Enjoying great popularity in decision theory, epistemology, and philosophy of science, Bayesianism as understood here is fundamentally concerned with epistemically ideal rationality. It assumes a tight connection between evidential probability and ideally rational credence, and usually interprets evidential probability in terms of such credence. Timothy Williamson challenges Bayesianism by arguing that evidential probabilities cannot be adequately interpreted as the credences of an ideal agent. From this and his assumption that evidential probabilities cannot be interpreted (...) as the actual credences of human agents either, he concludes that no interpretation of evidential probabilities in terms of credence is adequate. I argue to the contrary. My overarching aim is to show on behalf of Bayesians how one can still interpret evidential probabilities in terms of ideally rational credence and how one can maintain a tight connection between evidential probabilities and ideally rational credence even if the former cannot be interpreted in terms of the latter. By achieving this aim I illuminate the limits and prospects of Bayesianism. (shrink)

In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.

This paper is a comparison of how first-order Kyburgian Evidential Probability (EP), second-order EP, and objective Bayesian epistemology compare as to the KLM system-P rules for consequence relations and the monotonic / non-monotonic divide.

Calibration inductive logics are based on accepting estimates of relative frequencies, which are used to generate imprecise probabilities. In turn, these imprecise probabilities are intended to guide beliefs and decisions — a process called “calibration”. Two prominent examples are Henry E. Kyburg's system of Evidential Probability and Jon Williamson's version of Objective Bayesianism. There are many unexplored questions about these logics. How well do they perform in the short-run? Under what circumstances do they do better or worse? What (...) is their performance relative to traditional Bayesianism? -/- In this article, we develop an agent-based model of a classic binomial decision problem, including players based on variations of Evidential Probability and Objective Bayesianism. We compare the performances of these players, including against a benchmark player who uses standard Bayesian inductive logic. We find that the calibrated players can match the performance of the Bayesian player, but only with particular acceptance thresholds and decision rules. Among other points, our discussion raises some challenges for characterising “cautious” reasoning using imprecise probabilities. Thus, we demonstrate a new way of systematically comparing imprecise probability systems, and we conclude that calibration inductive logics are surprisingly promising for making decisions. (shrink)

Most legal evidence scholars agree that proof of guilt beyond a reasonable doubt requires the belief that the defendant probably committed the alleged acts. However, they also agree that this is not a sufficient condition, as this belief may be unreasonable. I focus on two popular proposals for additional conditions: (i) that the degree of belief should be robust and (ii) that it should be reasonable given the available evidence (should be an evidential probability). Both criteria face difficulties (...) regarding their meaning and utility. I propose an interpretation which overcomes these difficulties and which combines the two conditions into a single criterion on which both conditions are about the possibility of overlooked, exculpatory information. Proof of guilt beyond a reasonable doubt requires that we are sufficiently certain that we did not miss such information for the sake of achieving accurate beliefs and a just error distribution. How certain we can be that we did not miss anything relevant depends on our higher-order evidence. One important type of higher-order evidence is whether the hypotheses under consideration are sufficiently detailed. This point leads to a novel way of combining the Bayesian and explanation-based accounts of rational legal proof. (shrink)

This paper defends David Hume's "Of Miracles" from John Earman's (2000) Bayesian attack by showing that Earman misrepresents Hume's argument against believing in miracles and misunderstands Hume's epistemology of probable belief. It argues, moreover, that Hume's account of evidence is fundamentally non-mathematical and thus cannot be properly represented in a Bayesian framework. Hume's account of probability is show to be consistent with a long and laudable tradition of evidential reasoning going back to ancient Roman law.

NOTE: This paper is a reworking of some aspects of an earlier paper – ‘What else justification could be’ and also an early draft of chapter 2 of Between Probability and Certainty. I'm leaving it online as it has a couple of citations and there is some material here which didn't make it into the book (and which I may yet try to develop elsewhere). My concern in this paper is with a certain, pervasive picture of epistemic justification. On (...) this picture, acquiring justification for believing something is essentially a matter of minimising one’s risk of error – so one is justified in believing something just in case it is sufficiently likely, given one’s evidence, to be true. This view is motivated by an admittedly natural thought: If we want to be fallibilists about justification then we shouldn’t demand that something be certain – that we completely eliminate error risk – before we can be justified in believing it. But if justification does not require the complete elimination of error risk, then what could it possibly require if not its minimisation? If justification does not require epistemic certainty then what could it possibly require if not epistemic likelihood? When all is said and done, I’m not sure that I can offer satisfactory answers to these questions – but I will attempt to trace out some possible answers here. The alternative picture that I’ll outline makes use of a notion of normalcy that I take to be irreducible to notions of statistical frequency or predominance. (shrink)

