Probabilistic Frameworks, Misc

We leave in a beautiful and uniform world, a world where everything probable is possible. Since the epic theory of relativity many scientists have embarked in a pursuit of astonishing theoretical fantasies, abandoning the prudent and logical path to scientific inquiry. The theory is a complex theoretical framework that facilitates the understanding of the universal laws of physics. It is based on the space-time continuum fabric abstract concept, and it is well suited for interpreting cosmic events. However, it is not (...) well suited for handling of small, local topics as global warming, local energy issues, and overall common humanity matters. We now forward may fancy theories and spend unimaginable effort to validate them, even when we are perhaps headed in a wrong direction. For example, in our times matters of climate changes are debated by politicians based on economical considerations that are as illogical as they come. The venerable paths of scientific method developed during centuries by prominent scientists and philosophers has been willingly ignored and abandoned for various and prejudiced purpose. Contact email: [email protected] . (shrink)

NOTE: This paper is a reworking of some aspects of an earlier paper – ‘What else justification could be’ and also an early draft of chapter 2 of Between Probability and Certainty. I'm leaving it online as it has a couple of citations and there is some material here which didn't make it into the book (and which I may yet try to develop elsewhere). My concern in this paper is with a certain, pervasive picture of epistemic justification. On this (...) picture, acquiring justification for believing something is essentially a matter of minimising one’s risk of error – so one is justified in believing something just in case it is sufficiently likely, given one’s evidence, to be true. This view is motivated by an admittedly natural thought: If we want to be fallibilists about justification then we shouldn’t demand that something be certain – that we completely eliminate error risk – before we can be justified in believing it. But if justification does not require the complete elimination of error risk, then what could it possibly require if not its minimisation? If justification does not require epistemic certainty then what could it possibly require if not epistemic likelihood? When all is said and done, I’m not sure that I can offer satisfactory answers to these questions – but I will attempt to trace out some possible answers here. The alternative picture that I’ll outline makes use of a notion of normalcy that I take to be irreducible to notions of statistical frequency or predominance. (shrink)

In “Combining probability with qualitative degree-of-certainty metrics in assessment,” Helgeson et al. present a mathematical model of the confidence-likelihood relationship in the IPCC uncertainty framework. Their goal is to resolve ambiguities in the framework and clarify the roles of “confidence” and “likelihood” in decision-making. In this paper, I provide a conceptual evaluation of their proposal. I argue that the IPCC cannot implement the model coherently and that adopting it could result in unclear and potentially misleading communication of uncertainty.

Carl Hempel (1965) argued that probabilistic hypotheses are limited in what they can explain. He contended that a hypothesis cannot explain why E is true if the hypothesis says that E has a probability less than 0.5. Wesley Salmon (1971, 1984, 1990, 1998) and Richard Jeffrey (1969) argued to the contrary, contending that P can explain why E is true even when P says that E’s probability is very low. This debate concerned noncontrastive explananda. Here, a view of contrastive causal (...) explanation is described and defended. It provides a new limit on what probabilistic hypotheses can explain; the limitation is that P cannot explain why E is true rather than A if P assign E a probability that is less than or equal to the probability that P assigns to A. The view entails that a true deterministic theory and a true probabilistic theory that apply to the same explanandum partition are such that the deterministic theory explains all the true contrastive propositions constructable from that partition, whereas the probabilistic theory often fails to do so. (shrink)

In abductive reasoning, scientific theories are evaluated on the basis of how well they would explain the available evidence. There are a number of subtly different accounts of this type of reasoning, most of which are inspired by the popular slogan 'Inference to the Best Explanation.' However, these accounts disagree about exactly how to spell out the slogan so as to avoid various problems for abductive reasoning. This Element aims, firstly, to give an opinionated overview both of the many accounts (...) of abductive reasoning that have been proposed and the problems that have motivated them; and, secondly, to critically evaluate these accounts in a way that points toward a systematic view of the nature and purpose of abductive reasoning in science. This title is also available as Open Access on Cambridge Core. (shrink)

Sometimes you are unreliable at fulfilling your doxastic plans: for example, if you plan to be fully confident in all truths, probably you will end up being fully confident in some falsehoods by mistake. In some cases, there is information that plays the classical role of evidence—your beliefs are perfectly discriminating with respect to some possible facts about the world—and there is a standard expected‐accuracy‐based justification for planning to conditionalize on this evidence. This planning‐oriented justification extends to some cases where (...) you do not have transparent evidence, in the sense that your beliefs are not perfectly discriminating with respect to any non‐trivial facts. In other cases, accuracy considerations do not tell you to plan to conditionalize on any information at all, but rather to plan to follow a different updating rule. Even in the absence of evidence, accuracy considerations can guide your doxastic plan. (shrink)

