Rational agents seem more confident in any possible event than in an impossible event. But if rational credences are real-valued, then there are some possible events that are assigned 0 credence nonetheless. How do we differentiate these events from impossible events then when we order events? de Finetti (1975), Hájek (2012) and Easwaran (2014) suggest that when ordering events, conditional credences and subset relations are as relevant as unconditional credences. I present a counterexample to all their proposals in this (...) paper. While their proposals order possible and impossible events correctly, they deliver the wrong verdict for disjoint possible events assigned equal positive credence. (shrink)

There is no set Δ of probability axioms that meets the following three desiderata: (1) Δ is vindicated by a Dutch book theorem; (2) Δ does not imply regularity (and thus allows, among other things, updating by conditionalization); (3) Δ constrains the conditional probability q(·,z) even when the unconditional probability p(z) (=q(z,T)) equals 0. This has significant consequences for Bayesian epistemology, some of which are discussed.

This paper (based on joint work with M.J.Schervish and J.B.Kadane) discusses some differences between the received theory of regular conditional distributions, which is the countably additive theory of conditional probability, and a rival theory of conditional probability using the theory of finitely additive probability. The focus of the paper is maximally "improper" conditional probability distributions, where the received theory requires, in effect, that P{a: P(a|a) = 0} = 1. This work builds (...) upon the results of Blackwell and Dubins (1975). (shrink)

This paper discusses some differences between the received theory of regular conditional distributions, which is the countably additive theory of conditional probability, and a rival theory of conditional probability using the theory of finitely additive probability. The focus of the paper is maximally "improper" conditional probability distributions, where the received theory requires, in effect, that P{a: P = 0} = 1. This work builds upon the results of Blackwell and Dubins.

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional probability (...) _3.3_ The final axiom of NAP _3.4_ Infinite sums _3.5_ Definition of NAP functions via infinite sums _3.6_ Relation to numerosity theory _4_ Objections and Replies _4.1_ Cantor and the Archimedean property _4.2_ Ticket missing from an infinite lottery _4.3_ Williamson’s infinite sequence of coin tosses _4.4_ Point sets on a circle _4.5_ Easwaran and Pruss _5_ Dividends _5.1_ Measure and utility _5.2_ Regularity and uniformity _5.3_ Credence and chance _5.4_ Conditional probability _6_ General Considerations _6.1_ Non-uniqueness _6.2_ Invariance Appendix. (shrink)

A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity , is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a (...) natural weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example. (shrink)

Bayesians since Savage (1972) have appealed to asymptotic results to counter charges of excessive subjectivity. Their claim is that objectionable differences in prior probability judgments will vanish as agents learn from evidence, and individual agents will converge to the truth. Glymour (1980), Earman (1992) and others have voiced the complaint that the theorems used to support these claims tell us, not how probabilities updated on evidence will actually}behave in the limit, but merely how Bayesian agents believe they will behave, (...) suggesting that the theorems are too weak to underwrite notions of scientific objectivity and intersubjective agreement. I investigate, in a very general framework, the conditions under which updated probabilities actually converge to a settled opinion and the conditions under which the updated probabilities of two agents actually converge to the same settled opinion. I call this mode of convergence deterministic, and derive results that extend those found in Huttegger (2015b). The results here lead to a simple characterization of deterministic convergence for Bayesian learners and give rise to an interesting argument for what I call strong regularity, the view that probabilities of non-empty events should be bounded away from zero. (shrink)

This paper contributes to a recent research program that extends arguments supporting elementary conditionalization to arguments supporting conditionalization with general, measure-theoretic conditional probabilities. I begin by suggesting an amendment to the framework that Rescorla (2018) has used to characterize regular conditional probabilities in terms of avoiding Dutch book. If we wish to model learning scenarios in which an agent gains complete membership knowledge about some subcollection of the events of interest to her, then we should focus on (...) updating policies that are what I shall call proper. I go on to characterize regular conditional probabilities in proper learning scenarios using what van Fraassen (1999) calls The General Reflection Principle. (shrink)

