offered an alternative reply to the Fine-Graining Response on Paul’s behalf (Moss 2017) — we’ll call it the No Knowledge Reply. This appeals to the knowledge norm of action, together with Moss’ novel and intriguing account of probabilistic knowledge. In this paper, I consider Moss’ reply and argue that it fails. I argue first that it fails as a reply made on Paul’s behalf, since it forces us to abandon many of the features of Paul’s challenge that make it distinctive and with which Paul herself is particularly concerned. Then I argue that it fails as a reply independent of its fidelity to Paul’s intentions.

  2019

  • Internalism, externalism, and the KK principle (with Alexander Bird). Erkenntnis. PDF Abstract
    This paper examines the relationship between the KK principle and the epistemological theses of externalism and internalism. There is often thought to be a very close relationship between externalism and the rejection of the KK principle and between internalism and its acceptance. How strong are the connections? The stronger proposals are: externalism entails the denial of the KK principle; internalism entails the truth of the KK principle. We will consider a number of problems for the theses as stated; we will present two ways of amending them so that they avoid these problems.
  • What is conditionalization, and why should we do it? Philosophical Studies. PDF Abstract
    Conditionalization is one of the central norms of Bayesian epistemology. But there are a number of competing formulations, and a number of arguments that purport to establish it. In this paper, I explore which formulations of the norm are supported by which arguments. In their standard formulations, each of the arguments I consider here depends on the same assumption, which I call Deterministic Updating. I will investigate whether it is possible to amend these arguments so that they no longer depend on it. As I show, whether this is possible depends on the formulation of the norm under consideration.
  • On the Expected Utility Objection to the Dutch Book Argument for Probabilism. Noûs. PDF Abstract
    The Dutch Book Argument for Probabilism assumes Ramsey’s Thesis (RT), which purports to determine the prices an agent is rationally required to pay for a bet. Recently, a new objection to Ramsey’s Thesis has emerged (Hedden 2013, Wronski & Godziszewski 2017, Wronski 2018)–I call this the Expected Utility Objection. According to this objection, it is Maximise Subjective Expected Utility (MSEU) that determines the prices an agent is required to pay for a bet, and this often disagrees with Ramsey’s Thesis. I suggest two responses to Hedden’s objection. First, we might be permissive: agents are permitted to pay any price that is required or permitted by RT, and they are permitted to pay any price that is required or permitted by MSEU. This allows us to give a revised version of the Dutch Book Argument for Probabilism, which I call the Permissive Dutch Book Argument. Second, I suggest that even the proponent of the Expected Utility Objection should admit that RT gives the correct answer in certain very limited cases, and I show that, together with MSEU, this very restricted version of RT gives a new pragmatic argument for Probabilism, which I call the Bookless Pragmatic Argument.
  • Veritism, Epistemic Risk, and the Swamping Problem. Australasian Journal of Philosophy. PDF Abstract
    Veritism says that the fundamental source of epistemic value for a doxastic state is the extent to which it represents the world correctly—that is, its fundamental epistemic value is determined entirely by its truth or falsity. The Swamping Problem says that Veritism is incompatible with two pre-theoretic beliefs about epistemic value (Zagzebski 2003, Kvanvig 2003): (a) a true justified belief is more (epistemically) valuable than a true unjustified belief; (b) a false justified belief is more (epistemically) valuable than a false unjustified belief. In this paper, I consider the Swamping Problem from the vantage point of decision theory. I note that the central premise in the argument is what Stefansson and Bradley (2015) call Chance Neutrality in Richard Jeffrey’s decision-theoretic framework. And I describe their argument that it should be rejected. Using this insight, I respond to the Swamping Problem on behalf of the veritist.
  • On the accuracy of group credences. Oxford Studies in Epistemology. PDF Abstract
    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 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 judgment 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 paper, I give an argument for Linear Pooling based on considerations of accuracy. And I respond to two standard objections to the aggregation principle.

