(forthcoming in Oxford Readings in the Philosophy of Religion).

In his Truth and Probability (1926), Frank Ramsey provides foundations for measures of degrees of belief in propositions and preferences for worlds. Nonquantitative conditions on preferences for worlds, and gambles for worlds and certain near-worlds, are formulated which he says insure that a subject's preferences for worlds are represented by numbers, world values. Numbers, for his degrees of belief in propositions, probabilities, are then defined in terms of his world values. Ramsey does not also propose definitions of desirabilities for propositions, (...) though he is in a position to do this. Given his measures for probabilities of propositions and values of worlds, he can frame natural definitions for both evidential and causal desirabilities that would measure respectively the welcomeness of propositions as items of news, and as facts. His theory is neutral with respect to the evidential/causal division. In the present paper, as Ramsey's foundations are explained, several problems and limitations are noted. Their distinctive virtue – their evidential/causal neutrality – is demonstrated. Comparisons are made with other foundational schemes, and a perspective is recommended from which nonquantitative foundations are not the be all for quantitative theories of ideal preferences and credences. (shrink)

This is a wide-ranging 2004 book about arguments for and against beliefs in God. The arguments for the belief are analysed in the first six chapters and include ontological arguments from Anselm to Gödel, the cosmological arguments of Aquinas and Leibniz, and arguments from evidence for design and miracles. The next two chapters consider arguments against belief. The last chapter examines Pascalian arguments for and against belief in God. There are discussions of Cantorian problems for omniscience, of challenges to divine (...) omnipotence, and of the compatibility of everlasting complete knowledge of the world with free-will. There are appendices that present formal proofs in a system for quantified modal logic, a theory of possible worlds, notes on Cantorian set theory, and remarks concerning non-standard hyperreal numbers. This book will be a valuable resource for philosophers of religion and theologians and will interest logicians and mathematicians as well. (shrink)

J. Howard Sobel has long been recognized as an important figure in philosophical discussions of rational decision. He has done much to help formulate the concept of causal decision theory. In this volume of essays Sobel explores the Bayesian idea that rational actions maximize expected values, where an action's expected value is a weighted average of its agent's values for its possible total outcomes. Newcomb's Problem and The Prisoner's Dilemma are discussed, and Allais-type puzzles are viewed from (...) the perspective of causal world Bayesianism. The author establishes principles for distinguishing options in decision problems, and studies ways in which perfectly rational causal maximizers can be capable of resolute choices. Sobel also views critically Gauthier's revisionist ideas about maximizing rationality. This collection will be a desideratum for anyone working in the field of rational choice theory, whether in philosophy, economics, political science, psychology or statistics. Howard Sobel's work in decision theory is certainly among the most important, interesting and challenging that is being done by philosophers. (shrink)

A BAYESIAN ARTICULATION OF HUME’S VIEWS IS OFFERED BASED ON A FORM OF THE BAYES-LAPLACE THEOREM THAT IS SUPERFICIALLY LIKE A FORMULA OF CONDORCET’S. INFINITESIMAL PROBABILITIES ARE EMPLOYED FOR MIRACLES AGAINST WHICH THERE ARE ’PROOFS’ THAT ARE NOT OPPOSED BY ’PROOFS’. OBJECTIONS MADE BY RICHARD PRICE ARE DEALT WITH, AND RECENT EXPERIMENTS CONDUCTED BY AMOS TVERSKY AND DANIEL KAHNEMAN ARE CONSIDERED IN WHICH PERSONS TEND TO DISCOUNT PRIOR IMPROBABILITIES WHEN ASSESSING REPORTS OF WITNESSES.

Two partition-theorems are proved for a particular causal decision theory. One is restricted to a certain kind of partition of circumstances, and analyzes the utility of an option in terms of its utilities in conjunction with circumstances in this partition. The other analyzes an option's utility in terms of its utilities conditional on circumstances and is quite unrestricted. While the first form seems more useful for applications, the second form may be of theoretical importance in foundational exercises. Comparisons are made (...) with theorems of Richard Jeffrey, Brad Armendt, and Peter Fishburn. (shrink)

According to a familiar argument, iterated prisoner's dilemmas of known finite lengths resolve for ideally rational and well-informed players: They would defect in the last round, anticipate this in the next to last round and so defect in it, and so on. But would they anticipate defections even if they had been cooperating? Not necessarily, say recent critics. These critics "lose" the backward-induction paradox by imposing indicative interpretations on rationality and information conditions. To regain it I propose subjunctive interpretations. To (...) solve it I stress that implications for ordinary imperfect players are limited. (shrink)

