The standard axiomatization of quantum mechanics (QM) is not fully explicit about the role of the time-parameter. Especially, the time reference within the probability algorithm (the Born Rule, BR) is unclear. From a probability principle P1 and a second principle P2 affording a most natural way to make BR precise, a logical conflict with the standard expression for the completeness of QM can be derived. Rejecting P1 is implausible. Rejecting P2 leads to unphysical results and to a conflict with a (...) generalization of P2, a principle P3. All three principles are shown to be without alternative. It is thus shown that the standard expression of QM completeness must be revised. An absolutely explicit form of the axioms is provided, including a precise form of the projection postulate. An appropriate expression for QM completeness, reflecting the restrictions of the Gleason and Kochen-Specker theorems is proposed. (shrink)

James Joyce's article “A Nonpragmatic Vindication of Probabilism” introduced an approach to arguing for credal norms by appealing to the epistemic value of accuracy. The central thought was that credences ought to accurately represent the world, a guiding thought that has gone on to generate an entire research paradigm on the rationality of credences. Recently, a number of epistemologists have begun to apply this same thought to full beliefs, attempting to explain and argue for norms of belief in terms of (...) epistemic value. This paper examines these recent attempts, showing how they interact with work on the accuracy of credences. It then examines how differing judgments about epistemic value give rise to distinct rational requirements for belief, concluding by considering some of the fundamental questions and issues yet to be fully explored. (shrink)

In this paper, I examine Plantinga’s (1993, 2000, 2011) Evolutionary Argument Against Naturalism (EAAN). While there has been much discussion about Plantinga’s use of probabilities in the argument, I contend that insufficient attention has been paid to the question of how we are to interpret those probabilities. In this paper, I argue that views Plantinga defends elsewhere limit the range of interpretations available to him here. The upshot is that the EAAN is more limited in its applicability than Plantinga alleges.

We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework.

Rescorla explores the relation between Reflection, Conditionalization, and Dutch book arguments in the presence of a weakened concept of sure loss and weakened conditions of self‐transparency for doxastic agents. The literature about Reflection and about Dutch Book arguments, though overlapping, are distinct, and its history illuminates the import of Rescorla's investigation. With examples from a previous debate in the 70s and results about Reflection and Conditionalization in the 80s, I propose a way of seeing the epistemic enterprise in the light (...) of practical requirements to be met by demands for synchronic coherence and probability updating policies. This includes a defense of principles rejected by Rescorla, while allowing for the value of his results in the borderland between theories of cognition and formal epistemology. (shrink)

Personal probability is now a familiar subject in epistemology, together with such more venerable notions as knowledge and belief. But there are severe strains between probability and belief; if either is taken as the more basic, the other may suffer. After explaining the difficulties of attempts to accommodate both, I shall propose a unified account which takes conditional personal probability as basic. Full belief is therefore a defined, derivative notion. Yet we will still be able to picture opinion as follows: (...) my subjective probability is only a grading of the possibilities left open by my beliefs. My conditional probabilities generally derive from the strongest belief I can maintain when admitting the relevant condition. (shrink)

Legal probabilism is the view that juridical fact-finding should be modeled using Bayesian methods. One of the alternatives to it is the narration view, according to which instead we should conceptualize the process in terms of competing narrations of what happened. The goal of this paper is to develop a reconciliatory account, on which the narration view is construed from the Bayesian perspective within the framework of formal Bayesian epistemology.

This paper focuses on the view that rationality requires that our credences be regular. I go through different formulations of the requirement, and show that they face several problems. I then formulate a version of the requirement that solves most of, if not all, these problems. I conclude by showing that an argument thought to support the requirement as traditionally formulated actually does not; if anything, the argument, slightly modified, supports my version of the requirement.Send article to KindleTo send this (...) article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.REGULARITY REFORMULATEDVolume 9, Issue 4Weng Hong TangDOI: https://doi.org/10.1017/epi.2012.23Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. REGULARITY REFORMULATEDVolume 9, Issue 4Weng Hong TangDOI: https://doi.org/10.1017/epi.2012.23Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. REGULARITY REFORMULATEDVolume 9, Issue 4Weng Hong TangDOI: https://doi.org/10.1017/epi.2012.23Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

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)

