Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise.

José Benardete developed a famous paradox involving a beginningless set of items each member of which satisfies some predicate just in case no earlier member satisfies it. The Grim Reaper version of this paradox has recently been employed in favour of various finitist metaphysical theses, ranging from temporal finitism to causal finitism to the discrete nature of time. Here, I examine a new challenge to these finitist arguments—namely, the challenge of implying that the future cannot be endless. In particular, I (...) develop future-oriented Benardete paradoxes and examine their epistemic symmetry with past-oriented paradoxes. (shrink)

Causal finitism, the view that the causal history of any event must be finite, has garnered much philosophical interest recently—especially because of its applicability to the Kalām cosmological argument. The most prominent argument for causal finitism is the Grim Reaper argument, which attempts to show that, if infinite causal histories are possible, then other paradoxical states of affairs must also be possible. However, this style of argument has been criticized on the grounds of (i) relying on controversial modal principles, and (...) (ii) providing a false diagnosis of the paradoxes involved. In this paper, I develop a new kind of Grim Reaper argument immune to these criticisms. I show that, by using insights from the literature on time travel, causal finitists should instead argue that infinite causal histories are problematically inexplicable, as they entail the possibility of unexplained foiling mechanisms. The fruits of this paper are that (i) a novel supporting argument for the Kalām is developed, and (ii) along the way of building this argument, it is shown that the literatures on time travel and causal finitism are deeply and intimately connected. (shrink)

We draw out significant consequences of a relatively popular theory of metaphysical modality—branching actualism—for cosmological arguments for God’s existence. According to branching actualism, every possible world shares an initial history with the actual world and diverges only because causal powers (or dispositions, or some such) are differentially exercised. We argue that branching actualism undergirds successful responses to two recent cosmological arguments: the Grim Reaper Kalam argument and a modal argument from contingency. We also argue that branching actualism affords a response (...) to one popular defense of the classic contingency argument. What results are new difficulties for several cosmological arguments arising from the metaphysics of modality. (shrink)

This dissertation is concerned with two of the largest questions that we can ask about the nature of physical reality: first, whether physical reality begin to exist and, second, what criteria would physical reality have to fulfill in order to have had a beginning? Philosophers of religion and theologians have previously addressed whether physical reality began to exist in the context of defending the Kal{\'a}m Cosmological Argument (KCA) for theism, that is, (P1) everything that begins to exist has a cause (...) for its beginning to exist, (P2) physical reality began to exist, and, therefore, (C) physical reality has a cause for its beginning to exist. While the KCA has traditionally been used to argue for God's existence, the KCA does not mention God, has been rejected by historically significant Christian theologians such as Thomas Aquinas, and raises perennial philosophical questions -- about the nature and history of physical reality, the nature of time, the nature of causation, and so on -- that should be of interest to all philosophers and, perhaps, all humans. While I am not a religious person, I am interested in the questions raised by the KCA. In this dissertation, I articulate three necessary conditions that physical reality would need to fulfill in order to have had a beginning and argue that, given the current state of philosophical and scientific inquiry, we cannot determine whether physical reality began to exist. -/- Friends of the KCA have sought to defend their view that physical reality began to exist in two distinct ways. As I discuss in chapter 2, the first way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a priori arguments meant to show that, as a matter of metaphysical necessity, the past must be finite. If the past is necessarily finite, then the past history of physical reality is necessarily finite. And if having a finite past suffices for having a beginning, then, since the past history of physical reality is necessarily finite, physical reality necessarily began to exist. I show that the arguments which have been offered thus far for the view that the past is necessarily finite do not succeed. Moreover, as I elaborate on in chapter 5, having a finite past does not suffice for having a beginning. -/- As I discuss in chapter 3, the second way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a posteriori arguments meant to show that we have empirical evidence that physical reality has a finite past history. For example, the big bang is sometimes claimed to have been the beginning of physical reality and, since we have excellent empirical evidence for the big bang, we have excellent empirical evidence for the beginning of physical reality. The big bang can be understood in two ways. On the one hand, the big bang can be understood as a theory about the history and development of the observable universe. Understood in that sense, then I agree that the big bang is supported by excellent empirical evidence and by a scientific consensus. On the other hand, some authors (particularly science popularizers, science journalists, and religious apologists) have wrongly interpreted big bang theory as a theory about the beginning of the whole of physical reality. As I argue, while a beginning of physical reality may be consistent with classical big bang theory, classical big bang theory does not provide good reason for thinking that physical reality began to exist. -/- In part II, I turn to discussing three necessary, but not necessarily sufficient, conditions for physical reality to have a beginning. Before discussing the three conditions, in chapter 4, I introduce three metaphysical accounts of the nature of time (A-theory, B-theory, and C-theory) as well as some formal machinery that will subsequently become useful in the dissertation. I introduce the first of the three conditions in chapter 5. According to the Modal Condition, physical reality began to exist only if, at the closest possible worlds without time, physical reality does not exist. I show that this condition helps us to make sense of various views in both theology and philosophy of physics. In chapter 6}, I introduce the second of my three conditions, the Direction Condition, according to which, roughly, physical reality began to exist only if all space-time points agree about the direction of time, so that all space-time points can agree that physical reality's putative beginning took place in their objective past. In chapter 7, I discuss the third condition, the Boundary Condition, according to which physical reality began to exist only if there is a past temporal boundary such that physical reality did not exist before the boundary. I show that there are two senses in which physical reality could be said to have had a past temporal boundary. Lastly, in chapter 8, I show that there is a relationship between my three conditions and classical big bang theory, even though the relationship is not the one usually identified in the literature. -/- In part III, I present four arguments for the view that, at the present stage of philosophical and scientific inquiry, we cannot know whether physical reality satisfies the three necessary conditions to have had a beginning and, consequently, we cannot know whether physical reality had a beginning. As I will prove in chapter 9, no set of observations that we currently have, when conjoined with General Relativity, entails that physical reality satisfies the Direction or Boundary Conditions. As I show in chapter 10, considerations in the philosophical foundations of statistical mechanics entail either that the Cosmos violates the Modal Condition or else that there is a transcendental condition on the possibility of our knowledge of the past that prevents our access to data we would need to gather to determine whether physical reality satisfies the Boundary Condition. In chapter 11, I show that there are a number of live cosmological models according to which physical reality does not satisfy the Boundary Condition. As long as we don't know whether any of those cosmological models are correct, we do not know whether physical reality satisfies the Boundary Condition. Lastly, I turn to confirmation theory and show that, at our present stage of inquiry, ampliative inferences for the conclusion that physical reality satisfies the Modal, Direction, and Boundary Conditions are not successful. (shrink)

