I came to philosophy as a refugee from mathematics and statistics. I was impressed by their power at codifying and precisifying antecedently understood but rather nebulous concepts, and at clarifying and exploring their interrelations. I enjoyed learning many of the great theorems of probability theory—equations rich in ‘P’s of this and of that. But I wondered what is this ‘P’? What do statements of probability mean? When I asked one of my professors, he looked at me like I needed medication. (...) That medication was provided by philosophy, and I found it first during my Masters at the University of Western Ontario, working with Bill Harper, and then during my Ph.D. at Princeton, working with Bas van Fraassen, David Lewis, and Richard Jeffrey—all deft practitioners of formal methods. I found that philosophers had been asking my question about ‘P’ since about 1650, but they were still struggling to find definitive answers. I was also introduced to a host of other philosophical problems, and it became clear to me within nanoseconds of arriving at U.W.O. that I wanted to spend my life pursuing some of them. But I kept being drawn back to the formal methods of mathematics, and in particular of probability theory. It may be worthwhile to pause for a moment and to ask “What are formal methods?” Of course, it’s easy to come up with examples: the use of various logical systems, computational algorithms, causal graphs, information theory, probability theory and mathematics more generally. What do they have in common? They are all abstract representational systems. Sometimes the systems are studied in their own.. (shrink)
Counterfactuals are a species of conditionals. They are propositions or sentences, expressed by or equivalent to subjunctive conditionals of the form 'if it were the case that A, then it would be the case that B', or 'if it had been the case that A, then it would have been the case that B'; A is called the antecedent, and B the consequent. Counterfactual reasoning typically involves the entertaining of hypothetical states of affairs: the antecedent is believed or presumed to (...) be false, or contrary-to-the-fact, but its truth is imagined or supposed. Counterfactual reasoning is thus a form of modal reasoning, kindred to reasoning about necessity or possibility, and in contrast to reasoning about the way things actually are. The philosophical study of conditionals goes back at least as far as the Stoics of ancient Greece, although their systems of logic apparently did not accord the counterfactual any emphasis. The rise in interest in counterfactuals has been a rather recent phenomenon, as it started to become clear to philosophers that counterfactuals are implicated in a host of other important concepts—laws of nature, confirmation, causation, scientific explanation, knowledge, perception, dispositions, free action, etc. The significance of counterfactuals has also become increasingly appreciated in the.. (shrink)
“Probability logic” might seem like an oxymoron. Logic traditionally concerns matters immutable, necessary and certain, while probability concerns the uncertain, the random, the capricious. Yet our subject has a distinguished pedigree. Ramsey begins his classic “Truth and Probability” [44] with the words: “In this essay the Theory of Probability is taken as a branch of logic...”. De Finetti [7] speaks of “the logic of the probable”. And more recently, Jeffrey [25] regards probabilities as estimates of truth values, and thus probability (...) theory as a natural outgrowth of two-valued logic—what he calls “probability logic”. However we put the point, probability theory and logic are clearly intimately related. This chapter explores some of the multifarious connections between probability and logic, and focuses on various philosophical issues in the foundations of probability theory. Our survey begins in §2 with the probability calculus, what Adams [1, p. 34] calls “pure probability logic”. As we will see, there is a sense in which the axiomatization of probability presupposes deductive logic. Moreover, some authors see probability theory as the proper framework for inductive logic—a formal apparatus for codifying the degree of.. (shrink)
Arguably, Hume's greatest single contribution to contemporary philosophy of science has been the problem of induction (1739). Before attempting its statement, we need to spend a few words identifying the subject matter of this corner of epistemology. At a first pass, induction concerns ampliative inferences drawn on the basis of evidence (presumably, evidence acquired more or less directly from experience)—that is, inferences whose conclusions are not (validly) entailed by the premises. Philosophers have historically drawn further distinctions, often appropriating the term (...) “induction” to mark them; since we will not be concerned with the philosophical issues for which these distinctions are relevant, we will use the word “inductive” in a catch-all sense synonymous with “ampliative”. But we will follow the usual practice of choosing, as our paradigm example of inductive inferences, inferences about the future based on evidence drawn from the past and present. A further refinement is more important. Opinion typically comes in degrees, and this fact makes a great deal of difference to how we understand inductive inferences. For while it is often harmless to talk about the conclusions that can be rationally believed on the basis of some.. (shrink)
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 decisiontheoretic 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)
Gonzales tells Mark Crimmins (1992) that Crimmins knows him under two guises, and that under his other guise Crimmins thinks him an idiot. Knowing his cleverness, but not knowing which guise he has in mind, Crimmins trusts Gonzales but does not know which of his beliefs to revise. He therefore asserts to Gonzales.