Rational actions reflect beliefs and preferences in certain orderly ways. The problem of theory is to explain which beliefs and preferences are relevant to the rationality of particular actions, and exactly how they are relevant. One distinction of interest here is between an agent's beliefs and preferences just before an action's time, and his beliefs and preferences at its time. Theorists do not agree about the times of beliefs and desires that are relevant to the rationality of action. Another distinction (...) is between actions that would, in one sense or another, maximize expected utilities given relevant beliefs and desires, and actions, decisions for which would, in one sense or another, be stable. There is disagreement about the relevance of these things to the rationality of actions. Here, in the first part below, several possible positions on these issues are explained and compared. In the second part, I contrast perspectives on these issues, and comment on arguments that might be brought against the position I favor. In the third part, I restate and elaborate on this position. (shrink)

A person who does not have good intellectual reasons for believing in God can, depending on his probabilities and values for consequences of believing, have good practical reasons. Pascalian wagers founded on a variety of possible probability/value profiles are examined from a Bayesian perspective central to which is the idea that states and options are pragmatically reasonable only if they maximize subjective expected value. Attention is paid to problems posed by representations of values by Cantorian infinities. An appendix attends to (...) Robinsonian hyperreals. Another appendix presents for comparison Newcomb's Problem and a problem in some ways like it suggested, I think, by ideas of John Calvin. (shrink)

Donald Regan's book is a study of forms of utilitarian theory and of properties to which these forms may aspire. He deals exclusively with “fully objective” theories that seek to systematize and set conditions for obligations that an agent is supposed to lie under as a consequence of features of situations in which he finds himself, with no dependence on his possibly defective beliefs, attitudes, and values. Metaethical questions concerning the nature of these obligations are not raised. And practical problems (...) that could be encountered in attempts to apply these theories receive very little attention. They are said to lie outside the main project of the book which is to explain and then to circumvent a certain apparent logical impasse that utilitarianisms ca. 1980 are supposed to confront. The book consists for the most part of abstract, often very long and difficult, arguments, and detailed elaborations of a small number of minimally specified, highly artificial cases of the sort now common in semi-formal utilitarian theory. In the present note, while leaving unquestioned and unstated the presuppositions of Regan's book, I first outline its central two-stage “argument”. Next I explain several distinctions important to issues it addresses and relate these distinctions to several of Regan's theses. I then discuss prima facie cases against his main claim, according to which the apparent impasse that utilitarianisms confront is circumvented by a new utilitarian theory designed for the task. (shrink)

The work is a charitable study on what the internationally renowned presenter and author, Howard Sobel, views to be largely the truth about moral thought and talk. Discussions and observations from David Humes own writings oftentimes reinforce and elaborate the authors notions and there is an assertive attempt to weave logical thinking into the book. Applications to such mathematical concepts as game theory, decision-making, and conditionals are dispersed throughout so as to enlighten the theory behind the ideas.

An example shows that 'pairwise preferences' (certain hypothetical choices) can cycle even when rational. General considerations entail that preferences tout court (certain relations of actual valuations) cannot cycle. A world-bayesian theory is explained that accommodates these two kinds of preference, and a theory for rational actions that would have them maximize and be objects of ratifiable choices. It is observed that choices can be unratifiable either because of troublesome credences or because of troublesome preferences. An appendix comments on a third (...) way in which efforts to maximize can be frustrated. (shrink)

To resolve putative liar paradoxes it is sufficient to attend to the distinction between liar-sentences and the propositions they would express, and to exercise the option of turning would-be deductions of paradox (of contradictions) into reductions of the existence of those propositions. Defending the coherence of particular resolutions along these lines, leads to recognition of the non-extensionality of some liar-sentences. In particular, it turns out that exchanges of terms for identicals in the open-sentence '- does not expression a true proposition' (...) are not invariably truth-preserving because they are not invariably proposition-expression preserving. All of this recommends propositions as fruitful subjects of interesting renewed research. (shrink)

David Gauthier stages a competition between two arguments, each of which purports to decide once for all transparent agents which is best, being a straight or being a constrained maximizer. The first argument, which he criticizes and rejects, is for the greater utility, on a certain weak assumption, of straight maximization for all transparent agents. The second, which he endorses, is for the greater utility on the same weak assumption of constrained maximization for all transparent agents.In Section I, Gauthier’s account (...) of constrained maximization is presented, and his use in the two arguments of the idea of choosing a disposition to choose actions is noted. Section II is about the unfortunate argument that Gauthier criticizes. This argument is flawed in ways additional to those he notices, but a less ambitious form of reasoning can, for individuals whose probabilities and values are right, be good for the greater expected utility of straight maximization. Section III takes up the argument that Gauthier endorses and maintains that it is wrong in a way specific to it as well as in ways closely related to all of the first argument’s noted flaws. An Appendix features a three-person prisoners’ dilemma and includes demonstrations of principal conclusions reached in the body of this paper. (shrink)

When causal decision theory was created in the 1970s, access to Howard Sobel’s contribution was available only in a narrowly circulated mimeographed manuscript. After some time, he allowed his ideas to appear in the form of articles. Here we finally have a book length exposition on Sobel’s causal Bayesian point of view consisting of collected, revised, and amplified papers spanning a period of twenty years.

