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)

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

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for (...) single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

The philosophy of probability has been alive and well for several decades in Australia and New Zealand. Some distinctive lines of thought have emerged, resonating with broader themes that have come to be associated with Australasian philosophers: realist/objectivist accounts of various theoretical entities; an ongoing concern with logic, including the development of non­classical logics; and enthusiasm for conceptual analysis, rooted in commonsense but informed by science. In this article I concentrate on work by philosophers on the interpretation of probability, its (...) logical foundations, and its philosophical applications.1 My nomination for the earliest major Australasian philosopher of probability may surprise some readers: Karl Popper. He counts as Australasian by dint of his employment at the University of Canterbury from 1937 until the end of World War II; he counts as a major philosopher of probability by any estimation. Two of his contributions have initiated research programs in the foundations of probability that are still thriving: his (1959a) axiomatization of primitive conditional probability functions (so­called ‘Popper functions’), and his ‘propensity’ interpretation of probability (1959b), intended to illuminate single­case attributions of objective probabilities, as are putatively found in quantum mechanics. (shrink)

This paper examines definitions of independence for events and variables in the context of full conditional measures; that is, when conditional probability is a primitive notion and conditioning is allowed on null events. Several independence concepts are evaluated with respect to graphoid properties; we show that properties of weak union, contraction and intersection may fail when null events are present. We propose a concept of “full” independence, characterize the form of a full conditional measure under full (...) independence, and suggest how to build a theory of Bayesian networks that accommodates null events. (shrink)

This chapter contains sections titled: Introduction The Mathematical Theory of Probability The Philosophical Theory of Probability Conclusion References.

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference (...) class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains. (shrink)

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions (...) and hence require the Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

Which and how much of our current drives –individually and as a global community– are driven by ancestral, inherited traits or imprinted on our animal condition? An attempt to approximate this intriguing query is explored here. It pertains to our identity, social constructions, and our ecological interaction. The origin of our species has its roots in ancestral habits, behaviors and a survival drive, transformed from changing environmental conditions. We were not born in a mother-of-pearl cradle nor were protected by magical (...) agents. We grew on the thread of time within the complex nature of bio-ecological interactions. They carved our identity, our genetic expression and the possible origin of our beliefs, resulting in an arch of various behaviors and cultural phenotypes. The biological nature of our construction dragged through millennia through species trials with variable rates of survival times provided traces of inconstant and multiple evolutive chains. Our modern cultural contexts –norms, priorities, values– appear as newly born. Yet, socio-cultural ecology upon which our brain organization and behavioral constructs derive include shared basic behavioral drives with non-human primates. The emergence of humans with a sophisticated language allowed the development of complex virtual constructs based on symbolism and the instruments of culture. This has enhanced cognitive capacity and emotional interaction supported by processes anchored in neural networks distributed within cortical and subcortical levels. Basic, essential neural connectivities were preserved during the evolutionary development of the species. The biological matrix and inheritance are usually segregated from the social and cultural construction. Sophistication of our cultural development tends to set up a divisive fault from primitive foundations of non-human animal behavior (survival, territory, reproduction, prevalence, access to nutrients). These are basic templates and underlie essential individual and group basic drives, and cultural constructs. Humans have not ceased from being territorial (whether applied to virtual or material dimensions). Various forms of social inequities have been expressed in our time and through human history. On evolutive terms, the notion of individual social status within the social structure of a gregarious community generated the probability of an individual ascending or descending that hierarchy. Thus, becoming a potential figure of leadership, or the subordinate or the marginalized. Is there an evolutive antecedent for human social inequities? How do we construct a different future? Post-industrial societies became increasingly dependent on material consumerism and technological cultures, to the point of embraining them; conceptually becoming technological hybrids. Are these a developmental must, or an uncontrolled spin-off of human inventiveness affecting our future? Construction of supernatural agents played a significant role in the socialization/domestication processes. Agents’ intentionality flourished through altered states of conscience, or under fear of natural phenomena attributed to supposed inhabitants of the Natural Kingdom or virtual beings. This imaginary universe, reinforced by ritual behaviors, contributed to control personal/collective distress of various possible origins and conditioned our degrees of emotional and cognitive freedom. (shrink)

This paper proves that certain supertasks constitute counterexamples to countable additivity even in the frame of an objective (not subjective, à la de Finetti) conception of probability. The argument requires taking conditional probability as a primitive notion.

