Anna Mahtani

In this paper I investigate an alternative to imprecise probabilism. Imprecise probabilism is a popular revision of orthodox Bayesianism: while the orthodox Bayesian claims that a rational agent’s belief-state can be represented by a single credence function, the imprecise probabilist claims instead that a rational agent’s belief-state can be represented by a set of such functions. The alternative that I put forward in this paper is to claim that the expression ‘credence’ is vague, and then apply the theory of supervaluationism to sentences containing this expression. This gives us a viable alternative to imprecise probabilism, and I end by comparing the two accounts. I show that supervaluationism has a simpler way of handling sentences relating the belief-states of two different people, or of the same person at two different times; that both accounts may have the resources to develop plausible decision theories; and finally that the supervaluationist can accommodate higher-order vagueness in a way that is not available to the imprecise probabilist.

There has been a recent surge of interest in normative internalism, according to which what you ought to do depends on your beliefs about morality. This view has potentially wide-reaching implications for practical action, policy choice, and societal norms. The view is motivated by certain compelling examples, but in this paper I argue that such examples may not support normative internalism after all, for the phenomenon of imaginative resistance – well-known to philosophers of fiction – is at work here, muddying our intuitions. Once these examples are set aside, we get a more realistic view of the limited interest and importance of an internalist sense of ‘ought’.

In ordinary conversation, we describe all sorts of different things as vague: you can have vague plans, vague ideas and vague aches and pains. In philosophy of language, in contrast, it is parts of language – words, expressions and so on – that are said to be vague. One classic example of a vague term is the word ‘heap’. A single grain clearly does not make a heap, and a million grains (when arranged in the right way) does make a heap, but where exactly does the boundary lie? How many grains do you need to make a heap? There seems to be no precise answer to this question, and because the term is imprecise in this way, we call it vague. Vague terms are extremely common in natural language. The term ‘bald’ is vague, because there is no precise number of hairs that mark the boundary between ‘bald’ and ‘not bald’; the term ‘hot’ is vague because there is no precise temperature that something must reach to count as hot – and so on. As we have seen, adjectives can be vague, but so can nouns, adverbs and perhaps all parts of language. To find terms which are precise rather than vague, we need to look to the languages of logic and mathematics. We can use vague terms to construct paradoxes known as sorites paradoxes. From an obviously true premise, such as that a collection of 1 million grains (in a certain arrangement) is a heap, together with the claim that ‘heap’ has no sharp boundary, we can derive the absurd conclusion that just 1 grain counts as a heap. Any theory of vagueness must offer some solution to this paradox. Some of the most popular theories of vagueness include supervaluationism, the degree theory of truth and the epistemic theory, and many of the available theories demand a radical rethink of classical accounts of logic and language.

Reflection Principle can be defended with a Diachronic Dutch Book Argument (DBA), but it is also defeated by numerous compelling counter-examples. It seems then that Diachronic DBAs can lead us astray. Should we reject them en masse—including Lewis’s Diachronic DBA for Conditionalization? Rachael Briggs’s “suppositional test” is supposed to differentiate between Diachronic DBAs that we can safely ignore (including the DBA for Reflection) and Diachronic DBAs that we should find compelling (including the DBA for Conditionalization). I argue that Brigg’s suppositional test is wrong: it sets the bar for coherence too high and places certain cases of self-doubt on the wrong side of the divide. Given that the suppositional test is unsatisfactory, we are left without any justification for discriminating between Diachronic DBAs and ought to reject them all—including the DBA for Conditionalization.

There has been a recent surge of interest in normative internalism, according to which what you ought to do depends on your beliefs about morality. This view has potentially wide-reaching implications for practical action, policy choice, and societal norms. The view is motivated by certain compelling examples, but in this paper I argue that such examples may not support normative internalism after all, for the phenomenon of imaginative resistance – well-known to philosophers of fiction – is at work here, muddying our intuitions. Once these examples are set aside, we get a more realistic view of the limited interest and importance of an internalist sense of ‘ought’.

We present a new argument for the claim that in the Sleeping Beauty problem, the probability that the coin comes up heads is 1/3. Our argument depends on a principle for the updating of probabilities that we call ‘generalized conditionalization’, and on a species of generalized conditionalization we call ‘synchronic conditionalization on old information’. We set forth a rationale for the legitimacy of generalized conditionalization, and we explain why our new argument for thirdism is immune to two attacks that Pust (Synthese 160:97–101, 2008) has leveled at other arguments for thirdism.

