Handbook of quantum logic and quantum structures; quantum logic

HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM LOGIC www.pdfgrip.com This page intentionally left blank www.pdfgrip.com HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM LOGIC Edited by KURT ENGESSER Universität Konstanz, Konstanz, Germany DOV M GABBAY King's College London, Strand, London, UK DANIEL LEHMANN The Hebrew University of Jerusalem, Jerusalem, Israel Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo North-Holland is an imprint of Elsevier www.pdfgrip.com North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2009 Copyright © 2009 Elsevier B.V All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+ 44) (0) 1865 843830; fax (+ 44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-444-52869-8 For information on all North-Holland publications visit our website at elsevierdirect.com Printed and bound in Hungary 09 10 11 12 10 www.pdfgrip.com CONTENTS Foreword Anatolij Dvureˇ censkij vii Editorial Preface Kurt Engesser, Dov Gabbay and Daniel Lehmann ix The Birkhoff–von Neumann Concept of Quantum Logic os Redei ´ e Mikl´ Is Quantum Logic a Logic? Mladen Paviˇ ci´ c and Norman D Megill 23 Is Logic Empirical? Guido Bacciagaluppi 49 Quantum Axiomatics Diederik Aerts 79 Quantum Logic and Nonclassical Logics 127 Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli Gentzen Methods in Quantum Logic Hirokazu Nishimura 227 Categorical Quantum Mechanics Samson Abramsky and Bob Coecke 261 Extending Classical Logic for Reasoning about Quantum Systems Rohit Chadha, Paulo Mateus, Am´ılcar Sernadas and Cristina Sernadas 325 Sol`er’s Theorem Alexander Prestel 373 Operational Quantum Logic: A Survey and Analysis David J Moore and Frank Valckenborgh 389 www.pdfgrip.com vi Contents Test Spaces Alexander Wilce 443 Contexts in Quantum, Classical and Partition Logic Karl Svozil 551 Nonmonotonicity and Holicity in Quantum Logic Kurt Engesser, Dov Gabbay and Daniel Lehmann 587 A Quantum Logic of Down Below Peter D Bruza, Dominic Widdows and John H Woods 625 A Completeness Theorem of Quantum Set Theory Satoko Titani 661 Index 703 www.pdfgrip.com vii FOREWORD More than a century ago Hilbert posed his unsolved (and now famous), 23 problems of mathematics When browsing through the Internet recently I found that Hilbert termed his Sixth Problem non-mathematical How could Hilbert call a problem of mathematics non-mathematical? And what does this problem say? In 1900, Hilbert, inspired by Euclid’s axiomatic system of geometry, formulated his Sixth Problem as follows: To find a few physical axioms that, similar to the axioms of geometry, can describe a theory for a class of physical events that is as large as possible The twenties and thirties of the last century were truly exciting times On the one hand there emerged the new physics which we call quantum physics today On the other hand, in 1933, N A Kolmogorov presented a new axiomatic system which provided a solid basis for modern probability theory These milestones marked the entrance into a new epoch in that quantum mechanics and modern probability theory opened new gates, not just for science, but for human thinking in general Heisenberg’s Uncertainty Principle showed, however, that the micro world is governed by a new kind of probability laws which differ from the Kolmogorovian ones This was a great challenge to mathematicians as well as to physicists and logicians One of the responses to this situation was the, now famous, 1936 paper by Garret Birkhoff and John von Neumann entitled “The logic of quantum mechanics”, in which they suggested a new logical model which was based on the Hilbert space formalism of quantum mechanics and which we, today, call a quantum logic G Mackey asked the question whether every state on the lattice of projections of a Hilbert space could be described by a density operator; and his young student A Gleason gave a positive answer to this question Although this was not part of Gleason’s special field of interest, his theorem, now known as Gleason’s theorem, had a profound impact and is rightfully considered one of the most important results about quantum logics and structures Gleason’s proof was non-trivial When John Bell became familiar with it, he said he would leave this field of research unless there would