The right interpretation of subjective probability is implicit in the theories of upper and lower odds, and upper and lower previsions, developed, respectively, by Cedric Smith (1961) and Peter Walley (1991). On this interpretation you are free to assign contingent events the probability 1 (and thus to employ conditionalization as a method of probability revision) without becoming vulnerable to a weak Dutch book.

This paper presents a formalism that combines useful properties of both logic and probabilities. Like logic, the formalism admits qualitative sentences and provides symbolic machinery for deriving deductively closed beliefs and, like probability, it permits us to express if-then rules with different levels of firmness and to retract beliefs in response to changing observations. Rules are interpreted as order-of-magnitude approximations of conditional probabilities which impose constraints over the rankings of worlds. Inferences are supported by a unique priority ordering (...) on rules which is syntactically derived from the knowledge base. This ordering accounts for rule interactions, respects specificity considerations and facilitates the construction of coherent states of beliefs. Practical algorithms are developed and analyzed for testing consistency, computing rule ordering, and answering queries. Imprecise observations are incorporated using qualitative versions of Jeffrey's rule and Bayesian updating, with the result that coherent belief revision is embodied naturally and tractably. Finally, causal rules are interpreted as imposing Markovian conditions that further constrain world rankings to reflect the modularity of causal organizations. These constraints are shown to facilitate reasoning about causal projections, explanations, actions and change. (shrink)

The orthodoxy that conditional probabilities reflect what are for a subject evidential bearings is seconded. This significance suggests that there should be principles equating rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. Several principles, two of which are endorsed, are considered. A book is made against a violator of these, and it is argued that there must be something wrong with a person against whom such books can be made. Appendices comment on Popper-functions, elaborate (...) on bets and odds, and relate dutch books and strategies to conditions of inconsistency (Ramsey's idea) and imperfection. (shrink)

A common objection to the Molinist account of divine providence states that counterfactuals of creaturely freedom lack grounds. Some Molinists appeal to brute counterfactual facts about the subject of the CCF in order to ground CCFs. Others argue that CCFs are grounded by the subject's actions in nearby worlds. In this article, I argue that Open Theism's account of divine providence employs would-probably conditionals that are most plausibly grounded by either brute facts about the subject of these conditionals or non-actual (...) entities. As a result, Open Theism's revision of the traditional notion of divine providence is unmotivated. The Molinist can ground CCFs just as easily as the Openist can ground would-probably conditionals but the Molinist has the advantage of maintaining a robust account of divine providence.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.On grounding God's knowledge of the probableVolume 49, Issue 1JENNIFER JENSEN DOI: https://doi.org/10.1017/S0034412512000182Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. On grounding God's knowledge of the probableVolume 49, Issue 1JENNIFER JENSEN DOI: https://doi.org/10.1017/S0034412512000182Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. On grounding God's knowledge of the probableVolume 49, Issue 1JENNIFER JENSEN DOI: https://doi.org/10.1017/S0034412512000182Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Beliefs come in degrees, and we often represent those degrees with numbers. We might say, for example, that we are 90% confident in the truth of some scientific hypothesis, or only 30% confident in the success of some risky endeavour. But what do these numbers mean? What, in other words, is the underlying psychological reality to which the numbers correspond? And what constitutes a meaningful difference between numerically distinct representations of belief? In this Element, we discuss the main approaches to (...) the measurement of belief. These fall into two broad categories—epistemic and decision-theoretic—with divergent foundations in the theory of measurement. Epistemic approaches explain the measurement of belief by appeal to relations between belief states themselves, whereas decision-theoretic approaches appeal to relations between beliefs and desires in the production of choice and preferences. -/- (This book is part of the Cambridge Elements in Decision Theory and Philosophy series.). (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single (...) events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many (...) systematic deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings. (shrink)

Subjective probability plays an increasingly important role in many fields concerned with human cognition and behavior. Yet there have been significant criticisms of the idea that probabilities could actually be represented in the mind. This paper presents and elaborates a view of subjective probability as a kind of sampling propensity associated with internally represented generative models. The resulting view answers to some of the most well known criticisms of subjective probability, and is also supported (...) by empirical work in neuroscience and behavioral psychology. The repercussions of the view for how we conceive of many ordinary instances of subjective probability, and how it relates to more traditional conceptions of subjective probability, are discussed in some detail. (shrink)

This paper is about the statics and dynamics of belief states that are represented by pairs consisting of an agent's credences (represented by a subjective probability measure) and her categorical beliefs (represented by a set of possible worlds). Regarding the static side, we argue that the latter proposition should be coherent with respect to the probability measure and that its probability should reach a certain threshold value. On the dynamic side, we advocate Jeffrey conditionalisation as the (...) principal mode of changing one's belief state. This updating method fits the idea of the Lockean Thesis better than plain Bayesian conditionalisation, and it affords a flexible method for adding and withdrawing categorical beliefs. We show that it fails to satisfy the traditional principles of Inclusion and Preservation for belief revision and the principle of Recovery for belief withdrawals{, as well as the Levi and Harper identities. We take this to be a problem for the latter principles rather than for the idea of coherent belief change. (shrink)

