Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth”. This is a subtle logical statement usually felt as an evident true. The problem we are trying to solve is the cognitive reason for such a feeling. We postulate here that we (...) accept Holmes’ maxim as true because our adult brains are equipped with neural modules that perform naturally modal logical computations. (shrink)

This paper addresses the problem of ‘ignorance’ in philosophy and science, particularly with respect to the conceptualization, study and solution of environmental problems. We begin by distinguishing between ‘risk’, ‘uncertainty’ and ‘ignorance’. We then offer a categorization of ignorance, and use these categories to assess the role of science as a means of reducing ignorance. We note that to proceed with science, several 'acts of faith' are necessary. We conclude with a discussion of the importance of an attitude of openness (...) in science and philosophy, especially regarding environmental problems. Throughout, we illustrate our discussion of ignorance, and the problems involved in its reduction, by reference to environmental issues. (shrink)

One of the most actively discussed aspects of Keynes's thought during the last decade has been his concern with uncertainty and probability theory. As the concerns of current macroeconomic theorists have turned increasingly to the effects of expectations and uncertainty, interest has grown in the fact that Keynes was the author of A Treatise on Probability and that uncertainty plays a prominent role in Chapter 12 of The General Theory as well as in three 1937 papers in which he summarized (...) The General Theory's main point. Not surprisingly, though, there has been very little agreement in this recent discussion about exactly what the significance of Keynes's early work in probability was to his later work as an economist, or about what the roles of uncertainty and expectations are in The General Theory. (shrink)

Epistemic theories of objective chance hold that chances are idealised epistemic probabilities of some sort. After giving a brief history of this approach to objective chance, I argue for a particular version of this view, that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. The main argument for this view is the demonstration that it entails all of the commonly-accepted properties of chance. For example, this analysis entails that chances (...) supervene on the physical facts, that the chance function is an expert probability, that the existence of stable frequencies results from invariant chances in repeated experiments, and that chances are probably close to long-run relative frequencies in repeated experiments. Despite these virtues, epistemic approaches to chance have been neglected in recent decades, on account of their conflict with accepted views about closely related topics such as causation, laws of nature, and epistemic probability. However, existing views on these topics are also very problematic, and I believe that the epistemic view of chance is a key piece of this puzzle that, once in place, allows all the other pieces to fit together into a new and coherent way. I also respond to some criticisms of the epistemic theory that I favour. (shrink)

The subject matter of this thesis is analytic ontology. Chapters II and III deal with two versions of trope theory, or moderate nominalism; these are defined as ontologies which recognise properties and relations but no (real) universals. The key notion of both theories, trope, is characterised as an abstract particular. What the abstractness amounts to differs between the two. Yet another difference is that simplicity is an essential trait of a trope according to one theory, but not according to the (...) other. Though exact similarity is said to play an important role in both theories, as it turns out, this does not seem to be the case. The ontology dealt with in chapter IV is a mixture of moderate nominalism concerning qualities and realism concerning relations. In it, quality instances (moments) and universal relations are the ultimate constituents of the universe. While relations and moments are considered to be constituents of states of affairs, which are characterised as objects of higher orders, complexes that are objects of the first order are made up of moments on their own. Among these complexes one finds the ordinary objects. Paradoxically, although relations are necessary for the existence of complex first order objects, relations are not thought to be among the contents of these objects. The main subject of chapter V is a particular version of moderate realism; it is an ontology which is realistic in its recognition of universals and moderate in its recognition of instances of these universals. Instances combine to form complex networks. A theoretically motivated claim is that although each instance has a predicational aspect as well as a universal one, it is simple in the sense of lacking internal predicative structure; though, this claim can be called into question. Keywords: analytic ontology, moderate nominalism, moderate realism, particular, universal, abstract, concrete, abstract particular, abstract universal, concrete particular, concrete universal, trope, moment, complex unity, collection, instance, unit attribute, intensional aspect, predicational aspect, continuous composite, articulated composite. (shrink)

Time may be infinite in both directions. If it is, then, if persons could live at most once in all of time, the probability that you would be alive now would be zero. But if persons can live more than once, the probability that you would be alive now would be nonzero. Since you are alive now, with certainty, either the past is finite, or persons can live more than once.