The number of independent messages a physical system can carry is limited by the number of its adjustable properties. In particular, systems with only one adjustable property cannot carry more than a single message at a time. We demonstrate that this is true for the photons in the double-slit experiment, and that this is what leads to the fundamental limit on measuring the complementary aspect of the photons. Next, we illustrate that systems with a single adjustable property exhibit other quantum (...) behaviors, such as noncommutativity and no-cloning. Finally, we formulate a mathematical theory to describe the dynamics of such systems and derive the standard Hilbert space formalism of quantum mechanics as well as the Born probability rule. Our derivation demonstrates the physical foundation of quantum theory. (shrink)

Chapter 10 turns to a range of issues concerning the _determinacy_ of rational belief. First, we consider cases in which the opinions of the “evaluating angel” do not consist in a _unique_ probability function, but only in a big _set_ of such functions. Here, it is argued that, even in these cases, a _perfectly_ rational thinker would in effect pick one of the probability functions in the set that constitutes the angel’s opinions, and would have a belief-system that matches the (...) result of conditionalizing _that_ probability function on the supposition that she has that belief-system. Secondly, we consider whether rational agents would sometimes have imprecise or indeterminate credences. Tentatively, it is argued that our account implies that perfectly rational thinkers never have imprecise credences of this sort: it is only _imperfectly_ rational thinkers who ever have such imprecise credences. Thirdly, it is argued that degrees of irrationality are sometimes incommensurable: the degrees of irrationality sometimes fail to rank alternative belief-systems in relation to each other. However, these incommensurabilities only affect irrational belief-systems: for every thinker and time, some available belief-systems are perfectly rational; these perfectly rational belief-systems are all equally rational, and determinately more rational than all irrational systems. (shrink)

This book gives a general theory of rational belief. Although it can be read by itself, it is a sequel to the author’s previous book, The Value of Rationality (Oxford, 2017). It takes the general conception of rationality that was developed in that earlier book and combines it with an account of the varieties of belief, and of what it is for these beliefs to count as “correct”, to provide an account of what it is for beliefs to count as (...) rational. According to this account, rationality comes in degrees: the degree to which a belief-system counts as rational is determined by its distance from a corresponding probability function—where this distance is measured by those beliefs’ “expected degree of incorrectness” according to the probability function; the account also explains what determines exactly which probability function plays this role in each case. In developing and defending this account, light is shed on several central epistemological issues. These issues include: the distinction between propositional and doxastic justification; the debates between internalism and externalism, and between foundationalism and coherentism; the relationship between credences, full belief, inference, and suspension of judgment; the kind of possibility that is presupposed by the relevant sort of probability; and whether rationality is “diachronic”—so that the beliefs that it is rational for us to have now depend, in part, on the beliefs that we held in the past. (shrink)

Chapter 5 begins to develop an inventory of the different kinds of belief that it is possible for thinkers to have. It is argued that these varieties of belief include: precise and imprecise levels of confidence or degrees of belief; full or outright beliefs; conditional or suppositional beliefs (which may be identified with inferences, on a certain understanding of what inferences are); and the attitude of suspension of judgment. The discussion of full or outright belief is postponed until the following (...) chapter; but, otherwise, it is argued that every belief-system consisting of beliefs of all these different kinds can be modelled by means of a _set_ of _partially defined_ credence functions (including both conditional and unconditional credence functions)—where these credence functions need not necessarily be probabilistically coherent. This approach allows for the thinker to be simply “attitudeless”—that is, to have no attitude whatsoever—towards some of the propositions that she is capable of entertaining; it allows for the thinker’s beliefs to be imprecise or indeterminate, and also to be probabilistically incoherent; and it allows for genuinely conditional or suppositional belief—attitudes of _conditionally_ believing a conclusion, conditionally on the assumption of a set of premises. (shrink)

This is the Introduction to the book—the goal of which it is to present a general theory of rational belief. This book is a sequel to the author’s previous book, _The Value of Rationality_ (Oxford University Press, 2017). It aims to take the general conception of rationality that was developed in that earlier book, and to combine this conception with an account of the nature of belief, to yield an account of what it is for beliefs to count as rational. (...) The central idea of this general conception of rationality is that rational thinking gives the thinker “good news” about this thinking’s degree of correctness—while the more irrational one’s thinking is, the worse the news that it gives one about its degree of correctness. A brief account is given of some of the epistemological questions that the book is designed to address. Then, after some comments on the methodological issues that any book on this subject must confront, the overall plan of the book is outlined. Finally, some of the limits of this project are discussed: in particular, despite the enormous importance of social epistemology, the focus of this book will be on what it is for an individual’s beliefs to count as rational. (shrink)