Moral properties would supervene upon non-moral properties and be conceptually autonomous. That, according to Simon Blackburn, would make them if not impossible at least mysterious, and evidence for them best explained by theorists who say they are not real. In fact moral properties would not challenge in ways Blackburn has contended. There is, however, something new that can be gathered from his arguments. What would the supervenience of moral properties and their conceptual autonomy from at least total non-moral properties entail (...) not only for Intuitionists, who ‘knew this all along,’ but for all moral realists, that there are synthetic necessary moral principles? There is for all moral realists the problem of explaining ‘what in the world’ makes possible these necessities. (shrink)

This paper is about David Gauthier’s concept of constrained maximization. Attending to his most detailed and careful account, I try to say how constrained maximization works, and how it might be changed to work better. In section I, that detailed account is quoted along with amplifying passages. Difficulties of interpretation are explained in section II. An articulation, a spelling out, of Gauthier's account is offered in section III to deal with these difficulties. Next, in section IV, constrained maximization thus articulated (...) is tested on several choice problems and shown to be seriously wanting. It appears that there are prisoners’ dilemmas in which constrained maximizers would not cooperate to mutual advantage, but would interact sub-optimally just as straight-maximizers would. ‘Coordination problems’ are described with which constrained maximizers might, especially if transparent to one another, not be able to cope–problems in which they might not be able to make up their minds to do anything at all. And I prove that there are prisoners’ dilemmas that, though possible for real agents and for straight maximizers, are not possible for constrained maximizers, so that agents’ internalising dispositions of constrained maximization could not be of help in connection with such possibly impending dilemmas. Taking constrained maximization as it stands, there are many problems for which it does not afford the ‘moral solutions’ with which Gauthier would have it replace Hobbesian political ones. After displaying these shortcomings of constrained maximization as presently designed, I sketch, in section V, possible revisions that would reduce them, stressing that these revisions would not be cost-free. Whether finishing the job of fixing up and making precise constrained maximization would be worth the considerable trouble it would involve lies beyond the issues taken up in this paper. So, of course, do substantive comparisons of constrained maximization, perfected and made precise, and straight maximization. (shrink)

It is plausible that Newcomb problems in which causal maximizers and evidential maximizers would do different things would not be possible for ideal maximizers who are attentive to metatickles. An objection to Eells’s first argument for this makes welcome a second. Against it I argue that even ideal evidential and causal maximizers would do different things in some non-dominance Newcomb problems; and that they would hope for different things in some third-person and non-action problems, which is relevant if a good (...) theory of rational choices of acts should fit smoothly into a good theory of rational desires for facts. (shrink)

Maximizers in isolated Prisoner's Dilemmas are doomed to frustration. But in Braybrooke's view maximizers might do better in a series, securing Pareto-optimal arrangements if not from the very beginning, at least eventually. Given certain favourable special conditions, it can be shown according to Braybrooke and shown even without question-begging motivational or value assumptions, that in a series of Dilemmas maximizers could manage to communicate a readiness to reciprocate, generate thereby expectations of reciprocation, and so give rise to optimizing reciprocations which, (...) in due course, would reinforce expectations, the net result of all this being an increasingly stable practice to mutual benefit. In this way neighbours could learn to be good neighbours: they could learn to respect each other's property, to engage in reciprocal assistance, and perhaps even to make and keep promises covering a range of activities. So maximizers are, Braybrooke holds, capable of a society of sorts. Out of the ashes they might build something. But even under favourable conditions they could not build much and most conditions would defeat them almost entirely, for many-person Dilemmas whether isolated or in series would be quite beyond them, question-begging motivational assumptions aside. In settings at all like those in which we live, utility maximization assuming only ‘traditional’ motivation is self-defeating without remedy, and most certainly not an adequate basis for social life.' The probable inference from as a conception of individual rationality: for surely truly rational agents would be if not perfectly well designed at least better designed for communal living than utility maximizers would be assuming only ‘traditional’, non-question-begging values or motivation. Or, if this inference is denied and maximization is not challenged as a conception of rationality, we can conclude instead, Braybrooke has more recently suggested, that the advice, ‘Be rational,’ is incomplete, and when addressed to groups should be supplemented by, ‘And be somewhat ‘untraditional’ in your values; respect at least some rules that would make coordination and cooperation possible even in many-person interaction problems.’. (shrink)

