As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, (...) in particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

Recent results in probability theory have cast doubt on the coherence theory of justification, allegedly showing that coherence cannot produce justification for beliefs in the absence of foundational justification, and that there can be no measure of coherence on which coherence is generally truth-conducive. I argue that the coherentist can reject some of the assumptions on which these theorems depend. Coherence can then be held to produce justification on its own, and truth-conducive measures of coherence can be (...) constructed. (shrink)

In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of (...) the density matrices induced by the experiments or `measurements' is the Lüders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM. (shrink)

We provide a formally rigorous framework for integrating singular causation, as understood by Nuel Belnap's theory of causae causantes, and objective single case probabilities. The central notion is that of a causal probability space whose sample space consists of causal alternatives. Such a probability space is generally not isomorphic to a product space. We give a causally motivated statement of the Markov condition and an analysis of the concept of screening-off. 1. Causal dependencies and probabilities1.1Background: causation in (...) branching space-times1.2What are probabilities defined for?2. Basic transitions2.1Basics of basic transitions2.2Sets of basic transitions3. Causal probability theory3.1Some simple cases3.2General causal probabilities3.3Application: probability of suprema of a chain. (shrink)

Epistemic closure under known implication is the principle that knowledge of "p" and knowledge of "p implies q", together, imply knowledge of "q". This principle is intuitive, yet several putative counterexamples have been formulated against it. This paper addresses the question, why is epistemic closure both intuitive and prone to counterexamples? In particular, the paper examines whether probability theory can offer an answer to this question based on four strategies. The first probability-based strategy rests on the accumulation (...) of risks. The problem with this strategy is that risk accumulation cannot accommodate certain counterexamples to epistemic closure. The second strategy is based on the idea of evidential support, that is, a piece of evidence supports a proposition whenever it increases the probability of the proposition. This strategy makes progress and can accommodate certain putative counterexamples to closure. However, this strategy also gives rise to a number of counterintuitive results. Finally, there are two broadly probabilistic strategies, one based on the idea of resilient probability and the other on the idea of assumptions that are taken for granted. These strategies are promising but are prone to some of the shortcomings of the second strategy. All in all, I conclude that each strategy fails. Probability theory, then, is unlikely to offer the account we need. (shrink)

We describe a computational model of two central aspects of people's probabilistic reasoning: descriptive probability estimation and inferential probability judgment. This model assumes that people's reasoning follows standard frequentist probability theory, but it is subject to random noise. This random noise has a regressive effect in descriptive probability estimation, moving probability estimates away from normative probabilities and toward the center of the probability scale. This random noise has an anti-regressive effect in inferential judgement, (...) however. These regressive and anti-regressive effects explain various reliable and systematic biases seen in people's descriptive probability estimation and inferential probability judgment. This model predicts that these contrary effects will tend to cancel out in tasks that involve both descriptive estimation and inferential judgement, leading to unbiased responses in those tasks. We test this model by applying it to one such task, described by Gallistel et al. ). Participants' median responses in this task were unbiased, agreeing with normative probability theory over the full range of responses. Our model captures the pattern of unbiased responses in this task, while simultaneously explaining systematic biases away from normatively correct probabilities seen in other tasks. (shrink)

This chapter contains sections titled: The Bayesian Orthodoxy Idealization Two Approaches to a Theory of Probability Adjustment for Nonclassical Logics Carnap's Confirmation Theory Proportional Syllogisms Kyburg's Fuzzy Probabilities Levi's Indeterminate Systems Qualitative Theories of Probability The Dynamics of Subjective Probability Probability Theory and Quantum Theory.

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and (...) the origin of the Schrödinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained. (shrink)

This paper presents a theory of probability based on the paraconsistent logic D4. The resulting probability functions are then used to define two sorts of Bayesian updating. One sort of updating merely uses the simple rule of conditionalisation. The other sort adds a wrinkle to the simple rule so that agents' beliefs become more consistent as well as more complete through updating.