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure (...) as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

There is consensus among computer scientists, logicians, and philosophers that good reasoning with qualitative beliefs must have the structural property of cumulative transitivity or, for short, cut. This consensus is typically explicitly argued for partially on the basis of practical and mathematical considerations. But the consensus is also implicit in the approach philosophers take to almost every puzzle about reasoning that involves multiple steps: philosophers typically assume that if each step in reasoning is acceptable considered on its own, the whole (...) chain of reasoning must also acceptable. In this paper I focus on whether there are good philosophical reasons for thinking that the consensus that good reasoning must satisfy cut is true. My central claim is that we should not accept the consensus—good reasoning might not satisfy cut. In particular, I consider four arguments for the consensus and explain why they are unpersuasive. (§2-5). I then show that the issue of whether good reasoning is cut turns on a substantive yet until now unnoticed question in epistemology (§6). (shrink)

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on (...) the qualitative part of probabilistic inference leads to a very broad class of 'proto-probability' functions. (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 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)

What is the proper attitude toward what is expressed by a vague sentence in the face of borderline evidence? Some call this attitude “ambivalence” and distinguish it from uncertainty. It has been argued that Classical Epistemicism conjoined with classical probability theory fails to characterize this attitude, and that we must therefore abandon classical logic or classical probabilities in the presence of vagueness. In this paper, I give a characterization of ambivalence assuming a supervaluationist semantics for vague terms that does not (...) revise either. The theory, which I call the theory of Superprobabilities, identifies the proper attitude toward a vague sentence, in the presence of exact borderline evidence, as the set of classical probabilities of the evidence on each member of the set of all precisifications of a vague sentence. I defend the use of sets of probabilities against objections by generalizing the theory of Superprobabilities to a decision theory called Superrationality. I then compare the merits of the theory of Superprobabilities to Classical Epistemicism and nonclassical probabilities theories with respect to the problem of ambivalence. (shrink)

One interpretation of the conditional If P then Q is as saying that the probability of Q given P is high. This is an interpretation suggested by Adams (1966) and pursued more recently by Edgington (1995). Of course, this probabilistic conditional is nonmonotonic, that is, if the probability of Q given P is high, and R implies P, it need not follow that the probability of Q given R is high. If we were confident of concluding Q from the fact (...) that we knew P, and we have stronger information R, we can no longer be confident of Q. We show nonetheless that usually we would still be justified in concluding Q from R. In other words, probabilistic conditionals are mostly monotonic. (shrink)

Sometimes epistemologists theorize about belief, a tripartite attitude on which one can believe, withhold belief, or disbelieve a proposition. In other cases, epistemologists theorize about credence, a fine-grained attitude that represents one’s subjective probability or confidence level toward a proposition. How do these two attitudes relate to each other? This article explores the relationship between belief and credence in two categories: descriptive and normative. It then explains the broader significance of the belief-credence connection and concludes with general lessons from the (...) debate thus far. (shrink)

This guide accompanies the following article(s): Jackson, E., Philosophy Compass 15/6 (2020) pp. 1-13 10.1111/phc3.12668.x.

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. (...) The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change. (shrink)

Vague subjective probability may be modeled by means of a set of probability functions, so that the represented opinion has only a lower and upper bound. The standard rule of conditionalization can be straightforwardly adapted to this. But this combination has difficulties which, though well known in the technical literature, have not been given sufficient attention in probabilist or Bayesian epistemology. Specifically, updating on apparently irrelevant bits of news can be destructive of one’s explicitly prior expectations. Stability of vague subjective (...) opinion appears to need a more complex model. (shrink)

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

Doxastic TheoriesThe application of formal tools to questions related to epistemology is of course not at all new. However, there has been a surge of interest in the field now known as “formal epistemology” over the past decade, with two annual conference series and an annual summer school at Carnegie Mellon University, in addition to many one-off events devoted to the field. A glance at the programs of these series illustrates the wide-ranging set of topics that have been grouped under (...) this name, ranging from rational choice theory and the foundations of statistics, to logics of knowledge and formal measures of coherence, with much more besides.In this paper I will ignore most of these topics, and just trace some parts of the history of two ideas about belief whose current interaction may lead to future progress. One idea is the idea of belief, disbelief, and suspension of judgment as a meaningful tripartite dis .. (shrink)

