### Internalism, externalism, and the KK principle (with Alexander Bird)

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.

### The Principal Principle does not imply the Principle of Indifference

*The British Journal for the Philosophy of Science*

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.

### Remaking the elite university: An experiment in widening participation in the UK (with Josie McLellan and Tom Sperlinger)

*Power and Education *8(1):54-72

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

### Risk, rationality, and expected utility theory

*Canadian Journal of Philosophy *45(5-6): 798-826

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.

### Pluralism about belief states

*Proceedings of the Aristotelian Society (Supp. Vol.)* 89(1):187-204 (Contribution to a symposium on Hannes Leitgeb’s Humean thesis on belief at the Joint Session of the Aristotelian Society and Mind Association 2015)

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.*

### What chance-credence norms should not be

*Noûs *49(1):177-196

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.

### Review of Mark Colyvan’s *An Introduction to the Philosophy of Mathematics*

*Bulletin of Symbolic Logic* 19(3): 396-397

### Identity and Discernibility in Philosophy and Logic (with James Ladyman and Øystein Linnebo)

*Review of Symbolic Logic* 5(1):162-186

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.

### The foundations of arithmetic in finite bounded Zermelo set theory

in Hinnion, R. and T. Libert (eds.) One Hundred Years of Axiomatic Set Theory, *Cahiers du Centre de Logique* 17: 99-118

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.

### On interpretations of bounded arithmetic and bounded set theory

*Notre Dame Journal of Formal Logic* 50(2): 141-152

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, *I*Δ0 + 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.