Some philosophers, most notably Hempel and Salmon, have tried to reduce explanation to probability by proposing analyses of explanation in probabilistic terms. Hempel claims, roughly, that a hypothesis H explains a datum D if and only if the conditional probability P is close to 1. It is well known that such an account fails in cases where H is irrelevant for D. Even though it is highly likely that Tom will not become pregnant, given that he regularly (...) takes his wife’s birth control pills, the latter does not explain the former. Neither does an idea work which is in the proximity of Salmon’s, namely, that H explains D if and only if P > P. Suppose Susan swallows a pound of arsenic in order to commit suicide. Shortly after, however, she dies because she is run over by a bus. The probability of dying, given that one ingests a pound of arsenic, is usually higher than the prior probability of dying. Nonetheless, it is not the arsenic but the collision with the bus which explains Susan’s death. The aforementioned objections are directed against …. (shrink)

Generics and frequency statements are puzzling phenomena: they are lawlike, yet contingent. They may be true even in the absence of any supporting instances, and extending the size of their domain does not change their truth conditions. Generics and frequency statements are parametric on time, but not on possible worlds; they cannot be applied to temporary generalizations, and yet are contingent. These constructions require a regular distribution of events along the time axis. Truth judgments of generics vary considerably across (...) speakers, whereas truth judgments of frequency statements are much more uniform. A generic may be false even if the vast majority of individuals in its domain satisfy the predicated property, whereas a frequency statement using, e.g., usually would be true. This paper argues that all these seemingly unrelated puzzles have a single underlying cause: generics and frequency statements express probability judgments, and these, in turn, are interpreted as statements of hypothetical relative frequency. (shrink)

Jeffrey conditionalization is a rule for updating degrees of belief in light of uncertain evidence. It is usually assumed that the partitions involved in Jeffrey conditionalization are finite and only contain positive-credence elements. But there are interesting examples, involving continuous quantities, in which this is not the case. Q1 Can Jeffrey conditionalization be generalized to accommodate continuous cases? Meanwhile, several authors, such as Kenny Easwaran and Michael Rescorla, have been interested in Kolmogorov’s theory of regular conditional distributions as (...) a possible framework for conditional probability which handles probability-zero events. However the theory faces a major shortcoming: it seems messy and ad hoc. Q2 Is there some axiomatic theory which would justify and constrain the use of rcds, thus serving as a possible foundation for conditional probability? These two questions appear unrelated, but they are not, and this paper answers both. We show that when one appropriately generalizes Jeffrey conditionalization as in Q1, one obtains a framework which necessitates the use of rcds. It is then a short step to develop a general theory which addresses Q2, which we call the theory of extensions. The theory is a formal model of conditioning which recovers Bayesian conditionalization, Jeffrey conditionalization, and conditionalization via rcds as special cases. (shrink)