be a simpler proof of Gleason’s theorem Fortunately, Bell did find a relatively simple proof of the partial result that there exists no two-valued measure on a three-dimensional Hilbert space An elementary proof of Gleason’s theorem was presented by R Cooke, M Keane and W Moran in 1985 In the eighties and nineties it was the American school that greatly enriched the theory of quantum structures For me personally Varadarajan’s paper and subsequently his book were the primary sources of inspiration for my work together with www.pdfgrip.com viii Foreword Gleason’s theorem The theory of quantum logics and quantum structures inspired many mathematicians, physicists, logicians, experts on information theory as well as philosophers of science I am proud that in my small country, Czechoslovakia and now Slovakia, research on quantum structures is a thriving field of scientific activity The achievements characteristic of the eighties and nineties are the fuzzy approaches which provided a new way of looking at quantum structures A whole hierarchy of quantum structures emerged, and many surprising connections with other branches of mathemtics and other sciences were discovered Today we can relate phenomena first observed in quantum mechanics to other branches of science such as complex computer systems and investigations on the functioning of the human brain, etc In the early nineties, a new organisation called International Quantum Structures Association (IQSA) was founded IQSA gathers experts on quantum logic and quantum structures from all over the world under its umbrella It organises regular biannual meetings: Castiglioncello 1992, Prague 1994, Berlin 1996, Liptovsky Mikulas 1998, Cesenatico 2001, Vienna 2002, Denver 2004, Malta 2006 In spring 2005, Dov Gabbay, Kurt Engesser, Daniel Lehmann and Jane Spurr had an excellent idea — to ask experts on quantum logic and quantum structures to write long chapters for the Handbook of Quantum Logic and Quantum Structures It was a gigantic task to collect and coordinate these contributions by leading experts from all over the world We are grateful to all four for preparing this monumental opus and to Elsevier for publishing it When browsing through this Handbook, in my mind I am wandering back to Hilbert’s Sixth Problem I am happy that this problem is in fact not a genuinely mathematical one which, once it is solved, brings things to a close Rather it has led to a new development of scientific thought which deeply enriched mathematics, the understanding of the foundations of quantum mechanics, logic and the philosophy of science The present Handbook is a testimony to this fact Those who bear witness to it are Dov, Kurt, Daniel, Jane and the numerous authors Thanks to everybody who helped bring it into existence Anatolij Dvureˇcenskij, President of IQSA July 2006 www.pdfgrip.com ix EDITORIAL PREFACE There is a wide spread slogan saying that Quantum Mechanics is the most successful physical theory ever And, in fact, there is hardly a physicist who does not agree with this However, there is a reverse of the medal Not only is Quantum mechanics unprecedently successful but it also raises fundamental problems which are equally unprecedented not only in the history of physics but in the history of science in general The most fundamental problems that Quantum Mechanics raises are conceptual in nature What is the proper interpretation of Quantum Mechanics? This is a question touching on most fundamental issues, and it is, at this stage, safe to say that there is no answer to this question yet on which physicists and philosophers of science could agree It is, moreover, no exaggeration to say that the problem of the conceptual understanding of Quantum Mechanics constitutes one of the great intellectual puzzles of our time The topic of the present Handbook is, though related to this gigantic issue, more modest in nature It can, briefly, be described as follows Quantum Mechanics owes is tremendous success to a mathematical formalism It is the mathematical and logical investigation of the various aspects of this formalism that constitutes the topic of the present Handbook This formalism the core of which is the mathematical structure of a Hilbert space received its final elegant shape in John von Neumann’s classic 1932 book “Mathematical