Originally published in 1981, this book forms volume 15 of the Encyclopedia of Mathematics and its Applications. The text provides a clear and thorough treatment of its subject, adhering to a clean exposition of the mathematical content of serious formulations of rational physical alternatives of quantum theory as elaborated in the influential works of the period, to which the authors made a significant contribution. The treatment falls into three distinct, logical parts: in the first part, the modern version of accumulated wisdom is presented, avoiding as far as possible the traditional language of classical physics for its interpretational character; in the second part, the individual structural elements for the logical content of the theory are laid out; in part three, the results of section two are used to reconstruct the usual Hilbert space formulation of quantum mechanics in a novel way.

These are the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2 - 9, 1979, at the Ettore Hajorana Centre for Scientific Culture. A conference of this sort was originally proposed by Giuliano Toraldo di Francia, who suggested the idea to Antonino Zichichi, and thus laid the foundation for the Workshop. To both of them we express our appreciation and thanks, also on behalf of the other participants, for having made this conference possible. There were approximately fifty participants; their names and institutions are listed in the text. Quantum logic, which has now a history of some forty or more years, has seen remarkable growth during the sixties and seventies. The papers in the present volume presuppose, by and large, some acquaintance with the elements of the subject. These may be found in the well-known books by J.H. Jauch (Foundations of Quantum Hechanics; Reading, 1968), V.S. Varadarajan (Geometry of Quantum Theory: Princeton, 1968), and C. Piron (Foundations of Quantum Theory; New York, 1976). The initial program for the conference listed about twenty-five invited papers. But in the context of a very active and qualified attendance, other contributions were offered. This volume contains all of them. The program listed six main topics: I. Classification or different areas of quantum logic, and open problems. II. Comparison and unification of different approaches to quantum theories; problems of interpretation. III. Formal quantum logic; axiomatics. IV. Hodal interpretations of quantum logic. v vi FOREWORD V. Quantum set theory.

These are the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2 - 9, 1979, at the Ettore Hajorana Centre for Scientific Culture. A conference of this sort was originally proposed by Giuliano Toraldo di Francia, who suggested the idea to Antonino Zichichi, and thus laid the foundation for the Workshop. To both of them we express our appreciation and thanks, also on behalf of the other participants, for having made this conference possible. There were approximately fifty participants; their names and institutions are listed in the text. Quantum logic, which has now a history of some forty or more years, has seen remarkable growth during the sixties and seventies. The papers in the present volume presuppose, by and large, some acquaintance with the elements of the subject. These may be found in the well-known books by J.H. Jauch (Foundations of Quantum Hechanics; Reading, 1968), V.S. Varadarajan (Geometry of Quantum Theory: Princeton, 1968), and C. Piron (Foundations of Quantum Theory; New York, 1976). The initial program for the conference listed about twenty-five invited papers. But in the context of a very active and qualified attendance, other contributions were offered. This volume contains all of them. The program listed six main topics: I. Classification or different areas of quantum logic, and open problems. II. Comparison and unification of different approaches to quantum theories; problems of interpretation. III. Formal quantum logic; axiomatics. IV. Hodal interpretations of quantum logic. v vi FOREWORD V. Quantum set theory.

2 EnricoBeltrametti 1andJean-MarcLevy-Leblond 1Dipartimentodi Fisica, Universitadi Genova ViaDodecaneso33,1-16146 Genova, Italy 2physiqueTheorique, Universitede NiceSophia-Antipolis Pare Valrose, F-06108 Nice Cedex, France As customary, the idea of this meeting was born out during another one. The present authors were attending some colloquium among the numerous meetings devoted to the endless discussion of the foundational problems of quantum theory, when it struck their minds that physics, after all, was also an experimental science. Maybe, they thought to themselves, our very understanding ofquantum concepts would gain from more thoroughly taking into account the impressive accumulation of modern experimental results on the fundamental aspectsofquantum behaviour. Experiments once deemed as 'gedankenexperiments' now become routinely feasible, and far-ranging progresses in the practical mastering of quantum phenomena have been accomplished in the recent years. The variety of domains where quantum phenomena are currently investigated is extending beyond expectations, while their essential unity is ever more obvious.As no physicist can seriously believe that "quantum" is the last word about Nature, thedrive tocheck thelimitsofvalidityofquantum ideas is a powerful motive behind the everincreasing precisionofitsexperimental tests. Butno experimental fact up to now has surfacedwhich seems tochallenge the main concepts of quantum theory. The robustness and reliabilityofquantum theory is all the moresurprising whenoneconsiders theextensionof its rangeofvalidityduring the nearcenturyofitsexistence - from the atomic scale down to the subnuclear one ("particle" physics), and up to the astronomical one (white dwarfs, neutron stars, etc. ).