I examine a range of popular solutions to the puzzle of imaginative resistance. According to each solution in this range, imaginative resistance occurs only when we are asked to imagine something that conflicts with what we believe. I show that imaginative resistance can occur without this sort of conflict, and so that every solution in the range under consideration fails. I end by suggesting a new explanation for imaginative resistance—the Import Solution—which succeeds where the other solutions considered fail.

This paper is about the standard Reflection Principle (van Fraassen, 1984) and the Group Reflection Principle (Elga, 2007; Bovens & Rabinowicz, 2011; Titelbaum, 2012; Hedden, 2015). I argue that these principles are incomplete as they stand. The key point is that deference is an intensional relation, and so whether you are rationally required to defer to a person at a time can depend on how that person and that time are designated. In this paper I suggest a way of completing the Reflection Principle and Group Reflection Principle, and I argue that so completed these principles are plausible. In particular, they do not fall foul of the Sleeping Beauty case (Elga, 2000), the Cable Guy Paradox (Hajek, 2005), Arntzenius' prisoner cases (Arntzenius, 2003), or the Puzzle of the Hats (Bovens & Rabinowicz, 2011).

In this paper I investigate an alternative to imprecise probabilism. Imprecise probabilism is a popular revision of orthodox Bayesianism: while the orthodox Bayesian claims that a rational agent’s belief-state can be represented by a single credence function, the imprecise probabilist claims instead that a rational agent’s belief-state can be represented by a set of such functions. The alternative that I put forward in this paper is to claim that the expression ‘credence’ is vague, and then apply the theory of supervaluationism to sentences containing this expression. This gives us a viable alternative to imprecise probabilism, and I end by comparing the two accounts. I show that supervaluationism has a simpler way of handling sentences relating the belief-states of two different people, or of the same person at two different times; that both accounts may have the resources to develop plausible decision theories; and finally that the supervaluationist can accommodate higher-order vagueness in a way that is not available to the imprecise probabilist.

In ordinary conversation, we describe all sorts of different things as vague: you can have vague plans, vague ideas and vague aches and pains. In philosophy of language, in contrast, it is parts of language – words, expressions and so on – that are said to be vague. One classic example of a vague term is the word ‘heap’. A single grain clearly does not make a heap, and a million grains (when arranged in the right way) does make a heap, but where exactly does the boundary lie? How many grains do you need to make a heap? There seems to be no precise answer to this question, and because the term is imprecise in this way, we call it vague. Vague terms are extremely common in natural language. The term ‘bald’ is vague, because there is no precise number of hairs that mark the boundary between ‘bald’ and ‘not bald’; the term ‘hot’ is vague because there is no precise temperature that something must reach to count as hot – and so on. As we have seen, adjectives can be vague, but so can nouns, adverbs and perhaps all parts of language. To find terms which are precise rather than vague, we need to look to the languages of logic and mathematics. We can use vague terms to construct paradoxes known as sorites paradoxes. From an obviously true premise, such as that a collection of 1 million grains (in a certain arrangement) is a heap, together with the claim that ‘heap’ has no sharp boundary, we can derive the absurd conclusion that just 1 grain counts as a heap. Any theory of vagueness must offer some solution to this paradox. Some of the most popular theories of vagueness include supervaluationism, the degree theory of truth and the epistemic theory, and many of the available theories demand a radical rethink of classical accounts of logic and language.

Richard Bradley and others endorse Reverse Bayesianism as the way to model awareness growth. I raise a problem for Reverse Bayesianism—at least for the general version that Bradley endorses—and argue that there is no plausible way to restrict the principle that will give us the right results. To get the right results, we need to pay attention to the (dispositional) attitudes that agents have towards propositions of which they are unaware. This raises more general questions about how awareness growth should be modelled. Classical Bayesian decision theory leaves no room for awareness growth. But in his book Decision Theory with a Human Face (Bradley 2017), Richard Bradley aims to give a decision theory for real agents, and it seems that real agents can grow in awareness. For example, I might be alerted to a possible state of affairs that had not occurred to me before; or I might discover that there is a further option available to me that I had not previously considered. Bradley gives an account of what rationality requires of agents without full awareness, and how rational agents’ credal states can change when they grow in awareness. In this paper I raise a problem for this account, and argue that we need to model awareness growth quite differently, taking into account an agent’s dispositional attitudes towards propositions of which (s)he is unaware.

