This paper considers a puzzling conflict between two positions that are each compelling: it is irrational for an agent to pay to avoid `free' evidence before making a decision, and rational agents may have imprecise beliefs and/or desires. Indeed, we show that Good's theorem concerning the invariable choice-worthiness of free evidence does not generalise to the imprecise realm, given the plausible existing decision theories for handling imprecision. A key ingredient in the analysis, and a potential source of controversy, (...) is the general approach taken for resolving sequential decision problems { we make explicit what the key alternatives are and defend our own approach. Furthermore, we endorse a resolution of the aforementioned puzzle { we privilege decision theories that merely permit avoiding free evidence over decision theories for which avoiding free evidence is uniquely admissible. Finally, we situate this particular result about free evidence within the broader `dynamic-coherence' literature. (shrink)

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)

Jim Joyce argues for two amendments to probabilism. The first is the doctrine that credences are rational, or not, in virtue of their accuracy or “closeness to the truth” (1998). The second is a shift from a numerically precise model of belief to an imprecise model represented by a set of probability functions (2010). We argue that both amendments cannot be satisfied simultaneously. To do so, we employ a (slightly-generalized) impossibility theorem of Seidenfeld, Schervish, and Kadane (2012), who (...) show that there is no strictly proper scoring rule for imprecise probabilities. -/- The question then is what should give way. Joyce, who is well aware of this no-go result, thinks that a quantifiability constraint on epistemic accuracy should be relaxed to accommodate imprecision. We argue instead that another Joycean assumption — called strict immodesty— should be rejected, and we prove a representation theorem that characterizes all “mildly” immodest measures of inaccuracy. (shrink)

Traditional Bayesianism requires that an agent’s degrees of belief be represented by a real-valued, probabilistic credence function. However, in many cases it seems that our evidence is not rich enough to warrant such precision. In light of this, some have proposed that we instead represent an agent’s degrees of belief as a set of credence functions. This way, we can respect the evidence by requiring that the set, often called the agent’s credal state, includes all credence functions that are in (...) some sense compatible with the evidence. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In this article I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid it without compromising the initial evidentialist motivation. _1_ Introduction _2_ Precision and Its Problems _3_ Imprecise Bayesianism and Respecting Ambiguous Evidence _4_ Local Belief Inertia _5_ From Local to Global Belief Inertia _6_ Responding to Global Belief Inertia _7_ Conclusion. (shrink)

In this paper I investigate an alternative to imprecise probabilism. Imprecise probabilism is a popular revision of orthodox Bayesianism: while the orthodox Bayesian claims that a rational agent’s belief-state can be represented by a single credence function, the imprecise probabilist claims instead that a rational agent’s belief-state can be represented by a set of such functions. The alternative that I put forward in this paper is to claim that the expression ‘credence’ is vague, and then (...) apply the theory of supervaluationism to sentences containing this expression. This gives us a viable alternative to imprecise probabilism, and I end by comparing the two accounts. I show that supervaluationism has a simpler way of handling sentences relating the belief-states of two different people, or of the same person at two different times; that both accounts may have the resources to develop plausible decision theories; and finally that the supervaluationist can accommodate higher-order vagueness in a way that is not available to the imprecise probabilist. (shrink)

This chapter explores the topic of imprecise probabilities as it relates to model validation. IP is a family of formal methods that aim to provide a better representationRepresentation of severe uncertainty than is possible with standard probabilistic methods. Among the methods discussed here are using sets of probabilities to represent uncertainty, and using functions that do not satisfy the additvity property. We discuss the basics of IP, some examples of IP in computer simulation contexts, possible interpretations of the IP (...) framework and some conceptual problems for the approach. We conclude with a discussion of IP in the context of model validation. (shrink)

In formal modelling, it is essential that models be supplied with an interpretative story: there must be a clear and coherent account of how the formal model relates to the phenomena it is supposed to model. The traditional representation of degrees of belief as mathematical probabilities comes with a clear and simple interpretative story. This paper argues that the model of degrees of belief as imprecise probabilities (sets of probabilities) lacks a workable interpretation. The standard interpretative story given in (...) the literature is shown to lead to unacceptable results -- and, it is argued, there is no way for imprecise probabilists to restrict, replace or finesse this story so as either to avoid or to be able to live with the consequences. Thus the imprecise probabilist lacks a viable account of how a set of probability functions models a belief state: a story about which properties of the model correspond to facts about the thing modelled and how exactly we are to extract information about an agent's belief state from a set of probabilities. (shrink)

