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Valuations for the quantum propositional structures and hidden variables for quantum mechanics Chernavska, Ariadna 1980

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VALUATIONS FOR THE QUANTUM PROPOSITIONAL STRUCTURES AND HIDDEN VARIABLES FOR QUANTUM MECHANICS by ARIADNA CHERNAVSKA B.A., The U n i v e r s i t y of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Philosophy) We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA •-July 1 1979 • @ Ariadna Chernavska In present ing t h i s t he s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re fe rence and study. I f u r t h e r agree that permiss ion f o r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n permi s s ion . Department of PHILOSOPHY The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 A b s t r a c t The t h e s i s i n v e s t i g a t e s the p o s s i b i l i t y of a c l a s s i c a l semantics f o r quantum p r o p o s i t i o n a l s t r u c t u r e s . A c l a s s i c a l semantics i s defined as a set of mappings each of which i s ( i ) b i v a l e n t , i . e . , the value 1 (true) or <p! ( f a l s e ) i s assigned t o each p r o p o s i t i o n , and ( i i ) t r u t h - f u n c t i o -n a l , i . e . , the l o g i c a l operations are preserved. I n a d d i t i o n , t h i s set must", be " f u l l " . , i . e . , any'pair of d i s t i n c t p r o p o s i t i o n s i s assigned d i f f e -rent values by some mapping i n the s e t . When the p r o p o s i t i o n s make asser-t i o n s about the p r o p e r t i e s of c l a s s i c a l or of quantum systems, the mappings should a l s o be ( i i i . ) " s tate-induced", i . e . , values assigned by the seman-t i c s should accord w i t h values assigned by. c l a s s i c a l or by quantum me-chanics. In c l a s s i c a l p r o p o s i t i o n a l l o g i c , (equivalence c l a s s e s . o f ) propo-s i t i o n s form a Boolean a l g e b r a , and each semantic;(mapping assigns the value 1 t o the members of a c e r t a i n subset of th e , a l g e b r a , namely, an u l t r a f i l t e r , and assigns/p'^to the members of the dual u l t r a i d e a l , where the union of these two subsets i s the e n t i r e algebra. The p r o p o s i t i o n a l s t r u c t u r e s o f c l a s s i c a l mechanics are l i k e w i s e Boolean a l g e b r a s , so one can s t r a i g h t f o r w a r d l y provide a c l a s s i c a l semantics, which a l s o s a t i s f i e s ( i i i ) . However, quantum p r o p o s i t i o n a l s t r u c t u r e s are non-Boolean, so i t i s an open question whether a semantics s a t i s f y i n g ( i ) , ( i i ) and ( i i i ) can be provided. Von Neumann f i r s t proposed (.1932) that the a l g e b r a i c s t r u c t u r e s of the subspaces (or p r o j e c t o r s ) of H i l b e r t space be regarded as the pro-p o s i t i o n a l s t r u c t u r e s P ^ of quantum mechanics. These s t r u c t u r e s have been f o r m a l i z e d i n two ways: as orthomodular l a t t i c e s which have the b i n a r y operations "and", " o r " , de f i n e d among a l l elements, compatible i and incompatible j£ ; and as p a r t i a l - B o o l e a n algebras which have the i i "binary operations d e f i n e d among only compatible elements. In the t h e s i s , two b a s i c senses i n which these s t r u c t u r e s are non-Boolean are d i s c r i -minated. And two notions of t r u t h - f u n c t i o n a l i t y are d i s t i n g u i s h e d : t r u t h -f u n c t i o n a l i t y ( ) a p p l i c a b l e t o the P n l a t t i c e s ; and t r u t h - f u n c t i o -n a l i t y ( A ) a p p l i c a b l e t o both the l a t t i c e s and p a r t i a l - B o o l e a n a l g e -bras . Then i t i s shown how the l a t t i c e d e f i n i t i o n s of "and", " o r " , among incompatibles r u l e out a b i v a l e n t , t r u t h - f u n c t i o n a l ( A,/^f) semantics f o r any l a t t i c e c o n t a i n i n g incompatible, elements. In c o n t r a s t , the Gleason and Kochen-Specker proofs of the i m p o s s i b i l i t y of h i d d e n - v a r i a b l e s f o r quantum mechanics show the i m p o s s i b i l i t y of a b i v a l e n t , t r u t h - f u n c -t i o n a l ( A ) semantics f o r three-or-higher dimensional H i l b e r t space s t r u c t u r e s ; and the presence of incompatible elements i s necessary but i s not s u f f i c i e n t t o r u l e out such a semantics. As f o r ( i i i ) , each quantum state-induced e x p e c t a t i o n - f u n c t i o n on a.PQM t r u t h - f u n c t i o n a l l y assigns 1 and'iS' values t o the elements i n a u l t r a f i l t e r and dual u l t r a i d e a l of P_„. where, i n general the union of an u l t r a f i l t e r and i t s dual u l t r a i d e a l i s smaller than.the e n t i r e s t r u c -t u r e . Thus i t i s argued t h a t each e x p e c t a t i o n - f u n c t i o n i s the quantum analog of a c l a s s i c a l semantic mapping, even though the domain where each e x p e c t a t i o n - f u n c t i o n i s b i v a l e n t and t r u t h - f u n c t i o n a l i s u s u a l l y a non-Boolean substructure of The f i n a l p o r t i o n of the t h e s i s surveys proposals f o r the i n t r o -duction of hidden v a r i a b l e s i n t o quantum mechanics, proofs of the im-p o s s i b i l i t y o f such h i d d e n - v a r i a b l e p r o p o s a l s , and c r i t i c i s m s of these i m p o s s i b i l i t y p r o o f s . And arguments i n favour of the p a r t i a l - B o o l e a n a l g e b r a , r a t h e r than the orthomodular l a t t i c e , f o r m a l i z a t i o n o f the quantum p r o p o s i t i o n a l s t r u c t u r e s are reviewed. ." i i i -TABLE OF CONTENTS Chapter 0 I n t r o d u c t i o n 1 Chapter I A l g e b r a i c P r e l i m i n a r i e s 8 A. Group and Ring S t r u c t u r e s 8 B. The Boolean Algebra and the Boolean L a t t i c e . . . . 10 C. Subsets o f a Boolean S t r u c t u r e 18 D. The Quantum P a r t i a l - B o o l e a n Algebra 20 E. The Quantum Orthomodular L a t t i c e . . 24 F. Subsets o f and p 28 QMA QML G. Mappings on a S t r u c t u r e '.' 30 Chapter I I The C l a s s i c a l Precedent f o r a B i v a l e n t T r u t h - f u n c t i o n a l Semantics 33 A. The Standard Semantics o f C l a s s i c a l P r o p o s i t i o n a l Logic 33 B. The Boolean S t r u c t u r e Determined by C l a s s i c a l P r o p o s i t i o n a l Logic . . . . . 34 C. B i v a l e n t Homomorphic Mappings on Any Boolean S t r u c t u r e 36 D. The A l g e b r a i c Semantics f o r the Lindenbaum Algebra . 39 Chapter I I I The C l a s s i c a l Precedent f o r a State-induced Semantics . 43 Preface 43 A. The States o f a C l a s s i c a l System Determine the Real Values o f That System's Magnitudes . . . . . . 44 B. The P r o p o s i t i o n a l S t r u c t u r e Determined by C l a s s i c a l Mechanics 46 C. The B i v a l e n t , T r u t h - f u n c t i o n a l , State-induced Semantics f o r the Boolean P_w S t r u c t u r e s . . . .-' 48 CM Chapter IV The Non-Boolean P r o p o s i t i o n a l S t r u c t u r e s Determined by Quantum Mechanics 57 A. The Fundamental P o s t u l a t e s of Quantum Mechanics . . 57 B. I n c o m p a t i b i l i t y 59 C. The P r o p o s i t i o n a l S t r u c t u r e s Determined by Quantum Mechanics 61 D. The P a r t i a l - B o o l e a n Algebra and the Orthomodular L a t t i c e Quantum P r o p o s i t i o n a l S t r u c t u r e s 66 E. R a m i f i c a t i o n s o f the Bas i c D i f f e r e n c e between P.M. and P_MT 72 QMA QML F. The Two Bas i c Senses i n Which the Quantum P r o p o s i t i o n a l S t r u c t u r e s Are Non-Boolean 77 i v Chapter V The Impossibility of a Bivalent, Truth-functional Semantics for the Non-Boolean Propositional Structures Determined by Quantum Mechanics . . . . . . . 90 Preface 90 A. The Impossibility of a Bivalent, Truth-functional(6,&) Semantics for Any P Which Contains Incompatible Elements 91 B. The Kochen-Specker Proof of the Impossibility of a Bivalent Homomorphism(o) on a Three Dimensional Hilbert Space PniuIA 97 c QMA C. Avoiding These Impossibility Proofs 101 D. The Meaning of the Hidden-Variable Impossibility Proofs for the Issue of a Classical Semantics for the Quantum Propositional Structures 103 Summary 109 Chapter VI A State-induced Semantics for the Non-Boolean Propositional Structures Determined by Quantum Mechanics 114-A. The Quantum State-induced Expectation Functions . . 114 B. The Quantum Expectation Function As an Ultravaluation on an Ultrasubstructure of P . .121 C. An Example . . . . 138 D. A State-induced Semantics for the ?.., Structures . 147 Summary 154 Chapter VII Hidden Variables Reconsidered 162 Preface 162 A. Criticisms of the Hidden-Variable Impossibility Proofs 167 B. Either P.,,. Preservation or Boolean Reconstruction 199 QMA — v 1 CHAPTER 0 INTRODUCTION In 1932, von Neumann published the f i r s t proofs o f the completeness of quantum mechanics and the i m p o s s i b i l i t y o f i n t r o d u c i n g hidden v a r i a b l e s i n t o quantum theory. T h i r t y years l a t e r i n 1963, Jauch and P i r o n published another proof o f the i m p o s s i b i l i t y o f hidden v a r i a b l e s which they regarded as a strengthening o f von Neumann's i m p o s s i b i l i t y r e s u l t . Although von Neumann's proofs were l a t e r challenged, the completeness o f quantum mechanics and the i m p o s s i b i l i t y of hidden v a r i a b l e s were proven anew by Gleason i n 1957. And t e n years l a t e r , Kochen and Specker published t h e i r v e r s i o n of Gleason's i m p o s s i b i l i t y proof. The proofs by Jauch-Piron and Kochen-Specker are e s p e c i a l l y i n t e r e s t i n g because they connect the e n t e r p r i s e o f i n t r o d u c i n g hidden v a r i a b l e s i n t o quantum theory w i t h the e n t e r p r i s e o f a s s i g n i n g 0, 1 values to the a l g e b r a i c s t r u c t u r e s o f the subspaces ( o r p r o j e c t o r s ) o f H i l b e r t space. Such a l g e b r a i c s t r u c t u r e s have been regarded as the p r o p o s i t i o n a l o r l o g i c a l s t r u c t u r e s o f quantum mechanics ever s i n c e von Neumann's proposals, i n 1932 and 1936, to consider the subspaces ( o r p r o j e c t o r s ) as the mathematical r e p r e s e n t a t i v e s o f quantum p r o p o s i t i o n s and to consider the operations and r e l a t i o n s among the subspaces as the mathematical r e p r e s e n t a t i v e s o f l o g i c a l operations and r e l a t i o n s . So these proofs o f the i m p o s s i b i l i t y o f a s s i g n i n g 0, 1 values to the a l g e b r a i c s t r u c t u r e s of quantum p r o p o s i t i o n s i n a manner which preserves the l o g i c a l operations and r e l a t i o n s among the p r o p o s i t i o n s are p r o o f s — o r a t l e a s t 2 suggestive of a p r o o f — o f the i m p o s s i b i l i t y of a c l a s s i c a l , that i s , a bivalent and t r u t h - f u n c t i o n a l , semantics f o r the quantum propositions. In Chapter I, the algebraic notions employed throughout t h i s t h e s i s are presented. In Chapter I I , the Boolean Lindenbaum algebra L of a set of well-formed formulae of c l a s s i c a l p r o p o s i t i o n a l l o g i c i s introduced, and the notion of a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r an L i s defined as a complete c o l l e c t i o n of algebraic valuations on L. We s h a l l see i n Chapter V how such a semantics f a i l s to work f o r the algebraic structures of quantum propositions. And we s h a l l see i n Chapter VI how such a semantics does work f o r c e r t a i n substructures of the quantum prop o s i t i o n a l structures. The quantum pr o p o s i t i o n a l structures, labeled P , have been formalized i n two ways: as orthomodular l a t t i c e s P.,,T which have the J QML operations A (and), V (or) defined among a l l elements, compatible i and incompatible & ; and as partial-Boolean algebras ^Q^A w n ^ c n have A, V defined among only compatible elements. In Chapter IV, some differences between the P „ „ T and the P « „ . formalizations are described. QML QMA Also, two notions of t r u t h - f u n c t i o n a l i t y are distinguished: t r u t h - f u n c t i o n a l i t y (<i,jfe) appropriate to a P and t r u t h - f u n c t i o n a l i t y ( i ) appropriate to a * And two basic senses i n which the quantum pro p o s i t i o n a l structures may be said to be non-Boolean are elaborated. Then i n Chapter V, i t i s shown how the l a t t i c e d e f i n i t i o n s of A, V among incompatibles cause t r u t h - f u n c t i o n a l i t y problems which r u l e out a b i v a l e n t , t r u t h - f u n c t i o n a l (O,JD) semantics f o r any quantum P Q M L containing incompatible elements. In contrast, the Kochen-Specker 1967 i m p o s s i b i l i t y proof, which semantically interpreted i s a proof of the i m p o s s i b i l i t y of a bi v a l e n t , t r u t h - f u n c t i o n a l (o) semantics f o r any three-or-higher 3 dimensional H i l b e r t space PQJJA structure, r e s t s upon t r u t h - f u n c t i o n a l i t y problems caused by the presence of overlapping maximal Boolean substructures i n » "the presence of incompatible elements i s necessary but not s u f f i c i e n t to r u l e out such a c l a s s i c a l semantics f o r a P?,7? . QMA In Chapter VI, another semantic proposal i s considered f o r the quantum p r o p o s i t i o n a l structures. This i s the proposal of a state-induced semantics, which i s p a r t l y motivated by the f a c t that, as described i n Chapter I I I , the state-induced semantics f o r a Boolean p r o p o s i t i o n a l structure P^ M determined by c l a s s i c a l mechanics i s exactly analogous to the c l a s s i c a l (algebraic) semantics f o r a Boolean Lindenbaum algebra L. So the notion of a state-induced semantics f o r a P , with the quantum state-induced expectation-functions Exp^, as semantic mappings, i s investigated. Each Exp^, on a P ^ t r u t h - f u n c t i o n a l l y assigns 0, 1 values to the elements i n a substructure of P i n a manner exactly analogous to the way algebraic valuations on an L t r u t h - f u n c t i o n a l l y assign 0, 1 values to the elements i n L. Thus Exp^, may he regarded as the quantum analog of the standard valuation of c l a s s i c a l p r o p o s i t i o n a l l o g i c , even though the domain where each Exp^. i s bivalent and t r u t h - f u n c t i o n a l i s only a sub-structure of P_ W rather than the e n t i r e QM P , and even though that substructure may be larger than any Boolean substructure of P.,, and so may be a non-Boolean substructure of P - W . QM J QM In short, the basic methodology of the quantum state-induced semantics fo r a PQ^ i s exactly l i k e the methodology of the c l a s s i c a l (algebraic) semantics f o r an L. Thus, when the c l a s s i c a l semantic method i s applied to a non-Boolean quantum , the r e s u l t i s a semantics (which happens to also be state-induced and) which i s n o n - c l a s s i c a l i n the sense that the domain of each semantic mapping Exp^, i s a non-Boolean substructure of 4 P„„ . In c o n t r a s t , i n the case of c l a s s i c a l mechanics, the domain of each QM state-induced semantic mapping i s the e n t i r e Boolean P ^  ; and l i k e w i s e , i n the case of c l a s s i c a l p r o p o s i t i o n a l l o g i c , the domain of each a l g e b r a i c v a l u a t i o n i s the e n t i r e Boolean L. Chapter VII surveys hidden v a r i a b l e (HV) pro p o s a l s , proofs o f 1 t h e i r i m p o s s i b i l i t y , and c r i t i c i s m s of these HV i m p o s s i b i l i t y p r o o f s . Kochen-Specker present the c l e a r e s t n o t i o n of the goal of proposed HV t h e o r i e s : t o give a c l a s s i c a l , Boolean r e c o n s t r u c t i o n of quantum mechanics, whereby the s t a t i s t i c a l r e s u l t s of quantum mechanics are reproduced by c l a s s i c a l p r o b a b i l i t y measures on a proposed Boolean s t r u c t u r e P of HV (subsets o f ) a proposed c l a s s i c a l phase space of hidden v a r i a b l e s . Kochen-Specker r e q u i r e that such a c l a s s i c a l HV r e c o n s t r u c t i o n of quantum mechanics preserve the f u n c t i o n a l r e l a t i o n s among quantum magnitudes and the l o g i c a l operations among compatible quantum p r o p o s i t i o n s ; i n other words, an HV r e c o n s t r u c t i o n must preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f eatures of the quantum P ^ . Such a requirement may be c a l l e d a s t r u c t u r a l c o n d i t i o n . Von Newmann and Jauch-Piron each impose an a d d i t i o n a l s t r u c t u r a l c o n d i t i o n r e q u i r i n g the p r e s e r v a t i o n of an operation among incompatibles. That i s , according t o von Newmann and Jauch-Piron, an HV theory must preserve some of the l a t t i c e f e a t u r e s of the quantum PQ^ ; t h i s view i s c r i t i c i z e d i n three notes at the end of Chapter V I I . Now Kochen-Specker show t h a t t h e i r n o t i o n of an HV r e c o n s t r u c t i o n of quantum mechanics i s p o s s i b l e IFF there e x i s t s what i n t h i s t h e s i s i s c a l l e d a complete c o l l e c t i o n o f b i v a l e n t , t r u t h - f u n c t i o n a l ( i ) mappings on P . In t h i s way, the problem of hidden v a r i a b l e s f o r quantum mechanics i s connected w i t h the problem of a c l a s s i c a l semantics f o r the 5 quantum p r o p o s i t i o n a l s t r u c t u r e s . And as mentioned above, Kochen-Specker prove t h a t f o r ^QJ^ s t r u c t u r e s , b i v a l e n t , t r u t h - f u n c t i o n a l (6) • mappings are i m p o s s i b l e , and so a c l a s s i c a l HV r e c o n s t r u c t i o n i s impossible f o r the quantum mechanics of three-or-higher dimensional H i l b e r t space. The other HV i m p o s s i b i l i t y proofs s i m i l a r l y i n v o l v e showing the i m p o s s i b i l i t y of proposed b i v a l e n t , operation-preserving HV mappings on the P ^ s t r u c t u r e s . C r i t i c s of these HV i m p o s s i b i l i t y proofs argue t h a t the proofs r e s t upon c o n t r a d i c t i o n s caused by r e q u i r i n g the proposed HV mappings to s a t i s f y the various s t r u c t u r a l c o n d i t i o n s imposed by the authors of the HV i m p o s s i b i l i t y p r o o f s . So whether or not the proofs are accepted depends upon whether or not the s t r u c t u r a l c o n d i t i o n s are accepted as j u s t i f i a b l y imposed requirements. And as Bub makes c l e a r , the l a t t e r depends upon how quantum mechanics i s i n t e r p r e t e d . In p a r t i c u l a r , we have the f o l l o w i n g dichotomy a r t i c u l a t e d by Bub: E i t h e r quantum mechanics i s taken t o be a ( p r i n c i p l e ) theory which p o s i t s a non-Boolean l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e f o r quantum phenomena, as given by the s t r u c t u r e a b s t r a c t e d from the fundamental p o s t u l a t e s of quantum mechanics; i n t h i s case, the quantum PQJ^ must be preserved, and as shown by Gleason and Kochen-Specker, quantum mechanics i s a complete theory of quantum phenomena and an HV r e c o n s t r u c t i o n of quantum mechanics i s impossible. Or the e n t e r p r i s e of p r o v i d i n g a c l a s s i c a l HV r e c o n s t r u c t i o n of quantum mechanics i s t r e a t e d as paramount, w i t h respect t o which the quantum need not be preserved; i n t h i s case, as proved by Kochen-Specker, the quantum PQJ^ cannot be preserved, and as e x e m p l i f i e d by the s o - c a l l e d c o n t e x t u a l HV t h e o r i e s , a c l a s s i c a l HV r e c o n s t r u c t i o n which does not preserve P „ W . i s ^ QMA p o s s i b l e and quantum mechanics i s incomplete r e l a t i v e t o such an HV 6 r e c o n s t r u c t i o n . Bub argues that behind each of these two p o s i t i o n s there i s a p r e s u p p o s i t i o n about l o g i c : The l a t t e r i s motivated by the pre s u p p o s i t i o n t h a t the l o g i c a l s t r u c t u r e of quantum phenomena and quantum theory must be a Boolean s t r u c t u r e l i k e the Boolean P „ „ s t r u c t u r e of J CM c l a s s i c a l phenomena and c l a s s i c a l mechanics. The former i s motivated by a n open acceptance o f the non-Boolean character of the l o g i c a l s t r u c t u r e of quantum phenomena and quantum theory, as manifested i n the non-Boolean Q^MA s t r u c t u r e (which i s abst r a c t e d from the quantum formalism by the same way th a t the Boolean P „ W s t r u c t u r e i s ab s t r a c t e d from the formalism CM of c l a s s i c a l mechanics). Thus one's views on l o g i c may co l o u r one's i n t e r p r e t a t i o n of quantum mechanics. But r e g a r d l e s s of the above l o g i c a l p o i n t , since 1967 i t has been c l e a r that a c l a s s i c a l HV r e c o n s t r u c t i o n of quantum mechanics which preserves the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of the quantum P „ „ i s QM impossible. And i t i s arguable that because the c o n t e x t u a l HV t h e o r i e s do not preserve the quantum PQJ^ » such t h e o r i e s are not r e a l l y r e c o n s t r u c t i o n s of quantum mechanics but r a t h e r are e n t i r e l y separate t h e o r i e s of quantum phenomena which, as Bub puts i t , w i l l have t o stand on t h e i r own f e e t . Their f e e t are shaky si n c e so f a r , experiments have f a l s i f i e d the d e v i a t i o n s from quantum mechanics p r e d i c t e d by the c o n t e x t u a l HV t h e o r i e s . Thus quantum mechanics, whose state-induced Exp^ mappings do preserve the p a r t i a l - B o o l e a n s t r u c t u r a l features of P and do s u c c e s s f u l l y p r e d i c t the r e s u l t s of experiments, marks a r a d i c a l departure from c l a s s i c a l p h y s i c a l t h e o r i e s and may a l s o mark a r a d i c a l departure from c l a s s i c a l l o g i c . 7 x In t h i s t h e s i s , two types o f HV t h e o r i e s are i n v e s t i g a t e d , namely, what are c a l l e d by B e l i n f a n t e HV t h e o r i e s o f the "zeroth k i n d " (proved impossible by von Neumann, Jauch-Piron, Gleason, Kochen-Specker) and HV t h e o r i e s of the " f i r s t k i n d " ( a l s o c a l l e d c o n t e x t u a l HV t h e o r i e s ) . What B e l i n f a n t e c a l l s HV t h e o r i e s o f the "second k i n d , " t h a t i s , the s o - c a l l e d l o c a l HV t h e o r i e s , are not discussed i n t h i s t h e s i s . And i n p a r t i c u l a r , the cel e b r a t e d paper by E i n s t e i n , Podolsky, and Rosen, i n which the n o n - l o c a l i t y of quantum phenomena i s h i g h l i g h t e d , i s not discussed i n t h i s t h e s i s . Bernard d'Espagnat, i n h i s paper "The Quantum Theory and R e a l i t y " i n a recent S c i e n t i f i c American ( V o l . 241, No. 5, November, 1979) presents a l u c i d and a c c e s s i b l e d e s c r i p t i o n o f the n o n - l o c a l i t y of quantum phenomena and o f the proposal of a l o c a l HV theory. Though d'Espagnat does not say so, h i s explanation of the d e r i v a t i o n o f B e l l ' s I n e q u a l i t y i n a l o c a l HV theory makes i t c l e a r t h a t the d e r i v a t i o n depends upon a s e t - t h e o r e t i c , i . e . , Boolean, manipulation o f the p r o p e r t i e s o f c o r r e l a t e d quantum systems. Bub makes a s i m i l a r p o i n t i n h i s book (Bub, 1974, pp. 79, 83); he argues t h a t the c r u c i a l assumption i n the d e r i v a t i o n of B e l l ' s I n e q u a l i t y i n a l o c a l HV theory i s not the assumption of l o c a l i t y but r a t h e r the assumption t h a t c e r t a i n quantum p r o b a b i l i t i e s are to be computed as though they were c l a s s i c a l c o n d i t i o n a l p r o b a b i l i t i e s on a c l a s s i c a l , i . e . , Boolean, p r o b a b i l i t y space. Thus the problems and i s s u e s r a i s e d by HV t h e o r i e s o f the "second k i n d " may i n f a c t be no d i f f e r e n t from the problems and iss u e s r a i s e d by HV t h e o r i e s o f the " f i r s t k i n d " which hinge upon attempted Boolean treatments o f quantum p r o p e r t i e s and p r o p o s i t i o n s . A f u l l e x p l i c a t i o n of these p o i n t s i s l e f t f o r f u t u r e work. 8 CHAPTER I ALGEBRAIC PRELIMINARIES Sec t i o n A. Group and Ring St r u c t u r e s Consider an a r b i t r a r y , nonempty c o l l e c t i o n o f elements E = { a , b , c , d , e , . . .} with a bin a r y ( u n i v a l e n t ) o p e r a t i o n plus + defined from E x E t o E such that E i s c l o s e d w i t h respect to + ; i . e . , f o r any b, c € E, b + c € E, and the f o l l o w i n g c o n d i t i o n s o b t a i n f o r any elements i n E: (1) + i s a s s o c i a t i v e , i . e . , b + (c + d) = (b + c) + d. (2) There e x i s t s a d i s t i n g u i s h e d element 0 € E such t h a t b + 0 = 0 + b = 0 , f o r any b € E. ( 3 ) For any b € E, there e x i s t s a c € E such t h a t b + c = 0. I t can be proven that c i s unique; i t i s designated as "-b" (the a d d i t i v e inverse of b) and s a t i s f i e s b + (-b) = 0 = (-b) +b. For any b, c € E, b + ( - c ) i s a l s o w r i t t e n as b - c. The ordered t r i p l e <E, +, 0> s a t i s f y i n g c l o s u r e and (1), (2), ( 3 ) i s an a d d i t i v e group. For example, <S, A, 0> i s a s e t - t h e o r e t i c r e a l i z a t i o n of an a d d i t i v e group, where S i s a set of subsets of some s e t , A i s symmetric d i f f e r e n c e , and 0 i s the empty s e t . I f an a d d i t i v e group <E, t , 0> i s such t h a t : ( 4 ) + i s commutative, i . e . , b + c = c + b , then <E, +, 0> i s an ab e l i a n or commutative a d d i t i v e group. Now l e t a second b i n a r y ( u n i v a l e n t ) operation dot • be defined from E x E t o E such that E i s cl o s e d w i t h respect t o •, i . e . , f o r any b, c € E, b • c € E (by convention, b • c i s a l s o w r i t t e n b e ) , and these two co n d i t i o n s o b t a i n : (5) • i s a s s o c i a t i v e . (6) • i s d i s t r i b u t i v e w i t h respect to + , i . e . , b * ( c + d ) = b c + b d and (c + d) • b = cb + db. The ordered quadruple <E, +, •, 0> s a t i s f y i n g the two clo s u r e c o n d i t i o n s and ( l ) - ( 6 ) i s a r i n g . For example, <S, A, f l , 0> i s a s e t - t h e o r e t i c r e a l i z a t i o n of a r i n g , where fl i s the i n t e r s e c t o p e r a t i o n . I f a r i n g i s such t h a t : (7) ••• i s commutative, then the r i n g i s a commutative r i n g . Consider a l s o t h i s c o n d i t i o n : (8) There e x i s t s a d i s t i n g u i s h e d element 1 € E such that b « l = l « b = b , f o r any b € E. The ordered f i v e - t u p l e <E, +, •, 0, 1> s a t i s f y i n g c l o s u r e , ( l ) - ( 6 ) , and (8) i s a r i n g - w i t h - u n i t ; and a r i n g - w i t h - u n i t which s a t i s f i e s (7) i s a commutative r i n g - w i t h - u n i t . Consider such a r i n g which a l s o s a t i s f i e s : (9) b • b = b, f o r any b 6 E, th a t i s , each element i n E i s idempotent. 2 , (By convention, b • b i s a l s o w r i t t e n b .) Two c o n d i t i o n s f o l l o w from ( 9 ) : (10) Each element b € E i s i t s own a d d i t i v e i n v e r s e . Proof: For any b € E, (b + b ) 2 = b 2 + b 2 + b 2 + b 2 . And by ( 9 ) , ( b + b ) 2 = b + b = b + b + b + b . So by ( 2 ) , b + b = 0 , and so by ( 3 ) , b = -b. Q.E.D. Thus f o r any b, c € E, b + c = b + (-c) = b -(7) • i s commutative. 10 Proof: For any b, c 6 E, (b + c) = b + be + cb + c . And by ( 9 ) , 2 (b + c) = b + c = b + be + cb + c. So by ( 2 ) , be + cb = 0, and so by ( 3 ) , be = - ( c b ) , and hence by ( 1 0 ) , be - cb. Q.E.D. (Halmos, 1963, p. 2) The ordered quadruple <E, +, •, 0> s a t i s f y i n g c l o s u r e , ( l ) - ( 6 ) , ( 9 ) , and hence (.10) and (7) i s a Boolean r i n g . And the ordered f i v e - t u p l e <E, +, •, 0, 1> s a t i s f y i n g c l o s u r e , (1)-(10) i s a Boolean r i n g - w i t h - u n i t . Or i n other words, the idempotent elements of a commutative r i n g form a Boolean r i n g , and the idempotent elements of a r i n g - w i t h - u n i t or a commutative r i n g - w i t h - u n i t form a Boolean r i n g - w i t h - u n i t . For example, <S, A, f l , 0 , X> i s a s e t - t h e o r e t i c r e a l i z a t i o n of a Boolean r i n g - w i t h - u n i t , where S i s the set of subsets of a given set X. S e c t i o n B. The Boolean Algebra and the Boolean L a t t i c e In a Boolean r i n g - w i t h - u n i t , two bi n a r y operations meet A and j o i n V are defined from E x E t o E and a unary operation complementation ' i s defined from E t o E i n terms of the r i n g operations +, • as f o l l o w s : f o r any b, c € E, b A c = b • c, b V c = b + c - ( b * c ) , b' = 1 - b. The r e s u l t i n g sextuple <E, A, V, ', 0, 1> i s a Boolean algebra. For example, <S, R, U, ', 0 , X> i s a s e t - t h e o r e t i c r e a l i z a t i o n of a Boolean a l g e b r a , where S i s the set of a l l subsets of a given set X, 0 i s the empty s e t , and 0 , 1 1 , ' are the set operations i n t e r s e c t , union, complementation, r e s p e c t i v e l y . From the above l i s t of c o n d i t i o n s (1)-(10) which a Boolean r i n g - w i t h - u n i t s a t i s f i e s with respect t o + and *, we can d e r i v e a lengthy l i s t of c o n d i t i o n s which a Boolean algebra s a t i s f i e s w i t h respect t o i t s operations. However, i n a d d i t i o n t o the c l o s u r e of E w.r.t. A, V, ', 11 and the existence of the d i s t i n g u i s h e d 0 and 1-elements i n E, the f o l l o w i n g f i v e c o n d i t i o n s are necessary and s u f f i c i e n t t o c h a r a c t e r i z e a Boolean algebra: f o r any b, c, d € E, ( B l ) Commutativity: b A c = c A b and b V c = c V b, by (4) and (7 ) . (B2) A s s o c i a t i v i t y : (b A c) A d - b A (c A d) and (b V c) V d = b V. (c V d ) , by ( 1 ) , ( 4 ) , ( 5 ) , ( 6 ) , (10). (B3) Absorption: b A (b V c) = b and (b A c) V c = c, by ( 2 ) , ( 3 ) , ( 4 ) , ( 6 ) , ( 9 ) . (B4) Complementation: b A b' = 0 and b v b* = 1, by ( 1 ) , ( 3 ) , ( 6 ) , ( 8 ) , ( 9 ) , (10). (B5) D i s t r i b u t i v i t y : b A (c V d) = (b A c ) V (b A d) and b V (c A d) = (b V c) A (b V d ) , by ( 3 ) , ( 6 ) , ( 9 ) . Among the many other i d e n t i t i e s and c o n d i t i o n s which can be derived from (1)-(10) we note the f o l l o w i n g : Idempotence: b A b = b and b V b = b. D i s t i n g u i s h e d elements: b A 0 = 0, b V 0 = b, b A 1 = b, and b V 1 = 1. I n v o l u t i o n of complementation: C h ' ) ' = b, by ( 1 ) , ( 2 ) , (10). Moreover, i n a Boolean algebra we may define a b i n a r y r e l a t i o n < i n terms of the meet or j o i n operations as: f o r any b, e € E, b S c IFF b A c - b, and b 5 c IFF b V c = c. I t f o l l o w s t h a t 0 < b 5 1, f o r every b € E. Then by ( 2 ) , ( 3 ) , ( 6 ) , ( 8 ) , and (.1), the ' operation a l s o s a t i s f i e s the c o n d i t i o n : f o r any b, c £ E, b 5 c IFF c' 2 b ' . This c o n d i t i o n together w i t h (B4) and the i n v o l u t i o n c o n d i t i o n define ' as orthocomplementation x . Since i n a Boolean a l g e b r a , complementation i s orthocomplementation, I h e r e a f t e r s u b s t i t u t e x f o r ' i n the ordered 12 s e x t u p l e d e s i g n a t i o n o f a B o o l e a n a l g e b r a <E, A, V , x , 0, 1>. Any B o o l e a n a l g e b r a a l s o s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s , f o r any b , c 6 E: De Morgan's l a w s : (b A c) = b V c and (b V c) = b A c . C o m p a t i b i l i t y : Cb A c ) V c = Cc A b ) V b, p r o v e n as f o l l o w s . By ( B 5 ) , (b A c A ) V c = (b V c) A ( c ^ V c ) , w h i c h by (B4) and by t h e d i s t i n g u i s h e d c h a r a c t e r o f t h e 1-element e q u a l s (b V c ) . And by t h e same c o n d i t i o n s , Cc A b"1") V b - Cc V b) A 1 = ( c V b ) . And by ( B l ) , b V c = c V b. Q.E.D. M o d u l a r i t y : I f b 5 c t h e n b V ( e A c ) = ( b V e ) A c , f o r any e 6 E. M o d u l a r i t y f o l l o w s from ( B 5 ) . O r t h o m o d u l a r i t y : I f b 5 c t h e n b = ( b v c J * ) A c and c = ( c A b"1") V b , w h i c h a g a i n f o l l o w s from ( B 5 ) . Any e l e m e n t s b, c € E a r e s a i d t o be d i s j o i n t o r o r t h o g o n a l I F F b < c , where b < c I F F c 5 b . Mo r e o v e r , b < c I F F b A c = 0, pr o v e n as f o l l o w s . Assume b 5 c , t h e n b A c 5 c A c , and so by (B4-) , b A c 5 0, i . e . , b A c = 0 s i n c e 0 2 e, f o r e v e r y e € E. Assume b A c = 0. Then s i n c e b : b A 1 - b A ( c V c x ) - (b A c) V (b A c"1") = 0 V (b A c 1 ) = b A c"1", we have b 5 c^. Q.E.D. The c o m p a t i b i l i t y , m o d u l a r i t y , and o r t h o m o d u l a r i t y c o n d i t i o n s and th e r e l a t i o n o f d i s j o i n t e d n e s s o r o r t h o g o n a l i t y a r e mentioned h e r e because t h e y a r e i m p o r t a n t f o r t h e quantum s t r u c t u r e s d e s c r i b e d i n S e c t i o n s D and E. W i t h t h e b i n a r y r e l a t i o n < d e f i n e d as above i n a B o o l e a n a l g e b r a , i t f o l l o w s f r o m t h e f i v e c o n d i t i o n s ( B 1 ) - ( B 5 ) t h a t w . r . t . - a B o o l e a n a l g e b r a i s a B o o l e a n l a t t i c e , as w i l l be shown below. A -Boolean. i . e . , an orthocomplemented and d i s t r i b u t i v e , l a t t i c e i s d e f i n e d as f o l l o w s . C o n s i d e r an a r b i t r a r y , nonempty c o l l e c t i o n o f el e m e n t s 13 E = {a,b,c,d,e, . . . } w i t h a bin a r y r e l a t i o n 2 c E x E which has the f o l l o w i n g p r o p e r t i e s , f o r any b,c,d € E: ( 5 a ) R e f l e x i v i t y : b 5 b. (2 b) Anti-symmetry: I f b 2 c and c 5 b, then b = c . ( 2 c ) T r a n s i t i v i t y : I f b 2 c and c 5 d, then b 2 d. The ordered p a i r <E,2> i s a p a r t i a l l y ordered s e t , a l s o c a l l e d a poset. With respect to 2, de f i n e the greatest lower bound ( g . l . b . ) and the l e a s t upper bound ( l . u . b . ) of any subset F c E as f o l l o w s . The g.l. b . of F i s that element b € E such t h a t , f o r every f € F, b 2 f , and f o r any e € E, i f e 2 f f o r every f € F then e 2 b. The l.u.b. i s defined d u a l l y , i . e . , s u b s t i t u t e > f o r 2. (The dual of any c o n d i t i o n i s the r e s u l t of interchanging 2 and >, A and V, and 0 and 1 (Halmos, 1963, pp. 7-8, 22).) The uniqueness of the g.l . b . and l.u.b. of any F c E f o l l o w s from ( 2 b ) . A l a t t i c e <E,2,A,V> i s a poset any two of whose elements b,c € E have a g . l . b . , w r i t t e n b A c and c a l l e d the meet of b, c, and have a l.u. b . , w r i t t e n b v e and c a l l e d the j o i n of b, c. For example, <S,c,n,U> i s a s e t - t h e o r e t i c r e a l i z a t i o n o f a l a t t i c e , where c i s the s e t - i n c l u s i o n r e l a t i o n . I f a l a t t i c e has a g . l . b . , 0, and a l.u.b., 1, and i f , f o r any b € E, there e x i s t s a t l e a s t one c € E such that b A c = 0 and b v c = 1 (such a c w i l l be c a l l e d the complement of b and be denoted " b " ' ) , then the l a t t i c e i s a complemented l a t t i c e <E,2,A,V,',0,1>. I f a l a t t i c e has a g . l . b . , 0, and a l.u.b., 1, and i f , f o r any b € E, there e x i s t s a unique orthocomplement b~ € E s a t i s f y i n g : b A b1" = 0, b V b~ = 1, (b"1")"1" = b, and b 2 c IFF 2 b^, then the l a t t i c e i s an orthocomplemented l a t t i c e <E,2,A,V,"L",0,1>. 14 I f the meet and j o i n operations are d i s t r i b u t i v e then the l a t t i c e i s d i s t r i b u t i v e . I f a l a t t i c e i s complemented and d i s t r i b u t i v e , then complementation i s unique and i s orthocomplementation ( B i r k h o f f , 1948, p. 152). An orthocomplemented, d i s t r i b u t i v e l a t t i c e i s c a l l e d a Boolean  l a t t i c e . I t i s easy to prove t h a t a Boolean algebra i s a Boolean l a t t i c e w i t h respect to the p a r t i a l - o r d e r i n g r e l a t i o n 2 d e f i n e d i n a Boolean algebra as above. For ( B l ) , (B2), and (B3) ensure t h a t 2 s a t i s f i e s ( 2 a ) , (2b), and ( 2 c ) . And ( B l ) , (B2), (B3) ensure that the element b A c i s a lower bound f o r the subset {b,c} because b A (b A c) = (b A b) A c = b A c, thus b A c 2 b, and c A (c A b) = (c A c) A b = c A b, thus b A c 2 c. And i f d i s any lower bound f o r {b,c}, i . e . , d A b = d and d A c = d, then ( b A c ) A d = b A ( c A d ) = b A d = d , and hence d 2 (b A c ) ; thus b A c i s the greatest lower bound of {b,c}. D u a l l y , the element b V c i s the l e a s t upper bound of the subset {b,c}. Moreover, b A c and b V c are each unique because A, V i n a Boolean algebra are o p e r a t i o n s , i . e . , they are u n i v a l e n t . Hence, a Boolean algebra i s a l a t t i c e w i t h respect to the 2 r e l a t i o n defined i n a Boolean algebra as above. And i n other words, the ( B l ) , (B2), (B3) c o n d i t i o n s completely c h a r a c t e r i z e a l a t t i c e ( B i r k h o f f , 1948, p. 18). I t f o l l o w s from (B4) that the d i s t i n g u i s h e d 0 and 1 elements of a Boolean algebra are the greatest lower bound of E and the l e a s t upper bound of E, r e s p e c t i v e l y , as shown next. For any b € E , b A l = b A ( b v b i ) = b, so b 2 1 , and b A 0 = b A (b Ab1") = (b Ab) AbJ"= b Ab~= 0, so 0 2 b ; t h a t i s , 1 i s an upper bound of E, and 0 i s a lower bound of E. I f there i s an e € E such t h a t 1 2 e, i . e . , 1 A e = 1, then e i t h e r e = 1 or (B3) 15 i s v i o l a t e d , and d u a l l y , i f there i s an e € E such t h a t e 5 0, i . e . , e V 0 = 0, then e i t h e r e = 0 or (B3) i s v i o l a t e d ; thus 1 i s the l e a s t upper bound of E and 0 i s the greatest lower bound of E. Q.E.D. So a Boolean algebra i s a complemented l a t t i c e . And by (B5), a Boolean algebra i s a d i s t r i b u t i v e l a t t i c e whose complementation i s unique and i s orthocomplementation . Thus a Boolean algebra i s a Boolean l a t t i c e w i t h respect t o the ^ r e l a t i o n defined i n a Boolean algebra as above. Conversely, i t i s easy t o prove t h a t a Boolean l a t t i c e i s a Boolean a l g e b r a . ( B e l l and Slomson, 1969, pp. 9-11). H e r e a f t e r , I use the phrase Boolean s t r u c t u r e and the sextuple B = <E,A,V,a",5,0,l> t o r e f e r t o both a Boolean algebra and a Boolean l a t t i c e i n d i s c r i m i n a t e l y . The Boolean s t r u c t u r e s determined by c l a s s i c a l mechanics, which I l a b e l P and describe in-Chapter I I I ( B ) , are cr-complete and atomic . The Boolean s t r u c t u r e s determined by c l a s s i c a l l o g i c , which I l a b e l L and describe i n Chapter 11(B), are complete and atomic i f they are f i n i t e . These a d d i t i o n a l c o n d i t i o n s are defined as f o l l o w s . Completeness: B i s complete i f every subset of elements i n B has a g.l.b. and a l.u.b. B i s cr-complete i f every denumerable subset of elements i n B has a g . l . b . and a l.u.b.. A t o m i c i t y : B i s atomic i f , f o r every non-zero element b € 8, there i s an.atom a € B such t h a t b > a, where an atom i s an element which covers the 0-element, i . e . , a > 0 and a > e > 0 i s not s a t i s f i e d by any e € B. (For any b,c € B, b > c IFF b > c and b t c .) B i s non-atomic i f i t has no atoms (Halmos, 1963, p. 69). 16 I t f o l l o w s t h a t i n an atomic 8 , every element i s the l.u.b. o f the atoms i t dominates (Halmos, 1963, p. 70). And i n an atomic 8 , two elements are equal IFF they dominate the same atoms (Rutherford, 1965, p. 83). Thus f o r any d i s t i n c t elements b ^ c i n an atomic 8 , there i s an atom a € 8 such that a 5 b but a £ c, or a < c but a £ b. Every f i n i t e 8 i s atomic and complete. Every f i n i t e 8 i s isomorphic to the c a r t e s i a n product (Z^)nt where n i s the number o f atoms i n 8 and i s the two-element Boolean s t r u c t u r e <E = {0 ,1},A,V,- L,2,0,l>. Every f i n i t e 8 i s isomorphic to the power set Boolean s t r u c t u r e <E = the set o f a l l subsets o f a given set X,fl,LI,' ,5,0,X>, where the number of elements i n X i s the same as the number o f atoms i n 8 . The diagram o f a f i n i t e 8 looks l i k e a two-dimensional r e p r e s e n t a t i o n o f an n dimensional cube, where n i s the number of atoms i n 8 . For example: n = 3 In these diagrams, the dots represent the elements of the structure and the l i n e s connecting the dots represent the operations and r e l a t i o n s among the elements, e.g., 18 3 ' Section C. Subsets of a Boolean Structure In conformity with standard mathematical parlance, I do not d i s t i n g u i s h between the structure 8 and i t s set of elements E. So members or subsets of the set E are also more simply r e f e r r e d to as members or subsets of 8. A subalgebra or s u b l a t t i c e of 8 i s a non-empty subset of 8 which i s closed with respect to the operations A,v, x of 8. A non-empty subset of 8, when closed with respect to the operations of 8, i s said to generate a subalgebra or s u b l a t t i c e of 8. A (proper) f i l t e r i n 8 i s a non-empty (proper) subset F of B which s a t i s f i e s : ( ) (a) For any b,c € F, b A c € F. ( ) (b) For any b € F and f o r any e € B, i f b 2 e then e € F. A (proper) i d e a l i n B i s defined d u a l l y , that i s , a (proper) i d e a l i s a non-empty (proper) subset I of B which s a t i s f i e s : (a') (a') For any b , c € I , b V c € I . c ) (b') For any b € I and f o r any e € B, i f b > e then e € I. The distinguished 1-element of 8 i s a member of every f i l t e r i n B and i s i t s e l f a f i l t e r i n 8. Dually, the 0-element i s a member of every i d e a l i n B and i s i t s e l f an i d e a l i n 8. Moreover, i t follows from de Morgan's laws 19 and from one of the c o n d i t i o n s d e f i n i n g ( i f b 2 c then c 2 b ) t h a t , f o r any f i l t e r F i n B, the set of elements {b € B : b^ € F} i s an i d e a l i n B; and d u a l l y , f o r any i d e a l I i n B, the set of elements {b• € B : b^ € 1} i s a f i l t e r i n . B ( S i k o r s k i , 1960, pp. 9, 11). Thus f o r any F and dual I i n B we have: (c) For any b € B, b"1" € F IFF b € I . (c*) For any b € B, b X € I IFF b € F. And i f b € F then b j f l , and i f b € I then b £ F. For example, a f i l t e r and dual i d e a l are designated i n the Boolean s t r u c t u r e diagrammed below by t r i a n g l e s around the elements i n the f i l t e r and squares around the elements i n the dual i d e a l : l=aVbVcV/d .0 The union of any f i l t e r and i t s dual i d e a l i n B i s a subalgebra or a s u b l a t t i c e of B ( B e l l and Slomson, 1969, p. 17). An u l t r a f i l t e r UF i n B i s a proper f i l t e r which i s not the proper subset of any proper f i l t e r i n B. An u l t r a i d e a l UI i n B i s a 20 proper i d e a l which i s not the proper subset of any proper i d e a l i n B. Every f i l t e r i n B i s contained i n a n . u l t r a f i l t e r ; every i d e a l i n B i s contained i n an u l t r a i d e a l ( S i k o r s k i , 1960, pp. 15, 17). Moreover, every u l t r a f i l t e r and every u l t r a i d e a l i n any B i s prime, t h a t i s , f o r any b,c € B: (d) : I f b V c € UF, then e i t h e r b € UF or c € UF. (d») I f b A c € 'Ul, then e i t h e r b € Ul or c € U l . And e q u i v a l e n t l y , i f b £ UF then b^ € UF, and i f b £ Ul then b~ € U l . ( B e l l and Slomson, 1969, p. 20). I t f o l l o w s that f o r any UF and i t s dual Ul i n any B and f o r any b € B, b 6 UF or b € U l , and thus B = UF U U l . Proof: For any UF i n any B and f o r any b € B, b € UF or b £ UF. I f b f. UF then b"1" € UF, and so by ( c ) , b € U l . Q.E.D. Each u l t r a f i l t e r and i t s dual u l t r a i d e a l i n a f i n i t e atomic 8 i s a p r i n c i p a l ' 1 u l t r a f i l t e r and a • p r i n c i p a l " u l t r a i d e a l d e fined w i t h respect t o a n atom a € B as f o l l o w s : UF = { b € B : b > a} and 3. UI = {b € B : b < a }. And i n an atomic B, there i s a one-to-one 3. correspondence between atoms ard u l t r a f i l t e r s (and dual u l t r a i d e a l s ) ( B e l l and Slomson, 1969, p. 27). For any p a i r o f d i s t i n c t elements b i- c i n any B, there i s an u l t r a f i l t e r UF i n B c o n t a i n i n g one but not the other. A l s o , each non-zero element i n a B i s contained i n some u l t r a f i l t e r i n B. ( B e l l and Slomson, 1969, p. 16). Section D. The Quantum P a r t i a l - B o o l e a n Algebra Kochen and Specker define a p a r t i a l - B o o l e a n algebra by f i r s t d e f i n i n g a p a r t i a l - a l g e b r a over a f i e l d . In s h o r t , a p a r t i a l - a l g e b r a over a f i e l d i s a set of elements E w i t h the u s u a l r i n g operations + and 21 defined from A t o E, where o S E xE i s c a l l e d the c o m p a t i b i l i t y r e l a t i o n . A commutative algebra i s a s p e c i a l case of a p a r t i a l - a l g e b r a , namely the case where t = E xE. The idempotent elements of a p a r t i a l - a l g e b r a form a p a r t i a l - B o o l e a n algebra <E ,A, A, V,-1-, 0,1 > which has the Boolean operations AjV,-1-, defined i n terms of the r i n g operations +, •, as u s u a l , but the bi n a r y operations A, V are again defined from only ^ to E. A Boolean algebra i s a s p e c i a l case of a p a r t i a l - B o o l e a n a l g e b r a , namely the case where ^ = E x E. (Kochen-Specker, 1965, pp. 180, 183; 1967, pp. 64-65). Using the terminology and s t y l e of Sections (A) and ( B ) , these s t r u c t u r e s are described as f o l l o w s . A p a r t i a l - r i n g - w i t h - u n i t <E ,A, + ,»,0,1> i s a non-empty set of elements E = {a,b,c,d,e, . . . ,} i n c l u d i n g the d i s t i n g u i s h e d 0 and 1, wi t h a b i n a r y r e l a t i o n of c o m p a t i b i l i t y <!