Foundations of Quantum Mechanics” In 1936 John von Neumann published, jointly with the Harvard matthematician Garret Birkhoff, a paper entitled “The logic of quantum mechanics” In the Introduction the authors say: ”The object of the present paper is to discover what logical structure we may hope to find in physical theories which, like quantum mechanics, not conform to classical logic” The idea of the paper, which was as ingenious as it was revolutionary, was that the Hilbert space formalism of Quantum Mechanics displayed a logical structure that could prove useful to the understanding of Quantum Mechanics Birkoff and von Neumann were the first to put forward the idea that there is a link between logic and (the formalism of) Quantum Mechanics, and their now famous paper marked the birth of a field of research which has become known as Quantum Logic The Birkhoff-von Neumann paper triggered, after some time of dormancy admittedly, a rapid development of quantum logical research Various schools of thought emerged Let us, in this Introduction, highlight just a few of the milestones in this development In his famous essay “Is logic empirical?” Putnam put forward the view that the role played by logic in Quantum Mechanics is similar to that played by geometry in www.pdfgrip.com 702 Satoko Titani put f0 def = {f | ∃y(♦(y ≺ x) ∧ T (f, y))} ∪ { y, H(f Sy ) | ♦(y ≺ x) ∧ T (f, y)} Then T (f0 , x) BIBLIOGRAPHY [Birkhoff and von Neumann, 1936] Garrett Birkhoff and John von Neumann, The logic of Quantum Mechanics, Ann Math., 37, 823, 1936 [Grayson, 1975] Robin J Grayson, A sheaf approach to models of set theory M.Sc thesis Oxford 1975 [Halmos, 1951] Paul R Halmos, Introduction to Hilbert Space, Chelsea Publishing Company, 1951 [Kodera and Titani, submitted] Heiji Kodera and Satoko Titani, The equivalence of two sequential calculi of quantum logic Submitted [von Neumann, 1955] John von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton University Press, 1955 [Piron, 1976] Constantin P Piron, Foundations of Quantum Physics, W.A Benjamin, Inc 1976 [Takano, 2002] Michio Takano, Strong Completeness of Lattice Valued Logic, Archive for Mathematical Logic, 41 (2002) 497-505 [Takeuti, 1978] Gaisi Takeuti, Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton (1978) [Takeuti, 1981] Gaisi Takeuti, Quantum Set Theory, Current Issues in Quantum Logic, eds E.Beltrametti and B.C.van Frassen, Plenum,New York (1981) pp.303-322 [Titani, 1999] Satoko Titani, Lattice Valued Set Theory, Archive for Mathematical Logic 386(1999) pp.395-421 www.pdfgrip.com 703 INDEX (α ✄ u1 ; u2 ), 339 ( α), 339 (∧G A), 343 (A, Σ), 502 C(x, y), C(x) = y, 681 F≺u , 678 L(˝), 450 M O2 , 561 P (H), 661, 663 V , 661 V L , 661 V Q , 670 W , 681 W Q , 688 -closed, 671 Dom, 677 F ld, 677 ⇔, 669 LZFZ, 662 =⇒, 663 Ord(α), 678 Π(A), 484 Q-valued universe, 670 QL, 661, 666 QZFZ, 663, 676 Rge, 677 ⇒, 663 | α GA , 343 [[ϕ]]W , 688 u, 675 u ˇ, 674 ck(x), 674, 681 ∀W , ∃W , 682 AB, 474 Gl(x), 674 HSuc(x), 682 ♦u, 676 ♦, 665 | ϕ |, 687 ¬, 665 ω, 682 ϕ, ψ, · · · ; Γ,Δ,· · · , 667 x, y , 677 | ψ S , 336 ρ, 687 Q, 687 Q-valued universe, 688 , 665 -closed, 667 Suc(x), 682 →, 661, 665 →τ , 661, 664, 676 T r(α), 678 , 341 ϕ◦| ψ, 676 , 346, 669, 676 {u, v}, 675 {x}, 677 {x ∈ u | ϕ(x)}, 675 b ◦| C, 664 b◦| c, 664 u p, 698 x ∈y, 674 C(X, ⊥), 463 L(A, Σ), 502 Ap , 506 P(u), 675 A1–A11, 675 AC-lattice, 413 additive conjunction, 178 adjoint operator, 216 Aerts’ orthogonality relation, 434 agent, 626 algebraic closure (of a test space), 492 algebraic model for BZL, 169 www.pdfgrip.com 704 Index algebraic model for OL, 135 algebraic PQL-model, 167 Amemiya-Araki Theorem, 461 analogical reasoning, 652 apriorism issue in logic, 51 atom, 115, 211, 452 atomic, 211 atomic formulas, 667 atomic holistic model, 204 atomistic, 211 atomistic lattice, 115 atomisticity, 115 atomless, 211 automaton model, 573 axioms, 81 B, 227, 228 BA-lattice, 171 Bacciagaluppi, G., 65 backward product, 514 Baer ∗-semigroup, 424, 428 Baez, J., 68 basic implication, 661 basic logic, 183 basic orthologic, 185 basis state, 643, 644 Bell inequalities, 53, 71 Bell states, 365 Bell, J S., 56, 71 Benatti, F., 74 benzene ring, 29 Bertrand’s random-chord paradox, 428 binary logics, 25 biorthogonal, 123 biproducts, 264, 290 Birkhoff, G., 1, 2, 4, 10, 12, 18–20, 68, 79, 227, 