Chapter 3 clarifies some structural features of the framework within which the account of rational belief will be developed. First, a broadly “holistic” view of rational belief is defended, according to which the fundamental items that are rational to some degree or other are entire _belief-systems_ that the thinker might have at the time—rather than the beliefs or doxastic attitudes that the thinker might have in any _individual_ proposition. Secondly, it is argued that, given our other assumptions, this kind of (...) holism commits us to the view that rationality fundamentally consists in _holistic coherence with the “given”_. This view is defended against a number of objections. Finally, it is argued that we can make sense of coherence _over time_, as well as coherence at a time, and that this is all that we need to make sense of how we can evaluate mental processes or transitions, such as belief revisions (as well as mental states, such as beliefs and other attitudes). (shrink)

Chapter 1 recapitulates the key ideas of the author’s previous book _The Value of Rationality_. The central idea is that rationality is an _evaluative concept_: rational thinking is thinking that is _good_ in a certain respect; and the more irrational one’s thinking is, the _worse_ one’s thinking is in that respect. The various belief-systems available to a thinker at a time can be compared as more or less rational than each other, on the basis of how they relate to certain (...) facts that are “given” to the thinker at the time—that is, facts that are beyond the control of the thinker to change through the way in which she reasons at the time. Specifically, a kind of _internalism_ holds of the relevant facts that are “given”: the facts that determine the degree to which an available belief-system is rational for a thinker at a time are all “internal” facts about the thinker’s mental states at that time. However, rationality is not a completely free-standing value: it has an essential connection to another value—the value of “correctness”, which unlike rationality is an “externalist” value. The general idea is that irrational thinking is thinking that gives one “bad news” about how correct it is. This idea of “bad news” is to be understood in _probabilistic_ terms. (shrink)

Chapter 4 turns to consider the notion of “evidence” which is central to many contemporary epistemological discussions. Here it is argued that talk of “evidence” is best avoided, at least in our most precise statements of our account of rational belief. Given the meaning that the term ‘evidence’ has in everyday English, the claim that rational beliefs must “respect the evidence” has some unfortunate results. In particular, it encourages some assumptions that are in fact inconsistent with the general conception of (...) rationality that was articulated in the previous chapter. According to the account that is developed here, the degree to which each of the available alternative belief systems is rational for a thinker at a time is determined by how this belief-system relates to what is “given”—that is, to the relevant internal mental facts that cannot be changed by the way in which the thinker deploys her reasoning capacities at that time. But this notion of “the given” does not in fact correspond to what in everyday English could at all naturally be called the thinker’s “evidence”. It is suggested that the concept of ‘evidence’ belongs, not in an account of the rationality of an individual’s beliefs, but in a kind of social epistemology: in effect, “evidence” is what one can use to _show_ that something is the case, to a particular audience. (shrink)

Chapter 6 takes up the topic of full or outright belief. What is puzzling about full belief is that its functional role seems to be that of treating the believed proposition as if it were _certain_, and yet it also seems possible to have a full belief in propositions of which we are _not_ absolutely certain. The solution to the puzzle lies in seeing that in addition to our _theoretical_ credences—beliefs of the kind whose role is, to put it roughly, (...) to gauge how strongly the relevant proposition is supported by one’s “evidence”—there are also _practical_ credences—beliefs of the kind whose role is to guide us in action and practical decision making. It is argued that, for limited finite believers like us, it is unavoidable that we will sometimes act on practical credences that are different from our theoretical credences. To have a full or outright belief in a proposition can be plausibly identified with being disposed to treat the proposition as _practically_ certain, for all _normal_ practical purposes. (shrink)

Chapter 8 turns to the question of how to measure the incorrectness of the _intermediate_ doxastic attitudes—that is, partial degrees of belief that do not involve believing (or disbelieving) anything with certainty. It is argued that the incorrectness score for each precise credence in a particular proposition must have the features that are known as symmetry, continuity, and strict propriety. It follows that this score is simply the well-known Brier score (the square of the difference between the credence and the (...) proposition’s real truth value); and, as argued in the previous chapter, the score for a whole system of such precise credences is simply a _weighted sum_ of its scores for the propositions over which it is defined. Then it is explained how to generalize this approach to deal with belief-systems involving imprecise credences, attitudes of suspended judgment, and conditional credences (or inferences). (shrink)

