The foundations of epistemological probability

Epistemological probability is the kind of probability relative to a body of evidence. Many philosophers, including Henry Kyburg and Roderick Chisholm, hold that all epistemological probabilities reflect a relation between an evidential body of propositions and other propositions. But this article argues that some epistemological probabilities for empirical propositions must be relative to non-propositional evidence, specifically the contents of non-propositional perceptual states. In doing so, the article distinguishes between internalism and externalism regarding epistemological probability, and argues for a version of awareness internalism. The article draws three main concluding lessons. First, epistemological probability is not to be identified with the sort of objective, experience-independent probability that is familiar from statistical and propensity interpretations of probability. Second, it is doubtful that epistemological probability is measurable, in any useful way, by real numbers, even if it admits of comparative assessments. Third, contrary to the familiar claim of C. I. Lewis, epistemological probability should not be viewed as requiring a basis of certainty.     

Gwiazda on the Bayesian Argument for God

Jeremy Gwiazda made two criticisms of my formulation in terms of Bayes’s theorem of my probabilistic argument for the existence of God. The first criticism depends on his assumption that I claim that the intrinsic probabilities of all propositions depend almost entirely on their simplicity; however, my claim is that that holds only insofar as those propositions are explanatory hypotheses. The second criticism depends on a claim that the intrinsic probabilities of exclusive and exhaustive explanatory hypotheses of a phenomenon must sum to 1; however it is only those probabilities plus the intrinsic probability of the non-occurrence of the phenomenon which must sum to 1.     

Reichenbach's metaphysical picture

Conclusion To recapitulate, then, for Reichenbach probability is the foundation of both metaphysics and epistemology. Metaphysically, probability is fundamental because it is the probability relations among the sequences of events in the world that gives rise to causality, time, and space. Epistemologically, probability is fundamental because empirical knowledge is simply knowledge of probabilities. Even knowledge of observation sentences is considered to be probabilistic knowledge by Reichenbach (EP, pp. 183–188), because Reichenbach's fallibilism leads him to hold that no observation sentence is absolutely incorrigible, and with the advance of scientific knowledge we need to inquire into the probability that our singular observation judgments may be in error.My aim here has not been to argue that Reichenbach succeeded in his magnificent attempt any more than Carnap succeeded in his. But I hope to have convinced you that is was one of the most magnificent attempts by any empiricist philosopher of this or of any other century, and I believe that the effort to understand it and to master its details will as richly repay us as the much greater effort which has been devoted to the study of Carnap's work has already repayed us.     

Contrastive Empiricism and Indispensability

The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this paper I defend the Quine-Putnam argument against Sober's objections. This revised version was published online in July 2006 with corrections to the Cover Date.     

The frequentist implications of optional stopping on Bayesian hypothesis tests

Null hypothesis significance testing (NHST) is the most commonly used statistical methodology in psychology. The probability of achieving a value as extreme or more extreme than the statistic obtained from the data is evaluated, and if it is low enough, the null hypothesis is rejected. However, because common experimental practice often clashes with the assumptions underlying NHST, these calculated probabilities are often incorrect. Most commonly, experimenters use tests that assume that sample sizes are fixed in advance of data collection but then use the data to determine when to stop; in the limit, experimenters can use data monitoring to guarantee that the null hypothesis will be rejected. Bayesian hypothesis testing (BHT) provides a solution to these ills because the stopping rule used is irrelevant to the calculation of a Bayes factor. In addition, there are strong mathematical guarantees on the frequentist properties of BHT that are comforting for researchers concerned that stopping rules could influence the Bayes factors produced. Here, we show that these guaranteed bounds have limited scope and often do not apply in psychological research. Specifically, we quantitatively demonstrate the impact of optional stopping on the resulting Bayes factors in two common situations: (1) when the truth is a combination of the hypotheses, such as in a heterogeneous population, and (2) when a hypothesis is composite—taking multiple parameter values—such as the alternative hypothesis in a t-test. We found that, for these situations, while the Bayesian interpretation remains correct regardless of the stopping rule used, the choice of stopping rule can, in some situations, greatly increase the chance of experimenters finding evidence in the direction they desire. We suggest ways to control these frequentist implications of stopping rules on BHT.     