Sections 1 through 3 present all of the main ideas behind the probabilistic logic of evidential support. For most readers these three sections will suffice to provide an adequate understanding of the subject. Those readers who want to know more about how the logic applies when the implications of hypotheses about evidence claims (called likelihoods) are vague or imprecise may, after reading sections 1-3, skip to section 6. Sections 4 and 5 are for the more advanced reader who wants a (...) detailed understanding of some telling results about how this logic may bring about convergence to the truth. (shrink)

The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the re-labelling (...) of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve labelling invariance is identified, and it is argued that, under the interpretation of Bertrand’s paradox suggested in the paper, the paradox does not undermine either the principle of indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena. It is shown in particular that violation of labelling invariance does not entail that labelling of random events affects the probabilities of random events. It also is argued, however, that the content of the principle of indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful. (shrink)

I argue for a theory that combines elements of reliabilism and evidentialism.

How to regard the weight we give to a proposition on the grounds of its being endorsed by an authority? I examine this question as it is raised within the epistemology of science, and I argue that “authority-based weight” should receive special handling, for the following reason. Our assessments of other scientists’ competence or authority are nearly always provisional, in the sense that to save time and money, they are not made nearly as carefully as they could be---indeed, they are (...) typically made on the basis of only a small portion of the available evidence. Consequently, we need to represent the authority-based elements of our epistemic attitudes in such a way as to allow the later revision of those elements, in case we decide in the light of new priorities that a more conscientious assessment is warranted. I look to the literature in confirmation theory, statistics, and economics for a semiformal model of this revision process, and make a particular proposal of my own. The discussion also casts some light on the question of why certain aspects of science’s epistemic state are not made public. (shrink)

This paper considers whether an analogy between distance and dissimilarlity supports the thesis that degree of dissimilarity is distance in a metric space. A straightforward way to justify the thesis would be to define degree of dissimilarity as a function of number of properties in common and not in common. But, infamously, this approach has problems with infinity. An alternative approach would be to prove representation and uniqueness theorems, according to which if comparative dissimilarity meets certain qualitative conditions, then it (...) is representable by distance in a metric space. I will argue that this approach faces equally severe problems with infinity. (shrink)

This essay addresses the methodology of philosophy of science and illustrates how formal and empirical methods can be fruitfully combined. Special emphasis is given to the application of experimental methods to confirmation theory and to recent work on the conjunction fallacy, a key topic in the rationality debate arising from research in cognitive psychology. Several other issue can be studied in this way. In the concluding section, a brief outline is provided of three further examples.

Commentators agree that simple conclusions about a general base rate fallacy are not appropriate. It is more constructive to identify conditions under which base rates are differentially weighted. Commentators also agree that improving the ecological validity of the research is desirable, although this is less important to those interested exclusively in psychological processes. The philosophers and ecologists among the commentators offer a kinder perspective on base rate reasoning than the psychologists. My own perspective is that the interesting questions (both psychological (...) and normative/prescriptive) are best answered through empirical studies that use real-world performance standards. (shrink)