> c E x E and two b i n a r y operations + and • defined from CD t o E such t h a t : (A a) i i s r e f l e x i v e , i . e . , f o r any b € E, b i b , symmetric, i . e . , f o r any b, c € E, i f B i c then c i> b, and n o n - t r a n s i t i v e , i . e . , f o r any b,c,d € E, i f b o c and c o d , i t does not f o l l o w t h a t b o d . ( O b ) For every b € E, b A 1 and b i 0. ( o c ) c> i s closed under + and • , i . e . , f o r any b,c,d € E, i f b, c, d are p a i r w i s e compatible then (b+c) i d , (b • c) i d, e t c . f o r a l l combinations. And f o r any subset F £ E, i f a l l the elements i n F are p a i r w i s e compatible, then by c l o s u r e they generate a commutative-ring-with-unit. (Kochen-Specker, 1965, p. 180; 1967, p. 64). F i n a l l y , s i n c e +, *, are defined from only C3 to E, r a t h e r than from E xE t o E, they are 22 c a l l e d p a r t i a l - o p e r a t i o n s or p a r t i a l - f u n c t i o n s by Kochen-Specker (1965, pp. 177, 178). Kochen and Specker do not s t a t e any other c o n d i t i o n s which the Q and 1 elements and the +, • operations must s a t i s f y . However, since any p a r t i a l - r i n g - w i t h - u n i t which has <^> = E x E i s a commutative-ring-with-unit and since any subset of mutually compatible elements i n a p a r t i a l - r i h g - w i t h -u n i t form a commutative-ring-with-unit, the 0 and 1 elements and the + , • operations of a p a r t i a l - r i n g - w i t h - u n i t presumably must s a t i s f y a l l the c o n d i t i o n s ( l ) - ( 8 ) which d e f i n e the 0, 1, +•, *, of a commutative-ring-w i t h - u n i t . The idempotent elements of a p a r t i a l - r i n g - w i t h - u n i t form a p a r t i a l - B o o l e a n - r i n g - w i t h - u n i t which i s a p a r t i a l - B o o l e a n algebra A = <E,i,A,v, J",0,l > when the AjV,-1- operations are defined i n terms of + and • as u s u a l and <b , 0, 1 are defined as above (Kochen-Specker, 1965, p. 183). In p a r t i c u l a r , E i s non-empty; 0, 1 are the d i s t i n g u i s h e d elements; i s r e f l e x i v e , symmetric, and n o n - t r a n s i t i v e ; f o r every b € E, b cb 1 and b o 0; i> i s close d under AjV,"1"; and f i n a l l y , f o r any subset F c E , i f a l l the elements i n F are p a i r w i s e compatible, then by c l o s u r e they generate a Boolean ( s u b ) s t r u c t u r e . Moreover, s i n c e a l l the elements i n a p a r t i a l - B o o l e a n - r i n g - w i t h - u n i t are idempotent, not only c o n d i t i o n s ( l ) - ( 8 ) but a l s o c o n d i t i o n (9) and hence c o n d i t i o n (10) are a l l s a t i s f i e d . Thus the bi n a r y operations A, v defined from >^ to E i n terms of + and • as u s u a l s a t i s f y the commutativity, a s s o c i a t i v i t y , a b s o r p t i o n , and d i s t r i b u t i v i t y c o n d i t i o n s which f o l l o w from the c o n d i t i o n s ( l ) - ( 7 ) , ( 9 ) , (10) s a t i s f i e d by + and • . And s i n c e A, V are defined from only t o E, r a t h e r than from E x E to E, they are i n f a c t 23 p a r t i a l - o p e r a t i o n s . The unary operation defined from E t o E i n terms of + and the 1-element as u s u a l s a t i s f i e s the complementation and i n v o l u t i o n c o n d i t i o n s and (assuming 5 i s defined i n terms of A or V as usual) the c o n d i t i o n : b 5 c IFF c^" < b X , which f o l l o w from the c o n d i t i o n s ( l ) - ( 3 ) , ( 6 ) , ( 8 ) - ( 1 0 ) . Thus X i s orthocomplementation. The p a r t i a l - o r d e r i n g r e l a t i o n 5 i s defined i n a p a r t i a l - B o o l e a n algebra i n terms of A or V as u s u a l , i . e . , b 5 c IFF b A c = b, and b 5 c IFF b V e = c. Since the meet b A c and the j o i n b V c are defined i n A IFF b ° c, we can be sure t h a t , f o r any b,c € A , i f b 5 c then b k> c. The p a r t i a l - o r d e r i n g r e l a t i o n so defined i n A i s r e f l e x i v e and anti-symmetric as u s u a l . However, i f b 5 c and c 5 d but b jo d, then b A d, b V d are not defined i n A and so i t does not f o l l o w t h a t b 5 d. So 5 may not be t r a n s i t i v e , i n which case 5 i s not a p a r t i a l - o r d e r i n g . But 5 i s t r a n s i t i v e i n the p a r t i a l - B o o l e a n algebras considered i n t h i s t h e s i s , namely the p a r t i a l - B o o l e a n algebras determined by quantum mechanics, which s h a l l be l a b e l e d = < E J ^ , > A , V , x , 5 , 0 , 1 > . The pQj^ s t r u c t u r e s are a s s o c i a t i v e , t r a n s i t i v e , and atomic. A p a r t i a l - B o o l e a n algebra A i s a s s o c i a t i v e IFF, f o r any b,c,d € A such th a t b o c and c o d: b i (c A d) IFF (b A c) cb d; and b d> (c A d) i m p l i e s b A (c A d ) = (b A c ) A d . A t r a n s i t i v e p a r t i a l - B o o l e a n algebra A s a t i s f i e s the c o n d i t i o n : For any b , c , d € A , i f b 5 c, and c 5 d, then b i d and b 5 d. And an atomic p a r t i a l - B o o l e a n algebra s a t i s f i e s the same a t o m i c i t y c o n d i t i o n as an atomic B. (An a d d i t i o n a l c o n d i t i o n on *QMA s t r u c t u r e s i s introduced i n Chapter V I ( D ) n o t h i n g - b e f o r e Chapter VI(D) i s a f f e c t e d by t h i s a d d i t i o n a l c o n d i t i o n . ) . • The notion of a p a r t i a l - B o o l e a n algebra i s .further> e l u c i d a t e d by the f o l l o w i n g c o n s t r u c t i o n due to Kochen-Specker. Consider a nonempty f a m i l y of 24 Boolean algebras { B . } . „ . such t h a t the i n t e r s e c t i o n of two algebras of 1 i€Index the f a m i l y i s i t s e l f an algebra of :the f a m i l y ; so - a l l the; B^  share the same d i s t i n g u i s h e d 0^  1 elements.! A n d i f - {e-^,-e ,. . . .} are elements, of the union E = U B. • such t h a t every rpair o*f'theitf l i e in-some'common algebra = B; ,. i then there i s a B , k € index such t h a t k {e^te2' * * *} ^ \ * Then a p a r t i a l - B o o l e a n algebra A i s defined on the union E as f o l l o w s . For any b,c,d € E, b i c i n A IFF there e x i s t s a B. • such t h a t b,c € B. ; b A c = d i n A IFF there e x i s t s a B. such l l l that b A c = d i n B.; b V c = d i n A IFF there e x i s t s a B. such x 1 that b V e = d i n B.; b = c i n A IFF there e x i s t s a B. such t h a t l l b = c in- B^ ; 1 and 0 i n A are the common d i s t i n g u i s h e d elements of a l l the 8^  . Kochen-Specker s t a t e and Hughes proves t h a t every A i s isomorphic to an A constructed on a f a m i l y of Boolean algebras as above. (Kochen-Specker, 1965, pp. 183-184; Hughes, 1978, pp. 113-114). Secti o n E. The Quantum Orthomodular L a t t i c e Jauch's d e f i n i t i o n of the l a t t i c e s t r u c t u r e s determined by quantum mechanics, which I l a b e l P , s t a r t s w i t h an orthocomplemented l a t t i c e <E,5,A,V,X,0,1> which i s complete i n the u s u a l sense t h a t every subset of E has a g . l . b . and a l.u.b. Then Jauch d e f i n e s the c o m p a t i b i l i t y r e l a t i o n A i n t h i s l a t t i c e as f o l l o w s : A subset F c E i s a compatible set i f the l a t t i c e generated by F i s a Boolean s u b l a t t i c e of the o r i g i n a l l a t t i c e . (Let {SL.}.,_ , be the f a m i l y of a l l the s u b l a t t i c e s which c o n t a i n F ; I i€Index J the s u b l a t t i c e S L Q = D SL. i s the l a t t i c e generated by F , (Jauch, 1968, i pp. 74-77, 80-81).) As a b i n a r y r e l a t i o n , c o m p a t i b i l i t y co £ E x E i s r e f l e x i v e , symmetric, and n o n - t r a n s i t i v e . And i t i s easy t o show t h a t , f o r any b € E , b « ^ 0 , b o 1, and b i b . 25 In order t o define PQ M T j » Jauch furthermore p o s t u l a t e s the c o n d i t i o n s : (P) I f b 5 c then b ^> c, f o r any b,c € E. Jauch c a l l s t h i s c o n d i t i o n weak modularity. (A l ) A t o m i c i t y (as u s u a l ) . (A2) I f a i s an atom and a A e = 0, then a V e covers e, f o r any e € E (Jauch, 1968, pp. 86-87). And i t f o l l o w s t h a t i f a i s any atom, then (a V e) A e"1" i s a l s o an atom, f o r any e € E ( P i r o n , 1976, p. 24). Thus P = <E,5,A, V,+,<^ ,0,1> i s a complete, orthocomplemented, weakly modular, atomic l a t t i c e . I t i s easy t o show th a t i n such a l a t t i c e , f o r any b,c £ E: b <^> c IFF (b A c"*") V c = (c A b x ) V b = b V c; b d> c IFF (b A c) V (b A cx) = b; and b <^> c IFF the elements b, bX, c , - c X , s a t i s f y the d i s t r i b u t i v e law f o r any combination (Jauch, 1968, p. 87; P i r o n , 1976, p. 26). Moreover, si n c e b < c IFF b A c = b, i t f o l l o w s from weak modularity t h a t , f o r any b,c € E , i f b 5 c then b = (b V c X ) A c, and i f b 5 c then c = (c A b"*") V b. This i s the orthomodularity c o n d i t i o n , according t o Rose (1964, p. 331) and according to P i r o n (1976, p. 24). The phrase "orthomodular" subsumes the two co n d i t i o n s of orthocomplementation and weak modularity; thus PQ^L ^ S A complete, atomic, orthomodular l a t t i c e . P i r o n develops h i s d e f i n i t i o n of the complete, atomic, orthomodular l a t t i c e P^„ T i n a d i f f e r e n t manner which r e v e a l s the f a c t t h a t each QML element b € PQ^^ may have non-unique complements defined i n besides the unique orthocomplement b"*". P i r o n s t a r t s w i t h a l a t t i c e which i s 26 complete i n the usu a l sense. Since completeness ensures t h a t the e n t i r e l a t t i c e has a g.l . b . which i s the d i s t i n g u i s h e d 0-element and a l.u.b. which i s the d i s t i n g u i s h e d 1-element, a complete l a t t i c e i s an. ordered sextuple <E,5,A,V,0,1>. In order to define PQ^ l »• P i r o n furthermore p o s t u l a t e s the c o n d i t i o n s : ( A l ) and (A2), as i n Jauch. (C) For each element b € E, there i s at l e a s t one compatible complement b^ € E, where b, b^ are complements s a t i s f y i n g the usu a l complementation c o n d i t i o n (b A b^ = 0 and b V b* = 1 ) , and b, b^ are compatible i n a sense which P i r o n d e f i n e s independently of the A, v, operations and 5 r e l a t i o n . Most simply, any b, c are compatible i n Piron*s sense i f they are as s o c i a t e d w i t h simultaneously measurable quantum p r o p o s i t i o n s . (P) For any b,c € E, i f b 5 c then the s u b l a t t i c e generated by b, b^, c, c^ i s d i s t r i b u t i v e . P i r o n c a l l s t h i s c o n d i t i o n weak modularity ( P i r o n , 1976, pp. 21-23). Two r e s u l t s f o l l o w from Piron's weak modularity. F i r s t , i f b < c, then by ( P ) , the elements b, b*, c are d i s t r i b u t i v e and so \ b V ( b l A c) = (b V b l ) A (b V c) = 1 A (b V c) = c; s i m i l a r l y , i f b < c then c A ( c ^ v b ) = b. This r e s u l t w i l l be mentioned again s h o r t l y . Secondly, according t o P i r o n , i t f o l l o w s immediately from (P) t h a t , f o r any b,c € E, i f b 5 c then c ^ 2 b^, and i t f o l l o w s that the compatible complement of each element i s unique. Thus the a s s o c i a t i o n of an element b w i t h i t s unique b* i s orthocomplementation s a t i s f y i n g : b A b^ = 0, b V b^ = 1, (b1)1 = b, and i f b 2 c then c A 5 b A ( P i r o n , 1976, pp. 23-24). 2 27 S u b s t i t u t i n g f o r ^ , the f i r s t r e s u l t f o l l o w i n g from (P) becomes the orthomodularity c o n d i t i o n . And P i r o n proves t h a t i f the orthocomplement i s i n t e r p r e t e d as a compatible complement, then any orthomodular l a t t i c e s a t i s f i e s h i s c o n d i t i o n s (C) and ( P ) . Moreover, Piron's weak modularity can be shown t o be equivalent to Jauch's weak modularity. P i r o n l a t e r d e f i n e s the c o m p a t i b i l i t y r e l a t i o n i c E x E i n a complete l a t t i c e s a t i s f y i n g (C) and (P) as f o l l o w s : b o c IFF the s u b l a t t i c e generated by b, b*, c, i s d i s t r i b u t i v e . With t h i s d e f i n i t i o n of c o m p a t i b i l i t y , Jauch's (P) i s equivalent t o Pi r o n ' s (P) w i t h s u b s t i t u t e d f o r ^  . Jauch a l s o says t h a t h i s weak modularity i s equivalent to the p o s t u l a t e t h a t the compatible complement i s unique, t h a t i s , the second r e s u l t which P i r o n d e r i v e s from h i s weak modularity (Jauch, 1968, p. 87). So l i k e Jauch, P i r o n d e f i n e s PQ^L a s a complete, atomic, orthomodular l a t t i c e , . Moreover, P i r o n makes i t c l e a r t h a t an element i n P Q M L may have non-unique complements which s a t i s f y the complementation c o n d i t i o n but which are not compatible complements and are not orthocomplements. Thus there a r i s e s i n P ^ M L the problem of a complementation which i s not unique (and hence i s not an o p e r a t i o n ) , as w i l l be discussed i n Chapter I V ( F ) . The Boolean L and P „ „ s t r u c t u r e s and the p a r t i a l - B o o l e a n algebra P . „ . CM 4 3 QMA each have only one complementation, namely, the orthocomplementation, which i s unique. F i n a l l y , as w i t h P Q M A > a Boolean s t r u c t u r e i s a s p e c i a l case of an orthomodular l a t t i c e P Q M L > namely, the case where CJ = E x E. Moreover, any quantum P ^ can be extended t o an orthomodular l a t t i c e P Q M L by d e f i n i n g the A, V operations among incompatible elements. The 28 two s t r u c t u r e s P „ „ . and P „ W T w i l l be f u r t h e r compared i n Chapter IV(E) QMA QML and ( F ) . Section F . Subsets o f P and ? The n o t i o n of a f i l t e r , i d e a l , u l t r a f i l t e r , u l t r a i d e a l , p r i n c i p a l u l t r a f i l t e r , and p r i n c i p a l u l t r a i d e a l are defined i n any l a t t i c e , e.g., i n the quantum » e x a c t l y as they are defined i n a Boolean l a t t i c e ( B i r k h o f f , 1967, pp. 25, 28). Bub mentions t h a t a f i l t e r and an u l t r a f i l t e r i n the quantum (and d u a l l y an i d e a l and an u l t r a i d e a l i n P Q J ^ A R E defined as i n a Boolean a l g e b r a , i . e . , any f i l t e r s a t i s f i e s ( a ) , ( b ) , any any i d e a l s a t i s f i e s ( a 1 ) , (b') (Bub, 1974, p. 120). However, R. Hughes modifies c o n d i t i o n (a) (and d u a l l y , ( a ' ) ) . The m o d i f i c a t i o n i s motivated by the f a c t t h a t , f o r any b, c i n any f i l t e r F c P Q M A i f b & c then b A c i s not defined i n PQJ^a • Hughes' s modified d e f i n i t i o n i s : A f i l t e r i n a P_ W . i s a non-empty subset F of P ^ „ A , t h a t , f o r any b,c,d € P«„ .• QMA r J QMA such J QMA (a ) I f b,c € F , then there i s a d € F such t h a t d < b any d 5 c. H And Hughes adds as a pr o v i s o the c o n d i t i o n : (c„) 0 £ F . n C o n d i t i o n (b) i s l e f t as before; that i s , ( b ) , (a,,), and (c ) de f i n e a H H f i l t e r i n a P_„ . . QMA According t o the d e f i n i t i o n of a f i l t e r i n a Boolean s t r u c t u r e 8, the e n t i r e 8 i s a f i l t e r , a l b e i t an improper f i l t e r . But according t o Hughes's d e f i n i t i o n of a f i l t e r i n P Q ^ A J "the e n t i r e P Q^ a i s not a' f i l t e r s i n c e 0 € P - „ . but according t o c o n d i t i o n ( c T T ) , 0 i s not a QMA H member of any f i l t e r . Conditions ( b ) , ( a u ) , ( c ) , a c t u a l l y d e f i n e a proper H H 29 f i l t e r i n P_„. • So we may drop c o n d i t i o n (c T T) and de f i n e a f i l t e r i n a QMA H Q^MA a S 3 N O N ~ E M P ' t y subset F of PQ^ a which s a t i s f i e s (a^) and ( b ) . The d i f f e r e n c e between (a) and (a ) may be c h a r a c t e r i z e d as f o l l o w s . For any b,c € F, according t o (a) and assuming t h a t b A c i s defined i n P „ „ . (i<?e., b i c ) , the element b A c i s a member of F, QMA where b A c i s the greatest lower bound of {b,c}, as shown i n Sectio n (D); while according t o (a„) and re g a r d l e s s of whether or not b A c i s H defined i n P Q ^ » a n Y o n e °f "the lower bounds of {b,c} i s a member of F. Now i f b A c i s defined i n P „ W . , then (a„) and (b) do ensure t h a t QMA H b A c € F i f b,c € F. For by (a ), some lower bound of {b,c} i s a member of F i f b,c € F, and so by ( b ) , the g.l . b . {b,c} = b A c i s a member of F i f b,c € F. That i s , though a f i l t e r i n a PQJ^ i S defined by c o n d i t i o n (a I T) r a t h e r than ( a ) , nevertheless a f i l t e r i n . P . „ . does H QMA s a t i s f y c o n d i t i o n (a) f o r those b,c € F such t h a t b ch c. The dual modified c o n d i t i o n (a') which, together w i t h the n unmodified ( b 1 ) , defines an i d e a l I i n a P^w. i s , of course, f o r any QMA b,c,d € P N „ . : ' ' QMA (a') I f b,c € I , then there i s a d € I such t h a t d > b and d > c. ri And as above, an. i d e a l i n a P Q ^ does s a t i s f y the unmodified c o n d i t i o n ( a 1 ) f o r those b,c € I such t h a t b o c. As i n the Boolean case, we define an u l t r a f i l t e r ( u l t r a i d e a l ) i n a PQJ^ A S A P R O P E R f i l t e r ( i d e a l ) which i s not the proper subset of any proper f i l t e r ( i d e a l ) i n P Q ^ • A n d a p r i n c i p a l u l t r a f i l t e r . and a p r i n c i p a l u l t r a i d e a l are defined w i t h respect t o an atom of P Q ^ A S ^ N Section C. Herea f t e r , P^ r e f e r s t o both P Q ^ A N I ' 'QML i n d i s c r i m i n a t e l y . 30 A substructure of P „ W i s a non-empty subset of elements of P_ W which QM QM i s c l o s e d w i t h respect t o the A, y,^ operations of P^ M (where the A, v operations of P^^ are p a r t i a l - o p e r a t i o n s , as described i n Secti o n (D).) Any non-empty subset of elements of P generates a substructure of P^ M when closed w i t h respect t o the operations of P^ M . A substructure o f PQ^ i s Boolean IFF i t s elements are mutually ( i . e . , p a i r w i s e ) compatible. Any non-empty subset of mutually compatible elements i n P generates a Boolean substructure of P when cl o s e d w i t h respect t o the operations o f P_„ • And f o r any P. # P„ i n P . „ , there i s no Boolean substructure i n QM J 1 2 QM P_„ which contains both P_ , P. . Any element P € P . W i s a member of QM 1 2 QM some Boolean substructure i n P.,, , at l e a s t the Boolean substructure QM c o n s i s t i n g of j u s t the elements { P ^ P ^ O j l } . A maximal Boolean substructure mBS of P_„ i s a Boolean substructure which i s not the proper subset of QM any other Boolean substructure of P . And by Zprn's lemma, any Boolean substructure of P i s contained i n a maximal one (Varadarajan, 1962, p. 204). The centre of a P... i s the subset of elements i n P . W which QM QM are compatible w i t h every element i n P ^ . This subset i s i n f a c t a c l o s e d substructure of P . W , and moreover, i t i s a Boolean sub s t r u c t u r e . The QM centre of any P „ , contains a t l e a s t the 0, 1 elements of P_„ s i n c e J QM QM 3 the 0, 1 elements are compatible w i t h every other element i n P ^ . S e c t i o n G. Mappings on a S t r u c t u r e Let X, y be any a l g e b r a i c and/or l a t t i c e - t h e o r e t i c s t r u c t u r e s which have A, V,-1- operations defined on a set of elements i n c l u d i n g the d i s t i n g u i s h e d 0-element and 1-element. Any mapping m : X V from any 31 s t r u c t u r e X t o any s t r u c t u r e / assigns values as f o l l o w s : Ma For any b,c,d € X, m(b) i s unique, that i s , i f b = c i n X then m(b) = m(c) i n V. For example, i f b A c = d i n X then m(b A c) = m(d) i n Yt i f b = c i n X then m(b ) = m(c) i n y. Mb m(0) = 0 i n /. Moreover, any n o n - t r i v i a l mapping m : X -»• V a l s o a s s i g n s : Mc m(l) = 1 i n V. I f y i s the two-element Boolean s t r u c t u r e » then m i s a b i v a l e n t mapping designated as m : X {0,1}. A homomorphic mapping h : X y preserves the operations defined i n X, i . e . , f o r any b,c € X, HI h(b A cO = h(b) A h ( c ) . H2 h(b V c) = h(b) v h ( c ) . H3 Hbx) =(h(b)) i. A mapping m : X -*• y i s s a i d t o be i n j e c t i v e IFF, f o r any b,c £ X, i f b t c then m(b) f m(c). C l e a r l y , an i n j e c t i v e mapping i s one-to-one i n t o y. A mapping m : X -*• y i s s a i d t o be s u r j e c t i v e IFF m(X) = y, i . e . , the image of X under m i s the e n t i r e y. An isomorphic mapping m : X -»• Y i s an i n j e c t i v e and sur j e c t i v e mapping, i . e . , a one-to-one mapping, which preserves the operations of X (Lang, 1971, pp. 87, 90, 106; B i r k h o f f , 1948, p. v i i ) . An imbedding of one s t r u c t u r e i n t o another i s a homomorphic mapping which i s i n j e c t i v e (Bub, 1974, p. 68). Notes 1 ' Hughes discusses the problem of the t r a n s i t i v i t y of 5 i n a p a r t i a l - B o o l e a n algebra and proves t h a t a quantum p a r t i a l - B o o l e a n algebra 32 of subspaces of a H i l b e r t space i s a s s o c i a t i v e and t r a n s i t i v e (Hughes, 1978, p. VI.18). 2 As described i n note 5 of chapter I V ( E ) , orthocomplementation i s defined as a type of mapping, namely, a dual automorphism. 3 P i r o n d e f i n e s the centre of a P n M ; the centre of a P- . can be defined i n e x a c t l y the same way. ( P i r o n , 1976, p. 29)-. 33 CHAPTER I I THE CLASSICAL PRECEDENT FOR A BIVALENT TRUTH-FUNCTIONAL SEMANTICS Section A. The Standard Semantics of C l a s s i c a l P r o p o s i t i o n a l Logic C l a s s i c a l p r o p o s i t i o n a l l o g i c assigns t r u t h values t o a set L = of well-formed formulae by semantic mappings, c a l l e d v a l u a t i o n s , which are b i v a l e n t and t r u t h - f u n c t i o n a l . A v a l u a t i o n v on an L i n i t i a l l y assigns the value 0 ( f a l s e ) or 1 ( t r u e ) t o each of the atomic (sub)formulae i n L. And then the v a l u a t i o n assigns 0, 1 values to every other formula i n L i n the f o l l o w i n g r e c u r s i v e manner: f o r any TF3 v ( f x ) = 1 IFF v ( f ) = 0, where "A" designates "and," "v" designates " o r , " and " - L " designates "not." This (redundant) l i s t of b i c o n d i t i o n a l s c h a r a c t e r i z e s the t r u t h - f u n c t i o n a l i t y c o n d i t i o n on the v a l u a t i o n s . The bi v a l e n c y c o n d i t i o n r e q u i r e s that every formula i n L be assigned a 0 or 1 value. there are as many v a l u a t i o n s f o r a set L of formulae as there are rows i n the t r u t h - t a b l e f o r L, where each row i n the t r u t h - t a b l e i s s p e c i f i e d by a d i f f e r e n t i n i t i a l assignment of 0, 1 values t o the atomic (sub)formulae o c c u r r i n g i n L. And i f n i s the number of atomic (sub)formulae i n L, TF1 v ( f A f ) = 1 TF2 v ( f V f ) = 1 IFF v ( f 1 ) = v ( f ) = 1 IFF v ( f ) = 1 or v ( f ) = 1 According t o the t r u t h - t a b l e method of schematizing v a l u a t i o n s , 34 then there are e x a c t l y 2 v a l u a t i o n s f o r L. Such a c o l l e c t i o n of v a l u a t i o n s can be regarded as a b i v a l e n t t r u t h - f u n c t i o n a l semantics f o r L. This n o t i o n of a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r an L w i l l be r e s t a t e d i n a l g e b r a i c terms i n Sectio n (D). Section B. The Boolean S t r u c t u r e Determined by C l a s s i c a l P r o p o s i t i o n a l Logic In the a l g e b r a i c approach t o c l a s s i c a l p r o p o s i t i o n a l l o g i c , we s t a r t w i t h a set L of formulae which i s cl o s e d w i t h respect t o the A, V, -1- operations. Such a closed L i s p a r t i t i o n e d i n t o equivalence c l a s s e s w i t h respect t o the standard, c l a s s i c a l proof t h e o r e t i c equivalence r e l a t i o n : f o r any f. ,f„ € L, f. ~ fn IFF \r f„ o f n and h f n D f , - ' 1 2 1 2 1 2 2 1 where ~^ i s c l a s s i c a l d e r i v a b i l i t y . The r e s u l t i n g set of equivalence c l a s s e s form a Boolean s t r u c t u r e , o f t e n c a l l e d the Lindenbaum a l g e b r a , which s h a l l be l a b e l e d L = <E = { / f ^ A / f ^ / , . . . },A, V, J _,5,0,l>. (The equivalence c l a s s c o n t a i n i n g f i s designated " / f ^ / " . ) For any f±,f2 € L, /f±/ A / f 2 / = /f± A f 2 / ; /f±/ V / f y = /f± V f ^ ; / f ^ " ^ 1^1 \ and If^l S If^l IFF f ^ Y f . The 0-element of L i s the equivalence c l a s s of anti-theorems or c o n t r a d i c t i o n s , w h i l e the 1-element i s the equivalence c l a s s of theorems or t a u t o l o g i e s . When the number n of atomic (sub)formulae i n L i s f i n i t e , then the L s t r u c t u r e of L i s f i n i t e and 2 n n atomic, w i t h e x a c t l y 2 elements and 2 atoms. But when the number of atomic (sub)formulae i n L i s i n f i n i t e , then the L s t r u c t u r e o f L i s i n f i n i t e and atomless. For example, the closed set L^ of p r o p o s i t i o n a l formulae i n j u s t two p r o p o s i t i o n a l v a r i a b l e s , say R and S, i s p a r t i t i o n e d i n t o e x a c t l y 16 equivalence c l a s s e s : 35 ^ /—N co CO \ co -l CO CO -1 ^ co co CO co CO Pi Pi < < Ml > > < < in >^  > > < > \ A A Pi -< A A Pi \ co Pi \ Pi \ Pi \ Pi \ v ' \ CO \ Pi Pi \ Pi \ Pi \ Pi \ Pi \ 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 l u 2 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 l u 3 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 l u 4 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 l These equivalence classes form the Lindenbaum algebra L diagrammed as 0 = A A R 7 Notice that the four atoms i n t h i s Lindenbaum algebra are not the equivalence classes of the atomic formulae R, S, but rather are the following: /R A S/, /R A S V , ' /R"1" A S/, /R x A S"""/. 36 Every Lindenbaum algebra of (equivalence c l a s s e s o f) formulae of c l a s s i c a l p r o p o s i t i o n a l l o g i c i s a Boolean s t r u c t u r e . The simplest ( n o n - t r i v i a l ) Boolean s t r u c t u r e has j u s t the two elements 0 and 1 and i s o f t e n c a l l e d ; i t s h a l l a l s o be l a b e l e d {0,1}. Any Boolean s t r u c t u r e can be homomorphically mapped onto t h i s simplest Boolean s t r u c t u r e , as described next. Sect i o n C. B i v a l e n t Homompr.phic Mappings on Any Boolean S t r u c t u r e Given any Boolean s t r u c t u r e B, there e x i s t b i v a l e n t , homomorphic mappings h : 8 -+ {0,1} which can be defined w i t h respect t o the u l t r a f i l t e r s i n B since there i s a one-to-one correspondence between u l t r a f i l t e r s i n B and b i v a l e n t homomorphisms on B. S i k o r s k i d e f i n e s each b i v a l e n t homomorphism h w i t h respect t o an u l t r a f i l t e r UF as f o l l o w s : f o r any element b ( B, h(b) = 1 i f b € UF and h(b) = 0 i f b j£ UF ( S i k o r s k i , 1960, p. 16). However, each h on a B may be e q u i v a l e n t l y defined w i t h respect t o UF as: f o r any b € 8, h(b) = 1 i f b € UF and h(b) = 0 i f b € UI, where UI i s the unique u l t r a i d e a l dual to UF. In t h i s t h e s i s , the l a t t e r i s taken as my usu a l d e f i n i t i o n of a b i v a l e n t homomorphism. For any B, my d e f i n i t i o n and S i k o r s k i ' s d e f i n i t i o n are equivalent because, f o r any UF and dual UI i n B and f o r any b € 8, b f. UF IFF b 6 UI, as shown i n Chapter 1(C). But when we consider the non-Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by quantum mechanics, i t i s not always the case t h a t i f b J? UF then b € UI. So the two d e f i n i t i o n s d i f f e r and i t i s argued i n Chapter VI(B) t h a t my d e f i n i t i o n i s more u s e f u l . Each mapping h : 8 -+ {0,1} i s c l e a r l y b i v a l e n t . And by d e f i n i t i o n , a homomorphism s a t i s f i e s the c o n d i t i o n s HI, HI, H3 l i s t e d i n 37 Chapter 1(G), where l A i = l v l = l , O A O = O v O = 0 , 1 A 0 = 0 A 1 = 0 , l v O = O v l = l , and 1 X = 0 , 0^=1. I t i s easy to show that a b i v a l e n t mapping on an a l g e b r a i c s t r u c t u r e i s homomorphic qua HI, H2, H3, IFF i t i s t r u t h - f u n c t i o n a l qua TF1, TF2, TF3 (Bub, 1974, p. 99). Thus each b i v a l e n t homomorphism on a Lindenbaum algebra i s b i v a l e n t and t r u t h - f u n c t i o n a l . A l t e r n a t e l y , the t r u t h - f u n c t i o n a l character of every b i v a l e n t homomorphism on a 8 can be shown as f o l l o w s . As mentioned i n Chapter 1(C), every u l t r a f i l t e r and dual u l t r a i d e a l i n any 8 i s prime, and each u l t r a f i l t e r together w i t h i t s dual u l t r a i d e a l completely exhaust 8, i . e . , 8 = UF U UI. Moreover, i t f o l l o w s from the eigh t c o n d i t i o n s ( a ) - ( d ) , ( a ' ) - ( d ' ) , l i s t e d i n Chapter 1(C), which d e f i n e prime UF and prime UI, th a t each b i v a l e n t homomorphism defined w i t h respect t o UF and UI i s t r u t h - f u n c t i o n a l . For the eight c o n d i t i o n s y i e l d the f o l l o w i n g b i c o n d i t i o n a l s , f o r any b,b 1 >b2 € 8: UI b„ 1 A b 2 € UF IFF b l € UF and b 2 6 UF, by (a) and ( b ) . b l A b 2 € UI IFF b l € UI or b 2 € UI, by ( bT ) and ( d ' ) . U2 b„ 1 v b 2 6 UF IFF b l € UF or b 2 € UF, by (b) and ( d ) . b l v b 2 € m IFF b l € UI and b 2 € UI, by (a') and ( b ' ) . U3 b X € UF IFF b € UI, by ( c ) . b X 6 UI IFF b € UF, by (c»). So by the d e f i n i t i o n of h : 8 -»• {0,1} w i t h respect t o UF and UI: TF1 h(b A b ) = 1 IFF h(b ) = h(b ) = 1 h ( b 1 A b ) = 0 IFF h(b ) = 0 or h(b ) = 0 TF2 h ( b 1 v b 2 ) = 1 IFF h(b ) = 1 or h(b ) = 1 h ( b 1 V b 2 ) = 0 IFF h ( b 1 ) = h ( b 2 ) = 0 38 TF3 h ( b X ) = 1 IFF . h(b) = 0 h ( b X ) = 0 IFF h(b) = 1. Thus each b i v a l e n t homomorphism on a 8 i s t r u t h - f u n c t i o n a l . Furthermore, any 8 admits many b i v a l e n t homomorphisms. I f b^ ^ b 2 are any p a i r o f d i s t i n c t elements i n B, then as mentioned i n Chapter 1(C), there i s some u l t r a f i l t e r i n 8 which contains one element but not the other. Hence there i s some b i v a l e n t homomorphism on B which assigns the value 1 to one element and 0 to the r o t h e r . In other words, f o r any p a i r o f d i s t i n c t elements b^ t i n a 8, , there i s some h such that h(b^) t h ( b 2 ) ; t h i s has been c a l l e d the s e m i - s i m p l i c i t y property o f 8 (Kochen-Specker, 1967, p. 67). And i n p a r t i c u l a r , as Halmos shows, f o r any nonzero b ^ 0 i n a B, there i s some h such t h a t h(b) ^  h(0) = 0, i . e . , such that h(b) = 1 sin c e every h assigns the value 0 to the 0-element (Halmos, 1963, p. 77). The former n o t i o n s h a l l be taken to define a complete c o l l e c t i o n o f b i v a l e n t homomorphisms on an a l g e b r a i c s t r u c t u r e X, t h a t i s , a c o l l e c t i o n o f b i v a l e n t homomorphisms on an X i s complete IFF, 1 f o r any d i s t i n c t b i c i n X, there i s an h such t h a t h(b) ± h ( c ) . C l e a r l y , the completeness o f the c o l l e c t i o n of b i v a l e n t homomorphisms on a Boolean s t r u c t u r e 8 i s ensured by the s e m i - s i m p l i c i t y property o f 8. When 8 i s atomic, then besides the above-mentioned one-to-one correspondence between b i v a l e n t homomorphisms and u l t r a f i l t e r s (and dual u l t r a i d e a l s ) there i s a l s o a one-to-one correspondence between u l t r a f i l t e r s and atoms. Each atom a € 8 i s a member of e x a c t l y one u l t r a f i l t e r i n 8, namely UF = { b € 8 : b > a}. And each atom a i s assigned the value cL 1 by e x a c t l y one b i v a l e n t homomorphism on B, namely the h defined w i t h a respect to UF and i t s dual U l . I t i s easy to show th a t a c o l l e c t i o n a a of b i v a l e n t homomorphisms on an atomic 8 i s complete IFF i t i s as l a r g e as 39 the number of atoms i n 8 . Proof: By definition, a complete collection is large enough so that every atom a ± 0 is assigned the value l = h ( a ) ^ h ( 0 ) = 0 by some h on 8. Since each atom is assigned the value 1 by exactly one bivalent homomorphism, the complete collection i s as large as the number of atoms. Conversely, consider the collection of bivalent homomorphisms on an atomic 8 which i s as large as the number of atoms in : 8. By definition, each bivalent homomorphism in, this collection is an h defined (via UF and Ul ) with respect to an atom a € 8. a a a Now by a theorem due to Rutherford (Chapter 1(B)), for any b f c in 8, there i s an atom a € 8 such that a 5 b but a fc c, or a 2 c but a fc b. If a 5 b but a fc c, b € UF and c £ UF , and so a a h (b) = 1 i h (c). Similarly, i f a < c but a fc b, then c € UF and a a a b % UF , and so h (c) = 1 i- h (b). Thus for any b / c in 8» there a a a is an h on 8 such that h (b) # h (c). Q.E.D. a a a Section D. The Algebraic Semantics for the Lindenbaum Algebra These facts about bivalent homomorphisms on a Boolean structure are relevant for the concept of a bivalent truth-functional semantics for the Lindenbaum algebras of classical propositional logic. Each u l t r a f i l t e r in the L structure of a (closed) set of formulae L is i t s e l f a subset of (equivalence classes of) formulae in L which i s deductively complete in the sense that, for any UF in the L of an L, and for any formulae f ^ ' f ^ ^  ^ ' ^ ^ ^  a n c^ ^1 1" 2^ ' then If I £ UF. And each u l t r a f i l t e r in L is maximally consistent in the sense that the meet of a l l the elements in any UF i s never the 0-element of L, i.e., the conjunction of a l l the (equivalence classes) of formulae in UF i s never a contradiction; but i f any element in L which i s outside a 40 given UF were added to that UF, then the meet of a l l the elements i n UF would be the 0-element of L. As described i n the previous section, each bivalent homomorphism on a Lindenbaum algebra i s bivalent and t r u t h - f u n c t i o n a l . Moreover, f o r any element If I € L and any UF c L, e i t h e r If/ € UF or /fV € UF but not both; hence, no bivalent homomorphism on L assigns the value 1 to both If I and / f "*"/ since every bivalent homomorphism assigns the value 1 to an u l t r a f i l t e r of elements i n L. And i f a bivalent homomorphism were to assign the value 1 to any other element i n L besides those elements in the u l t r a f i l t e r which defines h, then h would assign the value 1 to the 0-element of L. So each bivalent homomorphism can be said to be a maximally consistent mapping on L. Moreover, each bivalent homomorphism on the Lindenbaum algebra of an L i s the algebraic version of one of the standard valuations f o r L. That i s , f o r any given valuation v^ on an L, there i s a corresponding bivalent homomorphism h^ on the L of that L such that, f o r every formula f € L, v ( f ) = h Q ( / f / ) (Bub, 1974, p. 102). And f i n a l l y , i n t h i s thesis the complete c o l l e c t i o n of bivalent homomorphisms on a Lindenbaum algebra i s regarded as a bivalent, t r u t h - f u n c t i o n a l semantics. The analogy between the complete c o l l e c t i o n of bivalent homomorphisms on an L and the t r u t h table c o l l e c t i o n of valuations f o r an L may be elaborated as follows. I f we assume that the number n of atomic (sub)formulae i n L i s f i n i t e , then the L structure of L i s f i n i t e and atomic, with exactly 2 n atoms. Thus the complete c o l l e c t i o n of bivalent homomorphisms on L contains 2 n bivalent homomorphisms, ju s t as the truth-table c o l l e c t i o n of valuations f o r that L contains 2 n valuations. 41 Each v a l u a t i o n f o r L i s s p e c i f i e d by i t s i n i t i a l assignment of 0, 1 values t o the n atomic (sub)formulae i n L, and l i k e w i s e each b i v a l e n t homomorphism on the L s t r u c t u r e of L i s s p e c i f i e d by i t s i n i t i a l assignment o f 0, 1 values t o the n equivalence c l a s s e s o f atomic formulae i n L. For example, an i n i t i a l assignment of the values 0 t o R and 1 t o S s p e c i f i e s the v a l u a t i o n v i n the t r u t h t a b l e f o r L o Z given i n Sectio n (B). S i m i l a r l y , the i n i t i a l assignment of the values 0 to /R/ and 1 t o /S/ s p e c i f i e s the unique atom /RA/A S/ i n the Lindenbaum algebra of L ^ ; t h i s atom i n t u r n s p e c i f i e s the unique u l t r a f i l t e r U F / R J . A & / = { / R X A S/, /S/, /RV, /R V S/, /R x V S/, /R x V S x/, /(R = S ) X / , /R V R X/}; and t h i s u l t r a f i l t e r s p e c i f i e s a unique b i v a l e n t homomorphism h ^ R x A g ^ on , where ^ R x A g ^ ( / f / ) = v 3 ( f ) f°r every formula f € L^ • The concept of a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r a Boolean Lindenbaum algebra described i n t h i s chapter w i l l be t r e a t e d i m t h i s t h e s i s as a precedent f o r any proposed b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r the Boolean; p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics and the non-Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by quantum mechanics. In p a r t i c u l a r , subsequent chapters make use of the f o l l o w i n g : For any p r o p o s i t i o n a l s t r u c t u r e P, a mapping which assigns the value 1 t o an u l t r a f i l t e r UF of elements i n P and assigns the value 0 to the dual u l t r a i d e a l U l of elements i n P i s not only b i v a l e n t but a l s o t r u t h - f u n c t i o n a l w i t h respect t o the elements i n UF U U l . Such a b i v a l e n t , t r u t h - f u n c t i o n a l mapping defined w i t h respect t o an UF and dual U l may be c a l l e d an ul t r a v a - l u a t i o n because, on a Lindenbaum algebra of c l a s s i c a l p r o p o s i t i o n a l l o g i c , such a mapping i s the a l g e b r a i c v e r s i o n of a standard 42 v a l u a t i o n , which i s regarded i n t h i s t h e s i s as the paradigm semantic mapping. The 0, 1 values assigned by an u l t r a v a l u a t i o n on a p r o p o s i t i o n a l s t r u c t u r e may be i n t e r p r e t e d as the t r u t h - v a l u e s t r u e and f a l s e , again because, on a Lindenbaum a l g e b r a , an u l t r a v a l u a t i o n i s the a l g e b r a i c v e r s i o n of a standard v a l u a t i o n . And use i s e s p e c i a l l y made of the n o t i o n t h a t a b i v a l e n t t r u t h - f u n c t i o n a l semantics f o r a P i s a complete c o l l e c t i o n of b i v a l e n t , t r u t h - f u n c t i o n a l mappings. So i t i s c l e a r t h a t the existence of only one or s e v e r a l b i v a l e n t , t r u t h - f u n c t i o n a l mappings on a P does not yet c o n s t i t u t e a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r P. But i n order t o show the i m p o s s i b i l i t y of such a semantics, i t obviously s u f f i c e s t o show that there i s not even one b i v a l e n t , t r u t h - f u n c t i o n a l mapping on P. Notes This n o t i o n of a complete c o l l e c t i o n o f b i v a l e n t homomorphisms was suggested t o me by Kochen and Specker. In t h e i r 1967 Theorem 0, Kochen-Specker prove t h a t a p a r t i a l - B o o l e a n algebra of quantum p r o p o s i t i o n s , l a b e l e d ^ n M A » c a n be imbedded i n t o a Boolean algebra 8 IFF there e x i s t s what i n t h i s t h e s i s i s c a l l e d a complete c o l l e c t i o n of b i v a l e n t homomorphisms on P. . Kochen-Specker a l s o d e f i n e a weak imbedding of a P_ i n t o a 8; such an imbedding e x i s t s IFF there e x i s t s a l a r g e enough c o l l e c t i o n of b i v a l e n t homomorphisms on P so t h a t , f o r every nonpzero element P i- 0 i n P_„. , there i s some h : P -> {0,1} such t h a t h(P) t h ( 0 ) , i . e . , g h(P) = 1 since every h assigns the value 0 t o the 0-element (Kochen-Specker, 1967, pp. 67,884). Such a c o l l e c t i o n may be c a l l e d weakly complete. The n o t i o n of a weakly- complete c o l l e c t i o n of b i v a l e n t homomorphisms on a p r o p o s i t i o n a l s t r u c t u r e i s mentioned i n Chapters V and VI. 43 CHAPTER I I I THE CLASSICAL PRECEDENT FOR A STATE-INDUCED SEMANTICS Preface 1 We consider p r o p o s i t i o n s which make a s s e r t i o n s about the real-number values of the magnitudes, i . e . , measurable p r o p e r t i e s , of a c l a s s i c a l p h y s i c a l system, f o r example: 2 2 2 The k i n e t i c energy of a 1 kg swinging pendulum i s between 19-20 kg m /sec . (magnitude) ( system ) (value) As w i l l be described i n t h i s chapter, such p r o p o s i t i o n s and the l o g i c a l operations "and," " o r , " "not" among such p r o p o s i t i o n s can-be a s s o c i a t e d w i t h v a r i o u s mathematical machinery i n the formalism of c l a s s i c a l mechanics. These a s s o c i a t i o n s determine the s t r u c t u r e of a set of such p r o p o s i t i o n s . This s t r u c t u r e i s a cr-complete, atomic Boolean s t r u c t u r e P . Moreover, the formalism of c l a s s i c a l mechanics i n c l u d e s state-induced b i v a l e n t homomorphisms, or e q u i v a l e n t l y , state-induced d i s p e r s i o n - f r e e p r o b a b i l i t y measures, which can be regarded as performing the semantic task of a s s i g n i n g t r u t h - v a l u e s to the elements of P„„ . For each b i v a l e n t CM homomorphism or d i s p e r s i o n - f r e e p r o b a b i l i t y measure induced by the s t a t e of a c l a s s i c a l system i s an u l t r a v a l u a t i o n on the P s t r u c t u r e of p r o p o s i t i o n s d e s c r i b i n g the system, j u s t as each o f the standard v a l u a t i o n s f o r a set L of formulae of c l a s s i c a l l o g i c i s an u l t r a v a l u a t i o n on the L s t r u c t u r e of L. This s t r a i g h t f o r w a r d analogy i s a strong m o t i v a t i o n f o r s e r i o u s l y c o n s i d e r i n g the no t i o n of a state-induced semantics f o r the p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics and a l s o c o n s i d e r i n g the n o t i o n 44 of a state-induced semantics f o r the p r o p o s i t i o n a l s t r u c t u r e s determined by quantum mechanics, as s h a l l be proposed i n Chapter VI. S e c t i o n A. The States o f a C l a s s i c a l System Determine the Real Values of  That System's Magnitudes According to the Hamiltonian f o r m a l i z a t i o n of c l a s s i c a l mechanics, a p h y s i c a l system i s a s s o c i a t e d w i t h an a b s t r a c t phase space 2 which i s parameterized by p o s i t i o n and momentum coordinates and whose d i m e n s i o n a l i t y r e f l e c t s the degrees of freedom of the system. For example, a p h y s i c a l system w i t h only one degree of freedom, such as a b a l l f a l l i n g i n a s t r a i g h t l i n e , i s a s s o c i a t e d w i t h the sim p l e s t phase space which i s two dimensional and has one p o s i t i o n coordinate and one momentum coordinate. Each p o i n t w € £! represents a pure s t a t e of the system a s s o c i a t e d w i t h 2, f o r a pure s t a t e i s a s p e c i f i c a t i o n of the system's p o s i t i o n and momentum values. According to c l a s s i c a l mechanics, the values o f every other (mechanical) magnitude o f the system can be c a l c u l a t e d once the system's s t a t e i s s p e c i f i e d . In p a r t i c u l a r , the c l a s s i c a l formalism represents each magnitude 3 A by a r e a l - v a l u e d , measurable f u n c t i o n f : 2 R on the phase space a s s o c i a t e d w i t h the system such t h a t the image of any p o i n t w € 2 under the f u n c t i o n f i s the real-number value a € R (the real-number l i n e ) o f the magnitude A when the system i s i n the s t a t e w. The r e a l - v a l u e d f u n c t i o n s • • • r e p r e s e n t i n g the c l a s s i c a l magnitudes A,B, . . . have the r i n g operations + and • defined among them as the usual sum and product of f u n c t i o n s : f o r any f , f ^ on 2 and f o r every w € 2, ( f +f f i)(w) = f (w)+f (w), and ( f • f )(w) = f A (w) "fgCw). (Here + and • work l i k e the a d d i t i o n and m u l t i p l i c a t i o n of r e a l numbers.) 45 For example, consider as a system a 1 kg pendulum swinging so that i t s maximum height i s 2 m above i t s minimum height. Let w^  and w^  be the f o l l o w i n g s t a t e s . w^ : At the top of i t s swing, the pendulum's height p o s i t i o n i s 2 m and i t s momentum i s 0 kg m/sec. w^ : Near the bottom of i t s swing, the pendulum's height p o s i t i o n i s n e a r l y 0 m and i t s momentum i s n e a r l y maximal, say 6.2 kg m/sec. The magnitude k i n e t i c energy, K, i s represented i n the c l a s s i c a l formalism 1 2 by the r e a l - v a l u e d f u n c t i o n f„ = • (momentum) . So when the J K 2 • mass 2 2 pendulum's s t a t e i s w^  , the real-number value of K i s 0 kg m /sec . 2 2 And when the pendulum's s t a t e i s w^  , the value of K i s 19.2 kg m /sec . So the f a c t that the real-number values of a c l a s s i c a l system's magnitudes depend upon the system's s t a t e has been f o r m a l i z e d by r e p r e s e n t i n g each magnitude A by a r e a l - v a l u e d , measurable f u n c t i o n f : °J -* R on a c l a s s i c a l phase space whose p o i n t s represent the system's s t a t e s . A l t e r n a t e l y , each s t a t e w € Q can: i t s e l f be regarded as a mapping from a (closed) set F of f u n c t i o n s r e p r e s e n t i n g c l a s s i c a l magnitudes to the real-number l i n e , i . e . , w : F ->• R, such t h a t , f o r any poi n t w € & and f o r any f u n c t i o n f : Q -> R, w(f.) = f.(w). The A A A mapping w : R R may be c a l l e d the state-induced mapping. I t f o l l o w s that each state-induced mapping preserves the + and * operations defined among the f u n c t i o n s : f o r any given, f i x e d w € S2 and f o r any f u n c t i o n s f" A, f g on ffi, w(f + f ) = ( f + fg ) (w ) = f (w) + f (w) = w ( f A ) + w ( f B ) ; and w ( f A • f g ) = ( f f t • f g X w ) = f ^ w ) • f g ( w ) = w ( f A ) - w C ^ ) . 46 This mathematical machinery of real-valued functions and state-induced mappings not only formalizes the procedure by which real-number values are assigned to the magnitudes of a c l a s s i c a l system, but also i m p l i c i t l y formalizes a procedure by which t r u t h values can be assigned to the propositions which make assertions about the real-number values of a c l a s s i c a l system's magnitudes, as w i l l be made e x p l i c i t i n Section (C). Section B. The Propositional Structure Determined by C l a s s i c a l Mechanics When a set of real-valued, measurable functions on a Si i s a closed set with respect to the +, • operations, then the set forms a t commutative-ring-with-unit, labeled F ^ = <{f^,f^, . . .