Suppose we wish to decide which of a pair of actions has better consequences in a case in which both actions result in infinite utility. Peter Vallentyne and others have proposed that one action has better consequences than a second if there is a time after which the cumulative utility of the first action always outstrips the cumulative utility of the second. I argue against this principle, in particular I show how cases may arise in which (...) up to any point of time action a1 produces more utility than action a2, yet for each individual involved a2 produces more utility. (shrink)

In 1991 Mark Nelson argued that if time is infinitely long towards the future, then under certain easily met conditions traditional utilitarianism is unable to discriminate among actions. For under these conditions, each action produces the same infinite amount of utility, and thus it seems that utilitarianism must judge all actions permissible, judge all actions impermissible, or remain completely silent. In response to this criticism of utilitarianism, I argued that utilitarianism had the resources for dealing with at least (...) some cases of infinite utility. More specifically, I defended the following principle as being part of the "spirit" of traditional (total) utilitarianism: -/- PMU*: An action a1 has better consequences than an action a2 if and only if there is a time t such that for any later time t' the cumulative amount of utility produced by a1 up to t' is greater than that produced by action a2 up to t'. -/- For the usual finite cases, this principle agrees exactly with the standard approach of comparing totals. For infinite cases, however, it can distinguish among two actions each of which produces an infinite amount of utility. It says, for example, that an action that produces 2 units of utility at each time has better consequences than an action that produces 1 unit of utility at each time -- even though they both produce the same infinite amount of utility. Of course, cases where one action produces more utility than a second at each time are going to be rather rare. PMU* is not limited to such cases, however; it also has bite in cases such as the following: -/- Time a1: 1, 1, 1, 1, 2, 2, 2, 2, 2, ... a2: 3, 3, 3, 3, 1, 1, 1, 1, 1, ... -/- For although the cumulative utility of a1 up to and including the fourth time is only 4 utiles and a2's is 12, a1 starts catching up after that, ties a2 in the 12th time, and stays ahead after that. Thus, PMU* judges a1 as having better consequences than a2. I claimed that PMU* better captured the "spirit" of traditional utilitarianism than the usual sum total view -- on the grounds that almost anyone inclined to defend traditional utilitarianism would, upon reflection, want to hold, for example, that 2 utiles at every time is indeed better than 1 utile at each time. Two responses to my view have since been given in this journal. Luc Van Liederkerke has argued that PMU* violates an anonymity (neutrality, impartiality) condition that is central to utilitarianism. And James Cain has argued that PMU* is not person-centered, and that being so is central to utilitarianism. Here I will reply to the criticisms. Basically, I will acknowledge that each is right about an important point, but deny that this establishes that the core idea underlying PMU* is less plausible than the standard sum-ranking versions of utilitarianism. (shrink)

Suppose that time is infinitely long towards the future, and that each feasible action produces a finite amount of utility at each time. Then, under appropriate conditions, each action produces an infinite amount of utility. Does this mean that utilitarianism lacks the resources to discriminate among such actions? Since each action produces the same infinite amount of utility, it seems that utilitarianism must judge all actions permissible, judge all actions impermissible, or remain completely silent. If (...) the future is infinite, that is, the prospects for utilitarianism look bleak. (shrink)

The note addresses the problem of how utilitarianism and other finitely additive theories of value should evaluate infinitely long utility streams. We use the axiomatic approach and show that finite anonymity does not apply in an infinite framework. A stronger anonymity demand (fixed step anonymity) is proposed and motivated. Finally, we construct an ordering criterion that combines fixed step anonymity and strong monotonicity.

Traditional act utilitarianism judges an action permissible just in case it produces as much aggregate utility as any alternative. It is often supposed that utilitarianism faces a serious problem if the future is infinitely long. For in that case, actions may produce an infinite amount of utility. And if that is so for most actions, then utilitarianism, it appears, loses most of its power to discriminate among actions. For, if most actions produce an infinite amount of (...) utility, then few actions produce non-maximal utility, and so most actions are permissible.1 I will argue that potentially infinite futures create no major problems for utilitarianism. Utilitarianism has, I argue, the resources to distinguish among actions all of which produce infinite amounts of utility -- judging some permissible and some impermissible. For brevity of expression I will focus on act utilitarianism, but all the points apply equally well to many other traditional forms of utilitarianism. (shrink)

An infinite future thus threatens to paralyze utilitarianism. Utilitarians need principled ways to determine which possible infinite futures are better or worse. In this article, I discuss a recent suggestion of Peter Vallentyne and Shelly Kagan. I conclude that the best way forward for utilitarians is, in fact, to by-pass the infinite utility debate altogether. (edited).