  2018

  • What is justified credence? Episteme. PDF Abstract
    In this paper, we seek a reliabilist account of justified credence. Reliabilism about justified beliefs comes in two varieties: process reliabilism (Goldman, 1979, 2008) and indicator reliabilism (Alston, 1988, 2005). Existing accounts of reliabilism about justified credence comes in the same two varieties: Jeff Dunn’s is a version of process reliabilism (Dunn, 2015) while Weng Hong Tang offers a version of indicator reliabilism (Tang, 2016). As we will see, both face the same objection. If they are right about what justification is, it is mysterious why we care about justification, for neither of the accounts explains how justification is connected to anything of epistemic value. We will call this the Connection Problem. I begin by describing Dunn’s process reliabilism and Tang’s indicator reliabilism. I argue that, understood correctly, they are, in fact, extensionally equivalent. That is, Dunn and Tang reach the top of the same mountain, albeit by different routes. However, I argue that both face the Connection Problem. In response, I offer my own version of reliabilism, which is both process and indicator, and I argue that it solves that problem. Furthermore, I show that it is also extensionally equivalent to Dunn’s reliabilism and Tang’s. Thus, I reach the top of the same mountain as well.
  • What we talk about when we talk about numbers. Annals of Pure and Applied Logic. PDF Abstract
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
  • An accuracy-dominance argument for conditionalization (with R. A. Briggs). Noûs. PDF Abstract
    Epistemic decision theorists aim to justify Bayesian norms by arguing that these norms further the goal of epistemic accuracy—having beliefs that are as close as possible to the truth. The standard defense of probabilism appeals to accuracy-dominance: for every belief state that violates the probability calculus, there is some probabilistic belief state that is more accurate, come what may. The standard defense of conditionalization, on the other hand, appeals to expected accuracy: before the evidence is in, one should expect to do better by conditionalizing than by following any other rule. We present a new argument for conditionalization that appeals to accuracy-dominance, rather than expected accuracy. Our argument suggests that conditionalization is a rule of diachronic coherence: failing to conditionalize is not just a bad response to the evidence; it is also inconsistent.
  • Making things right: the true consequences of decision theory in epistemology. Epistemic Consequentialism (eds.) Jeffrey Dunn and Kristoffer Ahlstrom-Vij. PDF Abstract
    In his 1998 paper, ‘A Nonpragmatic Vindication of Probabilism’, Jim Joyce offered a novel argument for the credal principle of Probabilism. In this paper, I consider an objection to Joyce’s argument that has been raised by Hilary Greaves (‘Epistemic Decision Theory’, Mind, 2013); and I try to answer that objection.
  • Book symposium on Accuracy and the Laws of Credence. Philosophy and Phenomenological Research. Journal

  2017

  • The Principal Principle does not imply the Principle of Indifference. The British Journal for the Philosophy of Science. PDF Abstract
    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.
  • Aggregating incoherent agents who disagree. Synthese. PDF Abstract
    In this paper, we explore how we should aggregate the degrees of belief of a group of agents to give a single coherent set of degrees of belief, when at least some of those agents might be probabilistically incoherent. There are a number of way of aggregating degrees of belief, and there are a number of ways of fixing incoherent degrees of belief. When we have picked one of each, should we aggregate first and then fix, or fix first and then aggregate? Or should we try to do both at once? And when do these different procedures agree with one another? In this paper, we focus particularly on the final question.
  • Epistemic Utility and the Normativity of Logic. Logos and Episteme. PDF Abstract
    How does logic relate to rational belief? Is logic normative for belief, as some say? What, if anything, do facts about logical consequence tell us about norms of doxastic rationality? In this paper, we consider a range of putative logic-rationality bridge principles. These purport to relate facts about logical consequence to norms that govern the rationality of our beliefs and credences. To investigate these principles, we deploy a novel approach, namely, epistemic utility theory. That is, we assume that doxastic attitudes have different epistemic value depending on how accurately they represent the world. We then use the principles of decision theory to determine which of the putative logic-rationality bridge principles we can derive from considerations of epistemic utility.
  • Book symposium on Accuracy and the Laws of Credence. Episteme. Journal

