Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb's stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum (...) entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state - such as a plausibility ranking over hypotheses, or a belief set - the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generated AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of beliefs. (shrink)

In this article, I discuss three distinct but related puzzles involving lotteries: Kyburg’s lottery paradox, the statistical evidence problem, and the Harman-Vogel paradox. Kyburg’s lottery paradox is the following well-known problem: if we identify rational outright belief with a rational credence above a threshold, we seem to be forced to admit either that one can have inconsistent rational beliefs, or that one cannot rationally believe anything one is not certain of. The statistical evidence problem arises from the observation that people (...) seem to resist forming outright beliefs whenever the available evidence for the claim under consideration is purely statistical. We need explanations of whether it is in fact irrational to form such beliefs, and of whether a clear distinction can be drawn between statistical and non-statistical evidence. The Harman-Vogel paradox is usually presented as a paradox about knowledge: we tend to assume that we can know so-called ordinary propositions, such as the claim that I will be in Barcelona next spring. Yet, we hesitate to make knowledge claims regarding so-called lottery propositions, such as the claim that I won’t die in a car crash in the next few months, even if these lottery propositions are obviously entailed by the ordinary propositions we claim to know. Depending on one’s view about the relationship between rational belief and knowledge, the Harman-Vogel paradox has ramifications for a theory of rational outright belief. Formal theories of the relationship between rational credence and rational belief, such as Leitgeb’s stability theory, tend to focus mostly on handling Kyburg’s lottery paradox, but not the other two puzzles I mention. My aim in this article is to draw out relationships and differences between the puzzles, and to examine to what extent existing formal solutions to Kyburg’s lottery paradox help with answering the statistical evidence problem and the Harman-Vogel paradox. (shrink)

This paper addresses how multiple individual credences on logically related issues should be aggregated into collective binary beliefs. We call this binarizing belief aggregation. It is vulnerable to dilemmas such as the discursive dilemma or the lottery paradox: proposition-wise independent aggregation can generate inconsistent or not deductively closed collective judgments. Addressing this challenge using the familiar axiomatic approach, we introduce general conditions on a binarizing belief aggregation rule, including rationality conditions on individual inputs and collective outputs, and determine which rules (...) (if any) satisfy different combinations of these conditions. Furthermore, we analyze similarities and differences between our proofs and other related proofs in the literature and conclude that the problem of binarizing belief aggregation is a free-standing aggregation problem not reducible to judgment aggregation or probabilistic opinion pooling. (shrink)

The current special issue focuses on logical and probabilistic approaches to reasoning in uncertain environments, both from a formal, conceptual and argumentative perspective as well as an empirical point of view. In the present introduction we give an overview of the types of problems addressed by the individual contributions of the special issue, based on fundamental distinctions employed in this area. We furthermore describe some of the general features of the special issue.

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: consistency and deductive closure for belief sets, conformity to the axioms of probability for degrees of belief, and the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed on the aggregation (...) of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief. (shrink)

The present article illustrates a conflict between the claim that rational belief sets are closed under deductive consequences, and a very inclusive claim about the factors that are sufficient to determine whether it is rational to believe respective propositions. Inasmuch as it is implausible to hold that the factors listed here are insufficient to determine whether it is rational to believe respective propositions, we have good reason to deny that rational belief sets are closed under deductive consequences.

There is a deep tension between logical and probabilistic norms of belief. This article illustrates the normative force that is associated with these frameworks by showing how rather unrestricted belief bases can be improved by undergoing logical and probabilistic reflection. It is argued that probabilistic reasoning accounts for the reliability of the conclusions one can draw from the beliefs. Most importantly, reliability commands us to care for the increasing uncertainty of conjunctions of beliefs. Deductive logic captures the agent's commitment towards (...) all conclusions of her beliefs. In other words, it seems that commitment forces the agent to believe conclusions, which reliability tells her to view with scepticism. This conflict is resolved by arguing that commitment merely forces the agent to find the logical conclusions of her beliefs acceptable, not necessarily to believe them. This demands consistency but no deductive closure. Furthermore, it is shown that by restoring consistency, a straightforward connection between high subjective probability and full belief is undermined. Finally, the article delineates how this approach deals with the lottery paradox and the preface paradox. (shrink)