What if human joy went on endlessly? Suppose, for example, that each human generation were followed by another, or that the Western religions are right when they teach that each human being lives eternally after death. If any such possibility is true in the actual world, then an agent might sometimes be so situated that more than one course of action would produce an infinite amount of utility. Deciding whether to have a child born this year rather than (...) next is a situation wherein an agent may face several alternatives whose effects could well ramify endlessly on such suppositions, for the child born this year would be a different person—one who preferred different things, performed different actions, and had different descendants—from a child born next year. It has recently been suggested that traditional utilitarianism stumbles on such cases of infinite utility. Specifically, utilitarianism seems to require, for its application, that all experience of pleasure and pain cease at some time in the future or asymptotically approach zero.2 If neither of these conditions holds, then the utility produced by each of two alternative actions may turn out to be infinite, and utilitarianism thus loses its ability to discriminate morally between them. (shrink)

We examine the range of anonymity that is compatible with a Paretian social welfare relation (SWR) on infinite utility streams. Three alternative coherence properties of an SWR are considered, namely, acyclicity, quasi-transitivity, and Suzumura consistency. For each case, we show that a necessary and sufficient condition for a set of permutations to be the set of permissible permutations of some Paretian SWR is given by the cyclicity of permutations and a weakening of group structure. Further, for each case (...) of coherence property, we show that the extended Pareto rule is the least element of the class of anonymous Paretian SWRs. (shrink)

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)

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)

Theories that use expected utility maximization to evaluate acts have difficulty handling cases with infinitely many utility contributions. In this paper I present and motivate a way of modifying such theories to deal with these cases, employing what I call “Direct Difference Taking”. This proposal has a number of desirable features: it’s natural and well-motivated, it satisfies natural dominance intuitions, and it yields plausible prescriptions in a wide range of cases. I then compare my account to the most (...) plausible alternative, a proposal offered by Arntzenius :31–58, 2014). I argue that while Arntzenius’s proposal has many attractive features, it runs into a number of problems which Direct Difference Taking avoids. (shrink)

How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that worlds are made better if we increase the value in every region of space and time, or that they are made (...) better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most. (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such theory. (...) -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

The problem of how to rationally aggregate probability measures occurs in particular when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single ‘aggregate belief system’ and when an individual whose belief system is compatible with several probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory. We investigate this problem by first recalling some negative results from preference and (...) judgment aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected utility preferences. We describe how McConway’s :410–414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed. (shrink)

One of the standard approaches to the metaphysics of personal identity has some counter-intuitive ethical consequences when combined with maximising consequentialism and a plausible doctrine about aggregation of consequences. This metaphysical doctrine is the so-called ‘multiple occupancy’ approach to puzzles about fission and fusion. It gives rise to a new version of the ‘utility monster’ problem, particularly difficult problems about infinite utility, and a new version of a Parfit-style ‘repugnant conclusion’. While the article focuses on maximising consequentialism (...) for simplicity, the problems demonstrated apply more widely to a range of ethical views, especially flavours of consequentialism. This article demonstrates how these problems arise, and discusses a number of options available in the light of these problems for a consequentialist tempted by a multiple occupancy metaphysics. (shrink)

One of the standard approaches to the metaphysics of personal identity has some counter‐intuitive ethical consequences when combined with maximising consequentialism and a plausible (though not uncontroversial) doctrine about aggregation of consequences. This metaphysical doctrine is the so‐called ‘multiple occupancy’ approach to puzzles about fission and fusion. It gives rise to a new version of the ‘utility monster’ problem, particularly difficult problems about infinite utility, and a new version of a Parfit‐style ‘repugnant conclusion’. While the article focuses (...) on maximising consequentialism for simplicity, the problems demonstrated apply more widely to a range of ethical views, especially flavours of consequentialism. This article demonstrates how these problems arise, and discusses a number of options available in the light of these problems for a consequentialist tempted by a multiple occupancy metaphysics. (shrink)

A common argument for atheism runs as follows: God would not create a world worse than other worlds he could have created instead. However, if God exists, he could have created a better world than this one. Therefore, God does not exist. In this paper I challenge the second premise of this argument. I argue that if God exists, our world will continue without end, with God continuing to create value-bearers, and sustaining and perfecting the value-bearers he has already created. (...) Given this, if God exists, our world—considered on the whole—is infinitely valuable. I further contend that this theistic picture makes our world's value unsurpassable. In support of this contention, I consider proposals for how infinitely valuable worlds might be improved upon, focusing on two main ways—adding value-bearers and increasing the value in present value-bearers. I argue that neither of these can improve our world. Depending on how each method is understood, either it would not improve our world, or our world is unsurpassable with respect to it. I conclude by considering the implications of my argument for the problem of evil more generally conceived. (shrink)

