In population ethics, Narveson’s dictum states: morality favours making people happy, but is neutral about making happy people. The thought is intuitively appealing; for example, it prohibits creating new people at the expense of those who already exist. However, there are well-known obstacles to accommodating Narveson’s dictum within a standard framework of overall betterness: any attempt to do so violates very plausible formal features of betterness. Therefore, the prevailing view is that the dictum is off-limits to consequentialists, who are thereby (...) committed to the unsavoury normative consequences of denying it. We argue against the prevailing view, by showing that Narveson’s dictum can be accommodated within “multidimensional” consequentialism. The key move is to deny the normative preeminence of overall betterness, instead taking moral decision-making to rest directly on “respects” of betterness. The multidimensional approach permits a consequentialist account of Narveson’s dictum in which betterness is well-behaved. It also yields a new way to think of the connection between goodness and rightness, thus revealing new terrain in the space of possible moral theories. (shrink)

Many philosophers have been attracted to a restricted version of the principle of indifference in the case of self-locating belief. Roughly speaking, this principle states that, within any given possible world, one should be indifferent between different hypotheses concerning who one is within that possible world, so long as those hypotheses are compatible with one’s evidence. My first goal is to defend a more precise version of this principle. After responding to several existing criticisms of such a principle, I argue (...) that existing formulations of the principle are crucially ambiguous, and I go on to defend a particular disambiguation of the principle. According to the disambiguation I defend, how we should apply this restricted principle of indifference sensitively depends on our background metaphysical beliefs. My second goal is to apply this disambiguated principle to classical skeptical problems in epistemology. In particular, I argue that Eternalism threatens to lead us to external world skepticism, and Modal Realism threatens to lead us to inductive skepticism. (shrink)

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.

I argue that inferences from highly probabilifying racial generalizations are not solely objectionable because acting on such inferences would be problematic, or they violate a moral norm, but because they violate a distinctively epistemic norm. They involve accepting a proposition when, given the costs of a mistake, one is not adequately justified in doing so. First I sketch an account of the nature of adequate justification—practical adequacy with respect to eliminating the ~p possibilities from one’s epistemic statespace. Second, I argue (...) that inferences based on demographic generalizations tend to disproportionately expose group members to the risks associated with mistakenly assuming stereotypical propositions, and so magnify the wrong involved in relying on such inferences without adequate justification. (shrink)

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that cannot be captured within the orthodox Bayesian framework.

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. Further, it is often claimed that the Bayesian framework automatically implements Occam's razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that (...) cannot be captured within the orthodox Bayesian framework and that the automatic razor does not function for such problems. (shrink)

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

Are there plausible synchronic constraints on how a subject thinks of herself extended over time? At first glance, Bas van Fraassen’s principle of Reflection seems to prescribe the sort of epistemic authority one’s future self should be taken by one to have over one’s current epistemic states. (The gist of this principle is that I should now believe what I’m convinced I will believe tomorrow.) There has been a general consensus that, as a principle concerning epistemic authority, Reflection does not (...) apply to epistemically non-ideal agents. I agree with this, but argue here that it misses the point of Reflection. Rather than an epistemic principle concerning reasons for belief, Reflection concerns the semantics of belief avowal. I present a non-factual interpretation of Reflection, argue that the principle provides a constraint on the ways in which one can reflectively endorse one’s future epistemic self, and say something about the logic governing such an interpretation. (shrink)

We examine a distinctive kind of problem for decision theory, involving what we call discontinuity at infinity. Roughly, it arises when an infinite sequence of choices, each apparently sanctioned by plausible principles, converges to a ‘limit choice’ whose utility is much lower than the limit approached by the utilities of the choices in the sequence. We give examples of this phenomenon, focusing on Arntzenius et al.’s Satan’s apple, and give a general characterization of it. In these examples, repeated dominance reasoning (...) (a paradigm of rationality) apparently gives rise to a situation closely analogous to having intransitive preferences (a paradigm of irrationality). Indeed, the agents in these examples are vulnerable to a money pump set-up despite having preferences that exhibit no obvious defect of rationality. We explore several putative solutions to such problems, particularly those that appeal to binding and to deliberative dynamics. We consider the prospects for these solutions, concluding that if they fail, the examples show that money pump arguments are invalid. (shrink)

