This essay discusses the difficulty to reconcile two paradigms about beliefs: the binary or categorical paradigm of yes/no beliefs and the probabilistic paradigm of degrees of belief. The possibility for someone to hold both types of belief simultaneously is challenged by the lottery paradox, and more recently by a general impossibility theorem by Dietrich and List (2018, 2021). The nature, relevance, and implications of the tension are explained and assessed.

This symposium in the overlap of philosophy and decision theory is described well by its title “Beliefs in Groups”. Each word in the title matters, with one intended ambiguity. The symposium is about beliefs rather than other attitudes such as preferences; these beliefs take the form of probabilities in the first three contributions, binary yes/no beliefs (‘judgments’) in the fourth contribution, and qualitative probabilities (‘probability grades’) in the fifth contribution. The beliefs occur in groups, (...) which is ambiguous between beliefs of groups as a whole and beliefs of group members. The five contributions – all of them interesting, we believe – address several aspects of this general theme. (shrink)

This paper addresses how multiple individual credences on logically related issues should be aggregated into collective binary beliefs. We call this binarizing belief aggregation. It is vulnerable to dilemmas such as the discursive dilemma or the lottery paradox: proposition-wise independent aggregation can generate inconsistent or not deductively closed collective judgments. Addressing this challenge using the familiar axiomatic approach, we introduce general conditions on a binarizing belief aggregation rule, including rationality conditions on individual inputs and collective outputs, and determine (...) which rules (if any) satisfy different combinations of these conditions. Furthermore, we analyze similarities and differences between our proofs and other related proofs in the literature and conclude that the problem of binarizing belief aggregation is a free-standing aggregation problem not reducible to judgment aggregation or probabilistic opinion pooling. (shrink)

Beliefs like the Gambler's Fallacy and the Hot Hand have interested cognitive scientists, economists, and philosophers for centuries. We propose that these judgment patterns arise from the observer's mental models of the sequence-generating mechanism, moderated by the strength of belief in an a priori base rate. In six behavioral experiments, participants observed one of three mechanisms generating sequences of eight binary events: a random mechanical device, an intentional goal-directed actor, and a financial market. We systematically manipulated participants’ (...) class='Hi'>beliefs about the base rate probabilities at which different outcomes were generated by each mechanism. Participants judged 18 sequences of outcomes produced by a mechanism with either an unknown base rate, a specified distribution of three equiprobable base rates, or a precise, fixed base rate. Six target sequences ended in streaks of between two and seven identical outcomes. The most common predictions for subsequent events were best described as pragmatic belief updating, expressed as an increasingly strong expectation that a streak of identical signals would repeat as the length of that streak increased. The exception to this pattern was for sequences generated by a random mechanical device with a fixed base rate of.50. Under this specific condition, participants exhibited a bias toward reversal of streaks, and this bias was larger when participants were asked to make a dichotomous choice versus a numerical probability rating. We review alternate accounts for the anomalous judgments of sequences and conclude with our favored interpretation that is based on Rabin's version of Tversky & Kahneman's Law of Small Numbers. (shrink)

Binarizing belief aggregation tackles the problem of aggregating individuals’ probabilistic beliefs on logically connected propositions into the group’s binary beliefs. One common approach to associating probabilistic beliefs with binary beliefs would be applying thresholds to probabilities. This paper aims to introduce and classify a range of threshold-based binarizing belief aggregation rules while characterizing them based on different forms of monotonicity and other properties.

Hybrid belief aggregation addresses aggregation of individual probabilistic beliefs into collective binary beliefs. In line with the development of judgment aggregation theory, our research delves into the identification of precise agenda conditions associated with some key impossibility theorems in the context of hybrid belief aggregation. We determine the necessary and sufficient level of logical interconnection between the propositions in an agenda for some key impossibilities to arise. Specifically, we prove three characterization theorems about hybrid belief aggregation: (...) (i) Precisely the path-connected and pair-negatable agendas lead to the ‘oligarchy result’—only oligarchic rules satisfy universal domain, proposition-wise independence, respect for unanimity, and deductive closure of collective beliefs. (ii) Precisely the negation-connected agendas lead to the ‘triviality result’—only unanimity rules satisfy those conditions as well as anonymity. (iii) Precisely the blocked agendas lead to the ‘non-existence result’—no rules satisfy those conditions as well as completeness and consistency of collective beliefs. Furthermore, we compare these novel findings with existing agenda-theoretic characterization theorems in the domains of judgment aggregation and belief binarization. (shrink)