We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as 'Trumped' (Arntzenius and McCarthy 1997), 'Rouble trouble' (Arntzenius and Barrett 1999), 'The airtight Dutch book' (McGee 1999), and 'The two envelopes puzzle' (Broome 1995). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not (...) be countably additive. The resolution also shows that when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin. Finally, the resolution reveals a new way in which the ability to make binding commitments can save perfectly rational agents from sure losses. (shrink)

Aggregative moral theories face a series of devastating problems when we apply them in a physically realistic setting. According to current physics, our universe is likely _infinitely large_, and will contain infinitely many morally valuable events. But standard aggregative theories are ill-equipped to compare outcomes containing infinite total value so, applied in a realistic setting, they cannot compare any outcomes a real-world agent must ever choose between. This problem has been discussed extensively, and non-standard aggregative theories proposed to overcome (...) it. This paper addresses a further problem of similar severity. Physics tells us that, in our universe, how remotely in time an event occurs is _relative_. But our most promising aggregative theories, designed to compare outcomes containing infinitely many valuable events, are sensitive to how remote in time those events are. As I show, the evaluations of those theories are then relative too. But this is absurd; evaluations of outcomes must be absolute. So we must reject such theories. Is this objection fatal for all aggregative theories, at least in a relativistic universe like ours? I demonstrate here that, by further modifying these theories to fit with the physics, we can overcome it. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

Consider a subjective expected utility preference relation. It is usually held that the representations which this relation admits differ only in one respect, namely, the possible scales for the measurement of utility. In this paper, I discuss the fact that there are, metaphorically speaking, two additional dimensions along which infinitely many more admissible representations can be found. The first additional dimension is that of state-dependence. The second—and, in this context, much lesser-known—additional dimension is that of act-dependence. The simplest (...) implication of their usually neglected existence is that the standard axiomatizations of subjective expected utility fail to provide the measurement of subjective probability with satisfactory behavioral foundations. (shrink)

Some environmental ethicists and economists argue that attributing infinite value to the environment is a good way to represent an absolute obligation to protect it. Others argue against modelling the value of the environment in this way: the assignment of infinite value leads to immense technical and philosophical difficulties that undermine the environmentalist project. First, there is a problem of discrimination: saving a large region of habitat is better than saving a small region; yet if both outcomes have (...) infinite value, then decision theory prescribes indifference. Second, there is a problem of swamping probabilities: an act with a small but positive probability of saving an endangered species appears to be on par with an act that has a high probability of achieving this outcome, since both have infinite expected value. Our paper shows that a relative concept of infinite value can be meaningfully defined, and provides a good model for securing the priority of the natural environment while avoiding the failures noted by sceptics about infinite value. Our claim is not that the relative infinity utility model gets every detail correct, but rather that it provides a rigorous philosophical framework for thinking about decisions affecting the environment. (shrink)

The logical treatment of the nature of religious belief (here I will concentrate on belief in Christianity) has been distorted by the acceptance of a false dilemma. On the one hand, many (e.g., Braithwaite, Hare) have placed the significance of religious belief entirely outside the realm of intellectual cognition. According to this view, religious statements do not express factual propositions: they are not made true or false by the ways things are. Religious belief consists in a certain attitude toward the (...) world, life, or other human beings, or in what sorts of things one values. On the other hand, others (such as Swinburne, 1981, Chapers 1 and 4) have taken religious belief to include (at least) being certain of the truth of particular factual religious propositions. The strength of a person's religious belief is identified with his degree of confidence in the truth of those propositions, measured by the "subjective probability" which those propositions have for that person. I propose a third alternative, according to which, (1) contrary to the first view, religious belief does involve a relation to factual religious propositions, such as that God exists, that Jesus was God and man, etc., -- propositions which are made true or false by the way things actually are -- but, (2) contrary to the second view, the strength of religious belief is measured, not by the degree of one's confidence1 in the truth of these propositions, but rather by the way in which the value or desirability to oneself of the various ways the world could be is affected by their including or not including the truth of these religious propositions. Thus, religious belief does consist in what one values or prizes, not in what.. (shrink)

In the context of worlds with infinitely many bearers of utility, we argue that several collections of natural Utilitarian principles--principles which are certainly true in the classical finite Utilitarian context and which any Utilitarian would find appealing--are inconsistent.

Recently in the philosophical literature there has been some effort made to understand the proper application of the theory of utilitarianism to worlds in which there are infinitely many bearers of utility. Here, we point out that one of the best, most inclusive principles proposed to date contradicts fundamental utilitarian ideas, such as the idea that adding more utility makes a better world.

This paper discusses random utility representations of the Luce model. Earlier works, such as McFadden, Yellott, and Strauss have discussed random utility representations under the assumption that utilities are additively separable in a deterministic and a random part. Under various conditions, they have established that a separable and independent random utility representation exists if and only if the random terms are type III extreme value distributed. This paper analyzes independent random utility representations without the separability condition (...) and with an infinite universal set of alternatives. Under these assumptions, it turns out that the most general random utility representation of the Luce model is a utility function that is an arbitrary strictly increasing transformation of a separable utility function with extreme value distributed random terms. (shrink)

Cerreia-Vioglio et al. :341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory, is employed.

