Bayesian Reasoning, Misc

Trust is defined as a belief of a human H (‘the trustor’) about the ability of an agent A (the ‘trustee’) to perform future action(s). We adopt here dispositionalism and internalism about trust: H trusts A iff A has some internal dispositions as competences. The dispositional competences of A are high-level metacognitive requirements, in the line of a naturalized virtue epistemology. (Sosa, Carter) We advance a Bayesian model of two (i) confidence in the decision and (ii) model uncertainty. To trust (...) A, H demands A to be self-assertive about confidence and able to self-correct its own models. In the Bayesian approach trust can be applied not only to humans, but to artificial agents (e.g. Machine Learning algorithms). We explain the advantage the metacognitive trust when compared to mainstream approaches and how it relates to virtue epistemology. The metacognitive ethics of trust is swiftly discussed. (shrink)

The Sleeping Beauty problem has attracted considerable attention in the literature as a paradigmatic example of how self-locating uncertainty creates problems for the Bayesian principles of Conditionalization and Reflection. Furthermore, it is also thought to raise serious issues for diachronic Dutch Book arguments. I show that, contrary to what is commonly accepted, it is possible to represent the Sleeping Beauty problem within a standard Bayesian framework. Once the problem is correctly represented, the ‘thirder’ solution satisfies standard rationality principles, vindicating why (...) it is not vulnerable to diachronic Dutch Book arguments. Moreover, the diachronic Dutch Books against the ‘halfer’ solutions fail to undermine the standard arguments for Conditionalization. The main upshot that emerges from my discussion is that the disagreement between different solutions does not challenge the applicability of Bayesian reasoning to centered settings, nor the commitment to Conditionalization, but is instead an instance of the familiar problem of choosing the priors. (shrink)

Intellectual progress involves forming a more accurate picture of the world. But it also figuring out which concepts to use for theorizing about the world. Bayesian epistemology has had much to say about the former aspect of our cognitive lives, but little if at all about the latter. I outline a framework for formulating questions about conceptual change in a broadly Bayesian framework. By enriching the resources of Epistemic Utility Theory with a more expansive conception of epistemic value, I offer (...) a picture of our cognitive economy on which adopting new conceptual tools can sometimes be epistemically rational. (shrink)

It is often argued that while biases routinely influence the generation of scientific theories (in the ‘context of discovery’), a subsequent rational evaluation of such theories (in the ‘context of justification’) will ensure that biases do not affect which theories are ultimately accepted. Against this line of thought, this paper shows that the existence of certain kinds of biases at the generation-stage implies the existence of biases at the evaluation-stage. The key argumentative move is to recognize that a scientist who (...) comes up with a new theory about some phenomena has thereby gained an unusual type of evidence, viz. information about the space of theories that could be true of the phenomena. It follows that if there is bias in the generation of scientific theories in a given domain, then the rational evaluation of theories with reference to the total evidence in that domain will also be biased. (shrink)

According to certain normative theories in epistemology, rationality requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omniscience requirement in Bayesian epistemology. Building on a key insight by Ian Hacking (1967), I develop a version of Bayesianism that permits logical ignorance. This includes an account of the synchronic norms that govern a logically ignorant individual at any given time, (...) as well as an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations. (shrink)

It is sometimes claimed that the Bayesian framework automatically implements Ockham’s razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. It is shown here that the automatic razor doesn’t in fact cut it for certain mundane curve-fitting problems.

Resource rationality may explain suboptimal patterns of reasoning; but what of “anti-Bayesian” effects where the mind updates in a direction opposite the one it should? We present two phenomena — belief polarization and the size-weight illusion — that are not obviously explained by performance- or resource-based constraints, nor by the authors’ brief discussion of reference repulsion. Can resource rationality accommodate them?

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning by the (...) ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem. (shrink)

Chancy modus ponens is the following inference scheme: ‘probably φ’, ‘if φ, then ψ’, therefore, ‘probably ψ’. I argue that Chancy modus ponens is invalid in general. I further argue that the invalidity of Chancy modus ponens sheds new light on the alleged counterexample to modus ponens presented by McGee. I close by observing that, although Chancy modus ponens is invalid in general, we can recover a restricted sense in which this scheme of inference is valid.