As part of an exceptionally lucid analysis of the Lottery Paradox, Dana Nelkin castigates the solutions to that paradox put forward by Laurence Bonjour and Sharon Ryan. According to her, these are “so finely tailored to lottery-like cases that they are limited in their ability to explain [what seem the intuitively right responses to such cases]”. She then offers a solution to the Lottery Paradox that allegedly has the virtue of being independently motivated by our intuitions regarding certain non-lottery-like cases. (...) This note argues that Nelkin fails to show that her solution applies to other than lottery-like cases or in any event that it is more general than Bonjour’s and Ryan’s solutions. (shrink)

As part of an exceptionally lucid analysis of the Lottery Paradox, Dana Nelkin castigates the solutions to that paradox put forward by Laurence Bonjour and Sharon Ryan. According to her, these are “so finely tailored to lottery-like cases that they are limited in their ability to explain [what seem the intuitively right responses to such cases]”. She then offers a solution to the Lottery Paradox that allegedly has the virtue of being independently motivated by our intuitions regarding certain non-lottery-like cases. (...) This note argues that Nelkin fails to show that her solution applies to other than lottery-like cases or in any event that it is more general than Bonjour’s and Ryan’s solutions. (shrink)

It is widely believed that the so-called knowledge account of assertion best explains why sentences such as “It’s raining in Paris but I don’t believe it” and “It’s raining in Paris but I don’t know it” appear odd to us. I argue that the rival rational credibility account of assertion explains that fact just as well. I do so by providing a broadly Bayesian analysis of the said type of sentences which shows that such sentences cannot express rationally held beliefs. (...) As an interesting aside, it will be seen that these sentences also harbor a lesson for Bayesian epistemology itself. (shrink)

This paper aims to flesh out the celebrated notion of reflective equilibrium within a probabilistic framework for epistemic rationality. On the account developed here, an agent's attitudes are in reflective equilibrium when there is a certain sort of harmony between the agent's credences, on the one hand, and what the agent accepts, on the other hand. Somewhat more precisely, reflective equilibrium is taken to consist in the agent accepting, or being prepared to accept, all and only claims that follow from (...) a maximally comprehensive theory that is more probable than any other such theory. Drawing on previous work, the paper shows that when an agent is in reflective equilibrium in this sense, the set of claims they accept or are prepared to accept is bound to be logically consistent and closed under logical implication. The paper also argues that this account can explain various features of philosophical argumentation in which the notion of reflective equilibrium features centrally, such as the emphasis on evaluating philosophical theories holistically rather than in a piecemeal fashion. (shrink)

Deductive Cogency holds that the set of propositions towards which one has, or is prepared to have, a given type of propositional attitude should be consistent and closed under logical consequence. While there are many propositional attitudes that are not subject to this requirement, e.g. hoping and imagining, it is at least prima facie plausible that Deductive Cogency applies to the doxastic attitude involved in propositional knowledge, viz. belief. However, this thought is undermined by the well-known preface paradox, leading a (...) number of philosophers to conclude that Deductive Cogency has at best a very limited role to play in our epistemic lives. I argue here that Deductive Cogency is still an important epistemic requirement, albeit not as a requirement on belief. Instead, building on a distinction between belief and acceptance introduced by Jonathan Cohen and recent developments in the epistemology of understanding, I propose that Deductive Cogency applies to the attitude of treating propositions as given in the context of attempting to understand a given phenomenon. I then argue that this simultaneously accounts for the plausibility of the considerations in favor of Deductive Cogency and avoids the problematic consequences of the preface paradox. (shrink)

It is a prevalent, if not popular, thesis in the metaphysics of belief that facts about an agent’s beliefs depend entirely upon facts about that agent’s underlying credal state. Call this thesis ‘credal reductivism’ and any view that endorses this thesis a ‘credal reductivist view’. An adequate credal reductivist view will accurately predict both when belief occurs and which beliefs are held appropriately, on the basis of credal facts alone. Several well-known—and some lesser known—objections to credal reductivism turn on the (...) inability of standard credal reductivist views to get the latter, normative, results right. This paper presents and defends a novel credal reductivist view according to which belief is a type of “imprecise credence” that escapes these objections by including an extreme credence of 1. (shrink)

