The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
A physical theory, in the narrow sense of the word, is a logical structure based on assumptions and definitions which permits one to predict the outcome of a maximum number of different experiments on the basis of a minimum number of postulates. One usually desires the postulates to be as self-evident as possible; one demands simplicity, beauty, and even elegance. A physical theory seems to have only a finite lifetime. Newtonian mechanics was replaced by relativistic mechanics, thermodynamics by statistical mechanics, classical by quantum mechanics. There is a logical order in theoretical physics. Indeed, such an order is necessary if the endeavor of theoretical physics is to be meaningful and is to succeed in the direction of more and more general axiomatic deductive systems. The relationship between classical and quantum mechanics is especially interesting. The chapter also presents an overview of the key concepts discussed in this book.
The aim of this paper is to present and discuss a probabilistic framework that is adequate for the formulation of quantum theory and faithful to its applications. Contrary to claims, which are examined and rebutted, that quantum theory employs a nonclassical probability theory based on a nonclassical “logic,” the probabilistic framework set out here is entirely classical and the “logic” used is Boolean. The framework consists of a set of states and a set of quantities that are interrelated in a specified manner. Each state induces a classical probability space on the values of each quantity. The quantities, so considered, become statistical variables (not random variables). Such variables need not have a “joint distribution.” For the quantum theoretic application, there is a uniform procedure that defines and determines the existence of such “joint distributions” for statistical variables. A general rule is provided and it is shown to lead to the usual compatibility-commutivity requirements of quantum theory. The paper concludes with a brief discussion of interference and the misunderstandings that are involved in the false move from interference to nonclassical probability.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
Volume 1: Introduction P. Humphreys. Part I: Probability. Some Contributions to Formal Theory of Probability K. Popper, D. Miller. Elementary Non-Archimedean Representations of Probability for Decision Theory and Games P.J. Hammond. Random Sequences and Hypotheses Tests R. Chuaqui. Changing Probability Judgements I. Levi. Upper and Lower Probability T.L. Fine. Some Connections between Epistemic Logic and the Theory of Nonadditive Probability P. Mongin. On the Properties of Conditional Independence W. Spohn. Qualitative Probabilities Revisited Z. Domotor. The Monks' Vote: a Dialogue on Unidimensional Probabilistic Geometry J.-C. Falmagne. Part II: Probabilistic Causality. Probabilistic Causation without Probability P.W. Holland. Causal Tendency, Necessitivity and Sufficientivity: an Updated Review I.J. Good. Practical Causal Generalizations E.W. Adams. In Place of Regression C. Glymour, P. Spirtes, R. Scheines. Testing Probabilistic Causality D. Costantini. Psychologistic Aspects of Suppes' Definition of Causality P. Legrenzi, M. Sonino. Name Index. Subject Index. Volume 2: Part III: Philosophy of Physics. Probability and Quantum Theory B. Loewer. Schrodinger's Version of EPR, and its Problems A. Fine. Classical Field Magnitudes J. Vuillemin. Quantity, Representation and Geometry B. Mundy. Numerical Experimentation P. Humphreys. Part IV: Theory Structure. Theories and Theoretical Models R. Wojcicki. Suppes Predicates and the Construction of Unsolvable Problems in the Axiomatized Sciences N.C.A. da Costa, F.A. Doria. Structural Explanation J.D. Sneed. Part V: Measurement Theory. Fifteen Problems concerning the Representational Theory of Measurement R.D. Luce, L. Narens. The Meaningfulness of Ordinal Comparisons for General Order Relational Systems F.S. Roberts, Z.S. Rosenbaum. Theories as Nets: the Case of Combinatorial Measurement Theory C.U. Moulines, J.A. Diez. Name Index. Subject Index. Volume 3: Part VI: Philosophy of Language and Logic. Patrick Suppes' Contribution to the Philosophy of Language D. Follesdal. Open Problems in Relational Grammar M. Bottner. A Variable-Free Logic for Anaphora W.C. Purdy. Is Snow White? J. Moravcsik. Can there be Reasons for Putting Limitations on Classical Logic? P. Weingartner. Quantum Logic as a Logic of Identification J. Hintikka, I. Halonen. Logic and Probability in Quantum Mechanics M.L. dalla Chiara, R. Giuntini. Part VII: Learning Theory, Action Theory, and Robotics. From Stimulus-Sampling to Array-Similarity Theory W.K. Estes. Action as Seeing to it that Something is the Case R. Tuomela, G. Sandu. Command Satisfaction and the Acquisition of Habits C. Crangle. Part VIII: General Philosophy of Science. Some Observations on Patrick Suppes' Philosophy of Science M.C. Galavotti. Epilogue. Postscript P. Suppes. Chronological and Topical Bibliography of Patrick Suppes' Publications. Name Index. Subject Index.
Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.
Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.