Bayesian prior probabilities have an important place in probabilistic and statistical methods. In spite of this fact, the analysis of where these priors come from and how they are formed has received little attention. It is reasonable to excuse the lack, in the foundational literature, of detailed psychological theory of what are the mechanisms by which prior probabilities are formed. But it is less excusable that there is an almost total absence of a detailed discussion of the highly differentiating nature (...) of past experience in forming a prior. The focus here is on what kind of account, even if necessarily schematic, can be given about the psychological mechanisms back of the formation of Bayesian priors. The last section examines a detailed experiment relevant to how priors are learned. (shrink)

Theoria, Volume 87, Issue 1, Page 247-251, February 2021.

Satisficing is a central concept in both individual and social multiagent decision making. In this paper we first extend the notion of satisficing by formally modeling the tradeoff between costs and decision failure. Second, we extend this notion of “neo”-satisficing into the context of social or multiagent decision making and interaction, and model the social conditioning of preferences in a satisficing framework.

This paper defends two related claims about belief. First, the claim that unlike numerical degrees of belief, comparative beliefs are primitive and psychologically real. Second, the claim that the fundamental norm of Probabilism is not that numerical degrees of belief should satisfy the probability axioms, but rather that comparative beliefs should satisfy certain constraints.

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen and (...) Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

This paper discusses a challenge for Comparativists about belief, who hold that numerical degree of belief (in particular, subjective probability) is a useful fiction, unlike comparative belief, which they regard as real. The challenge is to make sense of claims like ‘I am twice as confident in A as in B’ in terms of comparative belief only. After showing that at least some Comparativists can meet this challenge, I discuss implications for Zynda’s [2000] and Stefánsson’s [2017] defences of Comparativism.

We explore which types of probabilistic updating commute with convex IP pooling. Positive results are stated for Bayesian conditionalization, imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of externally Bayesian pooling operators due to Wagner :336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

I focus my discussion on the well-known Ellsberg paradox. I find good normative reasons for incorporating non-precise belief, as represented by sets of probabilities, in an Ellsberg decision model. This amounts to forgoing the completeness axiom of expected utility theory. Provided that probability sets are interpreted as genuinely indeterminate belief, such a model can moreover make the “Ellsberg choices” rationally permissible. Without some further element to the story, however, the model does not explain how an agent may come to have (...) unique preferences for each of the Ellsberg options. Levi holds that the extra element amounts to innocuous secondary “risk” or security considerations that are used to break ties when more than one option is rationally permissible. While I think a lexical choice rule of this kind is very plausible, I argue that it involves a greater break with xpected utility theory than mere violation of the ordering axiom. (shrink)

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis for Bayesian (...) reasoning with idealized models in science. (shrink)

If we think, as Ramsey did, that a degree of belief that P is a stronger or weaker tendency to act as if P, then it is clear that not only uncertainty, but also vagueness, gives rise to degrees of belief. If I like hot coffee and do not know whether the coffee is hot or cold, I will have some tendency to reach for a cup; if I like hot coffee and know that the coffee is borderline hot, I (...) will have some tendency to reach for a cup. Suppose that we take degrees of belief arising from uncertainty to obey the laws of probability and that we model vagueness using degrees of truth. We then encounter a problem: it does not look as though degrees of belief arising from vagueness should obey the laws of probability. One response would be to countenance two different sorts of degrees of belief: degrees of belief arising from uncertainty, which obey the laws of probability; and degrees of belief arising from vagueness, which obey a different set of laws. I argue, however, that if a degree of belief that P is a stronger or weaker tendency to act as if P, then this option is not open. Instead, I propose an account of the behaviour of degrees of belief that integrates subjective probabilities and degrees of truth. On this account, degrees of belief are expectations of degrees of truth. The account explains why degrees of belief behave in accordance with the laws of probability in cases involving only uncertainty, while also allowing degrees of belief to behave differently in cases involving only vagueness, and in mixed cases involving both uncertainty and vagueness. Justifications of the account are given both via Dutch books and in terms of epistemic accuracy. (shrink)