  2016

  • Illness as transformative experience (with Havi Carel and Ian James Kidd). The Lancet. PDF Abstract
    Imagine that you need to decide whether to adopt a child or not. It’s the only avenue to parenthood that is open to you. If you adopt a child, you will become a parent. You will experience the (currently unknown) highs and lows of being a parent. If you decide not to adopt, you will never know what being a parent is like. The decision you are asked to make is doubly risky. This problem has been discussed recently by philosopher L A Paul in her book Transformative Experience. Paul suggests that experiences such as becoming a parent are doubly transformative. First, they are epistemically transformative: you can only learn what it is like to be a parent by becoming one. Second, experiences such as becoming a parent are existentially transformative: you don’t know how such an experience will change you and your preferences. We suggest that serious illness is a transformative experience and that Paul’s framework usefully characterises central aspects of it.
  • The population ethics of belief: in search of an epistemic Theory X. Noûs. PDF Abstract
    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 in the following sense. 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.
  • Jamesian epistemology formalised: an explication of ‘The Will to Believe’. Episteme. PDF Abstract
    Famously, William James held that there are two commandments that govern our epistemic life: Believe truth! Shun error! In this paper, I give a formal account of James’ claim using the tools of epistemic utility theory. I begin by giving the account for categorical doxastic states – that is, full belief, full disbelief, and suspension of judgment. Then I will show how the account plays out for graded doxastic states – that is, credences. The latter part of the paper thus answers a question left open in (Pettigrew 2014).
  • Remaking the elite university: An experiment in widening participation in the UK (with Josie McLellan and Tom Sperlinger). Power and Education. PDF Abstract
    This article analyses and critiques the discourse around widening participation in elite universities in the UK. One response, from both university administrators and academics, has been to see this as an ‘intractable’ problem which can at best be ameliorated through outreach or marginal work in admissions policy. Another has been to reject the institution of the university completely, and seek to set up alternative models of autonomous higher education. The article presents a different analysis, in which the university is still seen as central and participation is seen as an aspect of pedagogy rather than as an administrative process. This is illustrated through a description of how a Foundation Year in Arts and Humanities was conceived, designed and implemented at the University of Bristol. This model is used to consider the problems, risks and successes in challenging received notions of how (and whether) widening participation can be achieved, and whether it can reach those who are currently most excluded from elite universities, such as those without qualifications. The article suggests how academics can utilise their expertise to solve key challenges faced by universities and reclaim autonomy in central aspects of university administration. At the same time, it demonstrates how change to the current model of student recruitment can also bring welcome – and transformative – change to the nature of elite higher education institutions in the UK and elsewhere.
  • Accuracy, Risk, and the Principle of Indifference. Philosophy and Phenomenological Research. PDF Abstract
    In Bayesian epistemology, the problem of the priors is this: How should we set our credences (or degrees of belief) in the absence of evidence? That is, how should we set our prior or initial credences, the credences with which we begin our credal life? The Principle of Indifference gives a very restrictive answer. It demands that such an agent divide her credences equally over all possibilities. That is, according to the Principle of Indifference, only one initial credence function is permissible, namely, the uniform distribution. In this paper, we offer a novel argument for the Principle of Indifference. I call it the Argument from Accuracy.