Chapter 11 is the first of three chapters devoted to investigating the principles that explain how the facts about the internal mental states that are “given” to the thinker at the time determine what the “rational probability” function is for each case that is available to the thinker at the time. Specifically, this chapter focuses on two features of this probability function: (a) the field of propositions and (b) the space of possible worlds that the probability function is defined over. (...) To play the role that our account assigns to this rational probability function, the field of propositions must include all propositions that are built up out of concepts that the thinker possesses. The space of possible worlds must consist of all and only those worlds that are, in a sense that is explained, _epistemically possible_ for the thinker in the case. (shrink)

Chapter 9 turns to explaining what is here being referred to as the “rational probability function” for each case. According to the account that is being developed here, the degree of irrationality of the belief-system that the thinker has in each case is in effect determined by the belief-system’s _distance_ from the rational probability function for the case. But are the degrees of irrationality of all belief-systems that are available to the thinker at the time determined by how distant they (...) are from the _same_ probability function? A consideration of so-called “Moore-paradoxical” propositions reveals that the answer to this question is: No. The internal mental facts “given” to the thinker at the time determine an appropriate _prior_ probability function; and the degree of irrationality of each of these belief-systems is determined by how distant it is from the result of _conditionalizing_ this prior probability on the supposition _that the thinker has that very belief-system_. This appropriate prior probability can be conceived by imagining an “evaluating angel”. This angel knows all the internal mental facts about what is given to the thinker at the time, and relies solely on these facts to form a body of opinions about what the world is like; the appropriate prior probability function may be conceived as this angel’s body of opinions about the world. (shrink)

Chapter 2 is concerned with the distinction that most contemporary epistemologists express by distinguishing between “propositional” and “doxastic” justification. The goal is to develop an account of this distinction that applies, not just to full or outright beliefs, but also to partial credences—and indeed, in principle, to attitudes of all kinds. The standard way of explaining this distinction, in terms of the “basing relation”, is criticized, and an alternative account—the “virtue manifestation” account—is proposed in its place. This account has a (...) certain kinship with the epistemological approach that is known as “reliabilism”, but it also has certain key differences—above all, it is compatible with an internalist view of rationality. (shrink)

Chapter 14 is an epilogue—a sketch of how the author’s theory of rational belief can address the remaining great questions of epistemology. First, this theory can give an account of the significance of _inference_—including _non-deductive_ inference as well as deductive inference; it can also explain how non-deductive inferences are _defeasible_—allowing for a version of the distinction between “rebutting” and “undercutting” defeaters. Secondly, the theory can give an account of the significance of beliefs—like perceptual beliefs—that endorse the contents of non-doxastic mental (...) states like _sensory experiences_ and _episodic memories_. Thirdly, the theory can be developed to explain what is wrong with the most challenging arguments for _radical scepticism_. Finally, the theory can be developed to give an analysis of _knowledge_—what has elsewhere been called the “internalist virtue theory of knowledge”. (shrink)

Chapter 13 turns to _diachronic_ constraints on the rational probability function—constraints that the thinker’s _past_ beliefs impose on which probability functions can be in the set that represents the opinions of the “evaluating angel”. Some philosophers—such as Brian Hedden—object that the idea of such diachronic constraints is inconsistent with internalism about rationality. These objections are answered here. It seems clear that the past continues to exert an influence over the present. This lingering influence of the thinker’s past beliefs will be (...) referred to as the “doxastic shadow” that those past beliefs cast into the present. The angel will be guided by this shadow in the following way: first, she will find the “shadow priors”—the probability functions that, of all those that meet all the synchronic constraints, are _closest_ to this doxastic shadow. Then the probability functions that constitute the angel’s opinions are those that result from _conditionalizing_ these shadow priors on the facts that are “given” to the thinker now. This chapter ends with a proposal about what explains why this view of diachronic rationality is correct. (shrink)

Chapter 12 is the first of two chapters devoted to exploring the further constraints that must be met by all probability functions in the set that constitutes the opinions that the “evaluating angel” has about the world. Specifically, this chapter investigates whether there are further _synchronic_ constraints on which probability functions are in this set—that is, constraints that are based on the mental states that are in the thinker’s mind at the very time in question—over and above the first two (...) features of the probability functions that were explored in the previous chapter. Some reasons are given for doubting the idea of a special privileged Ur-prior probability function, of the sort that is appealed to in Timothy Williamson’s idea of “evidential probability”. It is tentatively suggested that these synchronic constraints include certain further principles, such as the so-called “Principal Principle”. However, it is argued in conclusion that these synchronic constraints are not enough: there must also be diachronic constraints on which probability functions are in the set that represents the opinions of the “evaluating angel”. (shrink)