I develop a strategy for representing epistemic states and epistemic changes that seeks to be sensitive to the difference between voluntary and involuntary aspects of our epistemic life, as well as to the role of pragmatic factors in epistemology. The model relies on a particular understanding of the distinction between full belief and acceptance , which makes room for the idea that our reasoning on both practical and theoretical matters typically proceeds in a contextual way. Within this framework, I discuss (...) how agents can rationally shift their credal probability functions so as to consciously modify some of their contextual acceptances; the present account also allows us to represent how the very set of contexts evolves. Voluntary credal shifts, in turn, might provoke changes in the agent’s beliefs, but I show that this is actually a side effect of performing multiple adjustments in the total lot of the agent’s acceptance sets. In this way we obtain a model that preserves many pre-theoretical intuitions about what counts as adequate rationality constraints on our actual practices—and hence about what counts as an adequate, normative epistemological perspective. (shrink)

An important trend in contemporary epistemology centers on elaborating an old idea of pragmatist pedigree: theory selection (and in general the process of changing view and fixing beliefs) presupposes epistemic values. This article focuses on analyzing the case where epistemic values are indeterminate or when the sources of valuation are multiple (epistemic values like coherence and simplicity need not order options in compatible ways). According to the theory that thus arises epistemic alternatives need not be fully ordered by an underlying (...) notion of information-value and therefore the usual economic techniques of optimization cannot be applied in order to compute optimal contractions. But in cases of this sort it is still rational to maximize, i.e. to deem an option as choosable when it is not known to be worse that any other. We present here basic results about a notion of liberal contraction based on maximizing quasi-orderings. This requires the previous solution of some open problems in the theory of rational choice functions, namely a full characterization of choice functions rationalizable in terms of maximization of quasi-transitive relations. We conclude by discussing the problem of what is the adequate feasible set for calculating maximizing solutions for contraction problems and by considering the epistemological roots of some counterexamples against the most fundamental axioms on choice functions (like α). While the first part of the paper shows how economic insights can be used to improve our understanding of the principles of belief formation and change, this final section reverses this strategy by showing the utility of epistemological insights and techniques for providing invariance conditions capable of regulating the applicability of the pure principles of choice. (shrink)

The recent literature offers several models of the notion of matter of fact supposition1 revealed in the acceptance of the so-called indicative conditionals. Some of those models are qualitative [Collins 90], [Levi 96], [Stalnaker 84]. Other probabilistic models appeal either to infinitesimal probability or two place probability functions. Recent work has made possible to understand which is the exact qualitative counterpart of the latter probabilistic models. In this article we show that the qualitative notion of change that thus arises is (...) hypothetical revision, a notion previously axiomatized in [Arló-Costa 97] and [Arló-Costa & Thomason 96]. This notion is incompatible with AGM as well as with other standard methods of theory change. The way in which matter-of-fact supposition is modeled by hypothetical revision is illustrated via examples. The model is compared with other qualitative accounts of the notion of supposition encoded in two-place probability functions, with models of subjunctive supposition, as well as with some of the well know models of learning. Applications in knowledge representation and in the theory of games and decisions are summarized. (shrink)

In this paper, I develop a syntactic framework for the analysis ofstrategic form games that is based on a straightforward combination ofstandard systems of doxastic, probabilistic and conditionalpropositional logic. In particular, for the probabilistic part I makeuse of the axiomatization provided in Fagin and Halpern (1994). The use ofconditionals allows to represent a strategic form game by a logicalformula in a very natural way. Also expected utility maximization can benaturally captured. I use this framework to prove a version of a (...) resulton Nash equilibrium conjectures first presented in Aumann and Brandenburger (1995). (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)

Here I explore a new line of evidence for belief–credence dualism, the thesis that beliefs and credences are distinct and equally fundamental types of mental states. Despite considerable recent disagreement over this thesis, little attention has been paid in philosophy to differences in how our mindreading systems represent the beliefs and credences of others. Fascinatingly, the systems we rely on to accurately and efficiently track others’ mental states appear to function like belief–credence dualists: Credence is tracked like an emotional state, (...) composed of both representational and affective content, whereas belief is tracked like a bare representational state with no affective component. I argue on a preliminary basis that, in this particular case, the mechanics of mentalizing likely pick out a genuine affective dimension to credence that is absent for belief, further strengthening the converging case for belief–credence dualism. (shrink)

