This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

The first edition of the Handbook of Philosophical Logic (four volumes) was published in the period 1983-1989 and has proven to be an invaluable reference work ...

2014 Reprint of 1962 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. In "Algebraic Logic" Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. This book addresses some of the problems of mathematical logic and the theory of polyadic Boolean algebras in particular. It is intended to be an efficient (...) way of treating algebraic logic in a unified manner. (shrink)

John von Neumann (1903-1957) was undoubtedly one of the scientific geniuses of the 20th century. The main fields to which he contributed include various disciplines of pure and applied mathematics, mathematical and theoretical physics, logic, theoretical computer science, and computer architecture. Von Neumann was also actively involved in politics and science management and he had a major impact on US government decisions during, and especially after, the Second World War. There exist several popular books on his personality and various collections (...) focusing on his achievements in mathematics, computer science, and economy. Strangely enough, to date no detailed appraisal of his seminal contributions to the mathematical foundations of quantum physics has appeared. Von Neumann's theory of measurement and his critique of hidden variables became the touchstone of most debates in the foundations of quantum mechanics. Today, his name also figures most prominently in the mathematically rigorous branches of contemporary quantum mechanics of large systems and quantum field theory. And finally - as one of his last lectures, published in this volume for the first time, shows - he considered the relation of quantum logic and quantum mechanical probability as his most important problem for the second half of the twentieth century. The present volume embraces both historical and systematic analyses of his methodology of mathematical physics, and of the various aspects of his work in the foundations of quantum physics, such as theory of measurement, quantum logic, and quantum mechanical entropy. The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics. The general part of the Yearbook contains papers emerging from the Institute's annual lecture series and reviews of important publications of philosophy of science and its history. (shrink)

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. (shrink)

The idea of quantum logic first appears explicitly in the short Section 5 of Chapter III. in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics [31]; however, the real birthplace of quantum logic is commonly identified with the 1936 seminal paper co-authored by G. Birkhoff and J. von Neumann [5]. The aim of this review is to recall the main idea of the Birkhoff-von Neumann concept1 of quantum logic as this was put forward in the 1936 paper. (...) The review is motivated partly by two facts related to quantum logic: one, peculiar, is that the 1936 von Neumann concept is an almost totally neglected2 topic in the enormous quantum logic literature [17]; the other, not very well-known, is that von Neumann was never completely satisfied with how he had worked out quantum logic. (shrink)

Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculusOMC is proposed for the orthomodular-valid formulas.OMC is based on two primitive connectives — the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra ofOMC is constructed, and it is shown to be (...) an orthomodular lattice whose unit element is the equivalence class of theses ofOMC. (shrink)

The invitation of the Organizing Committee for me to speak about “Unsolved problems in mathematics” fills me as it should with considerable trepidation and a prevailing feeling of personal inadequacy. Hilbert gave a talk on this subject at the similar congress about 50 years ago and this is a very formidable precedent. He stated about a dozen unsolved problems in another widely separated areas of mathematics, and they proved to be prototypical for much of the development that followed in the (...) next decades. It would be absolutely foolish, if I tried to emulate this quite singular feat. In addition I do not know the future and the future at any rate can only be predicted ex post with any degree of reliability. I will, therefore, define what I am trying to do in a much more narrow way, hoping that in this manner I have a better chance of not failing. I will limit myself to a particular area of mathematics which I think I know and I will talk about it and about what its open ends appear to be, particularly in some directions which are not the ones that the evolution so far has mainly emphasized and which are, I think, quite important. (shrink)