Pascal's Wager

In "Pascal's Mugging" (Bostrom 2009), Pascal gives away his wallet for an extremely tiny chance of an extremely large reward. In this continuation of Bostrom's story, Pascal's friend counsels him to take into account the possibility of making mistakes about his true expected utilities, and they consider to what extent this will help Pascal make plans to avoid future muggings.

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)

Epistemic consequentialism is the view that epistemic goodness is more fundamental than epistemic rightness. This paper examines the relationship between epistemic consequentialism and theistic belief. I argue that, in an epistemic consequentialist framework, there is an epistemic reason to believe in God. Imagine having an unlimited amount of time to ask an omniscient being anything you wanted. The potential epistemic benefits would be enormous. Considerations like these point to an epistemic version of Pascal’s wager. I compare and contrast the epistemic (...) wager with the traditional wager, and argue that the epistemic wager has several notable advantages. (shrink)

I examine the relationship between taking Pascal’s wager, faith, and hope. First, I argue that many who take Pascal’s wager have genuine faith that God exists. The person of faith and the wagerer have several things in common, including a commitment to God and positive cognitive and conative attitudes toward God’s existence. If one’s credences in theism are too low to have faith, I argue that the wagerer can still hope that God exists, another commitment-justifying theological virtue. I conclude with (...) two upshots of the argument, including how it provides responses to common objections to Pascal’s wager. (shrink)

Traditional theistic arguments conclude that God exists. Pragmatic theistic arguments, by contrast, conclude that you ought to believe in God. The two most famous pragmatic theistic arguments are put forth by Blaise Pascal (1662) and William James (1896). Pragmatic arguments for theism can be summarized as follows: believing in God has significant benefits, and these benefits aren’t available for the unbeliever. Thus, you should believe in, or ‘wager on’, God. This article distinguishes between various kinds of theistic wagers, including finite (...) vs. infinite wagers, premortem vs. postmortem wagers, and doxastic vs. acceptance wagers. Then, we’ll turn to the epistemic-pragmatic distinction, and discusses the nuances of James’ argument, and how views like epistemic permissivism and epistemic consequentialism provide unique “hybrid” wagers. Finally, we’ll cover outstanding objections and responses. (shrink)

Epistemic permissivism is the thesis that the evidence can rationally permit more than one attitude toward a proposition. Pascal’s wager is the idea that one ought to believe in God for practical reasons, because of what one can gain if theism is true and what one has to lose if theism is false. In this paper, I argue that if epistemic permissivism is true, then the defender of Pascal’s wager has powerful responses to two prominent objections. First, I argue that (...) if permissivism is true, then permissivism is true about theistic belief. Second, I show how epistemic permissivism about theistic belief dispels two objections to Pascal’s wager: the objection that wagering is impossible, and the objection that wagering is epistemically impermissible. (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)

A review of Matt Cook's book Sleight of Mind: 75 Ingenious Paradoxes in Mathematics, Physics, and Philosophy.

Knowledge is analyzed in terms of the cost incurred when mistakes are made — things we should have known better, but didn’t. Following Wittgenstein at the end of On Certainty, an Epistemic Wager, similar to Pascal’s Wager, is set up to represent the cost differences not in belief vs. disbelief, but in knowledge vs. skepticism. This leads to a core class of absolutely certain knowledge, related to Moorean Facts, that is integrated into our everyday lives. This core knowledge is resistant (...) to both radical skepticism, such as a Cartesian Demon, and Gettier style objections, while providing a small, but sufficient base for future epistemological research. (shrink)

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the (...) decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

One version of Pascal’s Wager says we should commit to, or cultivate belief in, whatever religion we think is most likely to bring us eternal joy. I pose a reductio for this version of the Wager. After exploring some ways the Pascalian might respond, the verdict is that it provides some reason to suspect that somewhere, somehow, the Wager goes wrong.