This paper is a reply to some of Scott Soames ' comments on my colloquium paper Marcus, Kripke, and the Origin of the New Theory of Reference. Except for the indicated parts added in May, 1995, this paper was written on December 16th–25th, 1994 as my reply to Soames for the APA colloquium in Boston, December 28, 1994. In this paper, I argue that Soames ' contention that Marcus is not one of the primary founders of contemporary nondescriptivist theories of (...) reference is false. Soames presents numerous arguments for his thesis that Marcus did not originate ideas later elaborated upon by Kripke, but his arguments are unsound; they are based in part on a misunderstanding of Marcus' theory and in part on an inadequate grasp of some of the key notions of the New Theory of Reference, such as the notion of a posteriori necessities and the notion of reference-fixing descriptions. (shrink)

This paper presents a new solution to the problems for orthodox decision theory posed by the Pasadena game and its relatives. I argue that a key question raised by consideration of these gambles is whether evaluative compositionality (as I term it) is a requirement of rationality: is the value that an ideally rational agent places on a gamble determined by the values that she places on its possible outcomes, together with their mode of composition into the gamble (i.e. the probabilities (...) assigned to them)? The paper first outlines a certain simple response to the Pasadena game and identifies two problems with this response, the second of which is that it leads to a wholesale violation of evaluative compositionality. I then argue that rationality does not require decision makers to factor in outcomes of arbitrarily low probability. A method for making decisions which flows from this basic idea is then developed, and it is shown that this decision method (Truncation) leads to a limited — as opposed to wholesale — violation of evaluative compositionality. The paper then argues that the truncation method yields solutions to the problems posed by the Pasadena game and its relatives that are both attractive in themselves and superior to those yielded by alternative proposals in the literature. (shrink)

The question of coherence of rules for changing degrees of belief in the light of new evidence is studied, with special attention being given to cases in which evidence is uncertain. Belief change by the rule of conditionalization on an appropriate proposition and belief change by "probability kinematics" on an appropriate partition are shown to have like status.

In contemporary works on conditionals, the Ramsey test is a procedure for the evaluation of conditional sentences. There are several versions of the test, all inspired by a footnote by the British philosopher and mathematician Frank Ramsey, in his General Propositions and Causality. However, no study on Ramsey's own account of conditionals has been put forth so far. Furthermore, the footnote seems to cover indicative conditionals only, and this has led to the belief that no account of counterfactuals can be (...) found in Ramsey's work. In this paper, I recover Ramsey's account of counterfactuals and show that it is sketched in the footnote too. The result is a well-developed account of counterfactuals that resembles many contemporary ones. But Ramsey uses the same approach also for other types of conditionals, and this casts doubts on the current criteria for the classification of this type of sentences. (shrink)

The problem of creeping minimalism concerns how to tell the difference between metaethical expressivism and its rivals given contemporary expressivists’ acceptance of minimalism about truth and related concepts. Explanationism finds the difference in what expressivists use to explain why ethical language and thought has the content it does. I argue that two recent versions of explanationism are unsatisfactory and offer a third version, subject matter explanationism. This view, I argue, captures the advantages of previous views without their disadvantages and gives (...) us a principled and general characterisation of non-representational views about all kinds of language and thought. (shrink)

In this paper the strategy for the eliminative reduction of the alethic modalities suggested by John Venn is outlined and it is shown to anticipate certain related contemporary empiricistic and nominalistic projects. Venn attempted to reduce the alethic modalities to probabilities, and thus suggested a promising solution to the nagging issue of the inclusion of modal statements in empiricistic philosophical systems. However, despite the promise that this suggestion held for laying the ‘ghost of modality’ to rest, this general approach, tempered (...) modal eliminativism, is shown to be inadequate for that task.<br><br>. (shrink)

Theories of subjective probability are viewed as formal languages for analyzing evidence and expressing degrees of belief. This article focuses on two probability langauges, the Bayesian language and the language of belief functions (Shafer, 1976). We describe and compare the semantics (i.e., the meaning of the scale) and the syntax (i.e., the formal calculus) of these languages. We also investigate some of the designs for probability judgment afforded by the two languages.

Probabilism, the view that agents have numerical degrees of beliefs that conform to the axioms of probability, has been defended by the vast majority of its proponents by way of either of two arguments, the Dutch Book Argument and the Representation Theorems Argument. In this paper I argue that both arguments are flawed. The Dutch Book Argument is based on an unwarranted, ad hoc premise that cannot be dispensed with. The Representation Theorems Argument hinges on an invalid implication.

