Does the strength of a particular belief depend upon the significance we attach to it? Do we move from one context to another, remaining in the same doxastic state concerning p yet holding a stronger belief that p in one context than in the other? For that to be so, a doxastic state must have a certain sort of context-sensitive complexity. So the question is about the nature of belief states, as we understand them, or as we (...) think a theory should model them. I explore the idea and how it relates to work on imprecise probabilities and second-order confidence. (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)

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)

There is a trade-off between specificity and accuracy in existing models of belief. Descriptions of agents in the tripartite model, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian model, which requires real-valued credences, is perfectly specific, but often inaccurate: we often lack precise credences. I argue, first, that a popular attempt to fix the Bayesian model by using sets of functions is also inaccurate, since it requires (...) us to have interval-valued credences with perfectly precise endpoints. We can see this problem as analogous to the problem of higher order vagueness. Ultimately, I argue, the only way to avoid these problems is to endorse Insurmountable Unclassifiability. This principle has some surprising and radical consequences. For example, it entails that the trade-off between accuracy and specificity is in-principle unavoidable: sometimes it is simply impossible to characterize an agent’s doxastic state in a way that is both fully accurate and maximally specific. What we can do, however, is improve on both the tripartite and existing Bayesian models. I construct a new model of belief—the minimal model—that allows us to characterize agents with much greater specificity than the tripartite model, and yet which remains, unlike existing Bayesian models, perfectly accurate. (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)

Our ordinary assertions are often imprecise, insofar as the way we represent things as being only approximates how things are in the actual world. The phenomenon of assertoric imprecision raises a challenge to standard accounts of both the norm of assertion and the connection between semantics and the objects of assertion. After clarifying these problems in detail, I develop a framework for resolving them. Specifically, I argue that the phenomenon of assertoric imprecision motivates a rejection of the widely held (...) belief that a semantic theory for a language associates a single semantic value with each of the simple and complex expressions of that language, relative to the contexts in which they occur. Instead, I propose that we adopt a framework I call semantic pluralism. (shrink)

Ambiguity occurs insofar as a reasoner lacks information about the relevant physical probabilities. There are objections to the application of standard Bayesian inductive logic and decision theory in contexts of significant ambiguity. A variety of alternative frameworks for reasoning under ambiguity have been proposed. Two of the most prominent are Imprecise Bayesianism and Dempster–Shafer theory. We compare these inductive logics with respect to the Ambiguity Dilemma, which is a problem that has been raised for Imprecise Bayesianism. We develop (...) an agent-based model comparison that isolates the difference between the two inductive logics in their updating methods. We find that Dempster–Shafer theory does not avoid the Ambiguity Dilemma. We discuss the implications of this result. (shrink)

A number of Bayesians claim that, if one has no evidence relevant to a proposition P, then one's credence in P should be spread over the interval [0, 1]. Against this, I argue: first, that it is inconsistent with plausible claims about comparative levels of confidence; second, that it precludes inductive learning in certain cases. Two motivations for the view are considered and rejected. A discussion of alternatives leads to the conjecture that there is an in-principle limitation on formal representations (...) of belief: they cannot be both fully accurate and maximally specific. (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)

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)

Gordon Belot has recently developed a novel argument against Bayesianism. He shows that there is an interesting class of problems that, intuitively, no rational belief forming method is likely to get right. But a Bayesian agent’s credence, before the problem starts, that she will get the problem right has to be 1. This is an implausible kind of immodesty on the part of Bayesians. My aim is to show that while this is a good argument against traditional, precise Bayesians, (...) the argument doesn’t neatly extend to imprecise Bayesians. As such, Belot’s argument is a reason to prefer imprecise Bayesianism to precise Bayesianism. (shrink)

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations (...) where the decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