In recent years, the psychology of reasoning has been undergoing a paradigm shift, with general Bayesian, probabilistic approaches replacing the older, much more restricted binary logic paradigm. At the same time, dual processing theories have been gaining influence. We argue that these developments should be integrated and moreover that such integration is already underway. The new reasoning paradigm should be grounded in dual processing for its algorithmic level of analysis just as it uses Bayesian theory for its computational (...) level of analysis. Moreover, we propose that, within the new paradigm, these levels of analysis reflect on each other. Bayesianism suggests a specific theoretical understanding of dual processing. Just as importantly, the duality in processing carries over to duality in function; although both types of processes compute degrees of belief, they generate different functions. (shrink)

English abstract: This paper discusses the delicate relationship between traditional epistemology and the increasingly influential probabilistic (or ‘Bayesian’) approach to epistemology. The paper introduces some of the key ideas of probabilistic epistemology, including credences or degrees of belief, Bayes’ theorem, conditionalization, and the Dutch Book argument. The tension between traditional and probabilistic epistemology is brought out by considering the lottery and preface paradoxes as they relate to rational (binary) belief and credence respectively. It is then argued (...) that this tension can be alleviated by rejecting the requirement that rational (binary) beliefs must be consistent and closed under logical entailment. Instead, it is suggested that this logical requirement applies to a different type of binary propositional attitude, viz. acceptance. (shrink)

How should we revise our beliefs in response to the expressed probabilistic opinions of experts on some proposition when these experts are in disagreement? In this paper I examine the suggestion that in such circumstances we should adopt a linear average of the experts’ opinions and consider whether such a belief revision policy is compatible with Bayesian conditionalisation. By looking at situations in which full or partial deference to the expressed opinions of others is required by Bayesianism I (...) show that only in trivial circumstances are the requirements imposed by linear averaging compatible with it. (shrink)

It is tempting to analyse causality in terms of just one of the indicators of causal relationships, e.g., mechanisms, probabilistic dependencies or independencies, counterfactual conditionals or agency considerations. While such an analysis will surely shed light on some aspect of our concept of cause, it will fail to capture the whole, rather multifarious, notion. So one might instead plump for pluralism: a different analysis for a different occasion. But we do not seem to have lots of different concepts of (...) cause – just one eclectic notion. The resolution of this conundrum, I think, requires us to accept that our causal beliefs are generated by a wide variety of indicators, but to deny that this variety of indicators yields a variety of concepts of cause. This focus on the relation between evidence and causal beliefs leads to what I call epistemic causality. Under this view, certain causal beliefs are appropriate or rational on the basis of observed evidence; our notion of cause can be understood purely in terms of these rational beliefs. Causality, then, is a feature of our epistemic representation of the world, rather than of the world itself. This yields one, multifaceted notion of cause. (shrink)

This article compares inference to the best explanation with Bayes’s rule in a social setting, specifically, in the context of a variant of the Hegselmann–Krause model in which agents not only update their belief states on the basis of evidence they receive directly from the world, but also take into account the belief states of their fellow agents. So far, the update rules mentioned have been studied only in an individualistic setting, and it is known that in such a setting (...) both have their strengths as well as their weaknesses. It is shown here that in a social setting, inference to the best explanation outperforms Bayes’s rule according to every desirable criterion. 1 What Is Inference to the Best Explanation?2 Judging the Rules—By Which Lights?3 From an Individualistic to a Social Perspective 3.1 The Hegselmann–Krause model 3.2 A probabilistic extension of the Hegselmann–Krause model 3.3 Simulations4 Results and Discussion5 Interpretation6 Conclusion. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed (...) that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

