Alexander R. Pruss

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.

Bayesian epistemology has struggled with the problem of regularity: how to deal with events that in classical probability have zero probability. While the cases most discussed in the literature, such as infinite sequences of coin tosses or continuous spinners, do not actually come up in scientific practice, there are cases that do come up in science. I shall argue that these cases can be resolved without leaving the realm of classical probability, by choosing a probability measure that preserves “enough” regularity. This approach also provides a resolution to the McGrew, McGrew and Vestrum normalization problem for the fine-tuning argument.

Classical real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value—namely, zero—to cases that are impossible as well as to cases that are possible. There are three non-classical approaches to probability that can avoid this drawback: full conditional probabilities, qualitative probabilities and hyperreal probabilities. These approaches have been criticized for failing to preserve intuitive symmetries that can be preserved by the classical probability framework, but there has not been a systematic study of the conditions under which these symmetries can and cannot be preserved. This paper fills that gap by giving complete characterizations under which symmetries understood in a certain “strong” way can be preserved by these non-classical probabilities, as well as by offering some results to make it plausible that the strong notion of symmetry here is the right one. Philosophical implications are briefly discussed, but the main purpose of the paper is to offer technical results to help make further philosophical discussion more sophisticated.

It is widely thought that it can be permissible to persuade someone set on a greater evil to commit a lesser evil instead, though the question is not without controversy. I argue that a version of this kind of Principle of Counseling Lesser Evil can be derived from the Principle of Double Effect and some considerations about the way human choices work. As an application, I argue that giving bribes to officials who otherwise would not do their job might be considered a special case of this counseling principle.

It is often loosely said that Ramsey The foundations of mathematics and other logical essays, Routledge and Kegan Paul, Abingdon, pp 156–198, 1931) and de Finetti Studies in subjective probability, Kreiger Publishing, Huntington, 1937) proved that if your credences are inconsistent, then you will be willing to accept a Dutch Book, a wager portfolio that is sure to result in a loss. Of course, their theorems are true, but the claim about acceptance of Dutch Books assumes a particular method of calculating expected utilities given the inconsistent credences. I will argue that there are better ways of calculating expected utilities given a potentially inconsistent credence assignment, and that for a large class of credences—a class that includes many inconsistent examples—these ways are immune to Dutch Books and single-shot domination failures. The crucial move is to replace Finite Additivity with Monotonicity, then \\le P\)) and then calculate expected utilities for positive U via the formula \\, dy\). This shows that Dutch Book arguments for probabilism, the thesis that one’s credences should be consistent, do not establish their conclusion. Finally, I will consider a modified argument based on multi-step domination failure that does better, but nonetheless is not as compelling as the Dutch Book arguments appeared to be.

The Principle of Sufficient Reason must be justified dialectically: by showing the disastrous consequences of denying it. We formulate a version of the Principle that is restricted to basic natural facts, which entails the obtaining of at least one supernatural fact. Denying this principle results in extreme empirical skepticism. We consider six current theories of empirical knowledge, showing that on each account we cannot know that we have empirical knowledge unless we all have a priori knowledge of the PSR. We consider objections based on Neo-Humeanism and the essentiality of origins, and we consider the possibility that we have empirical knowledge without knowing it.

Our aim in this paper is to bring to light two sources of tension for Christian theists who endorse the principle of unrestricted composition, that necessarily, for any objects, the xs, there exists an object, y, such that the xs compose y. In Value, we argue that a composite object made of wholly valuable parts is at least as valuable as its most valuable part, and so the mereological sum of God and a wholly valuable part would be at least as valuable as God; but Christian theism arguably demands that no concrete object other than God can be as valuable as God. And in Creation, we argue that the conjunction of theism and unrestricted composition, together with the claim that every concrete entity that is numerically distinct from God is created by God, implies that God is created by God. We conclude by examining the prospects of restricting the thesis of unrestricted composition to the domain of material or spatiotemporal objects as a way to sidestep the above arguments against the conjunction of Christian theism and unrestricted composition.

The Adams Thesis holds for a conditional → and a probability assignment P if and only if P=P whenever P>0. The restriction ensures that P is well defined by the classical formula P=P/P. Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results, any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.

