LOGIC AND PROBABILITY IN QUANTUM MECHANICS

LOGIC AND PROBABILITY IN QUANTUM MECHANICS

SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES Managing Editor: J AAKKO HINTIKKA, Academy of Finland and Stanford University Editors: ROBERT S. COHEN, Boston University DONALD DAVIDSON, Rockefeller University and Princeton University GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, University of Arizona VOLUME 78

LOGIC AND PROBABILITY IN QUANTUM MECHANICS Edited by PATRICK SUPPES Stanford University SPRlNGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging in Publication Data Main entry under title: Logic and probability in quantum mechanics. (Synthese library ; v. 78) Bibliography : p. Includes index. I. Quantum theory- Addresses, essays, lectures. 2. Physics-Philosophy-Addresses, essays, lectures. I. Suppes, Patrick Colonel, 1922- QC174.l25.L62 530.1 '2 75-30834 ISBN 978-90-277-1200-4 ISBN 978-94-010-9466-5 (ebook) DOI 10.1007/978-94-010-9466-5 All Rights Reserved Copyright 1976 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1976 Softcover reprint of the hardcover 1st edition 1976 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner

PREFACE During the academic years 1972-1973 and 1973-1974, an intensive seminar on the foundations of quantum mechanics met at Stanford on a regular basis. The extensive exploration of ideas in the seminar led to the o r g ~ i zof a a t idouble o n issue of Synthese concerned with the foundations of quantum mechanics, especially with the role of logic and probability in quantum mechanics. About half of the articles in the volume grew out of this seminar. The remaining articles have been solicited explicitly from individuals who are actively working in the foundations of quantum mechanics. Seventeen of the twenty-one articles appeared in Volume 29 of Synthese. Four additional articles and a bibliography on -the history and philosophy of quantum mechanics have been added to the present volume. In particular, the articles by Bub, Demopoulos, and Lande, as well as the second article by Zanotti and myself, appear for the first time in the present volume. In preparing the articles for publication I am much indebted to Mrs. Lillian O'Toole, Mrs. Dianne Kanerva, and Mrs. Marguerite Shaw, for their extensive assistance. PA TRICK SUPPES

TABLE OF CONTENTS PREFACE INTRODUCTION V IX PART 1/ LOGIC G. KREISEL / A Notion of Mechanistic Theory 3 ROLAND FRAisSE / Essai sur la logique de l'indeterminisme et la ramification de l'espace-temps 19 HILARY PUTNAM / How to Think Quantum-Logically 47 GAR Y M. HARDEGREE / The Conditional in Quantum Logic 55 D. J. FOULIS and c. H. RANDALL / Empirical Logic and Quantum Mechanics 73 RICHARD J. GREECHIE / Some Results from the Combinatorial Approach to Quantum Logic 105 PART II / PROBABILITY J. M. JAUCH / The Quantum Probability Calculus 123 ZOL T AN DOMOTOR / The Probability Structure of Quantum- Mechanical Systems 147 TERRENCE L. FINE / Towards a Revised Probabilistic Basis for Quantum Mechanics 179 TED BASTIN / Probability in a Discrete Model of Particles and Observations 195 NANCY DELANEY CARTWRIGHT / Superposition and Macroscopic Observation 221 0ISTEIN BJ0RNESTAD / A Note on the So-Called Yes-No Experiments and the Foundations of Quantum Mechanics 235 PART III/COMPLETENESS AR THUR FINE / On the Completeness of Quantum Theory 249

VIII TABLE OF CONTENTS BAS C. V A N FRAASSEN / The Einstein-Podolsky-Rosen Paradox 283 PA TRICK SUPPES and MARIO ZANOTTI/Stochastic Incompleteness of Quantum Mechanics 303 ROBERT W. LATZER / Errors in the No Hidden Variable Proof of Kochen and Specker 323 DAVID J. ROSS / Operator-Observable Correspondence 365 JEFFREY BUB / Randomness and Locality in Quantum Mechanics 397 WILLIAM DEMOPOULOS / Fundamental Statistical Theories 421 ALFRED LANDE / Why the World Is a Quantum World 433 PA TRICK SUPPES and MARIO ZANOTTI / On the Determinism of Hidden Variable Theories with Strict Correlation and Conditional Statistical Independence of Observables 445 BIBLIOGRAPHY ON THE HIS TOR Y AND PHILOSOPHY OF QUAN- TUM PHYSICS: Compiled by Donald Richard Nilson 457 INDEX OF NAMES 521 INDEX OF SUBJECTS 529