In the present research we investigated gender-specific beliefs about emotional behaviour. In Study 1, 180 respondents rated the extent to which they agreed with different types of beliefs (prescriptive, descriptive, stereotypical, and contra-stereotypical) regarding the emotional behaviour of men and women. As anticipated, respondents agreed more with descriptive than with prescriptive beliefs, and more with stereotypical than with contra-stereotypical beliefs. However, respondents agreed more with stereotypical beliefs about the emotional behaviour of women than with those (...) about men. These results were replicated in Study 2 with a sample of 75 students and 80 nonstudents. In Study 3, a sample of 279 respondents rated the extent of agreement with the same items, this time with respect to then own emotional behaviour. A similar pattern of results was obtained, although agreement rates were higher than in Study 1 and 2. (shrink)

A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something (...) that is intrinsically vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness. (shrink)

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, (...) we seek to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

In this paper I respond to Rebecca Kukla's (2014) "Commentary on Karyn Freedman, "Testimony and Epistemic Risk: The Dependence Account."".

Over the past decades or so the probabilistic model of rational belief has enjoyed increasing interest from researchers in epistemology and the philosophy of science. Of course, such probabilistic models were used for much longer in economics, in game theory, and in other disciplines concerned with decision making. Moreover, Carnap and co-workers used probability theory to explicate philosophical notions of confirmation and induction, thereby targeting epistemic rather than decision-theoretic aspects of rationality. However, following Carnap’s early applications, philosophy has (...) more recently seen an increased popularity of probabilistic models in other areas concerned with the philosophical analysis of belief: there are models targeting coherence, informativeness, simplicity, and so on.In brief, the probabilistic model of belief comprises of a language, detailing the propositions about which an agent is supposed to have beliefs, and a function over the language that expresses beliefs: .. (shrink)

This Element is on new developments in the psychology of reasoning that raise or address philosophical questions. In traditional studies in the psychology of reasoning, the focus was on inference from arbitrary assumptions and not at all from beliefs, and classical binary logic was presupposed as the only standard for human reasoning. But recently a new Bayesian paradigm has emerged in the discipline. This views ordinary human reasoning as mostly inferring probabilistic conclusions from degrees of beliefs, (...) or from hypothetical premises relevant to a purpose at hand, and as often about revising or updating degrees of belief. This Element also covers new formulations of dual-process theories of the mind, stating that there are two types of mental processing, one rapid and intuitive and shared with other animals, and the other slow and reflective and more characteristic of human beings. The final topic covered is the new developments and rationality. (shrink)

There are three well-known models of how to account for perceptual belief within a probabilistic framework: (a) a Cartesian model; (b) a model advocated by Timothy Williamson; and (c) a model advocated by Richard Jeffrey. Each of these models faces a problem—in effect, the problem of accounting for the defeasibility of perceptual justification and perceptual knowledge. It is argued here that the best way of responding to this the best way of responding to this problem effectively vindicates the Cartesian (...) model. Finally, it is argued that, given the best interpretation of the probabilistic framework, that the Cartesian model is not vulnerable to the main criticisms that have been raised against it. (shrink)

In this paper, we explore the relation between an agent’s doxastic attitude and her arguments in support of a given claim. Our main contribution is the design of a logical setting that allows us reason about binary arguments which are either in favour or against a certain claim. This is a setting in which arguments and propositions are the basic building blocks so that the concept of argument-based belief emerges in a straightforward way. We work against the background of (...) Dung’s abstract argumentation framework, extending it to a new setting in which we can study the formal properties of binary arguments as well as the larger structures they establish. This paper introduces a formal ‘two-dimensional’ language to talk about propositions and arguments, for which a sound and complete axiom system is provided. (shrink)

This study examines the discursive structures and strategies used by Nigerian female blog users to legitimise gender inequality. Through a Critical Discourse Analysis (CDA) of 253 comments from five Nigerian-authored blogs, the study complicates celebratory approaches that conceptualise digital spaces as places for marginalised women to fight against sexism since they unravel ideological conceptions within female ‘counter-publics’ that rationalise un-equal gender relations. This refutes the belief that the struggle against gender discrimination only involves binary categories such as ‘men (...) class='Hi'>versus women’ or ‘women versus patriarchy’, spotlighting the self-limiting role of women in gender activism. The study demonstrates that understudying intra-group discrimination helps to understand and define inequality within specific contexts. Specifically, othering, classifying and categorising are inherent meaning-making traits that define intra-group dynamics and must be embraced and studied as contextual peculiarities that aid the understanding of given socio-cultural practices. This research underscores the usefulness of CDA in unpacking the intersections of culture, ideology, identity and language within multi-layered counter-public discourse, highlighting implications in social justice campaigns and the attainment of gender parity. (shrink)