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face of (...) plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

A common argument in support of a beginning of the universe used by advocates of the kalām cosmological argument (KCA) is the argument against the possibility of an actual infinite, or the “Infinity Argument”. However, it turns out that the Infinity Argument loses some of its force when compared with the achievements of set theory and it brings into question the view that God predetermined an endless future. We therefore defend a new formal argument, based on the nature of time (...) (just as geometrical reasoning is based on the nature of space), which addresses more directly the question of beginningless time. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. I (...) then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

We use two logical resources, namely, the notion of recursively defined function and the Benardete-Yablo paradox, together with some inherent features of causality and time, as usually conceived, to derive two results: that no ungrounded causal chain exists and that time has a beginning.

Various arguments have been put forward to show that Zeno-like paradoxes are still with us. A particularly interesting one involves a cube composed of colored slabs that geometrically decrease in thickness. We first point out that this argument has already been nullified by Paul Benacerraf. Then we show that nevertheless a further problem remains, one that withstands Benacerraf s critique. We explain that the new problem is isomorphic to two other Zeno-like predicaments: a problem described by Alper and Bridger in (...) 1998 and a modified version of the problem that Benardete introduced in 1964. Finally, we present a solution to the three isomorphic problems. (shrink)

This paper proves that certain supertasks constitute counterexamples to countable additivity even in the frame of an objective (not subjective, à la de Finetti) conception of probability. The argument requires taking conditional probability as a primitive notion.

A detailed consideration of the Trojan fly supertask reveals certain unsuspected characteristics relating to determinism and causation. I propose here a solution to the new difficulty in terms of bare dispositions.

The paper analyses Benardete's paradox of the gods from a more general perspective (the convergence approach) than several of the most important proposals made to date, but in close relation (and sharp contrast) with them. The new theory, based on the notion of limit, is systematically applicable in different possible scenarios involving a denumerable infinity of objects. In particular, it reveals in what way ω-consistency can be compromised in an otherwise consistent description of such "infinitary" situations.

The structure of Yablo’s paradox is analysed and generalised in order to show that beginningless step-by-step determination processes can be used to provoke antinomies, more concretely, to make our logical and our on-tological intuitions clash. The flow of time and the flow of causality are usually conceived of as intimately intertwined, so that temporal causation is the very paradigm of a step-by-step determination process. As a conse-quence, the paradoxical nature of beginningless step-by-step determina-tion processes concerns time and causality as usually (...) conceived. (shrink)

The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the infinite series of gods will be mysteriously halted despite none of the gods erecting any barriers. Using a revision-theoretic analysis of Benardete’s puzzle, four distinct possible outcomes that might occur given Benardete’s set-up are distinguished. This analysis provides additional insight into the puzzle at hand, via identifying heretofore unnoticed possible outcomes, but it also serves as an example of how the revision theoretic framework can be used to (...) construct exhaustive taxonomies of potential outcomes in apparently contradictory situations. (shrink)