We consider probability theories in general. In the first part of the paper, various constraints are imposed and classical probability and quantum theory are recovered as special cases. Quantum theory follows from a set of five reasonable axioms. The key axiom which gives us quantum theory rather than classical probability theory is the continuity axiom, which demands that there exists a continuous reversible transformation between any pair of pure states. In the second part (...) of this paper, we consider in detail how the measurement process works in both the classical and the quantum case. The key differences and similarities are elucidated. It is shown how measurement in the classical case can be given a simple ontological interpretation which is not open to us in the quantum case. On the other hand, it is shown that the measurement process can be treated mathematically in the same way in both theories even to the extent that the equations governing the state update after measurement are identical. The difference between the two cases is seen to be due not to something intrinsic to the measurement process itself but, rather, to the nature of the set of allowed states and, therefore, ultimately to the continuity axiom. (shrink)

I argue that to the extent to which philosophical theories of objective probability have offered theoretically adequate conceptions of objective probability , they have failed to satisfy a methodological standard -- roughly, a requirement to the effect that the conception offered be specified with the precision appropriate for a physical interpretation of an abstract formal calculus and be fully explicated in terms of concepts, objects or phenomena understood independently of the idea of physical probability. The significance of (...) this, and of the suggested methodological standard, is then briefly discussed. (shrink)

I argue that to the extent to which philosophical theories of objective probability have offered theoretically adequateconceptions of objective probability (in connection with such desiderata as causal and explanatory significance, applicability to single cases, etc.), they have failed to satisfy amethodological standard — roughly, a requirement to the effect that the conception offered be specified with the precision appropriate for a physical interpretation of an abstract formal calculus and be fully explicated in terms of concepts, objects or phenomena (...) understood independently of the idea of physical probability. The significance of this, and of the suggested methodological standard, is then briefly discussed. (shrink)

Various proposals for generalizing event spaces for probability functions have been put forth in the mathematical, scientific, and philosophic literatures. In cognitive psychology such generalizations are used for explaining puzzling results in decision theory and for modeling the influence of context effects. This commentary discusses proposals for generalizing probability theory to event spaces that are not necessarily boolean algebras. Two prominent examples are quantum probability theory, which is based on the set of closed subspaces (...) of a Hilbert space, and topological probability theory, which is based on the set of open sets of a topology. Both have been applied to a variety of cognitive situations. This commentary focuses on how event space properties can influence probability concepts and impact cognitive modeling. (shrink)

John Leslie has published an argument that our own birth rank among all who have lived can be used to make inferences about all who will ever live, and hence about the expected survival time for the human race. It is found to be shorter than usually supposed. The assumptions underpinning the argument are criticized, especially the unwarranted one that the argument's sampling is equiprobable from among all who ever live. A mathematical derivation shows that Leslie's argument is correct only (...) if there exists a correlation of our birth rank to the event of doomsday. Such correlation is highly improbable. (shrink)

This special issue of the International Journal of Approximate Reasoning grew out of the 8th International Symposium on Imprecise Probability: Theories and Applications. The symposium was organized by the Society for Imprecise Probability: Theories and Applications at the Université de Technologie de Compiègne in July 2013. The biennial ISIPTA meetings are well established among international conferences on generalized methods for uncertainty quantification. The first ISIPTA took place in Gent in 1999, followed by meetings in Cornell, Lugano, Carnegie Mellon, (...) Prague, Durham and Innsbruck. Compiègne proved to be a very nice location for ISIPTA 2013, offering wonderful opportunities for collaborations and discussions, as well as sightseeing places such as its imperial palace. (shrink)

Book Information Probability Theory and Probability Semantics. By P. Roeper and H. Leblanc. University of Toronto Press. Toronto. 1999. Pp. xii + 240. Hardback, US$65.00.

Recent results in the literature appear to show that it is impossible for two independent testimonies to jointly raise the probability of a proposition if neither testimony individually has any impact on that probability. I show that these impossibility results do not apply when testimonies agree on incidental details.

Book Information Probability Theory and Probability Semantics. By P. Roeper and H. Leblanc. University of Toronto Press. Toronto. 1999. Pp. xii + 240. Hardback, US$65.00.

This book sheds light on some recent discussions of the problems in probability theory and their history, analysing their philosophical and mathematical significance, and the role pf mathematical probability theory in other sciences.