INTRODUCTION The philosophy of physics has occupied an important place in philosophy since ancient times, and a wide spectrum of philosophers know something about the historical development of the fundamental concepts of space, time, matter, and motion. On the other hand, to many philosophers the problems that are discussed in the foundations of quantum mechanics seem specialized and esoteric in relation to the classical tradition in the philosophy of physics, and the relevance of analysis of the basic concepts of quantum mechanics to general philosophy seems restricted. The problems raised by this issue of relevance warrant further examination. On the one hand, the case is overwhelming that quantum mechanics is the most important scientific theory of the twentieth century. It is hard to believe that the new and surprising concepts that have arisen in the theory are not of major importance to philosophy and our fundamental conception of the world we live in. Yet the philosophical literature dealing specifically with quantum mechanics is, like the literature of physics on the theory, difficult and technical. It is admittedly no easy matter for an outsider not specifically concerned with the philosophical issues raised by quantum mechanics to get an overview of the subject and to be able to appreciate the general philosophical significance of the conceptual analyses made by a variety of philosophers, physicists, and mathematicians. I also hasten to add that the present volume does not in any sense fill this gap. It is meant to be a contribution to the continuing relatively specific and relatively technical discussion of the philosophical foundations of quantum mechanics. The twenty-one articles included in this volume cover many topics and issues, but I have simplified the range of issues and concepts covered in order to organize them in three parts. Each of the three parts is meant to represent a group of closely related topics pertinent not only to the general philosophy of science but to epistemology and metaphysics as well. Part I concerns logical issues raised by quantum mechanics. The first

x INTRODUCTION article, by Kreisel, is of a general nature and assumes no specific knowledge of quantum mechanics. Kreisel raises the philosophically interesting question of whether quantum mechanics will lead to yet another surprise in that it is in an essential sense a nonmechanistic theory. Here nonmechanistic means having nonrecursive solutions to differential equations describing fundamental natural processes. As Kreisel points out, the issue must be stated with some care and it is the sense of his article to make this care explicit, for classical mechanics is meant to hold for arbitrary nonrecursive measures of distances, masses, and forces. In the classical case, however, the usual rational approximations have a recursive or mechanistic character in most of the applications of apparent interest. The important question that he raises is whether this is true of quantum mechanics. Kreisel also makes clear the kind of problem in classical mechanics which may be nonrecursive in character. The second article by Fraisse raises general issues about the logic of indeterminism and the extent to which the fundamental results of quantum mechanics force a change in our classical conception oflogic. Fraisse is especially concerned to examine the philosophical consequences of Everett's bold hypothesis about the ramification of space-time or what is sometimes called the many-universes interpretation of quantum mechanics. His purpose is to examine the concept of i n d e t e that ~ i n i s m results from Everett's view with a minimum of dependence on technical details of quantum mechanics. The epistemological status of the laws of classical logic has been besieged by more than one sustained attack in the last hundred years. The rejection of the law of excluded middle by intuitionistic philosophers of mathematics is probably the most salient example. The striking and distinguishing feature of the attack that has been launched from a quantum mechanical base is that it is an attack that rests upon an empirical scientific theory of an advanced and complicated nature. That a challenge to classical logic could arise from highly specialized empirical concepts in physics dealing with the motion of very small particles seems to run counter to almost all the epistemological tradition in logic from Aristotle to Frege - by the 'epistemological tradition' I mean of course the philosophical analysis of the grounds for accepting a law of logic as valid. The third article, by Putnam, deals most directly with the quantum mechanical challenge to the classical epistemological tradition that de-