Confronted with the possibility of severe environmental harms, such as catastrophic climate change, some researchers have suggested that we should abandon the principle at the heart of standard decision theory—the injunction to maximize expected utility—and embrace a different one: the Precautionary Principle. Arguably, the most sophisticated philosophical treatment of the Precautionary Principle is due to Steel. Steel interprets PP as a qualitative decision rule and appears to conclude that a quantitative decision-theoretic statement of PP is both impossible and unnecessary. In (...) this article, we propose a decision-theoretic formulation of PP in terms of lexical utilities. We show that this lexical model is largely faithful to Steel’s approach, but also that it corrects three problems with Steel’s account and clarifies the relationship between PP and standard decision theory. Using a range of examples, we illustrate how the lexical model can be used to explore a variety of issues related to precautionary reasoning. (shrink)

Morally supererogatory acts are those that go above and beyond the call of duty. More specifically: they are acts that, on any individual occasion, are good to do and also both permissible to do and permissible to refrain from doing. We challenge the way in which discussions of supererogation typically consider our choices and actions in isolation. Instead we consider sequences of supererogatory acts and omissions and show that some such sequences are themselves problematic. This gives rise to the following (...) puzzle: what problem can we have with a sequences of actions if each individual act or omission is itself permissible? In this paper, we develop a response to this question, by exploring whether solutions analogous to those proposed in the rational choice literature are available in the case of supererogatory sequences. Our investigation leads us to the view that making sense of the supererogatory requires accepting that there are global moral norms that apply to sequences of acts alongside the local moral norms that apply to individual acts. (shrink)

ABSTRACT While pragmatic arguments are traditionally seen as supporting decision theory, recent discussions suggest the possibility of pragmatic arguments against this theory. I respond to two such arguments, and clarify what it would take for arguments of this sort to succeed.

I argue that standard decision theories, namely causal decision theory and evidential decision theory, both are unsatisfactory. I devise a new decision theory, from which, under certain conditions, standard game theory can be derived.

The idea that beliefs may be stake-sensitive is explored. This is the idea that the strength with which a single, persistent belief is held may vary and depend upon what the believer takes to be at stake. The stakes in question are tied to the truth of the belief—not, as in Pascal’s wager and other cases, to the belief’s presence. Categorical beliefs and degrees of belief are considered; both kinds of account typically exclude the idea and treat belief as stake-invariant (...) , though an exception is briefly described. The role of the assumption of stake-invariance in familiar accounts of degrees of belief is also discussed, and morals are drawn concerning finite and countable Dutch book arguments. (shrink)

One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on (...) the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)

Countenancing unbounded utility in ethics gives rise to deep puzzles in formal decision theory. Here, these puzzles are taken as an invitation to assess the most fundamental principles relating probability and value, with the aim of demonstrating that unbounded utility in ethics is compatible with a desirable decision theory. The resulting theory frames further discussion of Expected Utility Theory and of principles concerning symmetries of utility.

Suppose that you prefer A to B, B to C, and C to A. Your preferences violate Expected Utility Theory by being cyclic. Money-pump arguments offer a way to show that such violations are irrational. Suppose that you start with A. Then you should be willing to trade A for C and then C for B. But then, once you have B, you are offered a trade back to A for a small cost. Since you prefer A to B, you (...) pay the small sum to trade from B to A. But now you have been turned into a money pump. You are back to the alternative you started with but with less money. This Element shows how each of the axioms of Expected Utility Theory can be defended by money-pump arguments of this kind. The Element also defends money-pump arguments from the standard objections to this kind of approach. This title is also available as Open Access on Cambridge Core. (shrink)

This paper will be concerned with hard choices—that is, choice situations where an agent cannot make a rationally justified choice. Specifically, this paper asks: if an agent cannot optimize in a given situation, are they facing a hard choice? A pair of claims are defended in light of this question. First, situations where an agent cannot optimize because of incompleteness of the binary preference or value relation constitute a hard choice. Second, situations where agents cannot optimize because the binary preference (...) or value relation violates acyclicity do not constitute a hard choice. (shrink)

The 'Why ain'cha rich?' argument for one-boxing in Newcomb's problem allegedly vindicates evidential decision theory and undermines causal decision theory. But there is a good response to the argument on behalf of causal decision theory. I develop this response. Then I pose a new problem and use it to give a new 'Why ain'cha rich?' argument. Unlike the old argument, the new argument targets evidential decision theory. And unlike the old argument, the new argument is sound.