Understanding probabilities as something other than point values has often been motivated by the need to find more realistic models for degree of belief, and in particular the idea that degree of belief should have an objective basis in “statistical knowledge of the world.” I offer here another motivation growing out of efforts to understand how chance evolves as a function of time. If the world is “chancy” in that there are non-trivial, objective, physical probabilities at the macro-level, then the (...) chance of an event e that happens at a given time is \ until it happens. But whether the chance of e goes to one continuously or not is left open. Discontinuities in such chance trajectories can have surprising and troubling consequences for probabilistic analyses of causation and accounts of how events occur in time. This, coupled with the compelling evidence for quantum discontinuities in chance’s evolution, gives rise to a “continuity bind” with respect to chance probability trajectories. I argue that a viable option for circumventing the continuity bind is to understand the probabilities “imprecisely,” that is, as intervals rather than point values. I then develop and motivate an alternative kind of continuity appropriate for interval-valued chance probability trajectories. (shrink)

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic knowledge or her semantics for epistemic (...) modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch bookability iff they have consistent probabilistic beliefs. (shrink)

Cosmology raises novel philosophical questions regarding the use of probabilities in inference. This work aims at identifying and assessing lines of arguments and problematic principles in probabilistic reasoning in cosmology. -/- The first, second, and third papers deal with the intersection of two distinct problems: accounting for selection effects, and representing ignorance or indifference in probabilistic inferences. These two problems meet in the cosmology literature when anthropic considerations are used to predict cosmological parameters by conditionalizing the distribution of, e.g., the (...) cosmological constant on the number of observers it allows for. However, uniform probability distributions usually appealed to in such arguments are an inadequate representation of indifference, and lead to unfounded predictions. It has been argued that this inability to represent ignorance is a fundamental flaw of any inductive framework using additive measures. In the first paper, I examine how imprecise probabilities fare as an inductive framework and avoid such unwarranted inferences. In the second paper, I detail how this framework allows us to successfully avoid the conclusions of Doomsday arguments in a way no Bayesian approach that represents credal states by single credence functions could. -/- There are in the cosmology literature several kinds of arguments referring to self- locating uncertainty. In the multiverse framework, different "pocket-universes" may have different fundamental physical parameters. We don’t know if we are typical observers and if we can safely assume that the physical laws we draw from our observations hold elsewhere. The third paper examines the validity of the appeal to the "Sleeping Beauty problem" and assesses the nature and role of typicality assumptions often endorsed to handle such questions. -/- A more general issue for the use of probabilities in cosmology concerns the inadequacy of Bayesian and statistical model selection criteria in the absence of well-motivated measures for different cosmological models. The criteria for model selection commonly used tend to focus on optimizing the number of free parameters, but they can select physically implausible models. The fourth paper examines the possibility for Bayesian model selection to circumvent the lack of well-motivated priors. (shrink)

Evidentialists say that a necessary condition of sound epistemic reasoning is that our beliefs reflect only our evidence. This thesis arguably conflicts with standard Bayesianism, due to the importance of prior probabilities in the latter. Some evidentialists have responded by modelling belief-states using imprecise probabilities (Joyce 2005). However, Roger White (2010) and Aron Vallinder (2018) argue that this Imprecise Bayesianism is incompatible with evidentialism due to “inertia”, where Imprecise Bayesian agents become stuck in a state of ambivalence (...) towards hypotheses. Additionally, escapes from inertia apparently only create further conflicts with evidentialism. This dilemma gives a reason for evidentialist imprecise probabilists to look for alternatives without inertia. I shall argue that Henry E. Kyburg’s approach offers an evidentialist-friendly imprecise probability theory without inertia, and that its relevant anti-inertia features are independently justified. I also connect the traditional epistemological debates concerning the “ethics of belief” more systematically with formal epistemology than has been hitherto done. (shrink)

Rational credence should be coherent in the sense that your attitudes should not leave you open to a sure loss. Rational credence should be such that you can learn when confronted with relevant evidence. Rational credence should not be sensitive to irrelevant differences in the presentation of the epistemic situation. We explore the extent to which orthodox probabilistic approaches to rational credence can satisfy these three desiderata and find them wanting. We demonstrate that an imprecise probability approach does better. (...) Along the way we shall demonstrate that the problem of “belief inertia” is not an issue for a large class of IP credences, and provide a solution to van Fraassen’s box factory puzzle. (shrink)

According to van Fraassen, inference to the best explanation (IBE) is incompatible with Bayesianism. To argue to the contrary, many philosophers have suggested hybrid models of scientific reasoning with both explanationist and probabilistic elements. This paper offers another such model with two novel features. First, its Bayesian component is imprecise. Second, the domain of credence functions can be extended.

Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming suitable tripartitions of (...) the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers. (shrink)

Many philosophers regard the imprecise credence framework as a more realistic model of probabilistic inferences with imperfect empirical information than the traditional precise credence framework. Hence, it is surprising that the literature lacks any discussion on how to update one’s imprecise credences when the given evidence itself is imprecise. To fill this gap, I consider two updating principles. Unfortunately, each of them faces a serious problem. The first updating principle, which I call “generalized conditionalization,” sometimes forces an (...) agent to change her imprecise degrees of belief even though she does not have new evidence. The second updating principle, which I call “the generalized dynamic Keynesian model,” may result in a very precise credal state although the agent does not have sufficiently strong evidence to justify such an informative doxastic state. This means that it is much more difficult to come up with an acceptable updating principle for the imprecise credence framework than one might have thought it would be. (shrink)

The return rate is considered here as a fuzzy probabilistic set. Then the expected return is obtained as a fuzzy subset in the real line. This result is a theoretical foundation for new investment strategies. All considered strategies result of comparison profit fuzzy index and limit value. In this way we obtain an imprecise investment recommendation. Financial equilibrium criteria are a special case of comparison of the profit index and the limit value. The following criteria are generalized here: the (...) Sharpe's Ratio, the Jensen's Alpha and the Treynor's Ratio. Moreover, the safety-first criteria are generalized here for the fuzzy case. The Roy Criterion, the Kataoka Criterion and the Telser Criterion are also generalized. Obtained results show that proposed theory is useful for the investment applications. (shrink)

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

The basic Bayesian model of credence states, where each individual’s belief state is represented by a single probability measure, has been criticized as psychologically implausible, unable to represent the intuitive distinction between precise and imprecise probabilities, and normatively unjustifiable due to a need to adopt arbitrary, unmotivated priors. These arguments are often used to motivate a model on which imprecise credal states are represented by sets of probability measures. I connect this debate with recent work in Bayesian cognitive (...) science, where probabilistic models are typically provided with explicit hierarchical structure. Hierarchical Bayesian models are immune to many classic arguments against single-measure models. They represent grades of imprecision in probability assignments automatically, have strong psychological motivation, and can be normatively justified even when certain arbitrary decisions are required. In addition, hierarchical models show much more plausible learning behavior than flat representations in terms of sets of measures, which—on standard assumptions about update—rule out simple cases of learning from a starting point of total ignorance. (shrink)

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Finally, we present the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases.

We study a probabilistic logic based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence). We examine probabilistic conditional knowledge bases associated with imprecise probability assessments defined on arbitrary families of conditional events. We introduce a notion of conditional interpretation defined directly in terms of precise probability assessments. We also examine a property of strong satisfiability which is related to the notion of toleration well known in default reasoning. In our framework we give (...) more general definitions of the notions of probabilistic consistency and probabilistic entailment of Adams. We also recall a notion of strict p-consistency and some related results. Moreover, we give new proofs of some results obtained in probabilistic default reasoning. Finally, we examine the relationships between conditional probability rankings and the notions of g-coherence and g-coherent entailment. (shrink)

The modus ponens (A -> B, A :. B) is, along with modus tollens and the two logically not valid counterparts denying the antecedent (A -> B, ¬A :. ¬B) and affirming the consequent, the argument form that was most often investigated in the psychology of human reasoning. The present contribution reports the results of three experiments on the probabilistic versions of modus ponens and denying the antecedent. In probability logic these arguments lead to conclusions with imprecise probabilities. In (...) the modus ponens tasks the participants inferred probabilities that agreed much better with the coherent normative values than in the denying the antecedent tasks, a result that mirrors results found with the classical argument versions. For modus ponens a surprisingly high number of lower and upper probabilities agreed perfectly with the conjugacy property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the participants do not ignore irrelevant (“silent”) boundary probabilities. The results show that human mental probability logic is close to predictions derived from probability logic for the most elementary argument form, but has considerable difficulties with the more complex forms involving negations. (shrink)