I follow Hájek (Synthese 137:273–323, 2003c) by taking objective probability to be a function of two propositional arguments—that is, I take conditional probability as primitive. Writing the objective probability of q given r as P(q, r), I argue that r may be chosen to provide less than a complete and exact description of the world’s history or of its state at any time. It follows that nontrivial objective probabilities are possible in deterministic worlds and about the past. (...) A very simple chance–credence relation is also then natural, namely that reasonable credence equals objective probability. In other words, we should set our actual credence in a proposition equal to the proposition’s objective probability conditional on available background information. One advantage of that approach is that the background information is not subject to an admissibility requirement, as it is in standard formulations of the Principal Principle. Another advantage is that the “undermining” usually thought to follow from Humean supervenience can be avoided. Taking objective probability to be a two-argument function is not merely a technical matter, but provides us with vital flexibility in addressing significant philosophical issues. (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)

A naturalistic scheme of primitive conceptual representations is proposed using the statistical measure of mutual information. It is argued that a concept represents, not the class of objects that caused its tokening, but the class of objects that is most likely to have caused it (had it been tokened), as specified by the statistical measure of mutual information. This solves the problem of misrepresentation which plagues causal accounts, by taking the representation relation to be determined via ordinal relationships between (...) conditional probabilities. The scheme can deal with statistical biases and does not rely on arbitrary criteria. Implications for the theory of meaning and semantic content are addressed. (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)

A naturalistic scheme of primitive conceptual representations is proposed using the statistical measure of mutual information. It is argued that a concept represents, not the class of objects that caused its tokening, but the class of objects that is most likely to have caused it (had it been tokened), as specified by the statistical measure of mutual information. This solves the problem of misrepresentation which plagues causal accounts, by taking the representation relation to be determined via ordinal relationships between (...) conditional probabilities. The scheme can deal with statistical biases and does not rely on arbitrary criteria. Implications for the theory of meaning and semantic content are addressed. (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)

By inferences from fossil records and circumstantial evidences, it is now generally postulated that angiosperms have a much longer history than hitherto believed and that they have already existed probably in Jurassic time. Studies in vascular tissues and reproductive, structures have negated the possibility of originating angiosperms from various gymnosperm groups. Chronologically, this derivation will be also an impossibility.From a consideration of various aspects in the life history of angiosperms, a hypothesis is here presented postulating that protangiosperms originated in an (...) aquatic or subaquatic environment from an algal ancestry. The enclosure of ovules in the ovary is interpretated as a means of protection necessitated for the transmigration from an aquatic to an aerial habitat for the developing reproductive parts in the bud underwater. Trimery is suggested as a primitive condition, one that affords maximum efficiency in carrying out the purpose under the conditions.In the life history of angiosperms, the endosperm is of great significance in phylogeny. From morphological, physiological, cytological as well as genetical considerations, the endosperm should be regarded as of equal status with the gametophytic and sporophytic generations. There is no comparable structure in the gymnosperm groups and the ferns and “fern-allies”, while more or less similar conditions are present in the thallophytes. In the long life histories of many red algae, however, there is an especially notable resemblance to angiosperms, with certain details approaching the unique and characteristic features of the endosperms.Land plants are presumably all evolved from aquatic ancestors but instead of the usual concept of a monophyletic origin and lineal development of all vascular plants, a more plausible suggestion is the multiple and recurrent derivation of such plants from various algal ancestral groups. The angiosperms probably represent one of such major lines that originated more or less directly from a thallophytic ancestry. (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)

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)

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

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)

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)

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)

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)

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)

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 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.

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 (...) class='Hi'>conditional, this analysis might help to explain some of Douven and Verbrugge’s empirical observations. (shrink)

In Ch. 8, I distinguished epistemic probability from objective probability and then pointed out some debilitating problems with the three main accounts of epistemic probability. In this chapter, I propose my own account of epistemic probability. I first distinguish between two sides to epistemic probability, which I call the objective component and the normative component. In typical cases where we assert that some proposition is epistemically probable, two things get asserted: that the proposition is objectively probable with respect to the (...) evidence, and that it is rational to place a high degree of confidence in the proposition, given the circumstances of having as evidence what in fact we do have as evidence. Attempting to unpack these two components, I sketch the view that the normative component of the epistemic conditional probability of A on B is the interval containing the degrees of belief a rational person could have in A, provided she believed B and was aware that she believed B, considered the evidential bearing of B on A, had no other source of warrant for B or its denial, and had no defeater for the warrant, if any, accruing to A or its denial by virtue of being thus believed on the basis of B. (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 (...) class='Hi'>conditional probabilities are sufficient to trivialize the semantics. (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)

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.

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)

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)

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)

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)

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)

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.

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.

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)

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.

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

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)

We argue that quantum theory does not allow for a generalization of the notion of classical conditional probability by showing that the probability defined by the Lüders rule, standardly interpreted in the literature as the quantum-mechanical conditionalization rule, cannot be interpreted as such.

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)