In a fair finite lottery with n tickets, the probability assigned to each ticket winning is 1/n and no other answer. That is, 1/n is unique. Now, consider a fair lottery over the natural numbers. What probability is assigned to each ticket winning in this lottery? Well, this probability value must be smaller than 1/n for all natural numbers n. If probabilities are real-valued, then there is only one answer: 0, as 0 is the only real and (...) non-negative value that is smaller than 1/n for all natural numbers n. However, this answer seems counter-intuitive, as it violates Regularity: for all possible events A that are assigned a probability, A is assigned a positive probability. It is possible for ticket i to win in the second lottery. So, the set {i} should be assigned a positive probability and not 0. Consequently, in order to satisfy Regularity, probabilities must be allowed to take on `non-real' values, e.g. a positive infinitesimal x that is smaller than 1/n for all natural numbers n. So, what regular probability is assigned to {i} in the second lottery? This probability value must be positive but smaller than 1/n for all natural numbers n. Given the definition of x, x is a correct answer. But x isn't unique, because every other positive infinitesimal is a correct answer as well and there are uncountably many positive infinitesimals. After all, every positive infinitesimal is strictly between 0 and 1/n for all natural numbers n. Nothing about a fair lottery over the natural numbers uniquely determines x as the correct answer and nothing about the lottery rules out other positive infinitesimals as a correct answer as well. What probability to assign to {i} is underdetermined. -/- In this thesis, I explore this difference between fair finite lotteries and fair infinite lotteries. The constraints governing the former uniquely determine the probabilities assigned to events in those lotteries, but this isn't true for the latter lotteries, when Regularity is a constraint on probabilities. Now, although there are uncountably many correct probability values that can be assigned to {i} in the second lottery, there are some applications of probability theory, for which at least one correct probability value must be chosen in order to accomplish that application, e.g. calculation of expected utilities. In this thesis, I spell out sufficient conditions for when a choice like that is problematic. These conditions apply to precise probabilities, imprecise probabilities, and qualitative probabilities. So, I will consider Benci et al. (2018)'s and Pruss (2021)'s suggestion of allowing probabilities to be imprecise, as well as resorting to qualitative probabilities, to overcome the choice problem. I will argue that both don't solve the problem. Furthermore, there's a hitherto unnoticed difficulty in deriving imprecise probabilities by supervaluating over precise `non-real' probability values. To avoid this difficulty, precise probability values should be real-valued. Thus, Regularity cannot be a constraint on probabilities. But as it turns out, dropping Regularity as a constraint and requiring precise probabilities to be real-valued solve the choice problem. (shrink)

This doctoral dissertation investigates the notion of physical necessity. Specifically, it studies whether it is possible to account for non-accidental regularities without the standard assumption of a pre-existent set of governing laws. Thus, it takes side with the so called deflationist accounts of laws of nature, like the humean or the antirealist. The specific aim is to complement such accounts by providing a missing explanation of the appearance of physical necessity. In order to provide an explanation, I recur to fields (...) that have not been appealed to so far in discussions about the metaphysics of laws. Namely, I recur to complex systems’ theory, and to the foundations of statistical mechanics. The explanation proposed is inspired by how complex systems’ theory has elucidated the way patterns emerge, and by the probabilistic explanations of the 2nd law of thermodynamics. More specifically, this thesis studies how some constraints that make no direct reference to the dynamics can be a sufficient condition for obtaining in the long run, with high probability, stable regular behavior. I hope to show how certain metaphysical accounts of laws might benefit from the insights achieved in these other fields. According to the proposal studied in this thesis, some regularities are not accidental not in virtue of an underlying physical necessity. The non-accidental character of certain regular behavior is only due to its overwhelming stability. Thus, from this point of view the goal becomes to explain the stability of temporal patterns without assuming a set of pre-existent guiding laws. It is argued that the stability can be the result of a process of convergence to simpler and stable regularities from a more complex lower level. According to this project, if successful, there would be no need to postulate a (mysterious) intermediate category between logical necessity and pure contingency. Similarly, there would be no need to postulate a (mysterious) set of pre-existent governing laws. Part I of the thesis motivates part II, mostly by arguing why further explanation of the notions of physical necessity and governing laws should be welcomed (chapter 1), and by studying the plausibility of a lawless fundamental level (chapters 2 and 3). Given so, part II develops the explanation of formation of simpler and stable behavior from a more complex underlying level. (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)

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)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

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)

Philosophers investigating the interpretation and use of conditional sentences have long been intrigued by the intuitive correspondence between the probability of a conditional `if A, then C' and the conditional probability of C, given A. Attempts to account for this intuition within a general probabilistic theory of belief, meaning and use have been plagued by a danger of trivialization, which has proven to be remarkably recalcitrant and absorbed much of the creative effort in the area. (...) But there is a strategy for avoiding triviality that has been known for almost as long as the triviality results themselves. What is lacking is a straightforward integration of this approach in a larger framework of belief representation and dynamics. This paper discusses some of the issues involved and proposes an account of belief update by conditionalization. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. (...) A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P be the probability of B on the subjunctive supposition that A. Is P equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do (...) not succeed. On the other hand, I argue that even if we can equate P with P, we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped. (shrink)