What are quantum entities? Is the quantum domain deterministic or probabilistic? Orthodox quantum theory (OQT) fails to answer these two fundamental questions. As a result of failing to answer the first question, OQT is very seriously defective: it is imprecise, ambiguous, ad hoc, non-explanatory, inapplicable to the early universe, inapplicable to the cosmos as a whole, and such that it is inherently incapable of being unified with general relativity. It is argued that probabilism provides a very natural solution (...) to the quantum wave/particle dilemma and promises to lead to a fully micro-realistic, testable version of quantum theory that is free of the defects of OQT. It is suggested that inelastic interactions may induce quantum probabilistic transitions. (shrink)

We study abductive, causal, and non-causal conditionals in indicative and counterfactual formulations using probabilistic truth table tasks under incomplete probabilistic knowledge (N = 80). We frame the task as a probability-logical inference problem. The most frequently observed response type across all conditions was a class of conditional event interpretations of conditionals; it was followed by conjunction interpretations. An interesting minority of participants neglected some of the relevant imprecision involved in the premises when inferring lower or upper probability bounds on the (...) target conditional/counterfactual ("halfway responses"). We discuss the results in the light of coherence-based probability logic and the new paradigm psychology of reasoning. (shrink)

In this paper I offer an alternative - the ‘dispositional account’ - to the standard account of imprecise probabilism. Whereas for the imprecise probabilist, an agent’s credal state is modelled by a set of credence functions, on the dispositional account an agent’s credal state is modelled by a set of sets of credence functions. On the face of it, the dispositional account looks less elegant than the standard account – so why should we be interested? I argue (...) that the dispositional account is actually simpler, because the dispositional choice behaviour that fixes an agent’s credal state is faithfully depicted in the model of that agent’s credal state. I explore some of the implications of the account, including a surprising implication for the debate over dilation. (shrink)

Imprecise probabilism—which holds that rational belief/credence is permissibly represented by a set of probability functions—apparently suffers from a problem known as dilation. We explore whether this problem can be avoided or mitigated by one of the following strategies: (a) modifying the rule by which the credal state is updated, (b) restricting the domain of reasonable credal states to those that preclude dilation.

There has been much recent interest in imprecise probabilities, models of belief that allow unsharp or fuzzy credence. There have also been some influential criticisms of this position. Here we argue, chiefly against Elga (2010), that subjective probabilities need not be sharp. The key question is whether the imprecise probabilist can make reasonable sequences of decisions. We argue that she can. We outline Elga's argument and clarify the assumptions he makes and the principles of rationality he is implicitly (...) committed to. We argue that these assumptions are too strong and that rational imprecise choice is possible in the absence of these overly strong conditions. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the (...) evidence calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is (...) stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. (shrink)

We introduce an independence notion for choice functions, which we call ‘epistemic independence’ following the work by De Cooman et al. [17] for lower previsions, and study it in a multivariate setting. This work is a continuation of earlier work of one of the authors [29], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [11]. We obtain the many-to-one independent natural extension in this (...) framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less expressive imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [28], which De Bock and De Cooman [2] have called S-independence. We show that neither is more general than the other. (shrink)

We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware (...) of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other. (shrink)

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study (...) selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults. (shrink)

Analyses of singular causation often make use of the idea that a cause increases the probability of its effect. Of particular salience in such accounts are the values of the probability function of the effect, conditional on the presence and absence of the putative cause, analysed around the times of the events in question: causes are characterized by the effect’s probability function being greater when conditionalized upon them. Put this way, it becomes clearer that the ‘behaviour’ of probability functions in (...) small intervals about the times in question ought to be of concern. In this article, I make an extended case that causal theorists employing the ‘probability raising’ idea should pay attention to the continuity question. Specifically, if the probability functions are ‘jumping about’ in ways typical of discontinuous functions, then the stability of the relevant probability increase is called into question. The rub, however, is that sweeping requirements for either continuity or discontinuity are problematic and, as I argue, this constitutes a ‘continuity bind’. Hence more subtle considerations and constraints are needed, two of which I consider: utilizing discontinuous first derivatives of continuous probability functions, and abandoning point probability for imprecise probability. _1_ Introduction _2_ Probability Trajectories and Continuity _2.1_ Probability trajectories _2.2_ Causation as discontinuous jumps _2.3_ Against systematic discontinuity _3_ Broader Discontinuity Concerns _4_ The Continuity Bind _4.1_ Retaining continuity with discontinuous first derivatives _4.2_ Imprecise probability trajectories _5_ Concluding Remarks Appendix. (shrink)