Suppositions can be introduced in either the indicative or subjunctive mood. The introduction of either type of supposition initiates judgments that may be either qualitative, binary judgments about whether a given proposition is acceptable or quantitative, numerical ones about how acceptable it is. As such, accounts of qualitative/quantitative judgment under indicative/subjunctive supposition have been developed in the literature. We explore these four different types of theories by systematically explicating the relationships canonical representatives of each. Our representative qualitative accounts of (...) indicative and subjunctive supposition are based on the belief change operations provided by AGM revision and KM update respectively; our representative quantitative ones are offered by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as `provisional belief revision' wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we rely on a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update. (shrink)

This paper presents a probabilist paraconsistent theory of belief revision. This theory is based on a very general theory of probability, that fits with a wide range of classical and nonclassical logics. The theory incorporates a version of Jeffrey conditionalisation as its method of updating. A Dutch book argument is given, and the theory is applied to the problem of choosing a logical system.

Polarization simplifies and deforms language, ideas, and people and reduces social life into an oppositional binary based on harmful “us versus them” narratives. What can we do to bring about a transformation away from polarity to peace? What are the polarities obscuring the path to peace? Is it a question of belief versus belief? Does it make sense to appeal to reason, discourse, and compromise in a polarized climate? What is the difference between harmful and helpful polarities? (...) In the pursuit of peace and justice, the authors in this volume answer these and other questions relating to polarity and politics. (shrink)

The properties of antisymmetry and linearity are easily seen to be sufficient for a recursively enumerable binary relation to be recursively isomorphic to a recursive relation. Removing either condition allows for the existence of a structure where no recursive isomorph exists, and natural examples of such structures are surveyed.

Inwardness and interiority are concepts that have a multifaceted currency within many areas of medieval studies. These fields include, but are not limited to, historical studies, theology and religious studies, literary studies, and art history. Studies on inwardness, interiority, and selfhood intersect with an interest in what has often been called “popular religion” and in devotional behavior, both clerical and lay, to produce an engagement, across many fields, with inward or private aspects of religious belief and practice. “Popular religion” has (...) sometimes been presented as generally distinct or separate from official, ecclesiastical, or institutional ritual, and, as such, it is associated with other concepts like “private devotion” and even “interior piety.” Unhelpful binary oppositions are implied by qualifying terms like “private” and “popular,” because those qualifiers invoke the unsaid “public” or “official” or “outward.” More recently, terms like “vernacular theology” and Frömmigkeitstheologie have been preferred for their attempt to break down the “high-low” implications of terms like “popular religion” and “private devotion.” Though these terms themselves are not without potential problems or implications, the consonance or dissonance between individual religiosity and official or corporate expressions of religious belief can, and should, be examined in a more subtle manner than has often been the case. Medieval religious devotion was not a universally or solely private activity that was in some way opposite to the public, structured religiosity of the church's liturgy. It has been increasingly acknowledged in recent years that liturgy and devotion should not be opposed in this way, as private versus public, free and expressionist versus structured and defined. That having been said, it is nevertheless the case that the kind of piety that we are accustomed to call “devotional,” or contemplative, or meditative, presupposes and requires a certain inwardness and self-awareness, even if such activity is carried out in the company of others, or even during liturgical services. (shrink)

Coherentism in epistemology has long suffered from lack of formal and quantitative explication of the notion of coherence. One might hope that probabilistic accounts of coherence such as those proposed by Lewis, Shogenji, Olsson, Fitelson, and Bovens and Hartmann will finally help solve this problem. This paper shows, however, that those accounts have a serious common problem: the problem of belief individuation. The coherence degree that each of the accounts assigns to an information set (or the verdict it gives (...) as to whether the set is coherent tout court) depends on how beliefs (or propositions) that represent the set are individuated. Indeed, logically equivalent belief sets that represent the same information set can be given drastically different degrees of coherence. This feature clashes with our natural and reasonable expectation that the coherence degree of a belief set does not change unless the believer adds essentially new information to the set or drops old information from it; or, to put it simply, that the believer cannot raise or lower the degree of coherence by purely logical reasoning. None of the accounts in question can adequately deal with coherence once logical inferences get into the picture. Toward the end of the paper, another notion of coherence that takes into account not only the contents but also the origins (or sources) of the relevant beliefs is considered. It is argued that this notion of coherence is of dubious significance, and that it does not help solve the problem of belief individuation. (shrink)