}, + ,• ,0,1>. The 0-element i s the constant function f ^ which assigns the real-number 0 to every w € Si, and the 1-element i s the constant function f which assigns the real-number 1 to every point i n Si. Some of the functions i n F ^ are idempotent functions fp s a t i s f y i n g : f p • f p = f p , i . e . , f o r every w € Si, ( f p • f p)(w) = f p ( w ) . Since the product fp • fp i s defined as, f o r every w € Si, (fp • fp)(w) = fp(w) *fp(w), i t follows that the real-number value r = f p(w) of an idempotent function i s eit h e r 0 or 1. In other words, each fp i s a function from Si to {0,1}. A set of idempotent functions which i s closed with respect to the +, • operations forms a Boolean-ring-with-unit, or i n other words, the idempotent elements of F form a Boolean-ring-with-u n i t , as defined i n Chapter 1(A). And_in t h i s Boolean-ring-with-unit, the Boolean operations A, V,-1-, and the l a t t i c e p a r t i a l - o r d e r i n g r e l a t i o n 5 can be defined i n terms of the r i n g operations + and • as usual, y i e l d i n g a Boolean structure of idempotent functions on a Si . 47 Each idempotent f u n c t i o n on a c l a s s i c a l phase space i s a c h a r a c t e r i s t i c f u n c t i o n defined w i t h respect t o a unique subset Wp c 2 as f o l l o w s : f o r any p o i n t w € 2, f p ( w ) = 1 i f w € Wp and f p ( w ) = 0 i f w ^ Wp , i . e . , w € W^. Each Wp i s a measurable ( i . e . , B o r e l ) -1 A. subset o f 2- and Wp = f ({{.1 •})• = {w € 2 : f(w) = 1}; and each Wp i s the s e t - t h e o r e t i c (ortho)complement of Wp , i . e . , Wp = 2 - Wp . Thus the idempotent f u n c t i o n s on a 2 are i n a one-to-one correspondence w i t h the B o r e l subsets of 2; each B o r e l subset uniquely defines an idempotent f u n c t i o n (qua c h a r a c t e r i s t i c f u n c t i o n ) and each idempotent f u n c t i o n uniquely -1 s p e c i f i e s a B o r e l subset ( v i a i t s in v e r s e image f p ('{I})-)'. The B o r e l subsets of a 2 form a Boolean-ring-with-unit ( w i t h + , *, 0, 1, i n t e r p r e t e d as symmetric d i f f e r e n c e , s e t - i n t e r s e c t i o n , the empty s e t , and the e n t i r e space 2, r e s p e c t i v e l y ) , which i s isomorphic t o the Boolean-ring-with-unit of idempotent f u n c t i o n s on 2. And the Boolean-ring-w i t h - u n i t of B o r e l subsets of a 2 i s a l s o a Boolean s t r u c t u r e ( w i t h A, V,"1", 5, i n t e r p r e t e d as s e t - i n t e r s e c t i o n , set-union, set-(orthoComplementation, and s e t - i n c l u s i o n , r e s p e c t i v e l y ) , which i s isomorphic t o the Boolean s t r u c t u r e of idempotent f u n c t i o n s on 2 (Bub, 1974, p. 105). The Boolean s t r u c t u r e o f idempotent f u n c t i o n s on a c l a s s i c a l phase space, or i s o m o r p h i c a l l y , the Boolean s t r u c t u r e o f B o r e l subsets o f the phase space, have each been regarded as a p r o p o s i t i o n a l s t r u c t u r e determined by cl a s s i c a l ' m e c h a n i c s , l a b e l e d P Q ^ • For in:one-way or .another, p r o p o s i t i o n s which make a s s e r t i o n s about the real-number values o f a c l a s s i c a l system's magnitudes have been a s s o c i a t e d w i t h e i t h e r the idempotent f u n c t i o n s on the system's phase space or the uniquely corresponding B o r e l subsets of the system's phase space. For example, i n h i s 1932 book, von Neumann argues 48 that p r o p o s i t i o n s which make a s s e r t i o n s about the values of a system's magnitudes can themselves be regarded as idempotent magnitudes whose 0, 1 values can be i n t e r p r e t e d as the " v e r i f i c a t i o n " and the n o n - v e r i f i c a t i o n of the p r o p o s i t i o n s . Mentioning von Neumann's argument, Kochen-Specker l i k e w i s e regard p r o p o s i t i o n s as idempotent magnitudes whose 0, 1 values are i n t e r p r e t e d as f a l s i t y and t r u t h . There i s a b e t t e r reason, given i n Section ( C ) , why the 0, 1 values e x h i b i t e d by the idempotent magnitudes may be i n t e r p r e t e d as the t r u t h - v a l u e s of p r o p o s i t i o n s . Nevertheless, i n the c l a s s i c a l formalism, idempotent magnitudes are represented by the above-described idempotent f u n c t i o n s on a phase space. On the other hand, i n t h e i r 1936 paper, von Neumann and B i r k h o f f a s s o c i a t e p r o p o s i t i o n s which make a s s e r t i o n s about a c l a s s i c a l system's magnitudes w i t h the subsets of the 4 system's phase space. S i m i l a r l y , Jauch a s s o c i a t e s such p r o p o s i t i o n s w i t h the B o r e l subsets of the system's phase space. E i t h e r a s s o c i a t i o n y i e l d s the Boolean p r o p o s i t i o n a l s t r u c t u r e P = <E = {P'^*^' " ' * ^ ' A' v'~L»->0>-'->* The elements o f P ^ may be thought o f e i t h e r as idempotent f u n c t i o n s or as B o r e l subsets of the phase space; the elements of P represent or are ass o c i a t e d w i t h p r o p o s i t i o n s . The P s t r u c t u r e of any S2 i s a cr-complete atomic, Boolean s t r u c t u r e . And each atom P i n a P „ , i s a one-point w CM idempotent f u n c t i o n f uniquely corresponding w i t h the s i n g l e t o n B o r e l w subset {w}. Section C". The B i v a l e n t , T r u t h - F u n c t i o n a l , State-Induced Semantics f o r the Boolean P_ W S t r u c t u r e s CM Ju s t as the real-number values of a system's magnitudes depend upon the system's s t a t e ( i . e . , upon the values o f the system's p o s i t i o n and momentum), l i k e w i s e the t r u t h values of p r o p o s i t i o n s which make a s s e r t i o n s 49 about the real-number values of a system's magnitudes depend upon th a t system's s t a t e . For example, when the pendulum described i n Section A i s i n the s t a t e w^  , the t r u t h value of the f o l l o w i n g p r o p o s i t i o n i s f a l s e : 2 2 The k i n e t i c energy of the pendulum i s between 19-20 kg m /sec . And when the pendulum i s i n the s t a t e w 2 , the t r u t h value of that p r o p o s i t i o n i s t r u e . The f a c t t h a t a system's s t a t e determines the t r u e values of p r o p o s i t i o n s which make a s s e r t i o n s about the real-number values of the system's magnitudes may be f o r m a l i z e d by d e f i n i n g state-induced u l t r a v a l u a t i o n s on the P^ s t r u c t u r e of these p r o p o s i t i o n s , and such u l t r a v a l u a t i o n s may be described i n two ways. tBoth ways s h a l l be elaborated, 1 even though each y i e l d s the same n o t i o n of state-induced u l t r a v a l u a t i o n s on a P^yj . For one way makes use of concepts introduced i n Section A and thus shows the c o n t i n u i t y between the s t a t e ' s determining the real-number values of magnitudes and the s t a t e ' s determining the t r u t h values of p r o p o s i t i o n s . And the other way makes use of the concept of a d i s p e r s i o n - f r e e p r o b a b i l i t y measure, which recurs i n Chapters V, VI and V I I . As described i n Section A, each state-induced mapping w : F -> R preserves + and *, i . e . , each state-induced mapping on an F i s r e a l - v a l u e d and homomorphic. I t f o l l o w s t h a t each state-induced mapping on the Boolean s t r u c t u r e P„„ of idempotent elements of an F..„ i s b i v a l e n t CM CM and homomorphic. For by the d e f i n i t i o n of the mapping w, f o r any w € S2 and f o r any f on S, w ( f p ) = f p ( w ) = 1 i f w € Wp , and w ( f p ) = f p ( w ) = 0 i f w € W^, where Wp U W^= 2. Thus w : P^ -+ {0,1}, and i n other words, each pure s t a t e of a c l a s s i c a l system induces a b i v a l e n t , t r u t h - f u n c t i o n a l mapping w : P ^ ->- {0,1} on the p r o p o s i t i o n a l s t r u c t u r e ^CM °^ P ^ a s e s P a c e a s s o c i a t e d w i t h the system. ; 50 In f a c t , as we would expect, each state-induced mapping on a P^M i s an u l t r a v a l u a t i o n which assigns the value 1 t o an u l t r a f i l t e r of elements i n P and assigns the value 0 t o the dual u l t r a i d e a l of elements i n P ^ , as shown next. Each p o i n t w € S2 s p e c i f i e s a unique atom i n the P Q H s t r u c t u r e of 52, namely, the one-point idempotent f u n c t i o n f or the corresponding s i n g l e t o n B o r e l subset {w}.; And the w set {P € P „ „ : P > P } i s an u l t r a f i l t e r i n P_„ , namely the unique CM w CM u l t r a f i l t e r UF defined by the atom P ; d u a l l y , the set W W "{P € P „ : P < P"1"} i s the unique u l t r a i d e a l UI dual to UF . Now f o r CM w ^ w w any p o i n t w € S2 and f o r any B o r e l subset Wp c S2, w € Wp IFF 5 P, t h a t i s , f • 5 f_ or {w} c W_ , and a l s o w € W IFF P < P , i . e . , W ir r r W IFF P < P"1" . So by s u b s t i t u t i o n i n t o the above d e f i n i t i o n of the mapping w w, ' f o r any element P € P „ „ , wCP) =1 i f P € UF and w(P) = 0 i f J CM w P € UI . Hence, each state-induced mapping on a P_„ i s an u l t r a v a l u a t i o n w CM and so i s the c l a s s i c a l - m e c h a n i c a l analogue of a standard v a l u a t i o n of 5 c l a s s i c a l - p r o p o s i t i o n a l l o g i c . And thus the 0, 1 values assigned by the state-induced u l t r a v a l u a t i o n s t o the elements of P „ W can be i n t e r p r e t e d CM as the t r u t h values f a l s e and t r u e . This n o t i o n of the state-induced u l t r a v a l u a t i o n s on a V CM s t r u c t u r e may a l s o be developed as f o l l o w s . According to the mathematical formalism of c l a s s i c a l mechanics and c l a s s i c a l s t a t i s t i c a l mechanics, each pure s t a t e w of a system can be regarded as inducing a d i s p e r s i o n - f r e e p r o b a b i l i t y measure on the P s t r u c t u r e of the system's phase space.. A measure n i s a r e a l - v a l u e d f u n c t i o n on a Boolean a l g e b r a , e.g., on ? ^ , which s a t i s f i e s the f o l l o w i n g c o n d i t i o n s : 51 (ua) For any countable set {P.}.- T , of d i s j o i n t elements of P „ W 1 i€Index CM M-CV P.) = 1 |i.(P.). This i s the a d d i t i v i t y c o n d i t i o n , i i (|fb) 0 < LI(P) 5 °°, f o r every P € P C M . (LIC) M-(O) = 0. I t f o l l o w s t h a t H i s i s o t o n e , i . e . , (M-i) I f P 1 5 P 2 , then \J,(?±) < n(P 2), f o r any € P ^ ( S i k o r s k i , 1960, p. 10). A p r o b a b i l i t y measure i s a normed measure s a t i s f y i n g : (|in) LI(1) = 1. And hence, f o r every element P € ? C M , 0 5 (i(P) 5 1, tha t i s , u : P „ „ -»• [0,1] , where [0,1] i s the closed i n t e r v a l from 0 to 1 on CM the real-number l i n e . And f i n a l l y , a d i s p e r s i o n - f r e e p r o b a b i l i t y measure s a t i s f i e s the c o n d i t i o n : (Lid) n(P 2) - ( y ( P ) ) 2 = 0, f o r every P C . A d i s p e r s i o n - f r e e p r o b a b i l i t y measure on a P i s b i v a l e n t . 2 Proof: Since every element P € P C M i s idempotent, i . e . , P = P, 2 c o n d i t i o n (|od) y i e l d s the equation: LICP) = d-i(P)) , f o r every P € P C M . Thus ji(P) = 1 or 0. Q.E.D. (Bub, 1974, p. 60). So each d i s p e r s i o n - f r e e p r o b a b i l i t y measure, h e r e a f t e r l a b e l e d \i , i s a b i v a l e n t mapping H : P -*• {0,1}. Moreover, each d i s p e r s i o n - f r e e p r o b a b i l i t y measure on a w CM P i s a l s o a homomorphic mapping, as shown by the f o l l o w i n g proof due t o Gudder (though Gudder does not r e f e r t o a Boolean s t r u c t u r e l i k e P ). F i r s t , i t i s easy t o show t h a t , f o r any P 1 > p 2 ^ ^CM ' H (P„ V P J = [i. (P,) + n C P j - V. CP, A P ). Proof: The j o i n P V P of W l 2 W l W z W l ^ J- ^ 52 any P ,P € P can be w r i t t e n as the j o i n of three mutually d i s j o i n t elements, e.g., P 1 V ? 2 = P g V P^ V P 5 , where P g = ?± A P^; P^ = P 2 A P^ "; and P g = P A P 2 . Then by a d d i t i v i t y , [i (P„ V P ) = |J. (P_) + | i (P, ) + | i CP r). And by s u b s t i t u t i o n and a d d i t i v i t y : w 1 2 w 3 w 4 w 5 J J •VV + = ^w ( P3 V P 5 ^ + ^w ( P4 Y V = ^w (V + ^w (V + ^w (V + = ^ w ( P l V V + ^w ( P5 ) = ^ w C P l V V + ^ w C P l A V* T h u S (P„ V P ) = (i (P ) + CP ) - (i (P A P„). Q.E.D. With t h i s r e s u l t , i t w 1 2 w 1 w 2 "w 1 2 ' i s easy to prove t h a t any d i s p e r s i o n - f r e e p r o b a b i l i t y measure p. : ?^ {0,1} i s homomorphic, i . e . , f o r any P, P„ ,P„ € 'P_„ , ' ii (P*") = (u ( P ) ) X and 1 2 CM w w pi (P„ V P ) = (j. (.P ). V u. C P J . Proof: For any P € P „ M , w 1 2 w 1 w 2 CM p, (P V P""") = | i (1) = 1; and by a d d i t i v i t y , |i, CP V P x) = u. (P) + u. (P"4"). W W w w w Hence 1 = u, ( P ) + n ( P x ) , and so u. (P^) = 1 - u. (P) = (u. ( ? ) ) X . Now W W W W W (J^CP) = 0 or 1, f o r every P € P ^ so i n the next p a r t of t h i s .proof, there are two cases, one of which has two subcases. For Case 1, assume ^ w ^ P l V P 2 ^ = 1 , a n d i n E d i t i o n , f o r Subcase l a , assume (-^(P^ A P 2 ) = 1 . Then si n c e P^ A P 2 < P^ and P^ A P 2 < P 2 , by c o n d i t i o n 1 (pi.) we have u. (P„ ) = 1 and a l s o p. (P.) = 1. Hence p, (P. v P ) = p, (P ) v p, ( P . ) . w 1 w 2 *w 1 2 *w 1 w 2 For Subcase l b , assume p, (P. A P . ) = 0. Then si n c e u, (P. v P.) w 1 2 w 1 2 = p. ( P . ) + p. (P.) - p, (P. A P.), e i t h e r p, (P. ) = 1 and p ( P 0 ) = 0 or w 1 w 2 w 1 2 w 1 w 2 e l s e p (P„) = 0 and p ( P j = 1. Hence, p (P. v P ) = p (P„) V p (P_) = 1. "w 1 w^ 2 ' *w 1 2 *w 1 *w 2 For case 2, assume u, (P„ V P„) = 0. Then si n c e P < P. V P. and *w 1 2 1 1 2 P„ < P, V P„ , by c o n d i t i o n ( p i ) we have u, (P.) = p (P.) = 0. Hence 2 1 2 . w l ^ w 2 u (P. V P j = Li ( P J V u (P£) = 0. Q.E.D. (based on Gudder, 1970, pp. 433-434). 1 2 w 1 w 2 Thus each pure s t a t e of a c l a s s i c a l system induces a d i s p e r s i o n - f r e e p r o b a b i l i t y measure p, : P ^ -* {0,1} which i s a b i v a l e n t homomorphism on the P „ „ s t r u c t u r e of the phase space a s s o c i a t e d w i t h the system. CM 53 Moreover, each u, : P„„ {0,1} i s an u l t r a v a l u a t i o n on P_„ w CM CM and i s i n f a c t the u l t r a v a l u a t i o n w : P -* {0,1} described above, as shown next. A d i s p e r s i o n - f r e e p r o b a b i l i t y measure on a Boolean algebra of B o r e l subsets of a 2 i s an atomic measure concentrated on a s i n g l e p o i n t i n 2 (Bub, 1974, p. 47), namely the p o i n t w re p r e s e n t i n g the s t a t e which i s s a i d to induce the measure. That i s , each u, on the P_„ s t r u c t u r e w CM of a 2 assigns p r o b a b i l i t y 1 t o the s i n g l e t o n subset {w} (which i s the atom i n P ) and assigns p r o b a b i l i t y 0 t o every other s i n g l e t o n subset of p o i n t s i n 2. Now si n c e u, (P ) = 1 and since u, preserves w w w the ~*~ operation as shown above, i t f o l l o w s t h a t u. (?x) = 1X - 0. Then r w w sin c e u, i s i s o t o n e , we have, f o r any P € P_w , i f P > P then "w . CM w u, (P) = 1, and i f P < P X then u, CP) = 0. Thus each state-induced, w w w d i s p e r s i o n - f r e e p r o b a b i l i t y measure on a P assigns values as f o l l o w s : f o r any P € P„„ , H (P) = 1 i f P € -UF = {P € ? n t t : P > P } and J CM w w CM w u, (P) = 0 i f P € Ul = {P € "P„, : P 5 P x}. So each (J, i s an u l t r a v a l u a t i o n w^ w CM w w on P^M . And c l e a r l y , each [i.^ i s the very mapping w : P^ •+ {0,1} described above; conversely, each mapping w : P -* {0,1} i s a d i s p e r s i o n - f r e e p r o b a b i l i t y measure on P ^ . A l s o , s i n c e each i s an u l t r a v a l u a t i o n , the 0,1 values assigned by (i t o the elements of P w CM can be i n t e r p r e t e d as the t r u t h values f a l s e and t r u e . So the f a c t t h a t a system's s t a t e determines the t r u t h values of the p r o p o s i t i o n s which make a s s e r t i o n s about the real-number values of the system's magnitude i s f o r m a l i z e d v i a the n o t i o n o f state-induced u l t r a v a l u a t i o n s on the P . s t r u c t u r e of the phase space a s s o c i a t e d w i t h the system. And t h i s state-induced procedure of a s s i g n i n g t r u t h values t o the elements of a p r o p o s i t i o n a l s t r u c t u r e P works e x a c t l y l i k e the procedure by which 54 t r u t h values are assigned to the elements of an L s t r u c t u r e determined by c l a s s i c a l p r o p o s i t i o n a l l o g i c . The s t r a i g h t f o r w a r d analogy between the state-induced u l t r a v a l u a t i o n s on a P and the u l t r a v a l u a t i o n s on an L suggests, f o r example, t h a t we may p o s t u l a t e a p h y s i c a l system w i t h an as s o c i a t e d phase space un d e r l y i n g the s t r u c t u r e diagrammed i n Chapter 11(B) so that each u l t r a v a l u a t i o n on I.^ i s induced by a s t a t e of the p o s t u l a t e d system. Consider a t e t r a h e d r a l d i e w i t h the numbers 1, 2, 3, 4 marked on each s i d e , r e s p e c t i v e l y , and with' the convention t h a t we read the bottom face of the d i e as the outcome of a throw and thus as the s t a t e of the d i e . The phase space a s s o c i a t e d with- the d i e c o n s i s t s of four p o i n t s Si = {w ,w ,w ,w }, each r e p r e s e n t i n g one of the f o u r d i s c r e t e s t a t e s of U 1 2. O *+ the d i e . In order that be the p r o p o s i t i o n a l s t r u c t u r e of t h i s , we may i n t e r p r e t the element /R/ € as the p r o p o s i t i o n : "A number l e s s 7 than three appears (on the bottom face of the d i e ) . " This p r o p o s i t i o n i s ass o c i a t e d w i t h the idempotent f u n c t i o n f : °J -> {0,1} defined as R U f o l l o w s : f o r any w. € S2. , f_(.w.) = 1 i f w. € {w, ,w0} and 1 0 K 1 1 1 /. f (w.) = 0 i f w. € {w, ,w =•' {w ,w }. And we may i n t e r p r e t the element R i i X 2 3 • H /S/ € as the p r o p o s i t i o n : "An odd number appears." This p r o p o s i t i o n i s a s s o c i a t e d w i t h the idempotent f u n c t i o n f defined as f o l l o w s : f o r any w. ( S , f e(w.) = 1 i f w. € {w ,w } and f (w.) = 0 i f W i ^ ^ W 2 9 V % ^ " Each, of the four u l t r a v a l u a t i o n s on i s state-induced 55 because i t i s the s t a t e of the die which s p e c i f i e s an atom i n which-: i n t u r n s p e c i f i e s an u l t r a f i l t e r and dual u l t r a i d e a l d e f i n i n g an u l t r a v a l u a t i o n on . Thus each s t a t e of the po s t u l a t e d system i s the c l a s s i c a l - m e c h a n i c a l analogue of the i n i t i a l assignment of 0, 1 values t o R and S which s p e c i f i e s an atom i n , as described i n Chapter 11(D). F i n a l l y , by the s e m i - s i m p l i c i t y of the Boolean s t r u c t u r e P , the c o l l e c t i o n of state-induced u l t r a v a l u a t i o n s on a i s complete. Thus the complete c o l l e c t i o n of state-induced u l t r a v a l u a t i o n s on a P can be regarded as a state-induced, b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r P . This state-induced semantics f o r P^ s h a l l be regarded as the precedent f o r a proposed state-induced semantics f o r the quantum p r o p o s i t i o n a l s t r u c t u r e s , as developed i n Chapter VI. Notes 1 I use the term " p r o p o s i t i o n " i n a p h i l o s o p h i c a l l y u n s o p h i s t i c a t e d way; "sentence" or "statement" could serve as w e l l . 2 2 2 As suggested by R. E. Robinson, the u n i t s kg m /sec , which help make sense o f the real-number v a l u e s , may be considered t o be par t of the magnitude. 3 The m e a s u r a b i l i t y c o n d i t i o n on the f u n c t i o n s r e p r e s e n t i n g c l a s s i c a l magnitudes r e q u i r e s t h a t , f o r any measurable ( i . e . , B o rel) subset R c R, the set W of a l l p o i n t s w € 2 such that f ^ w ^ ^ ^  ^ s x t s e l f a B o r e l subset o f 2. (This s et W i s the in v e r s e image o f R under f .) The m e a s u r a b i l i t y r e s t r i c t i o n on the subsets R £ R and W c 2 r u l e s out sets such as the set of i r r a t i o n a l numbers between 0 and 1, which i s a non-denumerable i n f i n i t y o f d i s j o i n t p o i n t s so t h a t the measure of t h i s set cannot be expressed as a countable union or sum of the measures of each o f the set's elements. A s i n g l e t o n , one-point set i s a B o r e l set of measure 0. B i r k h o f f and von Neumann a c t u a l l y s p e c i f y a more r e s t r i c t e d c l a s s o f measurable subsets of 2 than the c l a s s o f B o r e l subsets, see (Jauch, 1968, p. 79). 56 J Bub describes t h i s connection between c l a s s i c a l s t a t e s , u l t r a f i l t e r s , and b i v a l e n t homomorphisms, see (Bub, 1974, pp. 97-106). However, Bub defines a b i v a l e n t homomorphism by the S i k o r s k i d e f i n i t i o n , as discussed i n Chapter I l C C ) . The domain of a c l a s s i c a l p r o b a b i l i t y measure i s u s u a l l y s p e c i f i e d t o be a Boolean r i n g , f i e l d , or algebra of s e t s , i n p a r t i c u l a r , the Boolean algebra of B o r e l subsets of c l a s s i c a l phase space. However, M. S t r a u s s , I . Segal, and others argue t h a t the (isomorphic) Boolean algebra of idempotent random v a r i a b l e s ( i . e . , idempotent, r e a l - v a l u e d , measurable f u n c t i o n s ) i s p r e f e r a b l e as the domain of the measures of p r o b a b i l i t y theory ( S t r a u s s , 1973, p.,. 268; Segal, 1954, p. 721). S i m i l a r l y , Gleason proposes th a t we may regard h i s quantum measures as being defined on the set of idempotent operators on a H i l b e r t space r a t h e r than the set of subspaces of H i l b e r t space (Gleason, 1957, p. 885). 7 I t may seem i n i t i a l l y more p l a u s i b l e t o i n t e r p r e t the p r o p o s i t i o n a l v a r i a b l e s . R, S as p r o p o s i t i o n s a s s o c i a t e d w i t h idempotent f u n c t i o n s on S2Q . Thus any p r o p o s i t i o n s P which makes a s s e r t i o n s about what appears a f t e r a throw of the die i s a molecular combination of R, S. Let L 2 l a b e l the c l o s e d , denumerable set of a l l molecular combinations of R, S. We have no equivalence r e l a t i o n w i t h which t o p a r t i t i o n L 2 i n order t o get the equivalence c l a s s e s which are the elements of L 2 . S t r i c t l y speaking, i t i s the elements of L 2 which I want t o i n t e r p r e t as p r o p o s i t i o n s a s s o c i a t e d w i t h idempotent f u n c t i o n s on S20 . However, l e t every P € L 2 be d i r e c t l y a s s o c i a t e d w i t h an idempotent f u n c t i o n fp (where fp(w) i s the t r u t h value of P given w), and say t h a t , f o r any P, Q € L 2 , P ~ Q IFF f p ( w ) = fq(w) f o r every w € &Q , where i f fp(w) = fq(w) f o r every w € '&„ , then fp = fQ . Thus a l l the members of the equivalence c l a s s /?/ are a s s o c i a t e d w i t h a s i n g l e idempotent f u n c t i o n f , as we want. And i n other words, P ~ Q IFF P, Q have the same t r u t h t a b l e , which i s the semantic counterpart of the proof-t h e o r e t i c equivalence r e l a t i o n s t a t e d i n Chapter 11(B). 57 CHAPTER IV THE NON-BOOLEAN PROPOSITIONAL STRUCTURES DETERMINED BY QUANTUM MECHANICS Se c t i o n A. The Fundamental P o s t u l a t e s o f Quantum Mechanics What f o l l o w s i s an extremely s i m p l i f i e d e x p o s i t i o n o f some of the mathematical formalism o f quantum mechanics. I t i s p o s t u l a t e d t h a t a p h y s i c a l system i s a s s o c i a t e d w i t h a H i l b e r t space H whose d i m e n s i o n a l i t y r e f l e c t s the degrees o f freedom of the system. Each magnitude A of the system i s represented by a s e l f - a d j o i n t operator A on the system's H. The operator A has a s p e c t r a l r e p r e s e n t a t i o n ( f o r the case o f a d i s c r e t e spectrum): ( I ) A = / a.P. , where f o r each i € Index, P. = !>!/•. ><\lr. I I T i Y i and A|\|A.> = a.|\|f.>. • 1 x 1 T i The r e a l numbers ^ a £ ^ g j n ( j e x a r e c a l l e d the eigenvalues o f A and of A. They are the real-number values and the only real-number values e x h i b i t e d by the 1 magnitude A. A pure s t a t e \Jr of a quantum system i s represented by a vector |\|f> i n the system's H or by a de n s i t y operator P^ = |\|/><\|/| on H. The operator P .^ i s s e l f - a d j o i n t and idempotent, t h a t i s , P ,^ i s a p r o j e c t i o n operator which i s a l s o c a l l e d a p r o j e c t o r , more g e n e r a l l y designated P, P^, P^, e t c . Each p r o j e c t o r P on an H corresponds uniquely to a subspace H o f H, where a subspace i s a set of vectors which form a cl o s e d l i n e a r manifold (see Bub, 1974, pp. 10, 12). The p r o j e c t o r s > i 6 I n d e x a n d t h e v e c t o r s { k i > > i € I n d e x appearing i n A. the s p e c t r a l r e p r e s e n t a t i o n o f any operator A represent the (pure) eigenstates of A and of A. The set o f eigenstates o f any A are mutually orthogonal (as defined i n S e c t i o n C) and s a t i s f y V |\|r.> = H and £ P =1 . ( i i s the i 1 i ^ i i d e n t i t y operator which s a t i s f i e s l|^> = l|^> f o r every |\|/-> € H.) 58 The s t a t e of a quantum system determines the real-number values of the system's magnitudes v i a the f o l l o w i n g formalism. When a system i s i n an eigenstate |YJ>» f o r some j € Index, of the magnitude A , then the real-number value of A i s the eigenvalue a. a f f i l i a t e d w i t h t h a t eigenstate | > | r b y the equation A|YJ> = a,. J.1'Y_.> • But when a system i s i n an a r b i t r a r y pure s t a t e \|/ which i s not an eigenstate of A , then upon measurement the magnitude A may e x h i b i t any of i t s real-number eigenvalues; the quantum formalism does not s p e c i f y which eigenvalue A w i l l e x h i b i t . However, f o r any pure s t a t e \|/y the p r o b a b i l i t y that the real-number value of A i s the eigenvalue a.. , f o r some j € Index, i s determined by the quantum formalism: ( I I ) P t j A ( a j ) = l ^ j l ^ l 2 = ^ k j ' x t j U> = o H ^ . I ^ • This p r o b a b i l i t y i s a real-number i n the closed i n t e r v a l [0 ,1 ] of the real-number l i n e . The p r o b a b i l i t y equals 1 ( c e r t a i n t y ) IFF the system i s i n the eigenstate \ | / y a f f i l i a t e d w i t h the eigenvalue a.. , i . e . , p' (a.) = |<\|f. |\|r.>|2 = 1 . And t h i s p r o b a b i l i t y equals 0 ( i m p o s s i b i l i t y ) % , A 1 : 1 IFF the system i s i n any one of the other eigenstates of A . The average v a l u e , i . e . , the expectation v a l u e , o f A when the system i s i n an a r b i t r a r y pure s t a t e Y i s defined as the f o l l o w i n g weighted sum of eigenvalues of A : ( I I I ) Exp, (A ) = I a.p: A ( a . ) = £ ( a . ) o | f l ^ ^ j ^ = < Y | I ( a i ) | Y i > < ^ 1 l | i > i ='<Y|A|Y>. 59 And when the system i s i n an eigenstate of A, then the expectation value of A i s the eigenvalue a_. . C l e a r l y , the expression f o r the p r o b a b i l i t y ( I I ) i s equal t o the expectation value of a p r o j e c t o r according t o ( I I I ) , i . e . , f o r any pure A A s t a t e \Jr and f o r any magnitude A, ACa_.) = E x p ^ C P ^ ) , where P ^ i s the p r o j e c t o r r e p r e s e n t i n g the eig e n s t a t e a f f i l i a t e d w i t h the eigenvalue a_. . In f a c t , i n s t e a d of moving from the p r o b a b i l i t y expression ( I I ) t o the e x p e c t a t i o n value expression ( I I I ) , the former expression ( I I ) can be derived from ( I I I ) , as i s done, f o r example, by Messiah (1966, pp. 176-179). In other words, ( I ) together w i t h e i t h e r ( I I ) or ( I I I ) are regarded as the fou n d a t i o n a l p o s t u l a t e s o f quantum mechanics. For example, von Neumann considers ( I I ) t o be the more general p r o b a b i l i t y expression but he regards ( I I I ) to be p r e f e r a b l e as a fundamental p o s t u l a t e (von Neumann, 1932, pp. pp. 200-206). Section B. I n c o m p a t i b i l i t y In both, c l a s s i c a l and quantum mechanics, a s u f f i c i e n t c o n d i t i o n f o r the simultaneous m e a s u r a b i l i t y of any set of magnitudes ^ i ^ i c i n d e x i s t h a t each magnitude i s equal t o a ( B o r e l ) f u n c t i o n of some common magnitude, say B; t h a t i s , f o r each- i € Index, A^ = g^(B) f o r some B o r e l f u n c t i o n g^ (Kochen-Specker, 1967, p. 64). Now f o r any magnitude B and any B o r e l f u n c t i o n g, the magnitude g(B) i s by d e f i n i t i o n t h a t magnitude which e x h i b i t s the value g(b) when B e x h i b i t s the value b. So when the real-number value of the common magnitude B i s b, then the real-number value of each A. = g.(B) i s g.(b). Hence a s i n g l e measurement i i l of B s u f f i c e s to determine the real-number values of a l l the magnitudes 60 ^ i ^ i € l n d e x " For example, as mentioned i n Chapter I I I ( A ) , every c l a s s i c a l magnitude i s a (B o r e l ) f u n c t i o n of the p o s i t i o n and/or momentum magnitudes, and so a l l c l a s s i c a l magnitudes are simultaneously measurable. But i t i s not the case t h a t every quantum magnitude i s a f u n c t i o n of the p o s i t i o n and/or momentum magnitudes. Moreover, the quantum p o s i t i o n and momentum magnitudes are themselves not simultaneously measurable. And i n g e n e r a l , the set of magnitudes d e s c r i b i n g a quantum system i n c l u d e s magnitudes which are not simultaneously measurable. With respect t o the ( s e l f - a d j o i n t ) operator r e p r e s e n t a t i o n of the quantum magnitudes, a necessary and s u f f i c i e n t c o n d i t i o n f o r the simultaneous m e a s u r a b i l i t y of any magnitudes i s the commutativity of t h e i r r e p r e s e n t a t i v e A* A\ A A* operators. Any operators A, B commute IFF A, B have a l l t h e i r e i g e n s t a t e s i n common. Moreover, any set ^ A i ^ i g i n < i e x °^ ° P e r a " t o r s ^ s mutually commutative IFF there i s an operator B and B o r e l f u n c t i o n s {g.}., T , such t h a t A. = g.(B) = g.(B), f o r every i € Index (von Neumann, 6 x J x € I n d e x x x x J 1932, p. 173). Now f o r any magnitude B and f o r any B o r e l f u n c t i o n g, i f As -^^ Vs- As B has the operator B, then g(B) has the operator g(B) = g(B). (von Neumann, 1932, p. 204; Fano, 1971, p. 394). Thus i t f o l l o w s that any quantum magnitudes are simultaneously measurable IF t h e i r r e p r e s e n t a t i v e operators are mutually commutative; the converse i s a l s o shown by von Neumann (1932, pp. 223-228). Commuting operators and simultaneously measurable magnitudes are s a i d t o be compatible; such operators or magnitudes have a l l t h e i r e igenstates i n common. Operators which do not commute and magnitudes which are not simultaneously measurable .are s a i d t o be incompatible; such operators or magnitudes may nevertheless have one or s e v e r a l eigenstates i n common so 61 t h a t one or s e v e r a l of t h e i r eigenvalues may be simultaneously determined by measurement. When, we t a l k of a p r o p o s i t i o n a l s t r u c t u r e determined by quantum mechanics, the p r o p o s i t i o n s we consider are p r o p o s i t i o n s which make a s s e r t i o n s about the real-number eigenvalues of quantum magnitudes. P r o p o s i t i o n s which make a s s e r t i o n s about the eigenvalues of compatible magnitudes are s a i d t o be compatible. P r o p o s i t i o n s which make a s s e r t i o n s about the eigenvalues of incompatible magnitudes are s a i d ' t o be incompatible w i t h the f o l l o w i n g exception. I f the eigenvalues happen to be associated. w i t h eigenstates which are shared i n common by the incompatible magnitudes, then p r o p o s i t i o n s which make a s s e r t i o n s about such eigenvalues of 2 incompatible magnitudes are s a i d t o be compatible. The attempt to a s s i g n t r u t h - v a l u e s t o incompatible quantum p r o p o s i t i o n s i s a problematic e n t e r p r i s e , as w i l l be shown i n Chapter V ( A ) . S e c t i o n C. The P r o p o s i t i o n a l S t r u c t u r e Determined by Quantum Mechanics As i n the c l a s s i c a l case described i n Chapter I I I , the s e l f - a d j o i n t operators r e p r e s e n t i n g quantum magnitudes have the bi n a r y r i n g operations A A + and • defined among them as f o l l o w s : f o r any A, B on H and f o r every |\|/> 6 H, (A+B)|\|/> = &|\|r> + B|\|r>, and (A«B)|\|/> = A(B|\|/>). The + operation so defined i s a s s o c i a t i v e and commutative, as u s u a l . And the • operation so defined i s a s s o c i a t i v e and d i s t r i b u t i v e w i t h respect to +, as u s u a l . But • i s not commutative, i . e . , A(B|\|/>) need not equal B(A|\|f>) f o r every A,B,|y>. In p a r t i c u l a r , i f A(B|\|A>) = B(A|\|r>) f o r every |\|r> € H, then A, B are s a i d to commute or t o be compatible. This suggests t h a t a cl o s e d set. of s e l f - a d j o i n t operators on. a H i l b e r t space has 62 the s t r u c t u r e of a non-commutative r i n g - w i t h - u n i t whose 0-element i s the constant operator 0 s a t i s f y i n g 0,|t> = 0, f o r every \ty> € tf, and whose 1-element i s the constant operator I s a t i s f y i n g 'l|^> = 1, f o r every |^ > € H. However, a set of s e l f - a d j o i n t operators i s not c l o s e d w i t h respect t o • unless • i s r e s t r i c t e d t o commuting, i . e . , compatible, operators. For although the sum of any two s e l f - a d j o i n t operators i s i t s e l f a s e l f - a d j o i n t operator, the product of two s e l f - a d j o i n t operators i s not i t s e l f a s e l f - a d j o i n t operator unless the two commute (von Neumann, 1932, p. 98). So r a t h e r than a non-commutative r i n g - w i t h - u n i t , a set of s e l f - a d j o i n t operators r e p r e s e n t i n g quantum magnitudes which i s cl o s e d with respect t o + and • form a s t r u c t u r e which may be c a l l e d a p a r t i a l - d o t - r i n g - w i t h - u n i t <E = {A,B, . . .}, + , • ,0,1>, where cb c E * E , + i s defined from E><E to E, and * i s defined from only o t o E. Taking t h i s n o t i o n of r e s t r i c t i n g the b i n a r y • operation to a p a r t i a l - o p e r a t i o n d efined from only i t o E one step f u r t h e r , we may define the s t r u c t u r e of the s e l f - a d j o i n t operators t o be a p a r t i a l - r i n g - w i t h - u n i t which has both the + and the • operations defined from only A> t o E, where again, o c E.xE. As mentioned i n Chapter 1(D), Kochen-Specker c a l l such a s t r u c t u r e a p a r t i a l algebra. And they d e f i n e the s t r u c t u r e of a quantum system's magnitudes, which are represented by and presumably r e f l e c t the s t r u c t u r e of s e l f - a d j o i n t operators on the system's H i l b e r t space, as a p a r t i a l - a l g e b r a ; i . e . , as a p a r t i a l - r i n g - w i t h - u n i t i n my terminology. But r e g a r d l e s s of the exact s t r u c t u r i n g o f the s e l f - a d j o i n t o perators, i t i s c l e a r t h a t the s t r u c t u r e of the operators r e p r e s e n t i n g the magnitudes of quantum mechanics i s d i f f e r e n t from the s t r u c t u r e o f the r e a l - v a l u e d f u n c t i o n s r e p r e s e n t i n g the magnitudes of c l a s s i c a l mechanics. 63 Thus i t i s reasonable t o expect t h a t the s t r u c t u r e of the quantum p r o p o s i t i o n s which make assertions: about the real-number eigenvalues of the quantum magnitudes i s d i f f e r e n t from the Boolean P s t r u c t u r e o f c l a s s i c a l p r o p o s i t i o n s . Nevertheless, the procedure by which a quantum p r o p o s i t i o n a l s t r u c t u r e i s a b s t r a c t e d from the mathematical formalism of quantum mechanics i s e x a c t l y analogous t o the procedure by which a P s t r u c t u r e i s abst r a c t e d from the c l a s s i c a l formalism, as described i n Chapter I I I ( B ) . For quantum p r o p o s i t i o n s have h i s t o r i c a l l y been a s s o c i a t e d e i t h e r w i t h the p r o j e c t o r s ( i . e . , idempotent, s e l f - a d j o i n t operators) on an H or w i t h the uniquely corresponding subspaces of H. And the l o g i c a l operations "and," " o r , " "not," e i t h e r have been i n d i r e c t l y defined i n terms of the p r o j e c t o r + and • operations or have been d i r e c t l y a s s o c i a t e d w i t h the subspace i n t e r s e c t A, span v, and orthocomplementation _ L o p e r a t i o n s , as w i l l be described s h o r t l y . These a s s o c i a t i o n s determine the s t r u c t u r e of a set of quantum p r o p o s i t i o n s , or i n von Neumann's terms, these a s s o c i a t i o n s determine "a s o r t of l o g i c a l c a l c u l u s " or a " p r o p o s i t i o n a l c a l c u l u s " f o r quantum mechanics (von Neumann, 1932, p. 253). In h i s 1932 book, von Neumann discusses c l a s s i c a l and quantum p r o p o s i t i o n s under the c a t e g o r i c a l l a b e l : p r o p e r t i e s of the s t a t e of the system. That i s , von Neumann's p r o p e r t i e s are i n f a c t p r o p o s i t i o n s which make a s s e r t i o n s about the real-number (eigen)values o f a system's magnitudes (von Neumann, 1932, p. 249). For example: The s P i n x o r" a n e l e c t r o n i s +3gn. Von Neumann argues t h a t each s u e h x p r o p o s i t i o n canf.be a s s o c i a t e d w i t h a magnitude which i s defined such, t h a t i t s value i s 1 i f the p r o p o s i t i o n i s v e r i f i e d and 0 i f the p r o p o s i t i o n i s not v e r i f i e d . In other words, each p r o p o s i t i o n which makes a s s e r t i o n s about the real-number eigenvalues of 64 a quantum system's magnitudes can i t s e l f be regarded as or as s o c i a t e d w i t h an idempotent magnitude o f the system. Since an idempotent magnitude i s represented by a p r o j e c t o r on the system's H i l b e r t space and each p r o j e c t o r i n t u r n corresponds u n i q u e l y t o a subspace of that H i l b e r t space, namely, t o the subspace onto which-the p r o j e c t o r p r o j e c t s every ve c t o r i n H i l b e r t space, von Neumann concludes t h a t quantum p r o p o s i t i o n s can be ass o c i a t e d e i t h e r w i t h p r o j e c t o r s on a H i l b e r t space or e q u a l l y w e l l w i t h subspaces of a H i l b e r t space. For example, consider a p r o p o s i t i o n which a s s e r t s that the value of the magnitude A i s i n some B o r e l subset R of the real-number l i n e . Such p r o p o s i t i o n s are regarded by most authors as the paradigm quantum (or c l a s s i c a l ) , p r o p o s i t i o n s . As described above, the only values e x h i b i t e d by A are i t s eigenvalues, and each eigenvalue i s uniquely a s s o c i a t e d w i t h a p r o j e c t o r P. = |\^ .><^ .|. So depending upon how many eigenvalues of A ^ i are i n the B o r e l subset R, the above paradigm p r o p o s i t i o n s p e c i f i e s e i t h e r the unique p r o j e c t o r P. and i t s corresponding subspace H. when ™i n Y i only one a. € R, or the unique p r o j e c t o r £ p i , a n d 1 T S corresponding 1 n i = l V i subspace V H, when s e v e r a l a ,a ( R, ._, t . 1' n i = l T i A l l other authors who di s c u s s a quantum p r o p o s i t i o n a l s t r u c t u r e or a quantum l o g i c a l calculus', a l s o somehow or other a s s o c i a t e quantum p r o p o s i t i o n s w i t h e i t h e r the p r o j e c t o r s on a H i l b e r t space or the subspaces of a H i l b e r t space. So the s t r u c t u r e of the p r o j e c t o r s on a H i l b e r t space, or i s o m o r p h i c a l l y , the s t r u c t u r e of the subspaces of th a t H i l b e r t space, i s regarded as the p r o p o s i t i o n a l s t r u c t u r e determined by quantum mechanics, l a b e l e d P Q M = <E - { P . P ^ P ^ . . . . },A,5,A, v,+,0,l>. The elements of P Q M may be thought o f e i t h e r as p r o j e c t o r s or as subspaces of a H i l b e r t space; 65 the elements of P represent or are as s o c i a t e d w i t h p r o p o s i t i o n s PSP^JP 2 The P s t r u c t u r e s have been f o r m a l i z e d i n two d i f f e r e n t ways. But before d e s c r i b i n g these two ways i n the next s e c t i o n , the fea t u r e s o f P_w which are common t o both f o r m a l i z a t i o n s are f i r s t d e s c r i b e d , QM ' as f o l l o w s . A P^ ,« i s an atomic s t r u c t u r e whose atoms, w r i t t e n P, or QM ' Y sometimes P , are the one-dimensional p r o j e c t o r s on H, e.g., P. = | Y > < Y I » c l y or the corresponding one-dimensional subspaces of H, e.g., the subspace which i s the range of P^ . The d i s t i n g u i s h e d 0-element of a P^ M i s the n u l l p r o j e c t o r 0 or the corresponding zero-subspace o f H; the d i s t i n g u i s h e d 1-element i s the i d e n t i t y p r o j e c t o r I or the corresponding e n t i r e H. As w i t h the L and the P ^  s t r u c t u r e s described i n Chapters I I and I I I , the 0-element of a P i s ass o c i a t e d w i t h impossible or c o n t r a d i c t o r y quantum p r o p o s i t i o n s , and the 1-element i s a s s o c i a t e d w i t h c e r t a i n or t a u t o l o g i c a l quantum p r o p o s i t i o n s . The c o m p a t i b i l i t y r e l a t i o n A of P . i s r e f l e x i v e , symmetric, and n o n - t r a n s i t i v e , and i s defined i n terms of the A , v, operations as f o l l o w s . For any P,,P. € P_„ P„ A P IFF there e x i s t three mutually J 1 2 QM 1 2 J d i s j o i n t ( i . e . , orthogonal) elements P ^ J P 2 2 ' P 3 S U G ^ ^at P^ = P ^ V P^ and P 2 = P 2 2 V Pg .. And assuming t h a t P^ cb P 2 , i t can be shown th a t P l l = P l A ?2 9 P22 = P 2 A P l " ' P 3 = P l A P 2 ( J A U C H > 1 9 6 8 > PP* 2 8 » 9 7 J Kochen-Specker, 1967, p. 65). Any P^> p 2 ^ ^QM a r e d i s j o i n t or orthogonal w r i t t e n P„ _L P„ IFF P„ < P X ( P i r o n , 1976, p. 29). I t f o l l o w s t h a t , 1 2 1 2 f o r any P 1»? 2 e P Q M ' i f P i ' P 2 a r e D I S J O L N T t h e n P i ^  p 2 ' a n d P 1cb.P 2 IFF P 1 i -P^ • The b i n a r y r e l a t i o n 5 of P^M , defined i n terms o f A or V as u s u a l ( i . e . , P 1 < P 2 IFF PP 1 A P 2 = ? ± . and P ± 5 P 2 IFF ?± V P 2 = P 2 ) , 66 i s a p a r t i a l - o r d e r i n g C i .e., i t i s r e f l e x i v e , anti-symmetric, and t r a n s i t i v e ) . Moreover, the c o m p a t i b i l i t y of any p 1 s P 2 ^ ^QM i s a n e c e s s a r y c o n d i t i o n f o r t h e i r being r e l a t e d to the p a r t i a l - o r d e r i n g 5, that i s , i f 5 P 2 then P 2 , f o r any P ^ 6 P Q M . 3 And f i n a l l y , the operations A, V,-1- of P ^ M are defined and discussed i n the next s e c t i o n . Section D. The P a r t i a l - B o o l e a n Algebra and the Orthomodular L a t t i c e Quantum  P r o p o s i t i o n a l Structures The P „ „ s t r u c t u r e has been fo r m a l i z e d i n two ways: as a QM J t r a n s i t i v e , atomic, p a r t i a l - B o o l e a n algebra P Q M A a n d as a complete, atomic, orthomodular l a t t i c e P „ „ T . These s t r u c t u r e s are defined i n Chapter 1(D) QML and (E). I r e t a i n the l a b e l P_„ t o r e f e r t o a P„__ or a P QM QMA QML i n d i s c r i m i n a t e l y . The b a s i c d i f f e r e n c e between a a n d a 1 S t h a t the former has the b i n a r y operations A, v d e f i n e d among only compatible elements w h i l e the l a t t e r has A, V defined among a l l elements, compatible and incompatible. The two f o r m a l i z a t i o n s do not d i f f e r w i t h respect t o any of the other e n t r i e s i n the ordered octuple P . That the quantum p r o p o s i t i o n a l s t r u c t u r e s have been f o r m a l i z e d i n these two ways i s at l e a s t p a r t l y due t o d i f f e r e n c e s between the p r o j e c t o r s and the subspaces of H. For d e s p i t e the one-to-one correspondence between the p r o j e c t o r s and the subspaces, the a s s o c i a t i o n of quantum p r o p o s i t i o n s w i t h p r o j e c t o r s n a t u r a l l y y i e l d s a P Q M A while the a s s o c i a t i o n of quantum p r o p o s i t i o n s w i t h subspaces suggests a P Q ^ L J a s w i l l be shown i n t h i s s e c t i o n . In h i s 1932 book, von Neumann proposes a l o g i c a l c a l c u l u s of 67 quantum p r o p o s i t i o n s which has "and" and "or" r e s t r i c t e d t o compatible p r o p o s i t i o n s . F i r s t von Neumann define s "not." For any quantum p r o p o s i t i o n p a s s o c i a t e d w i t h the'projector. Pwhose corresponding subspace i s H, the A S\ A J_ p r o p o s i t i o n "not p" i s ass o c i a t e d w i t h the p r o j e c t o r I - P = P whose corresponding subspace i s Ex. Next, f o r any compatible p r o p o s i t i o n s ^1 ' ^ 2 ' P r o P o s i t : i - o n "Pj_ a n d P2" ^ s a s s o c i a ' t e d w i t h the p r o j e c t o r A A P^ • P^ whose corresponding subspace i s A , where A i s i n t e r p r e t e d among subspaces as the s e t - t h e o r e t i c i n t e r s e c t operation. C l a s s i c a l l y fp or p^" i s equivalent to "not ((not p^) and (not p 2 ) ) " ; analogously, von Neumann a s s o c i a t e s "p^ or p 2 ," f o r any compatible p^ , p 2 , with A A A A A A A A the p r o j e c t o r I - (C.I-P ).