On some religious traditions, there are obligations to believe certain things. However, this leads to a puzzle, since many philosophers think that we cannot voluntarily control our beliefs, and, plausibly, ought implies can. How do we make sense of religious doxastic obligations? The papers in this issue present four responses to this puzzle. The first response denies that we have doxastic obligations at all; the second denies that ought implies can. The third and fourth responses maintain that we have either (...) indirect or direct control over our beliefs. This paper summarizes each response to the puzzle and argues that there are plausible ways out of this paradox. (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)

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory (...) to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives. (shrink)

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

Etycy nie poświęcali dotąd wiele uwagi niepewności, koncentrując się często na skrajnie wyidealizowanych hipotetycznych sytuacjach, w których zarówno kwestie empiryczne (np. stan świata, spektrum możliwych decyzji oraz ich konsekwencje, związki przyczynowe między zdarzeniami), jak i normatywne (np. treść norm, skale wartości) były jasno określone i znane podmiotowi. W poniższym artykule – który jest rezultatem projektu dotyczącego różnych typów decyzji w sytuacjach niepewności związanej z postępem w naukach i technologiach biomedycznych – przedstawię analizę sytuacji niepewności normatywnej, czyli takich, w których podmiot (...) musi podjąć decyzję, nie mając pewności co do tego, jaka norma postępowania jest właściwa, np. dlatego, że ma sprzeczne intuicje normatywne na temat dopuszczalności danej decyzji. Stanowisko, które określa się niekiedy mianem komparatywizmu, głosi, że w tego typu sytuacjach należy uwzględniać nie tylko stopnie przekonań podmiotu w kwestii różnych norm, ale także to, na ile możliwe decyzje warte są podjęcia wedle tych norm. Przedstawię tu analizę kilku przypadków niepewności normatywnej, a także charakterystyczny argument przeciwko dotychczasowym wersjom komparatywizmu, wskazujący, że nie uwzględniają one różnych nastawień do ryzyka w sytuacjach niepewności normatywnej, choć dopuszczają jednocześnie różne nastawienia do ryzyka w sytuacjach niepewności empirycznej. Pokażę też, że zasada ostrożności dotycząca statusu moralnego w upraszczający sposób przedstawia sytuacje niepewności normatywnej. (shrink)

Many think that Pascal’s Wager is a hopeless failure. A primary reason for this is because a number of challenging objections have been raised to the wager, including the “many gods” objection and the “mixed strategy” objection. We argue that both objections are formal, but not substantive, problems for the wager, and that they both fail for the same reason. We then respond to additional objections to the wager. We show how a version of Pascalian reasoning succeeds, giving us a (...) reason to pay special attention to the infinite consequences of our actions. (shrink)

Pascal’s wager has received the attention of philosophers for centuries. Most of its criticisms arise from how the wager is often framed. We present Pascal’s wager three ways: in isolation from any further apologetic arguments, as leading toward a regimen intended to produce belief, and finally embedded in a larger apology that includes evidence for Christianity. We find that none of the common objections apply when the wager is presented as part of Pascal’s larger project. Pascal’s wager is a successful (...) argument in its proper place. However, the most interesting features of our first two presentations of the wager turn out to be either irrelevant or missing from our reading: infinite utility and the relativity of evidence. The successful wager is a boring wager. Still, this study shows us how the wager might profitably be incorporated into different apologetic contexts and why it often can’t. (shrink)

Pascal’s Wager does not exist in a Platonic world of possible gods, abstract probabilities and arbitrary payoffs. Real decision-makers, such as Pascal’s “man of the world” of 1660, face a range of religious options they take to be serious, with fixed probabilities grounded in their evidence, and with utilities that are fixed quantities in actual minds. The many ingenious objections to the Wager dreamed up by philosophers do not apply in such a real decision matrix. In the situation Pascal addresses, (...) the Wager is a good bet. In the situation of a modern Western intellectual, the reasoning of the Wager is still powerful, though the range of options and the actions indicated are not the same as in Pascal’s day. (shrink)

Some recent work in philosophy of religion addresses what can be called the “axiological question,” i.e., regardless of whether God exists, would it be good or bad if God exists? Would the existence of God make the world a better or a worse place? Call the view that the existence of God would make the world a better place “Pro-Theism.” We argue that Pro-Theism is not implausible, and moreover, many Theists, at least, (often implicitly) think that it is true. That (...) is, many Theists think that various good outcomes would arise if Theism is true. We then discuss work in cognitive science concerning human cognitive bias, before discussing two noteworthy attempts to show that at least some religious beliefs arise because of cognitive bias: Hume’s, and Draper’s and Nichols’s. We then argue that, as a result of certain cognitive biases that result when good outcomes might be at stake, Pro-Theism causes many Theists to inflate the epistemic probability that God exists, and as a result, Theists should lower the probability they assign to God’s existence. Finally, based our arguments, we develop a novel objection to Pascal’s wager. (shrink)

This is a short review of Thomas S. Hibbs' book: *Wagering on an Ironic God: Pascal on Faith and Philosophy*.

