I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had (...) B as their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

In circumstances where one proposition A is propositional evidence for another proposition B, my believing A can confer warrant upon B. I use the term “epistemic probability” to refer to the relationship between a pair of propositions A and B when A is propositional evidence for B; more precisely, in those cases, I shall say that the epistemic conditional probability of B on A is high. In this chapter and the next, I concern myself with an analysis of (...) epistemic conditional probability. The first thing to see, in trying to get a general grasp of this topic, is to note the divide between epistemic probability and objective probability. In this chapter, I distinguish the former from the latter and point out some debilitating problems with the three main accounts of the former ; in the next chapter, I propose what I hope is a better substitute. (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)

Many philosophers think that part of what makes an event lucky concerns how probable that event is. In this paper, I argue that an epistemic probability account of luck successfully resists recent arguments that all theories of luck, including probability theories, are subject to counterexample (Hales 2016). I argue that an event is lucky if and only if it is significant and sufficiently improbable. An event is significant when, given some reflection, the subject would regard the event as significant, (...) and the event’s epistemic probability, determined by the subject’s evidence, is the only kind of probability that directly bears on whether or not the event is lucky. I conclude with some lessons that are applicable to probability theorists of luck generally, including those defending non-epistemic probability theories. (shrink)

Epistemology in modern times has been much influenced by the growth of the mathematical theory of probability. The reverse influence is less obvious, but perhaps equally important. This interrelation affords, moreover, an excellent example of the way philosophic ideas, while directing attention in one fruitful line of inquiry, at the same time may inhibit other discoveries until long overdue.

Is consistency always epistemically virtuous? In this paper, I examine one threat to the traditional view that consistency is a minimum requirement for rational belief. Central to the argument is the notion of epistemic probability, understood as the degree of support or confirmation provided by the total available evidence. My strategy in examining this argument is to apply analogous reasoning to carefully tailored examples. The conclusions which emerge are substantive, informative and utterly implausible. I conclude, first, that the argument (...) for inconsistency fails and, second, that it fails because epistemic probability does not conform to the axioms of the probability calculus. A plausible alternate model for determining degree of support is briefly considered. (shrink)

Epistemic probability theories of luck come in two versions. They are easiest to distinguish by the epistemic property they claim eliminates luck. One view says that the property is knowledge. The other view says that the property is being guaranteed by a subject’s evidence. Asbjørn Steglich-Petersen defends the Knowledge Account (KA). He has recently argued that his view is preferable to my Epistemic Analysis of Luck (EAL), which defines luck in terms of evidential probability. In this paper, (...) I defend EAL against Steglich-Petersen’s arguments, clarify the view, and argue for the explanatory significance of EAL with respect to some core epistemological issues. My overall goal is to show that an epistemic probability account of luck rooted in the concepts of evidence and evidential support remains a viable and fruitful overall account of luck. (shrink)

Philosophers have often attempted to use counterfactual conditionals to analyze probability. This article focuses on counterfactual analyzes of epistemic probability by Alvin Plantinga and Peter van Inwagen. I argue that a certain type of counterfactual situation creates problems for these analyses. I then argue that Plantinga's intuition about the role of warrant in epistemic probability is mistaken. Both van Inwagen's and Plantinga's intuitions about epistemic probability are flawed.

In Ch. 8, I distinguished epistemic probability from objective probability and then pointed out some debilitating problems with the three main accounts of epistemic probability. In this chapter, I propose my own account of epistemic probability. I first distinguish between two sides to epistemic probability, which I call the objective component and the normative component. In typical cases where we assert that some proposition is epistemically probable, two things get asserted: that the proposition is objectively probable (...) with respect to the evidence, and that it is rational to place a high degree of confidence in the proposition, given the circumstances of having as evidence what in fact we do have as evidence. Attempting to unpack these two components, I sketch the view that the normative component of the epistemic conditional probability of A on B is the interval containing the degrees of belief a rational person could have in A, provided she believed B and was aware that she believed B, considered the evidential bearing of B on A, had no other source of warrant for B or its denial, and had no defeater for the warrant, if any, accruing to A or its denial by virtue of being thus believed on the basis of B. (shrink)