I argue that the use of a social discount rate to assess the consequences of climate policy is unhelpful and misleading. I consider two lines of justification for discounting: (i) ethical arguments for a "pure rate of time preference" and (ii) economic arguments that take time as a proxy for economic growth and the diminishing marginal utility of consumption. In both cases I conclude that, given the long time horizons, distinctive uncertainties, and particular costs and benefits at stake in the (...) climate context, discount rates are at best a poor proxy for the normative considerations they are meant to represent. (shrink)

Value-based epistemology sees epistemic norms as explained by or grounded in distinctively epistemic values. This paper argues that, no matter what epistemic value is, credences or beliefs about some topics have at most infinitesimal amounts of this value. This makes it hard to explain why epistemic norms apply at all to credences or beliefs on these topics. My argument is inspired by a recent series of papers on epistemic versions of Parfit’s Repugnant Conclusion. The discussion in those papers parallels work (...) in ethics, because it focuses on an epistemic value—accuracy—that shares features with ethical values like well-being. I argue that, because of this focus, this discussion is inconclusive and only relevant to accounts of epistemic value that share those features. My argument is more general and more conclusive. It uses four types of problem cases that have no parallel in ethics. It applies to all extant accounts of the value of individual beliefs or credences and all extant accounts of the value of total doxastic states. (shrink)

I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.

In her recent book, L. A. Paul presses a serious problem for normative decision theory. Normative decision theory seems to be inapplicable when the values of potential outcomes are unknown, or when our preferences may change as a result of our choice. Paul then offers a framework for overcoming these problems, known as therevelation approach. I argue that, contrary to what Paul suggests, this approach is unhelpful in the large class of cases where the decision at hand centrally concerns persons (...) other than the decision-maker. Unless Paul’s account is supplemented in order to handle such cases, many decisions will remain rationally intractable. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

Should we make significant sacrifices to ever-so-slightly lower the chance of extremely bad outcomes, or to ever-so-slightly raise the chance of extremely good outcomes? *Fanaticism* says yes: for every bad outcome, there is a tiny chance of extreme disaster that is even worse, and for every good outcome, there is a tiny chance of an enormous good that is even better. I consider two related recent arguments for Fanaticism: Beckstead and Thomas's argument from *strange dependence on space and time*, and (...) Wilkinson's *Indology* argument. While both arguments are instructive, neither is persuasive. In fact, the general principles that underwrite the arguments (a *separability* principle in the first case, and a *reflection* principle in the second) are *inconsistent* with Fanaticism. In both cases, though, it is possible to rehabilitate arguments for Fanaticism based on restricted versions of those principles. The situation is unstable: plausible general principles tell *against* Fanaticism, but restrictions of those same principles (with strengthened auxiliary assumptions) *support* Fanaticism. All of the consistent views that emerge are very strange. (shrink)

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call *Countable (...) Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

It is now the majority view amongst philosophers and theologians that any world could have been better. This places the choice of which world to create into an especially challenging class of decision problems: those that are discontinuous in the limit. I argue that combining some weak, plausible norms governing this type of problem with a creator who has the attributes of the god of classical theism results in a paradox: no world is possible. After exploring some ways out of (...) the paradox, I conclude that the classical theist should accept Marilyn Adams’s view that no norms apply to gods. (shrink)

Currently, the most popular views about how to update de se or self-locating beliefs entail the one-third solution to the Sleeping Beauty problem.2 Another widely held view is that an agent‘s credences should be countably additive.3 In what follows, I will argue that there is a deep tension between these two positions. For the assumptions that underlie the one-third solution to the Sleeping Beauty problem entail a more general principle, which I call the Generalized Thirder Principle, and there are situations (...) in which the latter principle and the principle of Countable Additivity cannot be jointly satisfied. The most plausible response to this tension, I argue, is to accept both of these principles, and to maintain that when an agent cannot satisfy them both, she is faced with a rational dilemma. (shrink)

This paper discusses conditional probability $$P$$ P , or the probability of A given B. When $$P>0$$ P > 0 , the ratio formula determines $$P$$ P . When $$P=0$$ P = 0 , the ratio formula breaks down. The Borel–Kolmogorov paradox suggests that conditional probabilities in such cases are indeterminate or ill-posed. To analyze the paradox, I explore the relation between probability and intensionality. I argue that the paradox is a Frege case, similar to those that arise in many (...) probabilistic and non-probabilistic contexts. The paradox vividly illustrates how an agent’s way of representing an entity can rationally influence her credal assignments. I deploy my analysis to defend Kolmogorov’s relativistic treatment of conditional probability. (shrink)

This paper discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

Larry Temkin famously argues that what he calls spectrum arguments yield strong reason to reject Transitivity, according to which the ‘all-things-considered better than’ relation is transitive. Spectrum arguments do reveal that the conjunctions of independently plausible claims are inconsistent with Transitivity. But I argue that there is very strong independent reason to reject such conjunctions of claims, and thus that the fact that they are inconsistent with Transitivity does not yield strong reason to reject Transitivity.