According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities are permitted to have (...) conditions with zero unconditional probability. We reconsider probabilistic independence in this more general setting. We argue that a less orthodox but more general (Popperian) theory of conditional probability should be used, and that much of the conventional wisdom about probabilistic independence needs to be rethought. (shrink)
Much is asked of the concept of chance. It has been thought to play various roles, some in tension with or even incompatible with others. Chance has been characterized negatively, as the absence of causation; yet also positively—the ancient Greek τυχη´ reifies it—as a cause of events that are not governed by laws of nature, or as a feature of the laws themselves. Chance events have been understood epistemically as those whose causes are unknown; yet also objectively as a distinct (...) ontological kind, sometimes called ‘pure’ chance events. Chance gives rise to individual unpredictability and disorder; yet it yields collective predictability and order—stable long-run statistics, and in the limit, aggregate behavior susceptible to precise mathematical theorems. Some authors believe that to posit chances is to abjure explanation; yet others think that chances are themselves explanatory. During the Enlightenment, talk of ‘chance’ was regarded as unscientific, unphilosophical, the stuff of superstition or ignorance; yet today it is often taken to be a fundamental notion of our most successful scientific theory, quantum mechanics, and a central concept of contemporary metaphysics. (shrink)
Arguably, Hume's greatest single contribution to contemporary philosophy of science has been the problem of induction (1739). Before attempting its statement, we need to spend a few words identifying the subject matter of this corner of epistemology. At a first pass, induction concerns ampliative inferences drawn on the basis of evidence (presumably, evidence acquired more or less directly from experience)—that is, inferences whose conclusions are not (validly) entailed by the premises. Philosophers have historically drawn further distinctions, often appropriating the term (...) “induction” to mark them; since we will not be concerned with the philosophical issues for which these distinctions are relevant, we will use the word “inductive” in a catch-all sense synonymous with “ampliative”. But we will follow the usual practice of choosing, as our paradigm example of inductive inferences, inferences about the future based on evidence drawn from the past and present. (shrink)
We introduce a St. Petersburg-like game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your pay-offs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we (...) want, simply by reordering the pay-off table, yet the game remains unchanged throughout. Formally speaking, the expectation does not exist; but we contend that this presents a.. (shrink)
The philosophy of probability has been alive and well for several decades in Australia and New Zealand. Some distinctive lines of thought have emerged, resonating with broader themes that have come to be associated with Australasian philosophers: realist/objectivist accounts of various theoretical entities; an ongoing concern with logic, including the development of nonclassical logics; and enthusiasm for conceptual analysis, rooted in commonsense but informed by science. In this article I concentrate on work by philosophers on the interpretation of probability, its (...) logical foundations, and its philosophical applications.1 My nomination for the earliest major Australasian philosopher of probability may surprise some readers: Karl Popper. He counts as Australasian by dint of his employment at the University of Canterbury from 1937 until the end of World War II; he counts as a major philosopher of probability by any estimation. Two of his contributions have initiated research programs in the foundations of probability that are still thriving: his (1959a) axiomatization of primitive conditional probability functions (socalled ‘Popper functions’), and his ‘propensity’ interpretation of probability (1959b), intended to illuminate singlecase attributions of objective probabilities, as are putatively found in quantum mechanics. (shrink)
Four different conditionals were known to the Stoics. The so-called ‘first’ (Philonian) conditional has been interpreted fairly uncontroversially as an ancient counterpart to the material conditional of modern logic; the ‘fourth’ conditional is obscure, and seemingly of little historical interest, as it was probably not held widely by any group in antiquity. The ‘second’ (Diodorean) and ‘third’ (Chrysippean) conditionals, on the other hand, pose challenging interpretive questions, raising in the process issues in philosophical logic that are as relevant today as (...) they were then. This paper is a critical survey of some modern answers to four of the most tantalizing of these questions; the issues that I will discuss arise out of interpretations of the Diodorean and Chrysippean conditionals as expressions of natural law, and as strict implications. I will reject these interpretations, concluding with my own proposal for where they should be located on a ‘ladder’ of logical strength. The following passage from Sextus will form the basis of my discussion (from Outlines of Pyrrhonism [Pyrrhoneae Hypotyposes], as presented by Long and Sedley 1987b, 211). He has just introduced Philo’s account of “a sound conditional”—by which I understand a true conditional—with the example “when it is day and I am talking, ‘If it is day, I am talking’”. He then continues. (shrink)
I.1. Introduction Confirmation theory is intended to codify the evidential bearing of observations on hypotheses, characterizing relations of inductive “support” and “countersupport” in full generality. The central task is to understand what it means to say that datum E confirms or supports a hypothesis H when E does not logically entail H.