Formal epistemologists often claim that our credences should be representable by a probability function. Complete probabilistic coherence, however, is only possible for ideal agents, raising the question of how this requirement relates to our everyday judgments concerning rationality. One possible answer is that being rational is a contextual matter, that the standards for rationality change along with the situation. Just like who counts as tall changes depending on whether we are considering toddlers or basketball players, perhaps what counts as rational (...) shifts according to whether we are considering ideal agents or creatures more like ourselves. Even though a number of formal epistemologists have endorsed this type of solution, I will argue that there is no way to spell out this contextual account that can make sense of our everyday judgments about rationality. Those who defend probabilistic coherence requirements will need an alternative account of the relationship between real and ideal rationality. (shrink)

The Doomsday argument and anthropic reasoning are two puzzling examples of probabilistic confirmation. In both cases, a lack of knowledge apparently yields surprising conclusions. Since they are formulated within a Bayesian framework, they constitute a challenge to Bayesianism. Several attempts, some successful, have been made to avoid these conclusions, but some versions of these arguments cannot be dissolved within the framework of orthodox Bayesianism. I show that adopting an imprecise framework of probabilistic reasoning allows for a more adequate representation (...) of ignorance in Bayesian reasoning and explains away these puzzles. (shrink)

(This is for the series Elements of Decision Theory published by Cambridge University Press and edited by Martin Peterson) -/- Our beliefs come in degrees. I believe some things more strongly than I believe others. I believe very strongly that global temperatures will continue to rise during the coming century; I believe slightly less strongly that the European Union will still exist in 2029; and I believe much less strongly that Cardiff is east of Edinburgh. My credence in something is (...) a measure of the strength of my belief in it; it represents my level of confidence in it. These are the states of mind we report when we say things like ‘I’m 20% confident I switch off the gas before I left' or ‘I’m 99.9% confident that it is raining outside'. -/- There are laws that govern these credences. For instance, I shouldn't be more confident that sea levels will rise by over 2 metres in the next 100 years than I am that they'll rise by over 1 metre, since the latter is true if the former is. This book is about a particular way we might try to establish these laws of credence: the Dutch Book arguments (For briefer overviews of these arguments, see Alan Hájek’s entry in the Oxford Handbook of Rational and Social Choice and Susan Vineberg’s entry in the Stanford Encyclopaedia.) -/- We begin, in Chapter 2, with the standard formulation of the various Dutch Book arguments that we'll consider: arguments for Probabilism, Countable Additivity, Regularity, and the Principal Principle. In Chapter 3, we subject this standard formulation to rigorous stress-testing, and make some small adjustments so that it can withstand various objections. What we are left with is still recognisably the orthodox Dutch Book argument. In Chapter 4, we set out the Dutch Strategy argument for Conditionalization. In Chapters 5 and 6, we consider two objections to Dutch Book arguments that cannot be addressed by making small adjustments. Instead, we must completely redesign those arguments, replacing them with ones that share a general approach but few specific details. In Chapter 7, we consider a further objection to which I do not have a response. In Chapter 8, we'll ask what happens to the Dutch Book arguments if we change certain features of the basic framework in which we've been working: first, we ask how Dutch Book arguments fare when we consider credences in self-locating propositions, such as It is Monday; second, we lift the assumption that the background logic is classical and explore Dutch Book arguments for non-classical logics; third, we lift the assumption that an agent's credal state can be represented by a single assignment of numerical values to the propositions she considers. In Chapter 9, we present the mathematical results that underpin these arguments. (shrink)

Merging of opinions results underwrite Bayesian rejoinders to complaints about the subjective nature of personal probability. Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Here, we establish a merging result for sets of probability measures that are updated by Jeffrey conditioning. This generalizes a number of different merging results in the literature. (...) We also show that such sets converge to a shared, maximally informed opinion. Convergence to a maximally informed opinion is a (weak) Jeffrey conditioning analogue of Bayesian “convergence to the truth” for conditional probabilities. Finally, we demonstrate the philosophical significance of our study by detailing applications to the topics of dynamic coherence, imprecise probabilities, and probabilistic opinion pooling. (shrink)