Evidential support is often equated with confirmation, where evidence supports hypothesis H if and only if it increases the probability of H. This article argues against this received view. As the author shows, support is a comparative notion in the sense that increase-in-probability is not. A piece of evidence can confirm H, but it can confirm alternatives to H to the same or greater degree; and in such cases, it is at best misleading to conclude that the (...) evidence supports H. The author puts forward an alternative view that defines support in terms of measures of degree of confirmation. The proposed view is both sufficiently comparative and able to accommodate the increase-in-probability aspect of support. The author concludes that the proposed measure-theoretic approach to support provides a superior alternative to the standard confirmatory approach. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- (...) ditional counterparts) cannot generate a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

We show in this paper that some expressions indicating source of evidence are part of propositional content and are best analyzed as special kind of epistemic modal. Our evidence comes from the Japanese evidential system. We consider six evidentials in Japanese, showing that they can be embedded in conditionals and under modals and that their properties with respect to modal subordination are similar to those of ordinary modals. We show that these facts are difficult for existing theories of evidentials, (...) which assign evidentials necessarily widest scope, to explain. We then provide an analysis using a logical system designed to account for evidential reasoning; this logic is the first developed system of probabilistic dynamic predicate logic. This analysis is shown to account for the data we provide that is problematic for other theories. (shrink)

A familiar adage in the philosophy of science is that general hypotheses are better supported by varied evidence than by uniform evidence. Several studies suggest that young children do not respect this principle, and thus suffer from a defect in their inductive methodology. We argue that the diversity principle does not have the normative status that psychologists attribute to it, and should be replaced by a simple rule of probability. We then report experiments designed to detect conformity to the (...) latter rule in children's inductive judgment. The results suggest that young children in both the United States and Taiwan are sensitive to the constraints imposed by the rule on judgments of probability and evidential strength. We conclude with a suggested reinterpretation of the thesis that children's inductive methodology qualifies them as “little scientists.”. (shrink)

We show in this paper that some expressions indicating source of evidence are part of propositional content and are best analyzed as special kind of epistemic modal. Our evidence comes from the Japanese evidential system. We consider six evidentials in Japanese, showing that they can be embedded in conditionals and under modals and that their properties with respect to modal subordination are similar to those of ordinary modals. We show that these facts are difficult for existing theories of evidentials, (...) which assign evidentials necessarily widest scope, to explain. We then provide an analysis using a logical system designed to account for evidential reasoning; this logic is the first developed system of probabilistic dynamic predicate logic. This analysis is shown to account for the data we provide that is problematic for other theories. (shrink)

Evidential reasoning is hard, and errors can lead to miscarriages of justice with serious consequences. Analytic methods for the correct handling of evidence come in different styles, typically focusing on one of three tools: arguments, scenarios or probabilities. Recent research used Bayesian networks for connecting arguments, scenarios, and probabilities. Well-known issues with Bayesian networks were encountered: More numbers are needed than are available, and there is a risk of misinterpretation of the graph underlying the Bayesian network, for instance as (...) a causal model. The formalism presented here models presumptive arguments about coherent hypotheses that are compared in terms of their strength. No choice is needed between qualitative or quantitative analytic styles, since the formalism can be interpreted with and without numbers. The formalism is applied to key concepts in argumentative, scenario and probabilistic analyses of evidential reasoning, and is illustrated with a fictional crime investigation example based on Alfred Hitchcock’s film ‘To Catch A Thief’. (shrink)

Peter Achinstein has argued at length and on many occasions that the view according to which evidential support is defined in terms of probability-raising faces serious counterexamples and, hence, should be abandoned. Proponents of the positive probabilistic relevance view have remained unconvinced. The debate seems to be in a deadlock. This paper is an attempt to move the debate forward and revisit some of the central claims within this debate. My conclusion here will be that while Achinstein may (...) be right that his counterexamples undermine probabilistic relevance views of what it is for e to be evidence that h, there is still room for a defence of a related probabilistic view about an increase in being supported, according to which, if p > p, then h is more supported given e than it is without e. My argument relies crucially on an insight from recent work on the linguistics of gradable adjectives. (shrink)

