Recently, Nathan criticized Bar-Hillel and Falk's analysis of some classical probability puzzles on the grounds that they wrongheadedly applied mathematics to the solving of problems suffering from “ambiguous informalities”. Nathan's prescription for solving such problems boils down to assuring in advance that they are uniquely and formally soluble—though he says little about how this is to be done. Unfortunately, in real life problems seldom show concern as to whether their naturally occurring formulation is or is not ambiguous, does or does (...) not allow for unique formalization, etc. One step towards dealing with such problems intelligently is to recognize certain common cognitive pitfalls to which solvers seem vulnerable. This is discussed in the context of some examples, along with some empirical results. (shrink)

This paper presents a new solution to the well-known exchange paradox, or what is sometimes referred to as the two-envelope paradox. Many recent commentators have analyzed the paradox in terms of the agent’s biased concern for the contents of his own arbitrarily chosen envelope, claiming that such bias violates the manifest symmetry of the situation. Such analyses, however, fail to make clear exactly how the symmetry of the situation is violated by the agent’s hypothetical conclusion that he ought to switch (...) envelopes on the assumption that his own envelope contains some specific amount of money. In this paper, I offer an explanation of this fact based on the idea that the agent’s deliberations are not only constrained by the epistemic symmetry reflected in the agent’s uniform ignorance as to the contents of the two envelopes, but also by a deeper methodological symmetry that manifests itself in the intuition that the two envelopes constitute equally legitimate sources of potential information. I interpret this intuition as implying that the agent’s final decision should reflect a point of rational equilibrium arrived at through an iterated process of counterfactual self-reflection, whereby the agent takes account of what he would have thought had he instead chosen the other envelope. I provide a formal model of this method of counterfactual self-reflection and show, in particular, that it cannot issue in the paradoxical conclusion that the agent ought to switch regardless of how much money is in his envelope. In this way, by correcting for the bias in the agent’s reasoning, the paradox is resolved. (shrink)

In this response, we clarify several misunderstandings of the understanding/acceptance principle and defend our specific operationalization of that principle. We reiterate the importance of addressing the problem of rational task construal and we elaborate the notion of computational limitations contained in our target article. Our concept of thinking dispositions as variable intentional-level styles of epistemic and behavioral regulation is explained, as is its relation to the rationality debate. Many of the suggestions of the commentators for elaborating two-process models are easily (...) integrated into our generic dual-process account. We further explicate how we view the relation between System 1 and System 2 and evolutionary and normative rationality. We clarify our attempt to fuse the contributions of the cognitive ecologists with the insights of the original heuristics and biases researchers. (shrink)

Cognitive scientists employ brainteasers, or “cognitive illusions” in service of cognitive research. In some cases, rewording or paraphrasing a question can change the correct answer, without the experimenter realizing. In recent published articles describing cognitive research, participants have pondered brainteasers, including what is here called the “Two Children” problem. Although many accounts of this problem in the academic literature navigate its nuances correctly, in the popular press it is usually presented such that the wording of the question does not actually (...) compel the answer that is offered as correct. Unfortunately, cognitive scientists who have used this problem in research have adopted the more informal and ambiguous version that has become common in the popular press. Recent experiments have used the problem to demonstrate that people make common intuitive errors, but they unfortunately employ an ambiguous form of the question and the answer to the problem is not determinate, as they imply. (shrink)

This paper employs epistemic logic to investigate the philosophical foundations of Bayesian updating in belief revision. By Bayesian updating, we understand the tenet that an agent's degrees of belief—assumed to be encoded as a probability distribution—should be revised by conditionalization on the agent's total knowledge up to that time. A familiar argument, based on the construction of a diachronic Dutch book, purports to show that Bayesian updating is the only rational belief-revision policy. We investigate the conditions under which the premises (...) of this argument might be satisfied. Specifically, we consider the case of an artificial agent whose language (of thought) features a modal operator TK, where TK ψ has the interpretation “My total knowledge is ψ”. We show that every proposition of the form TK ψis epistemically categorical: it determines, for every proposition φ in the agent's language, whether the agent knows that φ. We argue that, for certain artificial agents employing such a language, the diachronic Dutch book argument for Bayesian updating is on firm ground. (shrink)

I discuss three puzzles of probability theory which seem connected with problems of direct reference and rigid designation. The resolution of at least one of them requires referential use of definite descriptions in probability statements. I argue that contrary to common opinion all these puzzles are in a way still unsolved: They seem to exemplify cases in which a change of probabilities is rationally required, even though any specific change presupposes unjustified assumptions.