Bartha (2012) conjectures that, if we meet all of the other objections to Pascal’s wager, then the many-Gods objection is already met. Moreover, he shows that, if all other objections to Pascal’s wager are already met, then, in a choice between a Jealous God, an Indifferent God, a Very Nice God, a Very Perverse God, the full range of Nice Gods, the full range of Perverse Gods, and no God, you should wager on the Jealous God. I argue that his (...) requirement of [strongly] stable equilibrium is not well-motivated. There are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, which are intuitively no worse wagers than the Jealous God. In particular, I have suggested that one does no worse to wager on a jealous cartel than one does to wager on a Jealous God. Moreover, I argue that there are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, that make trouble for Pascal’s wager, but not because one would do better to wager on them rather than on a Jealous God. Finally, I argue that, if we suppose that infinitesimal credences are in no worse standing the infinite utilities, then we cannot accept the assumption—built into the relative utilities framework—that there cannot be infinitesimal credences. (shrink)

Lately, there has been an explosion of literature exploring the the relationship between one’s practical situation and one’s knowledge. Some involved in this discussion have suggested that facts about a person’s practical situation might affect whether or not a person knows in that situation, holding fixed all the things standardly associated with knowledge (like evidence, the reliability of one’s cognitive faculties, and so on). According to these “pragmatic encroachment” views, then, one’s practical situation encroaches on one’s knowledge. Though we won’t (...) endorse pragmatic encroachment here, we find the view intriguing, and it’s popularity warrants carefully considering it’s implications. One potential avenue of exploration concerns religious epistemology, in particular, whether pragmatic encroachment has consequences concerning the epistemic requirements of atheism. We begin the journey down that avenue by connecting Pascal’s Wager to pragmatic encroachment in order to defend this conditional: If there is pragmatic encroachment, then it is ceteris paribus more difficult to know that atheism is true (if it is) than it is to know that God exists (if God does exist). (shrink)

This paper defends a principle I call Equal Treatment, according to which the rationality of a belief is determined in precisely the same way as the rationality of any other state. For example, if wearing a raincoat is rational just in case doing so maximizes expected value, then believing some proposition P is rational just in case doing so maximizes expected value. This contrasts with the popular view that the rationality of belief is determined by evidential support. It also contrasts (...) with the common idea that in the case of belief, there are two different incommensurable senses of rationality, one of which is distinctively epistemic. I present considerations that favor Equal Treatment over these two alternatives, reply to objections, and criticize some arguments for Evidentialism. I also show how Equal Treatment opens the door to a distinctive kind of response to skepticism. (shrink)

This paper presents a corrected version of Pascal's wager that makes it consonant with modern decision theory. The corrected wager shows that not committing to God's existence is the rational choice.

J.L. Schellenberg argues that divine hiddenness provides an argument for the conclusion that God does not exist, for if God existed he would not allow non-resistant non-belief to occur, but non-resistant non-belief does occur, so God does not exist. In this paper, I argue that the stakes involved in theistic considerations put pressure on Schellenberg’s premise that non-resistant non-belief occurs. First, I specify conditions for someone’s being a resistant non-believer. Then, I argue that many people fulfill these conditions because, given (...) some plausible assumptions, there is a very good pragmatic reason to be a theist rather than an atheist. I assume it is more likely that theists go to heaven than atheists, and I argue there is a non-zero probability that one can receive infinite utility and a method of comparing outcomes with infinite utilities in which the probability of each outcome affects the final expected values. Then, I show how this argument entails there is no good reason to think that there are very many non-resistant non-believers. (shrink)

