This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy (...) is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory. (shrink)

The Society for the Advancement of the Scientific World Conception has done me a great honor by inviting me to be the Sixth Vienna Circle Lecturer. The invitation has also stirred some deep emotions. A central figure of the Vienna Circle, Rudolf Carnap, was my revered teacher of philosophy at the University of Chicago in 1948–9 and later an informal adviser when I wrote a doctoral thesis at Yale University on inductive logic, and he was a friend during those years (...) and thereafter. I was not a disciple, but Carnap did not demand discipleship as a condition for admission to his seminars or to his friendship. He seemed to be baffled by the fact that despite my interest in mathematical logic and theoretical physics I proclaimed myself a metaphysicician and had even published an article in the first issue of The Review of Metaphysics. Carnap formulated a “principle of tolerance” as a philosophical maxim concerning rules of language, but he practiced a human and highly moral version of the principle of tolerance in his profoundly liberal social commitments and in his relations with his students. If he were here tonight, I would wish for his tolerance of the lapses of rigor and the flights of speculation to which he would be exposed. (shrink)

From the very beginning it was realised that quantum physics involves radically new interpretative and epistemological consequences. While hitherto there has been no satisfactory philosophical analysis of these consequences, recent years have witnessed the accomplishment of many experiments to test the foundations of quantum physics, opening up vistas to a completely novel technology: quantum technology. The contributions in the present volume review the interpretative situation, analyze recent fundamental experiments, and discuss the implications of possible future technological applications. Readership: Analytic philosophers (...) (logical empiricists), scientists (especially physicists), historians of logic, mathematics and physics, philosophers of science, and advanced students and researchers in these fields. Can be used for seminars on theoretical and experimental physics and philosophy of science, and as supplementary reading at advanced undergraduate and graduate levels. (shrink)

In this new edition, Arthur Fine looks at Einstein's philosophy of science and develops his own views on realism. A new Afterword discusses the reaction to Fine's own theory. "What really led Einstein . . . to renounce the new quantum order? For those interested in this question, this book is compulsory reading."--Harvey R. Brown, American Journal of Physics "Fine has successfully combined a historical account of Einstein's philosophical views on quantum mechanics and a discussion of some of the philosophical (...) problems associated with the interpretation of quantum theory with a discussion of some of the contemporary questions concerning realism and antirealism. . . . Clear, thoughtful, [and] well-written."--Allan Franklin, Annals of Science "Attempts, from Einstein's published works and unpublished correspondence, to piece together a coherent picture of 'Einstein realism.' Especially illuminating are the letters between Einstein and fellow realist Schrodinger, as the latter was composing his famous 'Schrodinger-Cat' paper."--Nick Herbert, New Scientist "Beautifully clear. . . . Fine's analysis is penetrating, his own results original and important. . . . The book is a splendid combination of new ways to think about quantum mechanics, about realism, and about Einstein's views of both."--Nancy Cartwright, Isis. (shrink)

ince the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition is dual to the notion of a subset. Hence the Boolean logic of (...) subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...) a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...) all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how finite probability theory can be interpreted using #2 abstracts as “superposition (...) events” in addition to the ordinary events. The goal is to use the second notion of abstraction to shed some light on the notion of an indefinite superposition in quantum mechanics. (shrink)

Categorical logic has shown that modern logic is essentially the logic of subsets (or "subobjects"). Partitions are dual to subsets so there is a dual logic of partitions where a "distinction" [an ordered pair of distinct elements (u,u′) from the universe U ] is dual to an "element". An element being in a subset is analogous to a partition π on U making a distinction, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite (...) probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered pairs |U|² from the finite universe. That yields a notion of "logical entropy" for partitions and a "logical information theory." The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon's theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon's theory based on the logical notion of "distinctions.". (shrink)

First-ever comprehensive introduction to the major new subject of quantum computing and quantum information.

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...) the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2. (shrink)