325, 661 Birkhoff-von Neumann Theorem, 459 bivalence, 56, 57 block, 464, 559 Bohm, D., 70, 71 Bohr, N., 59 Boolean algebra, 29, 56, 80, 210, 449, 552 Boolean lattice, 56 Boolean logic, 80, 652 Born rule, 129, 195, 567 bounded involution lattice, 168, 210 bounded involution poset, 209 bounded operator, 216 bounded poset, 208 bra, 642 Brouwer Zadeh logic, 169 Brouwer Zadeh poset, 155 Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic, 176 Brouwerian modal logic, 227 Brown, H., 60 Bruza, P D., 625 Bub, J., 69 Bueno, O., 51 C*-algebra, 315 Cartan map, 85, 402 categories in physics, 266 category of relations, 273, 276, 282, 286, 292 category theory, 263 center of a test space, 506 of an OML, 452 of an orthoalgebra, 458, 506 centerium, 458 chain, 208 chain-connected, 526 characterisation theorem, 614 check set, 674, 681 chinese lantern, 561 Choi,S., 76 classical logic, 327 tautology, 346 valuation, 334 classical equivalence, 28 classical Hilbert lattice, 613 classical implication, 28 classical logic, 80, 227, 228, 231 www.pdfgrip.com 705 Index classical observables, 68, 69 classical properties, 403 classical property, 103 classical propositions, 66, 67 classical test, 102, 103, 403 Clifton, R K., 69 closed projection, 424 closed subspace, 80, 215 closure operator, 120 co-diagram, 476 cognitive economy, 633, 635 cognitive systems, 626, 628 cognitive target, 630, 635 Cohen-Svetlichny Theorem, 511 coherent space, 176 collapse, 53, 71, 73 Colyvan, M., 51, 76 comensurability, 28 commutative ring, 213 commutativity, 28 compact closure, 275 compatibility, 54, 417 compatible, 664, 676 compatible complement, 404, 405 complementary events, 476 complementary observables, 316 complementary sentence, 174 complemented lattice, 208 complete lattice, 94 completeness, 93 dEQPL, 364 completeness of classical logic, 39 completeness of quantum logic, 36 composite physical system, 433 compositional QC-model, 198 compositional quantum computational semantics, 198 compositional semantics, 193 concept combination, 637, 649 concept generalization, 653 conceptual space, 628, 636, 639 conditional, 57 conditioning map, 508 configuration space, 70, 71 conjunction, 80 consciousness, 633, 634 consequence relation, 590 consequence revision system, 593 context, 551, 636, 640, 645, 648 contextual hidden variable theories, 150 contextual hidden variables, 65 contextual holistic model, 206 contextual meaning, 205 contextuality, 569 continuous geometry, 12, 13 continuous spontaneous localisation, 73 contraction, 180 contraposition theorem, 227, 230, 232, 235 conventionalism, 50, 72 convex polytope, 553 counterfactual conditional, 143 counterfactual connective, 57 covariance principle, 100 covering law, 119 covering property, 211, 409 Curry-Howard isomorphism, 176 cut, 232, 253, 254 cut elimination, 186 cut-elimination theorem, 227, 228, 232, 250, 251 Cutland, N J., 227, 232 Dacey space, 464 Dacey sum, 475 Dacey’s Theorem, 463 dagger category, 424 Dalla Chiara, M L., 24, 54, 57, 58 de Broglie’s matter waves, 71 de Broglie, L., 70, 71 de Broglie–Bohm theory, 51, 65, 70– 72, 74 decidability dEQPL, 364 decision theory, 75 decoherence, 66, 72, 74 www.pdfgrip.com 706 Index decomposition, 102 decomposition, spectral, 645 deduction theorem, 25 deductive quantum logic, 24 definite, 673 definite experimental project, 397 definite expermental project, 396 density matrix, 644, 647, 651 density operator, 129, 218 densly defined operator, 216 dEQPL, 339 axioms, 346 decidability, 364 dynamic extension, 367 language, 338 logic amplitude, 335 metatheorem of deduction, 351 model existence theorem, 362 semantics, 341 soundness, 350 weak completeness, 364 deterministic evolution, 425 Deutsch, D., 75 Dickson, M., 49, 63, 64, 66, 68, 76 different posets, 158 dimension, 215 Dirac notation, 286 Dirac, P A M., 59 direct sum Hilbert space, 219 direct sum of test spaces, 473 direct union, 113 Dishkant implication, 24 Dishkant, H., 24, 227, 228 disjunction, 80 dispersion-free state, 471 disposition, 60, 71 distributive lattice, 210 distributive law, 54, 80 distributivity, 3, 40 division ring, 213 domain theory, 176 dual intuitionistic logic, 185 dual modular pair, 408, 414 duality theorem, 232, 239, 240, 249 Dummett, M., 50 Dunn, J M., 2, 24 dynamics, 71, 73, 74 effect algebra, 157, 403, 456 effects, 153 eigenstate, 643, 