• (.I-Pj)) = P^ + P 2 - (P^ * P 2) whose corresponding subspace i s the cl o s e d l i n e a r sum of H , H , i . e . , V H 2 , where V i s i n t e r p r e t e d as the subspace span op e r a t i o n . Thus von Neumann's 1932 l o g i c a l c a l c u l u s has the "and," " o r , " "not" operations among p r o p o s i t i o n s defined i n terms of the +, • operations among p r o j e c t o r s i n the usu a l way tha t the Boolean operations A, V , 1 , are defined i n terms of the r i n g operations +, •. But the b i n a r y "and," " o r " operations are defined among only compatible p r o p o s i t i o n s . A s i m i l a r c a l c u l u s of quantum p r o p o s i t i o n s i s developed and discussed by Strauss under the a p p e l l a t i o n "complementary l o g i c " ( S t r a u s s , 1936, p. 196). and l a t e r by Kochen-Specker under the appellation " p a r t i a l - B o o l e a n a l g e b r a . " A l a t t i c e s t r u c t u r e of c a l c u l u s o f quantum p r o p o s i t i o n s was f i r s t proposed by B i r k h o f f and von Neumann i n t h e i r c e l e b r a t e d 1936 paper. There, i n a d i s c u s s i o n of t h e i r i n i t i a l a s s o c i a t i o n o f experimental p r o p o s i t i o n s w i t h the subsets of a phase space, B i r k h o f f and von Neumann are e s p e c i a l l y concerned t o preserve the r e l a t i o n of l o g i c a l i m p l i c a t i o n among the 68 p r o p o s i t i o n s . L o g i c a l i m p l i c a t i o n i s r e f l e x i v e , anti-symmetric, and t r a n s i t i v e , and so can be regarded as a p a r t i a l - o r d e r i n g . So B i r k h o f f and von Neumann p o s t u l a t e t h a t a p r o p o s i t i o n a l c a l c u l u s , determined by e i t h e r c l a s s i c a l mechanics or quantum mechanics, i s a p a r t i a l l y ordered s e t . They then assume th a t a p r o p o s i t i o n a l c a l c u l u s has a d i s t i n g u i s h e d 0-element, i n t e r p r e t e d as the " i d e n t i c a l l y f a l s e " or "absurd" p r o p o s i t i o n , and a d i s t i n g u i s h e d 1-element, i n t e r p r e t e d as the " i d e n t i c a l l y t r u e " or " s e l f - e v i d e n t " p r o p o s i t i o n . Next B i r k h o f f and von Neumann c l a i m t h a t : "In any c a l c u l u s of p r o p o s i t i o n s , i t i s n a t u r a l t o imagine t h a t there i s a weakest p r o p o s i t i o n i m p l y i n g , and a strongest p r o p o s i t i o n i m p l i e d by, a given p a i r o f p r o p o s i t i o n s " ( B i r k h o f f and von Neumann, 1936, pp. 828-829). In other words, w i t h respect t o the p a r t i a l - o r d e r i n g o f l o g i c a l i m p l i c a t i o n , B i r k h o f f and von Neumann assume th a t any given p a i r of p r o p o s i t i o n s p^ , P 2 , i n a p r o p o s i t i o n a l s t r u c t u r e has a g. l . b . (the meet p^ A p 2 ) and a l.u.b. (the j o i n p^ v P 2)> which they i n t e r p r e t as l o g i c a l c o njunction and d i s j u n c t i o n , r e s p e c t i v e l y . Hence, B i r k h o f f and von Neumann po s t u l a t e t h a t a p r o p o s i t i o n a l s t r u c t u r e i s a l a t t i c e which has A, V defined f o r every p a i r o f p r o p o s i t i o n s . But B i r k h o f f and von Neumann immediately mention the problematic character o f the meets and j o i n s of incompatible p r o p o s i t i o n s . They say th a t the meet or the j o i n o f incompatible experimental p r o p o s i t i o n s cannot i t s e l f be defined as an experimental p r o p o s i t i o n but r a t h e r must be expressed as a c l a s s o f l o g i c a l l y equivalent experimental p r o p o s i t i o n s which they c a l l a p h y s i c a l q u a l i t y . Nevertheless, B i r k h o f f and von Neumann go on t o a s s o c i a t e quantum p r o p o s i t i o n s w i t h the subspaces o f a H i l b e r t space, and they a s s o c i a t e "not," "and," " o r , " among compatible and incompatible 69 p r o p o s i t i o n s qua subspaces w i t h the subspace ~ht A , V , as defined by von Neumann i n 1932. x I t i s noteworthy that the orthocomplement H of any subspace H o f a H i l b e r t space i s i t s e l f a subspace, and l i k e w i s e the s e t - t h e o r e t i c i n t e r s e c t H A H^ and the closed l i n e a r sum H. V H„ of any p a i r of 1 2 1 2 J * subspaces H^ , H^ of a H i l b e r t space are themselves subspaces. So i t i s c l e a r t h a t the meets and j o i n s of incompatible p r o p o s i t i o n s qua subspaces are at l e a s t sure t o e x i s t , whether as experimental p r o p o s i t i o n s or as " p h y s i c a l q u a l i t i e s . " B i r k h o f f and von Neumann conclude that the orthocomplemented, modular, n o n - d i s t r i b u t i v e l a t t i c e o f subspaces of a H i l b e r t space may be regarded as the l o g i c a l s t r u c t u r e or p r o p o s i t i o n a l c a l c u l u s o f quantum mechanics. L a t e r , Jauch shows that the subspaces of an i n f i n i t e dimensional H i l b e r t space are not modular, and so Jauch weakens the modularity c o n d i t i o n on the quantum l a t t i c e of subspaces to weak modularity (see Chapter 1 ( E ) ) . Consequently, authors who favour the l a t t i c e f o r m a l i z a t i o n of quantum p r o p o s i t i o n s i n i t i a t e d by B i r k h o f f and von Neumann consider the p r o p o s i t i o n a l s t r u c t u r e or c a l c u l u s of quantum mechanics to be a complete, atomic, orthomodular ( i . e . , orthocomplemented and weakly modular) l a t t i c e . However, when quantum p r o p o s i t i o n s are as s o c i a t e d w i t h the p r o j e c t o r s on a H i l b e r t space r a t h e r than the subspaces, then the existence of the meets and j o i n s of incompatible p r o p o s i t i o n s qua p r o j e c t o r s i s more problematic. As mentioned i n Section CB), the operators and p r o j e c t o r s on a H i l b e r t space have + and • i n t e r p r e t e d as a d d i t i o n and m u l t i p l i c a t i o n defined among them. But a theorem s t a t e s t h a t the product of any two p r o j e c t o r s i s i t s e l f a p r o j e c t o r IFF the two are compatible; the sum of any 70 two p r o j e c t o r s i s i t s e l f a p r o j e c t o r IFF the two are orthogonal (von Neumann, A A A 1932, p. 81). In a d d i t i o n , any P i s a p r o j e c t o r IFF I-P i s a p r o j e c t o r (von Neumann, 1932, p. 79). So a set of p r o j e c t o r s i s closed w i t h respect t o + and * only i f the + operation i s r e s t r i c t e d to orthogonal p r o j e c t o r s and the • operation i s r e s t r i c t e d t o compatible p r o j e c t o r s r e s u l t i n g i n a s o r t of p a r t i a l - B o o l e a n r i n g - w i t h - u n i t <E = { P 1 » P 2 » • • • }s-L»is + > • »0,1>, where JL c i c E x E , + i s defined from JL t o E , and • i s defined from i t o E. I w r i t e _L c i because, f o r any P_ , P„ , i f P. X P„ then P„ <t> P„ , but not the converse. 1 2 1 2 1 2 Now although P + P i s a p r o j e c t o r IFF P^_L ? 2 > i f i s easy As As As As AS AS to show th a t the sum l e s s the product: P^ + p 2 ~ P i P 2 ' °^ a n Y P i ' P 2 ' i s a p r o j e c t o r IFF P i P . For .p + ? 2 - P ^ g = I - I + P 1 + P 2 - $ ± ? 2 A As As As As As A As As As As As As A s. , ^ ^ . -= i - ( ( I - P ) - ( P 2 - P 1 P 2 ) ) = 1 - C ( I - P 1 ) I - ( I - P 1 ) P 2 ) ) = i - ( ( I - P ^ - ( I - P 2 ) ) . And by the theorem and a d d i t i o n a l r e s u l t s t a t e d i n the previous paragraph, A A A A As As ^ f o r any P i > p 2 » (I-P^). and (I-Pg) are each p r o j e c t o r s ; and I - ( ( I - P ) • ( i-P 2>) i s a p r o j e c t o r IFF ( ( I - P ^ ) • -(I-P^)) i s a p r o j e c t o r ; the l a t t e r i s a p r o j e c t o r IFF tI-$ ).<b ( t - P j ) , which i s the case IFF P_ji P 2 . Q . E . D . So when the A, V,-1" operations are defined among p r o j e c t o r s i n terms of + and • as u s u a l , then a set of p r o j e c t o r s i s closed w i t h respect t o A, V , x only i f A and V are r e s t r i c t e d t o compatible p r o j e c t o r s r e s u l t i n g i n a p a r t i a l - B o o l e a n algebra of quantum p r o p o s i t i o n s qua p r o j e c t o r s . The subspace r e p r e s e n t a t i o n of quantum p r o p o s i t i o n s e a s i l y lends i t s e l f t o a p a r t i a l - B o o l e a n algebra s t r u c t u r i n g , as f i r s t suggested by von Neumann i n 1932. Merely r e s t r i c t the above defined A, V operations among subspaces t o compatible subspaces (e.g., see Kochen-Specker, 1967, 71 p. 65). On the other hand, the p r o j e c t o r r e p r e s e n t a t i o n of quantum p r o p o s i t i o n s may be s t r u c t u r e d as an orthomodular l a t t i c e , but the A , V operations can be defined i n terms of p r o j e c t o r a d d i t i o n + and m u l t i p l i c a t i o n • i n the usu a l way among only compatibles. Among incompatible p r o p o s i t i o n s qua p r o j e c t o r s , the A, V operations are defined by Jauch as f o l l o w s : P A P = lim(P„ • P „ ) n and P„ V P^ = (p^* A P^) J' 1 2 1 2 1 2 1 2 n = I - l i m ( ( I - P ) • (I-P )) CJauch, 1968, pp. 38, 219). These d e f i n i t i o n s of A , v reduce t o the usu a l d e f i n i t i o n s of A, v i n terms o f +, • when P^ A P 2 . So an orthomodular l a t t i c e PQ M I j °f quantum p r o p o s i t i o n s qua p r o j e c t o r s i s a l s o defined. Thus r e g a r d l e s s of whether quantum p r o p o s i t i o n s are ass o c i a t e d w i t h the p r o j e c t o r s or the subspaces of a H i l b e r t space, both a l t e r n a t i v e s have been s t r u c t u r e d as a and both have been s t r u c t u r e d as a P - „ T . QMA QML I have described how the a l t e r n a t i v e s have been f o r m a l i z e d as a P-„, and QMA as a P N M T i n order t o h i g h l i g h t the problematic character of the meets and QML j o i n s of incompatibles defined i n P Q ^ • x n summary, when quantum p r o p o s i t i o n s are a s s o c i a t e d w i t h the subspaces of a H i l b e r t space, then the meets and j o i n s of incompatibles are at l e a s t sure to e x i s t and the p r o p o s i t i o n s qua subspaces can be s t r u c t u r e d as a PQ^L • However, B i r k h o f f and von Neumann, f o r example, do not regard the meets and j o i n s of incompatible p r o p o s i t i o n s as p r o p o s i t i o n s but r a t h e r as " p h y s i c a l q u a l i t i e s . " When quantum p r o p o s i t i o n s are as s o c i a t e d w i t h the p r o j e c t o r s on a H i l b e r t space and the A, V,"1" operations are defined i n terms o f p r o j e c t o r a d d i t i o n + and m u l t i p l i c a t i o n • as u s u a l , then the r e s u l t i n g s t r u c t u r e i s a r a t h e r than a P „ W T . In order t o define a P . W T of quantum p r o p o s i t i o n s qua QML QML — — p r o j e c t o r s , Jauch must introduce d e f i n i t i o n s o f A and V which i n v o l v e the l i m i t s of i n f i n i t e products. 72 S e c t i o n E. R a m i f i c a t i o n s of the Bas i c D i f f e r e n c e between P _ „ . and P _ „ T — ; •• QMA QML The f a c t t h a t a P _ has A, v defined among incompatible QML elements while a PQ^ a does not have A , V defined among incompatible elements may suggest t h a t a PQ^ a l S i n some sense mi s s i n g elements compared t o a P N „ , . • For example, given an i n i t i a l set of one-dimensional QML subspaces of a H i l b e r t space, both, a P Q J^ a n d a PQ^L c a n he generated by c l o s i n g the i n i t i a l set w i t h respect t o the A , V,-1- operations as defined i n each s t r u c t u r e . When the i n i t i a l set i s f i n i t e , the ' QMA generated by c l o s i n g the i n i t i a l set i s a l s o f i n i t e . In c o n t r a s t , the l a t t i c e d e f i n i t i o n s of A, V among incompatibles o f t e n r e s u l t s i n a p r o l i f e r a t i o n of l a t t i c e elements so t h a t the P Q J^ generated by c l o s i n g a f i n i t e i n i t i a l set may be denumerably i n f i n i t e . An example o f t h i s p r o l i f e r a t i o n of elements i s given i n Chapter VI(C). This p r o l i f e r a t i o n of l a t t i c e elements does not occur i n the P « „ T s t r u c t u r e s of subspaces of QML two-dimensional H i l b e r t space. And i t does not occur i n higher dimensional H i l b e r t space s t r u c t u r e s when there are c e r t a i n angular r e l a t i o n s among the subspaces i n the i n i t i a l s e t . An example i s given i n note 8 below. In these cases when the p r o l i f e r a t i o n of l a t t i c e elements does not occur, both the PQML A N < ^ Q^MA generated by c l o s i n g an i n i t i a l set have e x a c t l y the same elements. And i n any case, i t i s not c o r r e c t t o consider a P „ W . t o be J ' QMA missing elements compared w i t h a PQ M^ • For given any f i n i t e or i n f i n i t e ^QML ' ^here l S a corresponding f i n i t e or i n f i n i t e PQ^A which has e x a c t l y the same elements as P „ „ T but i s missing some of the l a t t i c e r e l a t i o n s QML among these same elements. S p e c i f i c a l l y , an element P € P Q ^ m a v he the meet or j o i n o f two incompatible elements i n PQ^L » e ' g « s ^ = A ^2 ' 73 w i t h P„ ^ Pn , but the same three elements P, P„ , P„ , i n the P „ W . 1 2 ' ' 1 ' 2 QMA which corresponds t o t h a t P„,„. w i l l not be so r e l a t e d because A and v QML are not defined among incompatibles i n a P Q ^ A • Strauss makes a s i m i l a r p o i n t when he argues that the l a t t i c e i n t e r p r e t a t i o n of an element P as the meet of two incompatible elements P^ A P 2 i s a m i s i n t e r p r e t a t i o n because P i- P • P 2 . In other words, the A operation cannot be defined i n terms of the • operation as u s u a l when P^ & P 2 . Strauss concludes t h a t , compared w i t h a (orthomodular) l a t t i c e , a p a r t i a l - B o o l e a n algebra does not omit any elements but r a t h e r prevents the m i s i n t e r p r e t a t i o n o f elements ( S t r a u s s , 1936, p. 203). Of course, authors who favour the l a t t i c e s t r u c t u r e can argue t h a t the i n t e r p r e t a t i o n of an element P € P . „ T as the meet of two incompatible elements P„ A P~ QML 1 2 i s not a m i s i n t e r p r e t a t i o n , i n s p i t e o f the f a c t that P i- P^ • P 2 > since Jauch has created the i n f i n i t e - p r o d u c t d e f i n i t i o n of the meet of two incompatible elements i n ^Q^L • Regardless o f whether or not the l a t t i c e d e f i n i t i o n s o f A and V among incompatibles r e s u l t s i n m i s i n t e r p r e t a t i o n s , the l a t t i c e meets and j o i n s o f incompatibles do cause t r u t h - f u n c t i o n a l i t y problems which are p e c u l i a r t o the P_,,T s t r u c t u r e s but are avoided i n the P N.,. s t r u c t u r e s . R QML QMA For a t r u t h - f u n c t i o n a l mapping on a PQ^ a must preserve the unary x operation and the b i n a r y A, V operations among only compatibles; while a t r u t h - f u n c t i o n a l mapping on a ^ Q ^ ^ must preserve the unary x operation and the bin a r y A, V operations among compatibles and incompatibles. H e r e a f t e r , l e t t r u t h - f u n c t i o n a l (j>) r e f e r t o the former c o n d i t i o n and l e t t r u t h - f u n c t i o n a l (A,£>) r e f e r t o the l a t t e r c o n d i t i o n . The l a t t e r c o n d i t i o n i s not a p p l i c a b l e t o a mapping on a PQ^A s i n c e a ' ' Q I ^ B A S N O 74 operations defined among incompatibles. However, both c o n d i t i o n s can be ap p l i e d t o a mapping on a P Q M L » though a t r u t h - f u n c t i o n a l (d>) mapping on a P A M T ignores the l a t t i c e meets and j o i n s of incompatibles and thus QML preserves only the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of • x n Chapter V(A), i t i s shown how the l a t t i c e meets and j o i n s of incompatibles cause t r u t h - f u n c t i o n a l i t y problems which- r u l e out a b i v a l e n t , t r u t h - f u n c t i o n a l (<^ >,&) semantics f o r any PQ M L which contains incompatible elements. The f a c t that an orthomodular l a t t i c e P . W T has A, V defined QML among incompatibles a l s o a f f e c t s the n o t i o n of a complement i n PQJJL • For as mentioned i n Chapter 1(E), any element P i n a c o n t a i n i n g incompatible elements may have non-unique, incompatible complements. That i s , f o r any P^ € PQ M L » there may be an element P 2 ^ P l s u c n t h a t P 1 A ?2 = 0 and P y P = 1; so P 1 ., P 2 are complements, but P 1 , P 2 are not compatible and are not orthocomplements. For example, consider the orthomodular l a t t i c e diagrammed as f o l l o w s , w i t h P^ & P 2 (and hence P 1 ^ P 2 ) : In t h i s l a t t i c e , P^ , . are compatible and are orthocomplements; l i k e w i s e , P 2 , P^ are compatible and are orthocomplements. But moreover, as i s c l e a r from the diagram: P A P 2 = 0, P 1 V P 2 = 1, and P i A P 2 = °» P^ V P^ = 1. So besides i t s unique Orthocomplement P^ , the element P^ a l s o has two other complements, namely, the element P 2 and the element 75 P X j which are not compatible w i t h and are not orthocomplements of However, when we consider the corresponding p a r t i a l - B o o l e a n algebra which has e x a c t l y the same elements as the above orthomodular l a t t i c e but does not have A, V defined among incompatibles, these same elements P^ > P 2 , P^ are not r e l a t e d v i a the A, V operations and so they are not complements. The only complements i n a p a r t i a l - B o o l e a n algebra are the orthocomplements which are compatible and unique, j u s t as the only complements i n a Boolean s t r u c t u r e such as the c l a s s i c a l L and are orthocomplements which are compatible and unique. In c o n t r a s t , the elements i n an orthomodular l a t t i c e may have other complements. The presence of these other complements i n a P ^ M L c o n t r i b u t e s t o the l a t t i c e t r u t h -f u n c t i o n a l i t y (<!>,'&). problems, as s h a l l be shown i n Chapter V(A). And the presence of these other complements i n a PQ M L r a i s e s the question of whether the l o g i c a l "not" operation should be a s s o c i a t e d w i t h orthocomplementation or with complementation. The f a c t t h a t the "not" of c l a s s i c a l l o g i c i s an op e r a t i o n , t h a t i s , i s a f u n c t i o n which i s u n i v a l e n t , provides a precedent f o r a s s o c i a t i n g "not" with- orthocomplementation r a t h e r than the other . • 5 non-unique complementation. I t i s a l s o worth n o t i n g t h a t i n a p a r t i a l - B o o l e a n algebra PQ M A , the m a t e r i a l c o n d i t i o n a l 3 of ( c l a s s i c a l ) formal l o g i c can be defined i n terms of V "or" and _ L "not" as u s u a l ; moreover, as so d e f i n e d , the m a t e r i a l c o n d i t i o n a l i n PRT.„. i s t r a n s i t i v e as u s u a l . But i n a P , QMA QML the m a t e r i a l c o n d i t i o n a l cannot be defined as u s u a l , which r a i s e s the question of how t o de f i n e ^ i n PQ^L * In c l a s s i c a l l o g i c , the m a t e r i a l c o n d i t i o n a l i s defined as, f o r 76 any formulae f . , f . € L, f. ? f = df. f. v f . And the m a t e r i a l jr 1' 2 1 2 1 2 c o n d i t i o n a l i s t r a n s i t i v e , i . e . , f o r any f ^ , f 2 , f g •€ L, i f H f 3 f ^ and h f 3 f , then t~ f, ? f ., or e q u i v a l e n t l y , i f \= f 3 f and |= f 3 f , then J= r* 3 f ... A l g e b r a i c a l l y , f o r any elements If /, /f„/ i n the L s t r u c t u r e o f L, If^l => ff^l = If^ ? f^l i s an element i n L, namely the element / f ^ / X V If^l. And the r e l a t i o n s of l o g i c a l i m p l i c a t i o n r or semantic entailment 1= are i n t e r p r e t e d as the p a r t i a l - o r d e r i n g r e l a t i o n 5, where f o r any /fl € 1, < / f / IFF IfI = the 1-element. Then the above t r a n s i t i v i t y c o n d i t i o n can be r e s t a t e d a l g e b r a i c a l l y as f o l l o w s : For any / f / , / f / , / f g / € L, i f / y 1 v ' / y = 1 and J. / f 2 / X v / f / = 1, then / f ^ v / f 3 / = 1. With respect t o a quantum PQJ^ » "the m a t e r i a l c o n d i t i o n a l defined i n terms of and V as above does s a t i s f y t h i s t r a n s i t i v i t y c o n d i t i o n , i . e . , f o r any P 1 > P 2 ' P 3 € PQMA ' i f P l " V P 2 = 1 a n d P 2 X V P g = 1, then P^ V P g .= 1. But w i t h respect t o a quantum > i f the m a t e r i a l c o n d i t i o n a l i s defined i n terms o f ~^ and v as u s u a l , then the m a t e r i a l c o n d i t i o n a l i s t r a n s i t i v e IFF the l a t t i c e i s Boolean, as shown by Fay (1967, p. 267). According t o Jauch and others who worry about how to define the m a t e r i a l c o n d i t i o n a l i n a non-Boolean quantum » "the t r a n s i t i v i t y o f the m a t e r i a l c o n d i t i o n a l i s necessary f o r a l o g i c . And so these l a t t i c e t h e o r e t i c i a n s conclude t h a t => cannot be defined i n terms of x , V as usua l i n a quantum ?Q^L (Jauch-Piron, 1970, p. 174). So the c o r r e c t d e f i n i t i o n of the m a t e r i a l c o n d i t i o n a l and even the p o s s i b i l i t y of a r u l e l i k e modus ponens have been c o n t r o v e r s i a l i s s u e s among l a t t i c e - t h e o r e t i c i a n s . Yet another r a m i f i c a t i o n of the b a s i c d i f f e r e n c e between PQJ^ 77 and P„„, i s described i n the next s e c t i o n . QML Section F. The Two Bas i c Senses i n Which the Quantum P r o p o s i t i o n a l Structures Are Non-Boolean In c o n t r a s t t o the Boolean p r o p o s i t i o n a l or l o g i c a l s t r u c t u r e s determined by c l a s s i c a l mechanics and c l a s s i c a l p r o p o s i t i o n a l l o g i c , the quantum p r o p o s i t i o n a l s t r u c t u r e s are s a i d t o be non-Boolean. However, both an orthomodular l a t t i c e PQ^L a n d a P a r t i a i - B o o l e a n algebra c a n ^ e non-Boolean i n various senses. In t h i s s e c t i o n four senses are de s c r i b e d , three of which are eq u i v a l e n t . The most cel e b r a t e d sense i s the f a i l u r e of d i s t r i b u t i v i t y . I f an algebra or l a t t i c e i s Boolean, then i t s b i n a r y A, V operations are d i s t r i b u t i v e . So i f the A, v operations i n an algebra or l a t t i c e are not d i s t r i b u t i v e , then the s t r u c t u r e i s non-Boolean. In p a r t i c u l a r , any quantum P«., T which contains incompatible elements e x h i b i t s at l e a s t one instance QML of the f a i l u r e of d i s t r i b u t i v i t y . For as mentioned i n Chapter 1 ( E ) , f o r _1_ JL any P 1»l 2 e Q^ML ' t h e f o u r e l e m e n t s P i » P i •» P 2 ' P 2 ' s a t i s f y the d i s t r i b u t i v e i d e n t i t y f o r any combinations of these elements IFF P^ <b • I t f o l l o w s t h a t d i s t r i b u t i v i t y f a i l s i n PQ M L i f " P j _ ^ P 2 ' f o r a n y P.. »P~ 6 P„,„ • Most authors who favour the l a t t i c e f o r m a l i z a t i o n of the 1 2 QML quantum p r o p o s i t i o n a l s t r u c t u r e s , e.g., von Neumann and B i r k h o f f (1936, p. 831), Jauch (1963, p. 831), Putnam (1969, p. 226), Friedman and Glymour (1972, pp. 18, 20), focus upon the f a i l u r e of d i s t r i b u t i v i t y as the p e c u l i a r l y non-Boolean fea t u r e of the quantum p r o p o s i t i o n a l s t r u c t u r e s which d i s t i n g u i s h e s them from the Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics. Moreover, i t i s a theorem that a l a t t i c e i s d i s t r i b u t i v e 78 IFF every p a i r o f elements i n i t i s compatible (Jauch-Piron, 1963, p. 831). I t f o l l o w s t h a t a P ^ „ T i s non-Boolean i n the f a i l u r e of d i s t r i b u t i v i t y QML sense IFF i t contains incompatible elements. And hence, we can be sure t h a t instances o f the f a i l u r e of d i s t r i b u t i v i t y i n a PQJ^ always i n v o l v e the meets and j o i n s of incompatibles. Since a ^ a s A> v defined among only compatibles these operations are d i s t r i b u t i v e i n a PQ^A • Thus the f a i l u r e o f d i s t r i b u t i v i t y can n e i t h e r capture the sense i n which a ^Q^a i s non-Boolean nor d i s t i n g u i s h a ^Q^a from the Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics. However, P i r o n defines another sense of non-Boolean f o r the P_,„ QML s t r u c t u r e s which i s equivalent t o the f a i l u r e o f d i s t r i b u t i v i t y sense and which can a l s o be a p p l i e d t o the s t r u c t u r e s . P i r o n d e f i n e s the centre of a l a t t i c e as stat e d i n Chapter 1 ( F ) . And i t i s a theorem t h a t a l a t t i c e i s Boolean IFF i t s centre i s the e n t i r e l a t t i c e ( P i r o n , 1976, p. 29). So i f the centre of a l a t t i c e i s smaller than the e n t i r e l a t t i c e , i . e . , i f there i s an element i n the l a t t i c e which i s not compatible with a l l other elements, then the l a t t i c e i s non-Boolean. Any quantum c o n t a i n i n g incompatible elements i s non-Boolean i n t h i s sense. And P i r o n takes t h i s f a c t to be the p e c u l i a r l y non-Boolean feature o f the quantum PQJ^ s t r u c t u r e s . By the d e f i n i t i o n of the c e n t r e , a PQ^L i s non-Boolean i n the P i r o n sense IFF i t contains incompatible elements. So we expect that a P-.„ i s non-Boolean i n the P i r o n sense IFF i t i s non-Boolean i n the f a i l u r e QML of d i s t r i b u t i v i t y sense, as i t i s easy t o show. I f d i s t r i b u t i v i t y f a i l s i n a P r twT 9 then as mentioned above, not a l l p a i r s o f elements i n P-.... are QML ylXlL compatible. And so by the d e f i n i t i o n o f c e n t r e , the centre o f i s smaller than the e n t i r e PQ M L . Conversely, i f the centre of P ^ M L i s 79 smaller than the entire PQML » "that i s , i f there i s an element P € PQ^ L which is not in the centre of P Q ^ > then that P is incompatible with at least one other element in PQ m l • Hence not a l l pairs of elements in P^.T a r e compatible, and so distributivity f a i l s in P_WT . Q.E.D. QML J QML • But unlike the failure of distributivity sense of non-Boolean, the Piron sense of non-Boolean does not involve the meets and joins of incompatibles. So the Piron sense of non-Boolean can be applied to a P^w. » with the centre of a P_,,. defined exactly as the centre of a QMA ' QMA J Pp M L . And as defined in Chapter 1(D), a partial-Boolean algebra i s in fact a Boolean algebra IFF i t s elements are a l l mutually compatible, i.e., IFF i t s centre i s the entire algebra. Thus a P_„. is non-Boolean in the 6 QMA Piron sense i f i t s centre i s smaller than the entire P»„. . And as before, QMA a PQJ^ is non-Boolean in this Piron sense IFF i t contains incompatible elements. Similarly, the mere presence of incompatible elements in a PQJ/^  or a P „ W „ is a necessary and sufficient condition for the u l t r a f i l t e r s QMA J (and dual ultrideals) in PQ M L O R ''QMA t C > ^ e n 0 t P r^ m e» this provides us with a third sense of non-Boolean. For as mentioned in Chapter 1(C), the u l t r a f i l t e r s (and dual ultraideals) in a Boolean structure are a l l prime. So i f the u l t r a f i l t e r s in a ?n..T or a P.... are not a l l prime, then that QML QMA * ' structure can be said to be non-Boolean. As shown in Chapter VI(B), i f a PQm contains incompatible elements, then there i s at least one u l t r a f i l t e r in PQM which i s not prime, where a prime u l t r a f i l t e r satisfies the condition (d) stated in Chapter 1(C). Hence the presence of incompatible elements in a P„, is a sufficient condition for P_„ to be non-Boolean QM QM in the sense that i t s u l t r a f i l t e r s are not a l l prime. Moreover, this condition -80 i s a l s o necessary. For i f a l l the elements of a P^ are mutually compatible, then t h a t P^ i s i n f a c t a Boolean s t r u c t u r e whose u l t r a f i l t e r s are a l l prime. So a P i s non-Boolean i n the sense t h a t i t s u l t r a f i l t e r s are not a l l prime IFF P contains incompatible elements. In summary, w i t h respect t o a PQ^L » t n e f a i l u r e o f d i s t r i b u t i v i t y sense, the P i r o n sense, and the not-prime u l t r a f i l t e r sense of non-Boolean are a l l e q u i v a l e n t . And w i t h respect t o a » * n e P i r o n sense and the not-prime u l t r a f i l t e r sense o f non-Boolean are e q u i v a l e n t . For these senses of non-Boolean are each b i c o n d i t i o n a l l y connected w i t h the mere presence- o f incompatible elements i n a ^Q^L O R A Q^MA * However, there i s an e n t i r e l y d i f f e r e n t sense o f non-Boolean which i s not b i c o n d i t i o n a l l y connected w i t h the mere presence of incompatible elements. This sense i s suggested by Kochen-Specker, who r e f e r s p e c i f i c a l l y "to P ~„ . s t r u c t u r e s although t h e i r r e s u l t s a l s o apply t o P_ W T s t r u c t u r e s . QMA & * J QML According t o Kochen-Specker, a i s d i s t i n g u i s h e d from the Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics i f the PQJ^ cannot be imbedded i n t o a Boolean a l g e b r a . And i n t h e i r Theorem 0, Kochen-Specker prove t h a t a PQ^a c a n ^ e imbedded i n t o a Boolean algebra 8 IFF there e x i s t s a s u f f i c i e n t l y l a r g e c o l l e c t i o n of b i v a l e n t homomorphisms on PQJ^ s o "that, f o r any p a i r o f d i s t i n c t elements P^ i P^ i n ^Q^A » there i s at l e a s t one b i v a l e n t homomorphism h : {0,1} such t h a t h(P^) i- h ( P 2 ) . Next Kochen-Specker produce a f i n i t e , three-dimensional H i l b e r t space P.,,. which i s shown i n t h e i r Theorem 1 t o admit no b i v a l e n t * QMA homomorphisms. Kochen-Specker conclude t h a t the three-or-higher dimensional H i l b e r t space PQ^A s t r u c t u r e s of quantum mechanics l i k e w i s e admit no b i v a l e n t homomorphisms, and thus, by Theorem 0, these s t r u c t u r e s cannot be 81 imbedded i n t o a IS. Theorem 1 w i l l be discussed i n Chapter V; the Kochen-Specker proof o f Theorem 0 i s discussed here. The "only i f " h a l f of the b i c o n d i t i o n a l statement of Theorem 0 fo l l o w s immediately from the s e m i - s i m p l i c i t y property of Boolean s t r u c t u r e s . Let % be the proposed imbedding. An imbedding i s by d e f i n i t i o n i n j e c t i v e , i . e . , f o r any elements 4- i n J t %(?2)« A n d assuming t h a t the imbedding % : P Q J ^ 8 e x i s t s , the s e m i - s i m p l i c i t y property o f 8 guarantees t h a t t h ere i s a b i v a l e n t homomorphism h : 8 -*• {0,1} such th a t h ( % ( P 1 ) ) 4 h ( % ( P 2 ) ) f o r any •?± 4 Y 2 i n P . Thus the composition h o % : P ^ W A •+ {0,1} i s the d e s i r e d homomorphism o f P.... onto {0,1} QMA QMA which separates P 1 4 P 2 , f o r any •P 1»P 2 € P ^ M A . Kochen-Specker's proof of the converse h a l f o f Theorem 0 i s a l s o worth r e s t a t i n g here because i t suggests how t o construct a C a r t e s i a n product Boolean s t r u c t u r e i n t o which, a ?-.,. can be imbedded i f the r e q u i s i t e set QMA — of b i v a l e n t homomorphisms on e x i s t . Assume th a t t h i s set e x i s t s : l e t {h.}.,, , be the set and l e t s be the c a r d i n a l i t y o f t h i s s e t . I lCIndex J Then an imbedding o f PQJ^ i n t o the C a r t e s i a n product Boolean s t r u c t u r e s s ( Z 2 > , i . e . , % : PQJ^ ( z 2 ) » i s given by the a s s o c i a t i o n of each element P € P ^ M A with- the f u n c t i o n g p : { n i } i e i n d e x f 0 * 1 } defined so th a t Sp(h^) = l u ( P ) f o r every i € Index, where of course h^(P) € {0,1} f o r every i € Index. So f o r example, the image o f any given P € P Q ^ A under the imbedding i s %(P) = ^ ( P ) ,h ( P ) , . . . >\CP)> * ( Z 2 ) S (Kochen-Specker, 1967, p. 67). This c o n s t r u c t i o n w i l l be r e f e r r e d t o again s h o r t l y . The Kochen-Specker imbeddings and homomorphisms preserve the operations and r e l a t i o n s of a P Q J^ s t r u c t u r e . More e x a c t l y , a homomorphism 82 h : X -*• y between any p a r t i a l - B o o l e a n algebras X, y, s a t i s f i e s , f o r any compatible elements b A c € X : h ( b ) i h ( c ) , h(b A c) = h(b) A h ( c ) , h(b~) = (h(b))"\ h ( l ) = 1 (Kochen-Specker, 1967, pp. 66-67). In my terminology, an h s a t i s f y i n g the above i s a homomorphism(A) from X t o yt an i n j e c t i v e h s a t i s f y i n g the above i s an imbedding(A) of X i n t o y, and when y i s {0,1}, i . e . , when h i s b i v a l e n t , an h s a t i s f y i n g the above i s t r u t h - f u n c t i o n a l ( A ) . Thus Kochen-Specker's Theorem 0 b i c o n d i t i o n a l l y connects the p o s s i b i l i t y of imbedding(o) a . P Q ^ i n t o a Boolean s t r u c t u r e w i t h the existence of what I c a l l a complete c o l l e c t i o n o f b i v a l e n t , t r u t h - f u n c t i o n a l (A ) mappings on P ^ ^ , or i n other words, a b i v a l e n t , truth-functional(6) semantics f o r PQJ^ • And i t i s the i m p o s s i b i l i t y of imbedding(o) i n t o a Boolean s t r u c t u r e , or e q u i v a l e n t l y , the i m p o s s i b i l i t y of a b i v a l e n t , t r u t h - f u n c t i o n a l ( o ) semantics f o r J which Kochen-Specker focus upon as the d i s t i n g u i s h i n g non-Boolean feature of the quantum P Q J^ s t r u c t u r e s . Of course, t h i s sense of non-Boolean can a l s o be a p p l i e d t o the quantum P ^ M L s t r u c t u r e s , although an imbedding (A) of a P„,„ i n t o a Boolean s t r u c t u r e or a b i v a l e n t , t r u t h - f u n c t i o n a l ( c - ) QML semantics f o r a P _ M T ignore the l a t t i c e meets and j o i n s of incompatibles QML and preserve only the p a r t i a l - B o o l e a n s t r u c t u r a l features of T Q M L . L i k e the other senses of non-Boolean described above, the presence of incompatible elements i n a P ^ M A or P ^ M L i s a necessary c o n d i t i o n f o r tha t s t r u c t u r e t o be non-Boolean i n the Kochen-Specker sense. For as mentioned i n Chapter 1(D) and ( E ) , i f a l l the elements of a P Q M A or a ^QML 3 r e m u t u a l l y compatible, then t h a t P^ M i s a Boolean s t r u c t u r e as defined i n Chapter 1(B). And any Boolean s t r u c t u r e admits a complete c o l l e c t i o n of b i v a l e n t , homomorphic(o) mappings, i . e . , any Boolean s t r u c t u r e 83 can be imbedded(cO i n t o another Boolean s t r u c t u r e . (The s u f f i x ( i ) i s redundant and harmless s i n c e a l l elements i n a Boolean s t r u c t u r e are mutually compatible.) So i f a P i s non-Boolean i n the Kochen-Specker sense, then the elements of t h a t s t r u c t u r e cannot be mutually compatible, t h a t i s , the s t r u c t u r e must co n t a i n incompatible elements. However, u n l i k e the other senses of non-Boolean described above, the mere presence of incompatible elements i n a P ^ M i s not a s u f f i c i e n t c o n d i t i o n f o r the s t r u c t u r e ' s being non-Boolean i n the Kochen-Specker sense. In p a r t i c u l a r , r e g a r d l e s s of the presence of incompatible elements, a P s t r u c t u r e of two-dimensional H i l b e r t space does admit a complete c o l l e c t i o n o f b i v a l e n t , homomorphic(o.) mappings, i . e . , a PQ^ s t r u c t u r e o f two-dimensional H i l b e r t space can be imbedded (A) i n t o a Boolean s t r u c t u r e , as shown below. The p e c u l i a r s t r u c t u r a l f e a t u r e of three-or-higher dimensional H i l b e r t space P L W s t r u c t u r e s which makes them non-Boolean i n the R QM Kochen-Specker sense i s the presence of overlapping maximal Boolean sub s t r u c t u r e s . Any Boolean s t r u c t u r e has only one maximal Boolean s u b s t r u c t u r e , 2 namely, i t s e l f . And the two-dimensional H i l b e r t space P s t r u c t u r e diagrammed below has many maximal Boolean s u b s t r u c t u r e s , but they do not overlap: 0 84 Except f o r the t r i v i a l Boolean substructure c o n t a i n i n g j u s t the 0, 1 2 2 elements of P_„ , every other Boolean substructure of P_w contains the QM QM four elements , , 0, 1, f o r some i = 1,2, . . . , and i s a maximal Boolean substructure mBS. . The mBS's of Prtl, do share the 0, 1 elements l QM ' but do not share any other elements and so are non-overlapping. As shown next, any two-dimensional H i l b e r t space P2^ can be imbedded(eb) i n t o the Car t e s i a n product Boolean s t r u c t u r e ( Z A ' ) 2 T where r i s the c a r d i n a l i t y of the set of mBS's i n P and 2 i s the d i m e n s i o n a l i t y of the H i l b e r t space. 2 Each mBS.. i n P i s isomorphic t o the Cart e s i a n product Boolean 2 s t r u c t u r e (Z^) diagrammed i n Chapter 1(B). And by s e m i - s i m p l i c i t y each mBS^ has e x a c t l y two b i v a l e n t homomorphisms, f o r example, the h^ , h^ on mBS, and the h , h, on mBS_ l i s t e d i n the f o l l o w i n g t a b l e : 1 c d 2 P l P l P2 P 2 0 1 \ 1 0 0 1 \ 0 1 0 1 h c 1 0 0 1 h d 0 1 0 1 Since the mBS's of P^M do not o v e r l a p , i t i s p o s s i b l e t o define at l e a s t 2 2 »r b i v a l e n t homomorphisms(i) on the e n t i r e P^M by simply adding together the b i v a l e n t homomorphisms on each mBS^ . For example, assume t h a t r i s j u s t 2, i . e . , consider the six-element fragment o f P' QM c o n s i s t i n g o f j u s t mBS^ and mBS^ together. The above four b i v a l e n t 85 homomorphisms h a » ^  » n c 9 h d o n m B S i ' M B S 2 ' r e s p e c t i v e l y , can be added together as f o l l o w s t o y i e l d 2 • 2 b i v a l e n t homomorphisms(o) on ,2 t h i s six-element fragment of ? QM h + h = h, a c 1 P l P l P 2 P 2 0 1 1 0 1 0 0 1 h b + h d = H 2 0 1 0 1 0 1 h b + h c = h 3 0 1 1 0 0 1 h a + h d = \ 1 0. 0 1 0 1 S i m i l a r l y , f o r any P Q M w i t h r mBS's, i t i s p o s s i b l e t o de f i n e 2 • r b i v a l e n t homomorphisms Co) on the e n t i r e P „ „ . And thus, as 2 Kochen-Specker show i n t h e i r proof of Theorem 0, f o r each element P € P „ , , QM the mapping %CP) = <h ( P ) , h 2 ( P j , . . . ,h 2 > rCP)> defin e s an imbedding(o) 2 2 «r of P . W i n t o the Ca r t e s i a n product Boolean s t r u c t u r e (Z_) . The l a t t e r QM r 2 i s a l s o w r i t t e n : nT_^CZ 2)? ' P o r' e x a mP-'- e» "the six-element fragment of 2 P~„ c o n s i s t i n g o f j u s t mBS„ and mBS. can be imbedded(c>) i n t o the QM 1 2 2 • 2 4 C a r t e s i a n product Boolean.structure / ( Z 2 ) = ( Z 2 ) diagrammed i n Chapter 1(B) as f o l l o w s : %C p 1) = ^ ( P ^ h CP )..h CP ),h CP )> = <1,0,0,1>; % ( P j ) = <0,1,1,0>; %(P ) = <1,0,1,0>; % ( P X ) = <0,1,0,1>; %(1) = <1,1,1,1>: %(0) = <0,0,0,0>. I f the maximal Boolean substructures of any three-or-higher dimensional H i l b e r t space s t r u c t u r e d i d not o v e r l a p , then i t would s i m i l a r l y be p o s s i b l e t o imbed((i) t h a t s t r u c t u r e i n t o the C a r t e s i a n product \ n * r Boolean s t r u c t u r e (^2) , where again r i s the c a r d i n a l i t y of the set of mBS's i n the s t r u c t u r e and n i s the d i m e n s i o n a l i t y of the H i l b e r t 86 n>3 space. For each mBS^ i n a P s t r u c t u r e i s isomorphic t o the Boolean s t r u c t u r e (Z ) n a n c^ b y s e m i - s i m p l i c i t y has e x a c t l y n b i v a l e n t homomorphisms. So i f the mBS's of P d i d not o v e r l a p , then i t would be p o s s i b l e t o simply add together these b i v a l e n t homomorphisms on each mBS. y i e l d i n g at l e a s t n«r b i v a l e n t homomorphisms (A) on the e n t i r e r s t r u c t u r e . And thus by the Kochen-Specker Theorem 0, the could be imbedded(i) i n t o the Car t e s i a n product Boolean s t r u c t u r e (Z_) n«r T* Tl 1 which i s a l s o w r i t t e n : n. . (Z„). . i = l 2 1 However, the mBS's of a three-or-higher dimensional H i l b e r t n>3 space P may overlap and do overlap i n quantum mechanically r e l e v a n t PQ~ . Consequently, the attempt t o d e f i n e b i v a l e n t homomorphisms(A) on a n>3 P^jyj , by simply adding together the separate b i v a l e n t homomorphisms e x i s t i n g on each mBS o f P^~ , i s problematic and i n f a c t i s imp o s s i b l e . Kochen-Specker prove t h i s i m p o s s i b i l i t y ; t h e i r proof i s discussed i n Chapter V(B). An example of a t r i v i a l exception t o t h i s i m p o s s i b i l i t y i s 3n>: QM given i n the note below; such e x c e p t i o n a l ? ~ s t r u c t u r e s are not quantum g mechanically r e l e v a n t . In summary, there are two b a s i c senses i n which the quantum p r o p o s i t i o n a l s t r u c t u r e s may be s a i d to be non-Boolean and may be d i s t i n g u i s h e d from the Boolean p r o p o s i t i o n a l s t r u c t u r e s determined by c l a s s i c a l mechanics and c l a s s i c a l l o g i c . One b a s i c sense subsumes the f a i l u r e of d i s t r i b u t i v i t y , the P i r o n , and the not-prime u l t r a f i l t e r senses of non-Boolean; the presence o f incompatible elements i n a P^ M i s necessary and s u f f i c i e n t f o r the s t r u c t u r e t o be non-Boolean i n t h i s b a s i c sense. The other b a s i c sense i s suggested by Kochen-Specker's work; the mere presence of incompatible elements i n a P^ M i s necessary but i s not s u f f i c i e n t 87 f o r the s t r u c t u r e t o be non-Boolean i n t h i s second b a s i c sense. Notes ^ This f a c t i s a c t u a l l y d erived from one or the other of the fundamental p o s t u l a t e s ( I I ) or ( I I I ) which define p . and Exp,(A) (Messiah, 1966, pp. 178, 297). ' Y 2 According t o the terminology of h i s 1932 book, von Neumann c a l l s such p r o p o s i t i o n s simultaneously d e c i d a b l e . Von Neumann's n o t i o n of the simultaneous d e c i d a b i l i t y of p r o p o s i t i o n s i s a refinement of h i s notio n of the simultaneous m e a s u r a b i l i t y of magnitudes. The l a t t e r r e q u i r e s t h a t the s e l f - a d j o i n t operators r e p r e s e n t i n g the magnitudes commute. The former r e q u i r e s that only the p r o j e c t o r s r e p r e s e n t i n g the p r o p o s i t i o n s commute, but the magnitudes mentioned i n the p r o p o s i t i o n s need not be simultaneously measurable, i . e . , t h e i r operators need not commute. So while the operators r e p r e s e n t i n g simultaneously measurable magnitudes share a l l t h e i r e i g e n s t a t e s , the operators r e p r e s e n t i n g the magnitudes mentioned i n simultaneously decidable p r o p o s i t i o n s need share only the e i g e n s t a t e ( s ) s p e c i f i e d by the p r o p o s i t i o n s . Von Neumann has h i s own unusual use of the terms compatible and incompatible. Nevertheless, simultaneously decidable p r o p o s i t i o n s are compatible i n the u s u a l sense t h a t t h e i r r e p r e s e n t a t i v e p r o j e c t o r s commute (von Neumann, 1932, pp. 251, 253). 3 With respect t o an orthomodular l a t t i c e P^„. , t h i s c o n d i t i o n i s weak modular i t y , which c h a r a c t e r i z e s the quantum ^ P N M s t r u c t u r e s . With respect t o a p a r t i a l - B o o l e a n algebra PQ^A ' t h i s c o n d i t i o n holds because by d e f i n i t i o n , P. < P„ IFF P. A P = P. , but P A P„ i s defined i n P Q M IFF P 2 . 1 4. Von Neumann r e s t r i c t s "and" and "or" t o what he c a l l s simultaneously decidable p r o p o s i t i o n s . As mentioned i n Note 2 above, such p r o p o s i t i o n s are compatible i n the u s u a l sense that t h e i r r e p r e s e n t a t i v e p r o j e c t o r s commute. 5 This point was suggested by Dr. R. E. Robinson. In her d i s c u s s i o n of B i r k h o f f and von Neumann's quantum l a t t i c e s t r u c t u r e s , S. Haack i n c o r r e c t l y claims t h a t an element i n such a s t r u c t u r e may have more than one orthocomplement (Haack, 1974, p. 161). Though i t i s true that an element may have more than one complement, the orthocomplement of an element i s by d e f i n i t i o n unique. For according t o B i r k h o f f , the a s s o c i a t i o n of an element b w i t h i t s orthocomplement b i s a type of mapping (namely, a dual automorphism h : X ->• X which i s an isomorphism of a s t r u c t u r e w i t h i t s e l f s a t i s f y i n g , f o r every b,c € X, b < c IFF h(b) > h ( c ) ) ( B i r k h o f f , 1967, p. 3). And as s t a t e d i n Chapter 1(G), c o n d i t i o n Ma, the image of any element b € X under any mapping h : X -* S i s unique, e.g., h(b) = i s unique. 88 Likewise the f a i l u r e o f bi v a l e n c e sense of non-Boolean proposed by van Fraassen i s b i c o n d i t i o n a l l y connected w i t h the mere presence o f incompatible elements i n a quantum p r o p o s i t i o n a l s t r u c t u r e (van Fraassen, 1973, p. 89). 7 Bub makes a s i m i l a r p o i n t (1974, pp. 144-146). ^ P!L.? s t r u c t u r e s whose mBS's do not overlap and P!?,,^  s t r u c t u r e s which admit QMA b i v a l e n t homomorphisms(i) even though t h e i r QM mBS's do ove r l a p , may be generated by c l o s i n g c e r t a i n l i m i t e d sets o f one-dimensional subspaces ( o r t p r o j e c t o r s ) of H ,n>3 of P w i t h respect t o the A, v , x operations QMA or QM twelve-element QM For an example of the l a t t e r , consider the f o l l o w i n g s t r u c t u r e generated by c l o s i n g an i n i t i a l set of x f i v e one-dimensional subspaces of H with respect t o the A, v, operations of P^ M , where {P^,?^?^} are mutually compatible and l i k e w i s e {Pg,P^,P^} are mutually compatible. The f o l l o w i n g f i v e b i v a l e n t homomorphisms(cb) c o n s t i t u t e a complete c o l l e c t i o n of b i v a l e n t homomorphisms(6) on t h i s twelve-element P. : P l p l P 2 P 2 P 3 P 3 P 4 P4 P 5 P 5 0 1 h l 1 0 0 1 0 1 1 0 0 1 0 1 h 2 0 1 1 0 0 1 0 1 1 0 0 1 h 3 0 1 ft 0 1 1 0 0 1 0 1 0 1 \ 0 1 1 0 0 1 1 0 0 1 0 1 h 5 1 0 0 1 0 1 0 1 1 0 0 1 (Just the f i r s t three b i v a l e n t , homomorphisms(A) h^ , , h^ , c o n s t i t u t e a weakly complete c o l l e c t i o n . ) This twelve-element P ^ i s a l s o an example of a phenomenon mentioned i n Section (E) above, namely, an example of how the p r o l i f e r a t i o n of l a t t i c e elements due t o the l a t t i c e d e f i n i t i o n s of 89 A , V among incompatibles does not occur i n PQ^L s t r u c t u r e s generated by-c l o s i n g a f i n i t e i n i t i a l set o f one-dimensional subspaces of f/ n~ 3 when there are c e r t a i n angular r e l a t i o n s among the subspaces i n the i n i t i a l s e t ; most simply, i n t h i s case, P , P , P , P are a l l i n the same 3 3 two-dimensional subspace P„ . And i n t h i s case, the P „ „ T and the P _ „ . • . . 