The so-called ‘Adams’ Thesis’ is often understood as the claim that the assertibility of an indicative conditional equals the corresponding conditional probability—schematically: $${({\rm AT})}\qquad\qquad\quad As(A\rightarrow B)=P({B|A}),{\rm provided}\quad P(A)\neq 0.$$ The Thesis is taken by many to be a touchstone of any theorizing about indicative conditionals. Yet it is unclear exactly what the Thesis is . I suggest some precise statements of it. I then rebut a number of arguments that have been given in its favor. Finally, I offer a new (...) argument against it. I appeal to an old triviality result of mine against ‘Stalnaker’s Thesis’ that the probability of a conditional equals the corresponding conditional probability. I showed that for all finite-ranged probability functions, there are strictly more distinct values of conditional probabilities than there are distinct values of probabilities of conditionals, so they cannot all be paired up as Stalnaker’s Thesis promises. Conditional probabilities are too fine-grained to coincide with probabilities of conditionals across the board. If the assertibilities of conditionals are to coincide with conditional probabilities across the board, then assertibilities must be finer-grained than probabilities. I contend that this is implausible—it is surely the other way round. I generalize this argument to other interpretations of ‘ As ’, including ‘acceptability’ and ‘assentability’. I find it hard to see how any such figure of merit for conditionals can systematically align with the corresponding conditional probabilities. (shrink)
We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our (...) describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)
David Lewis was one of the most important philosophers of the 20th century working in the Anglo-American analytic tradition. His corpus is extraordinary for its breadth of subject matter and for its systematicity. For both these reasons, it is difficult to do justice to his work in a short space—there are rich interconnections among his myriad writings, and numerous possible entry points. This article approaches Lewis and his work in three passes: first, a biographical tracing of his intellectual influences; second, (...) a summary of his metaphilosophy; third, a survey of his more specific philosophical views, mostly following their order of conceptual dependence. (shrink)
According to finite frequentism, the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B. I present fifteen arguments against this position.
Bayesianism is our leading theory of uncertainty. Epistemology is defined as the theory of knowledge. So “Bayesian Epistemology” may sound like an oxymoron. Bayesianism, after all, studies the properties and dynamics of degrees of belief, understood to be probabilities. Traditional epistemology, on the other hand, places the singularly non-probabilistic notion of knowledge at centre stage, and to the extent that it traffics in belief, that notion does not come in degrees. So how can there be a Bayesian epistemology?