A frequentist interpretation of probability for model-based inductive inference

The main objective of the paper is to propose a frequentist interpretation of probability in the context of model-based induction, anchored on the Strong Law of Large Numbers (SLLN) and justifiable on empirical grounds. It is argued that the prevailing views in philosophy of science concerning induction and the frequentist interpretation of probability are unduly influenced by enumerative induction, and the von Mises rendering, both of which are at odds with frequentist model-based induction that dominates current practice. The differences between the two perspectives are brought out with a view to defend the model-based frequentist interpretation of probability against certain well-known charges, including [i] the circularity of its definition, [ii] its inability to assign ‘single event’ probabilities, and [iii] its reliance on ‘random samples’. It is argued that charges [i]–[ii] stem from misidentifying the frequentist ‘long-run’ with the von Mises collective. In contrast, the defining characteristic of the long-run metaphor associated with model-based induction is neither its temporal nor its physical dimension, but its repeatability (in principle); an attribute that renders it operational in practice. It is also argued that the notion of a statistical model can easily accommodate non-IID samples, rendering charge [iii] simply misinformed.     

Bayesian Confirmation Theory and The Likelihood Principle

The likelihood principle (LP) is a core issue in disagreements between Bayesian and frequentist statistical theories. Yet statements of the LP are often ambiguous, while arguments for why a Bayesian must accept it rely upon unexamined implicit premises. I distinguish two propositions associated with the LP, which I label LP1 and LP2. I maintain that there is a compelling Bayesian argument for LP1, based upon strict conditionalization, standard Bayesian decision theory, and a proposition I call the practical relevance principle. In contrast, I argue that there is no similarly compelling argument for or against LP2. I suggest that these conclusions lead to a restrictedly pluralistic view of Bayesian confirmation measures.     

On the Propositional Treatment of Anatmavāda in Early Buddhism and Ātmavāda in Hinduism

As propositions, Anatmavāda and ātmavāda are simply negations of one another. Thus whatever serves as a criterion for truth of the one must serve as a criterion for the other. When we treat them both as a priori propositions, I claim that we are unable to determine their truth value. But if we treat them both as a posteriori propositions, I argue, we are only able to determine their truth value if we attain unqualified omniscience. Because the Hindu account of knowing is far more conducive to the idea of unqualified omniscience, we might be tempted to assert that the empirical verification of these doctrines taken as propositions is far more likely in the Hindu tradition than the early Buddhist one. However, 'empirical omniscience' carries us very far from received views, thus I conclude that it makes no sense to treat these doctrines as truth-valued propositions.     

Walter the banker: the conjunction fallacy reconsidered

In a famous experiment by Tversky and Kahneman (Psychol Rev 90:293–315, 1983), featuring Linda the bank teller, the participants assign a higher probability to a conjunction of propositions than to one of the conjuncts, thereby seemingly committing a probabilistic fallacy. In this paper, we discuss a slightly different example featuring someone named Walter, who also happens to work at a bank, and argue that, in this example, it is rational to assign a higher probability to the conjunction of suitably chosen propositions than to one of the conjuncts. By pointing out the similarities between Tversky and Kahneman’s experiment and our example, we argue that the participants in the experiment may assign probabilities to the propositions in question in such a way that it is also rational for them to give the conjunction a higher probability than one of the conjuncts.     

Why we Believe

The radical probabilist counsels the prudent never to put away uncertainty, and hence always to balance judgment with probabilities of various sizes. Against this counsel I shall advise in favor of the practice of full belief — at least for some occasions. This advice rests on the fact that it is sometimes in a person's interests to accept certain propositions as a means of bringing it about that others recognize oneself as having accepted those propositions. With the pragmatists, therefore, I shall reject the view that belief formation must in every instance be a truth-directed affair. Unlike the pragmatists, however, I shall conclude that the enterprise of belief formation is not directed exclusively, or even primarily, at attaining knowledge. In other words, pursuit of that which it profits to believe, on the one hand, and pursuit of knowledge on the other, are distinct enterprises, which overlap (when they do) only accidentally. This revised version was published online in August 2006 with corrections to the Cover Date.     