3 QML QMA generated by c l o s i n g the i n i t i a l set of f i v e one-dimensional suspaces of H w i t h respect t o the A, V , - 1 - operations o f PQ^ l a n d P Q M A r e s p e c t i v e l y , each have e x a c t l y the same twelve elements, as diagrammed above. N > 3 Nevertheless, as e x e m p l i f i e d by Kochen-Specker, f o r H , the sets of one-dimensional subspaces r e p r e s e n t i n g quantum p r o p o s i t i o n s which describe a c t u a l quantum mechanical systems and s i t u a t i o n s y i e l d , upon N > 3 c l o s u r e , P _ „ s t r u c t u r e s whose mBS's do overlap and overlap i n such a n—3 way t h a t b i v a l e n t homomorphisms(£>) on P are r u l e d out. (Kochen-Specker, ^ n—3 1 9 6 7 , Section 4 ) . In other words, quantum mechanically r e l e v a n t P Q M s t r u c t u r e s have overlapping mBS's which r u l e out b i v a l e n t homomorphisms(6). 90 CHAPTER V THE IMPOSSIBILITY OF A BIVALENT, TRUTH-FUNCTIONAL SEMANTICS FOR THE NON-BOOLEAN PROPOSITIONAL STRUCTURES DETERMINED BY QUANTUM MECHANICS Preface As described i n Chapter I V ( F ) , there are two ba s i c senses i n which the quantum p r o p o s i t i o n a l s t r u c t u r e s may be s a i d to be non-Boolean. And as mentioned i n Chapter 11(C), any Boolean s t r u c t u r e admits a complete c o l l e c t i o n of b i v a l e n t homomorphisms, and t h i s c o l l e c t i o n i s a b i v a l e n t , t r u t h - f u n c t i o n a l semantics when the Boolean s t r u c t u r e i s a l o g i c a l or p r o p o s i t i o n a l s t r u c t u r e . But i f a p r o p o s i t i o n a l s t r u c t u r e i s i n some sense non-Boolean, then whether or not i t admits such a semantics i s an open question. With respect t o the non-Boolean quantum p r o p o s i t i o n a l s t r u c t u r e s , answers to t h i s question have already been given or at l e a s t suggested by von Neumann, Jauch-Piron, Gleason and Kochen-Specker.in t h e i r proofs and arguments against the p o s s i b i l i t y of hidden v a r i a b l e s i n quantum mechanics. For as s h a l l be described i n Section (D), when i n t e r p r e t e d s e m a n t i c a l l y Gleason's i m p o s s i b i l i t y proof and Kochen-Specker's Theorem 1 show the i m p o s s i b i l i t y of a b i v a l e n t , t r u t h - f u n c t i o n a l ( i > ) semantics f o r three-or-higher dimensional H i l b e r t space s t r u c t u r e s , whether PQ^A O R Q^ML. * A N ( ^ i n t e r p r e t e d s e m a n t i c a l l y , von Neumann's proof of the i m p o s s i b i l i t y of d i s p e r s i o n - f r e e quantum ensembles and Jauch-Piron's C o r o l l a r y 1 suggest the i m p o s s i b i l i t y of a b i v a l e n t , t r u t h - f u n c t i o n a l ( e i ,&) semantics f o r two-or-fhigher •OJ2 QML dimensional H i l b e r t space P ^ T s t r u c t u r e s . The proofs by von Neumann and Jauch-Piron must be i n t e r p r e t e d as r e f e r r i n g t o the orthomodular l a t t i c e s t r u c t u r e s and the t r u t h - f u n c t i o n a l i t y ( o , & ) c o n d i t i o n because von Neumann and Jauch-Piron do de f i n e operations among incompatibles and do r e q u i r e the p r e s e r v a t i o n of these operations. In the next s e c t i o n , von Neumann, Jauch-Piron suggestion i s pursued. S e c t i o n A. The I m p o s s i b i l i t y o f a B i v a l e n t , Truth-Functional(cj>,&) Semantics f o r any PQJ^ Wh i c n Contains Incompatible Elements 2 Consider the fragment o f the PQ M s t r u c t u r e o f two-dimensional H i l b e r t space diagrammed below, with- P^ H> P^ : 2 2 As mentioned i n Chapter IV (E ) , P Q M L and P Q M have the same elements. In both s t r u c t u r e s , the 0, 1 elements are equal t o the f o l l o w i n g meets and j o i n s o f compatibles: P^ A P^ " = 0, P 2 A P^ = 0, P± V ?x = 1, P v P X = 1. In a d d i t i o n , the 0, 1 elements are equal t o the f o l l o w i n g 2 ->-meets and j o i n s o f incompatibles i n P Q M L : P^ A p 2 = °» Pl A P2 = °* P 1 V P 2 = 1, . P 1 V P 2 = 1. Any n o n - t r i v i a l mapping m on P Q M L assigns the value 0 t o each meet which equals the 0-element, e.g., m ^ A P 2> = m(0) = 0. 92 1 Likewise, m assigns the value 1 to each join which equals the 1-element. And now i t i s easy to prove the following: Theorem 0. A bivalent, truth-functional (A,&) mapping on the 2 fragment of P diagrammed above i s impossible. Proof: Any bivalent m assigns either the value 1 or the value 0 to the element P. , so there are two cases. Case 1: Assume m(P„) = 1. 1 1 We have m(P^ A P ) = m(0) = 0, so by truth-functionality(i,&>), m(P1 A P 2) = m(P1) A m(P2) = 0. Hence m(P2) t 1 since 1 A 1 = 1, thus m(P ) = 0. So by truth-functionality(A,<!S) m(P2) = (m(P2))J" = 0 a = 1. Also we have m(P^ A F^) = m(0) = 0, so by truth- functionality (A ,<J6), m(P„ A P A) = m(P„) A m(Px) = 0. Hence m(Px) = 0. So we have a contradiction. 1 2 1 2 2 Case 2: Assume mCP^) = 0. Then as in Case 1, mCP^  V P 2) = m(l) = 1 and m(P^ V P 2) = m(l) = 1 yield contradictory assignments of values to the element P x . Q.E.D. Hence a bivalent, truth-functional(A,^>) semantics for this 2 fragment of P O V I T is impossible. This proof can be generalized to include QML n>2 any two-or-higher dimensional Hilbert space PQ M L structure which contains incompatible elements. (The t r i v i a l case of a one-dimensional Hilbert space structure i s excluded because that structure contains just a 0-element and a 1-element which are compatible.) The generalization makes use of the following lemmas: Lemma A. For any atom P^ in any P^ and for any element P € Pnr, , P A P IFF P < P or P < P"1". QM a a a Assume P 6 P. By definition of A , there exist three a J mutually disjoint elements P^P^Pg € •? such that ? a = P 1 V P g 93 and P = P„ V P . Since P = P V P i s an atom,' P. V P > 0 2. o a 1 o 1 3 and there i s no element P € P„„ such t h a t P_ v P„ > P > 0. x QM 1 3 x Since ?± V ?3> -P and -P V ?3 > Pg ., e i t h e r P = 0 and P = P ., or P = G. ' and P = P . I f the former, then H O O 3. 1 P = P V P > P . I f the l a t t e r , then P = P V 0 = P ; and . Z a. cL A z. s i n c e P. , P_ are d i s j o i n t , P = P„ and P = P„ are d i s j o i n t . 1 2 a 1 2 J Assume P < P, then P i P. (See note 3 o f Chapter IV.) a a L i k e w i s e , i f P 5 P X, then P A P"1", where P A P"1" IFF ' a a ' a P A P. Q.E.D. a Lemma B. For any atom P a i n any a n d ^ o r anY element P £ P A „ T , i f P £ P then P A P = 0. QML a a By assumption, P > 0 and there i s no element P € P . ,„ J c a x QML such t h a t P > P > 0. But P > P A P > 0. So e i t h e r a x a a P = P A P o r P A P = 0 . The former i s r u l e d out because a a a P = P A P IFF P 5 P, which c o n t r a d i c t s the antecedent of a a a Lemma B. Hence P A P = 0. Q.E.D. a Lemma C. Every element P i 0 i n P Q M L 1 S The j o i n o f the atoms i t dominates. Let P. be any atom i n P..,_ such t h a t P. 5 P, and l e t l J QML I V P. be the ( f i n i t e o r i n f i n i t e ) j o i n o f a l l such atoms. (This • X 1 j o i n e x i s t s because P R T.„ i s complete.) And l e t P = V P. . J QML ^ x ^ I We want t o show th a t P = P^ . C l e a r l y , P^ < P, and so P A P. Now i f P 1 A P = 0. then P = 1 A P = (P V P X) A P X X X X = (P A P) V ( P X A P) = (P A P) V 0 = P A P, i . e . , P < P X X X X X and thus P.. = P. Assume on the co n t r a r y that PJ" A P / 0. Then 94 since P r tWT is atomic, there is an atom P in P„„T such that QML a QML P S P ^ A P , so P ' 5 P 1 and P < P . Since P 5 P , P = P . , a x a x a a a x for some i , and so P < P , i . e . , P A P = P . And since a x a x a P S P X , P = P A P = P X A P = 0 , a contradiction. Q.E.D. a x a a x x x Lemma D. The join of a l l the atoms in any P Q J ^ is equal to the 1-element. Let P . be any atom in P.W T and let V P . be the l QML ^ l (finite or infinite) join of a l l the atoms i n P-WT . QML Assume on the contrary that V . P . j- 1. Then ( V P . ) " L t 0, i 1 i 1 and so ( V P . ) X > P . , for some atom P . . C l e a r l y , V P . > P . . i 1 3 D i 1 • 3 I t follows that 0 = ( V P J A C V P . ) X > (V P . ) A P . = P . , l l l J which is impossible. Hence ( V P.) = 1. Q.E.D. i Lemma E. Any proposed bivalent, truth-functional(<i,jb) mapping on any PA M T must assign the value 1 to at least one of the atoms QML in P-„T . QML Again, let P . . be any atom in P_,.T and let V P . be the & x J QML ^ l join of a l l the atoms in P.„T . I assume that the truth-J QML functionality(<!>,&) condition includes the preservation of infinite meets and joins. By Lemma D, V P . = 1, and so any (non-trivial) bivalent, i truth-functionaK 1,^) mapping m on P Q M L assigns the value V m ( P . ) = m ( V P.) = m(l) = 1. And for every P . , m ( P . ) = 0 # 1 * 1 1 3L X X or 1, since m is bivalent. I f m ( P ^ ) = 0 for every P ^ , then V m ( P . ) = 0 i 1. Thus at least one of the atoms in PA„T x QML x 95 2 i s assigned the value 1 by m. Q.E.D. Besides these lemmas, the generalization makes use of the distinction between irreducible and reducible P R T „ T structures, defined QML ' as follows. As defined in Chapter 1(F), the centre of any P Q J ^ contains at least the 0 and 1 elements of P-„.T • A Tn.,T whose centre QML QML contains just the 0, 1 elements is irreducible. A P_,,T whose centre J ' QML contains other elements besides the 0, 1 elements is reducible. As described in Chapter IV(F), any P Q ^ l contains incompatible elements IFF i t s centre is less than the entire structure. Clearly, any P Q J ^ containing incompatible elements is either irreducible or reducible. And any irreducible Q^ML c o n " t a i n s incompatible' elements. A-reducible PQJ^l may have a i l i t s elements mutually-compatible, but such a reducible P Q ^ is in fact, a Boolean structure. If the centre of a reducible P.... contains any atoms of P.„_ , QML J QML then the structure does admit some bivalent, truth-functional(<_!>,&) mappings, as shall be described'-in a-brief'-digression. Each central'.atom P of such a reducible P_,„ specifies an u l t r a f i l t e r UF and dual ultraideal Ul QML c c c by the usual definitions: UF = { P € P N., T : P > P } and J c QML c Ul = {P € P« .„ : P 5 P" 1"}. And since each central atom is by definition c QML c compatible with every other element in P Q ^ > i t follows by Lemma A that, for every element P € P Q ^ » and for any given central atom P C , either P > P or P*"> P . Since by definition of , P X > P IFF P < P 1 , c c c c either P > P or P 5 P 1 . So every element in P«., T is either a member c c QML of UF or a member of Ul ; thus P O V T T = UF U Ul . Then as w i l l be c c QML c c shown in Chapter VI(B), i t follows by the conditions defining an u l t r a f i l t e r and dual ultraideal that the bivalent homomorphism h c : P Q M L •+ {0,1}, defined with respect to UF and Ul as usual, truth-functionally(i,&) 96 assigns 0, 1 values to every element i n PQJVJL • x n p a r t i c u l a r , each h c assigns the value 1 t o i t s a f f i l i a t e d c e n t r a l atom P c and assigns the value 0 t o every other atom P i P i n P_„T . For every other atom J a c QML J P i s compatible w i t h P and so by Lemma A, P 2 P"^ (the a l t e r n a t i v e a c a c P 5 P i s r u l e d out s i n c e P i s an atom): thus P € UI . There are a c c a c as many such b i v a l e n t , t r u t h - f u n c t i o n a l ( o ,<&) mappings on a r e d u c i b l e PQ^L as there are c e n t r a l atoms i n P_WT . This ends the d i g r e s s i o n . QML Now the previous Theorem 0 i s g e n e r a l i z e d as f o l l o w s . Theorem A. A b i v a l e n t , t r u t h - f u n c t i o n a K i , ^ ) semantics i s impossible f o r any (two-or-higher dimensional H i l b e r t space) F*QML which contains incompatible elements. Case 1: I r r e d u c i b l e P Q M L • B v Lemma E, any proposed b i v a l e n t , t r u t h - f u n c t i o n a l ( i , i J S ) mapping on any i r r e d u c i b l e assigns the value 1 t o at l e a s t one atom i n P_„„T , say m(P ) = 1. Since P i s not i n QML J a a the centre o f the i r r e d u c i b l e P„„T , there i s some element P € P QML QML such t h a t P & P. Then by Lemma A, P £ P and P % P"*". So by Lemma B, 3. cl cL P A P = 0 and P A P X = 0. Then i t f o l l o w s by the reasoning given i n cL cL the case 1 of Theorem 0 th a t m assigns c o n t r a d i c t o r y values t o the element P X. So a proposed b i v a l e n t t r u t h - f u n c t i o n a l ( o , & ) mapping on an i r r e d u c i b l e ^QML l S i m P o s s i k l e * Hence a b i v a l e n t , t r u t h - f u n c t i o n a l ( o , & ) semantics f o r an i r r e d u c i b l e P„,„ i s impossible. QML Case 2: Reducible P A M T ( c o n t a i n i n g incompatible elements). Any r e d u c i b l e '•' QML P^ wr contains at l e a s t one non- c e n t r a l atom. For i f every atom i n P..,T QML J QML were c e n t r a l , then since the centre i s a s u b l a t t i c e c l o s e d w i t h respect t o the j o i n operation i t f o l l o w s by Lemma C th a t every element i n w o u l d be c e n t r a l , i . e . , there would be no incompatible elements i n P Q M L . A 97 non-central atom in x S clearly distinct from the 0-element, and so a complete collection of Bivalent, truth-functional(i , cK) mappings on P QML must include a mapping which assigns the value 1 to the non-central atom. But by the same reasoning given in case 1 of this proof, any proposed bivalent, truth-functionalComapping which assigns the value 1 to a non-central atom in w i l l assign contradictory values to some other element in PQJ^ l which is incompatible with that atom. So although a reducible P Q J^ m a v admit some bivalent, truth-functional(c}),j6) mappings, as shown in the digression above, a reducible P O V T T does not admit enough QML such mappings to constitute a bivalent, truth-functional(o,&) semantics for P Q M L ' Q - E ' D - 3 One way of avoiding the contradictions which, yield this impossibility proof i s to weaken the truth-functionality(o,<iS) condition to just truth-functionality(o), thus allowing the semantic mappings on a P QML to ignore the lattice meets and joins of incompatibles. Such bivalent, truth-functional (A) mappings which preserve the partial-Boolean structural features of a P Q ^ L O R A ^QMA a r e bivalent homomorphisms(A) and are considered by Kochen-Specker Section B. The Kochen-Specker Proof of the Impossibility of Bivalent Homomorphisms (A) on a Three-Dimensional Hilbert Space 2 As described in Chapter IV(F), two-dimensional Hilbert space P structures do admit a complete collection of bivalent homomorphisms (A), i.e., they do admit a bivalent, truth-functional(A) semantics, in spite of the fact that they contain incompatible elements. But three-or-higher dimensional Hilbert space F^Z? structures do not admit bivalent QM 98 homomorphisms(o). The peculiar structural feature of P A^ which rules out bivalent homomorphisms (i>) i s not just the presence of incompatible elements but rather the presence of overlapping maximal Boolean substructures ' (for which the presence of incompatibles is a necessary condition). In their Theorem 1, Kochen-Specker consider a particular f i n i t e 3 P«„« and show that this structure does not admit even a single bivalent QMA & homomorphism(o). By definition, a proposed bivalent homomorphism(<b) ,h 3 3 on PQJ^ satisfies, for any three mutually orthogonal atoms P ^ J P ^ J P ^ € P Q ^ > h(P 1) V h ( P 2 ) V h(P 3). = h(P 1 v P 2 v P G ) = h(l) = 1, and h(P.) A h.(Pj) = hCP. A Pj) = h(0) = 0 for 1 < i i j 5 3. Thus exactly 3 one of every three mutually orthogonal atoms in PQ^A ^ s assigned the 3 value 1 by a bivalent homomorphism(A>) on PQ^A (Kochen-Specker, 1967, p. 67). More generally, a bivalent homomorphism(cb) h on any n dimensional Hilbert space P" satisfies: QM (KS1) For any n mutually orthogonal atoms P^>P2' " * " ' Pn ^  ^QM ' h(P„) V h(P ) v . . . V h(P ) = h ( P , v P 0 V . . . v P ) = h(l) = 1, 1 2 n 1 2 n and h(P.) A h ( P . ) = h(P. A P . ) = h(0) =0 for 1 < i i j < n. By the Lemma A of the previous section, any two atoms in a P A^ are orthogonal IFF they are compatible. By closure with respect to the A, V , ^ operations of P A^ , a set of n mutually orthogonal (i . e . , compatible) atoms in a P ^ generates a Boolean substructure of P^M . In particular, such a set generates a maximal Boolean substructure of P ^ since the maximum number of mutually compatible atoms in a P ^ structure 14. of n dimensional Hilbert space i s n. Thus condition (KS1) refers to the (maximal) Boolean substructures of a P Q M and ensures that their Boolean 99 s t r u c t u r a l f e a t u r e s are preserved by a b i v a l e n t homomorphismCi). And j u s t the u s u a l d e f i n i t i o n of a b i v a l e n t homomorphism on a Boolean s t r u c t u r e ensures t h a t any b i v a l e n t homomorphism h : mBS -*• {0,1} s a t i s f i e s (KS1). So the f a c t t h a t a b i v a l e n t homomorphism(<b) on any .P^^ by d e f i n i t i o n s a t i s f i e s (KS1) does not focus a t t e n t i o n upon th a t p e c u l i a r l y non-Boolean s t r u c t u r a l f e a t u r e of s t r u c t u r e , namely, the presence of overlapping mBS's i n P ^ 3 . QM B i v a l e n t homomorphisms(i) on a PQ~ preserve not only the Boolean s t r u c t u r a l f e a t u r e s of every (maximal) Boolean substructure but a l s o the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of the e n t i r e PQ^» i n p a r t i c u l a r , n—3 the overlap p a t t e r n s of the mBS's i n PQ^ . One way i n which the overlap patterns can be v i o l a t e d i s by a l l o w i n g d i f f e r e n t values to be assigned t o a given element P which i s a member of two or more overlapping mBS's; the value assigned t o P i n the context o f one mBS may be d i f f e r e n t from the value assigned t o P i n the context of another mBS. Such proposed v i o l a t i o n s of the overlap p a t t e r n s are f u r t h e r discussed i n Chapter V I I . A b i v a l e n t homomorphism(i>) on a does not v i o l a t e the overlap patterns i n t h i s way (or i n any way). For a b i v a l e n t homomorphism(o) i s a mapping, i . e . , h(P) i s unique, as state d i n Chapter 1(G), and i n . p a r t i c u l a r , h(P) does not depend upon which mBS i s being considered. Kochen-Specker do not e x p l i c i t l y s t a t e t h i s aspect of a b i v a l e n t homomorphism(o); here and i n Chapter VII i t s h a l l be r e f e r r e d t o as: (KS2) The values assigned by a b i v a l e n t homomorphism(i) h : • ^ QM' are unique and do not vary w i t h or depend upon the mBS's of Pq^ . This aspect of the n o t i o n of a b i v a l e n t homomorphism(o) i s a r t i c u l a t e d , 100 s h a l l r e s t a t e Kochen-Specker's proof of t h i s i m p o s s i b i l i t y f o r t h e i r P A ^ i n a manner which e l u c i d a t e s the e f f e c t of the overlap p a t t e r n s and which e x p l i c i t l y r e f e r s t o (KS1) and (KS2). the value 1 t o e x a c t l y one atom i n the mBSQ s p e c i f i e d by the three atoms Kochen-Specker l a b e l p^ * q A , r n . Let us i n i t i a l l y assume t h a t h ( p Q ) = 1 and thus h ( q n ) = h ( r n ) = 0. This i n i t i a l assignment of values to the atoms i n mBSn of course determines the values assigned t o a l l the other non-atomic elements of mBSn . But i n a d d i t i o n , t h i s i n i t i a l assignment of values t o the atoms i n mBSQ places r e s t r i c t i o n s upon the values assigned by h t o the atoms i n every mBS which overlaps w i t h mBS- . For example, consider an mBS which contains p and two other 0 0 atoms; by assumption and by (KS2), h ( p Q ) = 1. And so by (KS1), the other two atoms i n every mmBS co n t a i n i n g p A are each assigned the value 0 by h. These value assignments i n t u r n determine the values assigned by h to the atoms i n every mBS which overlaps w i t h any o f the mBS which 3 overlap w i t h mBSg . And t h i s process continues through the PQJ^ s t r u c t u r e u n t i l we get h(qg) = 1, which c o n t r a d i c t s the statement t h a t h ( q n ) = 0 (which f o l l o w s by (KS1) from the i n i t i a l assumption t h a t h ( p A ) = 1). A s i m i l a r c o n t r a d i c t i o n r e s u l t s i f we in s t e a d i n i t i a l l y assume t h a t By (KS1), any b i v a l e n t homomorphism(cb) h on P QMA must assign 101 h ( q Q ) = 1 and h ( p Q ) = h.(.rQ) = 0. And l i k e w i s e a c o n t r a d i c t i o n r e s u l t s i f we i n i t i a l l y assume t h a t h-Cr-p). = 1 and h ( p Q ) = M q ^ ) = 0. Thus, si n c e any b i v a l e n t homomorphism(c>) on t h i s P must a s s i g n the value 1 t o one of the three atoms PQ , q ^ , r ^ i n mBSQ yet a l l three attempts lead to c o n t r a d i c t i o n , a b i v a l e n t homomorphism(i) on t h i s P Q M A considered by Kochen-Specker i s impossible. In other words, a b i v a l e n t , t r u t h - f u n c t i o n a l ( i ) mapping on t h i s 3 P_,„. i s im p o s s i b l e , and so a b i v a l e n t , t r u t h - f u n c t i o n a l ( o ) semantics f o r QMA 3 t h i s P Q J^ l S impossible. 3 Kochen-Specker a l s o consider a much sma l l e r P Q^ a which contains 3 27 atoms and 16 overlapping mBS's. This does admit some b i v a l e n t homomorphisms (i>), but as Kochen-Specker p o i n t out, there are two d i s t i n c t atoms i n t h i s s t r u c t u r e such t h a t no b i v a l e n t homomorphismCcb) assigns d i f f e r e n t values t o these two d i s t i n c t elements. That i s , the c o l l e c t i o n of b i v a l e n t homomorphisms(A) which do e x i s t on t h i s P Q ^ 1 S no* complete. So l i k e the r e d u c i b l e orthomodular l a t t i c e P_„_ s t r u c t u r e s discussed i n QML 3 the d i g r e s s i o n i n the previous s e c t i o n , t h i s PQ^ a does admit some b i v a l e n t , t r u t h - f u n c t i o n a l ( o ) mappings, but i t does not admit a b i v a l e n t , t r u t h -f u n c t i o n a l ( o ) semantics (Kochen-Specker, 1967, p. 67). The Kochen-Specker r e s u l t i s f u r t h e r discussed i n Sectio n (D) and i n Chapter V I I . Sectio n C. Avoiding These I m p o s s i b i l i t y Proofs There are a t l e a s t two ways o f a v o i d i n g not on l y Theorem A but a l s o the Kochen-Specker i m p o s s i b i l i t y proof. One way i s t o f u r t h e r weaken the truth-functionality (<M c o n d i t i o n ; another way i s t o r e s t r i c t the domains 102 of proposed semantic mappings on P N ^ to c e r t a i n substructures of P^ . With regard t o the l a t t e r , f o r example, i f the domain of the 2 mapping m on the P N M T discussed i n the beginning of Section (A) were QMLJ r e s t r i c t e d such that mCP^), mCP^) are not defined when mCP^) i s 7 d e f i n e d , then m would not as s i g n c o n t r a d i c t o r y values. ' In other words, i f the domain of each proposed semantic mapping on a P A ^ were r e s t r i c t e d to an mBS of P N ^ , then both i m p o s s i b i l i t y proofs would c l e a r l y be avoided. More i n t e r e s t i n g l y , semantic mappings which avoid both i m p o s s i b i l i t y proofs yet whose domains are substructures o f P which i n c l u d e overlapping mBS's are described i n Chapter V I ; they-are the quantum state-induced e x p e c t a t i o n - f u n c t i o n s . With regard to the f i r s t mentioned way of avoiding both i m p o s s i b i l i t y p r o o f s , Friedman and Glymour propose, f o r quantum P Q J ^ s t r u c t u r e s , semantic mappings which are r e q u i r e d to preserve the -~J~ ope r a t i o n and the 5 r e l a t i o n o f P Q M L but are allowed t o ignore the meets and j o i n s of both compatibles and incompatibles (Friedman-Glymour, 1972). However, as shown i n Chapter V I ( B ) , the Friedman-Glymour semantic mappings are i n f a c t b i v a l e n t and t r u t h - f u n c t i o n a l ( i , # ) on substructures of P.... QML which i n c l u d e overlapping mBS's; i n t h i s r e s p e c t , the Friedman-Glymour mappings are e x a c t l y l i k e the quantum state-induced e x p e c t a t i o n - f u n c t i o n s mentioned above. So a weakening of the t r u t h - f u n c t i o n a l i t y ( A , & ) c o n d i t i o n t o j u s t x , 5 p r e s e r v a t i o n nevertheless ensures the p r e s e r v a t i o n of the meets and j o i n s of compatible and incompatible elements i n c e r t a i n substructures o f P - „ T . QML More extremely, the s o - c a l l e d c o n t e x t u a l hidden-variable t h e o r i e s propose b i v a l e n t mappings f o r P^ which are not r e q u i r e d t o preserve even 103 the x operation and the < r e l a t i o n o f P and which avoid both i m p o s s i b i l i t y p r oofs. This proposal i s discussed i n Chapter V I I . Section D. The 'Meaning of the Hidden-Variable I m p o s s i b i l i t y Proofs f o r the  Issue o f a C l a s s i c a l Semantics f o r the Quantum P r o p o s i t i o n a l S t r u c t u r e s In h i s 1957 proof of the completeness o f quantum mechanics, Gleason r e f e r s t o the i n f i n i t e set of a l l subspaces (or p r o j e c t o r s ) o f a three-or-higher dimensional H i l b e r t space, but Gleason does not e x p l i c i t l y s t a t e whether the s t r u c t u r e of such a set i s an orthomodular l a t t i c e or a p a r t i a l - B o o l e a n algebra. As mentioned i n Chapter I V ( E ) , such i n f i n i t e ^ Q M L a n d c o r r e s P o n d i n g i n f i n i t e P Q J ^ s t r u c t u r e s have e x a c t l y the same elements, but the P Q M L has i t s A, V operations defined among compatible and incompatible elements while the P Q M A has i t s A , V operations defined among only compatible elements. Nevertheless, Gleason i s e f f e c t i v e l y ^ committed t o p a r t i a l - B o o l e a n algebra s t r u c t u r e s because the mapping LI which he defin e s on the subspaces must s a t i s f y h i s a d d i t i v i t y c o n d i t i o n : (Ga) For any denumerable.' c o l l e c t i o n ^ i ^ i e i n d e x ° f m u t u a l l y orthogonal subspaces, u.(v P.) = £ |x(P.); f o r example, • X • X 1 1 L L ( P 1 V p 2 ) = L L ( P 1 ) + n(P 2) (Gleason, 1957, p. 885). This a d d i t i v i t y c o n d i t i o n ensures t h a t when Gleason's mappings are d i s p e r s i o n - f r e e , i . e . , b i v a l e n t , then the mappings preserve the unary 8 operation and b i n a r y A , V operations among compatibles. But the mappings do not preserve the A , V operations among incompatibles. In other words, d i s p e r s i o n - f r e e h i dden-variable mappings which s a t i s f y (Ga) are b i v a l e n t 104 homomorphisms(o), and v i c e versa. Viewed s e m a n t i c a l l y , such mappings are b i v a l e n t and t r u t h - f u n c t i o n a l (A). Such mappings preserve the operations and r e l a t i o n s of a P Q J ^ • But such mappings on a P R T M T ignore the l a t t i c e meets and j o i n s of QML incompatibles. So Gleason i s e f f e c t i v e l y r e f e r r i n g t o P Q M a s t r u c t u r e s of subspaces, although h i s r e s u l t s a l s o do apply t o PQ^L s t r u c t u r e s . C l e a r l y , since d i s p e r s i o n - f r e e Gleason mappings ignore the meets and j o i n s of incompatibles, they do not run i n t o the t r u t h - f u n c t i o n a l i t y ( o , & ) problems which are the b a s i s of Theorem A. However, the mappings do run i n t o t r u t h - f u n c t i o n a l i t y ( o ) problems. For a c o r o l l a r y t o Gleason's completeness proof shows that proposed d i s p e r s i o n - f r e e hidden-variable mappings s a t i s f y i n g (Ga) are impossible on the i n f i n i t e set of a l l subspaces of a three-or-higher dimensional H i l b e r t space. This c o r o l l a r y i s known as Gleason's proof of the i m p o s s i b i l i t y of hidden v a r i a b l e s . The Kochen-Specker 1967 Theorem 1, described i n Section B i s a f i n i t e v e r s i o n of Gleason's i m p o s s i b i l i t y proof which makes e x p l i c i t the f a c t that Gleason's proof considers b i v a l e n t homomorphisms(o) on P N M A s t r u c t u r e s (although Gleason's r e s u l t a l s o a p p l i e s t o s t r u c t u r e s ) . Moreover, w i t h t h e i r orthohelium example, Kochen-Specker provide a concrete, 3 quantum mechanical r e a l i z a t i o n of t h e i r f i n i t e (1967, pp. 71-74). Thus Gleason's proof, which r e f e r s t o a l l subspaces or p r o j e c t o r s of a H i l b e r t space, i s protected from c r i t i c s who argue that only some f i n i t e set of operators i n f a c t represent quantum magnitudes or argue t h a t only some " e s s e n t i a l " magnitudes need be assigned values by proposed d i s p e r s i o n - f r e e hidden-variable mappings ( B e l i n f a n t e , 1973, pp. 48-49; B a l l e n t i n e , 1970, p. 376). 1 0 5 In c o n t r a s t t o the Gleason, Kochen-Specker p r o o f s , both the von Neumann and the Jauch-Piron i m p o s s i b i l i t y proofs consider mappings which are r e q u i r e d t o preserve an operation among incompatibles, and both proofs incl u d e the case of two-dimensional H i l b e r t space. In h i s 1 9 3 2 proofs of the completeness of quantum mechanics and the i m p o s s i b i l i t y of d i s p e r s i o n - f r e e hidden-variable ensembles i n quantum mechanics, von Neumann does not e x p l i c i t l y r e f e r t o b i v a l e n t , o p e r a t i o n -p r e s e r v i n g mappings on e i t h e r P Q ^ A O R ^ Q M L s t r u c t u r e s of subspaces or p r o j e c t o r s of H i l b e r t space. Rather, von Neumann considers e x p e c t a t i o n -f u n c t i o n s whose domain i s the ( i n f i n i t e ) set of quantum magnitudes represented by s e l f - a d j o i n t operators on a H i l b e r t space of any dimension, and he r e q u i r e s t h a t e x p e c t a t i o n - f u n c t i o n s preserve the + operation defined among the magnitudes represented by operators. However, d i s p e r s i o n - f r e e e x p e c t a t i o n - f u n c t i o n s which s a t i s f y von Neumann's requirements can be shown to be b i v a l e n t , operation-preserving mappings on quantum p r o p o s i t i o n a l s t r u c t u r e s as f o l l o w s . Consider only the idempotent quantum magnitudes represented by p r o j e c t o r s on a H i l b e r t space, and l e t E x p w he a d i s p e r s i o n - f r e e e x p e c t a t i o n - f u n c t i o n . As described i n Chapter IV(D), the s t r u c t u r e of the p r o j e c t o r s on a H i l b e r t space i s a P ^ M , w i t h the A, v,-1* operations of Q^M d e f i n e d i n terms of the r i n g operations +, *, as usu a l f o r compatible p r o j e c t o r s and by means of Jauch's d e f i n i t i o n s f o r incompatible p r o j e c t o r s . Moreover, w i t h respect t o the idempotent magnitudes represented by p r o j e c t o r s , the d i s p e r s i o n - f r e e c o n d i t i o n w i t h which, von Neumann c h a r a c t e r i z e s h i s hidden-variable E x p w mappings ensures t h a t the mappings are b i v a l e n t , as shown by a simple proof. Thus von Neumann's E*P w mappings are b i v a l e n t 106 mappings on Pn^ s t r u c t u r e s . Furthermore, von Neumann r e q u i r e s any ex p e c t a t i o n - f u n c t i o n Exp t o s a t i s f y h i s a d d i t i v i t y c o n d i t i o n , which may be s p l i t i n t o two p a r t s : (vNi>) For any compatible magnitudes A,B, . . . , Exp(A + B + . . .) = Exp(A) + Exp(B) + . . . (vN&) For any incompatible magnitudes A,B, . . . Exp(A + B + . . .) = Exp(A) + Exp(B) + . . . 10 In p a r t i c u l a r , an Exp must preserve the + operation among compatible and incompatible idempotent magnitudes represented by p r o j e c t o r s . Now l i k e c o n d i t i o n (Ga), the c o n d i t i o n (.vN<t) ensures that the b i v a l e n t Exp w mappings preserve the unary operation and the bin a r y A, V operations among compatible projectors. 1''' Thus the von Neumann Exp mappings are b i v a l e n t and truth-functional ( A ) mappings on PA^ s t r u c t u r e s , as are the Gleason, Kochen-Specker mappings. However, von Neumann's mappings are a l s o r e q u i r e d t o preserve the + operation among incompatibles. So c o n s i d e r i n g j u s t the idempotent magnitudes represented by p r o j e c t o r s , von Neumann i s e f f e c t i v e l y committed t o something l i k e a P^ ,„ s t r u c t u r e as the domain o f QML h i s Exp mappings, because he r e q u i r e s h i s mappings t o preserve a bin a r y w operation among incompatibles. In t h e i r 1963 proof of the i m p o s s i b i l i t y of hidden v a r i a b l e s i n quantum mechanics, Jauch-Piron do e x p l i c i t l y r e f e r t o d i s p e r s i o n - f r e e , i . e . , b i v a l e n t , mappings w on P s t r u c t u r e s . The mappings are r e q u i r e d t o s a t i s f y c e r t a i n c o n d i t i o n s , e s p e c i a l l y : (JPo) For any elements P 1»P 2 € ^ Q M L ' i f P l ^  P 2 t h e n w(P 1) + w(P ) = w(P V P 2) + wCP 1 A P 2 ) . 107 (JPiS) For any subset {P.}.,,. , of elements i n a P„ W T , i f J i i€Index QML w(P.) = 1 f o r every i € Index, then w(A P.) = 1; i f o r example, i f wCP^ = w(P 2) = 1, then w(P^ A P 2> = 1 (Jauch-Piron, 1963, p. 833). Lik e (Ga) and (vNo), the c o n d i t i o n ( J P i ) ensures that the b i v a l e n t mappings preserve the unary -1- operation and the b i n a r y A, v operations among 12 compatibles. So the Jauch-Piron mappings are b i v a l e n t and tr u t h - f u n c t i o n a l ( c S ) . But the mappings are i n a d d i t i o n r e q u i r e d to s a t i s f y (JP<&) which i n v o l v e s p r e s e r v i n g the A operation among compatible and incompatible elements of a • So Gleason, Kochen-Specker, von Neumann, and Jauch-Piron a l l r e q u i r e t h e i r proposed hidden-variable mappings t o be truth-functional(<o), but i n a d d i t i o n , von Neumann and Jauch-Piron r e q u i r e t h e i r mappings t o preserve an operation among incompatibles. And i t i s p r e c i s e l y these a d d i t i o n a l c o n d i t i o n s (vN#>) and (JP#) which a l l o w the von Neumann and the Jauch-Piron proofs t o work at a l l and which a l l o w t h e i r proofs t o in c l u d e the two-dimensional H i l b e r t space case which i s excluded from the Gleason, Kochen-Specker proofs. S p e c i f i c a l l y , u s i n g the trace-formalism developed i n h i s completeness proof, von Neumann shows th a t d i s p e r s i o n - f r e e e x p e c t a t i o n - f u n c t i o n s which s a t i s f y h i s c o n d i t i o n s are impossible on the ( i n f i n i t e ) set of operators on a H i l b e r t space of any dimension (von Neumann, 1932, pp. 320-321). And Jauch-Piron prove i n t h e i r C o r o l l a r y 1 t h a t , w i t h respect to any i r r e d u c i b l e 13 ^QML ' b i v a l e n t mappings which s a t i s f y t h e i r c o n d i t i o n s are impossible. Semantically i n t e r p r e t e d , since the truth-functionality(<J>,cH) c o n d i t i o n i s even stronger than the c o n d i t i o n s imposed by e i t h e r von Neumann or Jauch-Piron, t h e i r i m p o s s i b i l i t y proofs suggest that i n general and 108 i n c l u d i n g the two-dimensional H i l b e r t space case, quantum s t r u c t u r e s do not admit b i v a l e n t , truth-functional(c-,&) mappings and hence do not admit a b i v a l e n t , t r u t h - f u n c t i o n a l ( i > , ^ ) semantics; t h i s i s proven i n Secti o n A as Theorem A. There i s an i m p o s s i b i l i t y proof by Z i e r l e r and Schlessinger i n v o l v i n g a c o n d i t i o n which i s as strong as my tr u t h - f u n c t i o n a l i t y ( c b , & ) 14 c o n d i t i o n . In t h e i r Theorem 3.1, Z i e r l e r - S c h l e s s i n g e r show that i f there i s a s t r o n g l y a d d i t i v e embedding m of an orthomodular p a r t i a l l y ordered set P i n t o a Boolean alg e b r a , then the j o i n P^ V P^ e x i s t s i n P only when P^ commutes w i t h P^ ( i . e . , P^ p 2 ) ' A s t r o n g l y a d d i t i v e embedding preserves 5,J" , V , and moreover i s monomorphic, i . e . , i f m(P^) < nKP^) then P < ? 2 ( Z i e r l e r - S c h l e s s i n g e r , 1964, pp. 254-255, 260). I t i s easy to prove t h a t a monomorphic mapping m : P -»• 8 which preserves < i s i n j e c t i v e , i . e . , f o r any P^ t P^ i n P, m(P^) ^  mtP^). Proof: Assume on the co n t r a r y t h a t m(P^) = m(P 2) i n 8. Then si n c e < i s r e f l e x i v e , m(P„ ) < m(P.) and a l s o m(P^) <m(P„). Since m i s 1 2 2 1 monomorphic, i t f o l l o w s that P^ 5 P 2 and p 2 - P i ' t n u s P i = p 2 w n i ° n c o n t r a d i c t s P^ f ?2 . Q.E.D. And si n c e an imbedding i s an i n j e c t i v e homomorphism, i t f o l l o w s t h a t a s t r o n g l y a d d i t i v e embedding of P i n t o a 8 i s i n f a c t an imbedding (A,<&) of P i n t o B. So the c o n t r a p o s i t i v e of Theorem 3.1 says: I f the j o i n P„ V P„ e x i s t s i n P and P„ & P„ , then 1 2 1 2 an imbedding of P i n t o a 8 i s impossible. Or i n other words, with respect to an orthomodular l a t t i c e P_„T , which has V defined f o r QML any P. ,P„ € Prt,,_ , Theorem 3.1 y i e l d s : I f a P.,,. contains incompatible 1 2 QML QML elements, t h a t i s , i f the j o i n P^ v P 2 of incompatible P^ , P 2 e x i s t s i n Prt„_ , then an imbedding (o,&) of P_„T i n t o a B i s impossible. QML QML 109 Then assuming t h a t there i s a theorem f o r imbedding(6,<&) l i k e the Kochen-Specker Theorem 0 f o r imbeddingCt), i . e . , an imbedding(i,&) of a ^ Q M L ^ n t o a ^ e x i s t s IFF a complete c o l l e c t i o n of b i v a l e n t homomorphisms(<^,&) e x i s t s on P„.„ , the above restatement o f the c o n t r a p o s i t i v e of Q M L Z i e r l e r - S c h l e s s i n g e r ' s Theorem 3.1 i s equivalent to my Theorem A: I f a ^ Q M L C O T V t a i n s incompatible elements, then a b i v a l e n t , truth-functional(o=,&) semantics f o r P . W T i s impossible. Q M L Summary The general f a c t of the i m p o s s i b i l i t y of a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r the p r o p o s i t i o n a l s t r u c t u r e s determined by quantum mechanics should be more s u b t l y demarcated according to whether the s t r u c t u r e s are taken to be orthomodular l a t t i c e s P . „ T or Q M L p a r t i a l - B o o l e a n algebras P Q ^ a ' according to whether the semantic mappings are r e q u i r e d to be truth-functional(i>,<&) or t r u t h - f u n c t i o n a l ( i > ) ; and according to whether two-or-higher dimensional H i l b e r t space P s t r u c t u r e s or three-or-higher dimensional H i l b e r t space P ^ s t r u c t u r e s are being c o n s i d e r e d . X ^ I f the quantum P s t r u c t u r e s are taken t o be orthomodular l a t t i c e s , then b i v a l e n t mappings which preserve the operations and r e l a t i o n s o f a P A must be truth-functional(o,<i$). Then as suggested by von Neumann QML and Jauch-Piron and as proven i n Secti o n A, the mere presence of incompatible elements i n a P^„ T i s s u f f i c i e n t to r u l e out any semantical or hidden-Q M L v a r i a b l e proposal which imposes t h i s strong c o n d i t i o n , f o r any two-or-higher 2 dimensional H i l b e r t space P ~_ s t r u c t u r e . Thus from the orthomodular , Q M L l a t t i c e p e r s p e c t i v e , the p e c u l i a r l y n o n - c l a s s i c a l f e a t u r e o f quantum mechanics 110 and the p e c u l i a r l y non-Boolean f e a t u r e of the quantum p r o p o s i t i o n a l s t r u c t u r e s i s the existence of incompatible magnitudes and p r o p o s i t i o n s . However, the weaker t r u t h - f u n c t i o n a l i t y ( c ? ) c o n d i t i o n can i n s t e a d be imposed upon the semantic or hidden-variable mappings on the ^Q^L s t r u c t u r e s , although such mappings ignore the l a t t i c e meets and j o i n s o f incompatibles and preserve only the p a r t i a l - B o o l e a n algebra s t r u c t u r a l f e a t u r e s of the ^Q^L s t r u c t u r e s . 0r a l t e r n a t i v e l y , the quantum p r o p o s i t i o n a l s t r u c t u r e s can be taken to be p a r t i a l - B o o l e a n algebras, where b i v a l e n t mappings which preserve the operations and r e l a t i o n s of a need only be t r u t h - f u n c t i o n a l ( A ) . In e i t h e r case, the Gleason, Kochen-Specker proofs show th a t any semantical or hidden v a r i a b l e proposal which imposes t h i s t r u t h - f u n c t i o n a l i t y ( o ) c o n d i t i o n i s impossible f o r any three-or-higher dimensional H i l b e r t space P?~« or P ^ r s t r u c t u r e s . But such semantical R QMA QML or hidden-variable proposals are p o s s i b l e f o r any two-dimensional H i l b e r t 2 2 space P Q M A or PQ M I j s t r u c t u r e s , i n s p i t e o f the presence of incompatibles, and i n s p i t e of the f a c t t h a t these s t r u c t u r e s are non-Boolean i n the P i r o n 16 sense and i n the not-prime u l t r a f i l t e r sense. Notes 1 I t i s worth n o t i n g that these value assignments would be acceptable to the l a t t i c e t h e o r e t i c i a n s Jauch (1968, p. 76), Putnam (1969, p. 222), van Fraassen (1973, p. 90), Friedman and Glymour (1972, p. 18). For these authors do a s s o c i a t e the 0 element of a P w i t h c o n t r a d i c t o r y p r o p o s i t i o n s and the 1 element w i t h t a u t o l o g i c a l p r o p o s i t i o n s . So even though some of these authors do not d i s c u s s semantic proposals f o r P , a l l would accept the value assignments m(P A P ) = m(0) = 0 and QML 1 2 m(P V P ) = m(l) = 1, f o r any proposed semantic mapping m on a P . 1 2 QML For example, Putnam e x p l i c i t l y discusses the conjunction of two quantum p r o p o s i t i o n s a s s o c i a t e d w i t h two incompatible, one-dimensional subspaces I l l whose i n t e r s e c t i s the 0 subspace, e.g., our A = 0 and P^ A P^ = 0. Such a conjunction i s l o g i c a l l y f a l s e , according to Putnam, and so he i s committed to the value assignments m(P A P ) = 0 and m(P A p^) = 0. 2 Thanks to Edwin Levy, L. Peter B e l l u c e , and Richard E. Robinson f o r a u d i t i n g these proofs. Dr. B e l l u c e e s p e c i a l l y helped w i t h Lemmas A and B, and he proved Lemma C, adding t h a t i t i s a standard proof i n Boolean l a t t i c e theory. Dr.. Robinson suggested a more economical restatement of the proofs. 3 This i m p o s s i b i l i t y holds whether a semantics f o r a P i s taken to be a complete c o l l e c t i o n or a weakly complete c o l l e c t i o n o f b i v a l e n t , t r u t h - f u n c t i o n a l ( i ) mappings. The n o t i o n of a weakly complete c o l l e c t i o n i s defined i n note 1 of Chapter I I . The dimension of a H i l b e r t space H i s the maximum number of l i n e a r l y independent v e c t o r s i n the H i l b e r t space (Jauch, 1968, p. 20), and i s designated by the s u p e r s c r i p t n = 1,2 So any tf0 has n l i n e a r l y independent v e c t o r s ; t h i s set of v e c t o r s are a b a s i s f o r tfn (Lande, 1972, p. 47). A b a s i s may be orthogonalized (by the Gram-Schmidt process) and normalized (by d i v i d i n g each ve c t o r by i t s length) y i e l d i n g an orthonormal b a s i s . Thus the maximum number of mutually orthogonal v e c t o r s i n any H n i s n. Since each ve c t o r |y> corresponds uniquely w i t h the one-dimensional p r o j e c t o r P^ = | Y > < Y | and the one-dimensional subspace H^ which i s the range of P^ , the maximum number of mutually orthogonal, one-dimensional p r o j e c t o r s or subspaces of any H n i s n. .Each one-dimensional p r o j e c t o r or subspace i s an atom i n the P s t r u c t u r e of the H i l b e r t space, and by Lemma A, any two atoms i n a a r e orthogonal IFF they are compatible. Thus the maximum number o f mutually compatible atoms i n the P*V s t r u c t u r e of any f f n i s n. . • 0M And so I cl a i m without proof t h a t when a set of n mutually compatible or orthogonal atoms i n a P ^ i s c l o s e d w i t h respect to the A, v, "L" operations of P_„ we ob t a i n an mBS of P*\, . QM QM 5 3 The three orthogonal atoms i n each mBS of the PnMk which Kochen-Specker consider i n t h e i r Theorem 1 are represented by a t r i a n g l e i n the completion of the graph Kochen-Specker l a b e l V (Kochen-Specker, 1967, pp'. 68-69). Each i d e n t i c a l subportion of t h i s graph, which Kochen-Specker draw s e p a r a t e l y as I" , contains 13 p o i n t s (atoms) and e i g h t overlapping t r i a n g l e s (mBS's) upon completion. There are 15 such subportions i n r , so the completion of r contains 195 p o i n t s and 120 t r i a n g l e s . However, Kochen-Specker f u r t h e r i d e n t i f y the p o i n t s p Q = a, q = b, and r = c, so t h a t three p o i n t s and two t r i a n g l e s are redundant, o 2 o Thus the considered by Kochen-Specker contains 192 atoms and 118 mBS's. 112 This i m p o s s i b i l i t y holds whether a semantics f o r a P i s taken to be a complete or a weakly complete c o l l e c t i o n o f b i v a l e n t , t r u t h - f u n c t i o n a l ( i > ) mappings. 7 In p r e c i s e l y t h i s manner, the semantic mappings proposed by I. Hacking f o r the quantum P N M T s t r u c t u r e s , namely, the e v a l u a t i o n s , side-step the Theorem A i m p o s s i b i l i t y proof. This proposal was made i n an unpublished, 1974 paper which has s i n c e been rescinded. g A d i s p e r s i o n - f r e e Gleason mapping i s b i v a l e n t , as mentioned by Gudder; a proof i s given by Bub and r e s t a t e d i n note 9 below. And Gudder proves t h a t d i s p e r s i o n - f r e e mappings s a t i s f y i n g Gleason's a d d i t i v i t y c o n d i t i o n are b i v a l e n t homomorphisms(6) (Gudder, 1970, pp. 433-434). A v e r s i o n of t h i s proof i s r e s t a t e d i n Chapter I I I ( C ) . 9 By d e f i n i t i o n , an e x p e c t a t i o n - f u n c t i o n i s d i s p e r s i o n - f r e e IFF, f o r any quantum mechanical magnitude A, Exp(A 2) = ( E x p ( A ) ) 2 . So w i t h respect to any idempotent magnitude P which by d e f i n i t i o n s a t i s f i e s P = P, Exp(P) = (Exp(P)) . That i s , Exp(P) = 0 or 1. So an Exp on a P d i s p e r s i o n - f r e e IFF, f o r any element P € P , Exp(P) = 0 or 1 ^ (Bub, 1974, p. 60). Q M X ^ Condi t i o n (vNo) alone i s l a b e l e d (D) by von Neumann i n h i s book. And c o n d i t i o n s (vN<b), (vN#) together are subsumed by one c o n d i t i o n von Neumann l a b e l s (B') (von Neumann, 1932, pp. 309, 311). X X Kochen-Specker prove t h a t d i s p e r s i o n - f r e e e x p e c t a t i o n - f u n c t i o n s which preserve the + ope r a t i o n among compatible operators or p r o j e c t o r s a l s o preserve the • operation among compatibles (Kochen-Specker, 1967, p. 81). Since the A, v,-1- operations o f a P s t r u c t u r e can be defined i n terms o f the r i n g operations +, •, as usu a l among compatible p r o j e c t o r s , mappings on a P which preserve the +, • operations among compatibles a l s o preserve the A, V, x operations among compatibles. 