In this paper I offer an alternative - the ‘dispositional account’ - to the standard account of imprecise probabilism. Whereas for the imprecise probabilist, an agent’s credal state is modelled by a set of credence functions, on the dispositional account an agent’s credal state is modelled by a set of sets of credence functions. On the face of it, the dispositional account looks less elegant than the standard account – so why should we be interested? I argue that the dispositional account is actually simpler, because the dispositional choice behaviour that fixes an agent’s credal state is faithfully depicted in the model of that agent’s credal state. I explore some of the implications of the account, including a surprising implication for the debate over dilation.

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff (s)he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of whether we have a Dutch Book Argument for Conditionalization is left open.

This book brings to light a simple and deep insight with profound theoretical and policy implications for all who use the credence framework. There are concepts which need to be rethought, moves which turn out to be invalid, and principles which need to be rejected or transformed. The central aim is to give those who use the credence framework an awareness of the insight and its wide-reaching implications. The credence framework is used throughout the sciences and social sciences. On this framework, credences (degrees of belief, subjective probabilities, or Bayesian probabilities) are assigned to certain objects—but what objects? I argue that the objects of credence are ‘opaque’ or ‘hyperintensional’: to adapt an example from Gottlob Frege, a person’s credence that Hesperus is bright might be different from their credence that Phosphorus is bright, if that person does not know that Hesperus and Phosphorus are one and the same. We might put the point by saying that our credences are not about objects in themselves, but about objects under designators. This point has far-reaching implications for users of the credence framework. There are implications for principles of rationality, including deference principles and the Principal Principle, and practical implications for decision theory and welfare economics. Furthermore, there are implications for how the framework should be interpreted: this book explores both two-dimensionalism and impossible worlds, and assimilating either into the credence framework brings further significant repercussions.

The ex ante Pareto principle has an intuitive pull, and it has been a principle of central importance since Harsanyi’s defence of utilitarianism (to be found in e.g. Harsanyi, Rational behaviour and bargaining equilibrium in games and social situations. CUP, Cambridge, 1977). The principle has been used to criticize and refine a range of positions in welfare economics, including egalitarianism and prioritarianism. But this principle faces a serious problem. I have argued elsewhere (Mahtani, J Philos 114(6):303-323 2017) that the concept of ex ante Pareto superiority is not well defined, because its application in a choice situation concerning a fixed population can depend on how the members of that population are designated. I show in this paper that in almost all cases of policy choice, there will be numerous sets of rival designators for the same fixed population. I explore two ways that we might complete the definition of ex ante Pareto superiority. I call these the ‘supervaluationist’ reading and the ‘subvaluationist’ reading. I reject the subvaluationist reading as uncharitable, and argue that the supervaluationist reading is the most promising interpretation of the ex ante Pareto principle. I end by exploring some of the implications of this principle for prioritarianism and egalitarianism.

Dutch book and accuracy arguments are used to justify certain rationality constraints on credence functions. Underlying these Dutch book and accuracy arguments are associated theorems, and I show that the interpretation of these theorems can vary along a range of dimensions. Given that the theorems can be interpreted in a variety of different ways, what is the status of the associated arguments? I consider three possibilities: we could aggregate the results of the differently interpreted theorems in some way, and motivate rationality constraints based on this aggregation; we could be permissive, and accept the conclusions of the Dutch book and accuracy arguments under all interpretations of the associated theorems; or we could select one uniquely correct interpretation of the Dutch book or accuracy theorem, and use that to justify certain rationality constraints. I show that each possibility faces problems, and conclude that Dutch book and accuracy theorems cannot be used to justify any principle of rationality.

Sometimes a proposition is 'opaque' to an agent: (s)he doesn't know it, but (s)he does know something about how coming to know it should affect his or her credence function. It is tempting to assume that a rational agent's credence function coheres in a certain way with his or her knowledge of these opaque propositions, and I call this the 'Opaque Proposition Principle'. The principle is compelling but demonstrably false. I explain this incongruity by showing that the principle is ambiguous: the term 'know' as it appears in the principle can be interpreted in two different ways, as either basic-know or super-know. I use this distinction to construct a plausible version of the principle, and then to similarly construct plausible versions of the Reflection Principle and the Sure-Thing Principle.