This dissertation looks at a set of interconnected questions concerning the foundations of probability, and gives a series of interconnected answers. At its core is a piece of old-fashioned philosophical analysis, working out what probability is. Or equivalently, investigating the semantic question of what is the meaning of ‘probability’? Like Keynes and Carnap, I say that probability is degree of reasonable belief. This immediately raises an epistemological question, which degrees count as reasonable? To solve that in its full generality would (...) mean the end of human inquiry, so that won’t be attempted here. Rather I will follow tradition and merely investigate which sets of partial beliefs are coherent. -/- The standard answer to this question, what is commonly called the Bayesian answer, says that degrees of belief are coherent iff they form a probability function. I disagree with the way this is usually justified, but subject to an important qualification I accept the answer. The important qualification is that degrees of belief may be imprecise, or vague. Part one of the dissertation, chapters 1 to 6, looks largely at the consequences of this qualification for the semantic and epistemological questions already mentioned. It turns out that when we allow degrees of belief to be imprecise, we can discharge potentially fatal objections to some philosophically attractive theses. Two of these, that probability is degree of reasonable belief and that the probability calculus provides coherence constraints on partial beliefs, have been mentioned. Others include the claim, defended in chapter 4, that chance is probability given total history. -/- As well as these semantic and epistemological questions, studies of the foundations of probability usually include a detailed discussion of decision theory. For reasons set out in chapter 2, I deny we can gain epistemological insights from decision theory. Nevertheless, it is an interesting field to study on its own, and it might be expected that there would be decision theoretic consequences of allowing imprecise degrees of belief. As I show in part two, this expectation seems to be mistaken. Chapter 9 shows that there aren’t interesting consequences of this theory for decision theory proper, and chapters 10 and 11 show that Keynes’s attempt to use imprecision in degrees of belief to derive a distinctive theory of interest rates is unsound. -/- Chapters 7 and 8 provide a link between these two parts. In chapter 7 I look at some previous philosophical investigations into the effects of imprecision. In chapter 8 I develop what I take to be the best competitor to the theory defended here – a constructivist theory of probability. On this view degrees of belief are precise, but the relevant coherence constraint is a constructivist probability calculus. This view is, I think, mistaken, but the calculus has some intrinsic interest, and there are at least enough arguments for it to warrant a chapter-length examination. (shrink)

Epistemic states of uncertainty play important roles in ethical and political theorizing. Theories that appeal to a “veil of ignorance,” for example, analyze fairness or impartiality in terms of certain states of ignorance. It is important, then, to scrutinize proposed conceptions of ignorance and explore promising alternatives in such contexts. Here, I study Lerner’s probabilistic egalitarian theorem in the setting of imprecise probabilities. Lerner’s theorem assumes that a social planner tasked with distributing income to individuals in a population is (...) “completely ignorant” about which utility functions belong to which individuals. Lerner models this ignorance with a certain uniform probability distribution, and shows that, under certain further assumptions, income should be equally distributed. Much of the criticism of the relevance of Lerner’s result centers on the representation of ignorance involved. Imprecise probabilities provide a general framework for reasoning about various forms of uncertainty including, in particular, ignorance. To what extent can Lerner’s conclusion be maintained in this setting? (shrink)

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical (...) situations involving more than one maximal value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A, where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U- uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U = 1/h 0 h log [1+]d, where h is the largest membership value obtained by any element of A and is the measure of the -cut of A defined by the Lebesge integral of its characteristic function. (shrink)

We present a coherence-based probability semantics for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation rules of argument (...) forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three Figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a knowledge bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Finally, we show how the proposed semantics can be used to analyze syllogisms involving generalized quantifiers. (shrink)

Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions of the (...) square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers. (shrink)

We propose a model of uncertain belief. This models coherent beliefs by a filter, ????, on the set of probabilities. That is, it is given by a collection of sets of probabilities which are closed under supersets and finite intersections. This can naturally capture your probabilistic judgements. When you think that it is more likely to be sunny than rainy, we have{????|????(????????????????????)>????(????????????????????)}∈????. When you think that a gamble ???? is desirable, we have {????|Exp????[????]>0}∈????. It naturally extends the model of credal (...) sets; and we will show it captures all the expressive power of the desirable gambles model. It also captures the expressive power of sets of desirable gamble sets (with a mixing axiom, but no Archimadean axiom). (shrink)

After presenting a simple expressivist account of reports of probabilistic judgements, I explore a classic problem for it, namely the Frege-Geach problem. I argue that it is a problem not just for expressivism but for any reasonable account of ascriptions of graded judgements. I suggest that the problem can be resolved by appropriately modelling imprecise credences.

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)

In the standard formulation of game theory, agents use mixed strategies in the form of objective and probabilistically precise devices to conceal their actions. We introduce the larger set of probabilistically imprecise devices and study the consequences for the basic results on normal form games. While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes, as we illustrate by negotiation games with three players. We characterize Ellsberg equilibria in two-person conflict and coordination (...) games. These equilibria turn out to be related to experimental deviations from Nash equilibrium play. (shrink)