Originally published in 1981, this book forms volume 15 of the Encyclopedia of Mathematics and its Applications. The text provides a clear and thorough treatment of its subject, adhering to a clean exposition of the mathematical content of serious formulations of rational physical alternatives of quantum theory as elaborated in the influential works of the period, to which the authors made a significant contribution. The treatment falls into three distinct, logical parts: in the first part, the modern version of accumulated wisdom is presented, avoiding as far as possible the traditional language of classical physics for its interpretational character; in the second part, the individual structural elements for the logical content of the theory are laid out; in part three, the results of section two are used to reconstruct the usual Hilbert space formulation of quantum mechanics in a novel way.

These are the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2 - 9, 1979, at the Ettore Hajorana Centre for Scientific Culture. A conference of this sort was originally proposed by Giuliano Toraldo di Francia, who suggested the idea to Antonino Zichichi, and thus laid the foundation for the Workshop. To both of them we express our appreciation and thanks, also on behalf of the other participants, for having made this conference possible. There were approximately fifty participants; their names and institutions are listed in the text. Quantum logic, which has now a history of some forty or more years, has seen remarkable growth during the sixties and seventies. The papers in the present volume presuppose, by and large, some acquaintance with the elements of the subject. These may be found in the well-known books by J.H. Jauch (Foundations of Quantum Hechanics; Reading, 1968), V.S. Varadarajan (Geometry of Quantum Theory: Princeton, 1968), and C. Piron (Foundations of Quantum Theory; New York, 1976). The initial program for the conference listed about twenty-five invited papers. But in the context of a very active and qualified attendance, other contributions were offered. This volume contains all of them. The program listed six main topics: I. Classification or different areas of quantum logic, and open problems. II. Comparison and unification of different approaches to quantum theories; problems of interpretation. III. Formal quantum logic; axiomatics. IV. Hodal interpretations of quantum logic. v vi FOREWORD V. Quantum set theory.

These are the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2 - 9, 1979, at the Ettore Hajorana Centre for Scientific Culture. A conference of this sort was originally proposed by Giuliano Toraldo di Francia, who suggested the idea to Antonino Zichichi, and thus laid the foundation for the Workshop. To both of them we express our appreciation and thanks, also on behalf of the other participants, for having made this conference possible. There were approximately fifty participants; their names and institutions are listed in the text. Quantum logic, which has now a history of some forty or more years, has seen remarkable growth during the sixties and seventies. The papers in the present volume presuppose, by and large, some acquaintance with the elements of the subject. These may be found in the well-known books by J.H. Jauch (Foundations of Quantum Hechanics; Reading, 1968), V.S. Varadarajan (Geometry of Quantum Theory: Princeton, 1968), and C. Piron (Foundations of Quantum Theory; New York, 1976). The initial program for the conference listed about twenty-five invited papers. But in the context of a very active and qualified attendance, other contributions were offered. This volume contains all of them. The program listed six main topics: I. Classification or different areas of quantum logic, and open problems. II. Comparison and unification of different approaches to quantum theories; problems of interpretation. III. Formal quantum logic; axiomatics. IV. Hodal interpretations of quantum logic. v vi FOREWORD V. Quantum set theory.

2 EnricoBeltrametti 1andJean-MarcLevy-Leblond 1Dipartimentodi Fisica, Universitadi Genova ViaDodecaneso33,1-16146 Genova, Italy 2physiqueTheorique, Universitede NiceSophia-Antipolis Pare Valrose, F-06108 Nice Cedex, France As customary, the idea of this meeting was born out during another one. The present authors were attending some colloquium among the numerous meetings devoted to the endless discussion of the foundational problems of quantum theory, when it struck their minds that physics, after all, was also an experimental science. Maybe, they thought to themselves, our very understanding ofquantum concepts would gain from more thoroughly taking into account the impressive accumulation of modern experimental results on the fundamental aspectsofquantum behaviour. Experiments once deemed as 'gedankenexperiments' now become routinely feasible, and far-ranging progresses in the practical mastering of quantum phenomena have been accomplished in the recent years. The variety of domains where quantum phenomena are currently investigated is extending beyond expectations, while their essential unity is ever more obvious.As no physicist can seriously believe that "quantum" is the last word about Nature, thedrive tocheck thelimitsofvalidityofquantum ideas is a powerful motive behind the everincreasing precisionofitsexperimental tests. Butno experimental fact up to now has surfacedwhich seems tochallenge the main concepts of quantum theory. The robustness and reliabilityofquantum theory is all the moresurprising whenoneconsiders theextensionof its rangeofvalidityduring the nearcenturyofitsexistence - from the atomic scale down to the subnuclear one ("particle" physics), and up to the astronomical one (white dwarfs, neutron stars, etc. ).