INTRODUCTION XI fends logic as a collection of a priori truths. The following article, by Hardegree, deals with the way we may formulate conditional sentences or propositions in quantum logic. Hardegree is especially concerned with the philosophical controversy concerning the possibility of introducing a reasonable notion of implication in quantum logic. As opposed to the expressed views of Jauch, Piron, Greechie, and Gudder. Hardegree argues that the standard quantum logic as represented by the lattice of subspaces of a separable Hilbert space, does in fact admit an operation possessing the most essential properties of a material conditional. The conditional that Hardegree proposes is close to a Stalnaker conditional; it does not satisfy the laws of transivity or contraposition but it does satisfy modus ponens. To some extent, therefore, the differences with Jauch and the other authors mentioned above depend upon what one regards as essential properties of an operation of implication. In the fifth article, Foulis and Randall continue the development of empirical logic and apply it to quantum mechanics, relating at the same time their developments to some of the other technical papers in quantum logic. The next article, by Greechie, is concerned with some specific problems in quantum logic, especially with a problem posed by Jauch in his article in Part II on the quantum probability calculus. The articles by Foulis and Randall, and by Greechie, illustrate the extent to which the subject matter of quantum logic rapidly becomes a technical topic in its own right. Not only the character of quantum mechanics itself but also the mathematical level of contemporary work in logic make it hardly surprising that new specific results in quantum logic will necessarily be embodied in a framework of relatively new mathematical concepts. Closely following on questions about the nature of logic in quantum mechanics are a series of questions about probability in quantum mechanics. A case can be made for the claim that quantum mechanics is as disturbing to the classical concepts of probability as it is to the classical concepts of logic. The six articles I have placed in Part II are concerned with various aspects of probability in quantum mechanics. The first article of this part, by Jauch, gives an excellent general review of probability concepts in the context of quantum mechanics and makes clear the issues about probability central to quantum mechanics. The second article, by Domotor, provides a general analysis of probability structures that occur in quantum mechanics. Domotor's principal aim is to present a repre-

XII INTRODUCTION sentation of quantum logics, in particular, orthomodular, partially ordered sets, by means of structured families of Boolean algebras. He brings to his analysis of these structures methods that have been used in the study of manifolds by geometers. In the third article, Terrence Fine proposes a revised probabilistic basis for quantum mechanics based on his ideas of the proper approach to qualitative probability. The direction in which he strikes out in this article is conceptually different from most of the discussions of the nature of probability in quantum mechanics. Among the more interesting features of Fine's approach is the examination of new models of random phenomena that arise from consideration of qualitative probability and how these new models relate to the quantum mechanical concept of complementarity. In the fourth article, Bastin sets forth his ideas about the place of probability in the discrete model that he would use for formulating the fundamental principles of quantum mechanics. Bastin's frontal attack on continuity assumptions and use of a continuum in quantum mechanics is perhaps the most salient feature of his approach to foundations. He replaces the continuum by a discrete model that is hierarchical in cliaracter. Although he is concerned in this article to develop the place of probability in his approach, he also has a good deal to say about the kind of hidden-variable theory his approach represents, and for that reason his article also could properly be placed in Part III rather than in Part II. In the fifth article, Cartwright discusses a number of issues concerned with the relation between the behavior of microscopic and macroscopic objects and the pertinent statistical analysis of this relation. She examines in some detail the attempts to reconcile macroscopic physics and quantum mechanics by reducing superpositions to mixtures. As she puts it, the philosophical problem is not the replacement of superpositions by mixtures, but rather to explain why we mistakenly think that a mixture is called for. In the sixth article, Bjemestad discusses the central place of yes-no experiments in the conceptual foundations of quantum mechanics. He examines critically the use of such experiments by von Neumann, Mackey, Piron, and Jauch. Part III of this volume consists of nine articles organized around the issues concerning completeness of quantum mechanics. A more general title would have been hidden-variable theories, but the nine articles are sufficiently focused on questions of completeness and are not broadly