Two old problems in probability theory involving the concept of randomness are considered. Data obtained with one of them--Bertrand's chord problem--demonstrate the equivocality of this term in the absence of a definition or explication of assumptions underlying its use. They also support two propositions about probabilistic thinking: (1) upon obtaining an answer to a question of probability, people tend to see it as the answer, overlooking tacit assumptions on which it may be based, and tend not to consider the possibility (...) of other assumptions from which different answers would follow; (2) perceptual balance--moderate uniformity of density of distribution--is a determinant of the degree to which a two-dimensional pattern is perceived as random. (shrink)

Cognition is both empowered and limited by representations. The matrix lens model explicates tasks that are based on frequency counts, conditional probabilities, and binary contingencies in a general fashion. Based on a structural analysis of such tasks, the model links several problems and semantic domains and provides a new perspective on representational accounts of cognition that recognizes representational isomorphs as opportunities, rather than as problems. The shared structural construct of a 2 × 2 matrix supports a set of generic tasks (...) and semantic mappings that provide a unifying framework for understanding problems and defining scientific measures. Our model's key explanatory mechanism is the adoption of particular perspectives on a 2 × 2 matrix that categorizes the frequency counts of cases by some condition, treatment, risk, or outcome factor. By the selective steps of filtering, framing, and focusing on specific aspects, the measures used in various semantic domains negotiate distinct trade-offs between abstraction and specialization. As a consequence, the transparent communication of such measures must explicate the perspectives encapsulated in their derivation. To demonstrate the explanatory scope of our model, we use it to clarify theoretical debates on biases and facilitation effects in Bayesian reasoning and to integrate the scientific measures from various semantic domains within a unifying framework. A better understanding of problem structures, representational transparency, and the role of perspectives in the scientific process yields both theoretical insights and practical applications. (shrink)

Six recently discussed problems in discrete probabilistic sample space, which have been found puzzling and even paradoxical, are reexamined. The importance is stressed of a sharp distinction between the formalization of mathematical problems and their formal solution that, applied to probability theory, must lead through the explicit partitioning of a sample space. If this approach is consistently followed, such problems reveal themselves to be either inherently ambiguous, and therefore without solution, or quite straightforward. In both cases nothing remains of any (...) sense of paradox. (shrink)

The Sleeping Beauty problem—first presented by A. Elga in a philosophical context—has captured much attention. The problem, we contend, is more aptly regarded as a paradox: apparently, there are cases where one ought to change one’s credence in an event’s taking place even though one gains no new information or evidence, or alternatively, one ought to have a credence other than 1/2 in the outcome of a future coin toss even though one knows that the coin is fair. In this (...) paper we argue for two claims. First, that Sleeping Beauty does gain potentially new relevant information upon waking up on Monday. Second, his credence shift is warranted provided it accords with a calculation that is a result of conditionalization on the relevant information: “this day is an experiment waking day” (a day within the experiment on which one is woken up). Since Sleeping Beauty knows what days d could refer to, he can calculate the probability that the referred to waking day is a Monday or a Tuesday providing an adequate resolution of the paradox. (shrink)

The Bayesian model is used in psychology as the reference for the study of dynamic probability judgment. The main limit induced by this model is that it confines the study of revision of degrees of belief to the sole situations of revision in which the universe is static (revising situations). However, it may happen that individuals have to revise their degrees of belief when the message they learn specifies a change of direction in the universe, which is considered as changing (...) with time (updating situations). We analyze the main results of the experimental literature with regard to elementary qualitative properties of these two situations of revision. First, the order effect phenomenon is confronted with the commutative property. Second, an apparent new phenomenon is presented: the redundancy effect that is confronted with the idempotence property. Finally, results obtained in this kind of experimental situations are reinterpreted in the light of pragmatic analysis. (shrink)