This paper argues that the technical notion of conditional probability, as given by the ratio analysis, is unsuitable for dealing with our pretheoretical and intuitive understanding of both conditionality and probability. This is an ontological account of conditionals that include an irreducible dispositional connection between the antecedent and consequent conditions and where the conditional has to be treated as an indivisible whole rather than compositional. The relevant type of conditionality is found in some well-defined group of (...) conditional statements. As an alternative, therefore, we briefly offer grounds for what we would call an ontological reading: for both conditionality and conditional probability in general. It is not offered as a fully developed theory of conditionality but can be used, we claim, to explain why calculations according to the RATIO scheme does not coincide with our intuitive notion of conditional probability. What it shows us is that for an understanding of the whole range of conditionals we will need what John Heil (2003), in response to Quine (1953), calls an ontological point of view. (shrink)

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional (...) probabilities are sufficient to trivialize the semantics. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. (...) A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

Motivated by a recent proof of free choices in linking equations to the experiments they describe, I clarify some relations among purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. I relate conditional probabilities associated with projection-valued measures to conditional density operators identical, in some cases but not in others, to the usual reduced density operators. While (...) a fatal obstacle precludes associating conditional density operators with general non-projective measures, tensor products of general positive operator-valued measures (POVMs) are associated with conditional density operators. This association together with the free choice of probe particles allows a postulate of state reductions to be replaced by a theorem. An application shows an equivalence between one form of quantum key distribution and another with respect to certain eavesdropping attacks. (shrink)

The conditional probability of h given e is commonly claimed to be equal to the probability that h would have if e were learned. Here I contend that this general claim about conditional probabilities is false. I present a counter-example that involves probabilities of probabilities, a second that involves probabilities of possible future actions, and a third that involves probabilities of indicative conditionals. In addition, I briefly defend these counter-examples against charges that the probabilities they involve (...) are illegitimate. (shrink)

Claro R. Ceniza [1927-2001] is arguably one of the best philosophers that the Philippines has ever produced. However, it is quite unfortunate that some of his important contributions are not that well-known. This paper aims to rectify this by presenting an evaluation of his original insights on three outstanding problems in philosophy, viz., the paradoxes of material implication, the nature of probability, and the metaphysics of modality.

Dynamic update of information states is a new paradigm in logicalsemantics. But such updates are also a traditional hallmark ofprobabilistic reasoning. This note brings the two perspectives togetherin an update mechanism for probabilities which modifies state spaces.

Recent research (e.g., Evans & Over, 2004) has provided support for the hypothesis that people evaluate the probability of conditional statements of the form if p then q as the conditional probability of q given p , P( q / p ). The present paper extends this approach to pragmatic conditionals in the form of inducements (i.e., promises and threats) and advice (i.e., tips and warnings). In so doing, we demonstrate a distinction between the truth status (...) of these conditionals and their effectiveness as speech acts. Specifically, while probability judgements of the truth of conditional inducements and advice are highly correlated with estimates of P( q / p ), their perceived effectiveness in changing behaviour instead varies as a function of the conditional probability of q given not-p , P( q / ∼p ). Finally, we show that the conditional probability approach can be extended to predicting inference rates on a conditional reasoning task. (shrink)

Journal of Philosophical Logic 34, 97-119, 2005.

in Probability is the Very Guide of Life: The Philosophical Uses of Chance, eds. Henry Kyburg, Jr. and Mariam Thalos, Open Court. Abridged version in Proceedings of the International Society for Bayesian Analysis 2002.