Pragmatism, taken not just as a philosophical movement but as a way of addressing problems, strongly influenced the debate on the foundations of probability during the first half of the twentieth century. Upholders of different interpretations of probability such as Hans Reichenbach, Ernest Nagel, Rudolf Carnap, Frank Ramsey, and Bruno de Finetti, acknowledged their debt towards pragmatist philosophers, including Charles Sanders Peirce, William James, Clarence Irving Lewis, William Dewey and Giovanni Vailati. In addition, scientist-philosophers like Ernst Mach, Ludwig (...) Boltzmann, Henri Poincaré, Pierre Duhem, and Karl Pearson, who heralded a conception of science and knowledge at large that was close to pragmatism, were very influential in that debate. Among the main interpretations of probability – frequentism, propensionism, logicism and subjectivism -, the latter is no doubt the closest to the pragmatist outlook. This paper concentrates on three representatives of the subjective theory, namely Frank Ramsey, Bruno de Finetti and Émile Borel. (shrink)

In humans, conscious perception and cognition depends upon the thalamocortical complex, which supports perception, explicit cognition, memory, language, planning, and strategic control. When parts of the T-C system are damaged or stimulated, corresponding effects are found on conscious contents and state, as assessed by reliable reports. In contrast, large regions like cerebellum and basal ganglia can be damaged without affecting conscious cognition directly. Functional brain recordings also show robust activity differences in cortex between experimentally matched conscious and unconscious events. This (...) basic anatomy and physiology is highly conserved in mammals and perhaps ancestral reptiles. While language is absent in other species, homologies in perception, memory, and motor cortex suggest that consciousness of one kind or another may be biologically fundamental and phylogenetically ancient. In humans we infer subjective experiences from behavioral and brain evidence. This evidence is quite similar in other mammals and perhaps some non-mammalian species. On the weight of the biological evidence, therefore, subjectivity may be conserved in species with human-like brains and behavior. (shrink)

There is an increasing recognition by friends of personal probability that the standard systems of personal probability do not provide a fully adequate basis for the theories of scientific inference and rational decision making. This recognition has methodological and formal components. On the methodological side, Jeffrey [8] and Spielman [16], [17] have suggested that personal probabilities should be interpreted as judgments about thecredibilityof propositions, i.e., as appraisals of the degrees of confidence that are warranted by the information available (...) to the appraiser. The idea behind this interpretation is that there are criteria for appraising degrees of confidence as rational, plausible, silly or unwarranted. They have not been canonized, with the exception of the traditional informal fallacies of insufficient evidence, and if they were, would be highly vague and subject to revision as knowledge advances. (shrink)

How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences. However, standard representation theorems make strong rationality assumptions, in particular expected utility maximization. How do we ascribe subjective probability to agents which do not satisfy these strong rationality assumptions? I present a representation theorem with weak rationality assumptions which can (...) be used to define or measure subjective probability for partly irrational agents. (shrink)

This chapter is a philosophical survey of some leading approaches in formal epistemology in the so-called ‘Bayesian’ tradition. According to them, a rational agent’s degrees of belief—credences—at a time are representable with probability functions. We also canvas various further putative ‘synchronic’ rationality norms on credences. We then consider ‘diachronic’ norms that are thought to constrain how credences should respond to evidence. We discuss some of the main lines of recent debate, and conclude with some prospects for future research.

The current debate about peer disagreement has so far mainly focused on the question of whether peer disagreements provide genuine counterevidence to which we should respond by revising our credences. By contrast, comparatively little attention has been devoted to the question by which process, if any, such revision should be brought about. The standard assumption is that we update our credences by conditionalizing on the evidence that peer disagreements provide. In this paper, we argue that non-dogmatist views have good (...) reasons to reject conditionalization. Instead, peer disagreements should be understood to call for a revision of our prior conditional probabilities: rather than merely adding to our original evidence, they pose a challenge to the thought that we have properly assessed the probative force of our original evidence. (shrink)

This book offers a concise survey of basic probability theory from a thoroughly subjective point of view whereby probability is a mode of judgment. Written by one of the greatest figures in the field of probability theory, the book is both a summation and synthesis of a lifetime of wrestling with these problems and issues. After an introduction to basic probability theory, there are chapters on scientific hypothesis-testing, on changing your mind in response to generally (...) uncertain observations, on expectations of the values of random variables, on de Finetti's dissolution of the so-called problem of induction, and on decision theory. (shrink)

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A (...) third hypothesis, in line with dual-systems theory (e.g., Kahneman and Frederick, 2002 ) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others. (shrink)

This collection of essays is a Festschrift for Ernest W. Adams, and is based on a symposium that was held in his honor in 1993. As the title suggests, most of the essays focus on probability and the logic of conditionals, and the relationship between them; they draw their inspiration from Adams’s seminal work on the subject. As a computer scientist, I was struck by just how much the topics discussed play a major role in much recent work in (...) computer science, and how relevant much recent work in computer science is to some of the issues discussed here. (shrink)