It is a consequence of both Kennedy and McNally’s typology of the scale structures of gradable adjectives and Kennedy’s :1–45, 2007) economy principle that an object is clean just in case its degree of cleanness is maximal. So they jointly predict that the sentence ‘Both towels are clean, but the red one is cleaner than the blue one’ :259–288, 2004) is a contradiction. Surely, one can account for the sentence’s assertability by saying that the first instance of ‘clean’ is used (...) loosely: since ‘clean’ pragmatically conveys the property of being close to maximally clean rather than the property of being maximally clean, the sentence as a whole conveys a consistent proposition. I challenge this semantics–pragmatics package by considering the sentence ‘Mary believes that both towels are clean but that the red one is cleaner than the blue one’. We can certainly use this sentence to attribute a coherent belief to Mary: One of its readings says that she believes that the towels are clean by a contextually salient standard ; the other says that she believes that the towels are clean by her own standard. I argue that Kennedy’s semantics–pragmatics package can’t deliver those readings, and propose that we drop the economy principle and account for those readings semantically by assigning to the belief sentence two distinct truth conditions. I consider two ways to deliver those truth-conditions. The first one posits world-variables in the sentence’s logical form and analyzes those truth-conditions as resulting from two binding possibilities of those variables. The second one proposes that the threshold function introduced by the phonologically null morpheme pos is shiftable in belief contexts. (shrink)

Imprecise probabilities are often modelled with representors, or sets of probability functions. In the recent literature, two ways of interpreting representors have emerged as especially prominent: vagueness interpretations, according to which each probability function in the set represents how the agent's beliefs would be if any vagueness were precisified away; and comparativist interpretations, according to which the set represents those comparative confidence relations that are common to all probability functions therein. I argue that these interpretations have some important limitations. (...) I also propose an alternative—the functional interpretation—according to which representors are best interpreted by reference to the roles they play in the theories that make use of them. (shrink)

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to (...) be borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment. (shrink)

One recent topic of debate in Bayesian epistemology has been the question of whether imprecise credences can be rational. I argue that one account of imprecise credences, the orthodox treatment as defended by James M. Joyce, is untenable. Despite Joyce’s claims to the contrary, a puzzle introduced by Roger White shows that the orthodox account, when paired with Bas C. van Fraassen’s Reflection Principle, can lead to inconsistent beliefs. Proponents of imprecise credences, then, must either provide a (...) compelling reason to reject Reflection or admit that the rational credences in White’s case are precise. (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)

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)

First, ‘Is Justified True Belief Knowledge’ is imprecise but Gettier is explicit that ‘know’ is analysed as the definiendum is ‘S knows that P’. Second, Gettier does not misrepresent as Plato’s definition as the expressions used are ‘Plato considers’ and ‘seems to accept’. Third, Gettier is not mistaken to apply Plato’s definition to propositions since propositional knowledge is a species of Plato’s definition. Fourth, for Plato true belief temporally precedes an account. ‘Jones owns a Ford’ is never (...) a true opinion, hence no account for it can be given. The counterexample is reconstructed with temporality built into it. Fifth, Gettier does not fail to establish the equivalence of ‘believe’, ‘accepts’ and ‘sure’ in the three versions as this is implicitly established in the shifts made in the paper. Sixth, ‘entails’ logically is used only when the entailing proposition is true, but in the counterexamples a false proposition is taken to imply a true one. ‘Entail’ is to be taken in the ordinary sense of implies. Seventh, in Case I, the implication is preserved with the proper representation: → [Gy ↔ ], which implies { → [Gy ↔ ]}’. Eighth, the counterexample is reworked to avoid the objection that justification for p and justification for q may not be sufficient justification for ‘p & q’. (shrink)

I formulate and tentatively defend the view that we cannot be rationally required to have one type of doxastic attitude (e.g., beliefs, credences, imprecise credences, etc.) because we have another type; in other words, we can only be required to have, say, given credences because we have some other credences already. I explore an argument that appeals to the idea that there is no good reasoning from one type to the other type. I consider some important possible responses, and (...) conclude by sketching a different argument for the same conclusion, appealing to the nature of inquiry and belief-formation. (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)