645 Einstein, A., 72, 83 Einstein-Podolsky–Rosen paradox, 53 electron diffraction, 71 element of reality, 400 elementary properties, 62, 74 elements of reality, 392 emergence of the classical world, 72 encodedness, 602 Engesser, K., 76 entailment dEQPL, 341 entangled pure state, 203 entity, 84, 502 mapping, 503 standard, 504 equivalence operations, 32 event-state structure, 428 event-state system, 131 event-state-operation structure, 429 Everett, H., 51, 70, 72, 74–76 evolution deterministic, 425 maximally deterministic, 427 exchange property, 409 exogenous approach, 327, 329 exogenous quantum propositional logic axioms, 346 decidability, 364 language, 338 metatheorem of deduction, 351 model existence theorem, 362 semantics, 341 soundness, 350 weak completeness, 364 experiment, 88 first-kind, 418 ideal, 418 www.pdfgrip.com 707 Index experimental propositions, 50, 54, 59, 61, 71 extensional, 679 [F ], 339 field, 213 fifth Solvay conference, 70 filter, 497 boolean, 497 in an orthoalgebra, 501 local, 498 filtering, 56 Finch, P D., 24, 58 finite-dimensional holistic logic, 603 Finkelstein, D H., 587 first-kind experiment, 418 formula, 667 formula of dEQPL g-satisfiable, 359 s-satisfiable, 361 arithmetical, 341 classical, 338 comparison, 339 consistent, 356 extent, 340 global, 327 molecule, 356 quantum, 339 quantum atom, 340 quantum sub-, 340 satisfaction of quantum, 341 sub-system, 339 Foulis Compactness Theorem, 501 Foulis semigroup, 424 Foulis, D J., 28, 58 Foulis-Piron-Randall Theorem, 505 Foulis-Randall Theorem, 466 Frame manual, 468 free G-extension, 529 Friedman, M., 58, 65 functional, 678 fuzzy quantum logic, 58 fuzzy set, 160 Gă ardenfors, P., 627, 628, 636, 639 Gal, O., 76 Galois adjunction, 419 Gaukroger, S., 76 Gelfand quantale, 425 general relativity, 57, 72, 79 generalised Hilbert space, 81 generalized urn model, 572 generating set, 93 generic inference, 632, 633 Geneva school, 54, 394, 396 Gentzen method, 227 Gentzen, G., 227, 228, 232 geometry, 49, 57, 72 Ghiradi, G.C., 73 Gibbins, P F., 227, 232 Giuntini, R., 57, 58 GKLM model, 591 Gleason’s Theorem, 453, 469 unentangled, 521 Gleason;s Theorem, 567 global, 674, 678 logic, 327 valuation, 327 Glymour, C., 65 Goldblatt, R I., 24, 227, 228 Golfin’s Theorem, 519 Grassi, R., 74 greatest lower bound, 562 Greaves, H., 74 Greechie diagram, 43, 464, 470, 559 Greechie space, 464 Greechie, R J., 23 Grossman, J., 76 group, 212 group representation, 80 GRW theory, 73 Gudder, S P., 23, 58 Haase diagrams, 43 Halmos, P., Halpern, J Y., 326 Hardegree, G M., 2, 25, 57 Hasse diagram, 559 hasty generalization, 631–633 www.pdfgrip.com 708 Index Heisenberg, W., 79 hemimorphisms, 420 Hermitian product, 123 hexagon, 29 hexagon interpretation, 42 Heyting algebra, 418 Heywood, P., 65 hidden variable problem, 149 Hilbert lattices, 54, 130, 556 Hilbert quantale, 425 Hilbert space, 52, 79, 129, 191, 215, 329, 642 Hilbert space logic, 618 Hintikka, J., 626 Hoare triple, 367 holicity, 589 holism, 60 holistic logic, 589, 602 holistic model, 205 holistic semantics, 193, 204 Holland, S S., Jr., 29 horizontal sum of test spaces, 473 hull problem, 554 human agent, 629, 633 human reasoning, 626, 627 hyperspace analogue to language, 638 ideal experiment, 418 ideal in an orthoalgebra, 501 implication, 24, 87 imprimitivity, 529 set of, 530 incompatibility, 58 infimum, 87 infinitely many degrees of freedom, 57, 68 inner product, 214 inner product a` la Hilbert-Schmidt, 191 inner product space, 331 free, 332 institutional agent, 629, 630 interference and diffraction, 70 internal perspective, 74, 75 internalising connective, 597 intersection property, 412 intuitionistic logic, 50, 54, 141, 185, 227, 228 inverse test, 90 involution (on a poset), 449 involution bounded poset, 154 irreducible, 116, 212 orthoalgebra, 458 irreducible lattice, 116 irredundant, 468 isomorphic, 81 Jauch, J M., 23, 54, 56, 61 Jauch-Piron property, 408 jointly orthogonal, 458 HK(δ) (B), 333 K(δ), 332 Kalmbach implication, 24 Kalmbach, G., 24, 32 Keller, H A., 83, 613 ket, 642 Keynes , J M., 16 kinematics, 71, 74 Klă ay-Randall-Foulis Theorem, 521 Kleene lattice, 210 Kleene poset, 210 Kleene rule, 168 Kochen, S., 58, 65 Kochen-Specker theorem, 564 Kochen-Specker vector systems, 43 Kodera, H., 669 Kolmogoroff, A., 54 Kripke, S., 227 Kripkean model for BZL, 169 Kripkean