Some recent work in philosophy of religion addresses what can be called the “axiological question,” i.e., regardless of whether God exists, would it be good or bad if God exists? Would the existence of God make the world a better or a worse place? Call the view that the existence of God would make the world a better place “Pro-Theism.” We argue that Pro-Theism is not implausible, and moreover, many Theists, at least, (often implicitly) think that it is true. That (...) is, many Theists think that various good outcomes would arise if Theism is true. We then discuss work in cognitive science concerning human cognitive bias, before discussing two noteworthy attempts to show that at least some religious beliefs arise because of cognitive bias: Hume’s, and Draper’s and Nichols’s. We then argue that, as a result of certain cognitive biases that result when good outcomes might be at stake, Pro-Theism causes many Theists to inflate the epistemic probability that God exists, and as a result, Theists should lower the probability they assign to God’s existence. Finally, based our arguments, we develop a novel objection to Pascal’s wager. (shrink)

This paper develops an information-sensitive theory of the semantics and probability of conditionals and statements involving epistemic modals. The theory validates a number of principles linking probability and modality, including the principle that the probability of a conditional If A, then C equals the probability of C, updated with A. The theory avoids so-called triviality results, which are standardly taken to show that principles of this sort cannot be validated. To achieve this, we deny that rational agents update their (...) credences via conditionalization. We offer a new rule of update, Hyperconditionalization, which agrees with Conditionalization whenever nonmodal statements are at stake but differs for modal and conditional sentences. (shrink)

This paper contributes to the investigation of the nature of the relation between probability theory and ranking theory. The paper aims at explaining the structural harmony between the laws of probability theory and those of ranking theory in a way that respects the foundational dualistic attitude developed by Spohn in The Laws of Belief. The paper argues that the so called atomic translation family satisfies the desiderata and does so in the ‘best’ possible way. On the one hand, the atomic (...) translation can be seen as ensuring maximal order agreement between rankings and probability—more would lead to trivialising dualism. On the other hand, the atomic translation also is assumption minimal, i.e., a minimal set of justifiable correspondences or translation principles suffice to derive the translation family, and this result is relatively robust. (shrink)

The representational epistemic approach to the design of visual displays and notation systems advocates encoding the fundamental conceptual structure of a knowledge domain directly in the structure of a representational system. It is claimed that representations so designed will benefit from greater semantic transparency, which enhances comprehension and ease of learning, and plastic generativity, which makes the meaningful manipulation of the representation easier and less error prone. Epistemic principles for encoding fundamental conceptual structures directly in representational schemes are (...) described. The diagrammatic recodification of probability theory is undertaken to demonstrate how the fundamental conceptual structure of a knowledge domain can be analyzed, how the identified conceptual structure may be encoded in a representational system, and the cognitive benefits that follow. An experiment shows the new probability space diagrams are superior to the conventional approach for learning this conceptually challenging topic. (shrink)

Multiple epistemological programs make use of intuitive judgments pertaining to an individual’s ability to gain knowledge from exclusively probabilistic/statistical information. This paper argues that these judgments likely form without deference to such information, instead being a function of the degree to which having knowledge is representative of an agent. Thus, these judgments fit the pattern of formation via a representativeness heuristic, like that famously described by Kahneman and Tversky to explain similar probabilistic judgments. Given this broad insensitivity to probabilistic/statistical information, (...) it directly follows that these epistemic judgments are insensitive to a given agent’s epistemic status. From this, the paper concludes that, breaking with common epistemological practice, we cannot assume that such judgments are reliable. (shrink)

According to a standard view, quantum mechanics is a contextual theory and quantum probability does not satisfy Kolmogorov’s axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L, a classical probability measure on it and a family of classical probability measures on sets of μ-contexts, each element of the family (...) corresponding to a measurement procedure. By using only Kolmogorovian probability measures we can thus define mean conditional probabilities on the set of properties of any quantum system that admit an epistemic interpretation but are not bound to satisfy Kolmogorov’s axioms. The generalized probability measures associated with states in QM can then be seen as special cases of these mean probabilities, which explains how they can be non-classical and provides them with an epistemic interpretation. Moreover, the distinction between compatible and incompatible properties is explained in a natural way, and purely theoretical classical conditional probabilities coexist with empirically testable quantum conditional probabilities. (shrink)

This paper is concerned with representations of belief by means of nonadditive probabilities of the Dempster-Shafer (DS) type. After surveying some foundational issues and results in the D.S. theory, including Suppes's related contributions, the paper proceeds to analyze the connection of the D.S. theory with some of the work currently pursued in epistemic logic. A preliminary investigation of the modal logic of belief functions à la Shafer is made. There it is shown that the Alchourrron-Gärdenfors-Makinson (A.G.M.) logic of (...) belief change is closely related to the D.S. theory. The final section compares the critique of Bayesianism which underlies the present paper with some important objections raised by Suppes against this doctrine. -/- . (shrink)