This is the sequel to my “Fifteen Arguments Against Finite Frequentism” ( Erkenntnis 1997), the second half of a long paper that attacks the two main forms of frequentism about probability. Hypothetical frequentism asserts: The probability of an attribute A in a reference class B is p iff the limit of the relative frequency of A ’s among the B ’s would be p if there were an infinite sequence of B ’s. I offer fifteen arguments against this analysis. I (...) consider various frequentist responses, which I argue ultimately fail. I end with a positive proposal of my own, ‘hyper-hypothetical frequentism’, which I argue avoids several of the problems with hypothetical frequentism. It identifies probability with relative frequency in a hyperfinite sequence of trials. However, I argue that this account also fails, and that the prospects for frequentism are dim. (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. Introduction The Dutch Book Argument 2.1 Saving the Dutch Book argument 2.2 The Dutch Book argument merely dramatizes an inconsistency in the attitudes of an agent whose credences violate probability theory Representation Theorem-based Arguments The Calibration Argument The Gradational Accuracy Argument Conclusion CiteULike Connotea Del.icio.us What's this? (shrink)
Alan Hajek (2008). Dutch Book Arguments. In Paul Anand, Prasanta Pattanaik & Clemens Puppe (eds.), The Oxford Handbook of Rational and Social Choice. Oxford University Press.
in The Oxford Handbook of Corporate Social Responsibility, ed. Paul Anand, Prasanta Pattanaik, and Clemens Puppe, forthcoming 2007.
I analyze David Hume’s "Of Miracles". I vindicate Hume’s argument against two charges: that it (1) defines miracles out of existence; (2) appeals to a suspect principle of balancing probabilities. He argues that miracles are, in a certain sense, maximally improbable. To understand this sense, we must turn to his notion of probability as ’strength of analogy’: miracles are incredible, according to him, because they bear no analogy to anything in our past experience. This reveals as anachronistic various recent Bayesian (...) reconstructions of his argument. But it exposes him to other charges, with which I conclude. (shrink)
“Pascal's Wager” is the name given to an argument due to Blaise Pascal for believing, or for at least taking steps to believe, in God. The name is somewhat misleading, for in a single paragraph of his Pensées, Pascal apparently presents at least three such arguments, each of which might be called a ‘wager’ — it is only the final of these that is traditionally referred to as “Pascal's Wager”. We find in it the extraordinary confluence of several important strands (...) of thought: the justification of theism; probability theory and decision theory, used here for almost the first time in history; pragmatism; voluntarism (the thesis that belief is a matter of the will); and the use of the concept of infinity. (shrink)
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)
Frank Ramsey (1931) wrote: If two people are arguing 'if p will q?' and both are in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q. We can say that they are fixing their degrees of belief in q given p. Let us take the first sentence the way it is often taken, as proposing the following test for the acceptability of an indicative conditional: ‘If p then q’ (...) is acceptable to a subject S iff, were S to accept p and consider q, S would accept q. Now consider an indicative conditional of the form (1) If p, then I believe p. Suppose that you accept p and consider ‘I believe p’. To accept p while rejecting ‘I believe p’ is tantamount to accepting the Moore-paradoxical sentence ‘p and I do not believe p’, and so is irrational. To accept p while suspending judgment about ‘I believe p’ is irrational for similar reasons. So rationality requires that if you accept p and consider ‘I believe p’, you accept ‘I believe p’. (shrink)
Probabilism is committed to two theses: 1) Opinion comes in degrees—call them degrees of belief, or credences. 2) The degrees of belief of a rational agent obey the probability calculus. Correspondingly, a natural way to argue for probabilism is: i) to give an account of what degrees of belief are, and then ii) to show that those things should be probabilities, on pain of irrationality. Most of the action in the literature concerns stage ii). Assuming that stage i) has been (...) adequately discharged, various authors move on to stage ii) with varied and ingenious arguments. But an unsatisfactory response at stage i) clearly undermines any gains that might be accrued at stage ii) as far as probabilism is concerned: if those things are not degrees of belief, then it is irrelevant to probabilism whether they should be probabilities or not. In this paper we scrutinize the state of play regarding stage i). We critically examine several of the leading accounts of degrees of belief: reducing them to corresponding betting behavior (de Finetti); measuring them by that behavior (Jeffrey); and analyzing them in terms of preferences and their role in decision-making more generally (Ramsey, Lewis, Maher). We argue that the accounts fail, and so they are unfit to subserve arguments for probabilism. We conclude more positively: ‘degree of belief’ should be taken as a primitive concept that forms the basis of our best theory of rational belief and decision: probabilism. (shrink)