Epistemologists often appeal to the idea that a normative theory must provide useful, usable, guidance to argue for one normative epistemology over another. I argue that this is a mistake. Guidance considerations have no role to play in theory choice in epistemology. I show how this has implications for debates about the possibility and scope of epistemic dilemmas, the legitimacy of idealisation in Bayesian epistemology, uniqueness versus permissivism, sharp versus mushy credences, and internalism versus externalism.

We describe the relation between coherence and foundations approaches to belief change in terms of a correspondence between epistemic entrenchment relations(Gärdenfors and Makinson, 1988; Rott, 1992) and dependence consequence relations from Bochman (1999, 2000a).The general conclusion of the study is that dependence consequence relations are sufficiently expressive to subsume the notion of an epistemic entrenchment and its generalizations.

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both (...) the Shannon entropy measure as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails. (shrink)

In this paper the problem of the relation between belief and acceptance is discussed in view of recent literature on the topic. Belief and acceptance are characterized in terms of a number of properties, which show both the similarities and the dissimilarities between these notions. In particular it is claimed - contrary to some recently expressed views - that acceptance need not be intentional action and that the differences between belief and acceptance do not boil down to the simple view (...) that acceptance, contrary to belief, is based on the agent's direct exercise of his will. Acceptance which is not based on intentional action is shown in the paper to be especially closely related to belief, especially to linguistic belief. Thus if a person is in a non-intentional-ly acquired and held state of acceptance that something p, he also believes that p. Another general difference is that acceptance is language-dependent while there can be non-linguistic belief. Collective beliefs and acceptances are also briefly discussed in the paper Among other things, it is noted that the notion of wide as contrasted with truth-oriented, narrow acceptance is central in the collective case. (shrink)

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and (...) safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

We summarize the recent progress made by probabilistic programming as a unifying formalism for the probabilistic, symbolic, and data-driven aspects of human cognition. We highlight differences with meta-learning in flexibility, statistical assumptions and inferences about cogniton. We suggest that the meta-learning approach could be further strengthened by considering Connectionist and Bayesian approaches, rather than exclusively one or the other.

This paper shows that epidemiology, an approach developed to study the social communication of biological information, can be instructively applied to the diffusion of "endemic" cultural beliefs. In particular, I examine whether exposure to information, or susceptibility to belief is more important in determining the distribution of food taboos in an oral society from the Democratic Republic of Congo. Matrix regression techniques are used on optimally scaled cultural similarity data to infer which social and psychological characteristics of the participating (...) individuals are correlated with a higher probability of taboo transmission between them. Results indicate that, despite a lack of mechanized transportation, access to information in this population is nearly universal. Constraints on belief adoption rather than on information flow are much more important in determining the intra-cultural distribution of food taboos. At least for one class of taboo, belief susceptibility is a function of an individual's social role rather than any intrinsic quality of the individual or the believability of the cultural trait. Nevertheless, the processes underlying the dissemination of both cultural traits and pathogens, considered as replicating units of information, appear close enough to justify using epidemiology as a common framework for investigating cultural and biological diffusion. (shrink)

In this paper, I develop a syntactic framework for the analysis ofstrategic form games that is based on a straightforward combination ofstandard systems of doxastic, probabilistic and conditionalpropositional logic. In particular, for the probabilistic part I makeuse of the axiomatization provided in Fagin and Halpern (1994). The use ofconditionals allows to represent a strategic form game by a logicalformula in a very natural way. Also expected utility maximization can benaturally captured. I use this framework to prove a version (...) of a resulton Nash equilibrium conjectures first presented in Aumann and Brandenburger (1995). (shrink)