Peter Achinstein has argued at length and on many occasions that the view according to which evidential support is defined in terms of probability-raising faces serious counterexamples and, hence, should be abandoned. Proponents of the positive probabilistic relevance view have remained unconvinced. The debate seems to be in a deadlock. This paper is an attempt to move the debate forward and revisit some of the central claims within this debate. My conclusion here will be that while Achinstein may (...) be right that his counterexamples undermine probabilistic relevance views of what it is for e to be evidence that h, there is still room for a defence of a related probabilistic view about an increase in being supported, according to which, if p > p, then h is more supported given e than it is without e. My argument relies crucially on an insight from recent work on the linguistics of gradable adjectives. (shrink)

In this paper I will critically analyze the first part of David Hume’s argument against miracles, which has been traditionally referred to as the in-principle argument. However, unlike most critiques of Hume’s argument, I will (1) present a view of evidential epistemology and probability that will take into consideration Hume’s accurate observation that miracles are highly improbable events while(2) arguing that one can be within one’s epistemic rights in believing that a miracle has occurred. As for the proper (...) definition of a miracle, I offer the following, which I believe most religious people generally mean when they call an event miraculous: "A miracle is a divine intervention that occurs contrary to the regular course of nature within a significant historical-religious context". Although I am fully aware that this definition has its detractors, it will merely function in this paper as a working definition so that we can come to grips with Hume’s argument. This definition has been. (shrink)

_Philosophy of Probability: Contemporary Readings_ is the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others, the book looks in depth at the following key topics: subjective probability and credence probability updating: conditionalization and reflection Bayesian confirmation theory classical, logical, and evidential probability frequentism physical probability: (...) propensities and objective chances. The book features a useful primer on the mathematics of probability, and each section includes an introduction by the editor, as well as a guide to further reading. A broad-ranging and highly accessible exploration of the subject, _Philosophy of Probability_ is ideal for any student of formal epistemology, philosophy of science, metaphysics, or philosophy of mathematics. (shrink)

Is evidential support transitive? The answer is negative when evidential support is understood as confirmation so that X evidentially supports Y if and only if p(Y | X) > p(Y). I call evidential support so understood “support” (for short) and set out three alternative ways of understanding evidential support: support-t (support plus a sufficiently high probability), support-t* (support plus a substantial degree of support), and support-tt* (support plus both a sufficiently high probability and a (...) substantial degree of support). I also set out two screening-off conditions (under which support is transitive): SOC1 and SOC2. It has already been shown that support-t is non-transitive in the general case (where it is not required that SOC1 holds and it is not required that SOC2 holds), in the special case where SOC1 holds, and in the special case where SOC2 holds. I introduce two rather weak adequacy conditions on support measures and argue that on any support measure meeting those conditions it follows that neither support-t* nor support-tt* is transitive in the general case, in the special case where SOC1 holds, or in the special case where SOC2 holds. I then relate some of the results to Douven’s evidential support theory of conditionals along with a few rival theories. (shrink)

the symmetry of our evidential situation. If our confidence is best modeled by a standard probability function this means that we are to distribute our subjective probability or credence sharply and evenly over possibilities among which our evidence does not discriminate. Once thought to be the central principle of probabilistic reasoning by great..

I defend evidential decision theory and the theory of deliberation-probability dynamics from a recent criticism advanced by Jordan Howard Sobel. I argue that his alleged counterexample to the theories, called the Popcorn Problem is not a genuine counterexample.

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate (...) the usefulness of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)

Credences are beliefs about evidential probabilities. We give the view an assessment-sensitive formulation, show how it evades the standard objections, and give several arguments in support.

The thought that evil in the world poses a problem for belief in the existence of God is an ancient and very natural idea - going back at least to Job. But can that basic idea be converted into a sound argument for the non-existence of God? Arguments from evil against the existence of a deity come in two very different forms. On the one hand, one has what are known as incompatibility versions of the argument from evil. These are (...) typically directed against God conceived of as in classical monotheism––that is, as an omnipotent, omniscient, and morally perfect person––and such arguments attempt to show that the existence of such a deity is logically incompatible with certain true propositions about the existence of evil states of affairs. Then, on the other hand, one has what are known as evidential or inductive formulations of the argument from evil. These arguments have the more modest goal of showing that facts about evil states of affairs in the world make it unlikely that deities with certain properties exist. Arguments of the latter sort will be the focus of the present chapter. -/- As we shall see, evidential arguments from evil can be formulated in several ways, a full exposition of which would lead us into some quite technical waters. My goal, however, will be simply to make clear the basic ideas. Having done that, I shall describe three different responses to such arguments. One involves the idea of a theodicy, while a second appeals instead to countervailing evidence in support of the existence of God, and a third employs the idea of morally significant states of affairs that lie outside our ken. We shall see, I think, that all three responses are problematic. (shrink)