12 The proof by Gudder c i t e d i n note 8 above works w i t h c o n d i t i o n (JPo) as w e l l as w i t h c o n d i t i o n (Ga). 13 Jauch-Piron's C o r o l l a r y 1 speaks of coherent p r o p o s i t i o n systems; coherency i s i r r e d u c i b i l i t y and a p r o p o s i t i o n system i s an orthomodular l a t t i c e (Jauch-Piron, 1963, pp. 831, 834). 14 Z i e r l e r and Sch l e s s i n g e r ' s work was c a l l e d to my a t t e n t i o n by Prof. W. Demopoulos. 15 Some authors do not make these d i s t i n c t i o n s . For example, M. Gardner claims t h a t , "Kochen and Specker have proven t h a t there i s no homomorphism of P [ i . e . , P ^ ^ ] i n t o !' (Gardner, 1971, p. 519)«. Gardner does c l a r i f y that the homomorphisms considered by Kochen-Specker are homomorphisms(i). But Gardner does not mention t h a t two-dimensional ' ' 2 H i l b e r t space PQ^a s t r u c t u r e s are exempt from Kochen-Specker's proof; 113 that Kochen-Specker give an example of a P q ^ which does admit some homomorphisms ( i ) i n t o Z^ ( i . e . , b i v a l e n t homomorphisms(o)) but does not admit a complete c o l l e c t i o n o f such mappings; and t h a t the Kochen-Specker proof a l s o a p p l i e s to P s t r u c t u r e s , as explained i n Sections B and D. QML 16 Most of t h i s chapter i s t o be pu b l i s h e d as "The i m p o s s i b i l i t y o f a c l a s s i c a l semantics f o r the quantum p r o p o s i t i o n a l s t r u c t u r e s , " i n a •forthcoming i s s u e of P h i l o s o p h i a . 114 CHAPTER VI A STATE-INDUCED SEMANTICS FOR THE NON-BOOLEAN PROPOSITIONAL STRUCTURES DETERMINED BY QUANTUM MECHANICS Sec t i o n A. The Quantum State-induced Expectation-Functions As described i n Chapter IV(A), the quantum formalism a s s o c i a t e s a p h y s i c a l system w i t h a H i l b e r t space H, represents each pure s t a t e \|r o f the system by a one-dimensional p r o j e c t o r on H, and represents each magnitude A o f the system by a s e l f - a d j o i n t operator on H. The expectation value o f each o f the system's magnitudes A,B, , . . i s determined by the s t a t e ^ o f the system according to the expression Exp^.(A) = < ^ | A | \ ) / > , which i s one o f the real-number eigenvalues ^ ' i ^ i ^ index °^ A w n e n the s t a t e o f the system i s one of the eigenstates ^ j ^ i ^ i n d e x °^ A ' e*S'> Exp. (A) = a. . Thus, when the s t a t e o f a system i s an eigenstate of any of j : the system's magnitudes, then the s t a t e o f the system determines the exact values o f those magnitudes v i a the e x p e c t a t i o n - f u n c t i o n . When the s t a t e \|f of the system i s not an eigenstate of a given magnitude A, then the s t a t e determines the p r o b a b i l i t i e s o f t h a t magnitude A e x h i b i t i n g any one o f i t s eigenvalues according to the expression p. & ( a . ) = Exp(P. ), where Y »A 1 • P. i s the p r o j e c t o r r e p r e s e n t i n g the eigenstate a s s o c i a t e d w i t h the v i 1 . eigenvalue a^ . So f o r any o f the system's magnitudes, each pure s t a t e ^ of the system determines, v i a the e x p e c t a t i o n - f u n c t i o n Exp^. , e i t h e r the 115 exact real-number value o f the magnitude or the p r o b a b i l i t i e s o f the magnitude e x h i b i t i n g any one of i t s exact (eigen)values. And f o r any pure s t a t e the e x p e c t a t i o n - f u n c t i o n Exp^, i s unique to \|/, and conversely, Exp^, unambiguously defi n e s the s t a t e \|/ (Fano, 1971, p. 399). In f a c t , f o r any pure s t a t e the e x p e c t a t i o n - f u n c t i o n Exp^. ac t s as a mapping from the set of magnitudes represented by operators to the real-number l i n e , i . e . , Exp^. : {A,B, . . .} ->• R, which s a t i s f i e s : (Ea) Exp^(A + B + . . .) = Exp^.(A) + Exp^(B) + . . . (Eb) I f A > 6 then Exp^(AY>0. (Ec) Exp^(I) = 1. (Fano, 1971, p. 398; von Neumann, 1932, p. 308) For any pure s t a t e \|/, the uniquely a s s o c i a t e d mapping .,", T Exp^, : {A,B, . . .} -»• R may be c a l l e d the quantum state-induced mapping, j u s t as, f o r any pure s t a t e w of a c l a s s i c a l system, the uniquely a s s o c i a t e d mapping w : {f , f g , ...}-»- R i s c a l l e d the state-induced mapping i n Chapter I I I . As w i l l be shown i n t h i s s e c t i o n , c o n d i t i o n s ( E a ) , (Eb), ( E c ) , ensure t h a t , w i t h respect to the idempotent elements o f {A,B, . . .} which form a P ^ s t r u c t u r e , each Exp^ i s a p r o b a b i l i t y measure Exp^. : PQ^ [0,1]. C l a s s i c a l l y , the analogous r e s u l t i s t h a t each c l a s s i c a l state-induced mapping w :{f , f , . . . } - » • R i s a d i s p e r s i o n - f r e e p r o b a b i l i t y measure u.^  : P ^ -»• {0,1} w i t h respect to the P^ M s t r u c t u r e of idempotent elements of ^ A ' ^ B ' * ' Moreover, as i n the c l a s s i c a l case described i n Chapter I I I ( C ) , t h i s mathematical machinery of quantum state-induced mappings on the set of 116 operators on a H i l b e r t space not only f o r m a l i z e s the procedure by which real-number values and p r o b a b i l i t i e s are assigned to the magnitudes o f a quantum system, but a l s o i m p l i c i t l y f o r m a l i z e s a procedure by which t r u t h - v a l u e s and p r o b a b i l i t i e s may be assigned to the p r o p o s i t i o n s which make a s s e r t i o n s about the real-number values o f a quantum system's magnitudes, as s h a l l be shown i n t h i s chapter. As described i n Chapter IV(C), p r o p o s i t i o n s which make a s s e r t i o n s about the values o f a quantum system's magnitudes have been a s s o c i a t e d w i t h the p r o j e c t o r s or subspaces of the system's H i l b e r t space, and the l o g i c a l operations among the p r o p o s i t i o n s have been i n t e r p r e t e d as or defined i n terms o f operations among the p r o j e c t o r s or subspaces, y i e l d i n g a p r o p o s i t i o n a l s t r u c t u r e P.,. . In order to describe how the state-induced QM mappings Exp^, act w i t h respect to a P M^ , we focus t e m p o r a r i l y upon the elements of Pn^ as p r o j e c t o r s , which are by d e f i n i t i o n idempotent, s e l f - a d j o i n t , bounded operators whose o n l y eigenvalues are the real-numbers 0 and 1. With respect to a Pn]^  of p r o p o s i t i o n s qua p r o j e c t o r s , each state-induced Exp, on P_„ s a t i s f i e s the f i v e c o n d i t i o n s which d e f i n e a *Y QM p r o b a b i l i t y measure U-, as l i s t e d i n Chapter I I I ( C ) . For any Exp^, on a P Q M and f o r any € P Q M : (ua) As s t a t e d i n Chapter IV(D'), i f P^ 6 P*2 , then P 1 V P = P. + f - P • P 0 and P. A P = P • P . And i f 1 2 1 2 1 2 1 2 1 2 P^ , P*2 are d i s j o i n t , i . e . , P^ < P^ , then P 6 P and P. A P = 6 ( s i n c e P. A P < P„ A P^ = fl and P > 0 f o r every 1 2 1 2 2 2 J P € P_„). So i f P. , P are d i s j o i n t , then P V P = P + P . QM 1 2 J 1 2 1 2 Thus by (Ea), f o r any d i s j o i n t P^ , P 2 , Exp^ P ^ v P 2> = Exp^(P 1 + P 2) = Exp^CP^) + Exp^(P 2>. And f o r any countable set 117 {P. }." , of p a i r w i s e d i s j o i n t elements o f P_„ , 1 i€Index QM Exp,(V P.) = 2 Exp. (P.). Thus ( LQ) i s s a t i s f i e d , i i (Mb) Every element P € P i s by d e f i n i t i o n idempotent, i . e . , A **2 P = P ; i t f o l l o w s t h a t every element i s nonnegative, i . e . , P > 0 (von Neumann, 1932, p. 308). Hence by (Eb), f o r every P € , Exp^,(P) > 0. Moreover, s i n c e a p r o j e c t o r i s by A d e f i n i t i o n a bounded operator (Fano, 1971, p. 288), Exp^(P) 5 °°, A f o r every P € P^ (Fano, 1971, p. 396). So we have: 0 5 Exp^(P) < «», f o r every element P € P Q M . Thus (ub) i s s a t i s f i e d . (uc) By (Eb), i f P = 0 then Exp^(£) = 0, i . e . , Exp^(0) = 0. Thus Cue) i s s a t i s f i e d . ([il) In Halmos's d i s c u s s i o n o f the p r o b a b i l i t y measure U-, he says th a t the isotone character of u. f o l l o w s from the mon-riegative character of \i (Halmos, 1950, p. 37). This r e s u l t f o r any Exp^. on a P i s shown as f o l l o w s . For any P ^ » P 2 ^ P Q M ' Exp^(P 1) < Exp^(P 1) + Exp^(P 2 A P A ) because Exp^(P) > 0 f o r any P € P^ M , i n p a r t i c u l a r , f o r P = P 2 A P^X. And as s t a t e d i n Chapter IV(D), i f P± < P 2 , then $ o P 2 (where P A P 2 IFF P x d> £ 0) and P. A P = P ; and so by the mutual 1 2 1 2 1 A A A A c o m p a t i b i l i t y o f P^  , P 2 , P and by the d e f i n i t i o n o f I , ^ A A A A I A A A A I P 2 = P 2 A I = P 2 A (P 1 V P 1 ) = (P 2 A P 1) V (P 2 A P p = P V (P 2 A P ^ ) . So i f P± < P 2 , then Exp^(P 2) = Exp^CP^ V ( ? 2 A P ^ ) ) . Moreover, s i n c e I>2 A P X < p 1 , P^  and P 2 A P^  are d i s j o i n t , and so by (ua): Exp^(P^ V (P 2 A P^  )) = ExP^C^) + E xP^(P 2 A P 1 X ) * H E N C E ' F O R A N V € P Q M ' L F 118 V± 5 P 2 then E x p ^ ) 5 E x p ^ ) + Exp^(P 2 A P ^ ) = E x p ^ ) . So Exp^ i s an isotone mapping, t h a t i s , (|ii) i s s a t i s f i e d . (M-n) By ( E c ) , Exp^(I) = 1, thus ( u i ) i s s a t i s f i e d . And so f o r every P € P Q M , 0 < Exp^CP) < 1, t h a t i s , Exp^ : P Q M - [0,1]. So c o n d i t i o n s (Ea), (Eb), ( E c ) , ensure t h a t an Exp. on a P.„ s a t i s f i e s y QM the f i v e c o n d i t i o n s which d e f i n e a p r o b a b i l i t y measure u-. However, t h i s c l a s s i c a l p r o b a b i l i t y measure u. i s defined on a Boolean s t r u c t u r e , e.g., on a P N ^ , wh i l e the quantum Exp^ i s defined on a non-Boolean P N ^ s t r u c t u r e . So the quantum e x p e c t a t i o n - f u n c t i o n s on a P A ^ can be regarded as ge n e r a l i z e d p r o b a b i l i t y measures which s a t i s f y a l l the usual d e f i n i n g c o n d i t i o n s o f a c l a s s i c a l p r o b a b i l i t y measure but which are defined on a non-Boolean P .„ s t r u c t u r e r a t h e r than on a Boolean QM s t r u c t u r e . The n o t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure on a PQJ^ "*"S defined by Bub, and a d i f f e r e n t n o t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure on a P N M L i s defined by Jauch-Piron (Bub, 1974, p. 89; Jauch-Piron, 1963, p. 833). Bub and Jauch-Piron agree t h a t the c l a s s i c a l n o t i o n o f a p r o b a b i l i t y measure on a Boolean s t r u c t u r e must be g e n e r a l i z e d f o r PQ^A » PQ M L , i n such a way t h a t on every (maximal) Boolean substructure o f P A M A , *QML ' t h e § e n e r a l : J - z e d p r o b a b i l i t y measure reduces to the c l a s s i c a l p r o b a b i l i t y measure p.. In a d d i t i o n , w i t h respect to the e n t i r e non-Boolean P . „ . , P . „ T , both Bub and Jauch-Piron r e q u i r e t h a t a QMA QML ^ gen e r a l i z e d p r o b a b i l i t y measure be a d d i t i v e w i t h respect to orthogonal elements; t h i s a d d i t i v i t y c o n d i t i o n i s Gleason's (Ga) s t a t e d i n Chapter V(D) Bub does not s t a t e t h i s requirement e x p l i c i t l y , but i t i s c l e a r that he wants a g e n e r a l i z e d p r o b a b i l i t y measure t o s a t i s f y (Ga). In t h e i r 1963 119 paper, Jauch-Piron do e x p l i c i t l y r e q u i r e t h e i r g e n e r a l i z e d p r o b a b i l i t y measures to s a t i s f y an a d d i t i v i t y c o n d i t i o n which amounts to (Ga), namely, the c o n d i t i o n (JPo) s t a t e d i n Chapter V(D). And elsewhere, Jauch e x p l i c i t l y imposes (Ga) r a t h e r than (JPo) (1976, p. 135). Any quantum Exp^, on a PQM does s a t i s f y (Ga). For as shown above, any Exp^ on a P^ M s a t i s f i e s (pa) which i s equ i v a l e n t to (Ga) s i n c e d i s j o i n t e d n e s s and o r t h o g o n a l i t y are equivalent n o t i o n s , as s t a t e d i n Chapter IV(D). Besides (Ga), Jauch-Piron r e q u i r e t h e i r g e n e r a l i z e d p r o b a b i l i t y measures on a PQ^L t o s a t i S : f y t n e c o n d i t i o n (JP&) s t a t e d i n Chapter V(D), and Jauch-Piron c l a i m t h a t the quantum Exp^. mappings do s a t i s f y (JPi*) (1963, p. 833). Bub does not impose t h i s c o n d i t i o n . The n o t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure i s f u r t h e r discussed i n Chapter V I I ; i n p a r t i c u l a r , Jauch-Piron's i m p o s i t i o n of (JP{6) as p a r t o f the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure i s c r i t i c i z e d . Nevertheless, each state-induced mapping Exp^. : P^ -*• [0,1] i s a g e n e r a l i z e d p r o b a b i l i t y measure (as defined by e i t h e r Bub or Jauch-Piron) on PQM , j u s t as each c l a s s i c a l state-induced mapping w : -*• {0,1} i s a c l a s s i c a l p r o b a b i l i t y measure u. : P „ „ -*• {0,1}, as discussed i n w CM Chapter I I I ( C ) . But the c l a s s i c a l measures [i are d i s p e r s i o n - f r e e , where w a d i s p e r s i o n - f r e e measure s a t i s f i e s the c o n d i t i o n (u.d) which ensures b i v a l e n c y , w h i l e the quantum measures Exp^ a s s i g n d i s p e r s i v e , p r o b a b i l i t y values between 0 and 1 to some elements o f Pn., . Moreover, u n l i k e the QM c l a s s i c a l measures which are t r u t h - f u n c t i o n a l mappings on P^ M , the quantum Exp^ measures are not t r u t h - f u n c t i o n a l ( ( i ) or (o,&)) mappings on P^ M . Conditions (Ea) and (Ec) do ensure th a t the quantum measures preserve the -1 operation of P^ , i . e . , f o r any Exp^ on a P^ and f o r any P € P^ M , ExpCP 1) = , by s u b s t i t u t i o n , E x p ( l - P ) = , by (Ea), Exp(I) - Exp(P) = , 120 by (Ec) and s u b s t i t u t i o n , l - E x p ( P ) =, by d e f i n i t i o n o f , (Exp(P)) . But the quantum measures do not always preserve the A, v operations o f PQM . For example, consider the p r o j e c t o r s P^ = |y^><Y^I a n d P 2 = |^2><-^2 j such t h a t P^ cV ? 2 and P^ A P*2 = 0; and consider the pure s t a t e Y represented by the p r o j e c t o r P^ such t h a t P^ , 9 P^ and P 2 • The Exp^ induced by t h i s s t a t e Y assigns values as f o l l o w s : Exp. (P. A P 0) = Exp, (6) = 0, but Exp,(p\) A E x p , ( P ) t 0 because, f o r f 1 2 Y \|f 1 Y 2 each i = 1,2, Exp^(p\) = j l ^ x ^ l l ' 2 t 0. However, each quantum e x p e c t a t i o n - f u n c t i o n Exp^ : P^ -*• [0,1] induced by the pure s t a t e \|r o f a quantum system i s b i v a l e n t w i t h respect to a c e r t a i n subset o f elements o f the P n M s t r u c t u r e o f the system's H, namely, the subset o f elements i n P^ which are compatible w i t h the atom P^ , which qua p r o j e c t o r P^ = |Y > < :Y1 represents the pure s t a t e \|r which induced Exp^ . In p a r t i c u l a r , according to the quantum formalism, f o r any E X P Y ° N 3 P Q M ' E X P Y ( V = I ' ^ ' i 2 = X ' ^ } = ± - E X h i % ) = ° ' In a d d i t i o n , f o r any element P € P Q M , i f P & P^ , then Exp^(P) € (0,1): if P > P^ , then by ( n i ) , Exp^(P) = 1; and i f P < P X , then by (ui), Exp^(P) = 0. Rewritten: For any Exp^ on a P A M and f o r any element P { V ' E X P Y ( P ) = < 1 i f P > P, Y 0 i f P < Pf" Y £(0,1) i f P fa P, So each quantum e x p e c t a t i o n - f u n c t i o n Exp^ : PA^ -»• [0,1] i s b i v a l e n t w i t h respect to the subset {P € P^ : P > P^ or P < P^ } ; by Lemma A o f Chapter V(A), t h i s i s the subset o f elements i n PA^ which are compatible w i t h P^ . And each quantum Exp^, assigns p r o b a b i l i t y values between 0 and 1 to all other elements i n P_„ , i.e., to all elements i n P_„ Q M Q M which are incompatible w i t h P^ , the atom which qua p r o j e c t o r P^ 121 represents the s t a t e ty which induced Exp^, . In the next s e c t i o n , I s h a l l show t h a t , f o r any atom P. € P^„ , J ty QM the subset {P € P : P > P. or P 5 P \" } i s a cl o s e d substructure o f QM ty ty Q^M ' a n d t' i e ^ P 6 0 ^ 3 ^ 1 0 1 1 1 - ^ 1 1 1 1 0 1 1 ^ 0 1 1 E x P ^ -*-s n o t o n l y b i v a l e n t but a l s o t r u t h - f u n c t i o n a l ((A) or (o,&)) on t h a t substructure o f P^„ . QM Sect i o n B. The Quantum Expectation-Function As an U l t r a v a l u a t i o n on an U l t r a s u b s t r u c t u r e o f P_„ QM As described i n Chapter 1 ( F ) , the notions o f a f i l t e r and dual i d e a l are defined i n a ?..„ by the c o n d i t i o n s ( a ) , ( b ) , and the dual QML co n d i t i o n s ( a ' ) , ( b ' ) , l i s t e d i n Chapter 1(C). To d e f i n e these notions i n a PQJ^ which has the A, V operations defined among only compatibles, c o n d i t i o n s (a) and (a') are modified to the c o n d i t i o n s (a„) and (a') given H n i n Chapter 1 ( F ) ; n e v e r t h e l e s s , any f i l t e r i n a s t i l l s a t i s f i e s the unmodified c o n d i t i o n s (a) and ( b ) , and any i d e a l i n a P_„, s t i l l s a t i s f i e s J QMA (a') and ( b ' ) . As i n the case of a Boolean s t r u c t u r e , an u l t r a f i l t e r ( u l t r a i d e a l ) i n a quantum P i s a proper f i l t e r ( i d e a l ) which i s not the proper subset o f any proper f i l t e r ( i d e a l ) i n P^ . Making use o f Lemma B of Chapter V(A), i t i s easy to prove t h a t the subset o f elements {P € P^^ '• P - P^}, f o r any given atom P^ € P ^ M L i s an u l t r a f i l t e r UF^ i n P Q M L . For any P ^ € P Q M L , i f P± , ?2 are members of the set S = {P € : P > P^}, i . e . , P± > P^ , and P 2 > P^ , then P 1 A P 2 > P^ A P 2 = P^ and so P^ A P 2 € S; thus S s a t i s f i e s ( a ) . For any P 1 € P ^ and f o r any P 2 € S ( i . e . , P 2 > P^), i f P± - P 2 > then s i n c e P 2 > P^ we have P± > P^ , and so P_L 6 S; thus S s a t i s f i e s ( b ) . So S i s a f i l t e r i n P „ „ T . Moreover, S i s a QML proper f i l t e r , t h a t i s , S 4 P Q M L , e.g., 0 J? S s i n c e 0 Z P^ . And 122 f i n a l l y , S i s not the proper subset o f any proper f i l t e r i n P Q ^ • P O R assume on the contrary t h a t there i s a proper f i l t e r F i n PQ^L such t h a t S c F. Then there i s an element P € P R T „ T such t h a t P € F but P A S, QML i . e . , P £ P^ . Since P^ € S c F, both P^ , P are members of F and so by ( a ) , P A P^ € F. But sin c e P t P^ , by Lemma B, P A P^ = 0. Thus, 0 € F, and so by ( b ) , F = P A M T , which c o n t r a d i c t s the assumption QML that F i s a proper f i l t e r i n P_„_ . Q.E.D. QML The proof t h a t the subset o f elements {P € P . „ T : P 5 p f } i s an QML ty u l t r a i d e a l U I ^ i n PQ M L proceeds d u a l l y . S i m i l a r l y , the subset o f elements {P € PQJ^ P - P ^ > ^ o r a n v given atom P, € P_.,.A , i s an u l t r a f i l t e r UF. i n P„... , as shown next. s y QMA \|r QMA ' For any Pj_>P2 ^ PQMA ' ^ P l ' P2 a r e m e m h e r s °f the set S = {P € P „„ . : P > P.}, i . e . , P > P. and P„ > P, , then there i s a QMA ty 1 ty 2 ty d € S such t h a t d < and d 5 , namely, d = P^; thus S s a t i s f i e s ( a [ { ) . For any P 1 € P Q M and f o r any P 2 € S ( i . e . , P 2 > P^), i f P l ~ ? 2 ' t h e n s i n c e P 2 ~ P\|/ w e n a v e P l ~ ?ty ' a n d s o P l € S ; t h u s S s a t i s f i e s ( b ) . So S i s a f i l t e r i n Pn..A • Moreover, S i s a proper QMA f i l t e r , t h a t i s , S 4 P Q M , e.g., 0 £ S s i n c e 0 Z• P' . And f i n a l l y , S i s not the proper subset o f any proper f i l t e r i n PQJJa • P ° r assume on the contrary t h a t there i s a proper f i l t e r F i n P . such t h a t S c F. Then there i s an element P € P^m such t h a t P € F but P £ S, i . e . , P Z P^ . Since P^ , € S c F, both P^ , , P are members of F and so by (a H>, there i s a d € F such t h a t d 5 P^ and d 5 P. Since P^ i s an atom, only 0 5 P^ and P^ 5 P^ . However, P^ , 2 P, and so by ( a H ) , 0 € F. But then by ( b ) , F = PQJ^ » which c o n t r a d i c t s the assumption t h a t F i s a proper f i l t e r i n P Q M . Q.E.D. 123 The proof t h a t the subset of elements {P € P N M A : P < P^"} i s an u l t r a i d e a l UI^. i n P A M A proceeds d u a l l y . Any such u l t r a f i l t e r UF. and dual u l t r a i d e a l UI, de f i n e d w i t h respect to an atom o f P A ^ i s c a l l e d a p r i n c i p a l u l t r a f i l t e r and a p r i n c i p a l u l t r a i d e a l , r e s p e c t i v e l y , as mentioned i n Chapter 1(C) and ( F ) . In the case of an i n f i n i t e dimensional H i l b e r t space P^ , not every u l t r a f i l t e r and 2 not every dual u l t r a i d e a l i s p r i n c i p l e . Nevertheless, s i n c e a quantum pure s t a t e , as represented by a vector i n H i l b e r t space, i s an atom i n the P A ^ s t r u c t u r e of H i l b e r t space, each (pure) state-induced mapping is_ defined w i t h respect t o a p r i n c i p l e u l t r a f i l t e r and dual p r i n c i p l e u l t r a i d e a l i n Q^M " ^° W e n e e d o n l y consider p r i n c i p l e u l t r a f i l t e r s , l a b e l e d UF^. , and p r i n c i p l e u l t r a i d e a l s , l a b e l e d UF^ , i n t h i s d i s c u s s i o n o f a state-induced semantics f o r a P .„ . QM As mentioned above, any f i l t e r i n a P _ „ T by d e f i n i t i o n s a t i s f i e s QML ( a ) , ( b ) , and any i d e a l i n a by d e f i n i t i o n s a t i s f i e s ( a ' ) , ( b ' ) . Any f i l t e r i n a PQJ^ by d e f i n i t i o n s a t i s f i e s (b) and a l s o s a t i s f i e s ( a ) , as shown i n Chapter 1(F), and any i d e a l i n a PQ^A ^y d e f i n i t i o n s a t i s f i e s ( b 1 ) and a l s o s a t i s f i e s ( a ' ) . Moreover, i t i s easy to show th a t any u l t r a f i l t e r UF^ and dual u l t r a i d e a l UI^ i n a P Q M s a t i s f y the c o n d i t i o n s (c) and ( c 1 ) s t a t e d i n Chapter 1(C): (c) For any P € P Q M , ? S UF + IFF P € UI^ . (c') For any P € P^ , p X€ UI^ IFF P € UF^ . Proof: For any P € P^ , p X€ UF^ IFF P"*" > P^ IFF P ^ - P IFF P € UI^ . And f o r any P € P Q M , p X € UI^ IFF P^ < P ^ IFF P^ .< P IFF P € UF^. Q.E.D. These c o n d i t i o n s ( a ) , ( a ' ) , ( b ) , ( b ' ) , ( c ) , ( c ' ) , ensure t h a t , f o r any atom P^ € P q m , the union UF^ U UI^ = {P € .P Q | { , : P >-P+ or P <-PX } 124 3 i s c l o s e d w i t h respect to the A, v , o p e r a t i o n s o f Pn^ , as shown next. For any elements ?1*V2 € * Q M ' i f b o t h P l ' ? 2 € U F \ ) / ' t h e n b y ^ P 1 A P 2 € UF^ , by (b) P 1 v P 2 € UF^ ( s i n c e ?± < P± v P 2>, and by (c') P 1 € U I ^ and P 2 X € U I ^ . I f both P ^ € U I ^ , then by (a') P l V P 2 € % ' b y ( b ' } P l A P 2 € U I t ( s i n c e P i A P 2 ~ V ' a n d b y ( c ) P X € UF, and P X € UF; . I f P„ € UF, and P n € UI; , then by (b) 1 Y 2 y i f 2 Y P 1 v P 2 € UF^ , by (b') P 1 A P 2 6 U I ^ , by (c') P ^ € U I ^ , and by (c) P X € UF, . Since a f i l t e r F and an i d e a l I are each by d e f i n i t i o n 2 y nonempty and s i n c e , f o r any P € F, P 5 1, and f o r any P € I , P > 0, i t f o l l o w s by (b) th a t the 1 element o f P_„ i s a member of UF, , and Q M y i t f o l l o w s by (b') t h a t the 0 element o f P_w i s a member o f UI, . In J Q M \|/ other words, l e t t i n g US, l a b e l the union UF. U UI. , we have 0 € US, , ° Y Y Y Y 1 € US^ , and f o r any elements P^Pg € P Q M ' i f P l ' P 2 € U S y ' t h 6 n P„ A P„ € US, , P. V P € US, , and P X € US, . Thus, f o r any atom 1 2 Y 1 2 \|/ 1 2 \|c P, € P ™ > "the subset US, = UF-, U UI, i s a cl o s e d s u b s t r u c t u r e of P.„ Y Q M \(r Y Y 4 which may be c a l l e d an u l t r a s u b s t r u c t u r e . S p e c i f i c a l l y , US^ i s a subalgebra of P.„. , and US, i s a s u b l a t t i c e of P^„T . This r e s u l t Q M A Y Q M L i s analogous to the r e s u l t : In any Boolean s t r u c t u r e 8 (algebra or l a t t i c e ) , the union o f a f i l t e r and dual i d e a l form a substru c t u r e (subalgebra or s u b l a t t i c e ) of 8 ( B e l l and Slomson, 1969, p. 17). However, i t i s important to note t h a t t h i s c l o s u r e o f US^ = UF^ U UI^ w i t h respect to the A, V, "1~ operations of Pn^ guarantees f o r any elements P^Pg € P Q M , n e i t h e r t h a t i f P 1 v P 2 € US^ then P, € US; or P 0 € US. , nor t h a t i f P. A P 0 € US, then P. € US, or 1 \|r 2 \j/ 1 2 \|/ 1 Y P. € US . For any US, i n a P.„ , such meets and j o i n s which are 2 \|/ • ^ Q M J themselves members of US. but whose c o n s t i t u e n t elements P„ , P„ are not Y 1 2 both members o f US, are h e r e a f t e r c a l l e d U S - e x t r a meets and j o i n s . Y — Y 125 I t i s a l s o worth n o t i n g t h a t , f o r any atom i n a P ^ , the u l t r a s t r u c t u r e US. i s the union of a l l the Boolean substructures i n P.,, ty QM which c o n t a i n P^ . , and i n p a r t i c u l a r , US^, i s the union of a l l the _Q>3 overlapping mBS's i n P^^ which c o n t a i n P^ , . As mentioned i n the previous s e c t i o n , by Lemma A, {P € P^ M : P > P^ or P 5 } = {P € P Q M : P A P^}, t h a t i s , US^, i s the (unique) subset o f a l l elements i n P.., which are compatible w i t h P. . Let mBS. . be any mBS QM * ty ty,i J i n P^„ which contains P. , and l e t U mBS, . be the union o f a l l such QM ty . tyyl mBS. .'s i n P_„ . I t i s easy t o show t h a t , f o r any given atom P. € P.,, Y , i QM J ' J ° ty QM and f o r every element P € P „„ , P € US. IFF P € U mBS, . . I f P € US,, QM ty i ty,i ty9 then p i and so the set o f elements {P-,PJ",P^,P^X ,0,1} form a Boolean substructure i n P.,, which contains P. and which, by Zorri's lemma, i s QM ty J i t s e l f contained i n some maximal Boolean substructure mBS. . which contains P. ; thus P € U mBS. . . Conversely, i f P f U mBS, . , then ?6 P. , Y i . ty,i ty and so P € US^, . Q.E.D. So f o r any given atom P^ , (! P^ M and f o r every mBS^ ^ c o n t a i n i n g P^ , mBS^ ^ £ US^ c P ^ . i n p a r t i c u l a r , a l l the elements i n an mBS^ . ^  are compatible w i t h P^ , and are a l s o mutually compatible, w h i l e a l l the elements i n US^ are compatible w i t h P^ but 5 need not be mutually compatible. Since, as described i n Chapter I V ( F ) , 2 the mBS's i n a two-dimensional H i l b e r t space ^ do not o v e r l a p , e.g., 2 2 any atom P^ € P^ M i s a member of only one mBS i n P^ M , 2 US, = U mBS,, . = mBS, . That i s , an u l t r a s u b s t r u c t u r e i n a P „„ i s always Y . ty,± ty QM 2 j u s t a maximal Boolean subst r u c t u r e o f P^„ . But since the mBS's i n a QM thr e e - or higher-dimensional H i l b e r t space P^ M may o v e r l a p , e.g., any atom P, € P ^ 3 may be a member of many mBS's i n P^„ , US, = U mBS, . ty QM J QM ' ty . ty,i may be l a r g e r than any mBS. . . That i s , an u l t r a s u b s t r u c t u r e i n a Y, l 126 V may c o n t a i n incompatible elements and thus may i n some sense be a non-Boolean subst r u c t u r e o f P n>3 QM ' As s t a t e d i n the previous s e c t i o n , any Exp^, on a P^ M assigns values as f o l l o w s : For any P € P A ^ , Exp^,(P) = Since UF + = {P 6 P Q M : P > P^}, U I ^ {P € P QM 1 i f P > P; 0 i f P < P. Y 6(0,1) i f P ^ - P . P < P , x }, and Y Y, V US, = {P 6 P -„ : P i P , } , i t f o l l o w s t h a t any Exp. on a P_„ assigns Y QM y Y values as f o l l o w s : For any P ^'PQM > Exp^(P) = 1 i f P 6 UF 0 i f P € UI J ( 0 , 1 ) i f P^US^-UF^'UUI^ This r e s u l t suggests t h a t each Exp^_ on a P^ i s an u l t r a v a l u a t i o n on the u l t r a s u b s t r u c t u r e US^ . (H e r e a f t e r , an Exp^, and i t s US^ may be s a i d to be a f f i l i a t e d . ) Of course, an Exp. i s b i v a l e n t w i t h respect to the elements i n US', Y ^ . Moreover, i t s h a l l be shown below t h a t an Exp^, i s t r u t h - f u n c t i o n a l ((A) or (A,&)) w i t h respect to the elements i n US^ . Thus an Exp^. i s a b i v a l e n t , t r u t h - f u n c t i o n a l ((A) or (A,<&)) mapping on US^, defined w i t h respect to the u l t r a f i l t e r UF^, and the dual u l t r a i d e a l UI^, , t h a t i s , an Exp^ i s an u l t r a v a l u a t i o n on the a f f i l i a t e d u l t r a s u b s t r u c t u r e US. . Y The c o n d i t i o n s ( a ) , ( a ' ) , ( b ) , ( b ' ) , ( c ) , ( c ' ) , s a t i s f i e d by any UF^, and dual UI^. i n a P N M y i e l d the f o l l o w i n g b i c o n d i t i o n a l s and c o n d i t i o n a l s . For any UF^ and dual UI^ i n a P^ M , f o r any P € PQ M , and f o r any P ^ € P Q M (qua p ^ ) , f o r any F± £ ?2 , P Q M (qua P Q M ) UI P. A P. € UF, IFF P. € UF, and P„ € UF Y Y by (a) and ( b ) ; 127 P„ A P € U l . IF P, € U l , or P € U l , by ( b ' ) ; 1 2 y 1 y 2 y U2 P. V P , ( UF, IF P. € UF. or P 0 <E UF. , by ( b ) ; 1 2 y 1 y 2 y ' -' ' P 1 V P 2 € UI^ IFF P^^ € UI^ and P 2 € UI^ , by (a') and ( b ' ) ; U3 P-1- € UF^ IFF P € UI^ , by ( c ) ; P"1" € UI^ IFF P 6 UF^ , by Cc'). I t c l e a r l y f o l l o w s t h a t the Exp^, on P^ which assigns the value 1 to the elements i n UF. and assigns the value 0 to the elements i n U l , y y s a t i s f i e s a l l o f the c o n d i t i o n s TF1, TF2, TF3, which d e f i n e a t r u t h - f u n c t i o n a l mapping and are l i s t e d i n Chapter 11(C), except the f o l l o w i n g two: I f Exp^(P 1 V P 2) = 1, then Exp^,(P 1) = 1 or Exp^,(P 2) = 1; i f Exp^(P 1 A P 2) = 0, then Exp^CP^ = 0 or Exp^(P 2) = 0. These two c o n d i t i o n a l s are missing from the l i s t o f c o n d i t i o n s s a t i s f i e d by Exp^ because the f o l l o w i n g two c o n d i t i o n a l s are missing from the l i s t o f c o n d i t i o n s U l , U2, U3, s a t i s f i e d by UF^ , UI^, : I f P 1 V P 2 € UF^ , then P l € U F y o r P 2 € U F y ' I f P l A P 2 € U I y ' t h e n P l € U I y ° r P 2 € U I y ' These two c o n d i t i o n a l s i n f a c t c h a r a c t e r i z e a prime u l t r a f i l t e r and a prime u l t r a i d e a l , r e s p e c t i v e l y , as s h a l l be discussed next. Using the d e f i n i t i o n s t a t e d i n Chapter 1(C), we s h a l l say th a t an u l t r a f i l t e r UF^ i n a P Q M i s prime IFF, f o r any P . ^ € P Q M , (d) I f P 1 V P 2 € UF^ , then P± € UF^ or P 2 € UF^ . I f we take P Q M to be a P Q M and i f P 1 If ?2 , then P^^ v P 2 i s not defined and so t r i v i a l l y , the antecedent o f (d) does not o b t a i n , So no s p e c i a l p r o v i s i o n i s made f o r P^^ . D u a l l y , an u l t r a i d e a l UI^ i n a PQM i s P r i m e I F F> f o r a n y p ! ' p 2 * P Q M » 128 (d') I f P A P € U l , , then P. € U l . o r P. € U l . . 1 2 Y 1 Y 2 Y Every u l t r a f i l t e r ( u l t r a i d e a l ) i n a Boolean s t r u c t u r e i s prime. But as s t a t e d without proof i n Chapter I V ( F ) , i f a P^ M contains • r incompatible elements, then there i s some u l t r a f i l t e r i n PQ^ which i s not prime; i . e . , i f a P ^ contains incompatible elements, then not every u l t r a f i l t e r i n P^ M i s prime. This c l a i m s h a l l be proven w i t h the help o f the f o l l o w i n g p r o p o s i t i o n s . P r o p o s i t i o n A: For any UF^ i n a P^ M , i f UF^ i s prime, then UF. U U l , = P „ , . In other words, f o r any UF. i n a Y Y QM Y P Q M , i f UF^ s a t i s f i e s ( d ) , then, f o r any P € P Q M , e i t h e r P € UF, or P € U l . (where U l . i s the u l t r a i d e a l dual",to Y Y Y U V For any UF^ i n a P^ M , and f o r any P € P^ M , e i t h e r P € UF^. or P £ UF^ . Assuming t h a t UF^ s a t i s f i e s ( d ) , P £ UF i m p l i e s P"1" t UF. For P V P"1" = 1 € UF^ , and so by ( d ) , e i t h e r P € UF^ or P1" € UF^ ; so i f P £ UF then P X € UF^ . And by ( c ) , P"1" € UF^ i m p l i e s P € UI^ . So f o r any P € P_„ , e i t h e r P € UF or P € U l ; . Q.E.D. QM Y V P r o p o s i t i o n B: I f a l l the atoms i n a P Q J^ a r e mutually compatible, then every element P 4 0 i n PQ^A 1 S T N E J ° I N ° ^ the atoms i t dominates. Let P. be any atom i n P_„ . such t h a t P. < P, and l e t l J QMA l V P . be the ( f i n i t e or i n f i n i t e ) j o i n o f a l l such atoms. (This I I j o i n i s defined because by assumption, a l l the atoms i n P Q J ^ A R E 129 mutually compatible.) The r e s t o f the proof proceeds e x a c t l y as the proof o f Lemma C i n Chapter V(A), w i t h P Q M A s u b s t i t u t e d f o r P QML " Now the c l a i m s t a t e d above may be proven as f o l l o w s . Theorem B: I f a P . „ contains incompatible elements, then not QM every u l t r a f i l t e r UF^ i n P^ M i s prime. Proof: Assume on the contrary t h a t P N ^ contains incompatible elements and every u l t r a f i l t e r UF^ i n P^ M i s prime. Then by P r o p o s i t i o n A, f o r every UF. i n P.„ , UF, U UI, = P.„ , where UF. U UI, = US, J Y Q M Y Y Q M Y Y V = {P 6 P 0 „ : P A P,} f o r some atom P. € P.„ . Thus each atom P. i n QM \|/ Y Q M i P i s compatible w i t h every element i n P^ M ; i n p a r t i c u l a r , each atom i s compatible w i t h every other atom i n P N ^ , t h a t i s , the atoms i n P A ^ are mutually compatible. I t f o l l o w s t h a t the set o f atoms i n P^ M generates a Boolean substructure when closed w i t h respect to the A, V, -1-operations o f P^ M , f o r as s t a t e d i n Chapter 1(D), ( E ) , ( F ) , any set o f mutually compatible elements i n a P generate a Boolean subst r u c t u r e when clo s e d w i t h respect t o the operations o f P ^ . Moreover, f o r P qua PnMT » by Lemma C o f Chapter V(A), every element P ^ 0 i n P i s the QML QML j o i n o f the atoms i t dominates. And s i m i l a r l y , f o r P^ M qua P A M A » by P r o p o s i t i o n B, every element P t 0 i n i s f n e j o i n o f the atoms i t dominates, where a l l the atoms i n PQJ^ a r e mutually compatible. Thus every element P i n P^ i s a member of the Boolean subst r u c t u r e generated by c l o s i n g the set o f atoms i n P ^ w i t h respect t o the A, v, J _ operations of P N ^ . And so a l l elements i n P ^ are mutually compatible, which c o n t r a d i c t s the assumption t h a t P ^ c o n t a i n s incompatible elements. Q.E.D. 130 P r o p o s i t i o n A and Theorem B, w i t h " u l t r a i d e a l UI^," interchanged w i t h " u l t r a f i l t e r UF^. ," can a l s o be proven. In s h o r t , any P^^ which contains incompatible elements contains an u l t r a f i l t e r which does not s a t i s f y (d) and contains an u l t r a i d e a l which does not s a t i s f y ( d ' ) , and thus contains an u l t r a s u b s t r u c t u r e US^ = UF^ U UI^, which i s a proper subset of PQM ' However, f o r any US^ , c p ^ , i f we r e s t r i c t our a t t e n t i o n t o the elements of P.,, which are i n US. , then we do have, f o r any QM y J P,,P„ € US. = UF, U U l , : 1 2 Y Y Y (d) I f P 1 V P 2 € UF^ , then P± € UF^ or P 2 € UF^ . (d') I f P, A P € U l , , then P. € U l , or P„ € U l , . 1 2 Y 1 Y 2 Y Proof: Assume on the contrary t h a t P^ v p 2 i S a member o f the u l t r a f i l t e r UF^ but P 1 * UF^ and P 2 * UF^ . Then s i n c e P l f P 2 € US^ = UF^ U UI^ , p i ' p 2 € U I Y * S o b y ( c ) ' p i " ' P 2 L 6 U F Y ' a n d s o b y ( a ) ' P.1" A Px = (P, v P 0 ) X € UF. . ( I f P . V P , i s defined i n , then 1 2 1 2 Y 1 2 QMA P l ^ P 2 ' a n d x t f o l l o w s t h a t pi» P 2 ' P2' L' a r e m u t u a l l y compatible and so t h e i r meets and j o i n s are a l l defined i n PQJ^ Then by (a) again, 0 = (P± V P^ A (P± v P 2 ) € UF^ . And so by ( b ) , UF^ = P Q M , which c o n t r a d i c t s the assumption that UF^ i s an u l t r a f i l t e r , which i s a proper f i l t e r , i n P^ M . The proof of (d') proceeds d u a l l y . Q.E.D. I t i s noteworthy t h a t i f we take US^, to be an improper substr u c t u r e o f P^ M , i . e . , US^, = P ^ r a t h e r than US^ , c p ^ , then the above works as a proof of the converse of P r o p o s i t i o n A: For any UF^, i n a PQ M , i f UF^ U UI^ = PQ M , then UF^, i s prime. Proof: Assume on the c o n t r a r y t h a t , UF y U UI^ = P Q M , and f o r any P ^ * P Q M , P 1 V P 2 € UF^ 131 but P. * UF. and P . £ UF. . Then since P A M = UF. U UI. , P,,P„ € UI. I t y 2 y ty ty 1 2 ty The r e s t o f the proof continues as above to the end of the penultimate sentence. Q.E.D. Thus we have: For any UF^, i n a P^ , UF^, i s prime IFF UF^ U UI^ = P Q M . And e q u i v a l e n t l y , f o r any UF^ i n a P Q M , UF^ i s not prime IFF UF^ U UI^, + P Q M , i . e . , IFF UF^ U UI^ i s a proper substructure o f P .„ . QM Nevertheless, the p o i n t o f the proof given i n the paragraph preceding the previous paragraph i s to show t h a t , even when UF^, U UI^c p ^ i . e . , even when UF^, and UI^ are each not prime i n P ^ , w i t h respect to the elements i n the u l t r a s u b s t r u c t u r e US. = UF. U UI. c P , UF. ty ty ty QM Y does s a t i s f y (d) and UI^, does s a t i s f y ( d T ) , and so UF^, and UI^, may each be s a i d to be prime w i t h respect to US^ . Concordantly, f o r any atom P ' I € P~,. » even when the state-induced expectation f u n c t i o n Exp. on P . W ty QM ^ *ty QM does not s a t i s f y a l l o f the c o n d i t i o n s l i s t e d as TF1, TF2, TF3, nevertheles w i t h respect to the u l t r a s u b s t r u c t u r e US^ c P^ M , Exp^ does s a t i s f y a l l the c o n d i t i o n s : For any P € US c P^ , f o r any P-^Pj € US^ c ? ^ (qua p ) QML ' f o r any P^ d> P^ 6 US > c PQM ( q u a PQMA ) : TF1 Exp v k(P 1 V P 2) = 1 IFF E * P Y ( V = E x p t ( P 2 ) = 1 Exp^(P 1 V P 2) = 0 IFF Exp^CP^ - 0 or Exp^(P 2) = 0 TF2 E x t > ( P l A P 2 ) = 1 IFF E X P Y ( P 1 } = 1 or Exp^(P 2) = 1 Exp y (P 1 A P 2) = 0 IFF E X P Y ( V = E x p t ( P 2 ) = 0 TF3 Exp^P 1") = 1 IFF Exp^(P) = 0 Exp^(P i) = 0 IFF Exp^(P) = 1 Thus Exp^ i s an u l t r a v a l u a t i o n on the u l t r a s u b s t r u c t u r e US^ , ; th a t i s , Exp^, , as defined w i t h respect t o UF^ _ and the dual UI^, , i s a b i v a l e n t 132 t r u t h - f u n c t i o n a l ( ( i ) or (o,&)) mapping on US^ = UF^ U UI^. . More e x a c t l y , each quantum state-induced Exp^ on a P Q J^ " t r u t h - f u n c t i o n a l l y (A ) assigns 0, 1 values to the elements i n i t s a f f i l i a t e d US^ , which i s a subalgebra o f P Q ^ * A n d e a c n quantum state-induced Exp^, on a P t r u t h - f u n c t i o n a l l y (&,&) assigns 0, 1 values to the elements i n i t s a f f i l i a t e d u l t r a s u b s t r u c t u r e US. , which i s a s u b l a t t i c e o f P „ „ T . \j/ QML The t r u t h - f u n c t i o n a l (<}>,#) charac t e r of Exp^ on the domain US^ c P Q M L m a Y seem s u r p r i s i n g i n the l i g h t o f the Chapter V(A) d e s c r i p t i o n of the t r u t h - f u n c t i o n a l i t y (o,&) problems caused by the meets and j o i n s of incompatible elements i n P ^ M L . Yet Exp^ s a t i s f i e s TF1, TF2, TF3, f o r any elements P^ , P^ i n US^ c P Q ^ . Thus Exp^, preserves the A, V operations o f P Q ^ among any compatible and incompatible elements i n US^ ; i n other words, Exp^ i s t r u t h - f u n c t i o n a l ( A ,& ) on US^ c P Q M L . As mentioned i n Chapter V(C), Friedman and Glymour propose, f o r the quantum P A M T s t r u c t u r e s , semantic mappings which are r e q u i r e d to QML preserve the -L o p e r a t i o n and the S r e l a t i o n of P Q ^ but are not re q u i r e d to preserve the A, v operations among e i t h e r compatible or incompatible elements o f P Q ^ • However, i t i s easy to show that a Friedman-Glymour mapping i s i n f a c t b i v a l e n t and t r u t h - f u n c t i o n a l on an u l t r a s u b s t r u c t u r e o f P , j u s t l i k e the quantum state-induced QML Exp^, mapping. The Friedman-Glymour semantic mappings are c a l l e d S3-valuations v : PQ M^ {0,1} and need only s a t i s f y the f o l l o w i n g two c o n d i t i o n s : For any p 1 » p 2 ^ PQML ' ( i ) v ( P 1 ) = 1 IFF v t P ^ ) = 0 ( i i ) I f v ( P 1 ) = 1 and P 1 < ? 2 , then v ( P 2 ) = 1. I t f o l l o w s from ( i ) , ( i i ) , t h a t , f o r any S3-valuation v on a P^„ T and QML 133 f o r any given element P Q € P ^ M L , i f v ( P Q ) = 1, then f o r any P € P Q M L , v(P) = 1 i f P > P Q and v(P) = 0 i f P < P^". For i f v(P ) = 1 and P > P Q , then by ( i i ) , v(P) = 1. And si n c e P < P ^ IFF P Q < P X, i f v(P ) = 1 and P < P x , then P 2 P X and so by ( i i ) , v(P" L) = 1, and then by ( i ) , v(P) = v ( ( P x ) x ) = 0. When P Q i s an atom P^ i n PQ M L , then as shown i n t h i s s e c t i o n , the set {P € Pg^ : P - P^) i s a n u l t r a f i l t e r UF. i n P.„T and the set {P € PA M T : P 5 P\f~} i s the dual ty QML QML Y u l t r a i d e a l UI, i n P„WT . And i t f o l l o w s from the c o n d i t i o n s s a t i s f i e d Y QML by UF^, and UI^ th a t a mapping l i k e the S3-v a l u a t i o n which assigns the value 1 to the elements i n UF^ and assigns the value 0 to the elements i n UI^, i s not only b i v a l e n t but a l s o t r u t h - f u n c t i o n a l ( i , & ) on the u l t r a s u b s t r u c t u r e UF, U UI, o f P.„T . So besides being b i v a l e n t and Y Y Q M L 5 pr e s e r v i n g w i t h respect to the e n t i r e P Q ^ » f^e S3^valuations are a l s o b i v a l e n t , truth-functional(«b,^S) u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f PQ^L » a s a r e fhe quantum state-induced e x p e c t a t i o n - f u n c t i o n s . Of course, f o r any atom P^ , € P^ , i f the u l t r a s u b s t r u c t u r e US, = UF, U UI, i s an improper s u b s t r u c t u r e o f P_„ , i . e . , i f US, = P„. , ty ty ty QM ty QM then the quantum e x p e c t a t i o n - f u n c t i o n Exp^, » which i s induced by the pure s t a t e represented by P^ , , i s a b i v a l e n t , t r u t h - f u n c t i o n a l ((A) or (<!>,&)) u l t r a v a l u a t i o n on the e n t i r e P s t r u c t u r e . In p a r t i c u l a r , as described i n the d i g r e s s i o n p r i o r to the proof o f Theorem A i n Chapter V(A), i f P^M has a n o n t r i v i a l centre which inc l u d e s an atom P, ( l a b e l e d P i n the ty c d i g r e s s i o n ) o f P^ , so t h i s P^ i s compatible w i t h every P € P^ , then the u l t r a s u b s t r u c u t r e US, = {P 6 P_„ : P <!> P.} = p.., . And so the ty QM ty QM mapping ( l a b e l e d h c i n the d i g r e s s i o n ) which assigns the value 1 to the elements i n UF^, and assigns the value 0 to the elements i n UI^ , 134 namely, the state-induced Exp^ , truth-functionally ((<b) or (<J>,i4)) assigns 0, 1 values to every element in Pq^ = US^ = UF^ U UI^ . However, i f P^ contains incompatible elements, then as shown by Theorem B, there is some u l t r a f i l t e r UF^ in P^ which i s not prime, and so by the converse of Proposition A, UF^ U UI^ 4 Pq^ , i.e., UF^ . U UI^. c . i t is precisely because every quantum P^ containing incompatible elements has at least one ultrasubstructure which is smaller than the entire P„„ that I have chosen to assign 0, 1 truth-values to QM the elements of any propositional or logical structure P according to the definition: For any element P € P, v(P) = 1 i f P € UF and v(P) = 0 i f P € Ul, rather than according to Sikorski's definition of a bivalent homomorphism: for any element P € P, v(P) =1 i f P € UF and v(P) = 0 i f P £ UF. With respect to a Boolean propositional or logical structure B, e.g., L or P , the two definitions are equivalent because UF U Ul = 8 for every UF and dual Ul in 8, since every UF (and dual Ul) in a Boolean structure is prime. So each may be regarded as the definition of an ultravaluation on a Boolean propositional or logical structure 8. That i s , each definition defines a bivalent, truth-functional mapping on a B with respect to an UF and dual Ul in 8; such a mapping is called an ultravaluation because, with respect to the Lindenbaum algebra L of classical propositional logic, such a mapping is the algebraic version of a standard valuation. But the two definitions are not equivalent whenever UF U Ul c P. In particular, the two definitions are not equivalent with respect to a quantum Pq^ which contains incompatible elements and thus contains at least one ultrasubstructure UF, U Ul, c P„ w . ty ty QM According to both definitions, any P € P^ such that 135 P^€ UF^, U UI^ i s assigned the value 1 i f P € UF^ and i s assigned the value 0 i f P 6 UI^ , because f o r any such P € UF^ U UI^ , P € UI^ IFF P £ UF^, . So w i t h respect to a given u l t r a s u b s t r u c t u r e UF^, U UI^, c PQ M , both d e f i n i t i o n s are e q u i v a l e n t . In p a r t i c u l a r , a mapping which a s s i g n s 0, 1 values according t o e i t h e r d e f i n i t i o n i s b i v a l e n t and t r u t h - f u n c t i o n a l ( ( A ) or ( o , & ) ) on the u l t r a s u b s t r u c t u r e UF', U U l , c P^„ • But the two d e f i n i t i o n s d i f f e r w i t h respect to the y y QM elements o f P „ „ which are o u t s i d e o f a given UF, U U l , c P . „ . Every QM y y QM P € P_„, such t h a t P £ UF, II U l , c p i s assigned the value 0 according QM y y QM & to the S i k o r s k i d e f i n i t i o n s i n c e every such P i s not a member of UF^ . However, the assignment of the value 0 t o every P £ UF^ U UI^ according to the S i k o r s k i d e f i n i t i o n i s not a t r u t h - f u n c t i o n a l ((^ ») or (o,<iO) assignment, as shown by the f o l l o w i n g example. For any P € P^^ , i f P £ UF^ II UI^ then P^ £ UF^. U UI^ . For assume on the c o n t r a r y t h a t P £ UF. U U l . , i . e . , P £ UF. and P £ U l . , and P^ £ UF. U U l , , y y ' y y ' • y y ' i . e . , PX £ UF^ or P^ € UI^ . I f PX € UF^ , then by ( c ) , P € UI^ , which c o n t r a d i c t s the assumption P £ UI^, . And i f PX € UI^, , then by Cc'), P € UF^. , which c o n t r a d i c t s the assumption P £ UF^, . Thus i f P £ UF^ U UI^ , then a l s o P X £ UF^. U UI^ . In p a r t i c u l a r , both P, P"1* £ UF^ , and so according to the S i k o r s k i d e f i n i t i o n , v(P) = viP3') = 0. But P v P x = 1 € UF^ , and so v(P V P x) = 1. Hence, f o r any P £ UF^ U UI^ c P q m , V ( P V P X) = 1 j* 0 = v(P) V v ( P x ) . So although a mapping which assigns values according to the S i k o r s k i d e f i n i t i o n i s b i v a l e n t on the e n t i r e P.,, , i t i s not t r u t h - f u n c t i o n a l ( ( o ) or ( A & ) ) QM on the e n t i r e P . In c o n t r a s t , the other d e f i n i t i o n which uses the c o n d i t i o n " i f P 6 U l " r a t h e r than the c o n d i t i o n " i f P £ UF" leaves open the questions of how and what values are to be assigned to such elements 136 P f. UF, U UI, c P„„ . So the only d i f f e r e n c e between the two d e f i n i t i o n s y Y QM i s t h a t one leaves these questions open w h i l e the other assigns the value 0 to the elements o u t s i d e a given u l t r a s u b s t r u c t u r e . Since these 0 value assignments are not t r u t h - f u n c t i o n a l ((A) or (A,&)), i n c l u d i n g them as p a r t of the d e f i n i t i o n of an u l t r a v a l u a t i o n on a P^ M a c t u a l l y adds l i t t l e beyond s a t i s f y i n g i n a t r i v i a l way the b i v a l e n c y desideratum. Thus we have taken the d e f i n i t i o n which uses the c o n d i t i o n " i f P € UI" as the d e f i n i t i o n o f an u l t r a v a l u a t i o n on a P_„ . QM As described i n the preceding s e c t i o n s , a state-induced u l t r a v a l u a t i o n Exp^. assigns values between 0 and 1 t o the elements outside UF^, U UI^. c P ^ . And Exp^, does preserve the ^~ o p e r a t i o n and the 5 r e l a t i o n of P_„ as i t assigns these intermediate v a l u e s , but the QM A, V operations o f P ^ are not preserved. So an Exp^, i s n e i t h e r b i v a l e n t nor t r u t h - f u n c t i o n a l ((A) or (A,&)) on the e n t i r e P.,, . QM Friedman and Glymour propose t h a t t h e i r S3-valuations on a PQ M L ' which have been shown to be u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s of ^ n u T > a l s o a s s i g n 0, 1 values t o the elements o u t s i d e t h e i r a f f i l i a t e d u l t r a s u b s t r u c t u r e s . Most simply, the value 0 may be assigned to every atom (one-dimensional subspace) and the value 1 may be assigned t o the orthocomplement of every atom (two-dimensional subspace) o u t s i d e a given 3 u l t r a s u b s t r u c t u r e o f a three-dimensional H i l b e r t space P.,,T (Friedman-QML Glymour, 1972, p. 27). Again, the -1- o p e r a t i o n and the 5 r e l a t i o n o f F*QML are preserved by such 0, 1 value assignments to the elements o u t s i d e an u l t r a s u b s t r u c t u r e . And t h i s proposal avoids at l e a s t some o f the t r u t h - f u n c t i o n a l i t y ( A , & ) problems of the more simple proposal t h a t the value 0 be assigned to every element outside an u l t r a s u b s t r u c t u r e . But Friedman-Glymour do not describe how 0, 1 values may be assigned f o r , say, 137 4 a four-dimensional H i l b e r t space PQ M L which has not only one- and two-dimensional subspaces but a l s o three-dimensional subspaces o u t s i d e any 4 given u l t r a s u b s t r u c t u r e of PQ^l • And of course, t h i s Friedman-Glymour proposal, and any other proposal of a b i v a l e n t semantics f o r the quantum P0_.T s t r u c t u r e s , i n e v i t a b l y runs i n t o truth-functionality(c> , i S ) problems, as shown i n Chapter V(A), and a l s o t r u t h - f u n c t i o n a l i t y ( A ) problems, as shown by Kochen-Specker. While addressing the issu e o f a p r e d i c a t e c a l c u l u s f o r a Kochen-Specker P n M A type of quantum p r o p o s i t i o n a l l o g i c , Levy proposes QMA t h a t , besides the 0, 1 values assigned by a state-induced u l t r a v a l u a t i o n to the elements i n an u l t r a s u b s t r u c t u r e of » a t h i r d t r u t h v a l u e , i n a p p r o p r i a t e , l a b e l e d N, be assigned to the elements o u t s i d e a given ult r a s u b s t r u c t u r e , . Such a three-valued semantics f o r a quantum o r P„,,T I s , o f course, not b i v a l e n t and i s a l s o not t r u t h - f u n c t i o n a l (A) or QML (<b,&)), as Levy mentions. An example o f a v i o l a t i o n o f '.