Richard Pettigrew

An Objective Justification of Bayesianism I: Measuring Inaccuracy

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.

An Objective Justification of Bayesianism II: The Consequences of Minimizing Inaccuracy*

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.

Accuracy, Chance, and the Principal Principle

How should my degrees of credence in propositions about objective chances interact with my degrees of credence in other propositions? David Lewis (1980) formulated a precise answer. He called it the “Principal Principle.” And he took it to specify the role that a feature of the world must play if it is to count as chance. However, he also argued that the chances delivered by his best-system analysis of laws and chances could not possibly play the roles required of them by the Principal Principle: the Principal Principle is inconsistent with the best-system analysis. Later, Lewis (1994) came to accept a slightly different precise answer to the question. This is Michael Thau’s and Ned Hall’s amendment to the Principal Principle, which is known as the New Principle (Hall 1994; Thau 1994; Joyce 2007). The best-system analysis is consistent with the New Principle. Later still, Jenann Ismael (2008) proposed an alternative amendment to the Principal Principle, which she called the “Generalized Principal Principle.” Ismael’s principle is also consistent with the bestsystem analysis. The large literature that has grown up around these proposals includes a number of different and illuminating arguments in their favor, as well as subtle improvements in their formulation. However, no attempt has been made to give them a formal justification. That is, there has been no attempt to deduce any chance-credence norm from more

Identity and Discernibility in Philosophy and Logic

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 non-trivial discernibility relation.

Remaking the elite university: An experiment in widening participation in the UK:

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.

Aristotle on the subject matter of geometry

I offer a new interpretation of Aristotles 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 Aristotles second attempt to solve Zenos 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.

An Improper Introduction to Epistemic Utility Theory

In epistemic utility theory, we apply the tools of decision theory to justify epistemic norms. We treat the possible epistemic states of an agent as if they were epistemic actions between which she must choose. And we ask how we should measure the purely epistemic utility of being in such a state. We then apply the general apparatus of decision theory to determine which epistemic states are rational in a given situation from a purely epistemic point of view; and how our epistemic states should evolve over time. In this paper, I survey recent attempts to justify the tenets of Bayesian epistemology by appealing to epistemic utility theory. And I raise objections to arguments based on the technical notion of propriety.

What we talk about when we talk about numbers

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.