New arguments against Bayesian regularity and an otherwise plausible domination principle are offered on the basis of rotational symmetry. The arguments against Bayesian regularity work in very general settings.

The cosmological and teleological argument both start with some contingent feature of the actual world and argue that the best or only explanation of that feature is that it was produced by an intelligent and powerful supernatural being. The cosmological argument starts with a general feature, such as the existence of contingent being or the presence of motion and uses some version of the Principle of Sufficient Reason to conclude that this feature must have an explanation. The debate then focuses (...) on two points: first, whether the PSR in question is true, and second, whether the explanation must involve God or at least some God-like being. The teleological argument begins with a general feature of the cosmos judged to have value, such as the existence of intelligent life or the presence of order in the universe, and argues, usually inductively but sometimes deductively, that this feature is to be explained by the agency of a powerful supernatural being. Here, the debate tends to focus on whether there are alternate naturalistic explanations, such as Darwinian evolution. (shrink)

A simple argument is given that shows that conditional probabilities do not supervene on unconditional probabilities. In particular, one cannot in general define conditional probabilities using the ratio formula P ( U | V ) = P ( U & V )/ P ( U ), or using any more sophisticated method based on unconditional probabilities.

(No abstract is available for this citation).

The free-will defence holds that the value of significant free will is so great that God is justified in creating significantly free creatures even if there is a risk or certainty that these creatures will sin. A difficulty for the FWD, developed carefully by Quentin Smith, is that God is unable to do evil, and yet surely lacks no genuinely valuable kind of freedom. Smith argues that the kind of freedom that God has can be had by creatures, without a (...) risk of creatures doing evil. I shall show that Smith's argument fails – the case of God is disanalogous to the case of creatures precisely because creatures are creatures. (shrink)

Assuming S5, the main controversial premise in modal ontological arguments is the possibility premise, such as that possibly a maximally great being exists. I shall offer a new way of arguing that the possibility premise is probably true.

Some, notably Peter van Inwagen, in order to avoid problems with free will and omniscience, replace the condition that an omniscient being knows all true propositions with a version of the apparently weaker condition that an omniscient being knows all knowable true propositions. I shall show that the apparently weaker condition, when conjoined with uncontroversial claims and the logical closure of an omniscient being's knowledge, still yields the claim that an omniscient being knows all true propositions.

Alexander R. Pruss examines a large family of paradoxes to do with infinity - ranging from deterministic supertasks to infinite lotteries and decision theory. Having identified their common structure, Pruss considers at length how these paradoxes can be resolved by embracing causal finitism.

This important philosophical reflection on love and sexuality from a broadly Christian perspective is aimed at philosophers, theologians, and educated Christian readers. Alexander R. Pruss focuses on foundational questions on the nature of romantic love and on controversial questions in sexual ethics on the basis of the fundamental idea that romantic love pursues union of two persons as one body. _One Body_ begins with an account, inspired by St. Thomas Aquinas, of the general nature of love as constituted by (...) components of goodwill, appreciation, and unitiveness. Different forms of love, such as parental, collegial, filial, friendly, fraternal, or romantic, Pruss argues, differ primarily not in terms of goodwill or appreciation but in terms of the kind of union that is sought. Pruss examines romantic love as distinguished from other kinds of love by a focus on a particular kind of union, a deep union as one body achieved through the joint biological striving of the sort involved in reproduction. Taking the account of the union that romantic love seeks as a foundation, the book considers the nature of marriage and applies its account to controversial ethical questions, such as the connection between love, sex, and commitment and the moral issues involving contraception, same-sex activity, and reproductive technology. With philosophical rigor and sophistication, Pruss provides carefully argued answers to controversial questions in Christian sexual ethics. "This is a terrific—really quite extraordinary—work of scholarship. It is quite simply the best work on Christian sexual ethics that I have seen. It will become the text that anyone who ventures into the field will have to grapple with—a kind of touchstone. Moreover, it is filled with arguments with which even secular writers on sexual morality will have to engage and come to terms." —_Robert P. George, Princeton University __ "_One Body_ is an excellent piece of philosophical-theological reflection on the nature of sexuality and marriage. This book has the potential to become a standard go-to text for professors and students working on sex ethics issues, whether in philosophy or theology, both for the richness of its arguments, and the scope of its coverage of cases. " — Christopher Tollefsen, University of South Carolina_ "Alexander Pruss here develops sound and humane answers to the whole range of main questions about human sexual and reproductive choices. His principal argument for the key answers is very different from the one I have articulated over the past fifteen years. But his argumentation is at every point attractively direct, careful, energetic in framing and responding to objections, and admirably attentive to realities and the human goods at stake." —_John Finnis, University of Oxford _. (shrink)

Necessary Existence breaks ground on one of the deepest questions anyone ever asks: why is there anything? Pruss and Rasmussen present an original defence of the hypothesis that there is a necessarily existing being capable of providing an ultimate foundation for the existence of all things.

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. (shrink)

Plantinga’s Free Will Defense is an argument that, possibly, God cannot actualize a world containing significant creaturely free will and no wrongdoings. I will argue that if standard Molinism is true, there is a pair of worlds w1 and w2 each of which contains a significantly free creature who never chooses wrongly, and that are such that, necessarily, at least one of these worlds is a world that God can actualize.