The goal of standard semantics is to provide truth conditions for the sentences of a given language. Probabilistic Semantics does not share this aim; it might be said instead, if rather cryptically, that Probabilistic Semantics aims to provide belief conditions.The central and guiding idea of Probabilistic Semantics is that each rational individual has ‘within’ him or her a personal subjective probability function. The output of the function when given a certain sentence as input represents the degree of (...) likelihood which the individual would assign to that sentence. One can characterize these functions via a set of axioms, and in the terms of this defined structure develop probabilistic analogues of all important semantical notions. Then, dealing with ‘being given probability 1’ instead of ‘truth,’ one can proceed to give a completely adequate semantics for the particular language under consideration. The axioms delimiting the class of probability functions are, in fact, chosen with this goal and language in mind. (shrink)

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2×2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch-structures, these two visualizations (compared to the 2×2 table) can even simultaneously (...) display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2×2 tables and (double-) trees have a decisive disadvantage: While joint probabilities (e.g., P(AB)) are represented in 2×2 tables but no conditional probabilities (e.g., P(A|B)), it is exactly the other way around with (double-) trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2×2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2×2 table, double-tree, or net diagram with natural frequencies) are consistently superior to probability visualizations. The frequency net performs as well as the frequency double-tree. Only the 2×2 table with frequencies—the only visualization that participants were already familiar with —led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, and no uniform format effect becomes visible. Participants reached the highest performance in the versions with probability 2×2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability. (shrink)

There is a deep tension between logical and probabilistic norms of belief. This article illustrates the normative force that is associated with these frameworks by showing how rather unrestricted belief bases can be improved by undergoing logical and probabilistic reflection. It is argued that probabilistic reasoning accounts for the reliability of the conclusions one can draw from the beliefs. Most importantly, reliability commands us to care for the increasing uncertainty of conjunctions of beliefs. Deductive logic (...) captures the agent's commitment towards all conclusions of her beliefs. In other words, it seems that commitment forces the agent to believe conclusions, which reliability tells her to view with scepticism. This conflict is resolved by arguing that commitment merely forces the agent to find the logical conclusions of her beliefs acceptable, not necessarily to believe them. This demands consistency but no deductive closure. Furthermore, it is shown that by restoring consistency, a straightforward connection between high subjective probability and full belief is undermined. Finally, the article delineates how this approach deals with the lottery paradox and the preface paradox. (shrink)

This study assessed the extent to which within-individual variation in schizotypy and paranormal belief influenced performance on probabilistic reasoning tasks. A convenience sample of 725 non-clinical adults completed measures assessing schizotypy (Oxford-Liverpool Inventory of Feelings and Experiences; O-Life brief), belief in the paranormal (Revised Paranormal Belief Scale; RPBS) and probabilistic reasoning (perception of randomness, conjunction fallacy, paranormal perception of randomness, and paranormal conjunction fallacy). Latent profile analysis (LPA) identified four distinct groups: class 1, low schizotypy and low paranormal (...) belief (43.9% of sample); class 2, moderate schizotypy and moderate paranormal belief (18.2%); class 3, moderate schizotypy (high cognitive disorganization) and low paranormal belief (29%); and class 4, moderate schizotypy and high paranormal belief (8.9%). Identification of homogeneous classes provided a nuanced understanding of the relative contribution of schizotypy and paranormal belief to differences in probabilistic reasoning performance. Multivariate analysis of covariance revealed that groups with lower levels of paranormal belief (classes 1 and 3) performed significantly better on perception of randomness, but not conjunction problems. Schizotypy had only a negligible effect on performance. Further analysis indicated that framing perception of randomness and conjunction problems in a paranormal context facilitated performance for all groups but class 4. (shrink)

Past research reveals a tension between children's preferences for egalitarianism and ingroup favoritism when distributing resources to others. Here we investigate how children's evaluations and expectations of others' behaviors compare. Four- to 10-year-old children viewed events where individuals from two different groups distributed resources to their own group, to the other group, or equally across groups. Groups were described within a context of intergroup competition over scarce resources. In the Evaluation condition, children were asked to evaluate which resource distribution actions (...) were nicer. In the Expectation condition, children were asked to predict which events were more likely to occur. With age, children's evaluations and expectations of others' actions diverged: Children evaluated egalitarian actions as nicer yet expected others to behave in ways that benefit their own group. Thus, children's evaluations about the way human social actors should behave do not mirror their expectations concerning those individuals' actions. (shrink)