Orthodox Bayesianism is a highly idealized theory of how we ought to live our epistemic lives. One of the most widely discussed idealizations is that of logical omniscience: the assumption that an agent’s degrees of belief must be probabilistically coherent to be rational. It is widely agreed that this assumption is problematic if we want to reason about bounded rationality, logical learning, or other aspects of non-ideal epistemic agency. Yet, we still lack a satisfying way to avoid logical omniscience within (...) a Bayesian framework. Some proposals merely replace logical omniscience with a different logical idealization; others sacrifice all traits of logical competence on the altar of logical non-omniscience. We think a better strategy is available: by enriching the Bayesian framework with tools that allow us to capture what agents can and cannot infer given their limited cognitive resources, we can avoid logical omniscience while retaining the idea that rational degrees of belief are in an important way constrained by the laws of probability. In this paper, we offer a formal implementation of this strategy, show how the resulting framework solves the problem of logical omniscience, and compare it to orthodox Bayesianism as we know it. (shrink)

How were reliable predictions made before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? The book examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. Also included are the problem of induction before Hume, design arguments for (...) the existence of God, and theories on how to evaluate scientific and historical hypotheses. It is explained how Pascal and Fermat's work on chance arose out of legal thought on aleatory contracts. The book interprets pre-Pascalian unquantified probability in a generally objective Bayesian or logical probabilist sense. (shrink)

In this chapter I analyse an objection to phenomenal conservatism to the effect that phenomenal conservatism is unacceptable because it is incompatible with Bayesianism. I consider a few responses to it and dismiss them as misled or problematic. Then, I argue that this objection doesn’t go through because it rests on an implausible formalization of the notion of seeming-based justification. In the final part of the chapter, I investigate how seeming-based justification and justification based on one’s reflective belief that one (...) has a seeming interact with one another. (shrink)

We conceptualize observation selection effects (OSEs) by considering how a shift from one process of observation to another affects discrimination-conduciveness, by which we mean the degree to which possible observations discriminate between hypotheses, given the observation process at work. OSEs in this sense come in degrees and are causal, where the cause is the shift in process, and the effect is a change in degree of discrimination-conduciveness. We contrast our understanding of OSEs with others that have appeared in the literature. (...) After describing conditions of adequacy that an acceptable measure of degree of discrimination-conduciveness must satisfy, we use those conditions of adequacy to evaluate several possible measures. We also discuss how the effect of shifting from one observation process to another might be measured. We apply our framework to several examples, including the ravens paradox and the phenomenon of publication bias. (shrink)

This paper describes a cubic water tank equipped with a movable partition receiving various amounts of liquid used to represent joint probability distributions. This device is applied to the investigation of deductive inferences under uncertainty. The analogy is exploited to determine by qualitative reasoning the limits in probability of the conclusion of twenty basic deductive arguments (such as Modus Ponens, And-introduction, Contraposition, etc.) often used as benchmark problems by the various theoretical approaches to reasoning under uncertainty. The probability bounds imposed (...) by the premises on the conclusion are derived on the basis of a few trivial principles such as "a part of the tank cannot contain more liquid than its capacity allows", or "if a part is empty, the other part contains all the liquid". This stems from the equivalence between the physical constraints imposed by the capacity of the tank and its subdivisions on the volumes of liquid, and the axioms and rules of probability. The device materializes de Finetti's coherence approach to probability. It also suggests a physical counterpart of Dutch book arguments to assess individuals' rationality in probability judgments in the sense that individuals whose degrees of belief in a conclusion are out of the bounds of coherence intervals would commit themselves to executing physically impossible tasks. (shrink)

Fading Foundations: Probability and the Regress Problem. By Atkinson David, Peijnenburg Jeanne.

Conditionalization is one of the central norms of Bayesian epistemology. But there are a number of competing formulations, and a number of arguments that purport to establish it. In this paper, I explore which formulations of the norm are supported by which arguments. In their standard formulations, each of the arguments I consider here depends on the same assumption, which I call Deterministic Updating. I will investigate whether it is possible to amend these arguments so that they no longer depend (...) on it. As I show, whether this is possible depends on the formulation of the norm under consideration. (shrink)

As stochastic independence is essential to the mathematical development of probability theory, it seems that any foundational work on probability should be able to account for this property. Bayesian decision theory appears to be wanting in this respect. Savage’s postulates on preferences under uncertainty entail a subjective expected utility representation, and this asserts only the existence and uniqueness of a subjective probability measure, regardless of its properties. What is missing is a preference condition corresponding to stochastic independence. To fill this (...) significant gap, the article axiomatizes Bayesian decision theory afresh and proves several representation theorems in this novel framework. (shrink)