The emergence of economic inequality has often been linked to individual differences in mental or physical capacities. By means of an agent-based simulation this paper shows that neither of these is a necessary condition. Rather, inequality can arise from iterated interactions of fully rational agents. This bears consequences for our understanding of both inequality and rationality. In a setting of iterated bargaining games, we claim that expected utility maximizing agents perform suboptimally in comparison with other strategies. The reason for this (...) lies in complex feedback effects between an agents’ action and the quality of beliefs used to calculate expected utility. Consequentially, we argue that the standard notion of rationality as maximizing expected utility is insufficient, even for certain standard cases of economic interaction. (shrink)

The proposition that Tweety is a bird coheres better with the proposition that Tweety has wings than with the proposition that Tweety cannot fly. This relationship of contrastive coherence is the focus of the present paper. Based on recent work in formal epistemology we consider various possibilities to model this relationship by means of probability theory. In a second step we consider different applications of these models. Among others, we offer a coherentist interpretation of the conjunction fallacy.

The hypothesis that human reasoning and decision-making can be roughly modeled by Expected Utility Theory has been at the core of decision science. Accumulating evidence has led researchers to modify the hypothesis. One of the latest additions to the field is Dual Process theory, which attempts to explain variance between participants and tasks when it comes to deviations from Expected Utility Theory. It is argued that Dual Process theories at this point cannot replace previous theories, since they, among other things, (...) lack a firm conceptual framework, and have no means of producing independent evidence for their case. (shrink)

We point out a simple but hitherto ignored link between the theoryof updates, the theory of counterfactuals, and classical modal logic: update is a classicalexistential modality, counterfactual is a classical universalmodality, and the accessibility relations corresponding to these modalities are inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverse relationship. We use this fact to translate between rules for updates andrules for counterfactuals. Thus, Katsuno and Mendelzons postulatesU1--U8 are translated into counterfactual rules C1--C8(Table VII), and (...) many of the familiar counterfactual rulesare translated into rules for updates (Table VIII). Ourconclusions are summarised in Table V. (shrink)

The aim of this paper is to provide a philosophical interpretation of bilateralism in terms of probabilism. In particular, to interpret the main concepts of bilateralism –acceptance, rejection and incoherence– in terms of the probabilistic notions of degree of belief and coherence. According to bilateralism, the meaning of logical connectives is determined by the acceptance and rejection conditions of the sentences in which they are involved, where acceptance and rejection cannot be reduced to one another. I will focus on a (...) variant of bilateralism that understands logical consequence as the statement that it is incoherent to accept all the premises of a valid argument while rejecting all its conclusions. On the other hand, probabilism states that it is possible to interpret our degrees of belief in terms of probabilities. The aim of this work is then to interpret the concept of incoherence in terms of probability functions and determine when it is coherent to accept or to reject a proposition according to some threshold defined in terms of degrees of belief. To achieve this goal, we need both an interpretation of the concept of incoherence coined by the bilateralists as well as an interpretation of acceptance and rejection. I will show that a good interpretation of coherence in probabilistic terms can already be found in the literature. Then, I will give an interpretation of acceptance and rejection in terms of degrees of belief. In particular, I will show that it is possible to interpret these concepts in accordance with Locke’s thesis, the thesis that states that there is some threshold r such that if you believe some sentence in degree equal or higher than r you should accept it, without falling into epistemic paradoxes. (shrink)

This paper challenges Bird’s view that scientific progress should be understood in terms of knowledge, by arguing that unjustified scientific beliefs (and/or changes in belief) may nevertheless be progressive. It also argues that false beliefs may promote progress.

When do we agree? The answer might once have seemed simple and obvious; we agree that p when we each believe that p. But from a formal epistemological perspective, where degrees of belief are more fundamental than beliefs, this answer is unsatisfactory. On the one hand, there is reason to suppose that it is false; degrees of belief about p might differ when beliefs simpliciter on p do not. On the other hand, even if it is true, it is too (...) vague; for what it is to believe simpliciter ought to be explained in terms of degrees of belief. This paper presents several possible notions of agreement, and corresponding notions of disagreement. It indicates how the findings are fruitful for the epistemology of disagreement, with special reference to the notion of epistemic peerhood. (shrink)