The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference (...) class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains. (shrink)
There are two central questions concerning probability. First, what are its formal features? That is a mathematical question, to which there is a standard, widely (though not universally) agreed upon answer. This answer is reviewed in the next section. Second, what sorts of things are probabilities---what, that is, is the subject matter of probability theory? This is a philosophical question, and while the mathematical theory of probability certainly bears on it, the answer must come from elsewhere. To see why, observe (...) that there are many things in the world that have the mathematical structure of probabilities---the set of measurable regions on the surface of a table, for example---but that one would never mistake for being probabilities. So probability is distinguished by more than just its formal characteristics. The bulk of this essay will be taken up with the central question of what this “more” might be. (shrink)
This paper revisits the Pasadena game (Nover and Háyek 2004), a St Petersburg-like game whose expectation is undefined. We discuss serveral respects in which the Pasadena game is even more troublesome for decision theory than the St Petersburg game. Colyvan (2006) argues that the decision problem of whether or not to play the Pasadena game is ‘ill-posed’. He goes on to advocate a ‘pluralism’ regarding decision rules, which embraces dominance reasoning as well as maximizing expected utility. We rebut Colyvan’s argument, (...) offering several considerations in favour of the Pasadena decision problem being well posed. To be sure, current decision theory, which is underpinned by various preference axioms, leaves indeterminate how one should value the Pasadena game. But we suggest that determinacy might be achieved by adding further preference axioms. We conclude by opening the door to a far greater plurality of decision rules. We suggest how the goal of unifying these rules might guide future research. (shrink)
The Dutch Book argument, like Route 66, is about to turn 80. It is arguably the most celebrated argument for subjective Bayesianism. Start by rejecting the Cartesian idea that doxastic attitudes are ‘all-or-nothing’; rather, they are far more nuanced degrees of belief, for short credences, susceptible to fine-grained numerical measurement. Add a coherentist assumption that the rationality of a doxastic state consists in its internal consistency. The remaining problem is to determine what consistency of credences amounts to. The Dutch Book (...) argument, in a nutshell, says that if your credences do not obey the probability calculus, you are ‘incoherent’—susceptible to sure losses at the hands of a ‘Dutch Bookie’—and thus irrational. Conclusion: rationality requires your credences to obey the probability calculus. And like Route 66, the fortunes of the Dutch Book argument have been mixed. Opinions on the argument are sharply divided. The list of its proponents is quite a ‘who’s who’ of philosophers of probability; they include de Finetti (1937, 1980), Carnap (1950, 1962, and more fully, 1955), Kemeny (1955), Lehman (1955), Shimony (1955), Adams (1962), Mellor (1971), Rosenkrantz (1981), van Fraassen (1989), Jeffrey (1983, 1992). (shrink)
The Cable Guy is coming. You have to be home in order for him to install your new cable service, but to your chagrin he cannot tell you exactly when he will come. He will definitely come between 8.a.m. and 4 p.m. tomorrow, but you have no more information than that. I offer to keep you company while you wait. To make things more interesting, we decide now to bet on the Cable Guy’s arrival time. We subdivide the relevant part (...) of the day into two 4-hour long intervals, ‘morning’: (8, 12], and ‘afternoon’: (12, 4). You nominate an interval on which you will bet. If he arrives during your interval, you win and I will pay you $10; otherwise, I win and you will pay me $10. Notice that we stipulate that if he arrives exactly on the stroke of noon, then (8, 12] is the winning interval, since it is closed on the right; but we agree that this event has probability 0 (we have a very precise clock!). At first you think: obviously there is no reason to favour one interval over the other. Your probability distribution of his arrival time is uniform over the 8 a.m. – 4 p.m. period, and thus assigns probability 1/2 to each of the two 4-hour periods at issue. Whichever period you nominate, then, your expected utility is the same. The two choices are equally rational. But then you reason as follows. Suppose that you choose the morning interval. Then there will certainly be a period during which you will regard the other interval as.. (shrink)