Gdzieś w ciemnej uliczce... Bandyta: Ej ty, dawaj portfel! Pascal: A niby dlaczego miałbym to zrobić? Bandyta: Bo w przeciwnym razie cię zastrzelę. Pascal: Ale przecież nie masz broni. Bandyta: A niech to! Wiedziałem, że zapomniałem o czymś. Pascal: No to zapomnij też o moim portfelu. Miłego wieczoru. Bandyta: Stój! Pascal: Co znowu? Bandyta: Jest interes do zrobienia... Co ty na to, żebyś jednak oddał mi portfel? W zamian obiecuję przyjść do ciebie jutro i dać ci dwukrotność kwoty, którą w (...) nim masz. Nieźle, co? 200 procent zwrotu z inwestycji w 24 godziny. (shrink)

According to conventionalist accounts of personal identity, persons are constituted in part by practices and attitudes of certain sorts of care. In this paper, we concentrate on the most well-developed and defended version of conventionalism currently on offer (namely, that proposed by David Braddon-Mitchell, Caroline West, and Kristie Miller) and discuss how the conventionalist appears forced either (1) to accept arbitrariness concerning from which perspective to judge one's survival or (2) to maintain egalitarianism at the cost of making “transfiguring” decisions (...) such as Pascal's Wager rationally intractable. We consider three ways the egalitarian conventionalist could make these choices tractable and show that each one comes at significant cost to the view. We end the paper by considering whether accepting arbitrariness would be a better move for the conventionalist and conclude that, even here, she runs the risk of transfiguring choices being rationally intractable. (shrink)

Using “brute reason” I will show why there can be only one valid interpretation of probability. The valid interpretation turns out to be a further refinement of Popper’s Propensity interpretation of probability. Via some famous probability puzzles and new thought experiments I will show how all other interpretations of probability fail, in particular the Bayesian interpretations, while these puzzles do not present any difficulties for the interpretation proposed here. In addition, the new interpretation casts doubt on some concepts often taken (...) as basic and unproblematic, like rationality, utility and expectation. This in turn has implications for decision theory, economic theory and the philosophy of physics. (shrink)

Lankford’s (2013) essential empirical argument, which is based on evidence such as psychological autopsies, is that suicide attacks are caused by suicidality. By operationalizing this causal claim in a hypothetical experiment, I show the claim to be provable, and I contend that its truth is supported by Lankford’s data. However, I question the success of his follow-on arguments about beauty and goodness.

Pascal’s Wager holds that one has pragmatic reason to believe in God, since that course of action has infinite expected utility. The mixed strategy objection holds that one could just as well follow a course of action that has infinite expected utility but is unlikely to end with one believing in God. Monton (2011. Mixed strategies can’t evade Pascal’s Wager. Analysis 71: 642–45.) has argued that mixed strategies can’t evade Pascal’s Wager, while Robertson (2012. Some mixed strategies can evade Pascal’s (...) Wager: a reply to Monton. Analysis 72: 295–98.) has argued that Monton is mistaken. We show that Monton is correct, highlight the crucial assumptions that he relies on, and shed some light on the role of mixed strategies in decision theory. (shrink)

The Many Gods Objection (MGO) is widely viewed as a decisive criticism of Pascal’s Wager. By introducing a plurality of hypotheses with infinite expected utility into the decision matrix, the wagerer is left without adequate grounds to decide between them. However, some have attempted to rebut this objection by employing various criteria drawn from the theological tradition. Unfortunately, such defenses do little good for an argument that is supposed to be an apologetic aimed at atheists and agnostics. The purpose of (...) this paper is to offer a defensive strategy of a different sort, one more suited to the Wager’s apologetic aim and status as a decision under ignorance. Instead of turning to criteria independent of the Wager, it will be shown that there are characteristics already built into its decision theoretic structure that can be used to block many categories of theological hypotheses including MGO’s more outrageous “cooked-up” hypotheses and “philosophers’ fictions”. -/- Please note that there are editorial errors in the published version. They have been corrected in the attached. (shrink)

I defend Pascal's Wager from a particular way of evading it, the mixed strategy approach. The mixed strategies approach suggests that Pascal's Wager does not obligate one to believe in God, because one can get the same infinite expected utility from other strategies besides the strategy of believing in God. I will show that while there's nothing technically wrong with the mixed strategy approach, rationality requires it to be applied in such a way that Pascal's Wager doesn't lose any force.