Population ethics studies the tradeoff between the total number of people who will ever live, and their quality of life. But widely accepted theories in modern cosmology say that spacetime is probably infinite. In this case, its population is also probably infinite, so the quantity/quality tradeoff of population ethics is no longer meaningful. Instead, we face the problem of how to ethically evaluate an infinite population of people dispersed throughout time and space. I argue that axiologies based (...) on _spatiotemporal Cesàro average utility_ are the most appropriate way to make this evaluation, because their unique properties make them superior to other axiologies that have been proposed. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group (...) R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions on scoring rules on finite sample spaces to argue for both probabilism—the doctrine that credences ought to satisfy the axioms of probabilism—and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common (...) condition in the arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is not at present known what is to be said about probabilism in this case. (shrink)

Ion Soteropoulos reconciles the contradiction between the finite and infinite and transforms this reconciliation into the founding principle of motion. This book will appeal to readers interested in the logical mechanics of the physical universe, the hidden powers of our finite brain, and the utility of robots in the future.

Where there are infinitely many possible [equiprobable] basic states of the world, a standard probability function must assign zero probability to each state—since any finite probability would sum to over one. This generates problems for any decision theory that appeals to expected utility or related notions. For it leads to the view that a situation in which one wins a million dollars if any of a thousand of the equally probable states is realized has an expected value of zero (...) (since each such state has probability zero). But such a situation dominates the situation in which one wins nothing no matter what (which also has an expected value of zero), and so surely is more desirable. I formulate and defend some principles for evaluating options where standard probability functions cannot strictly represent probability—and in particular for where there is an infinitely spread, uniform distribution of probability. The principles appeal to standard probability functions, but overcome at least some of their limitations in such cases. (shrink)

Popper's well-known demarcation criterion has often been understood to distinguish statements of empirical science according to their logical form. Implicit in this interpretation of Popper's philosophy is the belief that when the universe of discourse of the empirical scientist is infinite, empirical universal sentences are falsifiable but not verifiable, whereas the converse holds for existential sentences. A remarkable elaboration of this belief is to be found in Watkins's early work on the statements he calls “all-and-some,” such as: “For every (...) metal there is a melting point.” All-and-some statements are both universally and existentially quantified in that order. Watkins argued that AS should be regarded as both nonfalsifiable and nonverifiable, for they partake in the logical fate of both universal and existential statements. This claim is subject to the proviso that the bound variables are “uncircumscribed” ; i.e., that the universe of discourse is infinite. (shrink)

This paper considers the trade-off between unanimity and anonymity in collective decision-making with an infinite population. This efficiency-equity trade-off is a fundamental difficulty in making a normative judgment in a conflict between generations. In particular, it is known that this trade-off is quite sensitive in the formulation of unanimity axioms. In this study, we consider the trade-off in a preference-aggregation framework instead of the standard utility-aggregation framework. We show that there exists a social welfare function that satisfies I-strong (...) Pareto, independence of irrelevant alternatives, and finite anonymity. This contrasts with an impossibility result in the standard utility-aggregation framework, and this means that the trade-off is also sensitive for background frameworks of aggregations. (shrink)

We extend de Finetti’s (1974) theory of coherence to apply also to unbounded random variables. We show that for random variables with mandated infinite prevision, such as for the St. Petersburg gamble, coherence precludes indifference between equivalent random quantities. That is, we demonstrate when the prevision of the difference between two such equivalent random variables must be positive. This result conflicts with the usual approach to theories of Subjective Expected Utility, where preference is defined over lotteries. In addition, (...) we explore similar results for unbounded variables when their previsions, though finite, exceed their expected values, as is permitted within de Finetti’s theory. In such cases, the decision maker’s coherent preferences over random quantities is not even a function of probability and utility. One upshot of these findings is to explain further the differences between Savage’s theory (1954), which requires bounded utility for non-simple acts, and de Finetti’s theory, which does not. And it raises a question whether there is a theory that fits between these two. (shrink)

This paper proposes a class of independence axioms for simple acts. By introducing the E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}-cominimum independence axiom that is stronger than the comonotonic independence axiom but weaker than the independence axiom, we provide a new axiomatization theorem of simple acts within the framework of Choquet expected utility. Furthermore, in order to provide the axiomatization of simple acts, we generalize Kajii et al. into an infinite state space. Our axiomatization (...) theorem relates Choquet expected utility to multi-prior expected utility through the core of a capacity that is explicitly derived within our framework. Our result in this paper also derives Gilboa, Eichberger and Kelsey, and Rohde as a corollary. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)