  2015

  • Risk, rationality, and expected utility theory. Canadian Journal of Philosophy. PDF Abstract
    There are decision problems where the preferences that seem rational to many people cannot be accommodated within orthodox decision theory in the natural way. In response, a number of alternatives to the orthodoxy have been proposed. In this paper, I offer an argument against those alternatives and in favour of the orthodoxy. I focus on preferences that seem to encode sensitivity to risk. And I focus on the alternative to the orthodoxy proposed by Lara Buchak’s risk-weighted expected utility theory. I will show that the orthodoxy can be made to accommodate all of the preferences that Buchak’s theory can accommodate.
  • Transformative experience and decision theory. Philosophy and Phenomenological Research. PDF Abstract
    I have never eaten Vegemite—should I try it? I currently have no children—should I apply to adopt a child? In each case, one might imagine, whichever choice I make, I can make it rationally by appealing to the principles of decision theory. Not always, says L. A. Paul. In Transformative Experience, Paul issues two challenges to decision theory based upon examples such as these. I will show how we might reformulate decision theory in the face of these challenges. Then I will consider the philosophical questions that remain after the challenges have been accommodated.
  • Pluralism about belief states. Proceedings of the Aristotelian Society (Supp. Vol.). PDF Abstract
    With his Humean thesis on belief, Leitgeb (2015) seeks to say how beliefs and credences ought to interact with one another. To argue for this thesis, he enumerates the roles beliefs must play and the properties they must have if they are to play them, together with norms that beliefs and credences intuitively must satisfy. He then argues that beliefs can play these roles and satisfy these norms if, and only if, they are related to credences in the way set out in the Humean thesis. I begin by raising questions about the roles that Leitgeb takes beliefs to play and the properties he thinks they must have if they are to play them successfully. After that, I question the assumption that, if there are categorical doxastic states at all, then there is just one kind of them—to wit, beliefs—such that the states of that kind must play all of these roles and conform to all of these norms. Instead, I will suggest, if there are categorical doxastic states, there may be many different kinds of such state such that, for each kind, the states of that type play some of the roles Leitgeb takes belief to play and each of which satisfies some of the norms he lists. As I will argue, the usual reasons for positing categorical doxastic states alongside credences all tell equally in favour of accepting a plurality of kinds of them. This is the thesis I dub pluralism about belief states.
  • Accuracy and the belief-credence connection. Philosophers’ Imprint. PDF Abstract
    Probabilism is the thesis that an agent is rational only if her credences are probabilistic. This paper will be concerned with what we might call the Accuracy Dominance Argument for Probabilism (Rosenkrantz, 1981; Joyce, 1998, 2009). In this paper, I wish to identify and explore a lacuna in this argument that arises for those who take there to be (at least) two sorts of doxastic states: beliefs and credences.
  • What chance-credence norms should not be. Noûs. PDF Abstract
    A chance-credence norm states how an agent’s credences in propositions concerning objective chances ought to relate to her credences in other propositions. The most famous such norm is the Principal Principle (PP), due to David Lewis. However, Lewis noticed that PP is too strong when combined with many accounts of chance that attempt to reduce chance facts to non-modal facts. Those who defend such accounts of chance have offered two alternative chance-credence norms: the first is Hall’s and Thau’s New Principle (NP); the second is Ismael’s General Recipe (IP). Thus, the question arises: Should we adopt NP or IP or both? In this paper, I argue that IP has unacceptable consequences when coupled with reductionism, so we must accept NP alone.

  2014

  • Deference done right (with Mike Titelbaum). Philosophers’ Imprint. PDF Abstract
    There are many kinds of epistemic experts to which we might wish to defer in setting our credences. These include: highly rational agents, objective chances, our own future credences, our own current credences, and evidential (or logical) probabilities. But how, precisely, ought we defer to these experts? Exactly what constraint does a deference requirement place on an agent’s credences at a particular time? In this paper we consider three possible answers, inspired by three different principles that have been proposed for deference to objective chances. We consider how these options fare when applied to the other kinds of epistemic experts mentioned above. Besides assuming a baseline probabilism about rational credences, we are particularly interested in the following two desiderata: (1) A deference principle should be consistent with both the agent’s and the experts’ updating by Conditionalization. (2) A deference principle should permit agents to have various kinds of doubts about what’s rationally required. Of the three deference principles we consider, we argue that two of the options face insuperable difficulties meeting these desiderata. The third, on the other hand, fares well — at least when it is applied in a particular way.
  • Two types of abstraction for structuralism (with Øystein Linnebo). Philosophical Quarterly. PDF Abstract
    If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other by Dedekind. We argue that both face problems.