How are we to understand the use of probability in corroboration functions? Popper says logically, but does not show we could have access to, or even calculate, probability values in a logical sense. This makes the logical interpretation untenable, as Ramsey and van Fraassen have argued. -/- If corroboration functions only make sense when the probabilities employed therein are subjective, however, then what counts as impressive evidence for a theory might be a matter of convention, or even whim. So isn’t (...) so-called ‘corroboration’ just a matter of psychology? -/- In this paper, I argue that we can go some way towards addressing this objection by adopting an intersubjective interpretation, of the form advocated by Gillies, with respect to corroboration. I show why intersubjective probabilities are preferable to subjective ones when it comes to decision making in science: why group decisions are liable to be superior to individual ones, given a number of plausible conditions. I then argue that intersubjective corroboration is preferable to intersubjective confirmation of a Bayesian variety, because there is greater opportunity for principled agreement concerning the factors involved in the former. (shrink)

In this article, I present some new group level interpretations of probability, and champion one in particular: a consensus-based variant where group degrees of belief are construed as agreed upon betting quotients rather than shared personal degrees of belief. One notable feature of the account is that it allows us to treat consensus between experts on some matter as being on the union of their relevant background information. In the course of the discussion, I also introduce a novel distinction between (...) intersubjective and interobjective interpretations of probability. (shrink)

This paper argues that representationalism of a Fodorian variety can accommodate the fact that beliefs come in degrees. First, it responds to two key arguments to the contrary. Second, it builds upon these responses and outlines a novel representationalist theory of degrees of beliefs. I call this theory dispositional representationalism, as it involves direct appeal to our dispositions to form representations and propositional attitudes concerning them.

People often act in ways that appear incompatible with their sincere assertions. But how might we explain such cases? On the shifting view, subjects’ degrees of belief may be highly sensitive to changes in context. This paper articulates and refines this view, after defending it against recent criticisms. It details two mechanisms by which degrees of beliefs may shift.

Much linguistic evidence supports the view believing something only requires thinking it likely. I assess and reject a rival view, based on recent work on homogeneity in natural language, according to which belief is a strong, demanding attitude. I discuss the implications of the linguistic considerations about ‘believe’ for our philosophical accounts of belief.

The paper provides a formal representation of goal systems. The focus is on three properties: consistency, conflict, and coherence. An aim is to attain conceptual clarity of these properties. It is argued that consistency is adequately regarded as a property relative to the decision situation or, more specifically, the set of alternatives that the agent faces. Moreover, as a condition of rationality, consistency is stronger than some writers have claimed. Conflict is adequately regarded as a relation over subsets of a (...) given goal system and should likewise be regarded as relative to the set of alternatives that the agent faces. Coherence is given a probabilistic interpretation, based on a support relation over subsets of goal systems. (shrink)

I I set out my view that all inference is essentially deductive and pinpoint what I take to be the major shortcomings of the induction rule.II The import of data depends on the probability model of the experiment, a dependence ignored by the induction rule. Inductivists admit background knowledge must be taken into account but never spell out how this is to be done. As I see it, that is the problem of induction.III The induction rule, far from providing a (...) method of discovery, does not even serve to detect pattern. Knowing that there is uniformity in the universe is no help to discovering laws. A critique of Reichenbach's justification of the straight rule is constructed along these lines.IV The induction rule, by itself, cannot account for the varying rates at which confidence in an hypothesis mounts with data. The mathematical analysis of this salient feature of inductive reasoning requires prior probabilities. We also argue, against orthodox statisticians, that prior probabilities make a substantive contribution to the objectivity of inductive methods, viz. to the design of experiments and the selection of decision rules.V Carnap's general criticisms of various estimation rules, like the straight rule and the ‘impervious rule’, are seen to be misguided when the prior densities to which they correspond are taken into account.VI Analysis of Hempel's definition of confirmation qua formalization of the enumerative (naive) conception of instancehood. We show that from the standpoint of the quantitative measure P(H/E):P(H) for the degree to which E confirms H, Hempel's classificatory concept yields correct results only for sampling at large from a finite population with a two-way classification all of whose compositions are equally probable. We extend the analysis to Goodman's paradox, finding cases in which grue-like hypotheses do receive as much confirmation as their opposite numbers. We argue, moreover, the irrelevancy of entrenchment, and maintain that Goodman's paradox is no more than a straightforward counter-example to the enumerative conception of instancehood embodied in Hempel's definition.VII We rebutt the objection that prior probabilities, qua inputs of Bayesian analysis, can only be obtained by enumerative induction (insofar as they are objective). The divergence in the prior densities of two rational agents is less a function of subjectivity, we maintain, than of vagueness.VIII Our concluding remarks stress that, for Bayesians, there is no problem of induction in the usual sense. (shrink)