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a (...) theory called the operator theory and that the rival restrictor theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

Relations between conditional probabilities, revisions of probabilities in the light of new information, and conditions of ideal rationality are discussed herein. The formal character of conditional probabilities, and their significance for epistemic states of agents is taken up. Then principles are considered that would, under certain conditions, equate rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. And lastly the possibility of kinds of ' books ' against known non-conditionalizers is explored, and the (...) question is taken up, What, if anything, would be wrong with a person against whom such a book could be made? (shrink)

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering (...) asymptotic conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

I present new counterexamples to the asymmetry of grounding: we have prima facie reason to think that some conditional probabilities partially ground their inverse conditional probabilities, and vice versa. These new counterexamples may require that we reject the asymmetry of grounding, or alternatively may require that we reject one or more of the assumptions which enable the counterexamples. Either way, by reflecting on these purported counterexamples to grounding asymmetry we learn something important, either about the formal properties of (...) grounding, or about the nature of probability. (shrink)

A simple argument is given that shows that conditional probabilities do not supervene on unconditional probabilities. In particular, one cannot in general define conditional probabilities using the ratio formula P ( U | V ) = P ( U & V )/ P ( U ), or using any more sophisticated method based on unconditional probabilities.

The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. (...) The semantics is also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended. (shrink)

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces (...) as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ϕ entails conditional-free ψ, “if ϕ then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional. (shrink)

Sutton ( 2010 ) claims that on our analysis (2007), the problem in the two-envelope paradox is an error in counterfactual reasoning. In fact, we distinguish two formulations of the paradox, only one of which, on our account, involves an error in conditional reasoning. According to Sutton, it is conditional probabilities rather than subjunctive conditionals that are essential to the problem. We argue, however, that his strategy for assigning utilities on the basis of conditional probabilities leads to (...) absurdity. In addition, we show that a crucial presupposition of Sutton’s argument — namely, that one can know that envelope A contains n simply on the basis of a stipulation — is mistaken. (shrink)

Relations between conditional probabilities, revisions of probabilities in the light of new information, and conditions of ideal rationality are discussed herein. The formal character of conditional probabilities, and their significance for epistemic states of agents is taken up. Then principles are considered that would, under certain conditions, equate rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. And lastly the possibility of kinds of ‘books’ against known non-conditionalizers is explored, and the question is (...) taken up, What, if anything, would be wrong with a person against whom such a book could be made?1. The standard mathematical treatment takes probability functions as basic, and conditional probabilities defined. If P(-) is a probability function, then is its conditional probability function if and only if P(h/e) is defined only for e such that P(e) > 0, and is then equal to P(e & h)/P(e). (shrink)

The appropriateness, or acceptability, of a conditional does not just ‘go with’ the corresponding conditional probability. A condition of dependence is required as well. In this paper a particular notion of dependence is proposed. It is shown that under both a forward causal and a backward evidential reading of the conditional, this appropriateness condition reduces to conditional probability under some natural circumstances. Because this is in particular the case for the so-called diagnostic reading of (...) the conditional, this analysis might help to explain some of Douven and Verbrugge’s empirical observations. (shrink)

We investigate the notion of conditional probability and the quantum mechanical concept of state reduction in the context of GL spaces satisfying the Alfsen-Shultz condition.

The Adams Thesis holds for a conditional → and a probability assignment P if and only if P=P whenever P>0. The restriction ensures that P is well defined by the classical formula P=P/P. Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results, any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. (...) This puts significant limits on how far triviality results can go. (shrink)

An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if . . . , then . . . ” by conditional probabilities and not by the probability of the material implication. An inference rule is probabilistically informative if the coherent probability interval of its conclusion is not (...) necessarily equal to the unit interval [0, 1]. Not all logically valid inference rules are probabilistically informative and vice versa. The relationship between logically valid and probabilistically informative inference rules is discussed and illustrated by examples such as the modus ponens or the affirming the consequent. We propose a method to evaluate the strength of CPL inference. (shrink)