  2013

  • Accuracy and Evidence. Dialectica. PDF Abstract
    In 'A Nonpragmatic Vindication of Probabilism', Jim Joyce argues that our credences should obey the axioms of the probability calculus by showing that, if they don't, there will be alternative credences that are guaranteed to be more accurate than ours. But it seems that accuracy is not the only goal of credences: there is also the goal of matching one's credences to one's evidence. I will consider four ways in which we might make this latter goal precise... All but the fourth way of making the evidential goal precise are pluralist about credal virtue: there is the virtue of being accurate and there is the virtue of matching the evidence and neither reduces to the other. The fourth way is monist about credal virtue: there is just the virtue of being accurate. The pluralist positions lead to problems for Joyce's argument; the monist position avoids them. I endorse the latter.
  • A New Epistemic Utility Argument for the Principal Principle. Episteme. PDF Abstract
    Jim Joyce has presented an argument for Probabilism based on considerations of epistemic utility. In a recent paper, I adapted this argument to give an argument for Probablism and the Principal Principle based on similar considerations. Joyce's argument assumes that a credence in a true proposition is better the closer it is to maximal credence, whilst a credence in a false proposition is better the closer it is to minimal credence. By contrast, my argument in that paper assumed (roughly) that a credence in a proposition is better the closer it is to the objective chance of that proposition. In this paper, I present an epistemic utility argument for Probabilism and the Principal Principle that retains Joyce's assumption rather than the alternative.
  • Epistemic utility and norms for credence. Philosophy Compass. PDF Abstract
    Beliefs come in different strengths. An agent’s credence in a proposition is a measure of the strength of her belief in that proposition. Various norms for credences have been proposed. Traditionally, philosophers have tried to argue for these norms by showing that any agent who violates them will be lead by her credences to make bad decisions. In this article, we survey a new strategy for justifying these norms. The strategy begins by identifying an epistemic utility function and a decision-theoretic norm; we then show that the decision-theoretic norm applied to the epistemic utility function yields the norm for credences that we wish to justify. We survey results already obtained using this strategy, and we suggest directions for future research.

  2012

  • Accuracy, Chance, and the Principal Principle. Philosophical Review. PDF Abstract
    In “A Nonpragmatic Vindication of Probabilism,” James M. Joyce attempts to “depragmatize” de Finetti’s prevision argument for the claim that our credences ought to satisfy the axioms of the probability calculus. This article adapts Joyce’s argument to give nonpragmatic vindications of David Lewis’s original Principal Principle as well as recent reformulations due to Ned Hall and Jenann Ismael. Joyce enumerates properties that a function must have if it is to measure the distance from a set of credences to a set of truth values; he shows that, on any such measure, and for any set of credences that violates the probability axioms, there is a set that satisfies those axioms that is closer to every possible set of truth values. This article replaces truth values with objective chances in this argument; it shows that for any set of credences that violates the probability axioms or the Principal Principle, there is a set that satisfies both that is closer to every possible set of objective chances and similarly for Ned Hall’s New Principle and Jenann Ismael’s Generalized Principal Principle. Along the way, the article provides new arguments for some of Joyce’s central conditions on distance measures, and it answers two pressing objections to Joyce’s strategy.
  • Indispensability arguments and instrumental nominalism. Review of Symbolic Logic. PDF Abstract
    In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that mathematical objects exist on the grounds that we make indispensable reference to such objects in our best scientific theories (Quine, 1981a; Putnam, 1979a) and in our everyday reasoning (Ketland, 2005). I wish to defend a particular objection to such arguments called instrumental nominalism. Existing formulations of this objection are either insufficiently precise or themselves make reference to mathematical objects or possible worlds. I show how to formulate the position precisely without making any such reference. To do so, it is necessary to supplement the standard modal operators with two new operators that allow us to shift the locus of evaluation for a subformula. I motivate this move and give a semantics for the new operators.
  • Identity and Discernibility in Philosophy and Logic (with James Ladyman and Øystein Linnebo). Review of Symbolic Logic. PDF Abstract
    Questions about the relation between identity and discernibility are important both in philosophy and in model theory. We show how a philosophical question about identity and discernibility can be ‘factorized’ into a philosophical question about the adequacy of a formal language to the description of the world, and a mathematical question about discernibility in this language. We provide formal definitions of various notions of discernibility and offer a complete classification of their logical relations. Some new and surprising facts are proved; for instance, that weak discernibility corresponds to discernibility in a language with constants for every object, and that weak discernibility is the most discerning nontrivial discernibility relation.

  2011

  • An Improper Introduction to Epistemic Utility Theory. EPSA Philosophy of Science: Amsterdam 2009 (eds.) Henk de Regt, Stephan Hartmann, and Samir Okasha. PDF Abstract
    A survey of accuracy-based arguments for Probabilism and Conditionalization.
  • Probability. Continuum Companion to Philosophical Logic (eds.) Leon Horsten and Richard Pettigrew. PDF Abstract
    An introductory survey article on different interpretations of probability.
  • Category theory as an autonomous foundation (with Øystein Linnebo). Philosophia Mathematica. PDF Abstract
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. We argue that, while a strong case can be made for its logical and conceptual autonomy, its justificatory autonomy turns on whether or not mathematical theories can be justified by appeal to mathematical practice. If they can, a category-theoretical approach will be fully autonomous; if not, the most natural route to justificatory autonomy is blocked.