This paper looks at three ways of addressing probabilism’s implausible requirement of logical omniscience. The first and most common strategy says it’s okay to require an ideally rational person to be logically omniscient. I argue that this view is indefensible on any interpretation of ‘ideally rational’. The second strategy says probabilism should be formulated not in terms of logically possible worlds but in terms of doxastically possible worlds, ways you think the world might be. I argue that, on the interpretation (...) of this approach that lifts the requirement of certainty in all logical truths, the view becomes vacuous, issuing no requirements on rational believers at all. Finally, I develop and endorse a new solution to the problem. This view proposes dynamic norms for reasoning with credences. The solution is based on an old proposal of Ian Hacking’s that says you’re required to be sensitive to logical facts only when you know they are logical facts. (shrink)

Suppose that two competing norms, N1 and N2, can be identified such that a given person’s response can be interpreted as correct according to N1 but incorrect according to N2. Which of these two norms, if any, should one use to interpret such a response? In this paper we seek to address this fundamental problem by studying individual variation in the interpretation of conditionals by establishing individual profiles of the participants based on their case judgments and reflective attitudes. To investigate (...) the participants’ reflective attitudes we introduce a new experimental paradigm called the Scorekeeping Task. As a case study, we identify the participants who follow the Suppositional Theory of conditionals (N1) versus Inferentialism (N2) and investigate to what extent internally consistent competence models can be reconstructed for the participants on this basis. After extensive empirical investigations, an apparent reasoning error with and-to-if inferences was found in one of these two groups. The implications of this case study for debates on the proper role of normative considerations in psychology are discussed. (shrink)

Rational analysis is an influential but contested account of how probabilistic modeling can be used to construct non-mechanistic but self-standing explanatory models of the mind. In this paper, I disentangle and assess several possible explanatory contributions which could be attributed to rational analysis. Although existing models suffer from evidential problems that question their explanatory power, I argue that rational analysis modeling can complement mechanistic theorizing by providing models of environmental affordances.

In recent years, permissivism—the claim that a body of evidence can rationalize more than one response—has enjoyed somewhat of a revival. But it is once again being threatened, this time by a host of new and interesting arguments that, at their core, are challenging the permissivist to explain why rationality matters. A version of the challenge that I am especially interested in is this: if permissivism is true, why should we expect the rational credences to be more accurate than the (...) irrational ones? My aim is to turn this challenge on its head and argue that, actually, those who deny permissivism will have a harder time responding to such a challenge than those who accept it. (shrink)

Whereas Bayesians have proposed norms such as probabilism, which requires immediate and permanent certainty in all logical truths, I propose a framework on which credences, including credences in logical truths, are rational because they are based on reasoning that follows plausible rules for the adoption of credences. I argue that my proposed framework has many virtues. In particular, it resolves the problem of logical omniscience.

A Bayesian mind is, at its core, a rational mind. Bayesianism is thus well-suited to predict and explain mental processes that best exemplify our ability to be rational. However, evidence from belief acquisition and change appears to show that we do not acquire and update information in a Bayesian way. Instead, the principles of belief acquisition and updating seem grounded in maintaining a psychological immune system rather than in approximating a Bayesian processor.

"The Personal is Political": This was an often-heard slogan of feminist groups in the late sixties and early seventies. The slogan is no doubt open to many interpretations. There is one interpretation which touches on the epistemology of social facts, viz. the slogan claims that in assessing the features of a political system, personal experiences have privileged evidentiary value. For instancte, in the face of third person reports about political corruption, I may remain unmoved in my belief that the political (...) powers are morally upstanding, and it is only when I myself am adversely affected, that I come to change my views. There are two standard patterns of explanation of this type of belief formation: (i) We know that third-person reports may be lessreliable than first-person experiences; (ii) If the third-person reports are no less reliable than first-person experiences, we may just be dealing with a standard pattern of epistemic irrationality. However, we argue that there is also a much more surprising pattern of explanation: under certain conditions, a Bayesian argument can be proffered to the effect that it is rational to change one's beliefs in the face of personal experiences and not in the face of third-person reports, even if these experiences and reports are equally reliable. Hence, the feminist slogan (at least on one particular interpretation of it) receives unexpected support from Bayesian comers. We also show that this pattern of explanation has surprising repercussions on the question of the evidentiary value of miracles in philosophy of religion. (shrink)