Chapter 7 turns to the question of the standard of _correctness_ for beliefs, beginning with the case of believing propositions with certainty. It is argued that, if you have such a belief, this belief is correct if and only if the proposition believed is _true_. We can represent this by giving each belief of this kind in a true proposition an incorrectness score of 0 (it is not incorrect at all), and each belief of this kind in a false proposition (...) an incorrectness score of 1 (it is completely incorrect). Then, in comparing two or more alternative systems of such beliefs that are defined over the same set of propositions, the way to measure the degree to which each system is incorrect is simply by a _weighted sum_ of the incorrectness scores that the belief system gets for each proposition. The consequences of this idea are explained, and the idea is defended against a number of objections—including objections that are due to Krister Bykvist and Anandi Hattiangadi, and Selim Berker, among others. (shrink)

It is a consequence of the theory of imprecise credences that there exist situations in which rational agents inevitably become less opinionated toward some propositions as they gather more evidence. The fact that an agent's imprecise credal state can dilate in this way is often treated as a strike against the imprecise approach to inductive inference. Here, we show that dilation is not a mere artifact of this approach by demonstrating that opinion loss is countenanced as rational by a substantially (...) broader class of normative theories than has been previously recognised. Specifically, we show that dilation-like phenomena arise even when one abandons the basic assumption that agents have (precise or imprecise) credences of any kind, and follows directly from bedrock norms for rational comparative confidence judgements of the form `I am at least as confident in p as I am in q'. We then use the comparative confidence framework to develop a novel understanding of what exactly gives rise to dilation-like phenomena. By considering opinion loss in this more general setting, we are able to provide a novel assessment of the prospects for an account of inductive inference that is not saddled with the inevitability of rational opinion loss. (shrink)

This paper aims to open a dialogue between philosophers working in decision theory and operations researchers and engineers working on decision-making under deep uncertainty. Specifically, we assess the recommendation to follow a norm of robust satisficing when making decisions under deep uncertainty in the context of decision analyses that rely on the tools of Robust Decision-Making developed by Robert Lempert and colleagues at RAND. We discuss two challenges for robust satisficing: whether the norm might derive its plausibility from an implicit (...) appeal to probabilistic representations of uncertainty of the kind that deep uncertainty is supposed to preclude; and whether there is adequate justification for adopting a satisficing norm, as opposed to an optimizing norm that is sensitive to considerations of robustness. We discuss decision-theoretic and voting-theoretic motivations for robust satisficing, and use these motivations to select among candidate formulations of the robust satisficing norm. (shrink)

Proponents of the predictive processing (PP) framework often claim that one of the framework’s significant virtues is its unificatory power. What is supposedly unified are predictive processes in the mind, and these are explained in virtue of a common prediction error-minimisation (PEM) schema. In this paper, I argue against the claim that PP currently converges towards a unified explanation of cognitive processes. Although the notion of PEM systematically relates a set of posits such as ‘efficiency’ and ‘hierarchical coding’ into a (...) unified conceptual schema, neither the frameworks’ algorithmic specifications nor its hypotheses about their implementations in the brain are clearly unified. I propose a novel way to understand the fruitfulness of the research program in light of a set of research heuristics that are partly shared with those common to Bayesian reverse engineering. An interesting consequence of this proposal is that pluralism is at least as important as unification to promote the positive development of the predictive mind. (shrink)

There are many things—call them ‘experts’—that you should defer to in forming your opinions. The trouble is, many experts are modest: they’re less than certain that they are worthy of deference. When this happens, the standard theories of deference break down: the most popular (“Reflection”-style) principles collapse to inconsistency, while their most popular (“New-Reflection”-style) variants allow you to defer to someone while regarding them as an anti-expert. We propose a middle way: deferring to someone involves preferring to make any decision (...) using their opinions instead of your own. In a slogan, deferring opinions is deferring decisions. Generalizing the proposal of Dorst (2020a), we first formulate a new principle that shows exactly how your opinions must relate to an expert’s for this to be so. We then build off the results of Levinstein (2019) and Campbell-Moore (2020) to show that this principle is also equivalent to the constraint that you must always expect the expert’s estimates to be more accurate than your own. Finally, we characterize the conditions an expert’s opinions must meet to be worthy of deference in this sense, showing how they sit naturally between the too-strong constraints of Reflection and the too-weak constraints of New Reflection. (shrink)