  2010

  • An Objective Justification of Bayesianism II: The Consequences of Minimizing Inaccuracy (with Hannes Leitgeb). Philosophy of Science (Chosen for the Philosophers’ Annual 2010). PDF Abstract
    In this article and its prequel, we derive Bayesianism from the following norm: Accuracy—an agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we make the norm mathematically precise; in this article, we derive its consequences. We show that the two core tenets of Bayesianism follow from Accuracy, while the characteristic claim of Objective Bayesianism follows from Accuracy together with an extra assumption. Finally, we show that Jeffrey Conditionalization violates Accuracy unless Rigidity is assumed, and we describe the alternative updating rule that Accuracy mandates in the absence of Rigidity.
  • An Objective Justification of Bayesianism I: Measuring Inaccuracy (with Hannes Leitgeb). Philosophy of Science. PDF Abstract
    In this article and its sequel, we derive Bayesianism from the following norm: Accuracy—an agent ought to minimize the inaccuracy of her partial beliefs. In this article, we make this norm mathematically precise. We describe epistemic dilemmas an agent might face if she attempts to follow Accuracy and show that the only measures of inaccuracy that do not create these dilemmas are the quadratic inaccuracy measures. In the sequel, we derive Bayesianism from Accuracy and show that Jeffrey Conditionalization violates Accuracy unless Rigidity is assumed. We describe the alternative updating rule that Accuracy mandates in the absence of Rigidity.
  • The foundations of arithmetic in finite bounded Zermelo set theory. Cahiers du Centre de Logique (eds.) Hinnion, R. and T. Libert One Hundred Years of Axiomatic Set Theory). PDF Abstract
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory. And in section 3, I sketch foundations for arithmetic in that theory and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of it. An equivalent theory of sets and an equivalent foundation for arithmetic was introduced by Mayberry and developed by the current author in his doctoral thesis. In that thesis and in the joint paper with Mayberry to which it gave rise, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry from the early 1980s.

  2009

  • On interpretations of bounded arithmetic and bounded set theory. Notre Dame Journal of Formal Logic. PDF Abstract
    In ‘On interpretations of arithmetic and set theory’, Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. Theorem: The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable. In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic, IΔ0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong’s interpretation of the arithmetic in the set theory. Instead, I am forced to produce a different interpretation.
  • Aristotle on the subject matter of geometry. Phronesis. PDF Abstract
    I offer a new interpretation of Aristotle’s philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle’s second attempt to solve Zeno’s Runner Paradox in Book VIII of the Physics, I explain how such objects exist in the sensibles in a special way. I conclude by considering the passages that lead Jonathan Lear to his fictionalist reading of Met. M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk.

  2008

  • Platonism and Aristotelianism in Mathematics. Philosophia Mathematica. PDF Abstract
    Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at ‘face value’ is one on which the expressions ‘N’, ‘0’, ‘1’, ‘+’, and ‘×’ are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.