We have been oversold on the base rate fallacy in probabilistic judgment from an empirical, normative, and methodological standpoint. At the empirical level, a thorough examination of the base rate literature (including the famous lawyer–engineer problem) does not support the conventional wisdom that people routinely ignore base rates. Quite the contrary, the literature shows that base rates are almost always used and that their degree of use depends on task structure and representation. Specifically, base rates play a relatively larger role (...) in tasks where base rates are implicitly learned or can be represented in frequentist terms. Base rates are also used more when they are reliable and relatively more diagnostic than available individuating information. At the normative level, the base rate fallacy should be rejected because few tasks map unambiguously into the narrow framework that is held up as the standard of good decision making. Mechanical applications of Bayes's theorem to identify performance errors are inappropriate when (1) key assumptions of the model are either unchecked or grossly violated, and (2) no attempt is made to identify the decision maker's goals, values, and task assumptions. Methodologically, the current approach is criticized for its failure to consider how the ambiguous, unreliable, and unstable base rates of the real world are and should be used. Where decision makers' assumptions and goals vary, and where performance criteria are complex, the traditional Bayesian standard is insufficient. Even where predictive accuracy is the goal in commonly defined problems, there may be situations (e.g., informationally redundant environments) in which base rates can be ignored with impunity. A more ecologically valid research program is called for. This program should emphasize the development of prescriptive theory in rich, realistic decision environments. (shrink)

Base rates have no necessary relation to judgments that are not themselves probabilities. There is no logical imperative, for instance, that behavioral base rates must affect causal attributions or that base rate information should affect judgments of legal liability. Decision theorists should be cautious in arguing that base rates place normative constraints on judgments of anything other than posterior probabilities.

Abstract(1) The miscitations of seminal experiments in the base rate literature adds to the existing database of systematic miscitations of wellknown psychological experiments. These miscitations may be caused by a process of reconstructive remembering. (2) Representative design should be the methodological core of Koehler's call for ecologically valid research. This approach can benefit both basic and applied research.

Koehler's summary and critique of research on the base rate fallacy is cogent and persuasive. However, he may have overstated the case, and his suggestions for future research may be too restrictive. We agree that methodological approaches to this topic should be broadened, but we argue that experimental laboratory research and the Bayesian normative standard are useful and should not be abandoned.

This commentary presents a self-assessment inventory that will allow readers to determine their own attitude toward the base rate fallacy and its literature. The inventory is scientifically valid but not Medicare/Medicaid reimbursable.

When base rates are learned and used in an experiential manner subjects show better base rate use, perhaps because the implicit learning system is engaged. A causal framework in which base rates are relevant might also be necessary. Humans might thus perform better on more ecologically valid tasks, which are likely to contain those three components.

Base rates are vital in predicting violent criminal recidivism. However, both lay people given simulated prediction tasks and professionals milking real life predictions appear insensitive to variations in the base rate of violent recidivism. Although there are techniques to help decision makers attend to base rates, increased decision accuracy is better sought in improved actuarial models as opposed to improved clinicians.

Consistent with Koehler's position, we propose a generalization of the base rate fallacy and earlier conservatism literatures. In studies using both traditional tasks and new tasks based on ecologically valid base rates, our subjects typically underweight individuating information at least as much as they underweight base rates. The implications of cue consistency for averaging heuristics are discussed.

Koehler is right that base rate information is used, to various degrees, both in laboratory tasks and in everyday life. However, it is not time to turn our backs on laboratory tasks and focus solely on ecologically valid decision making. Tightly controlled experimental data are still needed to understandhowbase rate information is used, and how this varies among groups.

Any instance (i.e., event, behavior, trait) belongs to infinitely many reference classes, hence there are infinitely many base rates from which to choose. People clearly do not entertain all possible reference classes, however, so something must be limiting the search space. We suggest some possible mechanisms that determine which reference class is evoked for the purpose of judgment and decision.

The fallacy beneath the base rate fallacy is that we know what a base rate is. We talk as if base rates and individuating information were two different kinds of information. From a Bayesian perspective, however, the only difference between base rate and individuating information is – which comes first.

Koehler is right to argue for more nuanced interpretation of base rate anomalies. These anomalies are best understood in relation to a broader class of cognitive anomalies, which are important for theory and practice. Recognizing a need for more nuanced analysis should not be taken as a license for treating the effects as “explained away.”.

Koehler's target article attempts a balanced view of the relevance of knowledge of base rates to judgments of subjective or credal probability, but he is not sensitive enough to the difference between requiring and permitting the equation of probability judgments with base rates, the interaction between precision of base rate and reference class information, and the possibility of indeterminate probability judgment.