Group beliefs in social networks are often construed as arising from individual beliefs through processes of update and aggregation. In this paper, we explore an alternative ‘arational’ perspective. More specifically, we focus on group attitudes as neutral tendencies toward alignment of opinions driven by influence patterns among agents modeled in a Markov dynamics. In addition, we investigate logical patterns in the resulting potential group beliefs or, in more neutral arational terminology: collective opinion structures.

There are two ways of representing rational belief: qualitatively as yes-or-no belief, and quantitatively as degrees of belief. Standard rationality conditions are: consistency and logical closure, for qualitative belief, satisfaction of the probability axioms, for quantitative belief, and a relationship between qualitative and quantitative beliefs in accordance with the Lockean thesis. In this paper, it is shown that these conditions are inconsistent with each of three further rationality conditions: fallibilism, open-mindedness, and invariance under independent conceptual expansions. Restrictions of the Lockean (...) thesis that have been suggested in the literature cannot remove the inconsistency. In the outlook we discuss two alternative ways of dealing with this problem: restricting conjunctive closure or going for a dual system account. (shrink)

It is certainly well accepted that formal models play a key role in scientific job. Its use goes from natural sciences like physics and even to social sciences like economics and politics. Using mathematics allows the researcher to consider more complicated scenarios involving several variables. Some models are developed to make predictions, others to describe a phenomena, or just to improve the explanation of events in the world. But what has all this to do with philosophy? The aim of the (...) present paper is to investigate debates on the role of formal models in a specific philosophical subject, precisely, the epistemology of rationality. Are we able to explain why models are needed in epistemological work? This answer will be addressed on the assumptions that epistemological theorizing is committed with normative statements. More specifically, epistemologists are concerned with normative questions about what rationality requires from epistemic agents. The first goal is to discuss some assumptions about the role of mathematical models in formal epistemology undertaking. And secondly, I will argue for the following two claims: formal models are useful tools for predicting consequences of normative assumption about what is intuitively required by rationality; and insofar rationality theory is normative in virtue of being instrumentalist and aiming at truth, formal models are means-end tools, therefore, for rationality, mathematical models are devices for maximizing truth in doxastic states. (shrink)

There are two prominent ways of formally modelling human belief. One is in terms of plain beliefs, i.e., sets of propositions. The second one is in terms of degrees of beliefs, which are commonly taken to be representable by subjective probability functions. In relating these two ways of modelling human belief, the most natural idea is a thesis frequently attributed to John Locke: a proposition is or ought to be believed just in case its subjective probability exceeds a contextually fixed (...) probability threshold \. This idea is known to have two serious drawbacks: first, it denies that beliefs are closed under conjunction, and second, it may easily lead to sets of beliefs that are logically inconsistent. In this paper I present two recent accounts of aligning plain belief with subjective probability: the Stability Theory of Leitgeb (2013, 2014, 2015, 2017) and the Probalogical Theory of Lin and Kelly (2012a, 2012b). I argue that Leitgeb’s theory may be too sceptical for the purposes of real life. (shrink)

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket (...) n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method B p , which specifies an initial belief state B p (T) that is revised to the new propositional belief state B(E) upon receipt of information E. An acceptance rule tracks Bayesian conditioning when B p (E) = B p|E (T), for (...) every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The "Lockean" rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1-2):221-262, 1975; Alchourrón et al., J Symb Log 50:510-530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987). (shrink)

This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, this essay will show that this combination of principles is satisfiable (and indeed nontrivially so) and that the principles (...) are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as will be argued in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc and that has various philosophical features that are plausible independently. In particular, this essay shows that the theory allows for a solution to the Lottery Paradox, and it has nice applications to formal epistemology. The price that is to be paid for this theory is a strong dependency of belief on the context, where a context involves both the agent's degree of belief function and the partitioning or individuation of the underlying possibilities. But as this essay argues, that price seems to be affordable. (shrink)