Robustness arguments hold that hypotheses are more likely to be true when they are confirmed by diverse kinds of evidence. Robustness arguments require the confirming evidence to be independent. We identify two kinds of independence appealed to in robustness arguments: ontic independence —when the multiple lines of evidence depend on different materials, assumptions, or theories—and probabilistic independence. Many assume that OI is sufficient for a robustness argument to be warranted. However, we argue that, as typically construed, OI is not a (...) sufficient independence condition for warranting robustness arguments. We show that OI evidence can collectively confirm a hypothesis to a lower degree than individual lines of evidence, contrary to the standard assumption undergirding usual robustness arguments. We employ Bayesian networks to represent the ideal empirical scenario for a robustness argument and a variety of ways in which empirical scenarios can fall short of this ideal. (shrink)

Data from medical research are typically summarized with various types of outcome measures. We present three arguments in favor of absolute over relative outcome measures. The first argument is from cognitive bias: relative measures promote the reference class fallacy and the overestimation of treatment effectiveness. The second argument is decision-theoretic: absolute measures are superior to relative measures for making a decision between interventions. The third argument is causal: interpreted as measures of causal strength, absolute measures satisfy a set of desirable (...) properties, but relative measures do not. Absolute outcome measures outperform relative measures on all counts. (shrink)

Accuracy-first accounts of rational learning attempt to vindicate the intuitive idea that, while rationally-formed belief need not be true, it is nevertheless likely to be true. To this end, they attempt to show that the Bayesian's rational learning norms are a consequence of the rational pursuit of accuracy. Existing accounts fall short of this goal, for they presuppose evidential norms which are not and cannot be vindicated in terms of the single-minded pursuit of accuracy. I propose an alternative account, according (...) to which learning experiences rationalize changes in the way you value accuracy, which in turn rationalize changes in belief. I show that this account is capable of vindicating the Bayesian's rational learning norms in terms of the single-minded pursuit of accuracy, so long as accuracy is rationally valued. (shrink)

Crispin Wright maintains that the architecture of perceptual justification is such that we can acquire justification for our perceptual beliefs only if we have antecedent justification for ruling out any sceptical alternative. Wright contends that this principle doesn’t elicit scepticism, for we are non-evidentially entitled to accept the negation of any sceptical alternative. Sebastiano Moruzzi has challenged Wright’s contention by arguing that since our non-evidential entitlements don’t remove the epistemic risk of our perceptual beliefs, they don’t actually enable us to (...) acquire justification for these beliefs. In this paper I show that Wright’s responses to Moruzzi are ineffective and that Moruzzi’s argument is validated by probabilistic reasoning. I also suggest that Wright cannot answer Moruzzi’s challenge without weakening the support available for his conception of the architecture of perceptual justification. (shrink)

Crupi et al. propose a generalization of Bayesian confirmation theory that they claim to adequately deal with confirmation by uncertain evidence. Consider a series of points of time t0, . . . , ti, . . . , tn such that the agent’s subjective probability for an atomic proposition E changes from Pr0 at t0 to . . . to Pri at ti to . . . to Prn at tn. It is understood that the agent’s subjective probabilities change for (...) E and no logically stronger proposition, and that the agent updates her subjective probabilities by Jeffrey conditionalization. For this specific scenario the authors propose to take the difference between Pr0 and Pri as the degree to which E confirms H for the agent at time ti , C0,i. This proposal is claimed to be adequate, because. (shrink)

Dogmatism is sometimes thought to be incompatible with Bayesian models of rational learning. I show that the best model for updating imprecise credences is compatible with dogmatism.