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to prove (...) the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

The Principle of Sufficient Reason says that all contingent facts must have explanation. In this 2006 volume, which was the first on the topic in the English language in nearly half a century, Alexander Pruss examines the substantive philosophical issues raised by the Principle Reason. Discussing various forms of the PSR and selected historical episodes, from Parmenides, Leibnez, and Hume, Pruss defends the claim that every true contingent proposition must have an explanation against major objections, including Hume's imaginability (...) argument and Peter van Inwagen's argument that the PSR entails modal fatalism. Pruss also provides a number of positive arguments for the PSR, based on considerations as different as the metaphysics of existence, counterfactuals and modality, negative explanations, and the everyday applicability of the PSR. Moreover, Pruss shows how the PSR would advance the discussion in a number of disparate fields, including meta-ethics and the philosophy of mathematics. (shrink)

We shall use Gödel’s Second Incompleteness Theorem to show that consistency is not possibility, and then argue that the argument does serious damage to some theories of modality where consistency plays a major but not exclusive role.

Cantor proved that no set has a bijection between itself and its power set. This is widely taken to have shown that there infinitely many sizes of infinite sets. The argument depends on the princip...

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. (shrink)

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. (shrink)

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. (shrink)

Scoring rules measure the deviation between a credence assignment and reality. Probabilism holds that only those credence assignments that satisfy the axioms of probability are rationally admissible. Accuracy-based arguments for probabilism observe that given certain conditions on a scoring rule, the score of any non-probability is dominated by the score of a probability. The conditions in the arguments we will consider include propriety: the claim that the expected accuracy of _p_ is not beaten by the expected accuracy of any other (...) credence _c_ by the lights of _p_ if _p_ is a probability. I argue that if we think through how a non-probabilist can respond to pragmatic arguments for probabilism, we will expect the non-probabilist to accept a condition _stronger_ than propriety for the same reasons that the probabilist gives for propriety, but this stronger condition is incompatible with the other conditions that the probabilist needs to run the accuracy argument. This makes it unlikely for the probabilist’s argument to be compelling. (shrink)

An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough.

Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms of the probability calculus, but Predd et al. [9] have shown that given a (...) finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal. (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)

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. (shrink)

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. (shrink)

Consider the reasonable axioms of subjunctive conditionals if p → q1 and p → q2 at some world, then p → at that world, and if p1 → q and p2 → q at some world, then → q at that world, where p → q is the subjunctive conditional. I show that a Lewis-style semantics for subjunctive conditionals satisfies these axioms if and only if one makes a certain technical assumption about the closeness relation, an assumption that is probably (...) false. I will then show how Lewisian semantics can be modified so as to assure and even when the technical assumption fails, and in fact in one sense the semantics actually becomes simpler then. (shrink)

I shall argue that, appearances to the contrary notwithstanding, current biological science does not rule out the possibility of miraculous intervention in the evolutionary history of human beings. This shows that it is possible to reconcile evolutionary science with the claim that we are designed by God.

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. (shrink)

A nested mode ontology allows one to make sense of apparently contradictory Christological claims such as that Christ knows everything and there are some things Christ does not know.

We will give a new cosmological argument for the existence of a being who, although not proved to be the absolutely perfect God of the great Medieval theists, also is capable of playing the role in the lives of working theists of a being that is a suitable object of worship, adoration, love, respect, and obedience. Unlike the absolutely perfect God, the God whose necessary existence is established by our argument will not be shown to essentially have the divine perfections (...) of omnipotence, omniscience, omnibenevolence, and sovereignty. Furthermore, it is not even shown that he is contingently omnipotent and omniscient, just powerful and intelligent enough to be the supernatural designer-creator of the exceedingly complex and wondrous cosmos that in fact.. (shrink)

Adhering to the traditional concept of omniscience lands Gale in the incoherence Grim’s Cantorian arguments reveal in talk of “all propositions.” By constructing variants and extensions of Grim’s arguments, I explain why various ways out of the incoherence are unacceptable, why theists would do better to adopt a certain revisionary concept of omniscience, and why the Cantorian troubles are so deep as to be troubles as well for Gale’s Weak PSR. I conclude with some brief reflections on method, suggesting that (...) we pursue the full implications of Gale’s own revisionary remarks and replace his method of analytic argumentation with non-analytic revisionary theory-construction. (shrink)

In this article, we develop and defend a new argument for animalism -- the thesis that we human persons are human animals. The argument takes this rough form: since our pets are animals, we are too. We’ll begin with remarks on animalism and its rivals, develop our main argument, and then defend it against a few replies.