Pragmatic arguments seek to demonstrate that you can be placed in a situation in which you will face a sure and foreseeable loss if you do not behave in accordance with some principle P. In this article I show that for every P entailed by the principle of maximizing expected utility you will not be better off from a pragmatic point of view if you accept P than if you don’t, because even if you obey the axioms of expected utility (...) theory it is possible to place you in a situation in which you will face a certain and foreseeable loss. This shows that for a large class of Ps, there is no pragmatic difference between people who accept P and those who don’t. (shrink)

A vexing question in Bayesian epistemology is how an agent should update on evidence which she assigned zero prior credence. Some theorists have suggested that, in such cases, the agent should update by Kolmogorov conditionalization, a norm based on Kolmogorov’s theory of regular conditional distributions. However, it turns out that in some situations, a Kolmogorov conditionalizer will plan to always assign a posterior credence of zero to the evidence she learns. Intuitively, such a plan is irrational and easily Dutch bookable. (...) In this paper, we propose a revised norm, Kolmogorov-Blackwell conditionalization, which avoids this problem. We prove a Dutch book theorem and converse Dutch book theorem for this revised norm, and relate our results to those of Rescorla (2018). (shrink)

In “Bayesianism, Infinite Decisions, and Binding”, Arntzenius et al. (Mind 113:251–283, 2004 ) present cases in which agents who cannot bind themselves are driven by standard decision theory to choose sequences of actions with disastrous consequences. They defend standard decision theory by arguing that if a decision rule leads agents to disaster only when they cannot bind themselves, this should not be taken to be a mark against the decision rule. I show that this claim has surprising implications for a (...) number of other debates in decision theory. I then assess the plausibility of this claim, and suggest that it should be rejected. (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

In the asymmetrical variant of the two-envelope paradox, the amount in envelope A is determined first, and then the amount in envelope B is determined to be either twice or half the amount in A by flipping a fair coin. Contra the common belief that B is preferable to A in this case, I show that the proposed arguments for this common belief all fail, and argue that B is not preferable to A if the expected values of the amounts (...) in the envelopes are infinite. Using the examples I deploy in my arguments against the common belief, I also refute certain proposed solutions to the two-envelope paradox and draw some general lessons. (shrink)

Consider the following claim: given the choice between saving a life and preventing any number of people from temporarily experiencing a mild headache, you should always save the life. Many moral theorists accept this claim. In doing so, they commit themselves to some form of ‘moral absolutism’: the view that there are some moral considerations that cannot be outweighed by any number of lesser moral considerations. In contexts of certainty, it is clear what moral absolutism requires of you. However, what (...) does it require of you when deciding under risk? What ought you to do when there is a chance that, say, you will not succeed in saving the life? In recent years, various critics have argued that moral absolutism cannot satisfactorily deal with risk and should, therefore, be abandoned. In this paper, we show that moral absolutism can answer its critics by drawing on—of all things—orthodox expected utility theory. (shrink)

How should deontologists approach decision-making under uncertainty, for an iterated decision problem? In this paper I explore the shortcomings of a simple expected value approach, using a novel example to raise questions about attitudes to risk, the moral significance of tiny probabilities, the independent moral reasons against imposing risks, the morality of sunk costs, and the role of agent-relativity in iterated decision problems.

