Part I. Foundations of Probability: 1. Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualititative probability ordering 2. Necessary and sufficient qualitative axioms for conditional probability 3. On using random relations to generate upper and lower probabilities 4. Conditions on upper and lower probabilities to imply probabilities 5. Qualitative axioms for random-variable representation of extensive quantifiers Part II. Causality and Quantum Mechanics: 6. Stochastic incompleteness of quantum mechanics 7. On the determinism of hidden variable theories with strict correlation and conditional statistical independence of observables 8. A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distribution 9. When are probabilistic explanations possible? 10. Causality and symmetry 11. New Bell-type inequalities for N>4 necessary for existence of a hidden variable 12. Existence of hidden variables having only upper probabilities Part III. Applications in Education: 13. Mastery learning of elementary mathematics: theory and data.
Part I. Foundations of Probability: 1. Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualititative probability ordering 2. Necessary and sufficient qualitative axioms for conditional probability 3. On using random relations to generate upper and lower probabilities 4. Conditions on upper and lower probabilities to imply probabilities 5. Qualitative axioms for random-variable representation of extensive quantifiers Part II. Causality and Quantum Mechanics: 6. Stochastic incompleteness of quantum mechanics 7. On the determinism of hidden variable theories with strict correlation and conditional statistical independence of observables 8. A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distribution 9. When are probabilistic explanations possible? 10. Causality and symmetry 11. New Bell-type inequalities for N>4 necessary for existence of a hidden variable 12. Existence of hidden variables having only upper probabilities Part III. Applications in Education: 13. Mastery learning of elementary mathematics: theory and data.
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.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.
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.
An examination is made of the differing implications from applying the two mainstream interpretations of probability, frequentist and Bayesian, to QM (quantum mechanics) theory for the Bohm-EPR experiment. The joint probability distribution for the possible spin outcomes for two particles with coupled spins is examined. Contrasting conclusions are made: (i) Under the frequentist interpretation, the QM spin distribution implies a widely-discussed non-locality because probabilistic conditioning on a spin measurement is viewed as corresponding to a causal influence. Under the Bayesian interpretation, this conditioning is viewed as providing information relevant to the spin probabilities and the argument for non-locality loses its force. (ii) The frequentist interpretation leads to the locality condition used by John Bell in 1964 to establish conditions for the existence of hidden variables behind the spin probability distribution but this condition is not consistent with the product (Bayes) rule of probability theory. Under the Bayesian interpretation, there is no motivation for this locality condition and a new stochastic hidden-variable model is given that reproduces the spin distribution. (iii) Bell's original 1964 inequality provides a necessary condition for the existence of a third-order joint probability distribution for three spin variables that is compatible through marginalization with the second-order joint distributions for the three possible spin pairs. Bell's inequality is logically equivalent to the 1969 CHSH inequality that involves expectations of four pairs of four spin variables; the CHSH inequality must be satisfied in order for a fourth-order joint probability distribution for the four spin variables to exist. When any of these inequalities are violated, a joint probability distribution for three spin outcomes fails to be valid because some probabilities are negative.
Great attention has been given to the exact formulation of locality conditions in the existence of hidden variables for quantum mechanical configurations that do not satisfy Bell's inequalities. In contrast, almost no attention has been given to examining what happens if the requirements for a probability measure are relaxed. The purpose of this paper is to show that local hidden variables do exist if the requirement of a probability measure is weakened to that of having only an upper probability measure. In the past several decades, upper and lower measures have been brought into the theory of statistical inference and the measurement of beliefs rather extensively, but in almost all cases considered in statistical theory or the belief framework, it is assumed that a probability measure exists, or in many cases, that an entire family of probability measures is given such that the sup and inf of the family defines the upper and lower measures. Sometimes still more is required, namely, that the upper–lower measure be a capacity (in the sense of Choquet) of infinite order. This is, for example, true of Dempster's theory of statistical inference. For an extensive recent analysis of the various theories of imprecise statistical models or belief functions using upper and lower probabilities, Walley (1991) is an excellent reference. In contrast and as is well known, in standard quantum mechanical cases the Bell inequalities cannot be satisfied and this implies that there exists no probability measure.