Branden Fitelson and Alan Hájek have suggested that it is finally time for a “revolution” in which we jettison Kolmogorov’s axiomatization of probability, and move to an alternative like Popper’s. According to these authors, not only did Kolmogorov fail to give an adequate analysis of conditional probability, he also failed to give an adequate account of another central notion in probability theory: probabilistic independence. This paper defends Kolmogorov, with a focus on this independence charge. I show that Kolmogorov’s sophisticated theory (...) of conditional distributions, advocated by authors such as Kenny Easwaran but often overlooked by philosophers, has the resources to define the notion of independence Fitelson and Hájek seek. I then argue that Kolmogorov’s original definition of independence, once suitably reinterpreted as capturing a slight logical weakening of this notion, is not so problematic after all. Indeed, rather than discarding Kolmogorov’s definition, we should make room for two distinct notions of probabilistic independence, one stronger and one weaker, which play different theoretical roles. (shrink)

The standard of proof applied in civil trials is the preponderance of evidence, often said to be met when a proposition is shown to be more than 50% likely to be true. A number of theorists have argued that this 50%+ standard is too weak – there are circumstances in which a court should find that the defendant is not liable, even though the evidence presented makes it more than 50% likely that the plaintiff’s claim is true. In this paper, (...) I will recapitulate the familiar arguments for this thesis, before defending a more radical one: The 50%+ standard is also too strong – there are circumstances in which a court should find that a defendant is liable, even though the evidence presented makes it less than 50% likely that the plaintiff’s claim is true. I will argue that the latter thesis follows naturally from the former once we accept that the parties in a civil trial are to be treated equally. I will conclude by sketching an alternative interpretation of the civil standard of proof. (shrink)

Is it right to convict a person of a crime on the basis of purely statistical evidence? Many who have considered this question agree that it is not, posing a direct challenge to legal probabilism – the claim that the criminal standard of proof should be understood in terms of a high probability threshold. Some defenders of legal probabilism have, however, held their ground: Schoeman (1987) argues that there are no clear epistemic or moral problems with convictions based on purely (...) statistical evidence, and speculates that our aversion to such convictions may be nothing more than an irrational bias. More recently, Hedden and Colyvan (2019, section VI) describe our reluctance to convict on the basis of purely statistical evidence as an ‘intuition’, but suggest that there may be no ‘in principle’ problem with such convictions (see also Papineau, forthcoming, section 6). In this paper, I argue that there is, in some cases, an in principle problem with a conviction based upon statistical evidence alone – namely, it commits us to a precedent which, if consistently followed through, could lead to the deliberate conviction of an innocent person. I conclude with some reflections on the idea that the criminal justice system should strive to maximise the accuracy of its verdicts – and the related idea that we should each strive to maximise the accuracy of our beliefs. (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)

Innovative research on decision making under ‘deep uncertainty’ is underway in applied fields such as engineering and operational research, largely outside the view of normative theorists grounded in decision theory. Applied methods and tools for decision support under deep uncertainty go beyond standard decision theory in the attention that they give to the structuring of decisions. Decision structuring is an important part of a broader philosophy of managing uncertainty in decision making, and normative decision theorists can both learn from, and (...) contribute to, the growing deep uncertainty decision support literature. (shrink)

As stochastic independence is essential to the mathematical development of probability theory, it seems that any foundational work on probability should be able to account for this property. Bayesian decision theory appears to be wanting in this respect. Savage’s postulates on preferences under uncertainty entail a subjective expected utility representation, and this asserts only the existence and uniqueness of a subjective probability measure, regardless of its properties. What is missing is a preference condition corresponding to stochastic independence. To fill this (...) significant gap, the article axiomatizes Bayesian decision theory afresh and proves several representation theorems in this novel framework. (shrink)