The article is a plea for ethicists to regard probability as one of their most important concerns. It outlines a series of topics of central importance in ethical theory in which probability is implicated, often in a surprisingly deep way, and lists a number of open problems. Topics covered include: interpretations of probability in ethical contexts; the evaluative and normative significance of risk or uncertainty; uses and abuses of expected utility theory; veils of ignorance; Harsanyi’s aggregation theorem; (...) population size problems; equality; fairness; giving priority to the worse off; continuity; incommensurability; nonexpected utility theory; evaluative measurement; aggregation; causal and evidential decision theory; act consequentialism; rule consequentialism; and deontology. (shrink)

has offered evidential decision theorists a defence against the charge that they make unintuitive recommendations for cases like Newcomb's Problem. He says that when conditional probabilities are assessed from the agent's point of view, evidential decision theory makes the same recommendation as intuition. I argue that calculating the probabilities in Price's way leads to no recommendation. It condemns the agent to perpetual oscillation between different options. Price's Argument Instability Objections Conclusion.

Richard Jeffrey is beyond dispute one of the most distinguished and influential philosophers working in the field of decision theory and the theory of knowledge. His work is distinctive in showing the interplay of epistemological concerns with probability and utility theory. Not only has he made use of standard probabilistic and decision theoretic tools to clarify concepts of evidential support and informed choice, he has also proposed significant modifications of the standard Bayesian position in order that it provide (...) a better fit with actual human experience. Probability logic is viewed not as a source of judgment but as a framework for explaining the implications of probabilistic judgments and their mutual compatability. This collection of essays spans a period of some 35 years and includes what have become some of the classic works in the literature. There is also one completely new piece, while in many instances Jeffrey includes afterthoughts on the older essays. (shrink)

According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so (...) on for other kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so (...) on for other kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

In a series of three behavioral experiments, we found a systematic distortion of probability judgments concerning elementary visual stimuli. Participants were briefly shown a set of figures that had two features (e.g., a geometric shape and a color) with two possible values each (e.g., triangle or circle and black or white). A figure was then drawn, and participants were informed about the value of one of its features (e.g., that the figure was a “circle”) and had to predict the (...) value of the other feature (e.g., whether the figure was “black” or “white”). We repeated this procedure for various sets of figures and, by varying the statistical association between features in the sets, we manipulated the probability of a feature given the evidence of another (e.g., the posterior probability of hypothesis “black” given the evidence “circle”) as well as the support provided by a feature to another (e.g., the impact, or confirmation, of evidence “circle” on the hypothesis “black”). Results indicated that participants’ judgments were deeply affected by impact, although they only should have depended on the probability distributions over the features, and that the dissociation between evidential impact and posterior probability increased the number of errors. The implications of these findings for lower and higher level cognitive models are discussed. (shrink)

In this paper, I argue that Islamic theism is best explained by the hypothesis of Divine Commission, whereby Muhammad is viewed as being divinely commissioned to serve the overall salvific purposes of God. To this end, I present three observation reports relating to Islamic theism and evaluate HDC against an alternative hypothesis, the hypothesis of Non-Commission whereby Muhammad is not viewed as being divinely commissioned. I argue that the probability of the observation reports is greater on the assumption that (...) HDC is true than on the assumption that NC is true. Accordingly, this gives us reason to prefer HDC as a better explanation of Islamic theism. (shrink)

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.

Terence Horgan defends the thirder position on the Sleeping Beauty problem, claiming that Beauty can, upon awakening during the experiment, engage in “synchronic Bayesian updating” on her knowledge that she is awake now in order to justify a 1/3 credence in heads. In a previous paper, I objected that epistemic probabilities are equivalent to rational degrees of belief given a possible epistemic situation and so the probability of Beauty’s indexical knowledge that she is awake now is necessarily 1, precluding (...) such updating. In response, Horgan maintains that the probability claims in his argument are to be taken, not as claims about possible rational degrees of belief, but rather as claims about “quantitative degrees of evidential support.” This paper argues that the most plausible account of quantitative degree of support, when conjoined with any of the three major accounts of indexical thought in such a way as to plausibly constrain rational credence, contradicts essential elements of Horgan’s argument. (shrink)