We call something a paradox if it strikes us as peculiar in a certain way, if it strikes us as something that is not simply nonsense, and yet it poses some difficulty in seeing how it could make sense. When we examine paradoxes more closely, we find that for some the peculiarity is relieved and for others it intensifies. Some are peculiar because they jar with how we expect things to go, but the jarring is to do with imprecision and (...) misunderstandings in our thought, failures to appreciate the breadth of possibility consistent with our beliefs. Other paradoxes, however, pose deep problems. Closer examination does not explain them away. Instead, they challenge the coherence of certain conceptual resources and hence challenge the significance of beliefs which deploy those resources. I shall call the former kind weak paradoxes and the latter, strong paradoxes. Whether a particular paradox is weak or strong is sometimes a matter of controversy—sometimes it has been realised that what was thought strong is in fact weak, and vice versa,— but the distinction between the two kinds is generally thought to be worth drawing. In this Cchapter, I shall cover both weak and strong probabilistic paradoxes. (shrink)

Spinoza’s interpretation of the election of the Hebrews in the third chapter of the Theological Political Treatise enraged quite a few Jewish readers of the nineteenth and twentieth centuries. The rise of nationalism, and the demand of loyalty to one’s own genos brought about a certain style of patriotic writing aimed at Spinoza’s “betrayal.” In a series of lectures on the eve of the Great War, Hermann Cohen portrayed Spinoza as a person of “demonic spirt” and as “the great enemy (...) who emerged from our midst.” In a stream of words more akin to shouting than to analytic discourse, Cohen protested against what he took to be the universal complacency regarding Spinoza’s treachery: “When Spinoza, with merciless severity, makes his own nation the object of contempt – at the time that Rembrandt lived on the same street and immortalized the ideal type of the Jew - no voice rises in protest against this humanly incomprehensible betrayal.” Cohen was right in identifying in Spinoza an absence of ethnic patriotism. I, for one, find this absence a virtue rather than a vice, and in this chapter I will argue, inter alia, that in rejecting ethnic normativity Spinoza was consistent with a dominant strand of rabbinic thought. What precisely was so offensive in Spinoza’s words in chapter three of the TTP? A common reading of this chapter suggests that Spinoza presents the election of the Hebrews as merely political and not spiritual in nature, thus downgrading the importance of the election. This reading is not absolutely groundless, but it is highly imprecise, for as we shall shortly see, for Spinoza, God’s (genuine) election of the Hebrews indicates the political weakness of their state, rather than its strength. Apart from pointing out this important corrective, I will also attempt in this essay to evaluate Spinoza’s critique of the election of the Hebrews, the result of which might lead us to some highly unexpected conclusions. In the first section of this chapter I present an outline of Spinoza’s interpretation of the connection between the Hebrews’ belief in being chosen, and the xenophobic nature of their ancient state. In the second section, I discuss Spinoza’s interpretation of the election of the Hebrews in chapter three of the TTP, and show that on Spinoza’s sardonic reading it was nothing but luck which allowed the Hebrew state to survive for a rather long time in spite of its poor political constitution. The third and final part provides a defense of Spinoza’s critique of the notion of chosenness. I will argue, first, that chosenness has never been granted the status of theological doctrine or principle of faith within rabbinic Judaism. I will then point out the significant religious problems with the notion of chosenness, suggesting however two exceptions in which belief in chosenness might still be defensible. I will conclude this section with a discussion of rabbinic views on the conversion of minors, arguing that according to the mainstream rabbinic view, being a Jew is a merit only on the condition that a person is pious, a view which is not far from Spinoza’s own claims in the third chapter of the TTP. (shrink)

I analyze a recent exchange between Adam Elga and Julian Jonker concerning unsharp (or imprecise) credences and decision-making over them. Elga holds that unsharp credences are necessarily irrational; I agree with Jonker's reply that they can be rational as long as the agent switches to a nonlinear valuation. Through the lens of computational complexity theory, I then argue that even though nonlinear valuations can be rational, they come in general at the price of computational intractability, and that this problematizes (...) their use in defining rationality. I conclude that the meaning of "rationality" may be philosophically vague. (shrink)

Epistemologists often appeal to the idea that a normative theory must provide useful, usable, guidance to argue for one normative epistemology over another. I argue that this is a mistake. Guidance considerations have no role to play in theory choice in epistemology. I show how this has implications for debates about the possibility and scope of epistemic dilemmas, the legitimacy of idealisation in Bayesian epistemology, uniqueness versus permissivism, sharp versus mushy credences, and internalism versus externalism.