THE TRIPLISM OF PRACTICAL REASON

There can be reasons for belief, for action, and for feeling. In each case, knowledge of such reasons requires non‐empirical knowledge of some truths about them: these will be truths about what there is reason to believe, to feel, or to do – either outright or on condition of certain facts obtaining. Call these a priori truths about reasons, ‘norms’. Norms are a priori true propositions about reasons. It's an epistemic norm that if something's a good explanation that's a reason to believe it. It's an evaluative norm that if someone's cheated you that's a reason to be annoyed with them. There are many evaluative norms, relating to a variety of feelings. Equally, there may be various epistemic norms, even though in this case they all relate to belief. My concern here, however, is with practical norms: a priori truths about what there is reason to do. I have a suggestion about what fundamental practical norms there are, which I would like to describe and explain. It is that there are just three distinct kinds of practical norm governing what there is reason to do – three categories or generic sources of practical normativity, one may say. I call them the Bridge principle, the principle of Good, and the Demand principle – Bridge, Good and Demand for short. I have said more about them in my book, The Domain of Reasons; 1 here my aim is simply to set them out and sketch some questions to which this ‘triplism of practical reason’ 2 gives rise. In particular, since these norms are about practical reasons, not about morality, a question I'll touch on is how moral obligation comes onto the scene.     

Against Probabilistic Measures of Coherence

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and if P(B)=P(C). But such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability.     

Content Validity and Cognitive Tests: Response to Kehoe (), Ployhart (), and Sackett ()

The most important fact emerging from the combination of my article and the three commentaries is the consensus judgment that content validity is appropriate scientifically and professionally for use with tests of specific cognitive skills used in job performance. This is important because the 1978 Uniform Guidelines on Employee Selection Procedures have typically been interpreted as not permitting such usage, and this is particularly the case in the interpretation given to the Guidelines by federal government enforcement agencies. Although the Society for Industrial and Organizational Psychology Principles and the Standards do not prohibit such usage, many industrial–organizational psychologists believe that it is not professionally or scientifically appropriate to employ content validity methods with cognitive measures. The hope is that this series will convince them otherwise. On this point, all four authors in the series are in agreement. The major disagreement among us concerns whether specific cognitive skills used in content valid tests must be considered constructs or not. My position, and apparently that of Kehoe, is that they need not be so considered. I argue that constructs must be invoked only in the context of a substantive theory. Sackett and Ployhart, on the other hand, argue that all measures taken on people must be viewed as constructs, regardless of whether any theoretical propositions and assumptions are involved. In this response, I present reasons why this need not be the case.     

The necessary a posteriori: A response to Tichý

Conclusion Some of Tichý's conclusions rest on an assumption about substitutivity which Kripke would not accept. If we grant the assumption, then Tichý successfully shows that we can discover true identity statements involving names a priori, but not that we can discover a priori what properties things have essentially. Many of Tichý's arguments require an implausible rejection of the possibility of indirect belief as described in Section III. 25Are there necessary a posteriori propositions? I have argued that we certainly can discover necessary propositions a posteriori, but have left it an open question whether there are necessary propositions which we can only discover a posteriori.What effect do the considerations here presented have on the positivist doctrine that the a priori and the necessary coincide? My explanation of how we discover necessary propositions a posteriori involves our believing them indirectly, in virtue of believing contingent propositions. I would argue that Kripke's examples of the contingent a priori involve, similarly, our believing the contingent propositions in directly, in virtue of believing necessary propositions.This suggests that a reformulation of the positivist thesis along something like the following lines may well be correct. Let us say that someone directly believes a proposition just in case he could not fail to believe it without being in a different cognitive state. Then perhaps one can directly believe a proposition on the basis of a priori evidence only if it is necessary, and can directly believe a proposition on the basis of a posteriori evidence only if it is contingent.     