This chapter starts with a definition of theism, and distinguishes two ways in which empirical evidence might be relevant to its truth. Seven evidential objections to theism are spelled out. They rely on many different empirical facts, such as so‐called divine hiddenness, features of our universe, biological evolution, the occurrence of gratuitous natural evil, and the elimination of religious explanations during scientific progress. Finally, it is argued that a purely secular explanation of all religious beliefs in human cultures is (...) vastly superior to religious accounts. (shrink)

William Roche and Elliott Sober argue that explanatoriness is evidentially irrelevant. This conclusion is surprising since it conflicts with a plausible assumption—the fact that a hypothesis best explains a given set of data is evidence that the hypothesis is true. We argue that Roche and Sober's screening-off argument fails to account for a key aspect of evidential strength: the weight of a body of evidence. The weight of a body of evidence affects the resiliency of probabilities in the light (...) of new evidence. Thus, Roche and Sober are mistaken. Explanatoriness is evidentially relevant. (shrink)

The Variety of Evidence Thesis (VET) says that (ceteris paribus) the more diverse (or varied) of two bodies of evidence is the more confirmatory of a hypothesis H. Two recent types of Bayesian explication of VET account for the intuitive force of VET by defining variety as some function of the probabilities of the propositions which collectively constitute a body of evidence. I show that these two accounts of VET are not tracking a meaningful property of bodies of evidence, but (...) rather are tracking artifacts of how those bodies of evidence are described. According to each account, whether a body of evidence is more varied than another depends on how the bodies are split into parts. Furthermore, for each type of account there exists a way to redescribe the total evidence such that any two totalities are equally varied. ‡I would like to thank Elliott Sober for comments on multiple drafts of this paper, and Malcolm Forster, John Koolage, and Joel Velasco for comments on a verbal delivery. †To contact the author, please write to: Department of Philosophy, University of Wisconsin-Madison, 5185 Helen C. White Hall, 600 North Park Street, Madison, WI 53706; e-mail: [email protected]. (shrink)

Probability theory promises to deliver an exact and unified foundation for inquiry in epistemology and philosophy of science. But philosophy of religion is also fertile ground for the application of probabilistic thinking. This volume presents original contributions from twelve contemporary researchers, both established and emerging, to offer a representative sample of the work currently being carried out in this potentially rich field of inquiry. Grouped into five parts, the chapters span a broad range of traditional issues in religious epistemology. (...) The first three parts discuss the evidential impact of various considerations that have been brought to bear on the question of the existence of God. These include witness reports of the occurrence of miraculous events, the existence of complex biological adaptations, the apparent 'fine-tuning' for life of various physical constants and the existence of seemingly unnecessary evil. The fourth part addresses a number of issues raised by Pascal's famous pragmatic argument for theistic belief. A final part offers probabilistic perspectives on the rationality of faith and the epistemic significance of religious disagreement. (shrink)

Carnap's inductive logic (or confirmation) project is revisited from an "increase in firmness" (or probabilistic relevance) point of view. It is argued that Carnap's main desiderata can be satisfied in this setting, without the need for a theory of "logical probability." The emphasis here will be on explaining how Carnap's epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap's goal of bridging confirmation and credence, in favor of (...) bridging confirmation and evidential support. (shrink)

Probability can be used to measure degree of belief in two ways: objectively and subjectively. The objective measure is a measure of the rational degree of belief in a proposition given a set of evidential propositions. The subjective measure is the measure of a particular subject’s dispositions to decide between options. In both measures, certainty is a degree of belief 1. I will show, however, that there can be cases where one belief is stronger than another yet both (...) beliefs are plausibly measurable as objectively and subjectively certain. In ordinary language, we can say that while both beliefs are certain, one belief is more certain than the other. I will then propose second, non probabilistic dimension of measurement, which tracks this variation in certainty in such cases where the probability is 1. A general principle of rationality is that one’s subjective degree of belief should match the rational degree of belief given the evidence available. In this paper I hope to show that it is also a rational principle that the maximum stake size at which one should remain certain should match the rational weight of certainty given the evidence available. Neither objective nor subjective measures of certainty conform to the axioms of probability, but instead are measured in utility. This has the consequence that, although it is often rational to be certain to some degree, there is no such thing as absolute certainty. (shrink)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of (...) probability through a heterogeneous sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