Among recent objections to Pascal's Wager, two are especially compelling. The first is that decision theory, and specifically the requirement of maximizing expected utility, is incompatible with infinite utility values. The second is that even if infinite utility values are admitted, the argument of the Wager is invalid provided that we allow mixed strategies. Furthermore, Hájek has shown that reformulations of Pascal's Wager that address these criticisms inevitably lead to arguments that are philosophically unsatisfying and historically unfaithful. Both the objections (...) and Hájek's philosophical worries disappear, however, if we represent our preferences using relative utilities rather than a one-place utility function. Relative utilities provide a conservative way to make sense of infinite value that preserves the familiar equation of rationality with the maximization of expected utility. They also provide a means of investigating a broader class of problems related to the Wager. (shrink)

I argue that a version of Pascal's Wager applies to the persistent vegetative state with sufficient force that it ought to part of advance directives.

This paper is a critical review of Howard Sobel’s ’Logic and Theism’. I discuss his analyses of ontological arguments, cosmological arguments, teleological arguments, and arguments from evil, and comment upon his accounts of Pascal’s wager and Hume on miracles. My overall judgment is that this is the very best book on arguments about the existence of God that has yet appeared.

Arthur Falk has proposed a new construal of faith according to which it is not a mere species of belief, but has essential components in action. This twist on faith promises to resurrect Pascal’s Wager, making faith compatible with reason by believing as the scientist but acting as the theist. I argue that Falk’s proposal leaves religious faith in no better shape; in particular, it merely reframes the question in terms of rational desires rather than rational beliefs.

How were reliable predictions made before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? The book examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. Also included are the problem of induction before Hume, design arguments for (...) the existence of God, and theories on how to evaluate scientific and historical hypotheses. It is explained how Pascal and Fermat's work on chance arose out of legal thought on aleatory contracts. The book interprets pre-Pascalian unquantified probability in a generally objective Bayesian or logical probabilist sense. (shrink)

Pascal’s wager and Leibniz’s theory that this is the best of all possible worlds are latecomers in the Faith-and-Reason tradition. They have remained interlopers; they have never been taken as seriously as the older arguments for the existence of God and other themes related to faith and reason.

In "To Bet The Impossible Bet", Harmon Holcomb III argues: (i) that Pascal's wager is structurally incoherent; (ii) that if it were not thus incoherent, then it would be successful; and (iii) that my earlier critique of Pascal's wager in "On Rescher On Pascal's Wager" is vitiated by its reliance on "logicist" presuppositions. I deny all three claims. If Pascal's wager is "incoherent", this is only because of its invocation of infinite utilities. However, even if infinite utilities are admissible, the (...) wager is defeated by the "many gods" and "many wagers" objections. Moreover, these objections do not rely on mistaken "logicist" presuppositions: atheists and agnostics traditionally and typically hold that they have no more--or at any rate, hardly any more--reason to believe in the traditional Christian God than they have to believe in countless alternative deities. (shrink)

In Pascal's Wager: A Study Of Practical Reasoning In Philosophical Theology ,[1] Nicholas Rescher aims to show that, contrary to received philosophical opinion, Pascal's Wager argument is "the vehicle of a fruitful and valuable insight--one which not only represents a milestone in the development of an historically important tradition of thought but can still be seen as making an instructive contribution to philosophical theology".[2] In particular, Rescher argues that one only needs to adopt a correct perspective in order to see (...) that Pascal's Wager argument is a good argument. Moreover, there seems to be a certain amount of contemporary support for Rescher's claim that Pascal's Wager argument can be seen to be a good argument when properly construed .[3] However, despite this recent trend to adopt a more sympathetic stance towards Pascal's Wager argument, I propose to defend the traditional view that Pascal's Wager argument is almost entirely worthless--at least from the theological standpoint. (No doubt, it has historical significance from the standpoint of decision theory; but that's a separate matter.). (shrink)