• t r u t h - f u n c t i o n a l i t y ((i) or (A,#)) i s given a t the end o f the next s e c t i o n . This Levy proposal of three-valued semantic mappings f o r P s t r u c t u r e s i s d i f f e r e n t from previous proposals o f a three-valued semantics f o r quantum p r o p o s i t i o n s . For example, Reichenbach assigns h i s t h i r d t r u t h value I (Indeterminate) to quantum p r o p o s i t i o n s which are meaningless according to the Bohr-Heisenberg i n t e r p r e t a t i o n of quantum mechanics. In p a r t i c u l a r , i f P^ then a t most one of P^ , P^ i s meaningful w h i l e the other i s meaningless, and a l s o P^ A P^ and P^ V P^ are each meaningless (Reichenback, 1965, pp. 143-145). However, even though P^ & P^ 9 they may both be together i n some u l t r a s u b s t r u c t u r e of P^M , i n which case both o f them, and t h e i r meet and t h e i r j o i n are a l l assigned the 138 u s u a l 0, 1 t r u t h values by the state-induced u l t r a v a l u a t i o n a f f i l i a t e d w i t h t h a t u l t r a s u b s t r u c t u r e . In s h o r t , although semantic mappings on a which a s s i g n values between 0 and 1 or which a s s i g n a t h i r d t r u t h - v a l u e l i k e N t o the elements o u t s i d e a given u l t r a s u b s t r u c t u r e o f P . „ are not b i v a l e n t QM semantic mappings on the e n t i r e P „„ when UF, U UI, c P^„ , nevertheless QM y \|A Q M such mappings are t r u t h - f u n c t i o n a l ((A) or (o,ciS)) wherever they are b i v a l e n t , namely, on UF^ U UI^ . Thus the proposal o f such semantic mappings f o r P^ M has the v i r t u e o f c l e a r l y demarcating the substructures o f PQ M w i t h respect to which b i v a l e n t , t r u t h - f u n c t i o n a l ( ( A ) or (<!>,&)) value assignments are p o s s i b l e , namely, the u l t r a s u b s t r u c t u r e s UF^ U UI^ , f o r any atom P. € P - „ . Y QM Se c t i o n C. An Example Consider the fragment o f the P^ M s t r u c t u r e o f subspaces ( o r p r o j e c t o r s ) o f three-dimensional H i l b e r t space diagrammed below: 1 0 139 This fragment contains f o u r maximal Boolean substructures: mBS^ generated by the atoms {P.,P 0,P 0}, mBS generated by the atoms {P 0,P, j P c } , mBS,. 1 z o o 3 4 5 6 generated by the atoms { P ^ P ^ P , ^ , and mBSg generated by the atoms {P 7 , P g , P g } . C l e a r l y , these f o u r mBS's overlap s i n c e they share atoms. I f we had s t a r t e d w i t h the i n i t i a l s et S = { P ^ * ^ ' ' ' ' , P 9 ^ of these nine one-dimensional subspaces of h^, then the p a r t i a l - B o o l e a n algebra generated by c l o s i n g S w i t h respect to the A, v, ^~ operations of PQ^A ^ s t n « f i n i t e fragment of 20 elements diagrammed above. However, the orthomodular l a t t i c e generated by c l o s i n g S .with respect to the A,. V, operations of PQJJ^  l S denumerably i n f i n i t e and so e x e m p l i f i e s the p r o l i f e r a t i o n o f l a t t i c e elements due to the l a t t i c e d e f i n i t i o n s of A, V among incompatible elements, as mentioned i n Chapter IV(E). Let us focus on the element P g which i s compatible w i t h p^ » p 2 ' P4 ' P 5 ' Consider the incompatible elements P 0 ty P_ , t h e i r j o i n P. V P = P."1-. This P.f' 0 0 3 o 4 4 i s a l s o equal t o P 0 V Pc and to Pc V Pn , where P„ A P c and P r A P_. 0 0 b / o o b / So the j o i n P„ V P does not introduce any new element. And the j o i n P„ V P i s an example of what Strauss would c a l l the l a t t i c e m i s i n t e r p r e t a -o b t i o n o f the element P . S i m i l a r l y , c onsider the incompatible elements Pg ty Py . Again t h e i r j o i n P 3 V P^ = P ^ , so no new element i s introduced by having the V o p e r a t i o n defined among these two incompatible elements. But now consider the two incompatible elements P ty P ; t h e i r j o i n o o P„ V P and the meet o f t h e i r orthocomplements P^~ A pj~ are each not 0 0 0 0 equal to any o f the twenty elements i n the above diagram. Let P3" A P8 = P u a n d S ° P 3 V P 8 = K A P 8 X ^ = ?u * C l e a r l y ' P 3 -and so P_ • A P and a l s o P X <A P 1 . Let P X A P ^  = P and so 3 u 3 u 3 u v P, V P = ( P X A P-*-)^ = P 1-. C l e a r l y , P X > P 0 and P X > P , and so 3 u 3 u v v 3 v u P y A P g and P v A ? u • Thus {P g,P .P^} are three mutually compatible 140 atoms which generate another maximal Boolean s u b s t r u c t u r e , say mBS^ . The r e l a t i o n s among these elements are diagrammed below; f o r c l a r i t y , a l l the elements of the f i r s t diagram have been omitted except the P , PJ*", P D , 3 3 8 P Q X , 0, 1 elements: i_ X X. X But besides P, 5 P , when we l e t P 0 A P n = P and so P, V P = P , 3 u 3 8 u 3 8 u we a l s o have: P n 5 P X , and so P. <^  P and a l s o P r A"d>p J' . Let 8 u 8 u 8 u P X A P X = P and so P. v P = ( P X A P1)"*- = P X . C l e a r l y , P X > Fn and 8 u w 8 u 8 u w J ' w 8 PX > P , and so P i P n and P i P . Thus {P„,P ,P } are three w u w 8 w u 8 u wJ mutually compatible atoms which generate y et another maximal Boolean s u b s t r u c t u r e , say mBS . Moreover: P. V P = P. V ( P X A P ' L) J w 8 w 8 8 u = ( P D .VP Q X) A ( P 0 V P 1 ) = 1 A P X = P X . So P X A P 1" = (P V P ) A P X 8 8 8 u u u u w 8 w w ( P n A P x ) V (P A P X ) = P„ v 1 = P„ , and thus 8 w w w 8 8 141 P = (P A P ) = P V P 8 u w u w A l l these r e l a t i o n s are diagrammed below; f o r c l a r i t y , a l l the elements of the f i r s t diagram have been omitted except the J- x P 3 ' P 3 ' P8 ' P 8 ' °' 1 e l e m e n t s : 0 S i m i l a r l y , consider the two incompatible elements P <iS P ; 3 9 t h e i r j o i n P V P and the meet of t h e i r orthocomplements PX A are 3 y 3 9 each not equal to any of the 26 elements i n the above (combined) diagrams. Let P^ A P X = P and so P 0 v P = (P 0 X A P ^ ^ P 1 . Thus two 3 9 x 3 9 3 9 x more elements have been introduced, and as described above, by c l o s u r e , f o u r more elements P = P AP ., P = P 0 VP \, P = P n .AP i , and.^  P =. P n VP w i l l b x. y •3 x ' z 9 x 9 x introduced, where {P„ ,P ,P } and {PQ,P ,P } w i l l be two s e t s o f mutually \j x y y x z compatible atoms, each generating two more mBS's, mBS and mBS , i n y z 142 3 the P generated by c l o s i n g the i n i t i a l set S with respect to the A, QML V operations of PQ^ . Likewise, the incompatible pa i r s P g ty , P <^J P may be joined and meeted to introduce even more elements. And, of o z course, when we focus upon another element besides P g , say P^ , which i s incompatible with P^ , P g , Pg , P ? , P Q , P g , P u , P v , P w , P x , P % , the j o i n s of P^ with each of these elements w i l l introduce even more 3 elements, etc. Thus the P N M T generated by c l o s i n g the i n i t i a l f i n i t e set QML S with respect to the A, V , o p e r a t i o n s of PQ M L w i l l contain a 3 denumerable i n f i n i t y of elements. Nevertheless, the i n f i n i t e P_ W T and the QML 3 3 corresponding i n f i n i t e PQJ^ ° ^ a ^ s u b S P A C E S of H each contain the same elements, so i t i s not correct to consider a partial-Boolean algebra of subspaces to be missing elements compared with an orthomodular l a t t i c e of subspaces. The point of the above example i s to show how, when an orthomodular l a t t i c e of subspaces i s generated from an i n i t i a l set of subspaces by c l o s i n g the i n i t i a l set with respect to the A, v, operations of > the l a t t i c e d e f i n i t i o n s of A, V among incompatibles may r e s u l t i n a p r o l i f e r a t i o n of elements which does not occur when a partial-Boolean algebra of subspaces i s generated from the same i n i t i a l set by c l o s i n g the set with respect to the A, v, -1- operations of P Q ^ ' Let us assume that the quantum system, which i s associated with 3 H and which P.,P0, . . . ,P Q , represent propositions about, i s i n the pure state y represented by the projector P which i s the atom P i n o 3 3 the (combined) diagram, which i s a fragment of the system's pr o p o s i t i o n a l 3 structure P . So we focus on the state-induced expectation-function Exp g 3 and i t s a f f i l i a t e d ultrasubstructure US = UF U Ul e P . With respect O O o v^JXl 3 to the twenty element P Q ^ generated by the i n i t i a l set S, we have: 143 U F 3 = { 1 ' P 3 ' P 3 V P 1 = 4> P 3 V P 2 = ?1 • P 3 V P 4 = *S ' ? 3 V P 5 = K} a n d 3 UIg = {OJP^ ' P 2 ' P l , P 5 ' P 4 ^ * With respect to the denumerably i n f i n i t e P ^ M L generated by the i n i t i a l set S, we have: U F 3 = ^ P 3 ' P 3 v P i = P 2 " ' P 3 V P 2 = *t> P 3 V P 4 = *£ ' V P 5 = P ^ = = P 3 V P 6 = P 3 V P 7 » P 3 V P 8 = P u = P 3 V P v » P 3 V P u = P v > P„ VP~ = P = P. VP , P„ VP = P"1", e t c . , . . . a denumerable 3 9 x 3 y ' 3 x y 3 i n f i n i t y of two-dimensional subspaces of H , each c o n t a i n i n g P3^* UI = { 0 , P X , P ,P ,P ,P ,P ,P ,P ,P , e t c . , . . . a denumerable i n f i n i t y o o ^ X j M " U v x y 3 J-of one-dimensional subspaces o f H , each contained i n P^ }. 3 3 3 And w i t h respect to the i n f i n i t e PQ M A a n d PQ^L °^ s u h s p a c e s o f H , UF i n both s t r u c t u r e s includes the 1 element, P , and the 3 nondenumerable i n f i n i t y of a l l two-dimensional subspaces o f H c o n t a i n i n g P . And UI i n both s t r u c t u r e s i n c l u d e s the 0 element, P and the o o o 3 nondenumerable i n f i n i t y o f a l l one-dimensional subspaces o f H contained j _ 3 i n Pg . Hereafter, l e t us j u s t focus on the twenty element PQJ^ 3 generated by S and the denumerably i n f i n i t e PQ M l generated by S. C l e a r l y , US = UF U UI i s l a r g e r than the two maximal Boolean O O o substructures mBS_ and mBS which c o n t a i n P i n the twenty element 3 P^.,. • And l i k e w i s e US„ i s l a r g e r than any of the maximal Boolean QMA 3 substructures mBS- , mBS_ , mBS , mBS , e t c . , which c o n t a i n P„ i n 2 D v y o 3 the denumerably i n f i n i t e fragment P N M T • Moreover, by i n s p e c t i o n i t i s QML 3 c l e a r t h a t i n the f i n i t e P _ „ . , US. = mBSn U mBS.. ; and by i n s p e c t i o n i t QMA 3 2 5 3 i s c l e a r t h a t i n the denumerable PQ^l » c o n s i d e r i n g j u s t the e x p l i c i t l y l i s t e d elements i n US^ and j u s t the e x p l i c i t l y l i s t e d mBS's c o n t a i n i n g P 0 , the l i s t e d elements i n US„ = mBS„ U mBS,. U mBS U mBS . That i s , 3 3 2 5 v y 144 US- equals the union of a l l the mBS's c o n t a i n i n g P ., as proven i n 3 3 Sec t i o n B. I t i s a l s o worth n o t i c i n g how, i f we had used c o n d i t i o n s ( a ) , ( b ) , r a t h e r than c o n d i t i o n s (a„), ( b ) , to define a f i l t e r i n a P , then the n yMA 3 set S' = UF„ U {P„} would be a proper f i l t e r i n the twenty element P N M . 3 7 (jMA diagrammed above. Using (a ), i t i s easy t o show t h a t S' i s not a f i l t e r n i n t h i s P ? „ A . I f S' i s a f i l t e r , then s i n c e P X , p i " € S', by (a„), QMA 2 1 n there i s an element d € S' such t h a t d < P 1 and d s P 1 . In the twenty element ? \ m , P X < , 0 < P X , 2 P X , P3 < P X , P ? X 5 P^ , 0 5 P ? \ P, < P X , < P X , P„ 5 P X , and P„ < P X ; so only 0 < P X and 0 < P X . 4 7 6 7 8 7 9 7 J 2 7 But 0 £ S', and so S' i s not a f i l t e r . Q.E.D. But us i n g ( a ) , i t turns 3 out t h a t S' i s a proper f i l t e r i n the twenty element P Q ^ ' I f S' i s a f i l t e r , then s i n c e P^jPy € s'» by ( a ) , the meet o f P X , P X i s a 3 member of S', but t h i s meet i s not defined i n the twenty element P Q ^ s i n c e P P^ • ( I f the meet were d e f i n e d , as i n a p 3 c o n t a i n i n g P X 2 7 QML 2 and P^", then the meet P X A P X = 0 ; thus s i n c e 0 £ S', S' would not be a f i l t e r . ) Moreover, except f o r the 1 element, every other element i n S' i s incompatible w i t h P X and so the meets of P X w i t h every other 3 element i n S' are not defined i n the twenty element P Q ^ * A N D 1 A P X = P X € S'. Thus S' s a t i s f i e s ( a ) . Also S' = UF. U {P X} I I O I s a t i s f i e s ( b ) ; f o r UF s a t i s f i e s ( b ) , and only 1 > P X , Pj" > P X , O I I I 1, P ? X € S'. So S' i s a f i l t e r i n the twenty element P . Q.E.D. 3 Moreover, s i n c e 0 £ S', S' i s a proper f i l t e r i n t h i s P Q ^ * S O ^ 3 3 i s the proper subset o f a proper f i l t e r i n t h i s PQ M^ • Thus UF g i s not 3 an u l t r a f i l t e r i n t h i s P Q ^ ' a v e r v undesirable r e s u l t o f us i n g (a) r a t h e r than (a T T) to def i n e a f i l t e r i n a P^„„ . H QMA Returning to the state-induced Exp,, , which assigns the value 1 and 145 to elements i n UF„ and assigns the value 0 to elements i n UI , i t i s o o easy to f i n d examples o f how Exp i s not a t r u t h - f u n c t i o n a l ( o ) mapping on o 3 the e n t i r e twenty element PQJJA • Consider the compatible elements P6' P7 * PQMA : P 6 A ?7 = P4 € U I 3 ' S ° E X P 3 ( P 6 A *7 > = °" B u t Pg,p X £ UF 3 U U I 3 , so E x p 3 ( P ^ ) t 0 (and t 1) and E x p ^ P 1 ) t 0 (and t 1 ) . Thus E x p . ( P c x ) A Exp ( P x ) t 0 = Exp 0(P. XA P x ). S i m i l a r l y , i t i s easy to f i n d examples of how Exp 3 i s not a t r u t h - f u n c t i o n a l ( m a p p i n g 3 on the e n t i r e denumerable P^,„ . Consider the incompatible elements QML P 3 ' P 8 € P Q M L : P 3 A v P B * 1 6 U F 3 ' S° V V = 1 ' B u t ^ € U I3 » so E x p Q ( P 0 X ) = 0, aand P. t UF 0 U UI , so E x p 0 ( P 0 ) t 1 (and t 0). O O O O O o o Thus E x p 3 ( P 3 X ) V E x p 3 ( P g ) = 0 V E x p 3 ( P g ) = Exp^Pg) ji 1 = E x p ^ P 1 V Pg). Since the elements p X , P X , P n t UF„ U U I 0 , the meet P^ A P X b / o o o b / and the j o i n P 3 V P x are examples o f what were c a l l e d US^-extra meets and j o i n s i n S e c t i o n B, where here, US^, i s US 3 . These are the meets and j o i n s which cause t r u t h - f u n c t i o n a l i t y ((b) or (o,^A)) problems f o r Exp 3 . Whether they are the meets and j o i n s of compatible elements o r of incompatible elements i s i r r e l e v e n t . What makes such meets and j o i n s problematic f o r Exp i s t h a t one or the other or both o f t h e i r subformulae are elements o f o PQM w n ^ c n a r e rcof members of US 3 . Moreover, every v i o l a t i o n o f t r u t h - f u n c t i o n a l i t y ((A>) or (i>,#>)) by an Exp^, on a P^ M i n v o l v e s such US^-extra meets and j o i n s . For as has been shown i n S e c t i o n B, any Exp^, i s t r u t h - f u n c t i o n a l ((<A) or (A,&) on the domain US^, , t h a t i s , Exp^, does preserve the meets and j o i n s o f the elements o f P - W which are members J QM of US + . As mentioned i n S e c t i o n B, the t r u t h - f u n c t i o n a l ( o c h a r a c t e r o f an Exp^. on US^, c PQ M L may seem s u r p r i s i n g i n the l i g h t o f the Chapter V (A) d e s c r i p t i o n o f the truth-functionality(i,<)6) problems caused by the 1 4 6 meets and j o i n s o f incompatible elements i n P Q ^ L • However, we can f i n d many examples o f the truth-functionality(h,k>) of Exp 3 on the u l t r a s u b s t r u c t u r e US. of the denumerable P.„„T considered i n t h i s s e c t i o n . 3 Q M L Consider the incompatible p a i r s P X jk P g , P X ty> P y , P± ty ?^ , and the f o l l o w i n g meets and j o i n s o f these incompatible p a i r s : P X A P^ , P^ V P^ , P A P . C l e a r l y , P X € UF , P € U l , and P X A P = 0 Z U l , thus E x p 3 ( P A ) A E x p 3 ( P 5 ) = 1 A 0 = 0 = E x p 3 ( P X A P 5 ) . C l e a r l y , P X <E UF 3 , P € UI„ , and P X V P = 1 6 UF„ ; thus Exv>APX ) V Exp„(P ) = 1 V 0 y 3 u y 3 3 u 3 y = i = E x p , ( P X V P ) . C l e a r l y , P € U l , P € Ul , and P A P o i i y x o x o x x = .0 € U l - ; thus Exp (P ) A Exp„(P ) = 0 A 0 = 0 = Exp (P A P ). O O 1 \j X o 1 X I t i s a l s o easy to f i n d examples o f v i o l a t i o n s o f t r u t h - f u n c t i o n a l i t y ^ , ^ ) by a semantic mapping v which a s s i g n s 0, 1 values to the elements' i n US and i n a d d i t i o n assigns 0, 1 values to the elements outside o f US g according to the Friedman-Glymour proposal mentioned J i n S e c t i o n B. Consider the three .mutually compatible elements x, x Pg,Pg,P^ £ US 3 i n the denumerable P Q M L • According to the Friedman-Glymour QML ^ ) = v ( P X ) = 1 and v ( P ? ) = 0. However, P X A P X = P ? PQML , and so va^A P X ) = v ( P ? ) = 0 / l A l = v ( P g L ) A Y ^ Q ) • F i n a l l y , as an example o f a v i o l a t i o n o f t r u t h - f u n c t i o n a l i t y ^ ) by a semantic mapping v which a s s i g n s 0, 1 values to the elements i n US and i n a d d i t i o n assigns the value N to the elements o u t s i d e o f US o o according to the Levy proposal mentioned i n S e c t i o n B, consider these two j o i n s o f compatible elements i n the twenty element P : P V P 1 and QMA 6 6 V V S l n C e P 6 ' P 6 ' P 7 * U S 3 C P Q M A ' V ( V = V ( P 6 J " ) = V ( P 7 ) = N' S i m i l a r l y , s i n c e P c V P_ = P X £ US 0 , v(P_ v ? n ) = N. But b / o o b y P c V P X = 1 € UF„ , so v(P- V P X ) = 1. In order to show t h a t v i s not b b o 6 6 t r u t h - f u n c t i o n a l ( A ) , assume on the con t r a r y t h a t i t i s t r u t h - f u n c t i o n a l ( A ) . 147 Then 1 = v ( P 6 V P 1 ) = v ( P g ) V v(P^) = N V N = v(P ) V v ( P ? ) = v ( P g V P ?) = N, i . e . , 1 = N, which c o n t r a d i c t s the p r e s u p p o s i t i o n that N t 1. S e c t i o n D. A State-induced Semantics f o r the P_„ S t r u c t u r e s QM As described i n Chapter I I , a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r a Lindenbaum Boolean algebra o f c l a s s i c a l p r o p o s i t i o n a l l o g i c i s a complete c o l l e c t i o n o f u l t r a v a l u a t i o n s on the Lindenbaum al g e b r a . And as described i n Chapter I I I , a state - i n d u c e d , b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r a Boolean P ^ o f c l a s s i c a l mechanics i s a complete c o l l e c t i o n o f state-induced u l t r a v a l u a t i o n s on the P_„ . With these c l a s s i c a l CM precedents i n mind, i n order to f u l l y elaborate the n o t i o n o f a state-induced semantics f o r a quantum P^ M , i t remains to be shown th a t the c o l l e c t i o n o f state-induced u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f a P_„ i s QM complete. We can e s t a b l i s h completeness i n the r e q u i r e d sense i f we can show t h a t , f o r any given p a i r o f d i s t i n c t elements P^ ? P^ i n a P , the. set of atoms dominated by P^ i s not the same as the set o f atoms dominated by P j • For c l e a r l y , i f P^ i s an atom dominated by P^ , i . e . , P^ . < P^ , but not dominated by P^ , i . e . P^ £ P 2 , then the state-induced mapping Exp^, by d e f i n i t i o n assigns the values Exp(P^) = 1 ^ E x p ( P j ) . Now 7 as pointed out by van Fraassen, i f the elements of a P^ M are regarded as subspaces of a H i l b e r t space, i t i s easy t o show t h a t , f o r any P^ i- P j i n a PQ^ , the set of atoms dominated by P^ d i f f e r s from the set o f atoms dominated by P 2 . For as s t a t e d i n Chapter IV(A), a subspace of a H i l b e r t space i s a set of ve c t o r s (which forms a c l o s e d l i n e a r m a n i f o l d ) . Thus any two subspaces of a H i l b e r t space are d i s t i n c t IFF the two subspaces do not c o n t a i n e x a c t l y the same v e c t o r s , where a ve c t o r i n a H i l b e r t space i s 148 uniquely a s s o c i a t e d w i t h an atom i n the P^ M s t r u c t u r e of the H i l b e r t space. However, we may a l s o consider supporting the completeness r e s u l t by an a l g e b r a i c proof which does not invoke the subspace char a c t e r o f the elements o f a P A . For the case o f a P , an a l g e b r a i c proof o f the QM QML completeness r e s u l t can e a s i l y be shown to f o l l o w from Lemma C of Chapter V(A). An a l g e b r a i c proof o f the completeness r e s u l t f o r a x s m o r e d i f f i c u l t . Nevertheless, the weak completeness o f the c o l l e c t i o n of state-induced u l t r a v a l u a t i o n s on a P_.,„ or a Pn.,T i s e a s i l y proved as f o l l o w s : QMA QML P r o p o s i t i o n C: For any P N ^ , the c o l l e c t i o n of state-induced u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f P A M i s weakly complete, i . e . , f o r any element P f 0 i n P A ^ , there i s an Exp^ such t h a t Exp^(P) 4 Exp^(O). By the a t o m i c i t y of P N ^ , f o r any P f 0 i n P N M there i s an atom P^ . € P N ^ such t h a t P^ 5 P, and so the u l t r a f i l t e r UF. = {P € P . „ : P > P,} contains P, w h i l e the dual u l t r a i d e a l Y QM \|r ' U l , = {P € Pnxt : P < P-f"} contains 0 s i n c e 0 5 P f~. Thus the Y QM y Y state-induced u l t r a v a l u a t i o n Exp^ which assigns the value 1 to the members o f UF^. and assigns the value 0 to the members o f UI^ s a t i s f i e s : Exp^(P) = 1 t 0 = Exp^(O). Q.E.D. For the case o f a P.„_ , the completeness r e s u l t i s an immediate QML consequence o f the f o l l o w i n g P r o p o s i t i o n D which f o l l o w s from Lemma C. P r o p o s i t i o n D: For any P . „ T and f o r any P. ,P„ € P^,,T , i f J QML J 12 QML P ; P _ , then there i s an atom P. € P_„ T such t h a t e i t h e r 1 2 Y QML P Y 5 p i a n d P Y * P2 ' O R P Y - P2 a n d P Y * P i ' Assume on the contrary t h a t P^ i ?2 and.for every atom € PQML » 149 P. < P„ IFF P, < P n . Y 1 2 Let { P . } . „ be the set of atoms o f P^„T which are dominated l i£Endex QML by P„ and l e t V P . be the j o i n o f a l l such atoms. Let J 1 . l i {P,}, „ , be the set o f atoms P..,T which are dominated k k €Index QML by P and l e t V P be the j o i n of a l l such atoms. By 2 k k assumption, f o r every atom P^ € P Q M L , 6 { P i } i € l n d e x ^ F P Y € { Pk>k€Index • t h u S ^ i > i € I n d e x = { Pk>k*Index ' a n d S ° V P. = V P, . But by Lemma C, P = V P. and P 0 = V P. ; . I . k l . i 2 . k l k l k thus P^ = P j , which c o n t r a d i c t s the assumption P^ ^ P j . Q.E.D. Now the d e s i r e d r e s u l t f o l l o w s as an immediate C o r o l l a r y to P r o p o s i t i o n D: For any > fhe c o l l e c t i o n o f state-induced u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f P.... QML i s complete, i . e . , f o r any P^ f P j i n » there i s an Exp^. such t h a t E x p ^ P ^ / Ex p ^ ( P j ) . I f P^ ^ P j , then by P r o p o s i t i o n D, there i s an atom PY € PQML S U C h t h a t e l t h e r P y " P l a n d P Y * P 2 ' ° R P Y 5 P 2 and P f ^ P 1 • I f P Y S P l and P^ $ P j , then P± 6 UF = {P € P Q M L : P > P^} but P 2 % UF^ . And so E x p ^ P ^ = 1 but Exp^(Pj) t 1; thus E x p ^ P ^ t E x p ^ ( P j ) . S i m i l a r l y , i f P^ < P j and P^ % ? ± , then P j € UF y but P l * U F Y ' A n d S ° E x p \ | r ^ P 2 ^ = 1 B U T E X P Y ^ P 1 ^ ^ 1 ; t h u S E x p y ( P j ) * Exp vj r(P 1). Q.E.D. For the case o f P„„„. , we now assume th a t P„„. s t r u c t u r e s are QMA QMA not only a s s o c i a t i v e , t r a n s i t i v e , and atomic p a r t i a l - B o o l e a n algebras (as defined i n Chapter 1(D)), but a l s o s a t i s f y the f o l l o w i n g 150 Cond i t i o n A: Every maximal Boolean substructure o f a PQ^ ^S atomic. And we make use of the f o l l o w i n g two lemmas proved by Edwin Levy: Lemma F: For any P 1 , ? 2 i n a P^ , i f P 1 d> ?2 , then there are these f o u r non-exclusive but j o i n t l y exhaustive p o s s i b i l i t i e s : P^ < ? 2 , or P 2 ~ P l ' ° r P 1 " L " P 2 ' ° r t n e r e a r e n o n _ z e r o d i s j o i n t elements P ^ . ? ^ € P Q M such t h a t P 1 1 < P 1 and ? 2 2 < P 2 < Proof: I f P A P 2 , then by the d e f i n i t i o n of c o m p a t i b i l i t y (Chapter IV(C)) there e x i s t three mutually orthogonal elements P n ' P 2 2 ' P 3 ^ *QM s u c n "that P = P V P and P = P V P . We have eig h t cases depending upon which of P , P 2 2 , Pg are or are not equal to the 0 element. (1) I f P ^ = 0 then P 1 = 0 V P 3 = P g and P 2 = P 2 2 P 2 2 V ^ 1 ; thus P^^ < P 2 . (2) I f P 2 2 = 0, then P 2 = 0 V P 3 = P 3 and ? ± = P ^ V Pg = P 1 1 V P 2 ; thus P 2 < P 1 . (3) I f ? ± ± = P 2 2 = 0, then ^ = p 2 = P 3 • I f P l l = P 3 = °» t h e n P l = ° ( a n d S ° P l 5 V ' ( 5 ) I f P22 = P 3 = °' t h e n P 2 = 0 (and so P 2 < P 1 ) . (6) I f P ^ = P 2 2 = P g = 0, then ? ± = ? 2 = 0. ( C l e a r l y , the r e s u l t s o f cases ( 3 ) , ( 4 ) , ( 5 ) , ( 6 ) , are subsumed by the r e s u l t s o f cases (1) and (2).) (7) I f P„ = 0, then P. = P.. v O = P . , and 3 1 11 11 P 2 = P22 V ° = P22 ; thuS V P 2 S l n c e P n - i - P22 * ( 8 ) I f P l l * 0 and P.. 4 0 and P 0 4 0, then s i n c e P = P., v P. , P. < P. . However, 22 3 1 11 3 11 1 P ^ = P^ i s r u l e d out as f o l l o w s , l e a v i n g j u s t P ^ < P^ . Since P 4 0, P l l = P l I F F P 3 ~ P l l ' A n d s i n c e b y assumption P g X P ^ s i . e . , P 3 5 P l l ' W e h a V e P 3 ~ P l l ° n l y i f P 3 = °' B u t f o r t h l s c a s e b y assumption P g 4 0. Thus P ^ < P^ . Mutatis mutandis f o r P 2 2 < p 2 • Q.E.D. 151 Lemma G: A l l the atoms o f a maximal Boolean subst r u c t u r e o f a P_„ are a l s o atoms of P_„ . QM QM Proof: Assume on the c o n t r a r y t h a t there i s a maximal Boolean substructure mBS„ i n P.., and an element Prt € Pniur such t h a t P„ i s an G QM 0 QM 0 atom of rnBSg but PQ i s not an atom o f P . Since P^ i s an atomic s t r u c t u r e , there i s an atom P € 'P„„ such t h a t P < P A (P t P n s i n c e a QM a 0 a 0 by assumption, P Q i s not an atom o f P M^ ; and P & j£ mBSQ s i n c e by assumption PQ is_ an atom o f mBSg .) Now si n c e a l l elements i n a maximal Boolean subst r u c t u r e are mutually compatible, f o r every element P € mBSg , P A PQ . I t f o l l o w s by Lemma F t h a t , f o r every element P € m^S^ s u c n t h a t P t 0 and P * P Q , e i t h e r (1) P < P Q , or (2) P Q < P, or (3) P Q 1 P, or (4) there are nonzero elements P',P' d mBS such t h a t P' < P and 0 0 P' < P . Since by assumption, P„ i s an atom o f mBS,, and P ^ 0, 0 0 0 0 p o s s i b i l i t y (1) P < P^ i s r u l e d out. S i m i l a r l y , s i n c e by assumption, PQ i s an atom o f mBS^ > p o s s i b i l i t y (4) i s r u l e d out. Now c o n s i d e r i n g p o s s i b i l i t y ( 2 ) , i f P < P, then s i n c e P < P. we have P < P, and so r 0 a 0 a p i p . S i m i l a r l y , c o n s i d e r i n g p o s s i b i l i t y ( 3 ) , i f P J_ P, i . e . , P- < P , then s i n c e P < P. we have P < P^ ", and so P k P^, hence P i P. So a G a a a f o r every element P € such t h a t P ^ 0 and P ^ PQ , we have P o P. Moreover, f o r P = 0, s i n c e P A 0 we l i k e w i s e have P i P. a a a And f o r P = P. , sin c e P ^ Pn we l i k e w i s e have P i P. That i s , f o r 0 a 0 a every element P € mBS- , P i P. and Pc>P . So the set o f mutually U v EL compatible elements mBSQ U {P^} generate a Boolean substructure o f P M^ which contains a l l the elements o f IIIBSQ p l u s P^ (and perhaps o t h e r s ) . Thus mBSg i s the proper subset o f a Boolean subst r u c t u r e o f P^ , which c o n t r a d i c t s the d e f i n i t i o n o f mBS as a maximal Boolean s u b s t r u c t u r e . 0 Q.E.D. 152 Furthermore, given the conjecture t h a t every mBS of a PA^ i s atomic, i t i s a t r i v i a l p o i n t that a l l atoms o f P_„ which are i n an mBS of P.,, QM QM are a l s o atoms of the mBS. For the only way an atom of P A M which i s i n an mBS o f Prt„ could not be an atom of mBS i s i f some other element QM P € mBS were between P and the 0-element i n mBS but not i n P.„ . a QM But s i n c e mBS i s a substructure o f P.,, , i f some P € mBS were such t h a t QM 0 < P < P_ i n mBS then a l s o 0 < P 5 P_ i n P_, , and so P would not QM be an atom o f P_„ . QM We a l s o make use o f the f o l l o w i n g r e s u l t s . As mentioned i n Chapter 1(D), Hughes has proven t h a t any p a r t i a l - B o o l e a n algebra i s isomorphic to a p a r t i a l - B o o l e a n algebra constructed on a f a m i l y o f Boolean algebras ^ i ^ i € l n d e x ' 3 S ^ e s c r > i b e d by Kochen-Specker. Among other c o n d i t i o n s , the constructed p a r t i a l - B o o l e a n algebra A s a t i s f i e s , f o r any elements b,c,d € A , b V c = d i n A IFF there i s a 8. such t h a t b V c = d i n l 8^  . Now as part o f h i s proof, Hughes shows t h a t any p a r t i a l - B o o l e a n algebra can be constructed on the f a m i l y o f i t s own Boolean subalgebras. So i n p a r t i c u l a r , any c a n b e constructed on the f a m i l y o f i t s own Boolean subalgebras. Thus we have the f o l l o w i n g P r o p o s i t i o n E: For any P Q M and f o r any P . P ^ P j € P Q M , P = P l V P 2 i n Q^MA I F F t n e r e i s a B o o l e a n s u b s t r u c t u r e BS o f P-„. such t h a t P = P, v P„ i n BS. QMA 1 2 We a l s o make use o f these two lemmas. Lemma H: For any P 1,P 2 € P , i f P J v P 2 i s de f i n e d i n PQMA ' i , e ' ' i f P l ^ P 2 ' t h e n P l V P 2 1 S t h e l e a s t upper bound o f {Pl,P2} i n P Q M . 153 Proof: C l e a r l y , P. vP„ > P. and P. v P > P. ; thus P. v P n i s an J ' 1 2 1 1 2 2 1 2 upper bound of { P ^ P j } . And f o r any P € PQ M A , i f P > P 1 , i . e . , P v P 1 = P (and P A P^), and P > P ' , i . e . , P v P j = P, then because PQMA i s an a s s o c i a t i v e p a r t i a l - B o o l e a n algebra which s a t i s f i e s : P, A (P 0V P) IFF (P„ v P0)<A P, we have (P. V P j o P s i n c e P , i P and 1 2 1 2 1 2 1 P = P V P 2 = P 2 V P ( i . e . , P 1 A ( P 2 V P ) ) . So P A (P v P 2 ) and moreover, P = P V P 2 = (P V P 1) V P 2 = P V ( P 1 v P 2 ) , i . e . , P > P v P 2 . Q.E.D. Halmos' Lemma: In an atomic Boolean a l g e b r a , every element i s the j o i n ( l e a s t upper bound) of the atoms i t dominates (Halmos, 1963, p. 70). Now we may prove the f o l l o w i n g Theorem C f o r P . , which QMA corresponds to the above P r o p o s i t i o n D f o r P^„ T . (The proof i s due to QML Levy, Robinson, Chernavska.) Theorem C: For any P Q M and f o r any P ± . P 2 € P Q M , i f P -;";^'P ., then the set o f atoms dominated by P^ i s not equal to the set o f atoms dominated by P 2 ( i . e . , there i s an atom P, € P„„„ such t h a t e i t h e r P. < P. and P, £ P. , or P. < P. ty QMA ty 1 ty 2 ty 2 and P ^ P , ) . Proof: Let A„ be the set o f atoms o f P.,,. which are dominated by P. , 1 QMA J 1 i . e . , A. = {P. € Pn... : P. 5 P.}; and l e t A„ be the set of atoms o f 1 ty QMA ty 1 2 PQMA w n i c n a r e dominated by P 2 . Assume A^ = A 2 . C l e a r l y , i f A^ = A 2 = 0 (the empty s e t ) , then P^ = P 2 = 0. Assume then t h a t A^ = A 2 t 0. Since A^ = A 2 i- 0, there i s a nonempty set Ai_ of mutually compatible atoms o f PQ M A J each o f which i s dominated by P^ and by P 2 . (1) Since P^ dominates each member of A^ , P^ i s compatible w i t h each 154 member o f . Thus, A^ U {P- } i s a set of mutually compatible elements of PQJJ^  • Hence there i s a Boolean subalgebra o f PQJ^ c o n t a i n i n g P^ and a l s o c o n t a i n i n g a l l members of A^ ; and t h i s Boolean subalgebra i s contained i n a maximal Boolean subalgebra mBS' of P „ W « • By Co n d i t i o n A, QMA mBS' i s atomic, and by Lemma G, a l l o f i t s atoms are atoms o f P N M F L • Let A' = {Pi, } i ^ i n c j e x be the set of a l l atoms of mBS' which are dominated by i P^ ; c l e a r l y , A' c A^ . Now by Halmos's Lemma, P i s the l e a s t upper bound o f A' i n mBS', i . e . , P = V P! i n mBS*. Then by P r o p o s i t i o n E i * i P l = V P ^ i n POMA ' A n d b y Lemma H, V P ' i s the l e a s t upper bound o f i i i i A' i n • N o w p 2 dominates every member of A^ = , and A' c A^ , so P^ dominates every member of A', and hence P^ dominates the l e a s t upper bound o f A', namely, P^ . (2) By a s i m i l a r argument i t can be shown th e t P^ dominates P^ • Thus, P 1 = P 2 . So i f P t P 2 , then h^t . Q.E.D. And as i n the PQ^L case, the d e s i r e d completeness r e s u l t f o l l o w s as an immediate C o r o l l a r y to Theorem C: For any PQJ^ » t n e c o l l e c t i o n of state-induced u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f P N . . A QMA i s complete, i . e . , f o r any P^ 4 P 2 i n PQ^ > there i s an Exp^ such t h a t Exp^CP^ 4 Exp^,(P 2). The proof o f the c o r o l l a r y proceeds as i n the PQJJL case. Summary As described i n Chapters I I and I I I , a b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r a Lindenbaum Boolean algebra L o f c l a s s i c a l p r o p o s i t i o n a l 155 l o g i c i s a complete c o l l e c t i o n o f u l t r a v a l u a t i o n s on L, and a state-induced, b i v a l e n t , t r u t h - f u n c t i o n a l semantics f o r a Boolean P_„ of c l a s s i c a l CM mechanics i s a complete c o l l e c t i o n o f state-induced u l t r a v a l u a t i o n s on P ^ . In both cases, an u l t r a v a l u a t i o n i s a mapping which assigns the value 1 to the elements i n an u l t r a f i l t e r UF and assigns the value 0 to the elements i n the dual u l t r a i d e a l U l ; thus an u l t r a v a l u a t i o n i s s a i d to be defined w i t h respect to an UF and dual U l . C l e a r l y , an u l t r a v a l u a t i o n i s a b i v a l e n t mapping on UF U U l , i . e . , every element i n UF U U l i s assigned a 0 or a 1 value. And i t f o l l o w s from the c o n d i t i o n s s a t i s f i e d by any UF and dual U l i n a Boolean s t r u c t u r e t h a t an u l t r a v a l u a t i o n i s a t r u t h - f u n c t i o n a l mapping on UF U U l . Moreover, because the L, Pn^ s t r u c t u r e s are Boolean, f o r any UF and dual Ul i n an L or a Pn^ , UF U U l = L and UF U U l = P„, , thus the domain of each u l t r a v a l u a t i o n i s CM the e n t i r e L, P ^ s t r u c t u r e . And the completeness of the c o l l e c t i o n o f u l t r a v a l u a t i o n s on an L or a PC M i s ensured by the s e m i - s i m p l i c i t y property of Boolean s t r u c t u r e s . Furthermore, f o r the case of c l a s s i c a l p r o p o s i t i o n a l l o g i c , each u l t r a v a l u a t i o n on the L s t r u c t u r e of equivalence c l a s s e s of well-formed formulae i n a (closed) set L i s an a l g e b r a i c v e r s i o n of one o f the standard v a l u a t i o n s f o r L, which i s pa r t of the reason u l t r a v a l u a t i o n s are so c a l l e d and i s the main reason why u l t r a v a l u a t i o n s on any other p r o p o s i t i o n a l or l o g i c a l s t r u c t u r e are regarded i n t h i s t h e s i s as semantic mappings. And f o r the case o f c l a s s i c a l mechanics, u l t r a v a l u a t i o n s are s a i d to be state-induced because i n f a c t i t i s the s t a t e s o f a c l a s s i c a l mechanical system which induce mappings, namely, d i s p e r s i o n - f r e e c l a s s i c a l p r o b a b i l i t y measures, each o f which u- i s an u l t r a v a l u a t i o n on the w UF w U UI^ = Pn M s t r u c t u r e o f p r o p o s i t i o n s which make a s s e r t i o n s about the values o f the system's magnitudes. 156 When we consider a P .„ s t r u c t u r e of p r o p o s i t i o n s which make QM a s s e r t i o n s about the values o f a quantum mechanical system's magnitudes, the st a t e s o f the system s i m i l a r l y induce mappings, namely, d i s p e r s i v e g e n e r a l i z e d p r o b a b i l i t y measures, each o f which E x P ^ i s a n u l t r a v a l u a t i o n on UF, U UI, c P . And as i n the c l a s s i c a l cases, each state-induced ty y ~ QM u l t r a v a l u a t i o n Exp^ i s a b i v a l e n t mapping on UF^ U UI^ ; and i t f o l l o w s from the c o n d i t i o n s s a t i s f i e d by any UF, and dual UI, i n a P_„ t h a t J J ty ty QM each state-induced u l t r a v a l u a t i o n i s a t r u t h - f u n c t i o n a l ((A) or (<t>,&)) mapping on UF^ U UI^ . But u n l i k e the c l a s s i c a l cases i n which UF U UI = L and UF U UI = P „ „ f o r every u l t r a f i l t e r and dual u l t r a i d e a l w w CM J i n L, P^y , f o r the quantum case, i f P contains incompatible elements, then not every u l t r a f i l t e r and dual u l t r a i d e a l i n P . „ i s such t h a t QM UF. U UI. = P.., , r a t h e r , f o r some UF, and dual UI, , UF, U UI, c P N M . ty ty QM ty ty ty ty QM When UF, U UI, i s l e s s than the e n t i r e P^„ , we can a t l e a s t be sure t h a t ty ty QM UF, U UI. i s a closed substructure o f P_„ , which may be c a l l e d an ty ty QM J u l t r a s u b s t r u c t u r e . However, the a f f i l i a t e d state-induced u l t r a v a l u a t i o n Exp^, i s a b i v a l e n t , t r u t h - f u n c t i o n a l ( ( A ) or (o,<J6)) mapping on j u s t t h a t u l t r a s u b s t r u c t u r e US^ o f P^^ . Thus w h i l e every u l t r a v a l u a t i o n on an L and every state-induced u l t r a v a l u a t i o n on a P „ „ i s a b i v a l e n t , t r u t h -J CM f u n c t i o n a l mapping on the e n t i r e s t r u c t u r e , a t l e a s t some of the state-induced u l t r a v a l u a t i o n s on a P^ M c o n t a i n i n g incompatible elements are b i v a l e n t , t r u t h - f u n c t i o n a l ( ( o ) or (A ,&)) mappings on j u s t u l t r a s u b s t r u c t u r e s o f P Q ^ r a t h e r than on the e n t i r e P^ M . Moreover, the completeness of the c o l l e c t i o n o f state-induced u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f a P Q M must be proven, as done i n Se c t i o n D. However, the f a c t t h a t UF, U UI, c p f o r some UF, and dual ty ty QM ty 157 UI^, i n a PA^ c o n t a i n i n g incompatible elements need not be a problematic f e a t u r e and _is_ not the only problematic f e a t u r e of the quantum Pn^ s t r u c t u r e s . As described i n Chapter V(B), i f we ignore the l a t t i c e meets and j o i n s of incompatibles and consider the proposal of a b i v a l e n t , truth-functional(<!?) semantics f o r a. P A M , the presence of incompatible elements i n Pn^ i s necessary but not s u f f i c i e n t to r u l e out a b i v a l e n t , t r u t h - f u n c t i o n a l (A) semantics f o r P A„ . For a two-dimensional H i l b e r t QM space Pn^ does admit a b i v a l e n t , t r u t h - f u n c t i o n a l ( i ) semantics i n s p i t e of the presence o f incompatible elements. The p e c u l i a r s t r u c t u r a l f e a t u r e _n—3 of three-or-higher dimensional H i l b e r t space V s t r u c t u r e s which does r u l e out a b i v a l e n t , truth-functional(A) semantics i s the presence of _n>3 overlapping maximal Boolean substructures i n y , f o r which the presence of incompatible elements i s a necessary (but not a s u f f i c i e n t ) c o n d i t i o n . The f o l l o w i n g s i m i l a r remarks apply here i n the Chapter VI d i s c u s s i o n of the proposal of a semantics f o r Pn^ c o n s i s t i n g of a complete c o l l e c t i o n o f (state-induced) u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s of P„„ : QM I f we ignore the l a t t i c e meets and j o i n s o f incompatibles and consider the proposal o f a b i v a l e n t , truth-functional(<b) semantics f o r P n M c o n s i s t i n g o f u l t r a v a l u a t i o n s , then the f a c t t h a t UF, U U l , c P„„ ( r a t h e r y y QM than UF. U U l , = P n M) f o r some UF, and dual U l , i n any p . „ . which y y QM y y J QM contains incompatible elements would by i t s e l f be harmlessly unproblematic i f UF, U U l , were equal t o j u s t a mBS o f P„„ and i f the mBS's o f — y y QM — 2 PqM were non-overlapping. For example, both these " i f ' s " o b t a i n i n a P n M 2 and as described i n Chapter I V ( F ) , a does admit a complete c o l l e c t i o n o f b i v a l e n t , truth-functional(o) mappings, where by i n s p e c t i o n i t i s c l e a r that each of the f o u r b i v a l e n t , t r u t h - f u n c t i o n a l ( i ) mappings on the 2 six-element P e x p l i c i t l y considered i n t h a t Chapter IV(F) i s i n f a c t a sum 158 of two u l t r a v a l u a t i o n s , each defined on one of the two u l t r a s u b s t r u c t u r e s of the six-element P;L . That i s , UF, = {P € P ? M : P > P. } = {P.,.1}. QM 1 QM 1 1 U I 1 = { P € PQM : P ~ P l } = <P1'0>' a n d U S 1 = U F 1 U U I 1 = ; UF 2 = { P 2 , l } , U I 2 = {P 2,0}, and US 2 = UF 2 U U I 2 = mBS2 . So each US. 2 equals an mBS. i n P . W , and as described e a r l i e r i n t h a t s e c t i o n , the ^ I QM 2 2 mBS's of P . „ do not ove r l a p . (And the six-element P_„ equals the union QM QM of the two u l t r a s u b s t r u c t u r e s US^ U US 2-) Moreover, the two mappings h a , , are u l t r a v a l u a t i o n s on US^ , the two mappings h c , h^ are u l t r a v a l u a t i o n s on US 2 , and each o f the f o u r b i v a l e n t , t r u t h - f u n c t i o n a l ( A ) 2 mappings h^ , h 2 , h^ , h^ , on the e n t i r e six-element P^^ i s the sum of an u l t r a v a l u a t i o n on US^ plus an u l t r a v a l u a t i o n on US 2 . Thus, the 2 2 six-element , and more g e n e r a l l y , any P does admit a b i v a l e n t , t r u t h - f u n c t i o n a l (A) semantics c o n s i s t i n g o f a complete c o l l e c t i o n o f 2 b i v a l e n t , t r u t h - f u n c t i o n a l ( o ) mappings on the e n t i r e P , each o f which 2 i s a sum o f u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s of P . „ . So the f a c t QM tha t UF. U UI, c P f o r some UF, and dual UI, i n any P . „ c o n t a i n i n g Y Y QM ty ty J QM incompatible elements need not be a problematic f e a t u r e . In p a r t i c u l a r , the 2 u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f a P c o n t a i n i n g incompatible elements may be added together t o y i e l d a complete c o l l e c t i o n of b i v a l e n t , 2 truth-functional(<!>) mappings on the e n t i r e P^ M , and thus a b i v a l e n t , 2 truth-functional(<i>) semantics f o r P^ , i n s p i t e o f t h a t f a c t . However, n e i t h e r of the above, und e r l i n e d " i f s " o b t a i n i n a _n>3 three-or-higher dimensional H i l b e r t space PQ M . That i s , the mBS's of a P^~^ may o v e r l a p , and the u l t r a s u b s t r u c t u r e s i n a m a v be l a r g e r than -rJi—3 any mBS, f o r each u l t r a s u b s t r u c t u r e US^ i n a P ^ i s equal t o the union o f a l l the overlapping mBS's i n which share the atom P^ , as shown n>3 i n S e c t i o n B. So the f a c t t h a t UF, U UI, c P f o r some UF, and dual Y ty QM ty 159 TJI—3 UI^ . in any P Q M containing incompatible elements is problematic. In particular, the best we can do for such a i s to define bivalent, truth-functional ((6) or (<!