Two distinct issues are sometimes confused in the base rate literature: Why do people make logical mistakes in the assessment of probabilities? and why do subjects not use base rates the way experimenters do? The latter problem may often reflect differences in an implicit reference class rather than a disinclination to update a base rate by Bayes' theorem. Also important are considerations concerning the interaction of several potentially relevant base rates.

The underutilization of base rates is a consistent finding. The strong claim that base rates are ignored has been rejected and this needs no further emphasis. Following the path of “normal science,” research examines the conditions predicting changes in the degree of underutilization. A scientific revolution that might dethrone the heuristics and biases paradigm is not in sight.

Koehler's call for a reanalysis of the base rate fallacy is particularly important in the applied domain of accounting, since base rate data appear to be an important input for many accounting tasks. In this commentary I discuss the use of base rates in accounting and explain why more flexible standards of performance are important when judging the use of base rates.

This commentary discusses three points: (1) The implications of the fact that it is rational to ignore base rates if probabilities are estimated by frequencies from samples without missing data (natural sampling); (2) second order probabilities distributions are a plausible way to model imprecise probabilities; and (3) Bayesian networks represent a normative reference for multi-cue models of probabilistic inference.

Failure to consider base rate is regarded as potentially hazardous, mainly because its consideration is assumed to be determined solely by P(H/D), the probability of the individuating data if the hypothesis is true, and not at all by P(D/˜H), the probability if the hypothesis is false. However, when P(D/˜H) is unconfounded from P(D/H), it turns out to be the stronger determinant of base rate consideration.

Setting the two hypotheses of complete neglect and full use of base rates against each other is inappropriate. The proper question concerns the degree to which base rates are used (or neglected), and under what conditions. We outline alternative approaches and recommend regression analysis. Koehler's conclusion that we have been oversold on the base rate fallacy seems to be premature.

A recent study showed physicians' reasoning about a realistic case to be ignorant of base rate. It also showed physicians interpreting information pertinent to base rate differently, depending on whether it was presented early or late in the case. Although these adult reasoners might do better if given hints through talk of relative frequencies, this would not prove that they had no problem of base rate neglect.

The environment in which humans evolved is strongly and positively autocorrelated in space and time. Probabilistic judgments based on the assumption of independence may not yield evolutionarily adaptive behavior. A number of “faults” of human reasoning are not faulty under fuzzy arithmetic, a nonprobabilistic calculus of reasoning under uncertainty that may be closer to that underlying human decision making.

In contrast to traditional research on base-rate neglect, an ecologically-oriented research program would analyze the correspondence between cognitive algorithms and the nature of information in the environment. Bayesian computations turn out to be simpler when information is represented in frequency formats as opposed to the probability formats used in previous research. Frequency formats often enable even uninstructed subjects to perform Bayesian reasoning.

The base rate literature has an opposite twin in the social psychological literature on stereotypes, which concludes that people use their preexisting beliefs about probabilistic category attributes too much, rather than not enough. This ironic discrepancy arises because beliefs about category attributes enhance accuracy when the beliefs are accurate and diminish accuracy when they are not. To determine the accuracy of base rate/stereotype beliefs requires research that addresses specific content.

Abstract(1) There is enough contradictory evidence regarding the role of base rates in category learning to confirm the nonexistence of biases in such learning. (2) It is not always possible to activate statistical reasoning through frequentist representation. (3) It is necessary to use the concept of systematic processing in reconsidering the published work on biases.

The important question is how people process probabilistic information, not whether they process it in accordance with a normative model that we never should have expected them to be capable of following. Experience is not the cure, as widely thought, to problems with utilizing base rate information.

As illustrated by research Koehler himself cites (Dawes et al. 1993), the purpose of experiments is to choose between contrasting explanations of past observations – rather than to seek statistical generalizations about the prevalence of effects. True external validity results not from sampling various problems that are representative of “real world” decision making, but from reproducing an effect in the laboratory with minimal contamination (including from real world factors).