Is it possible to give an explicit definition of belief in terms of subjective probability, such that believed propositions are guaranteed to have a sufficiently high probability, and yet it is neither the case that belief is stripped of any of its usual logical properties, nor is it the case that believed propositions are bound to have probability 1? We prove the answer is ‘yes’, and that given some plausible logical postulates on belief that involve a contextual “cautiousness” threshold, there (...) is but one way of determining the extension of the concept of belief that does the job. The qualitative concept of belief is not to be eliminated from scientific or philosophical discourse, rather, by reducing qualitative belief to assignments of resiliently high degrees of belief and a “cautiousness” threshold, qualitative and quantitative belief turn out to be governed by one unified theory that offers the prospects of a huge range of applications. Within that theory, logic and probability theory are not opposed to each other but go hand in hand. (shrink)

A new justification of probabilism is developed that pays close attention to the structure of the underlying space of possibilities. Its central assumption is that rational numerical degrees of bel...

Many moral theories are committed to the idea that some kinds of moral considerations should be respected, whatever the cost to ‘lesser’ types of considerations. A person's life, for instance, should not be sacrificed for the trivial pleasures of others, no matter how many would benefit. However, according to the decision-theoretic critique of lexical priority theories, accepting lexical priorities inevitably leads us to make unacceptable decisions in risky situations. It seems that to operate in a risky world, we must reject (...) lexical priorities altogether. This paper argues that lexical priority theories can, in fact, offer satisfactory guidance in risky situations. It does so by equipping lexical priority theories with overlooked resources from decision theory. (shrink)

This paper defends the view, put roughly, that to think that p is to guess that p is the answer to the question at hand, and that to think that p rationally is for one’s guess to that question to be in a certain sense non-arbitrary. Some theses that will be argued for along the way include: that thinking is question-sensitive and, correspondingly, that ‘thinks’ is context-sensitive; that it can be rational to think that p while having arbitrarily low credence (...) that p; that, nonetheless, rational thinking is closed under entailment; that thinking does not supervene on credence; and that in many cases what one thinks on certain matters is, in a very literal sense, a choice. Finally, since there are strong reasons to believe that thinking just is believing, there are strong reasons to think that all this goes for belief as well. (shrink)

The strongest possible Lewisian triviality result for the indicative conditional is proven.

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)

This essay discusses the difficulty to reconcile two paradigms about beliefs: the binary or categorical paradigm of yes/no beliefs and the probabilistic paradigm of degrees of belief. The possibility for someone to hold both types of belief simultaneously is challenged by the lottery paradox, and more recently by a general impossibility theorem by Dietrich and List (2018, 2021). The nature, relevance, and implications of the tension are explained and assessed.

We have reasons to want an epistemology of simple belief in addition to the Bayesian notion of belief which admits of degree. Accounts of simple belief which attempt to reduce it to the notion of credence all face difficulties. We argue that each conception captures an important aspect of our pre-theoretic thinking about epistemology; the differences between the two accounts of belief stem from two different conceptions of unlikelihood. On the one hand there is unlikelihood in the sense of improbability, (...) on the other hand there is unlikelihood in the sense of far-fetchedness. A non-reductive account of simple belief is outlined. Belief aims not just at truth, but at attaining the status of knowledge, and knowledge should satisfy the weak modal principle: If S knows that p then S is certain that there is no possibility very close to actuality at which p is false. The account faces a difficulty in dealing with statistical inductive cases. We sketch a speculative strategy for dealing with such cases, based on the pragmatic considerations that lead to an agent’s partition of the space of possibilities and a nonprobabilistic notion of the “estimated distance” of elements of such a partition from actuality. (shrink)