Moral properties would supervene upon non‐moral properties and be conceptually autonomous. That, according to Simon Blackburn, would make them if not impossible at least mysterious, and evidence for them best explained by theorists who say they are not real. In fact moral properties would not challenge in ways Blackburn has contended. There is, however, something new that can be gathered from his arguments. What would the supervenience of moral properties and their conceptual autonomy from at least total non‐moral properties entail (...) not only for Intuitionists, who ‘knew this all along,’ but for all moral realists, that there are synthetic necessary moral principles? There is for all moral realists the problem of explaining ‘what in the world’makes possible these necessities. (shrink)

"It is a general maxim...’ That no testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous, than the fact which it endeavors to establish; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior.’" A Bayesian interpretation of the first half is proved as a theorem. (...) A stronger conditional principle, and a biconditional theorem based on it, are substantiated. And a truth that the second part can express is explained. (shrink)

Relations between conditional probabilities, revisions of probabilities in the light of new information, and conditions of ideal rationality are discussed herein. The formal character of conditional probabilities, and their significance for epistemic states of agents is taken up. Then principles are considered that would, under certain conditions, equate rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. And lastly the possibility of kinds of ' books ' against known non-conditionalizers is explored, and the question is (...) taken up, What, if anything, would be wrong with a person against whom such a book could be made? (shrink)

After maintaining in that certain cyclical preferences can be reasonable, the following questions were tabled: “in what circumstances and under what assumptions … these … preferences of George's would turn him into a ”money pump,” and in what circumstances and under what assumptions they would not do that.” Two pumps afford answers. George, who is sufficiently reasonable and well-informed to use backward induction, has, for this reason, nothing to fear from the first pump, but the second, of a nonstandard design (...) by Wlodek Rabinowicz, ruins him just because he is subject to the constraints of backward induction. Could he be helped by extending his options from actions in a single round, to courses of actions starting in that round? Possibly not. He would be safe if he were capable of resolute choices of couses of action. Even so, he is in trouble through no fault of his, if, as I think, not having that capacity, though a misfortune, is not a fault. (shrink)

Abstract: A Liar would express a proposition that is true and not true. A Liar Paradox would, per impossibile, demonstrate the reality of a Liar. To resolve a Liar Paradox it is sufficient to make out of its demonstration a reductio of the existence of the proposition that would be true and not true, and to "explain away" the charm of the paradoxical contrary demonstration. Persuasive demonstrations of the Liar Paradox in this paper trade on allusive scope-ambiguities of English definite (...) descriptions, and can seem confirmed by symbolizations in a Fregean theory in which scopes of definite descriptions are determinate. Symbolizing instead in a Russellian description theory in which alternative scopes are possible reveals that however the scope-ambiguities of the demonstration are settled the result is unsound. (shrink)

There are narrowest bounds for P(h) when P(e) = y and P(h/e) = x, which bounds collapse to x as y goes to 1. A theorem for these bounds -- bounds for probable modus ponens -- entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ’probability kinematics’ or ’Jeffrey-conditioning’ is, (...) and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner -- bounds for probable modus tollens -- registers narrowest bounds for P(not h) when P(not e) = y and P(e/h) = x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ’contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory -- for example, a random-chanc. (shrink)

Moral properties would supervene upon non‐moral properties and be conceptually autonomous. That, according to Simon Blackburn, would make them if not impossible at least mysterious, and evidence for them best explained by theorists who say they are not real. In fact moral properties would not challenge in ways Blackburn has contended. There is, however, something new that can be gathered from his arguments. What would the supervenience of moral properties and their conceptual autonomy from at least total non‐moral properties entail (...) not only for Intuitionists, who ‘knew this all along,’ but for all moral realists, that there are synthetic necessary moral principles? There is for all moral realists the problem of explaining ‘what in the world’makes possible these necessities. (shrink)

It is shown that even if a process of ideal evidential deliberation that paid attention to its own progress would in every case lead to credences that made things probabilistically independent of actions of which they were believed to be causally independent; it would not in every case lead to agreement in the ultimate dictates of evidential and causal decision theories. This point is made by a decision problem in which the action prescribed by causal decision theory is not (as (...) it is in Newcomb's Problem) a dominant action. It is also shown that such non-dominance problems provide decisive objections to Ratificationism. (shrink)