In the biomedical context, policy makers face a large amount of potentially discordant evidence from different sources. This prompts the question of how this evidence should be aggregated in the interests of best-informed policy recommendations. The starting point of our discussion is Hunter and Williams’ recent work on an automated aggregation method for medical evidence. Our negative claim is that it is far from clear what the relevant criteria for evaluating an evidence aggregator of this sort are. What is the (...) appropriate balance between explicitly coded algorithms and implicit reasoning involved, for instance, in the packaging of input evidence? In short: What is the optimal degree of ‘automation’? On the positive side: We propose the ability to perform an adequate robustness analysis as the focal criterion, primarily because it directs efforts to what is most important, namely, the structure of the algorithm and the appropriate extent of automation. Moreover, where there are resource constraints on the aggregation process, one must also consider what balance between volume of evidence and accuracy in the treatment of individual evidence best facilitates inference. There is no prerogative to aggregate the total evidence available if this would in fact reduce overall accuracy. (shrink)

In this public debate with Philippe Deterre (research director in immunology at the CNRS) – held at l'Enclos Rey in Paris' 15th district during the biennial Conference of the Réseau Blaise Pascal in March 2017 –, I defended the usefulness of natural theology. I first clarify theology's nature and understanding, then I speak about a tradition that upheld the public and exterior knowledge of God, and make an effort to show the presence of a theme reminiscent of natural theology behind (...) attempts at the good life. I then ask whether natural theology would only exist for the Christian. In the reply to my opponent's own reaction, I insist on the incongruity of separating our knowledge of God from our knowledge of science's wonderful discoveries, I ask whether nature could be said to be crafty and "ingenious," and I conclude by building a case for the return of God in public conversation, as part of an effort that our world needs in terms of finding back its compass, and restoring an ideal of living rationally. (shrink)

to appear in Szabó Gendler, T. & J. Hawthorne (eds.) Oxford Studies in Epistemology volume 6 -/- We often ask for the opinion of a group of individuals. How strongly does the scientific community believe that the rate at which sea levels are rising increased over the last 200 years? How likely does the UK Treasury think it is that there will be a recession if the country leaves the European Union? What are these group credences that such questions request? (...) And how do they relate to the individual credences assigned by the members of the particular group in question? According to the credal judgment aggregation principle, Linear Pooling, the credence function of a group should be a weighted average or linear pool of the credence functions of the individuals in the group. In this paper, I give an argument for Linear Pooling based on considerations of accuracy. And I respond to two standard objections to the aggregation principle. (shrink)

Some modern cosmological models predict the appearance of Boltzmann Brains: observers who randomly fluctuate out of a thermal bath rather than naturally evolving from a low-entropy Big Bang. A theory in which most observers are of the Boltzmann Brain type is generally thought to be unacceptable, although opinions differ. I argue that such theories are indeed unacceptable: the real problem is with fluctuations into observers who are locally identical to ordinary observers, and their existence cannot be swept under the rug (...) by a choice of probability distributions over observers. The issue is not that the existence of such observers is ruled out by data, but that the theories that predict them are cognitively unstable: they cannot simultaneously be true and justifiably believed. (shrink)

The paper discusses the notion of reasoning with comparative moral judgements (i.e judgements of the form “act a is morally superior to act b”) from the point of view of several meta-ethical positions. Using a simple formal result, it is argued that only a version of moral cognitivism that is committed to the claim that moral beliefs come in degrees can give a normatively plausible account of such reasoning. Some implications of accepting such a version of moral cognitivism are discussed.

Hempel’s Converse Consequence Condition (CCC), Entailment Condition (EC), and Special Consequence Condition (SCC) have some prima facie plausibility when taken individually. Hempel, though, shows that they have no plausibility when taken together, for together they entail that E confirms H for any propositions E and H. This is “Hempel’s paradox”. It turns out that Hempel’s argument would fail if one or more of CCC, EC, and SCC were modified in terms of explanation. This opens up the possibility that Hempel’s paradox (...) can be solved by modifying one or more of CCC, EC, and SCC in terms of explanation. I explore this possibility by modifying CCC and SCC in terms of explanation and considering whether CCC and SCC so modified are correct. I also relate that possibility to Inference to the Best Explanation. (shrink)

An influential suggestion about the relationship between Bayesianism and inference to the best explanation holds that IBE functions as a heuristic to approximate Bayesian reasoning. While this view promises to unify Bayesianism and IBE in a very attractive manner, important elements of the view have not yet been spelled out in detail. I present and argue for a heuristic conception of IBE on which IBE serves primarily to locate the most probable available explanatory hypothesis to serve as a working hypothesis (...) in an agent’s further investigations. Along the way, I criticize what I consider to be an overly ambitious conception of the heuristic role of IBE, according to which IBE serves as a guide to absolute probability values. My own conception, by contrast, requires only that IBE can function as a guide to the comparative probability values of available hypotheses. This is shown to be a much more realistic role for IBE given the nature and limitations of the explanatory considerations with which IBE operates. (shrink)