David Lewis [1988; 1996] canvases an anti-Humean thesis about mental states: that the rational agent desires something to the extent that he or she believes it to be good. Lewis offers and refutes a decision-theoretic formulation of it, the 'Desire-as-Belief Thesis'. Other authors have since added further negative results in the spirit of Lewis's. We explore ways of being anti-Humean that evade all these negative results. We begin by providing background on evidential decision theory and on Lewis's negative results. We (...) then introduce what we call the indexicality loophole: if the goodness of a proposition is indexical, partly a function of an agent's mental state, then the negative results have no purchase. Thus we propose a variant of Desire-as-Belief that exploits this loophole. We argue that a number of meta-ethical positions are committed to just such indexicality. Indeed, we show that with one central sort of evaluative belief--the belief that an option is right--the indexicality loophole can be exploited in various interesting ways. Moreover, on some accounts, 'good' is indexical in the same way. Thus, it seems that the anti-Humean can dodge the negative results. (shrink)
Petersburg-like game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your pay-offs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we want, simply by reordering (...) the pay-off table, yet the game remains unchanged throughout. Formally speaking, the expectation does not exist; but we contend that this presents a serious problem for decision theory, since it goes silent when we want it to speak. We argue that the Pasadena game is more paradoxical than the St. Petersburg game in several respects. We give a brief review of the relevant mathematics of infinite series. We then consider and rebut a number of replies to our paradox: that there is a privileged ordering to the expectation series; that decision theory should be restricted to finite state spaces; and that it should be restricted to bounded utility functions. We conclude that the paradox remains live. (shrink)
in Probability is the Very Guide of Life: The Philosophical Uses of Chance, eds. Henry Kyburg, Jr. and Mariam Thalos, Open Court. Abridged version in Proceedings of the International Society for Bayesian Analysis 2002.
Gonzales tells Mark Crimmins (1992) that Crimmins knows him under two guises, and that under his other guise Crimmins thinks him an idiot. Knowing his cleverness, but not knowing which guise he has in mind, Crimmins trusts Gonzales but does not know which of his beliefs to revise. He therefore asserts to Gonzales.
According to finite frequentism, the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B. I present fifteen arguments against this position.
I vindicate Hume’s argument against belief in miracle reports against a prevalent objection. Hume has us balance the probability of a miracle’s occurrence against the probability of its being falsely attested to, and argues that the latter must inevitably be the greater; thus, reason requires us to reject any miracle report. The "flaw" in this reasoning, according to Butler and many others, is that it proves too much--it counsels us to never believe historians, newspaper reports of lottery results, and so (...) on; and this is clearly absurd. I show that this objection is misguided: far from providing counterexamples to Hume’s "balancing principle", as I call it, these cases actually confirm it, as some simple calculations of probabilities show. (shrink)
Much is asked of the concept of chance. It has been thought to play various roles, some in tension with or even incompatible with others. Chance has been characterized negatively, as the absence of causation; yet also positively—the ancient Greek τυχη´ reifies it—as a cause of events that are not governed by laws of nature, or as a feature of the laws themselves. Chance events have been understood epistemically as those whose causes are unknown; yet also objectively as a distinct (...) ontological kind, sometimes called ‘pure’ chance events. Chance gives rise to individual unpredictability and disorder; yet it yields collective predictability and order—stable long-run statistics, and in the limit, aggregate behavior susceptible to precise mathematical theorems. Some authors believe that to posit chances is to abjure explanation; yet others think that chances are themselves explanatory. During the Enlightenment, talk of ‘chance’ was regarded as unscientific, unphilosophical, the stuff of superstition or ignorance; yet today it is often taken to be a fundamental notion of our most successful scientific theory, quantum mechanics, and a central concept of contemporary metaphysics. Chance has both negative and positive associations in daily life. The old word in English for it, hazard, which derives from French and originally from Arabic, still has unwelcome connotations of risk; ‘chance’ evokes uncertainty, uncontrollability, and chaos. Yet chance is also allied with luck, fortune, freedom from constraint, and diversity. And it apparently has various practical uses and benefits. It forms the basis of randomized trials in statistics, and of mixed strategies in decision theory and game theory; it is appealed to in order to resolve problems of fair division and other ethical.. (shrink)