There is a basic dividing line in the philosophy of time. According to the B-theory, we can describe the temporal reality of the world with freely repeatable sentences, using designators of fixed times and relations such as "earlier" and "later." The A-theory contends that there is an ontological feature of the world which is described by explicitly tensed statements such as "I am now writing this review," and which is not captured by any B-theoretic statements such as "I write this review at t1," whereas the B-theory will say that ontologically there is nothing to my writing this review now but my writing this review at t1. Since most present-day B-theorists will grant that propositions formulated in A-theoretic terms cannot in general be translated into propositions formulated in B-theoretic terms, the debate is over ontology, not translation. In his accessible, interesting, provocative, and at times quite insightful book, David Cockburn argues that the debate cannot be resolved by the kind of ontological and linguistic analysis that is most commonly brought to bear upon the issue by analytic philosophy. Instead, the debate should be moved to a more anthropocentric field, and based upon the significance of time for our concerns, desires, feelings, and actions: "[the] fundamental issues in the philosophy of time... can only be properly grasped within a framework in which the significance of time in our ethical thinking has a central place". This bold claim forming the centrepiece of Cockburn's book will of course set off alarm bells for philosophers of a scientistic persuasion for whom "anthropocentric" is an insult term, and whose dearest desire is to move to a point of view that abstracts from human feelings and concerns.

A number of philosophers have attempted to solve the problem of null-probability possible events in Bayesian epistemology by proposing that there are infinitesimal probabilities. Hájek and Easwaran have argued that because there is no way to specify a particular hyperreal extension of the real numbers, solutions to the regularity problem involving infinitesimals, or at least hyperreal infinitesimals, involve an unsatisfactory ineffability or arbitrariness. The arguments depend on the alleged impossibility of picking out a particular hyperreal extension of the real numbers and/or of a particular value within such an extension due to the use of the Axiom of Choice. However, it is false that the Axiom of Choice precludes a specification of a hyperreal extension—such an extension can indeed be specified. Moreover, for all we know, it is possible to explicitly specify particular infinitesimals within such an extension. Nonetheless, I prove that because any regular probability measure that has infinitesimal values can be replaced by one that has all the same intuitive features but other infinitesimal values, the heart of the arbitrariness objection remains.

Sceptical theism claims that we have vast ignorance about the realm of value and the connections, causal and modal, between goods and bads. This ignorance makes it reasonable for a theist to say that God has reasons beyond our ken for allowing the horrendous evils we observe. But if so, then does this not lead to moral paralysis when we need to prevent evils ourselves? For, for aught that we know, there are reasons beyond our ken for us to allow the evils, and so we should not prevent them. This paralysis argument, however, shall be argued to rest on a confusion between probabilities and expected utilities. A connection between this paralysis argument and Lenman's1 discussion of the butterfly effect and chaos will be drawn, and the solution offered will apply in both cases.

This thesis examines the alethic modal concepts of possibility and necessity. It is argued that one cannot do justice to all our modal talk without possible worlds, i.e., complete ways that a cosmos might have been. I argue that not all of the proposed applications of possible worlds succeed but enough remain to give one good theoretical reason to posit them. The two central problems now are: What feature of reality makes correct alethic modal claims true and What are possible worlds? ;David Lewis makes possible worlds be concretely existing universes. Unfortunately, I show Lewis's account involves set-theoretic, ethical, inductive and probabilistic paradoxes, and commits Lewis to an objectionable form of primitive modality that governs the choice of the counterpart relation. The most promising contemporary alternatives to Lewis's theory have been the worlds of Adams and Plantinga constructed out of Platonic entities such as maximal collections of consistent propositions. However, these approaches fail to provide a satisfactory answer to the question of what makes true modal claims true. I also criticize some alternative accounts. ;Finally, I discuss and combine two historical approaches. The first is an Aristotelian approach that says a non-actual event is possible is to say that some actual substances could have initiated a causal chain that could lead up to the event in question. However, it can be shown that some plausible global possibility claims can be made true on this account only if there is a necessarily existent first cause capable of initiating very different universes. On the other hand, Leibniz made possible worlds be ideas in the mind of an omniscient necessarily existent deity. Leibniz fails to explain what it is that makes these possible worlds possible, but if we were willing to combine his story with the conclusion drawn from the Aristotelian one, we could get the following story: Possible worlds are ideas in the mind of an omniscient deity and what makes them possible is that this deity has the Aristotelian capability of initiating causal chains that can lead to them being actualized

Case 1 (transplant) . You are a surgeon doing an appendectomy on Fred, who is otherwise healthy. You know from his file that, just by chance, his heart, lungs, bone marrow, liver and two kidneys are a perfect match for fifteen patients in your hospital who need various organs or bone marrow, of both of which there is a severe shortage of these organs; Fred, however, has refused to donate anything. If the fifteen patients do not receive the transplants today, they will die. You skillfully use your scalpel to kill Fred in such a way that no one can tell it was no accident, and you use his organs to save fifteen patients. One dead, fifteen innocent people saved.