This collection of essays is a Festschrift for Ernest W. Adams, and is based on a symposium that was held in his honor in 1993. As the title suggests, most of the essays focus on probability and the logic of conditionals, and the relationship between them; they draw their inspiration from Adams’s seminal work on the subject. As a computer scientist, I was struck by just how much the topics discussed play a major role in much recent work in (...) computer science, and how relevant much recent work in computer science is to some of the issues discussed here. (shrink)

Gambles which induce the decision-maker to experience ambiguity about the relative likelihood of events often give rise to ambiguity-seeking and ambiguity-avoidance, which imply violation of additivity and Savage's axioms. The inability of the subjective Bayesian theory to account for these empirical regularities has determined a dichotomy between normative and descriptive views of subjective probability. This paper proposes a framework within which the two perspectives can be reconciled. First, a formal definition of ambiguity is given over a continuum (...) ranging from ignorance to risk, and including ambiguous contexts as subsets. Second, it is shown that the systems of inductive logic account for the effects of ambiguity. Then, Carnap's X-system is applied as a psychological model and compared to Einhorn and Hogarth's non-normative psychological model. Finally, the implications of this research to the modeling of subjective probability judgements are discussed. (shrink)

The scope of embryo research in the UK has been expanded by the Human Fertilisation and Embryology Regulations 2001. Two advisory bodies—the Chief Medical Officer’s Expert Group and the House of Lords’ Select Committee—presented various arguments in favour of embryo research. One of these is the view that, just as lottery tickets have relatively little value before the draw because of the low probability of their being the winning ticket, early embryos have relatively little value because of the presumed (...) low probability that they will mature into more developed embryos. This argument from probability is questioned in this paper, as well as the contention that allowing embryo destruction is incompatible with the view that embryos have full moral status. Although I challenge Savulescu’s view that early embryos should be entered into a lottery in which they are subjected to the probability of being destroyed , a revised version of Thomson’s analogy of the famous violinist defies the view that the position that the embryo has full moral status is incompatible with qualified support for embryo destruction. (shrink)

On an attractive, naturalistically respectable theory of intentionality, mental contents are a form of measurement system for representing behavioral and psychological dispositions. This chapter argues that a consequence of this view is that the content/attitude distinction is measurement system relative. As a result, there is substantial arbitrariness in the content/attitude distinction. Whether some measurement of mental states counts as characterizing the content of mental states or the attitude is not a question of empirical discovery but of theoretical utility. If correct, (...) this observation has ramifications in the theory of rationality. Some epistemologists and decision theorists have argued that imprecise credences are rationally impermissible, while others have argued that precise credences are rationally impermissible. If the measure theory of mental content is correct, however, then neither imprecise credences nor precise credences can be rationally impermissible. (shrink)

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, (...) iterated revision is investigated. The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change. (shrink)

We propose a theory that relates perceived evidence to numerical probability judgment. The most successful prior account of this relation is Support Theory, advanced in Tversky and Koehler. Support Theory, however, implies additive probability estimates for binary partitions. In contrast, superadditivity has been documented in Macchi, Osherson, and Krantz, and both sub- and superadditivity appear in the experiments reported here. Nonadditivity suggests asymmetry in the processing of focal and nonfocal hypotheses, even within binary partitions. We extend Support Theory (...) by revising its basic equation to allow such asymmetry, and compare the two equations’ ability to predict numerical assessments of probability from scaled estimates of evidence for and against a given proposition. Both between- and within-subject experimental designs are employed for this purpose. We find that the revised equation is more accurate than the original Support Theory equation. The implications of asymmetric processing on qualitative assessments of chance are also briefly discussed. (shrink)

Popper's writings cover a remarkably wide range of subjects. The spectrum runs from Plato's theory of politics to the foundations of quantum mechanics. Yet even amidst this variety the philosophy of probability occupies a prominent place. David Miller once pointed out to me that more than half of Popper's The Logic of Scientific Discovery is taken up with discussions of probability. I checked this claim using the 1972 6th revised impression of The Logic of Scientific Discovery , and (...) found that of the approximately 450 pages of text, approximately 250 are to do with probability. Thus Miller's claim is amply justified. It seems indeed that the philosophy of probability was one of Popper's favourite subjects, and, as we shall see, Popper certainly enriched the field with several striking innovations. In this area, as in others, Popper held very definite views, and criticized his opponents in no uncertain terms. Popper was an objectivist and anti-Bayesian, and his criticisms were directed against subjectivism and Bayesianism. (shrink)

We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.

The aim of this paper is to present and analyse Bruno de Finetti's view that the axiom of countable additivity of the probability calculus cannot be justified in terms of the subjective interpretation of probability. After presenting the core of the subjective theory of probability and the main de Finetti's argument against the axiom of countable additivity (the so called de Finetti's infinite lottery) I argue against de Finetti's view. In particular, I claim that de (...) Finetti does not prove the impossibility of using Dutch Book argument for the axiom of countable additivity. Consequently, we can use Dutch Book argument for the justification of the axiom of countable additivity and regard de Finetti's lottery as a special case when the axiom does not hold, or we can justify countable additivity by Dutch Book argument and reject de Finetti's lottery as irrational. The second strategy, represented especially by Jon Williamson, is much more compatible with the idea of subjective interpretation of probability. (shrink)