Any theory of confirmation must answer the following question: what is the purpose of its conception of confirmation for scientific inquiry? In this article, we argue that no Bayesian conception of confirmation can be used for its primary intended purpose, which we take to be making a claim about how worthy of belief various hypotheses are. Then we consider a different use to which Bayesian confirmation might be put, namely, determining the epistemic value of experimental outcomes, and thus to decide (...) which experiments to carry out. Interestingly, Bayesian confirmation theorists rule out that confirmation be used for this purpose. We conclude that Bayesian confirmation is a means with no end. 1 Introduction2 Bayesian Confirmation Theory3 Bayesian Confirmation and Belief4 Confirmation and the Value of Experiments5 Conclusion. (shrink)

Bayesian confirmation theory is rife with confirmation measures. Zalabardo focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other (...) three measures. Glass and McCartney, hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p and a certain other probability involving H. I then evaluate G&M’s argument in light of them. (shrink)

I argue elsewhere (Roche 2014) that evidence of evidence is evidence under screening-off. Tal and Comesaña (2017) argue that my appeal to screening-off is subject to two objections. They then propose an evidence of evidence thesis involving the notion of a defeater. There is much to learn from their very careful discussion. I argue, though, that their objections fail and that their evidence of evidence thesis is open to counterexample.

There are many scientific and everyday cases where each of Pr and Pr is high and it seems that Pr is high. But high probability is not transitive and so it might be in such cases that each of Pr and Pr is high and in fact Pr is not high. There is no issue in the special case where the following condition, which I call “C1”, holds: H 1 entails H 2. This condition is sufficient for transitivity in high (...) probability. But many of the scientific and everyday cases referred to above are cases where it is not the case that H 1 entails H 2. I consider whether there are additional conditions sufficient for transitivity in high probability. I consider three candidate conditions. I call them “C2”, “C3”, and “C2&3”. I argue that C2&3, but neither C2 nor C3, is sufficient for transitivity in high probability. I then set out some further results and relate the discussion to the Bayesian requirement of coherence. (shrink)

We argued that explanatoriness is evidentially irrelevant in the following sense: Let H be a hypothesis, O an observation, and E the proposition that H would explain O if H and O were true. Then our claim is that Pr = Pr. We defended this screening-off thesis by discussing an example concerning smoking and cancer. Climenhaga argues that SOT is mistaken because it delivers the wrong verdict about a slightly different smoking-and-cancer case. He also considers a variant of SOT, called (...) “SOT*”, and contends that it too gives the wrong result. We here reply to Climenhaga’s arguments and suggest that SOT provides a criticism of the widely held theory of inference called “inference to the best explanation”. (shrink)

There is a plethora of confirmation measures in the literature. Zalabardo considers four such measures: PD, PR, LD, and LR. He argues for LR and against each of PD, PR, and LD. First, he argues that PR is the better of the two probability measures. Next, he argues that LR is the better of the two likelihood measures. Finally, he argues that LR is superior to PR. I set aside LD and focus on the trio of PD, PR, and LR. (...) The question I address is whether Zalabardo succeeds in showing that LR is superior to each of PD and PR. I argue that the answer is negative. I also argue, though, that measures such as PD and PR, on one hand, and measures such as LR, on the other hand, are naturally understood as explications of distinct senses of confirmation. (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)

Book review of Paul Horwich, Probability and Evidence (Cambridge Philosophy Classics edition), Cambridge: Cambridge University Press, 2016, 147pp, £14.99 (paperback).

Traditional Bayesianism requires that an agent’s degrees of belief be represented by a real-valued, probabilistic credence function. However, in many cases it seems that our evidence is not rich enough to warrant such precision. In light of this, some have proposed that we instead represent an agent’s degrees of belief as a set of credence functions. This way, we can respect the evidence by requiring that the set, often called the agent’s credal state, includes all credence functions that are in (...) some sense compatible with the evidence. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In this article I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid it without compromising the initial evidentialist motivation. _1_ Introduction _2_ Precision and Its Problems _3_ Imprecise Bayesianism and Respecting Ambiguous Evidence _4_ Local Belief Inertia _5_ From Local to Global Belief Inertia _6_ Responding to Global Belief Inertia _7_ Conclusion. (shrink)