This paper rejects a traditional epistemic interpretation of conditional probability. Suppose some chance process produces outcomes X, Y,..., with probabilities P(X), P(Y),... If later observation reveals that outcome Y has in fact been achieved, then the probability of outcome X cannot normally be revised to P(X|Y) ['P&Y)/P(Y)]. This can only be done in exceptional circumstances - when more than just knowledge of Y-ness has been attained. The primary reason for this is that the weight of a piece of evidence varies (...) with the means by which it is provided, so knowledge of Y-ness does not have uniform impact on the probability of X. A better updating of the probability of X is provided by P(X|Y*), where Y* is not an outcome of the chance process being observed, but the sentence 'the outcome Y has been observed', an 'outcome' of the subsequent observation process. This alternative formula is widely endorsed in practice, but not well recognized in theory, where the oversight has generated some unsatisfactory consequences. (shrink)

It has been a vexing question in recent years whether concepts are fuzzy. In this paper several views on the fuzziness of concepts are pointed out to have stemmed from dubious concepts of fuzziness. The underlying notions of the roles feasibly played byprototype, set, andprobability in modeling concepts strongly suggest that the controversy originates from a vague relation between intuitive and mathematical ideas in the cognitive sciences. It is argued that the application of fuzzy sets cannot resolve this vagueness since (...) they are one sided,viz., defined on sets. An alternative definition based on classes (in the sense of axiomatic set theory) is proposed. (shrink)

You know that a two-child family has a son. What is the probability that the family has two sons? And what is this probability if you know that the family has a son born on a Tuesday? The former question has been widely discussed previously. The latter adds a new puzzling twist to the situation. In both cases the answer should depend on the specifics of the assumed underlying procedure by which the given information has been obtained. Quantitative analysis, assuming (...) one scenario, shows that the information on the son's day of birth changes the target probability. However, the relevance of being born on Tuesday to the question of the children's genders seems bizarre, since the same would be true for any other day. This apparent paradox is further probed in an attempt to alleviate the ensuing psychological difficulty. (shrink)

Bayesian Conditionalization is a widely used proposal for how to update one’s beliefs upon the receipt of new evidence. This is in part because of its attention to the totality of one’s evidence, which often includes facts about what one’s new evidence is and how one has come to have it. However, an increasingly popular position in epistemology holds that one may gain new evidence, construed as knowledge, without being in a position to know that one has gained this evidence. (...) These are cases of KK-Failure, cases where one knows p but is not in a position to know that one knows p. This paper assumes that certain KK-Failures are possible and argues that Conditionalization goes wrong in those cases. (shrink)

Colin Howson (1995 ) offers a counter-example to the rule of conditionalization. I will argue that the counter-example doesn't hit its target. The problem is that Howson mis-describes the total evidence the agent has. In particular, Howson overlooks how the restriction that the agent learn 'E and nothing else' interacts with the de se evidence 'I have learnt E'.

Generally, rational decision-making is conceived as arriving at a decision by a correct application of the rules of logic and statistics. If not, the conclusions are called biased. After an impressive series of experiments and tests carried out in the last few decades, the view arose that rationality is tough for all, skilled field experts not excluded. A new type of planner's counsellor is called for: the normative statistician, the expert in reasoning with uncertainty par excellence. To unravel this view, (...) the paper explores a specific practice of clinical decision-making, namely Evidence-Based Medicine. This practice is chosen, because it is very explicit about how to rationalize practice. The paper shows that whether a decision-making process is rational cannot be assessed without taking into account the environment in which the decisions have to be taken. To be more specific, the decision to call for new evidence should be rational too. This decision and the way in which this evidence is obtained are crucial to validate the base rates. Rationality should be model-based, which means that not only the isolated decision-making process should take a Bayesian updating process as its norm, but should also model the acquisition of evidence (priors and tests results) as a rational process. (shrink)

Rutherford criticizes the way some people have analyzed the 2-children problem, claiming that slight nuances in the problem’s formulation can change the correct answer. However, his own data demonstrate that even when there is a unique correct answer, participants give intuitive answers that differ from it systematically — replicating the data reported by those he criticizes. Thus, his critique reduces to an admonition to use care in formulating and analyzing this brainteaser — which is always a good idea — but (...) contributes little what is known, analytically or empirically, about the 2-children problem. (shrink)