The base rate fallacy is directly dependent on a particular judgment paradigm in which information may be unambiguously designated as either “base rate” or “individuating,” and in which subjects make two-stage sequential judgments. The paradigm may be a poor match for real world settings, and the fallacy may thus be undefined for natural ecologies of judgment.

There is a familiar risk of antinomy if fromxisEand p(xisH/xisE) =rit is permissible to infer p(xisH) =r, and what Carnap (1950) called “The requirement of total evidence” will not prevent such antinomies satisfactorily. What is needed instead is a properly developed theory of evidential weight.

Koehler's work will assist the effort to understand legal fact finding. It leaves two questions somewhat open: (i) the extent to which empirical research can measure correctness of fact-finding, a function that involves the resolution of normative questions and (ii) the standards judges should use in the absence of the research he advocates.

Cognitive algebra strongly disproved the representativeness heuristic almost before it was published; and therewith it also disproved the base rate fallacy. Cognitive algebra provides a theoretical foundation for judgment-decision theory through its joint solution to the two fundamental problems – true measurement of subjective values, and cognitive rules for integration of multiple determinants.

It seems obvious that when higher-order evidence makes it rational for one to doubt that one’s own belief on some matter is rational, this can undermine the rationality of that belief. This is known as higher-order defeat. However, despite its intuitive plausibility, it has proved puzzling how higher-order defeat works, exactly. To highlight two prominent sources of puzzlement, higher-order defeat seems to defy being understood in terms of conditionalization; and higher-order defeat can sometimes place agents in what seem like epistemic (...) dilemmas. This chapter draws attention to an overlooked aspect of higher-order defeat, namely that it can undermine the resilience of one’s beliefs. The notion of resilience was originally devised to understand how one should reflect the ‘weight’ of one’s evidence. But it can also be applied to understand how one should reflect one’s higher-order evidence. The idea is particularly useful for understanding cases where one’s higher-order evidence indicates that one has failed in correctly assessing the evidence, without indicating whether one has over- or underestimated the degree of evidential support for a proposition. But it is exactly in such cases that the puzzles of higher-order defeat seem most compelling. (shrink)

In everyday life and in science we acquire evidence of evidence and based on this new evidence we often change our epistemic states. An assumption underlying such practice is that the following EEE Slogan is correct: 'evidence of evidence is evidence' (Feldman 2007, p. 208). We suggest that evidence of evidence is best understood as higher-order evidence about the epistemic state of agents. In order to model evidence of evidence we introduce a new powerful framework for modelling epistemic states, Dyadic (...) Bayesianism. Based on this framework, we then discuss characterizations of evidence of evidence and argue for one of them. Finally, we show that whether the EEE Slogan holds, depends on the specific kind of evidence of evidence. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities are (...) not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

This paper presents a new argument for the Principle of Indifference. This argument can be thought of in two ways: as a pragmatic argument, justifying the principle as needing to hold if one is to minimise worst-case expected loss, or as an epistemic argument, justifying the principle as needing to hold in order to minimise worst-case expected inaccuracy. The question arises as to which interpretation is preferable. I show that the epistemic argument contradicts Evidentialism and suggest that the relative plausibility (...) of Evidentialism provides grounds to prefer the pragmatic interpretation. If this is right, it extends to a general preference for pragmatic arguments for the Principle of Indifference, and also to a general preference for pragmatic arguments for other norms of Bayesian epistemology. (shrink)

Uncertainty plays an important role in The General Theory, particularly in the theory of interest rates. Keynes did not provide a theory of uncertainty, but he did make some enlightening remarks about the direction he thought such a theory should take. I argue that some modern innovations in the theory of probability allow us to build a theory which captures these Keynesian insights. If this is the right theory, however, uncertainty cannot carry its weight in Keynes’s arguments. This does not (...) mean that the conclusions of these arguments are necessarily mistaken; in their best formulation they may succeed with merely an appeal to risk. (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.