The so-called Preface Paradox seems to show that one can rationally believe two logically incompatible propositions. We address this puzzle, relying on the notions of truthlikeness and approximate truth as studied within the post-Popperian research programme on verisimilitude. In particular, we show that adequately combining probability, approximate truth, and truthlikeness leads to an explanation of how rational belief is possible in the face of the Preface Paradox. We argue that our account is superior to other solutions of the paradox, including (...) a recent one advanced by Hannes Leitgeb (Analysis 74.1). (shrink)

This paper is about the statics and dynamics of belief states that are represented by pairs consisting of an agent's credences (represented by a subjective probability measure) and her categorical beliefs (represented by a set of possible worlds). Regarding the static side, we argue that the latter proposition should be coherent with respect to the probability measure and that its probability should reach a certain threshold value. On the dynamic side, we advocate Jeffrey conditionalisation as the principal mode of changing (...) one's belief state. This updating method fits the idea of the Lockean Thesis better than plain Bayesian conditionalisation, and it affords a flexible method for adding and withdrawing categorical beliefs. We show that it fails to satisfy the traditional principles of Inclusion and Preservation for belief revision and the principle of Recovery for belief withdrawals{, as well as the Levi and Harper identities. We take this to be a problem for the latter principles rather than for the idea of coherent belief change. (shrink)

According to the so-called Lockean thesis, a rational agent believes a proposition just in case its probability is sufficiently high, i.e., greater than some suitably fixed threshold. The Preface paradox is usually taken to show that the Lockean thesis is untenable, if one also assumes that rational agents should believe the conjunction of their own beliefs: high probability and rational belief are in a sense incompatible. In this paper, we show that this is not the case in general. More precisely, (...) we consider two methods of computing how probable must each of a series of propositions be in order to rationally believe their conjunction under the Lockean thesis. The price one has to pay for the proposed solutions to the paradox is what we call “quasi-dogmatism”: the view that a rational agent should believe only those propositions which are “nearly certain” in a suitably defined sense. (shrink)

I argue that we should solve the Lottery Paradox by denying that rational belief is closed under classical logic. To reach this conclusion, I build on my previous result that (a slight variant of) McGee’s election scenario is a lottery scenario (see Lissia 2019). Indeed, this result implies that the sensible ways to deal with McGee’s scenario are the same as the sensible ways to deal with the lottery scenario: we should either reject the Lockean Thesis or Belief Closure. After (...) recalling my argument to this conclusion, I demonstrate that a McGee-like example (which is just, in fact, Carroll’s barbershop paradox) can be provided in which the Lockean Thesis plays no role: this proves that denying Belief Closure is the right way to deal with both McGee’s scenario and the Lottery Paradox. A straightforward consequence of my approach is that Carroll’s puzzle is solved, too. (shrink)

According to the Lockean thesis, a proposition is believed just in case it is highly probable. While this thesis enjoys strong intuitive support, it is known to conflict with seemingly plausible logical constraints on our beliefs. One way out of this conflict is to make probability 1 a requirement for belief, but most have rejected this option for entailing what they see as an untenable skepticism. Recently, two new solutions to the conflict have been proposed that are alleged to be (...) non-skeptical. We compare these proposals with each other and with the Lockean thesis, in particular with regard to the question of how much we gain by adopting any one of them instead of the probability 1 requirement, that is, of how likely it is that one believes more than the things one is fully certain of. (shrink)

Many argue that absolutist moral theories -- those that prohibit particular kinds of actions or trade-offs under all circumstances -- cannot adequately account for the permissibility of risky actions. In this dissertation, I defend various versions of absolutism against this critique, using overlooked resources from formal decision theory. Against the prevailing view, I argue that almost all absolutist moral theories can give systematic and plausible verdicts about what to do in risky cases. In doing so, I show that critics have (...) overlooked: (1) the fact that absolutist theories -- and moral theories, more generally -- underdetermine their formal decision-theoretic representations; (2) that decision theories themselves can be generalised to better accommodate distinctively absolutist commitments. Overall, this dissertation demonstrates that we can navigate a risky world without compromising our moral commitments. (shrink)