This article explores the main similarities and differences between Derek Parfit’s notion of imprecise comparability and a related notion I have proposed of parity. I argue that the main difference between imprecise comparability and parity can be understood by reference to ‘the standard view’. The standard view claims that 1) differences between cardinally ranked items can always be measured by a scale of units of the relevant value, and 2) all rankings proceed in terms of the trichotomy of (...) ‘better than’, ‘worse than’, and ‘equally good’. Imprecise comparability, which can be understood in terms of the more familiar notions of cardinality and incommensurability, rejects only the first claim while parity rejects both claims of the standard view. -/- I then argue that insofar as those attracted to imprecise comparability assume that all rankings are trichotomous, as Parfit appears to, the view should be rejected. This is because imprecise equality is not a form of equality but is a sui generis ‘fourth’ basic way in which items can be ranked. We should, I argue, understand imprecise equality as parity, and imprecise comparability as entailing ‘tetrachotomy’ – that if two items are comparable, one must better than, worse than, equal to, or on a par with the other. Thus those attracted to the idea that cardinality can be imprecise should abandon trichotomy and accept parity and tetrachotomy instead. -/- Finally, I illustrate the difference between Parfit’s trichotomous notion of imprecise comparability and parity by examining how each notion might be employed in different solutions to the problem posed by the Repugnant Conclusion in population ethics. I suggest that parity provides the arguably more ecumenical solution to the problem. (shrink)

It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek, and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued (...) from a related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision- theoretic implications. -/- The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgments about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to nonmeasurable regions. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can (...) be difficult to apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

Recently, Derek Parfit has offered a novel solution to the “Repugnant Conclusion” that compared with the existence of many people whose quality of life would be very high, there is some much larger number of people whose existence would be better but whose lives would be barely worth living. On this solution, qualitative differences between two populations will often entail that the populations are merely “imprecisely” comparable. According to Parfit, this fact allows us to avoid the Repugnant Conclusion without violating (...) the transitivity of better than. In this paper, I argue that Parfit’s view nevertheless implies two objectionable conclusions. The first is an alternative version of the Repugnant Conclusion that, Parfit suggests, may not be all that repugnant. The second is a revised version of the first that is nearly identical to the Repugnant Conclusion. I conclude that Parfit’s view offers no escape from repugnance. (shrink)

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the usefulness (...) of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)

Many have argued that a rational agent's attitude towards a proposition may be better represented by a probability range than by a single number. I show that in such cases an agent will have unstable betting behaviour, and so will behave in an unpredictable way. I use this point to argue against a range of responses to the ‘two bets’ argument for sharp probabilities.

Many have claimed that epistemic rationality sometimes requires us to have imprecise credal states (i.e. credal states representable only by sets of credence functions) rather than precise ones (i.e. credal states representable by single credence functions). Some writers have recently argued that this claim conflicts with accuracy-centered epistemology, i.e., the project of justifying epistemic norms by appealing solely to the overall accuracy of the doxastic states they recommend. But these arguments are far from decisive. In this essay, we prove (...) some new results, which show that there is little hope for reconciling the rationality of credal imprecision with accuracy-centered epistemology. (shrink)

In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for example, the paper collected (...) in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates. (shrink)