Probability,Approximate Truth,and Truthlikeness: More Ways out of the Preface Paradox

The so-called Preface Paradox seems to show that one can rationally believe two logically incompatible propositions. We address this puzzle, relying on the notions of truthlikeness and approximate truth as studied within the post-Popperian research programme on verisimilitude. In particular, we show that adequately combining probability, approximate truth, and truthlikeness leads to an explanation of how rational belief is possible in the face of the Preface Paradox. We argue that our account is superior to other solutions of the paradox, including a recent one advanced by Hannes Leitgeb (Analysis 74.1).     

Motivated reasoning and verbal vs. numerical probability assessment: Evidence from an accounting context

A perplexing yet persistent empirical finding is that individuals assess probabilities in words and in numbers nearly equivalently, and theorists have called for future research to search for factors that cause differences. This study uses an accounting context in which individuals are commonly motivated to reach preferred (rather than accurate) conclusions. Within this context, I predict new differences between verbal and numerical probability assessments, as follows: first, individuals will justify an optimistic verbal assessment (e.g., somewhat possible) by retaining the option of re-defining it, in case of negative outcomes, as though the phrase really means something different, and, for that matter, means more things. This re-definition will maintain some connection to the original meaning of the phrase, but de-emphasized relative to the new meaning. Second, based on this behavior, I also predict individuals’ verbal probability assessments to be (1) more biased and yet (2) perceived as more justifiable than their numerical assessments. I find supportive evidence in an experiment designed to test the hypotheses. This study contributes to motivated reasoning and probability assessment theories (1) with new evidence of how individuals can word-smith in multiple attributes of a phrase to justify reaching a preferred conclusion, and (2) with new, reliable differences between verbal and numerical probability assessments. This study has important theoretical and practical implications relevant to organizational contexts in which people assess the likelihoods of uncertainties in words or numbers, and with motivations to reach a preferred conclusion.     

Acting and Believing Under the Guise of Normative Reasons

In this paper, I defend an account of the reasons for which we act, believe, and so on for any Ф such that there can be reasons for which we Ф. Such reasons are standardly called motivating reasons. I argue that three dominant views of motivating reasons (psychologism, factualism and disjunctivism) all fail to capture the ordinary concept of a motivating reason. I show this by drawing out three constraints on what motivating reasons must be, and demonstrating how each view fails to satisfy at least one of these constraints. I then propose and defend my own account of motivating reasons, which I call the Guise of Normative Reasons Account. On the account I defend, motivating reasons are propositions. A proposition is the reason for which someone Ф‐s when (a) she represents that proposition as a normative reason to Ф, and (b) her representation explains, in the right way, her Ф‐ing. As I argue, the Guise of Normative Reasons Account satisfies all three constraints on what motivating reasons must be, and weathers several objections that might be leveled against propositionalist views.     

The reference class problem is your problem too

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains.     

Getting fancy with probability

There are a number of reasons for being interested in uncertainty, and there are also a number of uncertainty formalisms. These formalisms are not unrelated. It is argued that they can all be reflected as special cases of the approach of taking probabilities to be determined by sets of probability functions defined on an algebra of statements. Thus, interval probabilities should be construed as maximum and minimum probabilities within a set of distributions, Glenn Shafer's belief functions should be construed as lower probabilities, etc. Updating probabilities introduces new considerations, and it is shown that the representation of belief as a set of probabilities conflicts in this regard with the updating procedures advocated by Shafer. The attempt to make subjectivistic probability plausible as a doctrine of rational belief by making it more flowery — i.e., by adding new dimensions — does not succeed. But, if one is going to represent beliefs by sets of distributions, those sets of distributions might as well be based in statistical knowledge, as they are in epistemological or evidential probability.