This paper offers a probabilistic treatment of the conditions for argument cogency as endorsed in informal logic: acceptability, relevance, and sufficiency. Treating a natural language argument as a reason-claim-complex, our analysis identifies content features of defeasible argument on which the RSA conditions depend, namely: change in the commitment to the reason, the reason’s sensitivity and selectivity to the claim, one’s prior commitment to the claim, and the contextually determined thresholds of acceptability for reasons and for claims. Results contrast with, and (...) may indeed serve to correct, the informal understanding and applications of the RSA criteria concerning their conceptual dependence, their function as update-thresholds, and their status as obligatory rather than permissive norms, but also show how these formal and informal normative approachs can in fact align. (shrink)

Bayesian confirmation theory is rife with confirmation measures. Many of them differ from each other in important respects. It turns out, though, that all the standard confirmation measures in the literature run counter to the so-called “Reverse Matthew Effect” (“RME” for short). Suppose, to illustrate, that H1 and H2 are equally successful in predicting E in that p(E | H1)/p(E) = p(E | H2)/p(E) > 1. Suppose, further, that initially H1 is less probable than H2 in that p(H1) < p(H2). (...) Then by RME it follows that the degree to which E confirms H1 is greater than the degree to which it confirms H2. But by all the standard confirmation measures in the literature, in contrast, it follows that the degree to which E confirms H1 is less than or equal to the degree to which it confirms H2. It might seem, then, that RME should be rejected as implausible. Festa (2012), however, argues that there are scientific contexts in which RME holds. If Festa’s argument is sound, it follows that there are scientific contexts in which none of the standard confirmation measures in the literature is adequate. Festa’s argument is thus interesting, important, and deserving of careful examination. I consider five distinct respects in which E can be related to H, use them to construct five distinct ways of understanding confirmation measures, which I call “Increase in Probability”, “Partial Dependence”, “Partial Entailment”, “Partial Discrimination”, and “Popper Corroboration”, and argue that each such way runs counter to RME. The result is that it is not at all clear that there is a place in Bayesian confirmation theory for RME. (shrink)

Abstract The Preface Paradox, first introduced by David Makinson (1961), presents a plausible scenario where an agent is evidentially certain of each of a set of propositions without being evidentially certain of the conjunction of the set of propositions. Given reasonable assumptions about the nature of evidential certainty, this appears to be a straightforward contradiction. We solve the paradox by appeal to stake size sensitivity, which is the claim that evidential probability is sensitive to stake size. The argument is that (...) because the informational content in the conjunction is greater than the sum of the informational content of the conjuncts, the stake size in the conjunction is higher than the sum of the stake sizes in the conjuncts. We present a theory of evidential probability that identifies knowledge with value and allows for coherent stake sensitive beliefs. An agent’s beliefs are represented two dimensionally as a bid – ask spread, which gives a bid price and an ask price for bets at each stake size. The bid ask spread gets wider when there is less valuable evidence relative to the stake size, and narrower when there is more valuable evidence according to a simple formula. The bid-ask spread can represent the uncertainty in the first order probabilistic judgement. According to the theory it can be coherent to be evidentially certain at low stakes, but less than certain at high stakes, and therefore there is no contradiction in the Preface. The theory not only solves the paradox, but also gives a good model of decisions under risk that overcomes many of the problems associated with classic expected utility theory. (shrink)

Coherentism maintains that coherent beliefs are more likely to be true than incoherent beliefs, and that coherent evidence provides more confirmation of a hypothesis when the evidence is made coherent by the explanation provided by that hypothesis. Although probabilistic models of credence ought to be well-suited to justifying such claims, negative results from Bayesian epistemology have suggested otherwise. In this essay we argue that the connection between coherence and confirmation should be understood as a relation mediated by the causal relationships (...) among the evidence and a hypothesis, and we offer a framework for doing so by fitting together probabilistic models of coherence, confirmation, and causation. We show that the causal structure among the evidence and hypothesis is sometimes enough to determine whether the coherence of the evidence boosts confirmation of the hypothesis, makes no difference to it, or even reduces it. We also show that, ceteris paribus, it is not the coherence of the evidence that boosts confirmation, but rather the ratio of the coherence of the evidence to the coherence of the evidence conditional on a hypothesis. (shrink)