In this paper, we propose the introduction of a neutrosophic chi-square-test for consistency, incorporating neutrosophic statistics. Our aim is to modify the existing chi-square -test for consistency in order to analyze imprecise data. We present a novel test statistic for the neutrosophic chi-square -test for consistency, which accounts for the uncertainties inherent in the data. To evaluate the performance of the proposed test, we compare it with the traditional chi-square -test for consistency based on classical statistics. By conducting a (...) comparative analysis, we assess the efficiency and effectiveness of our proposed neutrosophic chi-square-test for consistency. Furthermore, we illustrate the application of the proposed test through a numerical example, demonstrating how it can be utilized in practical scenarios. Through this implementation, we aim to provide empirical evidence of the improved performance of our proposed test when compared to the traditional chi-square-test for consistency based on classical statistics. We anticipate that the proposed neutrosophic chi-square-test for consistency will outperform its classical counterpart, offering enhanced accuracy and reliability when dealing with imprecise data. This advancement has the potential to contribute significantly to the field of statistical analysis, particularly in situations where data uncertainty and imprecision are prevalent. (shrink)

This paper is a discussion note on Isaacs et al. (2022), who have claimed to offer a new motivation for imprecise probabilities, based on the mathematical phenomenon of non-measurability. In this note, I clarify some consequences of their proposal. In particular, I show that if their proposal is applied to a bounded 3-dimensional space, then they have to reject at least one of the following: (i) If A is at most as probable as B and B is at most (...) as probable as C, then A is at most as probable as C. (ii) Let A ∩ C = B ∩ C = ∅. A is at most as probable as B iff (A ∪ C) is at most as probable as (B ∪ C). But rejecting either statement seems unattractive. (shrink)

I argue for the existence of imaginative beliefs: mental states that are imaginative in format and doxastic in attitude. I advance two arguments for this thesis. First, there are imaginings that play the functional roles of belief. Second, there are imaginings that play the epistemic roles of belief. These arguments supply both descriptive and normative grounds for positing imaginative beliefs. I also argue that this view fares better than alternatives that posit distinct imaginative and doxastic states to account (...) for the same phenomena. Along the way, I demonstrate the theoretical significance of imaginative beliefs by showing how they pose a challenge for standard taxonomies of the mind, clarify the sense in which imagination is occurrent and voluntary, explain the justificatory role of the imagination, and point to a deep but underappreciated symmetry between imagistic and non-imagistic thought. (shrink)

Belief-in reports of the form 'S believes in O' have been taken to have at least two senses: factual and evaluative. I begin by briefly suggesting that there is no evidence for two distinct senses, then spend most of the paper developing a general semantics for belief-in reports. I explore, and use my semantics to explain, several features of belief-in reports: the context-dependence of what belief-that reports they entail, their widespread lack of equivalence with belief-that (...) reports, and their neg-raising property. Put roughly, my semantics says that 'S believes in O' a) asserts that, for some contextually salient property F, S believes that O has F and b) presupposes that S either has a belief that O has F for which they meet an affective and evidential requirement or has a belief that O doesn’t have F for which they meet an affective and evidential requirement. (shrink)

You shouldn’t have done it. But you did. Against your better judgment you scrolled to the end of an article concerning the state of race relations in America and you are now reading the comments. Amongst the slurs, the get-rich-quick schemes, and the threats of physical violence, there is one comment that catches your eye. Spencer argues that although it might be “unpopular” or “politically incorrect” to say this, the evidence supports believing that the black diner in his section will (...) tip poorly. He insists that the facts don’t lie. The facts aren’t racist. In denying his claim and in believing otherwise, it is you who engages in wishful thinking. It is you who believes against the evidence. You, not Spencer, are epistemically irrational. -/- My dissertation gives an account of the moral-epistemic norms governing belief that will help us answer Spencer and the challenge he poses. We live in a society that has been shaped by racist attitudes and institutions. Given the effects of structural racism, Spencer’s belief could have considerable evidential support. Spencer notes that it might make him unpopular, but he cares about the truth and he is willing to believe the unpopular thing. But, Spencer’s belief seems racist. Spencer asks, however, how could his belief be racist if his beliefs reflect reality and are rationally justified? Moreover, how could he wrong anyone by believing what he epistemically ought to believe given the evidence? In answer, I argue that beliefs can wrong. (shrink)