  Reviews

  • Right Belief and True Belief, Daniel Singer. The British Journal for the Philosophy of Science. Webpage
  • Optimality Justifications, Gerhard Schurz. The British Journal for the Philosophy of Science. Webpage
  • The Good It Promises, the Harm It Does: Critical Essays on Effective Altruism, (eds.) Carol Adams, Alice Crary, and Lori Gruen. Mind. PDF
  • Normal Decisions, Edna Ullmann-Margalit (eds.) Avishai Margalit and Cass R. Sunstein. Notre Dame Philosophical Reviews. Webpage
  • Self-Control, Decision Theory and Rationality: New Essays, (ed.) José Luis Bermúdez. Notre Dame Philosophical Reviews. Webpage
  • Rigor and Structure, John P. Burgess. Philosophia Mathematica. PDF Abstract
    In this review, I focus on the possibility of giving a precise account of informal mathematical proof; and the lesson that Burgess draws from the indifference that mathematicians have towards questions about the subject matter of their discipline.
  • An Introduction to the Philosophy of Mathematics, Mark Colyvan. Bulletin of Symbolic Logic. PDF
  • Transformative Experience, L. A. Paul. Mind. PDF Abstract
    In this review, I focus mainly on Paul’s own solution to the problems that she raises for orthodox decision theory; and I consider the possibility of an alternative solution, which I originally proposed in ‘Transformative Experience and Decision Theory’ (Philosophy and Phenomenological Research, 2014).
  • Degrees of Belief, (eds.) Franz Huber and Christoph Schmidt-Petri. Grazer Philosophische Studien. PDF Abstract
    The book under review provides a stimulating, informative, and focussed collection of new articles that survey a topic in formal epistemology that is fast becoming one of the central topics in mainstream epistemology. The twelve articles, as well as Huber’s excellent Introduction, address the following two questions: How should we model or represent an agent’s epistemic state? What constraints does rationality impose on an agent’s epistemic states thus modelled? I treat each of these questions in turn, and conclude with a detailed consideration of an argument from Joyce’s article.

  Drafts that will probably remain drafts

  • A pragmatic characterisation of linear pooling. PDF Abstract
    How we should determine a group’s collective probabilistic judgments, given the probabilistic judgments of the individuals in the group? A standard answer is given by this condition: The group probability distribution over the propositions should be a weighted average of the individual probability distributions. Call this Linear Pooling. We provide a condition on aggregates that characterises linear pooling: Given a utility function shared by all members of the group, if each individual in the group expects one act to have greater utility than another, then the group expects the first act to have greater utility than the second. Call this Pareto. We prove that Linear Pooling and Pareto are equivalent.
  • Accuracy-domination arguments and credences as estimates of truth-values. PDF Abstract
    Branden Fitelson has recently raised an intriguing objection (Fitelson, 2012) to Jim Joyce’s accuracy-domination arguments for probabilism (Joyce 1998, 2009). He adapts an objection raised by David Miller against accounts of verisimilitude that make it a measure of the accuracy of a theory’s predictions (Miller, 1975). As Joyce presents his accuracy domination argument, it is based on a conception of credences as estimates of truth-values; and Fitelson’s objection is based on an alleged analogy between an agent’s estimate of a quantity (such as a truth-value) and a scientific theory’s prediction of the value of a quantity. I will offer two responses to Fitelson’s objection: I will show that, even if the alleged analogy does hold, it does not undermine Joyce’s argument; then I will argue that the analogy does not hold.
  • Self-locating beliefs and the goal of accuracy. PDF Abstract
    The goal of a partial belief is to be accurate, or close to the truth. By appealing to this norm, I seek norms for partial beliefs in self-locating and non-self-locating propositions. My aim is to find norms that are analogous to the Bayesian norms, which, I argue, only apply unproblematically to partial beliefs in non-self-locating propositions. I argue that the goal of a set of partial beliefs is to minimize the expected inaccuracy of those beliefs. However, in the self-locating framework, there are two equally legitimate definitions of expected inaccuracy. And, while each gives rise to the same synchronic norm for partial beliefs, they give rise to different, inconsistent diachronic norms. I conclude that both norms are rationally permissible. En passant, I note that this entails that both Halfer and Thirder solutions to the well-known Sleeping Beauty puzzle are rationally permissible.

  PhD dissertation

  • Natural, rational, and real arithmetic in a finitary theory of finite sets. PDF Abstract
    In the first part of this thesis, I describe a finitary theory of finite sets called Euclidean Arithmetic, which was introduced by Mayberry in his The Foundations of Mathematics in the Theory of Sets. In the second part, I develop the theory of simply infinite systems in this theory. Since Dedekind’s Isomorphism Theorem for Simply Infinite Systems does not hold in Euclidean Arithmetic, there is a vast and varied fauna of simply infinite systems with different and surprising properties. I describe many of these systems; I detail the properties they have and the relations that hold between them; and I give recipes by means of which new systems may be defined from old, and from which systems with desired properties may be defined. In the final part of this thesis, I describe a novel version of infinitesimal analysis in a intuitionistic extension of the basic theory of Euclidean Arithmetic. I prove analogues to many of the standard elementary theorems of real analysis and the calculus.