It is a curious fact that in the many writings on the theory of probability, one rarely finds an instance of a clear cut definition of the fundamental problem of the subject. On intuitive grounds, one grasps the intent of most of the authors: it is to define the concept of probability and to show its relationship to the other concepts of science; and yet since the criterion of adequacy in the answering of this question is rarely (...) stated, one must feel that the problem itself has remained undefined. (shrink)

This chapter covers the epistemic or information-based interpretations of probability: logical, subjective, objective Bayesian, and group level. It explains how these differ from aleatory or world-based interpretations of probability, presents each in detail, and then discusses its strengths and weaknesses.

We discuss a generalization of the standard notion of probability space and show that the emerging framework, to be called operational probability theory, can be considered as underlying quantal theories. The proposed framework makes special reference to the convex structure of states and to a family of observables which is wider than the familiar set of random variables: it appears as an alternative to the known algebraic approach to quantum probability.

In order adequately to refute the charge that relativism makes against any absolute answer to problems of science, it will be necessary for us to generalize upon the principal theme and attempt a characterization of experimental methodology. Only by thus determining what constitutes experimental inquiry in general can we hope to define that particular aspect of it which involves the concept of probability.

A stereotype is a belief or claim that a group of people has a particular feature. Stereotypes are expressed by sentences that have the form of generic statements, like “Canadians are nice.” Recent work on generics lends new life to understanding generics as statements involving probabilities. I argue that generics (and thus sentences expressing stereotypes) can take one of several forms involving conditional probabilities, and these probabilities have what I call a naturalness requirement. This is the natural probability (...) class='Hi'>theory of stereotypes. Each of the two components of the theory entails a family of fallacies that contributes to the spurious reinforcement of stereotypes: inferential slippage within and between the different generic forms, and inferential slippage from facts about frequencies of group traits to beliefs about natural propensities or dispositions of groups. Empirical research suggests that we often commit these fallacies. Moreover, this theory can referee a vitriolic debate between some psychologists, who hold that stereotypes are always false and stereotyping is always wrong, and other psychologists, who hold that stereotypes are often accurate and stereotyping is often reasonable. (shrink)

The reader will be a patient one indeed who has not long since raised in his mind what appear to be pertinent and pressing problems concerning the description of experimental method that has so far been given. These questions are mainly concerned with matters of omission: for example, we have said that if a lack of control is shown, then the image should be changed. But how changed? What principle or method guides the selection of a new image of nature (...) when the old is no longer permissible? Or again, we have said that “observations should be made“. But what is an observation? What principle or method guides the experimenter in deciding which of his activities are observations? (shrink)

We argue that approaches adhering to normative systems can be as fruitful as those by descriptive systems. In measuring people's perception of randomness, discrepancies between human behavior and normative models could have resulted from unknown properties of the models, and it does not necessarily lead to the conclusion that people are irrational or that the normative system has to be abandoned.

The story of quantum probability theory and quantum logic begins with von Neumann’s recognition1, that quantum mechanics can be regarded as a kind of “probability theory”, if the subspace lattice L of the system’s Hilbert space H plays the role of event algebra and the ‘tr’-s play the role of probability distributions over these events. This idea had been completed in the Gleason theorem 2.

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for (...) QM/sets. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features in a classical setting. (shrink)

I WHAT IS PROBABILITY? Style manuals advise us that the proper way to begin a piece of expository writing is to introduce and identify clearly the subject ...

Quantum probability theory (QP) is the best formal representation available of the most common form of judgment involving attribute comparison (inside judgment). People are capable, however, of judgments that involve proportions over sets of instances (outside judgment). Here, the theory does not do so well. I discuss the theory both in terms of descriptive adequacy and normative appropriateness.

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates (...) monotonicity, as its name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have (...) led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Baysian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations. (shrink)

Kolmogorov's account in his [1933] of an absolute probability space presupposes given a Boolean algebra, and so does Rényi's account in his [1955] and [1964] of a relative probability space. Anxious to prove probability theory ‘autonomous’. Popper supplied in his [1955] and [1957] accounts of probability spaces of which Boolean algebras are not and [1957] accounts of probability spaces of which fields are not prerequisites but byproducts instead.1 I review the accounts in question, showing (...) how Popper's issue from and how they differ from Kolmogorov's and Rényi's, and I examine on closing Popper's notion of ‘autonomous independence’. So as not to interrupt the exposition, I allow myself in the main text but a few proofs, relegating others to the Appendix and indicating as I go along where in the literature the rest can be found. (shrink)

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.