The belief norm of academic publishing states that researchers should believe certain claims they publish. The purpose of this paper is to defend the belief norm of academic publishing. In its defense, the advantages and disadvantages of the belief norm are evaluated for academic research and for the publication system. It is concluded that while the norm does not come without costs, academic research systemically benefits from the belief norm and that it should be counted among (...) those that sustain the practice of academic publishing. (shrink)

How does it come about that a person akratically believes that P, while at the same time believing that the available evidence speaks against that P? Among the current accounts, Scanlon offers an intuitive suggestion that one’s seeming experience that P may play an important role in the aetiology of their akratic belief that P. However, it turns out to be quite challenging to articulate what the role of seeming experience is. This paper will offer a novel development of (...) Scanlon’s intuitive suggestion, with a focus on clear-eyed epistemic akrasia. I will argue that the primary role of seeming experience is unlikely to act as the subject’s reason or to provide the subject with prima facie justification; instead, based on the recent work in dogmatism and Cartesian clarity, I will propose a causal account, according to which, when it seems clear to the subject that P, the seeming experience may exert a brute causal force to persistently compel the subject to believe that P. This causal account also has the advantage of helping some existing accounts to explain clear-eyed epistemic akrasia. (shrink)

Belief-credence dualism is the view that we have both beliefs and credences and neither attitude is reducible to the other. Pragmatic encroachment is the view that practical stakes can affect the epistemic rationality of states like knowledge or justified belief. In this paper, I argue that dualism offers a unique explanation of pragmatic encroachment cases. First, I explain pragmatic encroachment and what motivates it. Then, I explain dualism and outline a particular argument for dualism. Finally, I show how (...) dualism can explain the intuitions that underlie pragmatic encroachment. My basic proposal is that in high-stakes cases, it is not that one cannot rationally believe that p; instead, one ought not to rely on one's belief that p. One should rather rely on one's credence in p. I conclude that we need not commit ourselves to pragmatic encroachment in order to explain the intuitiveness of the cases that motivate it. (shrink)

In this essay we focus on our vast web of beliefs that serves us as a rough and ready map of reality, generated more to give us comfort and confidence in an intimidating world than to be accurate. Maps of reality can never be accurate in any ultimate sense since reality itself is a convoluted entity that can only be accessed in never- ending layers. Our repertoire of beliefs, generated compulsively in the mind, span a huge spectrum in respect of (...) ties to affect and emotions on the one hand and of their credibility in depicting reality on the other. The interaction between our affect-based psychological needs, our need for having a sound comprehension of reality, and the infinitely tangled reality itself, leads to the generation of a large diversity of types of beliefs, where these get revised under the impact of reality by overcoming the restrain- ing effect of emotions in diverse ways and over diverse time scales—interestingly, the ‘reality’ includes the complex repertoire of affect and emotions itself that may, in cer- tain cases, generate the impact for our beliefs to change. Beliefs represented by small clusters in our belief network (see below) are commonly prone to frequent revision in a more or less random manner, though there are beliefs acquired in early life that are intransigent to change because of overriding ties to emotions. Beliefs represented by larger clusters are comparatively durable, and their revision has a systematic aspect to it because of a greater role of evidence-based factors—however,the evidence may, once again, be tied with emotions and beliefs oriented away from reason. Finally, there are prepotent beliefs that are resistant to change even over a lifetime. Often, these beliefs get revised only if we engage with our own selves, looking at conflicting forces within ourselves. In this context, we refer to narratives that are self-reinforcing and exercise almost a mesmerizing influence on us. In all this, there is a dual role of ‘rationality’ that needs being attended to in describing and explaining belief revision—rationality is not exclusively based on evidence and reason, but includes the implicit operation of affect and emotions too. (shrink)

I explore how rational belief and rational credence relate to evidence. I begin by looking at three cases where rational belief and credence seem to respond differently to evidence: cases of naked statistical evidence, lotteries, and hedged assertions. I consider an explanation for these cases, namely, that one ought not form beliefs on the basis of statistical evidence alone, and raise worries for this view. Then, I suggest another view that explains how belief and credence relate to (...) evidence. My view focuses on the possibilities that the evidence makes salient. I argue that this makes better sense of the difference between rational credence and rational belief than other accounts. (shrink)

The purpose of this paper is to show that if one adopts conditional probabilities as the primitive concept of probability, one must deal with the fact that even in very ordinary circumstances at least some probability values may be imprecise, and that some probability questions may fail to have numerically precise answers.