This paper develops transfinite extensions of transitivity and acyclicity in the context of population ethics. They are used to argue that it is better to add good lives, worse to add bad lives, and equally good to add neutral lives, where a life's value is understood as personal value. These conclusions rule out a number of theories of population ethics, feed into an argument for the repugnant conclusion, and allow us to reduce different-number comparisons to same-number ones. Challenges to these (...) arguments are addressed, including the issue of comparing existence and non-existence in terms of personal value, the possibility of minimal quanta of time and life, and the meaningfulness of measuring closeness between outcomes with different population sizes. An asymmetry is uncovered between transfinite cycles of worseness and betterness, supporting a version of the weak procreative asymmetry. Transfinite transitivity principles are also favourably compared to the better-known principles of continuity. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

An infinite lottery experiment seems to indicate that Bayesian conditionalization may be inconsistent when the prior credence function is finitely additive because, in that experiment, it conflicts with the principle of reflection. I will show that any other form of updating credences would produce the same conflict, and, furthermore, that the conflict is not between conditionalization and reflection but, instead, between finite additivity and reflection. A correct treatment of the infinite lottery experiment requires a careful treatment of finite additivity. I (...) will show that the results of the experiment, paradoxical though they may be, are not inconsistent, but that they conflict with one particular version of reflection . I will reject that version and I will propose a slight modification of a different version of reflection such that the modified version does maintain the mutual consistency of finite additivity, reflection and conditionalization. As a result, I will strengthen the case for finite additivity by showing that Bayesian conditionalization is fully consistent with it. (shrink)

This paper explores the consequences of applying two natural ideas from epistemology to decision theory: (1) that knowledge should guide our actions, and (2) that we know a lot of non-trivial things. In particular, we explore the consequences of these ideas as they are applied to standard decision theoretic puzzles such as the St. Petersburg Paradox. In doing so, we develop a “knowledge-first” decision theory and we will see how it can help us avoid fanaticism with regard to the St. (...) Petersburg puzzle and related puzzles. The result will be a decision theory that gives a novel, but well-motivated, reason for discounting small probabilities when making decisions. We examine the merits and demerits of such a decision theory. (shrink)

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

Philosophical Perspectives, Volume 35, Issue 1, Page 189-205, December 2021.

In our 2004, we introduced two games in the spirit of the St Petersburg game, the Pasadena and Altadena games. As these latter games lack an expectation, we argued that they pose a paradox for decision theory. Terrence Fine has shown that any finite valuations for the Pasadena, Altadena, and St Petersburg games are consistent with the standard decision-theoretic axioms. In particular, one can value the Pasadena game above the other two, a result that conflicts with both our intuitions and (...) dominance reasoning. We argue that this result, far from resolving the Pasadena paradox, should serve as a reductio of the standard theory, and we consequently make a plea for new axioms for a revised theory. We also discuss a proposal by Kenny Easwaran that a gamble should be valued according to its 'weak expectation', a generalization of the usual notion of expectation. (shrink)

According to the so-called ‘deliberation crowds out prediction’ thesis, while deliberating about what you’ll do, you cannot rationally have credences for what you’ll do – you cannot rationally have option-credences. Versions of the thesis have been defended by authors such as Spohn, Levi, Gilboa, Price, Louise, and others. After registering a number of concerns about the thesis, I rehearse and rebut many of the main arguments for it, grouped according to their main themes: agency, vacuity, betting, and decision-theoretical considerations. I (...) go on to suggest many possible theoretical roles for option-credences. -/- I locate the debate about the thesis in a broader discussion: Are there rational credence gaps – propositions to which one cannot rationally assign credences? If there are, they spell trouble for various foundations of Bayesian epistemology, including the usual ratio formula for conditional probability, conditionalization, decision theory, and independence. According to the thesis, credence gaps are completely mundane; they arise every time someone rationally deliberates. But these foundations are safe from any threat here, I contend, since the thesis is false. Deliberation welcomes prediction. (shrink)

Four important arguments for probabilism—the Dutch Book, representation theorem, calibration, and gradational accuracy arguments—have a strikingly similar structure. Each begins with a mathematical theorem, a conditional with an existentially quantified consequent, of the general form: if your credences are not probabilities, then there is a way in which your rationality is impugned.Each argument concludes that rationality requires your credences to be probabilities.I contend that each argument is invalid as formulated. In each case there is a mirror-image theorem and a corresponding (...) argument of exactly equal strength that concludes that rationality requires your credences not to be probabilities. Some further consideration is needed to break this symmetry in favour of probabilism. I discuss the extent to which the original arguments can be buttressed.1. Introduction2. The Dutch Book Argument2.1. Saving the Dutch Book argument2.2. ‘The Dutch Book argument merely dramatizes an inconsistency in the attitudes of an agent whose credences violate probability theory’3. Representation Theorem-based Arguments4. The Calibration Argument5. The Gradational Accuracy Argument6. Conclusion. (shrink)