There is some consensus on the claim that imagination as suppositional thinking can have epistemic value insofar as it’s constrained by a principle of minimal alteration of how we know or believe reality to be – compatibly with the need to accommodate the supposition initiating the imaginative exercise. But in the philosophy of imagination there is no formally precise account of how exactly such minimal alteration is to work. I propose one. I focus on counterfactual imagination, arguing that this can (...) be modeled as simulated belief revision governed by Laplacian imaging. So understood, it can be rationally justified by accuracy considerations: it minimizes expected belief inaccuracy, as measured by the Brier score. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

ABSTRACT: I shall consider the phenomenon of believing ourselves to have at least one false belief: a phenomenon I call believed fallibility I shall first present a paradoxical argument which appears to show that believed fallibility is incoherent; second, note that this argument assumes that we are committed to the conjunction of all our beliefs; third, sketch a more intuitive notion of commitment in which we are not committed to the conjunction of all our beliefs and argue that the original (...) paradoxical argument is now defused; fourth, construct a new paradoxical argument showing that if we believe that we have at least one false belief we are committed to a contradiction, employing the preferable notion of commitment; and, fifth, suggest that perhaps we might avoid the new paradox by denying that closing our beliefs under conjunction is required by rationalityRÉSUMÉ: Je me pencherai ici sur le phénomène qui consiste à penser que nous avons au moins une opinion fausse: je nomme ce phénoméne «la croyance d’etre faillible». Je présente en premier lieu un argument paradoxal qui semble démontrer que la croyance d’être faillible est incohérente. Deuxièmement, je remarque que cet argument suppose que nous nous obligions à réaliser la conjonction de toutes nosopinions. Troisièmement, je propose une version plus intuitive de cette obligation selon laquelle nous ne sommes pas obligés de réaliser la conjonction de toutes nos opinions, et je soutiens que le paradoxe initial est alors désamorcé. Quatrièmement, je construis un nouvel argument paradoxal démontrant que, si nous croyons que nous avons au moins une opinion fausse, nous sommes obligés de nous contredire, selon la notion d’obligation que je préconise d’adopter. Cinquièmement, je suggère que nous pourrions peut-être éviter ce nouveau paradoxe en niant que la rationalité exige de soumettre nos opinions à la conjonction. (shrink)

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief (...) over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...) logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on hereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like (...) FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

ABSTRACT: I shall consider the phenomenon of believing ourselves to have at least one false belief: a phenomenon I call believed fallibility I shall first present a paradoxical argument which appears to show that believed fallibility is incoherent; second, note that this argument assumes that we are committed to the conjunction of all our beliefs; third, sketch a more intuitive notion of commitment in which we are not committed to the conjunction of all our beliefs and argue that the original (...) paradoxical argument is now defused; fourth, construct a new paradoxical argument showing that if we believe that we have at least one false belief we are committed to a contradiction, employing the preferable notion of commitment; and, fifth, suggest that perhaps we might avoid the new paradox by denying that closing our beliefs under conjunction is required by rationalityRÉSUMÉ: Je me pencherai ici sur le phénomène qui consiste à penser que nous avons au moins une opinion fausse: je nomme ce phénoméne «la croyance d’etre faillible». Je présente en premier lieu un argument paradoxal qui semble démontrer que la croyance d’être faillible est incohérente. Deuxièmement, je remarque que cet argument suppose que nous nous obligions à réaliser la conjonction de toutes nosopinions. Troisièmement, je propose une version plus intuitive de cette obligation selon laquelle nous ne sommes pas obligés de réaliser la conjonction de toutes nos opinions, et je soutiens que le paradoxe initial est alors désamorcé. Quatrièmement, je construis un nouvel argument paradoxal démontrant que, si nous croyons que nous avons au moins une opinion fausse, nous sommes obligés de nous contredire, selon la notion d’obligation que je préconise d’adopter. Cinquièmement, je suggère que nous pourrions peut-être éviter ce nouveau paradoxe en niant que la rationalité exige de soumettre nos opinions à la conjonction. (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)