model for OL, 134, 135 Kripkean model for PL, 140 Kripkean semantics, 132 Kripkian relational semantics, 227, 228 latent semantic analysis, 641 lattice, 81, 208, 562 atomistic, 452 irreducible, 452 www.pdfgrip.com 709 Index modular, 459 orthomodular, 450 projection-, 450 lattice valued logic, 661 lattice valued set theory, 662 lattice valued universe, 661 latttice O6, 29 law of excluded middle, 54 least upper bound, 562 Lewis’s Principal Principle, 75 Lewis, D., 75 limiting case theorem, 612 Lindenbaum algebra, 38, 40 linear implication, 178 linear logic, 168, 175 linear operator, 216 link observable, 563 local implication, 661, 676 local logic, 49, 50, 72 Lock dual, 516 logic classical, 327 global, 327 modal, 329 probabilistic, 327–329 quantum, 329 logic of a test space, 484 logical axiom, 667 loop, 465 Loop Lemma generalized, 496 Greechie’s, 465 lower bound, 94 Lukasiewicz operation, 160 Lukasiewicz quantum logic, 173 MacFarlane, J., 76 Mackey decomposition, 458 Mackey system, 455 Mackey’s axioms, 455 Mackey, G., 81 macro-world, 83 macroscopic state, 403 Malinowski, J., 25 manual, 495 many worlds, 51, 74 matrix mechanics, 79 maximality, 56 maximally deterministic evolution, 427 maximum, 101 Mayet, R., 614 McKay, B D., 43 McKinsey, J C., 227 meaning, 67 meaning collapse, 645–648 meaning of the connectives, 66 measurement problem, 49, 59, 60, 64, 66, 73 measurement theory, 71 measuring apparatus, 82 meet property, 87 Megill, N D., 25, 38 merged equivalence, 25 merged implications, 25 Merlet, J.-P., 43 microworld, 79 minimal logic, 141 minimal quantum logic, 25, 227, 228, 230 Minkoswki-Weyl representation, 554 Mittelstaedt, P., 24 mixec states, 391 mixed state, 128, 194 MMP (McKay, Megill-Paviˇci´c) diagrams, 43 modal operator, 662 modal translations, 54 modality probability, 343 quantum, 343 modular lattice, 211, 459 modular pair, 414 modularity, modus ponens, 33, 34 monoid, 212 monoidal category, 272 monoidal symmetric frame, 187 monoidal symmetric model, 188 www.pdfgrip.com 710 Index morphism, 100 multiplicative conjunction, 178 Murray, F J., 9, 11, 14, 19, 20 MV algebra, 162 νGA , 336 natural deduction, 137 natural kinds, 633 network of beliefs, 50 Newtonian mechanics, 82 Newtonian physics, 82 Nilsson, N J., 326 Nishimura, H., 24, 227 no go theorem, 149 no windows theorem, 609 non-contextual hidden variables, 65 non-equilibrium, 71 non-locality, 53, 397 non-tollens implication, 25 nonBoolean logic, 80 nonclassical components, 111 nonEuclidean geometry, 80 nonlocality, 83 nonmonotonic logic, 590 nonmonotonic reasoning, 636, 637 nonmonotonicity, 588 nonspatiality, 83 norm of a vector, 214 normed space, 332 numbers complex, 83 quaternionic, 83 real, 83 O’Connor, R., 76 objective chances, 75 observable, 416, 646 observables, 53, 63, 71, 392 observer, 82 of course, 180 operation, 428 operational proposition, 80 operational quantum logic, 389 operational resolution, 422 operationally separated systems, 434 operations of implication, 32 operator, 216 order-determining set of states, 456 ordinal, 678 ortho property, 98 ortho test, 96 ortho-adjoint, 422 orthoalgebra, 158, 457 atomistic, 507 regular, 458, 539 topological, 537 orthoarguesian law, 140 orthoclosed, 463 orthocoherent orthoalgebra, 457 partial abelian semigroup, 455 test space, 491 orthocomplement, 129 orthocomplementation, 97, 449 orthoframes, 133, 177 orthogonal events, 476 supports, 505 orthogonal projection operator, 80 orthogonal properties, 91 orthogonal states, 91 orthogonality, 90, 404 orthogonality relation, 462 orthogonality space, 404, 462 Dacey, 464 ortholattice, 27, 210, 227, 230, 449, 664 orthologic, 25, 133, 185 orthomodular, 373–378, 385, 450 lattice, 450 law, 450 poset, 450 orthomodular lattice, 28, 54, 57, 67, 68, 211, 598, 664 orthomodular law, 38 orthomodular logics, 24 orthomodular poset, 159, 211 orthomodular posets, 265 orthomodular property, 407 www.pdfgrip.com 711 Index orthomodular quantum logic, 133 orthomodular space, 460, 613 orthomodular space logic, 618 orthomodular vector space, 123 orthomodularity, 3, 28 orthomonoid, 465 orthomorphism, 102 orthonormal base, 84 orthonormal basis, 215 orthonormal set of vectors, 214 orthopartition, 469 orthoposet, 158, 210, 449 outer product, 642 P Lock’s Theorem, 492 paraconsistent logic, 173 paraconsistent quantum logic, 166 partial