In his Dialogues and Natural Religion, David Hume offered an inductive argument claiming that the observed mixture of good and evil in the world inductively justifies belief in indifferent first causes. The existence of a benevolent, omniscient God is rejected because it is much less probable. I show that a more comprehensive analysis of Hume’s argument applying Bayes’s Theorem indicates that if the good in our world greatly outweighs the evil, theists can then claim the inductive evidence actually provides confirmation (...) for classical theism. I provide some reasons for believing that good greatly exceeds evil. Views that good modestly exceeds evil or is balanced by evil are also considered. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities (...) are not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

Reasoning from inconclusive evidence, or ‘induction’, is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This Element explores the prospects of using probability theory to provide an inductive logic, a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, or (...) confirmation, indicated by an increased probability given the evidence than otherwise. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

Epistemic closure under known implication is the principle that knowledge of "p" and knowledge of "p implies q", together, imply knowledge of "q". This principle is intuitive, yet several putative counterexamples have been formulated against it. This paper addresses the question, why is epistemic closure both intuitive and prone to counterexamples? In particular, the paper examines whether probability theory can offer an answer to this question based on four strategies. The first probability-based strategy rests on the accumulation of (...) risks. The problem with this strategy is that risk accumulation cannot accommodate certain counterexamples to epistemic closure. The second strategy is based on the idea of evidential support, that is, a piece of evidence supports a proposition whenever it increases the probability of the proposition. This strategy makes progress and can accommodate certain putative counterexamples to closure. However, this strategy also gives rise to a number of counterintuitive results. Finally, there are two broadly probabilistic strategies, one based on the idea of resilient probability and the other on the idea of assumptions that are taken for granted. These strategies are promising but are prone to some of the shortcomings of the second strategy. All in all, I conclude that each strategy fails. Probability theory, then, is unlikely to offer the account we need. (shrink)

The phenomenon of mutual support presents a specific challenge to the foundationalist epistemologist: Is it possible to model mutual support accurately without using circles of evidential support? We argue that the appearance of loops of support arises from a failure to distinguish different synchronic lines of evidential force. The ban on loops should be clarified to exclude loops within any such line, and basing should be understood as taking place within lines of evidence. Uncertain propositions involved in mutual (...) support relations are conduits to each other of independent evidence originating ultimately in the foundations. We examine several putative examples of benign loops of support and show that, given the distinctions noted, they can be accurately modeled in a foundationalist fashion. We define an evidential "tangle," a relation among three propositions that appears to require a loop for modeling, and prove that all such tangles are trivial in a sense that precludes modeling them with an evidential circle. (shrink)

This book explores the probability of human resurrection based upon the existence of God and a set of evidential elements, namely, the Resurrection of Jesus, near-death experiences, and apparitions. William Hunt's argument employs subjective probability theory using Bayes' theorem in a cumulative process.

The purpose of this paper is to improve on the logical and measure-theoretic foundations for the notion of probability in the law of evidence, which were given in my contributions Åqvist [ (1990) Logical analysis of epistemic modality: an explication of the Bolding–Ekelöf degrees of evidential strength. In: Klami HT (ed) Rätt och Sanning (Law and Truth. A symposium on legal proof-theory in Uppsala May 1989). Iustus Förlag, Uppsala, pp 43–54; (1992) Towards a logical theory of legal evidence: (...) semantic analysis of the Bolding–Ekelöf degrees of evidential strength. In: Martino AA (ed) Expert systems in law. Elsevier Science Publishers BV, Amsterdam, North-Holland, pp 67–86]. The present approach agrees with the one adopted in those contributions in taking its main task to be that of providing a semantic analysis, or explication, of the so called Bolding–Ekelöf degrees of evidential strength (“proof-strength”) as applied to the establishment of matters of fact in law-courts. However, it differs from the one advocated in our earlier work on the subject in explicitly appealing to what is known as “Pro-et-Contra Argumentation”, after the famous Norwegian philosopher Arne Naess. It tries to bring out the logical form of that interesting kind of reasoning, at least in the context of the law of evidence. The formal techniques used here will be seen to be largely inspired by the important work done by Patrick Suppes, notably Suppes [(1957) Introduction to logic. van Nostrand, Princeton and (1972) Finite equal-interval measurement structures. Theoria 38:45–63]. (shrink)

The first part of this paper exposits William Rowe's latest version of the evidential argument from evil. Integral to this new version is what we can call the 'level-playing field' requirement, which regulates probability values. It is the argument of the second part of this paper that either the two premises of the new version are regulated by the level-playing-field requirement or they're not. If they are both regulated, then no one would be in position to rationally accept (...) one of those premises; if they're not both regulated, then the theist would have good reason to reject the one that is. Either way, Rowe's latest version of the evidential argument fails. (shrink)