This paper puts forward a novel pluralist theory of epistemic justification that brings together two competing views in the literature—probabilistic and non-probabilistic accounts of justification. The first part of the paper motivates the new theory by arguing that neither probabilistic nor non-probabilistic accounts alone are wholly satisfactory. The second part puts forward what I call the Functional Theory of Justification. The key merit of the new theory is that it combines the most attractive features of both probabilistic and non-probabilistic (...) accounts of justification while avoiding their most serious shortcomings. The paper also provides a blueprint for future pluralist projects in epistemology. (shrink)

A pluralistic scientific method is one that incorporates a variety of points of view in scientific inquiry. This paper investigates one example of pluralistic method: the use of weighted averaging in probability estimation. I consider two methods of weight determination, one based on disjoint evidence possession and the other on track record. I argue that weighted averaging provides a rational procedure for probability estimation under certain conditions. I consider a strategy for calculating ‘mixed weights’ which incorporate mixed information about agent (...) credibility. I address various objections to the weighted averaging technique and conclude that the technique is a promising one in various respects. (shrink)

In this article, the epistemological interpretation of the relationship between concepts of relativism, beliefs, and probability ensures a defense of two theses, namely, (i) epistemic relativism refers to attitudes that depend on the repetition and anchoring of probabilistic beliefs, and (ii) Popper’s propensity interpretation of probability discloses the connections between relativity, probability, and collective subjectivity. The propensity interpretation brings a framework for describing the role of collective subjectivity in epistemic systems. This approach, as an acceptable epistemological stance, is (...) related to the variability of anchored collective beliefs. The repetition and anchoring of subjective probabilistic beliefs becomes a relative reference frame for evaluating the elements of knowledge. With this in mind, each particular way of considering probability indicates a manner of distinguishing epistemic systems. In epistemic systems, the statements are collective points of view expressed in the form ‘p is justified when belief in p is probably reliable.’ The statements can be accurate in a particular epistemic system but inaccurate in other systems. As a result, by avoiding the ‘all statements must be true or false’ requirement, each proponent of a certain epistemic system can build a probabilistic picture of the world. In this picture, not every statement needs to be ‘true’ when and if the statement expresses a collective point of view that has a propensity to be reliable and justified. (shrink)

I present a straightforward objection to the view that what we know has epistemic probability 1: when combined with Bayesian decision theory, the view seems to entail implausible conclusions concerning rational choice. I consider and reject three responses. The first holds that the fault is with decision theory, rather than the view that knowledge has probability 1. The second two try to reconcile the claim that knowledge has probability 1 with decision theory by appealing to contextualism and sensitive invariantism, (...) respectively. I argue that each response fails, and that we can hold on to much of what was attractive in the responses while denying that what we know has probability 1. (shrink)

Recognizing that probability (the Greek doxa) was understood in pre-modern theories as the polar opposite of certainty (episteme), the author of this study elaborates the forms which these polar opposites have taken in some twentieth century writers and then, in greater detail, in the writings of Thomas Aquinas. Profiting from subsequent more sophisticated theories of probability, he examines how Aquinas’s judgments about everything from God to gossip depend on schematizations of the polarity between the systematic and the non-systematic: revelation/reason, science/opinion, (...) saints/philosophers, Aristotle/others. Post-medieval developments have provided the means to discern within Thomas’s thought both a logical theory and a kind of relative frequency theory of probability. The former depends on Aristotle’s theory of demonstration, the latter on his theory of an orderly cosmos; but each as used by Thomas lead to convictions that are sometimes naive, sometimes amusing or even terrifying. (shrink)

Second or higher order probabilities have commonly been viewed with scepticism by those working within the realm of probability and decision theory. The aim of the present note is to show how the notion of second order probabilities can add to our understanding of judgmental and decision processes and how the traditional framework of Bayesian decision theory can be extended in a fruitful way by taking such entities into account. Section one consists of a brief account of arguments (...) put forth against higher order probabilities as well as of counterarguments. In order to provide an example of the applicability of second order probabilities a decision theory encompassing such probabilities will be presented in section two. In section three I will try to emphasize the value of second order probabilities for a deeper and more complete understanding of the notion of risk. (shrink)