Levy (2021 Levy, N. (2021). Bad beliefs: Why they happen to good people. Oxford University Press.[Crossref], [Google Scholar]) argues that bad beliefs predominately stem from automatic (albeit rational) updating in response to testimonial evidence. To counteract such beliefs, then, we should focus on ridding our epistemic environments of misleading testimony. This paper responds as follows. First, I argue that the suite of automatic processes related to bad beliefs extends well beyond the deference-based processes that Levy identifies. Second, I push back (...) against Levy’s claim that bad beliefs stem from wholly rational processes, suggesting that, in many cases, such processes are better characterized as arational. Finally, I note that Levy is too quick to dismiss the role that individuals can play in cleaning up their own epistemic environments, and I suggest one route through which this is possible. (shrink)

Most work in Kant’s epistemology focuses on what happens “upstream” from experience, prior to the formation of conscious propositional attitudes. By contrast, this essay focuses on what happens "downstream": the formation of assent (Fuerwahrhalten) in its various modes. The mode of assent that Kant calls "Belief" (Glaube) is the main topic: not only moral Belief but also "pragmatic" and "doctrinal" Belief as well. I argue that Kant’s discussion shows that we should reject standard accounts of the extent (...) to which theoretical reason can provide justified assent about things-in-themselves, in favor of one that is much more liberal. Interpretive benefits are not the only results of the discussion, however. I also hope it will become clear along the way that there is such a thing as Kantian Belief, and that we often have quite a lot of it. -/- . (shrink)

In this paper, I argue that the relationship between belief and credence is a central question in epistemology. This is because the belief-credence relationship has significant implications for a number of current epistemological issues. I focus on five controversies: permissivism, disagreement, pragmatic encroachment, doxastic voluntarism, and the relationship between doxastic attitudes and prudential rationality. I argue that each debate is constrained in particular ways, depending on whether the relevant attitude is belief or credence. This means that epistemologists (...) should pay attention to whether they are framing questions in terms of belief or in terms of credence and the success or failure of a reductionist project in the belief-credence realm has significant implications for epistemology generally. (shrink)

What is it for an imprecise credence to be justified? It might be thought that this is not a particularly urgent question for friends of imprecise credences to answer. For one might think that its answer just depends on how a well-trodden issue in epistemology plays out—namely, that of which theory of doxastic justification, be it reliabilism, evidentialism, or some other theory, is correct. I’ll argue, however, that it’s difficult for reliabilists to accommodate imprecise credences, at least (...) if we understand such credences to be determinate first-order attitudes. If I’m right, reliabilists will have to reject imprecise credences, and friends of imprecise credences will have to reject reliabilism. Near the end of the paper, I’ll also consider whether reliabilism can accommodate indeterminate credences. (shrink)

There are currently two robust traditions in philosophy dealing with doxastic attitudes: the tradition that is concerned primarily with all-or-nothing belief, and the tradition that is concerned primarily with degree of belief or credence. This paper concerns the relationship between belief and credence for a rational agent, and is directed at those who may have hoped that the notion of belief can either be reduced to credence or eliminated altogether when characterizing the norms governing ideally rational (...) agents. It presents a puzzle which lends support to two theses. First, that there is no formal reduction of a rational agent’s beliefs to her credences, because belief and credence are each responsive to different features of a body of evidence. Second, that if our traditional understanding of our practices of holding each other responsible is correct, then belief has a distinctive role to play, even for ideally rational agents, that cannot be played by credence. The question of which avenues remain for the credence-only theorist is considered. (shrink)