abelian semigroup, 454 partial Boolean algebra, 58 partial Boolean algebras, 151 partial classical logic, 151, 172 partial homomorphism, 56, 57 partial order relation, 92 particular physical system, 396 partition logic, 572 Paviˇci´c, M., 25, 38 Pearle, P., 73 perfect, 418 perspective events, 476 Petri-Toffoli gate, 195 phase semantics, 187 phase space, 52, 70 phase structures, 188 phase-space, 128 phenomenology of measurement, 53 physical system, 396 classical, 86 quantum, 86 Pincock, C., 76 Piron’s axioms, 461 Piron’s Theorem, 461 Piron, C., 23, 54, 56, 61, 81 plane transitivity, 120 Poincar´e, H., 50 poset, 208 positive logic, 140 positive operator, 216 positive-operator-valued measure, 58 positive-valued measure, 58 PQL-Kripkean model, 166 practical agent, 628, 630, 635 practical reasoning, 626, 628, 635, 637 pre-geometry, 83 pre-Hilbert space, 214 pre-Lindenbaum Lemma, 146 pre-order relation, 86 pre-ordered state, 87 preclusivity space, 131 preparation procedure, 392 Price, H., 76 Primas, H., 76 principal ultrafilters, 401 Principia Mathematica, 33 probabilistic logic, 327–329 valuation, 328 probability, 13–17, 19 modality, 343 probability of a qumix, 195 product, 90 product test, 90 projectale, 418 projection operator, 217, 646, 647, 649 projective geometry, 80, 412 projective lattice, 412 projective law, 413 projective representation, 80 proofnets, 185 properties, 392 property, 502 detectable, 502 lattice (of an entity), 502 actual, 84, 400 classical, 403 equivalent, 86 potential, 84, 400 property lattice, 399 www.pdfgrip.com 712 Index property state, 95 proposition, 399 proposition system, 664 pure state, 128, 194 pure states, 391 Putnam, H., 49, 51, 54, 57–59, 61–67, 70, 72, 76 QMP, 345 QTaut, 345 qB, 333 quadratic space, 460 quantaloid, 421 quantum abbreviations, 342 atom, 340 connectives, 340 consistent formula, 356 disjunctive normal form, 356 formula, 339 literal, 356 logic, 329 modality, 343 molecular formula, 356 structure, 336 sub-formula, 340 tautology, 344 valuation, 334 quantum axiomatics, 79 quantum coherent space, 191 quantum collapse, 642 quantum computational logic QCL, 199 quantum computational logics, 193 quantum disjunction, 653 quantum entity, 80 quantum equivalence, 28 quantum implication, 28 quantum logic, 32, 80, 227, 228, 230, 263, 265, 287, 308, 652, 653, 655, 661, 666 quantum logical gates, 195 quantum mechanics, 79, 627, 637, 641, 642, 646 postulates, 330, 334, 336–338 quantum MV algebra, 160, 165 quantum MV algebras, 157 quantum propositional logic axioms, 346 decidability, 364 language, 338 metatheorem of deduction, 351 model existence theorem, 362 semantics, 341 soundness, 350 weak completeness, 364 quantum set theory, 663, 676 quantum state, 643 quantum teleportation, 261, 269, 281, 300, 367 quantum trees, 200 qubit, 194, 642 qubit tree of α, 201 qubit-model, 199 question, 397 qumix, 194 quregister, 194 RCF, 345 R Lock’s Theorem, 517 Raggio, G., 67, 68 Randall, C H., 58 rank, 670 rank of a test space, 468 ray, 80 real closed field, 330 algebraic closure, 331 reality, 82 Redhead, M L G., 65 regular paraconsistent quantum logic, 167 relation of congruence, 36, 39 relation of equivalence, 36, 39 representation theorem, 81, 619 residual mappings, 420 residuated mappings, 420 resource adjustment strategies, 630, 631 www.pdfgrip.com 713 Index resource interpretation, 180 Restriction, 698 reversible transformation, 195 revision of logic, 50, 54, 62 Rimini, A., 73 ring, 212 ring with unity, 213 Russell, B., 39 S4, 227, 228 S5, 228 Sach-Arellano, Z., 76 Sasaki hook, 24, 28, 137 Sasaki projection, 24, 411, 452 generalized, 510 Sato, M., 227 Saunders, S., 74 scalar product, 213 Schechter, E., 42 Schră odinger dynamics, 60 Schră odingers cat, 49, 60, 64 Schră odingers equation, 53, 60, 73 Schră odinger, E., 53, 79 scotian logic, 146 self-adjoint operator, 53, 217 semantic space, 637, 639–644, 653, 655 semi-interpreted language, 54, 56, 61 Semiclassical, 473 semimodular, 415 separable, 212 separating, 408 sequent, 667 sequent calculus, 137, 179 sequent systems suggested for quantum logics, 181 sharp QT, 152 sharp test, 405 Silsbee, F B., 17 similarity space, 131 simultaneous observability, 417 Sol`eer, M P., 461 Sol`er’s theorem, 373–387, 613 Sol`er, M P., 84 soundness dEQPL, 350 soundness and completeness, 56 soundness of classical logic, 35 soundness of quantum logic, 34 space conceptual, 628 Hilbert, 642 semantic, 639 state space, 640 special relativity, 72 Specker, E P., 58, 65, 76 spontaneous collapse theory, 51, 63, 70, 73, 74 square root of the negation, 195 stable identity, 74 stably complemented, 542 Stachow, E.