Does knowledge depend in any interesting way on our practical interests? This is the central question in the pragmatic encroachment debate. Pragmatists defend the affirmative answer to this question while purists defend the negative answer. The literature contains two kinds of arguments for pragmatism: principle-based arguments and case-based arguments. Principle-based arguments derive pragmatism from principles that connect knowledge to practical interests. Case-based arguments rely on intuitions about cases that differ with respect to practical interests. I argue that there are insurmountable (...) problems for both kinds of arguments, and that it is therefore unclear what motivates pragmatism. (shrink)

Purists think that changes in our practical interests can’t affect what we know unless those changes are truth-relevant with respect to the propositions in question. Impurists disagree. They think changes in our practical interests can affect what we know even if those changes aren’t truth-relevant with respect to the propositions in question. I argue that impurists are right, but for the wrong reasons, since they haven’t appreciated the best argument for their own view. Together with “Minimalism and the Limits of (...) Warranted Assertability Maneuvers,” “The Pragmatic Encroachment Debate,” and “Anti-Intellectualism” (below), this paper constitutes my attempt to refute the entire pragmatic encroachment debate. As I show in this paper, there is an argument for impurism sitting in plain sight that is considerably more plausible than any extant argument for pragmatism. (shrink)

Quasi-realists aspire to accommodate core features of ordinary normative thought and discourse in an expressivist framework. One apparent such feature is that we can be more or less confident in our normative judgments—they vary in credence. Michael Smith has argued that quasi-realists cannot plausibly accommodate these distinctions simply because they understand normative judgments as desires, but desires lack the structure needed to distinguish these three features. Existing attempts to meet Smith’s challenge have accepted Smith’s presupposition that the way to meet (...) the challenge is to show that normative judgments have more structure than they initially seem to have. I argue that accepting this presupposition is accepting too much. The orthodox view of certitude, insofar as there is one, understands certitude very roughly in terms of counterfactual betting behaviour. Counterfactual betting behaviour, though, is not in any useful sense a structural feature of a given judgment. It is rather a more holistic feature of a given agent’s cognitive system. Insofar as it can meet the other challenges it faces, quasi-realism can characterize credences in terms of counterfactual betting behaviour and effectively say exactly what many realists will want to say about credences, thus meeting Smith’s challenge much more directly. (shrink)

Conditionalization is a norm that governs the rational reallocation of credence. I distinguish between factive and non-factive formulations of Conditionalization. Factive formulations assume that the conditioning proposition is true. Non-factive formulations allow that the conditioning proposition may be false. I argue that non-factive formulations provide a better foundation for philosophical and scientific applications of Bayesian decision theory. I furthermore argue that previous formulations of Conditionalization, factive and non-factive alike, have almost universally ignored, downplayed, or mishandled a crucial causal aspect of (...) Conditionalization. To formulate Conditionalization adequately, one must explicitly address the causal structure of the transition from old credences to new credences. I offer a formulation of Conditionalization that takes these considerations into account, and I compare my preferred formulation with some prominent formulations found in the literature. (shrink)

This paper discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

Lewis proved a Dutch book theorem for Conditionalization. The theorem shows that an agent who follows any credal update rule other than Conditionalization is vulnerable to bets that inflict a sure loss. Lewis’s theorem is tailored to factive formulations of Conditionalization, i.e. formulations on which the conditioning proposition is true. Yet many scientific and philosophical applications of Bayesian decision theory require a non-factive formulation, i.e. a formulation on which the conditioning proposition may be false. I prove a Dutch book theorem (...) tailored to non-factive Conditionalization. I also discuss the theorem’s significance. (shrink)

There is a substantial literature on the Bayesian approach, and the application of Bayes'' theorem, to legal matters. However, I have found no discussion that explores fully the issue of how a Bayesian juror might be led from an initial "presumption of innocence" to the judgment (required for conviction in criminal cases) that the suspect is "guilty beyond a reasonable doubt". I shall argue here that a Bayesian juror, if she acts in accord with what the law prescribes, will virtually (...) never reach such a judgment. In section I, I discuss Bayesianism, Bayes'' rule and the relation between them. Section II addresses the two legal notions key to my worries about Bayesian jurors: the presumption of innocence and the reasonable doubt criterion. Section III explores what a Bayesian is to make of these notions, and how the legal system requires her to reason in their light. If I am right, there emerges a conflict between current legal prescriptions and the Bayesian approach. Section IV explores the import of this conflict and how it might be resolved. (shrink)