The recognition of striking regularities in the physical world plays a major role in the justification of hypotheses and the development of new theories both in the natural sciences and in philosophy. However, while scientists consider only strictly natural hypotheses as explanations for such regularities, philosophers also explore meta-natural hypotheses. One example is mathematical realism, which proposes the existence of abstract mathematical entities as an explanation for the applicability of mathematics in the sciences. Another example is theism, which offers the (...) existence of a supernatural being as an explanation for the design-like appearance of the physical cosmos. Although all meta-natural hypotheses defy empirical testing, there is a strong intuition that some of them are more warranted than others. The goal of this paper is to sharpen this intuition into a clear criterion for the admissibility of meta-natural explanations for empirical facts. Drawing on recent debates about the indispensability of mathematics and teleological arguments for the existence of God, I argue that a meta-natural explanation is admissible just in case the explanation refers to an entity that, though not itself causally efficacious, guarantees the instantiation of a causally efficacious entity that is an actual cause of the regularity. (shrink)

Recently, Del Ratzsch proposed a new version of the design argument. He argues that belief in a designer is often formed non-inferentially, much like perceptual beliefs, rather than formed by explicit reasoning. Ratzsch traces his argument back to Thomas Reid (1710-1796) who argues that beliefs formed in this way are also justified. In this paper, I investigate whether design beliefs that are formed in this way can be regarded as knowledge. For this purpose, I look closer to recent scientific study (...) of how design beliefs are formed. I argue that the science strongly suggest that people easily form false beliefs. As a result, design beliefs can only constitute knowledge if subjects have additional reasons or evidence for design. (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)

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require the (...) binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

Richard Swinburne, in his The Existence of God (2004), presents a cosmological argument in defence of theism (Swinburne 1991: 119, 135). God, Swinburne argues, is more likely to bring about an ordered universe than other states (ibid.: 144, 299). To defend this view, Swinburne presents the following arguments: (1) That this ordered universe is a priori improbable (2004: 49, 150, 1991: 304 et seq.), given the stringent requirements for life (cf. also Leslie 2000: 12), and the Second Law of Thermodynamics (...) (Giancoli 1990: 396); (2) That it seems as if this ordered universe can be explained by theism; (3) A theistic explanation for the universe is more probable because it is a simple explanation. To this end, Swinburne makes use of Bayes’ Theorem. Symbolically, this claim can be represented as (e) for the evidence of the existence of a complex universe, and (h) for a hypothesis. Swinburne’s argument is that theism has a higher prior probability, P(htheism) > P(hmaterialism), since theism is simpler than materialism. He concludes that P(e|htheism) > P(e|hmaterialism). In this paper I will address only this argument (3) above, and defend the view that it is false: theism is not simpler than materialism, nor it is more probably true. I conclude that theism is less probable than materialism, expressed by P(htheism) < P(hmaterialism) : 2/N(2n+1) < 1/n, where N is the number of possible universes and n the number of entities in existence. (shrink)

Given ht as the hypothesis of theism, hm as the hypothesis of materialism, and e as the evidence of a complex life-bearing universe, Richard Swinburne presents these arguments in The Existence of God: that this ordered universe is a priori improbable, given the stringent requirements for life and the Second Law of Thermodynamics; that this universe’s structure is evidence for theism, and that theism therefore explains this universe; Swinburne argues that because P > P, it follows that P > P; (...) and a theistic explanation for the universe is more probable because it is simpler; therefore it is more likely that God exists than not. As I have addressed in a prior paper, this paper will address the Bayesian argument that Swinburne offers in , i.e. that P > P. In the paper I draw a number of conclusions, most pertinently, that Hacking’s Total Probability Rule for cases of mutually exclusive hypotheses [ht vs hm] and evidence e entails that ht can only be confirmed if P is low. I also conclude that if we follow the TPR for Swinburne’s argument, we achieve the result that theism is at best slightly improbable, or equiprobable with materialism. (shrink)

The fundamental constants that are involved in the laws of physics which describe our universe are finely-tuned for life, in the sense that if some of the constants had slightly different values life could not exist. Some people hold that this provides evidence for the existence of God. I will present a probabilistic version of this fine-tuning argument which is stronger than all other versions in the literature. Nevertheless, I will show that one can have reasonable opinions such that the (...) fine-tuning argument doesn’t lead to an increase in one’s probability for the existence of God. (shrink)

Is the fact that our universe contains fine-tuned life evidence that we live in a multiverse? Ian Hacking and Roger White influentially argue that it is not. We approach this question through a systematic framework for self-locating epistemology. As it turns out, leading approaches to self-locating evidence agree that the fact that our own universe contains fine-tuned life indeed confirms the existence of a multiverse. This convergence is no accident: we present two theorems showing that, in this setting, any updating (...) rule that satisfies a few reasonable conditions will have the same feature. The conclusion that fine-tuned life provides evidence for a multiverse is hard to escape. (shrink)

The idea that there might be multiple universes with different parameters of nature is often considered an attractive response to the finding that various parameters appear to be delicately fine-tuned for life. The present paper investigates whether the appeal to fine-tuning can legitimately be combined with an appeal to independent empirical evidence for other universes or whether, as suggested by Cory Juhl, combining such appeals inevitably results in illegitimate double counting of the finding that the parameters are right for life. (...) In doing so, the paper takes into account the fact that the parameters' life-friendliness is old evidence for us, which makes Bayesianism's problem of old evidence relevant to Bayesian analysis of its evidential impact. Ultimately, the verdict reached is that the warning of double counting is helpful, but that double counting can in principle be avoided. The paper highlights remaining independent challenges against the fine-tuning argument for the multiverse. (shrink)

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

In this paper, we discuss three probabilistic arguments for the existence of multiple universes. First, we provide an analysis of total evidence and use that analysis to defend Roger White's "this universe" objection to a standard fine-tuning argument for multiple universes. Second, we explain why Rodney Holder's recent cosmological argument for multiple universes is unconvincing. Third, we develop a "Cartesian argument" for multiple universes. While this argument is not open to the objections previously noted, we show that, given certain highly (...) plausible assumptions about evidence and epistemic probability, the proposition which it treats as evidence cannot coherently be regarded as evidence for anything. This raises the question of whether to reject the assumptions or accept that such a proposition cannot be evidence. (shrink)

Given the laws of our universe, the initial conditions and cosmological constants had to be "fine-tuned" to result in life. Is this evidence for design? We argue that we should be uncertain whether an ideal agent would take it to be so—but that given such uncertainty, we should react to fine-tuning by boosting our confidence in design. The degree to which we should do so depends on our credences in controversial metaphysical issues.

It seems improbable that life would exist in a naturalistic universe. But if the universe were spatially infinite, then seemingly improbable events would be expected to happen; life would be expected to exist. It follows that the existence of life provides evidence that the universe is spatially infinite.

This paper examines probabilistic versions of the fine-tuning argument for design (FTA), with an emphasis on the interpretation of the probability statements involved in such arguments. Three categories of probability are considered: physical, epistemic, and logical. Of the three possibilities, I argue that only logical probability could possibly support a cogent probabilistic FTA. However, within that framework, the premises of the argument require a level of justification that has not been met, and, it is reasonable to believe, will not be (...) met anytime soon. (shrink)