-W., 24 Stairs, A., 65 standard interpretation of quantum mechanics, 50, 51, 61, 62, 73 state, 80, 84, 391, 452 σ-additive, 452 -on an OMP, 452 of an entity, 502 macroscopic, 403 mixed, 391 pure, 391 state identification problem, 573 state property space, 85 state property system, 99 statistical mechanics, 71 stnadard interpretation, 59 strong compact closure, 264, 285 strong negation, 54 strong negative, 60 strong set of states, 456 strongly order-determining, 402 subject-object problem, 82 subspace-, 450 subsymbolic reasoning, 627, 628, 635, 655 subsystem recognition problem, 434 www.pdfgrip.com 714 Index superposition, 120, 639, 646 superposition principle, 129 superselection, 68, 75, 121 superselection rule, 409 superselecxtion rule, 403 suport, 430 Support in an orthoalgebra, 501 support, 479, 499 central, 506 minimal, 510 of a local filter, 498 of an homomorphism, 478 supremum, 93 sybsymbolic reasoning, 652 symbolic reasoning, 635 symmetric frame, 187 symmetric monoidal category, 272 symmetry, 522 symmetry condition, 621 Takano, M., 227, 228, 232, 661 Takeuti, G., 661 Tamura, S., 227, 232 Tarski, A., 227, 326 tautology classical, 346 quantum, 346 tensor product, 52, 68, 265, 273, 296, 308, 334 and influence-free states, 520 of algebraic test spaces, 515 of boolean test spaces, 518 of frame manuals, 519 of orthoalgebras, 515 of property lattices, 518 of quantum logics, 466 of UDF test spaces, 516 tensor product Hilbert space, 218 term, 667 term of dEQPL alternative, 339 amplitude, 339 arithmetical, 341 denotation of, 341 probability, 339 test, 88, 397 test space, 467 compounding, 474 G-test space, 522 algebraic, 488 bilateral product, 514 Borel, 468 chain-connected, 510 coherent, 493 Dacey, 486 Fano, 470 forward product, 513 frame manual, 468 fully symmetric, 523 Greechie, 471 homomorphism, 478 interpretation, 478 locally finite, 468 logic of, 484 of orthopartitions, 469 ortho-symmetric, 526 orthocoherent, 491 pre-algebraic, 491 regular, 486 semi-classical, 468 Semiclassical cover, 473 strongly symmetric, 523 symmetric, 522 topological, 531 UDF, 471 Wright Triangle, 472 test space approach, 58 testable propositions, 61, 63 time-dependent equilibium, 71 time-space, 83 Titani, S., 661 topological OML (TOML), 537 trace functional, 217 transitive, 679 truth valuation, 56, 57, 61, 65, 67 truth-functional, 64–67, 69, 75, 76 twist, 529 www.pdfgrip.com 715 Index ultrafilter, 57, 61, 67 unary logics, 25 undecidability of propositional linear logic, 181 unit (or normalized) vector, 214 unit vector, 80 unitary operator, 218 universe, 82 unsharp partial quantum logic (UPaQL), 172 unsharp QT, 152 unsharp quantum logics, 166 Widdows D., 625 Woods, John, 625 word sense, 643–645 yes/no experiments, 81 Zeman, J., 24 Zorn’s lemma, 57 valid, 662, 677 valuation classical, 334 global, 327 probabilistic, 328 quantum, 334 van Fraassen, B., 54 vector space, 213 Vietoris topology, 532 von Mises, R., 13, 15, 19 von Neumann algebra, 5, 6, 9, 11, 12, 19, 51, 65, 68, 75, 76 von Neumann, J., 1, 2, 4–10, 12–20, 59, 60, 68, 79, 227, 325, 661 w, 336 Wallace, D., 74, 75 wave machine, 79 weak completeness dEQPL, 364 weak implication, 484 weak modularity, 116 weak observable, 416 weakening, 180 weakly orthomodular ortholattice, 28 Weber, T., 73 weight, 481 vector-valued, 483 well order, 679 well-founded, 677 Whitehead, A N., 39 why not, 180 www.pdfgrip.com This page intentionally left blank www.pdfgrip.com .. .HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM LOGIC www.pdfgrip.com This page intentionally left blank www.pdfgrip.com HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM LOGIC. .. www.pdfgrip.com HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM LOGIC Edited by K Engesser, D M Gabbay and D Lehmann © 2009 Elsevier B.V All rights reserved THE BIRKHOFF–VON NEUMANN CONCEPT OF QUANTUM. .. www.pdfgrip.com HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM LOGIC Edited by K Engesser, D M Gabbay and D Lehmann © 2009 Elsevier B.V 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