INTRODUCTION XIII concerned with many of the traditional problems of hidden-variable theories, so that the more special title of completeness seems appropriate. There are many different but closely related concepts of completeness in science and mathematics; for instance, some of the deepest results in modem logic are concerned with completeness. There is, on the one hand, the truth-functional completeness of classical sentential logic and Godel's theorem on the completeness of first-order predicate logic, and, on the other hand, GOdel's classic results on the incompleteness of arithmetic. Within quantum mechanics, various senses of completeness can be defined, and controversy continues to exist over both the appropriateness of definitions and the exact character of the results that obtain for a given definition. The paradox set forth by Einstein, Podolsky, and Rosen attempted to show that quantum mechanics is not complete in the sense that additional variables are required for the theory to have the appropriate features of causality and locality. The p a r ~ arises d o x from measurements made on two particles, for example, a pair of spin one-half particles that are moving freely in opposite directions. The fact that the results of measurement on one particle determine the results of measurement on the other particle is taken to violate our ordinary ideas of causality which exclude having instantaneous action at a distance. It is argued that these paradoxical results require a more complete specification of the state of a quantum mechanical system. The ideas surrounding the Einstein-Podolsky-Rosen paradox, as well as other related paradoxes, are examined in detail in the first article py Arthur Fine arid in the second by van Fraassen. Fine takes the bull by the horns and challenges the significance of the. recent work of Bell and Wigner that yields a solution to the Einstein-Podolsky-Rosen paradox that, as Bell puts it, Einstein would have liked least. Fine ends up advocating his theory of statistical variables whose joint distributions do not necessarily exist. Fine interprets the Bell-Wigner arguments to show that certain arbitrary assumptions on joint distributions cannot be consistently realized or satisfied by any hidden-variable theory. He argues that his theory of statistical variables provides just the right sort of completeness for quantum mechanics. Even if he has not decisively settled the many issues raised by the Bell and Wigner work, he has advanced the argument one more stage in what is sure to be a continuing controversy. Van Fraassen focuses almost entirely on the Einstein-Podolsky-

XIV INTRODUCTION Rosen paradox and attempts to resolve it by the modal interpretation of quantum mechanics he has been developing in recent articles. In the third article of this part, Zanotti and I argue for a different kind of incompleteness of quantum mechanics. We argue that quantum mechanics is stochastically incomplete. We mean by this that, when timedependent phenomena are examined, the predictions of the theory give only mean probability distributions as a function of time and do not determine a unique stochastic process governing the motion of particles. To illustrate how a stochastic approach may be applied in quantum mechanics, we examine some of the paradoxical results that may be derived for the linear harmonic oscillator and explain them in a natural physical way by looking at the motion of the oscillator as made up of a classical component together with a random fluctuation. In the fourth article, Latzer examines in detail the well-known 'hiddenvariable' proof of Kochen and Specker and finds several serious difficulties with their conceptual formulation and mathematical development of the problem of hidden variables. In the next article, Ross discusses in detail the operator-observable correspondence in quantum mechanics. His examination of a set of inconsistent axioms that underlie many elementary discussions of quantum mechanics brings into concrete focus the peculiar problems of operator-observable correspondence that exist in quantum mechanics and that are often central to discussions of completeness. The four articles that are included in the present volume and that were not included in the original issue of Synthese deal essentially with problems relevant to Part III. The article by Bub is directly concerned with randomness and locality in quantum' mechanics, especially in relation to hidden-variable theories. Demopoulos examines the sense in which quantum mechanics can be regarded as a fundamental statistical theory. He examines Bub's earlier account of completeness of quantum mechanics, which itself assumes knowledge of the earlier work of Kochen and Specker. The Kochen and Specker work, of course, is examined in great detail in an earlier article by Latzer in this volume. The next article, by Lande, summarizes in somewhat different form his well-known views on the foundations of quantum mechanics. The final article, which is the second article by Zanotti and me, is concerned to show that any hiddenvariable theory with strict correlation and conditional statistical inde-

INTRODUCTION xv pendence of observables must be deterministic. The central point is that conditional statistical independence of observables seems to be too strong a condition to impose on properly stochastic hidden-variable theories. The volume closes with an extensive bibliography prepared by Nilson on the history and philosophy of quantum mechanics. P A TRICK SUPPES