As an application of his Material Theory of Induction, Norton (2018; manuscript) argues that the correct inductive logic for a fair infinite lottery, and also for evaluating eternal inflation multiverse models, is radically different from standard probability theory. This is due to a requirement of label independence. It follows, Norton argues, that finite additivity fails, and any two sets of outcomes with the same cardinality and co-cardinality have the same chance. This makes the logic useless for evaluating multiverse models based (...) on self-locating chances, so Norton claims that we should despair of such attempts. However, his negative results depend on a certain reification of chance, consisting in the treatment of inductive support as the value of a function, a value not itself affected by relabeling. Here we define a purely comparative infinite lottery logic, where there are no primitive chances but only a relation of ‘at most as likely’ and its derivatives. This logic satisfies both label independence and a comparative version of additivity as well as several other desirable properties, and it draws finer distinctions between events than Norton's. Consequently, it yields better advice about choosing between sets of lottery tickets than Norton's, but it does not appear to be any more helpful for evaluating multiverse models. Hence, the limitations of Norton's logic are not entirely due to the failure of additivity, nor to the fact that all infinite, co-infinite sets of outcomes have the same chance, but to a more fundamental problem: We have no well-motivated way of comparing disjoint countably infinite sets. (shrink)

The resolving of the main problem of quantum mechanics about how a quantum leap and a smooth motion can be uniformly described resolves also the problem of how a distribution of reliable data and a sequence of deductive conclusions can be uniformly described by means of a relevant wave function “Ψdata”.

You have higher-order uncertainty iff you are uncertain of what opinions you should have. I defend three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is (...) not obvious. The Enkratic Intuition---that your first-order opinions must “line up” with your higher-order opinions---is incorrect; epistemic akrasia can be rational. If all this is right, then it leaves us without answers---but with a clear picture of the question, and a fruitful strategy for pursuing it. (shrink)

Assume that it is your evidence that determines what opinions you should have. I argue that since you should take peer disagreement seriously, evidence must have two features. (1) It must sometimes warrant being modest: uncertain what your evidence warrants, and (thus) uncertain whether you’re rational. (2) But it must always warrant being guided: disposed to treat your evidence as a guide. Surprisingly, it is very difficult to vindicate both (1) and (2). But diagnosing why this is so leads to (...) a proposal—Trust—that is weak enough to allow modesty but strong enough to yield many guiding features. In fact, I claim that Trust is the Goldilocks principle—for it is necessary and sufficient to vindicate the claim that you should always prefer to use free evidence. Upshot: Trust lays the foundations for a theory of disagreement and, more generally, an epistemology that permits self-doubt—a modest epistemology. (shrink)

The lesson to be learned from the paradoxical St. Petersburg game and Pascal’s Mugging is that there are situations where expected utility maximizers will needlessly end up poor and on death’s door, and hence we should not be expected utility maximizers. Instead, when it comes to decision-making, for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero, regardless of the utilities associated with those possibilities.

Searching for information is critical in many situations. In medicine, for instance, careful choice of a diagnostic test can help narrow down the range of plausible diseases that the patient might have. In a probabilistic framework, test selection is often modeled by assuming that people's goal is to reduce uncertainty about possible states of the world. In cognitive science, psychology, and medical decision making, Shannon entropy is the most prominent and most widely used model to formalize probabilistic uncertainty and the (...) reduction thereof. However, a variety of alternative entropy metrics (Hartley, Quadratic, Tsallis, Rényi, and more) are popular in the social and the natural sciences, computer science, and philosophy of science. Particular entropy measures have been predominant in particular research areas, and it is often an open issue whether these divergences emerge from different theoretical and practical goals or are merely due to historical accident. Cutting across disciplinary boundaries, we show that several entropy and entropy reduction measures arise as special cases in a unified formalism, the Sharma–Mittal framework. Using mathematical results, computer simulations, and analyses of published behavioral data, we discuss four key questions: How do various entropy models relate to each other? What insights can be obtained by considering diverse entropy models within a unified framework? What is the psychological plausibility of different entropy models? What new questions and insights for research on human information acquisition follow? Our work provides several new pathways for theoretical and empirical research, reconciling apparently conflicting approaches and empirical findings within a comprehensive and unified information‐theoretic formalism. (shrink)

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

In addition to beliefs, people have attitudes of confidence called credences. Combinations of credences, like combinations of beliefs, can be inconsistent. It is common to use tools from probability theory to understand the normative relationships between a person’s credences. More precisely, it is common to think that something is a consistency norm on a person’s credal state if and only if it is a simple transformation of a truth of probability (a transformation that merely changes the statement from one about (...) probability to one about credences). Though it is common to challenge the right-to-left direction of this biconditional, I argue in this paper that the left-to-right direction is false for standard versions of probability theory. That is, I make the case that there are consistency constraints on credal states that are not simple transformations of truths of standard versions of probability theory. I do so by drawing on a newly discovered type of credal attitude, a quantificational credence, and by showing how the consistency norms on this attitude can’t be represented as simple transformations of truths of standard versions of probability theory. I conclude by showing that a probability theory that could avoid the result would have to be strikingly different from the standard versions—so different that I suspect many would hesitate to call it a theory of probability at all. (shrink)