The strict dominance principle that a wager always paying better than another is rationally preferable is one of the least controversial principles in decision theory. I shall show that (given the Axiom of Choice) there is a contradiction between strict dominance and plausible isomorphism or symmetry conditions, by showing how in several natural cases one can construct isomorphic wagers one of which strictly dominates the other. In particular, I will show that there is a pair of wagers on the outcomes (...) of a uniform spinner which differ simply in where the zero degrees point of the spinner is defined to be but where one wager dominates the other. I shall also argue that someone who accepts Williamson’s famous argument that the probability of an infinite sequence of heads is zero should accept the symmetry conditions, and thus has reason to weaken the strict dominance principle, and I shall propose a restriction of the principle to “implementable” wagers. Our main result also has implications for social choice principles. (shrink)

A serious error in the proof of a recent characterization of the existence of full conditional probabilities invariant under symmetries is corrected.

Gödel's ontological argument is a formal argument for a being defined in terms of the concept of a positive property. I shall defend several versions of Gödel's argument, using weaker premises than Anderson's (1990) version, and avoiding Oppy's (1996 and 2000) parody refutations.

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. (shrink)

I shall offer some very plausible assumptions for the measure of confirmation and show that they imply that E confirms H relative to background K to degree f/PK), where f is a strictly increasing function. An additional assumption about how measures of confirmation combine then makes f be proportional to a logarithm.

There is an important sense in which one can be sure without being certain, i.e., without assigning unit probability. I will offer an explication of this sense of sureness, connecting it with the level of credence that a rational agent would need to have to be confident that she won’t ever lose her confidence. A simple formal result then gives us an explicit formula connecting the threshold α for credence needed for confidence with the threshold needed for being sure: one (...) needs 1−(1−α) to be sure. I then suggest that stepping between α and 1−(1−α) gives a procedure that generates an interesting hierarchy of credential thresholds. (shrink)

I argue that an examination of the analogy between the notion of a bug and that of a genetic defect supports an analogy not just between a computer program and DNA, but between a computer program designed by a programmer and DNA. This provides an analogical teleological argument for the existence of a highly intelligent designer.

The cosmos is filled with evil that seemingly has no redeeming value. Granted, some evils do lead to greater goods, sometimes goods that could not exist without the evils. Thus, the exercise of courage is a good that requires either an actual evil to stand firm in the face of or the illusion of an evil—and an illusion is a kind of evil, too. But many evils appear to serve no such purpose. Philosophers call an evil that a supremely good (...) God would have insufficient reason to permit to exist a gratuitous evil. A particularly powerful form of the argument from evil against the existence of the God of Western monotheism is, thus, that there.. (shrink)

This paper argues that if creatures are to have significant free will, then God's essential omni-benevolence and essential omnipotence cannot logically preclude Him from creating a world containing a moral evil. The paper maintains that this traditional conclusion does not need to rest on reliance on subjunctive conditionals of free will. It can be grounded in several independent ways based on premises that many will accept.

Our paper ‘A new cosmological argument’ gave an argument for the existence of God making use of the weak Principle of Sufficient Reason (W-PSR) which states that for every proposition p, if p is true, then it is possible that there is an explanation for p. Recently, Graham Oppy, as well as Kevin Davey and Rob Clifton, have criticized the argument. We reply to these criticisms. The most interesting kind of criticism in both papers alleges that the W-PSR can be (...) justifiably denied by the atheist, and constitutes no improvement on the strong Principle of Sufficient Reason (S-PSR) which claims that every true proposition in fact has an explanation. The criticism is predicated on the fact that it can be shown that the W-PSR entails the S-PSR. We argue that the W-PSR's plausibility remains despite the criticisms. From this it can be seen to follow that the entailment relation between the W-PSR and the S-PSR gives one reason to believe the S-PSR. (shrink)

The classical principle of double effect offers permissibility conditions for actions foreseen to lead to evil outcomes. I shall argue that certain kinds of closeness cases, as well as general heuristic considerations about the order of explanation, lead us to replace the intensional concept of intention with the extensional concept of accomplishment in double effect.

Substantive theories of diachronic identity have been offered for different kinds of entities. The kind of entity whose diachronic identity has received the most attention in the literature is person, where such theories as the psychological theory, the body theory, the soul theory, and animalism have been defended. At the same time, Wittgenstein's remark that ?to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say (...) nothing at all? suggests that the idea of further analysing identity is mistaken at root. I shall offer a simple, deflationary theory that reduces diachronic identity to quantification, synchronic identity and existence at a spacetime location (or at a time, for non-spatial entities). On logical grounds, the theory is guaranteed to have no counterexamples. Because the theory is guaranteed to have no counterexamples, all the imaginative examples offered as intuitive support for theories of personal identity are going to be either incorrect or compatible with the theory. I shall argue that the deflationary theory is preferable on simplicity grounds to typical substantive theories, and that various problems that are commonly thought to concern diachronic identity are better seen as about something else. (shrink)

This chapter contains sections titled: Introduction The PSR Nonlocal CPs Toward a First Cause The Gap Problem Conclusions and Further Research References.

I shall discuss the problem of the definition of lying and the formulation of the duty of truthtelling. I shall argue that the morality of assertion is a special case of the morality of endorsement, and that a criterion of adequacy for an account of lying is that it handles certain cases of dishonest endorsement as well. Standardviews of lying fail to do so. I shall offer an account of the duty of honest endorsement in terms of the intention to (...) avoid falsehood. But, in the end, we may simplyhave reason to go back to the naïve view that lying is saying falsehoods. (shrink)