This chapter will review selected aspects of the terrain of discussions about probabilities in statistical mechanics (with no pretensions to exhaustiveness, though the major issues will be touched upon), and will argue for a number of claims. None of the claims to be defended is entirely original, but all deserve emphasis. The first, and least controversial, is that probabilistic notions are needed to make sense of statistical mechanics. The reason for this is the same reason that convinced Maxwell, Gibbs, (...) and Boltzmann that probabilities would be needed, namely, that the second law of thermodynamics, which in its original formulation says that certain processes are impossible, must, on the kinetic theory, be replaced by a weaker formulation according to which what the original version deems impossible is merely improbable. Second is that we ought not take the standard measures invoked in equilibrium statistical mechanics as giving, in any sense, the correct probabilities about microstates of the system. We can settle for a much weaker claim: that the probabilities for outcomes of experiments yielded by the standard distributions are effectively the same as those yielded by any distribution that we should take as a representing probabilities over microstates. Lastly, (and most controversially): in asking about the status of probabilities in statistical mechanics, the familiar dichotomy between epistemic probabilities (credences, or degrees of belief) and ontic (physical) probabilities is insufficient; the concept of probability that is best suited to the needs of statistical mechanics is one that combines epistemic and physical considerations. (shrink)

In previous work, I defend the following disparity between moral and epistemic facts: whereas moral facts are irreducibly normative, epistemic facts – facts such as that some subject is epistemically justified in believing something – are reducible to facts from some other domain (such as facts about probabilities). This moral-epistemic disparity is significant because it undercuts an important kind of argument for robust moral realism. My defense of epistemic reductionism and of the moral-epistemic disparity (...) has been criticized by Richard Rowland (2013) and by Terence Cuneo and Christos Kyriacou (forthcoming). This paper aims to rebut these criticisms and, more generally, to clarify and strengthen the case for epistemic reductionism and the moral-epistemic disparity. (shrink)

_Probability: A Philosophical Introduction_ introduces and explains the principal concepts and applications of probability. It is intended for philosophers and others who want to understand probability as we all apply it in our working and everyday lives. The book is not a course in mathematical probability, of which it uses only the simplest results, and avoids all needless technicality. The role of probability in modern theories of knowledge, inference, induction, causation, laws of nature, action and decision-making makes an understanding of (...) it especially important to philosophers and students of philosophy, to whom this book will be invaluable both as a textbook and a work of reference. In this book D. H. Mellor discusses the three basic kinds of probability – physical, epistemic, and subjective – and introduces and assesses the main theories and interpretations of them. The topics and concepts covered include: * chance * frequency * possibility * propensity * credence * confirmation * Bayesianism. _Probability: A Philosophical Introduction_ is essential reading for all philosophy students and others who encounter or need to apply ideas of probability. (shrink)

Epistemic modal operators give rise to something very like, but also very unlike, Moore's paradox. I set out the puzzling phenomena, explain why a standard relational semantics for these operators cannot handle them, and recommend an alternative semantics. A pragmatics appropriate to the semantics is developed and interactions between the semantics, the pragmatics, and the definition of consequence are investigated. The semantics is then extended to probability operators. Some problems and prospects for probabilistic representations of content and context are (...) explored. (shrink)

Relations between conditional probabilities, revisions of probabilities in the light of new information, and conditions of ideal rationality are discussed herein. The formal character of conditional probabilities, and their significance for epistemic states of agents is taken up. Then principles are considered that would, under certain conditions, equate rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. And lastly the possibility of kinds of ' books ' against known non-conditionalizers (...) is explored, and the question is taken up, What, if anything, would be wrong with a person against whom such a book could be made? (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)

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. -/- Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion that (...) probabilities of conditionals violate standard probability theory. -/- This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. -/- We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. -/- We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. -/- Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (shrink)

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)

In two recent papers Barry Loewer (2001, 2004) has suggested to interpret probabilities in statistical mechanics as Humean chances in David Lewis’ (1994) sense. I first give a precise formulation of this proposal, then raise two fundamental objections, and finally conclude that these can be overcome only at the price of interpreting these probabilities epistemically.

This paper addresses the question of how we should regard the probability distributions introduced into statistical mechanics. It will be argued that it is problematic to take them either as purely ontic, or purely epistemic. I will propose a third alternative: they are almost objective probabilities, or epistemic chances. The definition of such probabilities involves an interweaving of epistemic and physical considerations, and thus they cannot be classified as either purely epistemic or purely ontic. (...) This conception, it will be argued, resolves some of the puzzles associated with statistical mechanical probabilities: it explains how probabilistic posits introduced on the basis of incomplete knowledge can yield testable predictions, and it also bypasses the problem of disastrous retrodictions, that is, the fact the standard equilibrium measures yield high probability of the system being in equilibrium in the recent past, even when we know otherwise. As the problem does not arise on the conception of probabilities considered here, there is no need to invoke a Past Hypothesis as a special posit to avoid it. (shrink)