Science advances by means of argument and debate. Based on a formal model of complex argumentation, this article assesses the interplay between evidential and inferential drivers in scientific controversy, and explains, in particular, why both evidence accumulation and argumentation are veritistically valuable. By improving the conditions for applying veritistic indicators, novel evidence and arguments allow us to distinguish true from false hypotheses more reliably. Because such veritistic indicators also underpin inductive reasoning, evidence accumulation and argumentation enhance the reliability of inductive (...) inference, for example, inference to the best explanation. 1 Introduction2 Theory of Dialectical Structures3 Debate Simulations4 From Evidence and Arguments to Truth: The First Route5 From Evidence to Truth: The Second Route6 Conclusion. (shrink)

Subjective probability plays an increasingly important role in many fields concerned with human cognition and behavior. Yet there have been significant criticisms of the idea that probabilities could actually be represented in the mind. This paper presents and elaborates a view of subjective probability as a kind of sampling propensity associated with internally represented generative models. The resulting view answers to some of the most well known criticisms of subjective probability, and is also supported by empirical work in neuroscience and (...) behavioral psychology. The repercussions of the view for how we conceive of many ordinary instances of subjective probability, and how it relates to more traditional conceptions of subjective probability, are discussed in some detail. (shrink)

Lotteries, and our intuitions about them, lie at the heart of a number of persistent puzzles in epistemology. These include the lottery paradox, the lottery driven sceptical puzzle, Harman’s puzzle about statistical vs testimonial evidence concerning lottery outcomes and Vogel’s puzzle about iterated lotteries. In this chapter, these puzzles are considered anew in light of the suggestion that justification requires normic support.

In this chapter, the risk minimisation conception of justification is criticised for its predictions about the force of statistical evidence, and it is argued that justification cannot be understood solely in terms of probability. A non-probabilistic support relation between evidence and propositions—termed _normic support_—is introduced and put forward as a necessary condition for justification. The distinction between normic support and probabilistic support turns out to be akin to the more widely recognised distinction between ceteris paribus laws and brute statistical generalisations. (...) A formal model of normic support, involving a normalcy ordering of possible worlds, is proposed. (shrink)

Say that two goals are _normatively coincident_ just in case one cannot aim for one goal without automatically aiming for the other. While knowledge and justification are distinct epistemic goals, this chapter begins from the suggestion that they are nevertheless normatively coincident—aiming for knowledge and aiming for justification are one and the same activity. A number of consequences are derived from this, including limitations on how we can ascribe justification and knowledge in lottery cases. Ultimately, the suggestion is shown to (...) be in tension with the widespread _risk minimisation conception_ of justification, according to which one has justification for believing a proposition just when it is highly probable, given one’s evidence. (shrink)

This chapter develops a formal theory of degrees of safety and of normic support. The theory predicts that the degrees of normic support imposed upon propositions by a body of evidence will meet the conditions for a _ranking function_. Degrees of normic support are used to develop an alternative epistemology of degrees of belief that contrasts sharply with the more standard Bayesian approach. A defence of what Lewis in _Counterfactuals_ (1973) calls the ‘limit assumption’ is outlined.

In this chapter the formal features of safety and of normic support are explored using sphere models for conditional logic. The formal structure of normic support is contrasted with the formal structure of probabilistic support, and it is argued that the former corresponds more closely with the formal structure of epistemic justification. It is argued, in particular, that normic support exhibits a number of crucial formal properties that probabilistic support lacks—properties such as agglomeration, amalgamation, cautious monotonicity, cumulative transitivity, and rational (...) monotonicity. (shrink)

A number of epistemologists have attempted to refine the risk minimisation conception of justification, proposing more complex probabilistic rules for justification that enable one to avoid the lottery paradox without giving up multiple premise closure. These attempts to refine the risk minimisation conception often appear ad hoc, and are also beset by formal difficulties. An important result, proved by Douven and Williamson (in 2006), comes close to showing that the ambition behind these refinements cannot be realised. In this chapter it (...) is shown that a significant limitation of Douven and Williamson’s proof—namely, its assumption of a finite, uniform probability space—can be overcome, and the result is available for infinite probability spaces as well. While a normic theory of justification is able to accommodate multiple premise closure within a fallibilist framework, it is concluded that no purely probabilistic theory of justification can do so. (shrink)