>,&)) mappings on i t s ultrasubstructures. We cannot add together these ultravaluations on the ultrasubstructures of a P Q 1 " 3 to get bivalent, truth-functional(A) mappings on the entire P Q J ^ ' But the fact that UF. U Ul. c Pl„ is not the only reason why we cannot add \|/ \|/ Q M - J the ultravaluations in the suggested manner; the other reason i s that an ultrasubstructure UF^ U UI^, in a P ^ is a union of overlapping mBS's pn>3 i n QM ' Other sorts of semantic mappings may be and have been proposed for the quantum P structures. But in this thesis, only two semantic proposals have been seriously considered: the proposal of a bivalent, truth-functional semantics for P A ^ and the proposal of a state-induced semantics for Pnx, . The former i s motivated by the success and usefulness Q M J of such a semantics for classical logical and propositional structures such as L, P n M . The latter is motivated by the fact that the state-induced semantics for a P , consisting of state-induced u. mappings already CM W present in the formalism of classical mechanics, works exactly like the algebraic version of the standard, bivalent, truth-functional semantics of classical propositional logic. And the proposal of a state-induced semantics for is motivated by the fact that the quantum formalism, like the classical formalism, includes state-induced mappings which assign 0, 1 values to representatives of quantum propositions, i.e., to the projectors or subspaces of a Hilbert space. So for a P n ^ , i t is worth considering the notion of a state-induced semantics consisting of the state-induced Exp^ , mappings already present in the quantum formalism. Like the classical semantic mappings on an L, l i k e the classical state-induced u, mappings w 160 on a P „ „ , and l i k e the Friedman-Glymour S3-valuations proposed f o r a CM P Q M L , the state-induced Exp^ mappings are u l t r a v a l u a t i o n s on the u l t r a s u b s t r u c t u r e s o f P A „ . So the b a s i c semantic method i n a l l these cases Q M i s the same. The c r u c i a l d i f f e r e n c e between the c l a s s i c a l and the quantum cases i s t h a t , f o r some UF^ and dual U I ^ i n any P A M c o n t a i n i n g incompatible elements, UF^ U UI^ i s smaller than P A M r a t h e r than being equal to the e n t i r e ' P A M , and moreover, UF^ U UI^ i s l a r g e r than any mBS i n a PQ M because UF^, U UI^ i s the union o f a l l o v e r l a p p i n g mBS's i n P^~ 3 which c o n t a i n the atom P, . Q M y Notes: 1 Though other c o n d i t i o n s are sometimes taken as d e f i n i n g the o r t h o g o n a l i t y r e l a t i o n , e.g., P 1 , P 2 are orthogonal IFF ^  ' p 2 = °' these c o n d i t i o n s are s a t i s f i e d by any P 1 ' P 2 * PQM I F F P l 5 P 2 ' i , e ' ' I F F p i » P 2 a r e d i s J o i n t ' A n d » f o r example, P i r o n takes the P^ < P^ c o n d i t i o n as d e f i n i n g the o r t h o g o n a l i t y r e l a t i o n ( P i r o n , 1 9 7 6 , p. 2 9 ) . 2 This was pointed out to me independently by Dr. L. P. B e l l u c e and Dr. J . V. Whittaker. 3 The f a c t t h a t P 1 A P 2 and P^^ V P 2 are not defined i n P when P^ ty P 2 does not mean t h a t the union UF U Ul i s i n any way not cl o s e d w i t h respect to the A , V operations o f PQMA ' T h* e A ' V °Perations o f P N M A = < E , C 3 , < , A , V , X , 0 , 1 > are d e f i n e d from i c E x E to E r a t h e r than from E xE to E. Thus Kochen-Specker c a l l them p a r t i a l - o p e r a t i o n s or p a r t i a l - f u n c t i o n s (1965, pp. 177, 178). By c l o s u r e w i t h respect to the A , V operations o f PA_.. , I mean c l o s u r e w i t h respect to these operations qua p a r t i a l - o p e r a t i o n s . H Thanks to Dr. Edwin Levy f o r suggesting the " u l t r a " terminology. 5 In an e a r l i e r d r a f t , I claimed t h a t each quantum e x p e c t a t i o n -f u n c t i o n Exp^ on a P_ M i s b i v a l e n t and t r u t h - f u n c t i o n a l w i t h respect to a Boolean subst r u c t u r e o f mutually compatible elements i n P A M . Thanks to J e f f r e y Bub and Edwin Levy f o r h e l p i n g c l a r i f y t h a t i n f a c t , the subset of elements i n a P which are assigned 0 , 1 values by an Exp^ on P A M may i n c l u d e incompatible elements and so may be l a r g e r than 161 any Boolean substructure of P-,„ . QM Page 1 of a manuscript by Edwin Levy circulated in December, 1977. 7 As the external examiner, van Fraassen pointed out this alternate proof of the completeness result in his report on the thesis. 162 CHAPTER V I I HIDDEN-VARIABLES RECONSIDERED Preface In c l a s s i c a l mechanics, a pure s t a t e w s p e c i f i e s an exact value f o r any magnitude. But i n quantum mechanics, a pure s t a t e Y s p e c i f i e s an exact (eigen)value f o r only those magnitudes whose eigenstates are compatible w i t h Y * Any magnitude A whose eigenstates are incompatible w i t h Y m ay> upon measurement, e x h i b i t any of i t s (eigen)values. In the quantum formalism, f o r the given s t a t e Y "the average value of A i s determined by Exp^(A) = a J l Y ^ y l l > and the p r o b a b i l i t y t h a t A w i l l e x h i b i t any one of i t s (eigen)values, say a_. , i s determined by Exp,(P, ) = [|Y~.><Y1|^, where P, represents the eigenstate of A a s s o c i a t e d Y Yj 3 Yj w i t h the (eigen)value a_. . But the quantum formalism does not determine which exact (eigen)value A w i l l e x h i b i t . In other words, quantum systems c h a r a c t e r i z e d by the same quantum s t a t e Y e x h i b i t , upon measurement, d i f f e r e n t values f o r the same magnitude A, yet the quantum formalism does not determine which of the d i f f e r e n t values of A w i l l be e x h i b i t e d . For t h i s reason, i t has been argued t h a t quantum mechanics i s incomplete and should be supplemented by a hidden v a r i a b l e theory which r e f l e c t s the d i f f e r e n t p o s s i b l e outcomes of a measurement of A f o r a given Y -In terms of the quantum p r o p o s i t i o n s , t h i s problem i s connected w i t h the f a c t , described i n Chapter V I, that any P^ which contains incompatible elements has at l e a s t one u l t r a s u b s t r u c t u r e US^ . = UF^, U UI^ 163 which i s smaller than the e n t i r e P-w , and each element of P.„ which i s QM QM outside UEy i s assigned a value between 0 and 1 by the a f f i l i a t e d state-induced Exp^ r a t h e r than being assigned an exact 0 or 1 value by Exp^, . At the very l e a s t , such a value between 0 and 1 i s i n t e r p r e t e d as the p r o b a b i l i t y t h a t an element P jE US^ , qua idempotent magnitude, w i l l upon measurement e x h i b i t i t s (eigen)value 1, that i s , as the p r o b a b i l i t y that P, qua p r o p o s i t i o n , i s tr u e of a system or ensemble of systems whose s t a t e i s ty. So again, quantum systems c h a r a c t e r i z e d by the same quantum s t a t e ty e x h i b i t upon measurement, sometimes the t r u t h - v a l u e 0 and sometimes the t r u t h - v a l u e 1 f o r the same p r o p o s i t i o n P A US^ , but which o f these t r u t h - v a l u e s w i l l be the outcome of a measurement i s not determined by the quantum formalism. Now i f we presume that the p h y s i c a l theory of quantum phenomena should i n c l u d e a formalism which does determine, given the s t a t e of a quantum system, e x a c t l y whether any P € P i s tr u e or f a l s e , then quantum mechanics i s indeed an incomplete theory and we must seek a supplementary formalism. The proposals o f such supplementary formalisms have been c a l l e d h idden-variable t h e o r i e s . Hidden-variable (HV) t h e o r i e s are extensions or r e c o n s t r u c t i o n s of quantum mechanics which introduce f u r t h e r s p e c i f i c a t i o n s of the s t a t e of a quantum system so t h a t the s o - c a l l e d hidden s t a t e determines the exact values o f magnitudes and p r o p o s i t i o n s which are assigned d i s p e r s i v e values by a quantum state-induced Exp^, . So w h i l e a quantum s t a t e ty induces the g e n e r a l i z e d p r o b a b i l i t y measure Exp^. : P^ ^ ->• [0,1] which i s d i s p e r s i v e w i t h respect to every P £ US^, , a hidden s t a t e induces or i s a s s o c i a t e d w i t h a d i s p e r s i o n - f r e e p r o b a b i l i t y measure which somehow assigns an exact 0 or 1 value to such P f. US^ . . And so i n an HV 164 r e c o n s t r u c t i o n o f quantum mechanics, the presumed incompleteness o f quantum mechanics i s r e f l e c t e d by the f a c t t h a t the set of quantum Exp^, measures i s a proper subset o f a l a r g e r set of measures which i n c l u d e s the d i s p e r s i o n - f r e e HV measures. The d i s p e r s i o n - f r e e measures added by an HV theory may be c l a s s i c a l p r o b a b i l i t y measures on some Boolean s t r u c t u r e proposed by the HV theory, or they may be some s o r t of g e n e r a l i z e d p r o b a b i l i t y measures defined on the quantum P A ^ ( o r on substructures o f PQ^)* Von Neumann, Jauch-Piron and Gleason-Kochen-Specker prove the i m p o s s i b i l i t y o f three kinds o f ge n e r a l i z e d d i s p e r s i o n - f r e e measures on the quantum P A ^ s t r u c t u r e s , as described i n Chapter V(D); they thus r u l e out three kinds of HV t h e o r i e s , as s h a l l be elaborated below. But besides these three proposed but impossible kinds of HV t h e o r i e s , c o n t e x t u a l HV t h e o r i e s whose d i s p e r s i o n - f r e e measures avoid the above i m p o s s i b i l i t y proofs have a l s o been proposed. In a l l , f o u r cases, each quantum Exp^ measure i s represented i n the proposed HV theory as a mixture or complex, e.g., a convex sum or weighted i n t e g r a l , of d i s p e r s i o n - f r e e HV measures. And a l l f o u r kinds o f HV proposals impose a s t a t i s t i c a l c o n d i t i o n r e q u i r i n g t h a t the complexes which represent the quantum Exp^, measures i n the HV theory must y i e l d s t a t i s t i c a l r e s u l t s which reproduce the r e s u l t s given by the quantum Exp^, measures (and so f a r observed by experiment) (Kochen-Specker, 1967, p. 59; B e l i n f a n t e , 1973, p. 9). However, as Kochen-Specker argue, the i m p o s i t i o n of t h i s s t a t i s t i c a l c o n d i t i o n alone does not yet take i n t o c o n s i d e r a t i o n the s t r u c t u r a l and f u n c t i o n a l r e l a t i o n s among the quantum magnitudes (and p r o p o s i t i o n s ) . These r e l a t i o n s are embodied i n the a l g e b r a i c s t r u c t u r e of the quantum magnitudes, and concordantly, i n the P A M s t r u c t u r e o f the 165 quantum p r o p o s i t i o n s . Von Neumann, Jauch-Piron, Gleason, and Kochen-Specker do take t h i s c o n s i d e r a t i o n i n t o account by r e q u i r i n g t h a t some or a l l o f the operations and r e l a t i o n s of P must be preserved i n an HV r e c o n s t r u c t i o n of quantum mechanics. Such requirements may be c a l l e d s t r u c t u r a l c o n d i t i o n s . As shown a t l e n g t h i n Chapter V(D), each of these authors imposes a s t r u c t u r a l c o n d i t i o n which b o i l s down to the requirement that d i s p e r s i o n - f r e e HV measures, qua g e n e r a l i z e d p r o b a b i l i t y measures on the quantum P , must preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s  of PQ^ ( i . e . , P ^ ^ - p r e s e r v a t i o n ) , or i n other words, proposed d i s p e r s i o n - f r e e HV measures must be b i v a l e n t homomorphisms(<b) on P . In a d d i t i o n , von Neumann and Jauch-Piron each impose a s t r u c t u r a l c o n d i t i o n , l a b e l e d (vNJ6) and (JP&) i n Chapter V(D), which r e q u i r e s t h a t proposed d i s p e r s i o n - f r e e HV measures preserve an op e r a t i o n among incompatibles. So von Neumann's n o t i o n and Jauch-Piron's n o t i o n o f what i s a g e n e r a l i z e d p r o b a b i l i t y measure on P^ M i s c l e a r l y d i f f e r e n t from Gleason's and Kochen-Specker's n o t i o n . Now a l l of these s t r u c t u r a l c o n d i t i o n s are s a t i s f i e d by the quantum Exp^ measures on P ^ . The contentious i s s u e i s whether or not the proposed d i s p e r s i o n - f r e e HV measures introduced by a proposed HV extension o r r e c o n s t r u c t i o n o f quantum mechanics must a l s o be re q u i r e d to s a t i s f y these s t r u c t u r a l c o n d i t i o n s . The three kinds of HV proposals which r e q u i r e t h e i r d i s p e r s i o n - f r e e HV measures to s a t i s f y the three d i f f e r e n t sets of s t r u c t u r a l c o n d i t i o n s imposed by von Neumann, Jauch-Piron, and Gleason-Kochen-Specker have been shown by these authors to be impossible; e i t h e r the d i s p e r s i o n - f r e e HV measures are themselves impossible or e l s e complexes of the d i s p e r s i o n - f r e e HV measures cannot reproduce the s t a t i s t i c a l r e s u l t s o f the quantum Exp^ 166 measures, as r e q u i r e d by the s t a t i s t i c a l c o n d i t i o n . However, c r i t i c s o f the above HV i m p o s s i b i l i t y proofs and advocates of the c o n t e x t u a l HV proposals have brought f o r t h the f o l l o w i n g three s o r t s of arguments ag a i n s t the i m p o s i t i o n of the s t r u c t u r a l c o n d i t i o n s upon proposed d i s p e r s i o n - f r e e HV measures: ( i ) The s t r u c t u r a l c o n d i t i o n s are i n c o n s i s t e n t w i t h the other c o n d i t i o n s which are imposed upon the proposed HV measures, and so the s t r u c t u r a l c o n d i t i o n s immediately r u l e out a HV theory. But r a t h e r than concluding t h a t an HV theory i s i m p o s s i b l e , we should r e j e c t the s t r u c t u r a l c o n d i t i o n s , ( i i ) The s t r u c t u r a l c o n d i t i o n s r e l a t e the r e s u l t s o f d i f f e r e n t measurements i n ways which are not j u s t i f i e d i f we take i n t o account the i n t e r a c t i o n between measuring instruments and quantum phenomena. Thus the i m p o s i t i o n of the s t r u c t u r a l c o n d i t i o n s begs the question, and these c o n d i t i o n s should be r e j e c t e d . ( i i i ) The i m p o s i t i o n of the s t r u c t u r a l c o n d i t i o n s and the development of the i m p o s s i b i l i t y proofs beg the HV  question i n other ways. So the s t r u c t u r a l c o n d i t i o n s should be r e j e c t e d and the von Neumann, Jauch-Piron, Gleason, and Kochen-Specker proofs do not i n f a c t show the i m p o s s i b i l i t y o f an HV r e c o n s t r u c t i o n of quantum mechanics. In S e c t i o n A, these c r i t i c i s m s are described i n d e t a i l . Then i n Se c t i o n B, another p e r s p e c t i v e on quantum mechanics and the problem of hidden-variables i s introduced, according to which the s t r u c t u r a l c o n d i t i o n s (vN^) and (JP&) (and thus the von Neumann and the Jauch-Piron proofs) succumb to the above c r i t i c i s m s , but the s t r u c t u r a l c o n d i t i o n of P n ^ - p r e s e r v a t i o n (and thus the Gleason and Kochen-Specker proofs) are rescued from these c r i t i c i s m s . 167 Section A. C r i t i c i s m s o f the Hidden-Variable I m p o s s i b i l i t y Proofs Von Neumann poses the question of whether the d i s p e r s i v e ensembles of quantum systems can be re s o l v e d i n t o sub-ensembles which are d i s p e r s i o n - f r e e f o r any quantum magnitude; i n h i s view, an HV r e c o n s t r u c t i o n of quantum mechanics i n v o l v e s such a r e s o l u t i o n . Ensembles o f quantum systems are c h a r a c t e r i z e d by e x p e c t a t i o n - f u n c t i o n s , and so the question i s whether the d i s p e r s i v e quantum Exp^ f u n c t i o n s can be represented as mixtures or weighted sums of d i f f e r e n t d i s p e r s i o n - f r e e HV Exp^ fu n c t i o n s (von Neumann, 1932, pp. 305-307, 324). Von Neumann defin e s an ex p e c t a t i o n - f u n c t i o n by a l i s t of c o n d i t i o n s , one of which subsumes the two co n d i t i o n s l a b e l e d (vNo) and (vN#>) i n Chapter V(D). The domain of an ex p e c t a t i o n - f u n c t i o n i s the set of quantum magnitudes as represented by operators on a H i l b e r t space. And the f u n c t i o n a l r e l a t i o n s among the magnitudes are given by the f u n c t i o n a l r e l a t i o n s among the operators, t h a t i s , by the a l g e b r a i c s t r u c t u r e of the operators. So i n terms of the quantum p r o p o s i t i o n s qua idempotent magnitudes, a necessary c o n d i t i o n f o r an HV r e c o n s t r u c t i o n o f quantum mechanics i s , i n von Neumann's view, the existence • 1 of d i s p e r s i o n - f r e e Exp^ f u n c t i o n s on the quantum s t r u c t u r e s . As mentioned i n Chapter V(D), usi n g h i s trace-formalism, von Neumann proves t h a t no such d i s p e r s i o n - f r e e E xp w e x i s t ; I r e f e r r e d to t h i s r e s u l t as von Neumann's i m p o s s i b i l i t y proof. In a d d i t i o n , von Neumann proves that homogeneous ex p e c t a t i o n - f u n c t i o n s do e x i s t and i n f a c t correspond to the quantum Exp^ f u n c t i o n s induced by the pure quantum ty s t a t e s . So the quantum Exp^, cannot be represented as mixtures o f d i s p e r s i o n - f r e e Exp^ , f i r s t because the quantum Exp^, are themselves homogeneous (where by d e f i n i t i o n a homogeneous Exp cannot be represented as a weighted sum of 168 d i f f e r e n t E x p - f u n c t i o n s ) , and second because the d i s p e r s i o n - f r e e E x P w d o not e x i s t (von Neumann, 1932, p. 324). I t i s thus t h a t an HV r e c o n s t r u c t i o n of quantum mechanics i s impossible, according to von Neumann. In 1966, B e l l d i s c r e d i t e d von Neumann's i m p o s s i b i l i t y proof by arguing that i t r e s t s upon an i n c o n s i s t e n c y between the requirement t h a t HV exp e c t a t i o n - f u n c t i o n s s a t i s f y (vNjS) and the requirement t h a t HV expe c t a t i o n - f u n c t i o n s be d i s p e r s i o n - f r e e . For (vN&) r e q u i r e s the a d d i t i v i t y of the expectation values of incompatible magnitudes and incompatible p r o p o s i t i o n s qua idempotent magnitudes, and the d i s p e r s i o n - f r e e e x p e c t a t i o n value of any magnitude or p r o p o s i t i o n i s an eigenvalue of the magnitude or p r o p o s i t i o n . But sin c e the eigenvalues of incompatible magnitudes or p r o p o s i t i o n s are not a d d i t i v e , an HV e x p e c t a t i o n - f u n c t i o n which s a t i s f i e s (vN(&) and i s d i s p e r s i o n - f r e e i s impossible ( B e l l , 1966, p. 449). The Kochen-Specker v e r s i o n of von Neumann's i m p o s s i b i l i t y proof shows c l e a r l y how (vN#) i s the c u l p r i t i n the proof and so f u r t h e r s u b s t a n t i a t e s B e l l ' s c r i t i c i s m (Kochen-Specker, 1967, pp. 81-82). Such HV proposals whose imposed c o n d i t i o n s are i n c o n s i s t e n t w i t h each other are c a l l e d HV t h e o r i e s of the zero-th k i n d by B a l i n f a n t e ; t h e i r i m p o s s i b i l i t y i s not s u r p r i s i n g . B e l l a l s o appeals to the problem of measurement i n t e r a c t i o n i n order to argue t h a t HV measures (or ex p e c t a t i o n - f u n c t i o n s ) need not s a t i s f y (vn^). The r e s u l t o f a measurement of the sum + P^ o f two incompatible p r o p o s i t i o n s cannot be c a l c u l a t e d by simply adding together the r e s u l t s of separate measurements f o r P ^ > P 2 ' F ° r 3 S e x e m P l i f i e d by von Neumann (1932, p. 310), a measurement of a sum P^ + P^ o f incompatibles i n v o l v e s an experimental arrangement which i s e n t i r e l y d i f f e r e n t from the arrangements by which P^  and P Q are each measured sep a r a t e l y . Now although the 169 expectation-value assigned to + P^ by any quantum Exp^. always does equal the sum of the expectation-values assigned by Exp^, to each of the P^ , P j s e p a r a t e l y , t h i s i s not a t r i v i a l or necessary f e a t u r e of the quantum Exp^ measures. Rather, i t i s a very p e c u l i a r f e a t u r e of the quantum Exp^, measures, e s p e c i a l l y when, as B e l l suggests, one remembers wi t h Bohr "the i m p o s s i b i l i t y of any sharp d i s t i n c t i o n between the behavior of atomic o b j e c t s and the i n t e r a c t i o n w i t h measuring instruments which serve to d e f i n e the c o n d i t i o n s under which the [quantum] phenomena appear" (Bohr quoted by B e l l , 1966, p. 447). B e l l concludes t h a t there i s no reason to demand that proposed d i s p e r s i o n - f r e e HV measures must be a d d i t i v e w i t h respect to incompatible magnitudes and p r o p o s i t i o n s , as (vN&) r e q u i r e s . So when von Neumann imposes h i s c o n d i t i o n and then proves t h a t d i s p e r s i o n - f r e e HV measures are impossible and thus proves that an HV r e c o n s t r u c t i o n of quantum mechanics i s i m p o s s i b l e , he i s open to the charge of begging the HV question since (vNi$) i s u n j u s t i f i e d . Furthermore, von Neumann's i m p o s i t i o n of (vNsK) on proposed d i s p e r s i o n - f r e e HV measures begs the HV question i n another way. One of the c o n d i t i o n s which von Neumann incorporates as p a r t of h i s l i s t o f c o n d i t i o n s d e f i n i n g an e x p e c t a t i o n - f u n c t i o n Exp i n general i s the f o l l o w i n g , which he l a b e l s (E): (E) I f A,B,... are a r b i t r a r y magnitudes, then there i s an a d d i t i o n a l magnitude A + B + ••• (which does not depend on the choice of the e x p e c t a t i o n - f u n c t i o n ) , such that Exp(A + B + •••) = Exp(A) + Exp(B) + ••• (von Neumann, pp. 309, 311). With t h i s c o n d i t i o n ( E ) , von Neumann l e t s the 170 e x p e c t a t i o n - f u n c t i o n s d e f i n e the sum of incompatible magnitudes, e.g., the sum of A, B i s that magnitude which s a t i s f i e s (E) f o r a l l e x p e c t a t i o n - f u n c t i o n s . Von Neumann motivates t h i s d e f i n i t i o n by two f a c t s : A A The sum of the operators A, B (re p r e s e n t i n g the magnitudes A, B) i s i t s e l f a s e l f - a d j o i n t operator which can represent a quantum magnitude; and f o r a l l quantum Exp^ e x p e c t a t i o n - f u n c t i o n s , Exp^(A + B) = Exp^(A) + Exp^(B). Now i f we assume that d i s p e r s i o n - f r e e HV E x P w e x p e c t a t i o n - f u n c t i o n s do e x i s t , then the sum of A, B as defined by a l l the quantum Exp^ and HV Exp w may be d i f f e r e n t from the sum of A, B as defined by j u s t a l l the A A quantum Exp^ . And f o r example, although the operator A + B does represent the magnitude which i s the sum of A, B as defined by a l l the A A quantum Exp^, , the operator A + B may not represent the sum of A, B as defined by a l l the quantum Exp^, and HV E x P w » ± n which case A A A A Exp (A + B) i Exp (A) + Exp (B), c o n t r a r y to von Neumann's (vN^) c o n d i t i o n , w w w Of course, i f the d i s p e r s i o n - f r e e HV E X P W a r e impossible, then the two sums are the same. However, von Neumann imposes (vN#) which presumes th a t the two sums are the same (and so presumes th a t d i s p e r s i o n - f r e e HV E x P w d o not e x i s t ) and which r e q u i r e s proposed d i s p e r s i o n - f r e e HV Exp to s a t i s f y w Exp (A + B) = Exp (A) + Exp (B), and then von Neumann proves t h a t the W W W proposed d i s p e r s i o n - f r e e HV E x P w a r e impossible. Thus von Neumann i s begging the HV question because the i m p o s i t i o n o f c o n d i t i o n (vN&) presumes what i s being proved, namely, the i m p o s s i b i l i t y or non-existence of 2 d i s p e r s i o n - f r e e HV E x P w f u n c t i o n s . As mentioned i n Chapter V(D), u s i n g the s t r u c t u r a l c o n d i t i o n (JBiO, Jauch-Piron prove i n t h e i r C o r o l l a r y 1 that d i s p e r s i o n - f r e e measures are impossible on any i r r e d u c i b l e orthomodular l a t t i c e . T h i s , they say, i s 1 7 1 von Neumann's o l d r e s u l t , i . e . , von Neumann's proof of the i m p o s s i b i l i t y of d i s p e r s i o n - f r e e measures, proven without the contentious c o n d i t i o n (vN^). However, Jauch-Piron argue t h a t the quantum s u p e r s e l e c t i o n r u l e s ensure t h a t the quantum orthomodular l a t t i c e P Q M L s t r u c t u r e s are not i r r e d u c i b l e but ra t h e r are r e d u c i b l e l a t t i c e s w i t h n o n - t r i v i a l c e n t r e s . So C o r o l l a r y 1 does not r u l e out d i s p e r s i o n - f r e e measures on the quantum P Q ^ • Now according to Jauch-Piron, a quantum P which does admit QML hidden-variables i s c h a r a c t e r i z e d by the f o l l o w i n g property: Every measure on a PQML which admits hidden-variables can be represented as a weighted sum o f d i s p e r s i o n - f r e e measures on » ^ n p a r t i c u l a r , every quantum Exp^ measure on P Q ^ l c a n be so represented. Then i n t h e i r C o r o l l a r y 3 and again i n t h e i r Theorem 2, Jauch-Piron prove t h a t an orthomodular l a t t i c e admits hidden-variables only i f a l l i t s elements are mutually compatible, i . e . , o n ly i f the l a t t i c e i s Boolean. So any quantum P Q ^ which contains incompatible elements does not admit h i d d e n - v a r i a b l e s , and hence hidden-variables are impossible i n quantum mechanics (Jauch-Piron, 1 9 6 3 , pp. 8 3 5 - 8 3 7 ) . Bub's e l u c i d a t i o n of Jauch-Piron's work shows c l e a r l y how c o n d i t i o n (JP&) i s the c u l p r i t i n t h e i r i m p o s s i b i l i t y p r o o f ( s ) . For Bub shows how the quantum Exp^, measures on a P Q J ^ cannot be represented as weighted sums of d i s p e r s i o n - f r e e measures on Pq^L when the d i s p e r s i o n - f r e e HV measures are r e q u i r e d to s a t i s f y (JP£>) (Bub, 1974-, pp. 6 1 - 6 2 ) . For example, consider a quantum Exp^. which assigns values to two incompatible atoms P^ , P^ ° F 3 P Q M L 3 3 f o l l o w s : E x P y ( p Y > = 1» E x P y ( V = h*M € (0»1)» and s i n c e P^ A p^ = 0 , Exp^(P^ A P^) = Exp^ (0 ) = 0 . According to the Jauch-Piron c h a r a c t e r i z a t i o n o f a hidden-variables p r o p o s a l , i f ? n admits QML 172 hidden-variables then t h i s Exp^, measure on P A M L can be represented as a weighted sum 2 \.w. , where 2 \. = 1 and each w. i s a 4 3 . 1 1 . i i I l d i s p e r s i o n - f r e e (HV) measure on PQ M L • N O W ± n order to reproduce the assignment Exp^(P^,) = 1, each w^ must a s s i g n the value 1 to P^ . , i . e . , f o r every w^ i n the sum r e p r e s e n t i n g Exp^, , w ^ ( p ^ . ^ = ^ s o t h a t 2 X.w.(P.) = 1 = Exp, ( P . ) . And s i n c e P. A P = 0, w.(P,» A P ) = w.(0) ^ l l Y Y V V cp l Y cp i = 0, f o r every w^ i n the sum re p r e s e n t i n g Exp^ . Moreover, none o f the w^ can a s s i g n the value 1 to P^ because by ( J P ^ ) , w . ^ P Y ^ = ^ A N D w.(P ) = 1 y i e l d s w.(P, A P ) = 1, which c o n t r a d i c t s w.(P, A P ) = 0; i c p J 1 Y 9 1 V <P so w ^ P ^ = ^ f ° r e v e r Y w £ i n the sum re p r e s e n t i n g Exp^ . Thus the nonzero value assigned by Exp^, to P^ cannot be reproduced by any weighted sum which reproduces the value assignment Exp^,(P^) = 1. That i s , a weighted sum o f d i s p e r s i o n - f r e e (HV) measures s a t i s f y i n g (JPii) cannot reproduce the value assignments o f t h i s quantum Exp^, measure. So we can view the i m p o s s i b i l i t y o f a Jauch-Piron type o f HV proposal as being due to an i n c o n s i s t e n c y between three c o n d i t i o n s imposed on proposed HV measures: the s t r u c t u r a l c o n d i t i o n (JP&), the d i s p e r s i o n -f r e e c o n d i t i o n , and the s t a t i s t i c a l c o n d i t i o n , which r e q u i r e s t h a t the value assignments o f the quantum Exp^, measures be reproduced by, e.g., a weighted sum of d i s p e r s i o n - f r e e HV measures. Thus, as B e l i n f a n t e says, r a t h e r than proving the i m p o s s i b i l i t y o f h i d d e n - v a r i a b l e s , Jauch-Piron have merely shown th a t t h e i r type of HV proposal i s o f the zero-th k i n d ( B e l i n f a n t e , 1973, p. 59). B e l l ' s o b j e c t i o n to the s t r u c t u r a l c o n d i t i o n (JP#) i s s i m i l a r to h i s o b j e c t i o n to (vN^). When P^ , P^ are incompatible, a measurement of t h e i r meet P^ , A P^ in v o l v e s an experimental arrangement which d i f f e r s from the arrangements by which P^ . and P^ are each measured s e p a r a t e l y . 173 yet (JP#>) r e q u i r e s a proposed d i s p e r s i o n - f r e e HV measure to a s s i g n the value 1 to P, A P i f i t assigns the value.- 1 to each P. , P , Y cp Y <P s e p a r a t e l y . In s p i t e o f the d i f f e r e n t experimental arrangements, the quantum Exp^ measures do s a t i s f y (JP&). And so i t i s reasonable to r e q u i r e t h a t the weighted sums of d i s p e r s i o n - f r e e HV measures which represent the quantum Exp^, measures i n an HV r e c o n s t r u c t i o n l i k e w i s e s a t i s f y (JPv*). But i t i s not reasonable to r e q u i r e t h a t each d i s p e r s i o n - f r e e HV measure must i t s e l f s a t i s f y (JP$>), e s p e c i a l l y when we r e c a l l the problem of measurement i n t e r a c t i o n . So when Jauch-Piron impose t h e i r s t r u c t u r a l c o n d i t i o n (JP<!0 on proposed d i s p e r s i o n - f r e e HV measures and then show tha t h i d d e n - v a r i a b l e s are impossible, they are open t o the charge o f begging the HV question s i n c e t h e i r i m p o s i t i o n o f (JP&) i s not j u s t i f i e d . Bub a l s o argues t h a t the Jauch-Piron i m p o s s i b i l i t y p r o o f ( s ) beg the HV question i n the f o l l o w i n g manner. Jauch-Piron prove the impossi-b i l i t y o f r e p r e s e n t i n g the quantum Exp^ measures on a P Q^ l a s mixtures of d i s p e r s i o n - f r e e HV measures on P _ „ T . That i s , the HV measures QML considered by Jauch-Piron are a s o r t o f g e n e r a l i z e d p r o b a b i l i t y measure defined on the quantum P Q^ l • But then the Jauch-Piron proof does not r u l e out the f u r t h e r p o s s i b i l i t y of r e p r e s e n t i n g the quantum Exp^, measures as mixtures o f d i s p e r s i o n - f r e e HV measures which are c l a s s i c a l p r o b a b i l i t y measures defined on a Boolean s t r u c t u r e (Bub, 1974, p. 63). The same c r i t i c i s m can be d i r e c t e d against the proofs and arguments by which von Neumann purports t o show the i m p o s s i b i l i t y o f hi d d e n - v a r i a b l e s . For von Neumann r e f e r s to d i s p e r s i o n - f r e e oHV ex p e c t a t i o n - f u n c t i o n s defined on the s e t o f quantum p r o p o s i t i o n s , qua idempotent magnitudes represented by p r o j e c t o r s , whose s t r u c t u r e i s a quantum P Q M . S i m i l a r l y , Gleason's i m p o s s i b i l i t y proof and the 174 Kochen-Specker Theorem 1 v e r s i o n o f Gleason's proof are a l s o subject to t h i s c r i t i c i s m . For Gleason's proof shows th a t h i s s o r t of g e n e r a l i z e d d i s p e r s i o n - f r e e HV measures (which s a t i s f y (Ga) and thus are P ^ ^ - p r e s e r v i n g ) n^3 are impossible on the quantum P N ~ s t r u c t u r e s , t h a t i s , i n Kochen-Specker's v e r s i o n , b i v a l e n t homomorphisms(i) are impossible on PQ^' But Gleason's proof and Kochen-Specker's Theorem 1 do not address the above-mentioned f u r t h e r p o s s i b i l i t y o f r e p r e s e n t i n g the quantum Exp^, measures as mixtures of c l a s s i c a l d i s p e r s i o n - f r e e HV measures defined on a Boolean s t r u c t u r e . Moreover, Bub argues t h a t t h i s f u r t h e r p o s s i b i l i t y p r e c i s e l y captures the HV e n t e r p r i s e which Kochen-Specker do c o r r e c t l y formulate and address y et which t h e i r Theorem 1 alone does not r u l e out. C o r r e c t l y formulated, the HV e n t e r p r i s e can be s a i d to be the attempt to r e c o n s t r u c t the s t a t i s t i c a l r e s u l t s given by <W,pQ^,Exp^.>, i . e . , the quantum ge n e r a l i z e d Exp^, p r o b a b i l i t y measures on the non-Boolean P s t r u c t u r e of the quantum phase space H ( H i l b e r t space), i n terms o f a c l a s s i c a l measure space <S2,P ,u->, i . e . , c l a s s i c a l u. p r o b a b i l i t y measures on the HV Boolean P M , s t r u c t u r e of a po s t u l a t e d HV c l a s s i c a l phase space 2 (Kochen-Specker, 1967, pp. 62, 75). Thus an HV theory may be s a i d to be a 3 Boolean r e c o n s t r u c t i o n o f quantum mechanics. More e x p l i c i t l y , as described by Kochen-Specker, a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics can be formulated as f o l l o w s . L i k e the formalism o f c l a s s i c a l mechanics described i n Chapter I I I , an HV theory p o s i t s a c l a s s i c a l phase space 2', each p o i n t w € 2 represents a pure hidden s t a t e , and each r e a l - v a l u e d ( B o r e l ) f u n c t i o n f . : 2 -*• R represents a magnitude i n the HV theory. The idempotent f u n c t i o n s on 2, or e q u i v a l e n t l y , the B o r e l subsets o f 2, form a Boolean s t r u c t u r e which may be l a b e l e d ? ^ . L i k e the P C M s t r u c t u r e , t h i s ? ^ i s regarded as the 175 p r o p o s i t i o n a l s t r u c t u r e of the HV theory. That i s , an idempotent f u n c t i o n fp : & ->• {0,1}, or corresponding B o r e l subset c Q} represents a p r o p o s i t i o n i n the HV theory. Each pure hidden s t a t e w induces a d i s p e r s i o n - f r e e c l a s s i c a l p r o b a b i l i t y measure u. : P ^ -»• {0,1} which i s a b i v a l e n t homomorphism on P0, 7 . The HV r e c o n s t r u c t i o n of quantum mechanics proceeds by r e p r e s e n t i n g or a s s o c i a t i n g each of the quantum magnitudes A,B,... w i t h a r e a l - v a l u e d f u n c t i o n f , f ,... on the HV phase space 2. Each quantum p r o p o s i t i o n P, qua idempotent magnitude, i s l i k e w i s e a s s o c i a t e d w i t h an idempotent f u n c t i o n f on 2 or corresponding B o r e l subset W o f 2. That i s , quantum P P p r o p o s i t i o n s are a s s o c i a t e d w i t h the elements o f ; and l e t % l a b e l HV t h i s a s s o c i a t i o n . Kochen-Specker take the s t r u c t u r e o f the quantum p r o p o s i t i o n s to be a p a r t i a l - B o o l e a n algebra P n M f l 5 t h i s f a c t i s f u r t h e r discussed below. Next, each quantum pure s t a t e ty i s represented i n the HV r e c o n s t r u c t i o n as a mixed s t a t e which induces a d i s p e r s i v e c l a s s i c a l p r o b a b i l i t y measure [i^ : P ^ -»• [0,1] on the Boolean P ^ s t r u c t u r e . In the HV theory, these d i s p e r s i v e u-^  measures represent the quantum Exp^. measures. And these u.^  measures are r e q u i r e d to s a t i s f y the s t a t i s t i c a l  c o n d i t i o n , which Kochen-Specker give as f o l l o w s : For any quantum ty and f o r any quantum P, f p ( w ) du.^({w}) = Exp^.(P) (Kochen-Specker, 1967, pp. 61, 75). Now by d e f i n i t i o n , f o r any fp on 2 and f o r any hidden s t a t e w 6 2, f p ( w ) = 1 i f w € Wp and f (w) = 0 i f w € WpX . So by s u b s t i t u t i o n , the s t a t i s t i c a l c o n d i t i o n reduces t o : Exp^(P) = = J 1 d^C {»} )• + / t f 0 d(^({w}) = u.^(Wp), where tfp = f ^ C {!})•. Thus f o r a quantum system (or ensemble of quantum systems) whose s t a t e i s given 176 by y i n quantum mechanics, the p r o b a b i l i t y that the quantum proposition P i s true i s equal to the p r o b a b i l i t y that the pure hidden state w of the quantum system i s a member of that subset Wp c <2 of hidden states with respect to which the HV representative of P, namely, fp , has the value 1. Besides the s t a t i s t i c a l condition, Kochen-Specker also impose the following s t r u c t u r a l condition: The as s o c i a t i o n % of the quantum propositions with the elements of P U „ must be an imbedding(o) which nv preserves the pQjyjA structure of the quantum propositions. That i s , an imbedding(o) % : P Q ^ P^y i s a necessary condition f o r an HV reconstruction of quantum mechanics, according to Kochen-Specker. The arguments by which Kochen-Specker motivate t h i s imbedding(A) condition are further discussed below. Next, Kochen-Specker prove i n t h e i r Theorem 0, discussed i n Chapter IV(F), that an imbedding(<J>) % : P A M . -»• P e x i s t s IFF a complete QMA nV c o l l e c t i o n of bivalent homomorphisms(c>) h : PQ^ A {°»1} e x i s t s . We can better understand the " i f " h a l f of t h i s b i c o n d i t i o n a l by noting that the c l a s s i c a l p r o b a b i l i t y measures u. : P I T T I -»• {0,1} and L L , : P „ „ -»• [0,1] of J w HV \|r HV the HV reconstruction can also be regarded, v i a the imbedding(d>) % : > a s generalized p r o b a b i l i t y measures on the quantum PQ M A • The r e l a t i o n s h i p s among these mappings can be schematized as follows: 177 The equivalence between the quantum Exp^. : PQ M A -»• [0,1] and the composition u-^ p % : PQJ^ [0,1] i s ensured by the s t a t i s t i c a l c o n d i t i o n . And f o r every pure hidden s t a t e w, the composition u^o % : PQJJA -*• {0,1} i s a g e n e r a l i z e d d i s p e r s i o n - f r e e HV measure on PQJJA which preserves the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f PQ^ A > o r i n other words, each composition u^o % i s a b i v a l e n t homomorphism(cb) on PQ^a • Moreover, as described i n Chapter I V ( F ) , an imbedding i s by d e f i n i t i o n an i n j e c t i v e mapping, i . e . , f o r any ? ± ? ? 2 i n P Q M , i % ( P 2 ) . And by the s e m i - s i m p l i c i t y property o f the Boolean s t r u c t u r e P , f o r any HV f., t f„ i n P , there i s a b i v a l e n t homomorphism on P , namely, a r 2 HV HV c l a s s i c a l d i s p e r s i o n - f r e e p r o b a b i l i t y measure u.^  : P -»• {0,1} f o r some w, such t h a t p, ( f _ ) * u. ( f ). So i f the imbedding(cb) % : P . U . •+ P U „ W W r 2 yMA nv e x i s t s , then f o r every pure hidden s t a t e w, the composition \irp % i s a w we b i v a l e n t homomorphism(A) on P.... . And f o r any P„ i P„ i n P„ QMA J 1 2 QMA can be sure t h a t f p = % ( P 1 ) t %(P 2> = f p i n P H V , and we can be sure 178 t h a t f o r some w, u. (%(P.)) 4 M- ( % ( P 0 ) ) , t h a t i s , we can be sure t h a t the W 1 W 2 c o l l e c t i o n o f b i v a l e n t homomorphisms(i) on PQJ^ ^ s complete. Conversely, i f a complete c o l l e c t i o n o f b i v a l e n t homomorphisms(i>) e x i s t on P ^ ^ , then as described i n Chapter I V ( F ) , P ^ ^ can be imbeddedCci) i n t o a C a r t e s i a n product Boolean s t r u c t u r e . For example, any 2 2 two-dimensional H i l b e r t space P.... ( o r P - „ T ) can be imbedded(A) i n t o QMA QML the C a r t e s i a n product Boolean s t r u c t u r e ( Z 2 ^ *?i o r e 9 . u i v a l e n t T y , r 2 II (Z^). > where r i s the c a r d i n a l i t y o f the set o f maximal Boolean 1 1 2 substructures o f the . This C a r t e s i a n product Boolean s t r u c t u r e can be taken to be the Boolean P s t r u c t u r e o f a proposed HV r e c o n s t r u c t i o n HV 2»r of quantum mechanics, e.g., (Z_) can be regarded as the P „ „ of a 2 HV 2 2 proposed HV r e c o n s t r u c t i o n o f the quantum mechanics o f <H ,pQM,Exp^>. Or i n other words, as described by Bub, the c l a s s i c a l measure space <S2,PHy,p.> = X which provides a Boolean HV r e c o n s t r u c t i o n o f the quantum 2 2 mechanical s t a t i s t i c a l r e s u l t s given by <H ,pQM»Exp^> can be regarded as a C a r t e s i a n product measure space X = n X. where i ranges over the set 2 1 1 of maximal Boolean substructures o f P . „ and each X. = <2.,P t T„,u.> i s a QM i a/ HV ^ i c l a s s i c a l measure space introduced f o r each maximal Boolean subst r u c t u r e ,2 QM mBS o f (Bub, 1974, p. 145). Since each mBS. of pj? i s 1 WM 1 QM 2 2 isomorphic t o (Z ) , each i s isomorphic t o (Z ) , and so 2 HV. 2 1 r 2 P „ „ = II .'P„„ i s the Ca r t e s i a n product FI , (Z„). mentioned above. HV . nv. 2 1 1 1 1 Now the Kochen-Specker proof o f the i m p o s s i b i l i t y o f such a proposed <Q,P„ Tt\i> r e c o n s t r u c t i o n o f the quantum mechanics o f HV <W n -^,pQ~ 3,Exp^> proceeds i n two stages. F i r s t i n Theorem 1, which i s t h e i r v e r s i o n o f Gleason's i m p o s s i b i l i t y proof, Kochen-Specker show that b i v a l e n t homomorphisms(i>), i . e . , g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures, are 179 impossible on any (and t h i s r e s u l t a l s o a p p l i e s to PQJJJ^ ' T N E N f o l l o w s by Kochen-Specker's Theorem 0 t h a t an imbedding(o) o f any PQ^A i n t o any proposed Boolean P ^ s t r u c t u r e o f an HV r e c o n s t r u c t i o n i s impossible, and hence, s i n c e such an imbedding(i) i s a necessary c o n d i t i o n f o r an HV r e c o n s t r u c t i o n , an HV r e c o n s t r u c t i o n o f the quantum mechanical s t a t i s t i c a l r e s u l t s o f <ff N - ^ , P ^ H 3J E xP,> i n terms of some c l a s s i c a l HV QM Y measure space <8,P r,u> i s impossible; t h i s i s the second stage of the HV Kochen-Specker proof o f the i m p o s s i b i l i t y o f an HV r e c o n s t r u c t i o n o f quantum mechanics. So while Gleason's i m p o s s i b i l i t y proof and Kochen-Specker's Theorem 1 j u s t show the i m p o s s i b i l i t y o f b i v a l e n t homomorphisms(cb), i . e . , g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures, on P^~ , Kochen-Specker's Theorem 0 and imbedding(A) c o n d i t i o n connect t h i s r e s u l t w i t h the f u r t h e r question of the p o s s i b i l i t y o f a <S^P,,,.,u> type of HV r e c o n s t r u c t i o n o f n v quantum mechanics. For the imbedding(i) c o n d i t i o n , according to which an imbedding(o) % : Pjjy ^ s a n e c e s s a r v c o n d i t i o n f o r such an HV r e c o n s t r u c t i o n , ensures t h a t proposed c l a s s i c a l d i s p e r s i o n - f r e e HV measures '"w : PHV ~* preserve the PQ^a s t r u c t u r e o f the quantum p r o p o s i t i o n s so t h a t , f o r each hidden s t a t e w, the composition u. o % i s a g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measure on P-. M A • And Theorem 0, which QMA b i c o n d i t i o n a l l y connects the existence o f a complete c o l l e c t i o n o f such measures on a P.,,. w i t h the existence o f an imbedding(<4) % : P „ „ A -»• P I T „ , QMA QMA HV thus e n t a i l s t h a t the existence o f a complete c o l l e c t i o n o f g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures i s a necessary c o n d i t i o n f o r a < S 2 , P „ „ , | J > HV type o f HV r e c o n s t r u c t i o n . In t h i s way, Kochen-Specker apply Gleason's r e s u l t to the c o r r e c t l y formulated HV q u e s t i o n ; w h i l e i n c o n t r a s t , the . von Neumann and the Jauch-Piron proofs do not even address the HV question 180 as so formulated. L i k e the s t r u c t u r a l c o n d i t i o n s (vN$) and (JP&), the s t r u c t u r a l c o n d i t i o n P ^ ^ - p r e s e r v a t i o n , whose i m p o s i t i o n upon proposed d i s p e r s i o n - f r e e HV measures i s e n t a i l e d by the i m p o s i t i o n o f the (vN6) c o n d i t i o n , or the (JPo) c o n d i t i o n , or the (Ga) c o n d i t i o n , or the Kochen-Specker imbedding(<!>) c o n d i t i o n , has a l s o been subject to the three s o r t s of c r i t i c i s m s l i s t e d i n the Preface above. These c r i t i c i s m s w i l l be elaborated next. But i n S e c t i o n B, another p e r s p e c t i v e on the problem o f hidden-variables i s introduced, according to which the s t r u c t u r a l c o n d i t i o n of P ^ ^ - p r e s e r v a t i o n emerges unscathed by these c r i t i c i s m s . In B e l i n f a n t e ' s view, the type of HV theory proved impossible by Gleason and Kochen-Specker i s l i k e the types proved impossible by von Neumann and Jauch-Piron; they are a l l HV t h e o r i e s of the z e r o t h k i n d whose i m p o s s i b i l i t y i s due to an i n c o n s i s t e n c y between the c o n d i t i o n s which the proposed d i s p e r s i o n - f r e e HV measures are r e q u i r e d to s a t i s f y ( B e l i n f a n t e , 1973, p. 17). However, the s t r u c t u r a l c o n d i t i o n of P ^ ^ - p r e s e r v a t i o n i s not simply i n c o n s i s t e n t w i t h the d i s p e r s i o n - f r e e c o n d i t i o n , i n the way t h a t i s . Nor i s P ^ ^ - p r e s e r v a t i o n i n c o n s i s t e n t w i t h the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h the s t a t i s t i c a l c o n d i t i o n , i n the way t h a t (JP&) i s . On the c o n t r a r y , the s t r u c t u r a l c o n d i t i o n o f P ^ ^ - p r e s e r v a t i o n f o l l o w s from  the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h (Ga), or the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h ( v N i ) , or the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h (JP<b), as described i n Chapter V(D). The t r o u b l e w i t h P ^ ^ - p r e s e r v a t i o n i s more s u b t l e than the t r o u b l e s w i t h (vN#>) and (JP^>). In f a c t , the t r o u b l e w i t h P ^ ^ - p r e s e r v a t i o n has to do w i t h the overlap p a t t e r n s among the mBS's o f a s t r u c t u r e , and as B e l i n f a n t e p o i n t s out, the t r o u b l e w i t h P ^ ^ - p r e s e r v a t i o n has to do w i t h the assumption t h a t HV 181 measures are noncontextual, as s h a l l be described below. B e l l ' s c r i t i c i s m of Gleason's i m p o s s i b i l i t y proof (and thus of Kochen-Specker's Theorem 1) hinges upon the d i f f e r e n c e between what are sometimes c a l l e d c o n t e x t u a l and noncontextual HV t h e o r i e s , though B e l l does not use these terms. B e l l presents a v e r s i o n o f Gleason's proof which focuses upon two s t r u c t u r a l c o n d i t i o n s which B e l l d e r i v e s from Gleason's a d d i t i v i t y (Ga). Both c o n d i t i o n s are subsumed by the s t r u c t u r a l c o n d i t i o n o f P„„„-preservation, which l i k e w i s e f o l l o w s from (Ga). B e l l shows how the QMA second c o n d i t i o n which he d e r i v e s from (Ga) r u l e s out ( g e n e r a l i z e d ) d i s p e r s i o n - f r e e HV measures on the set of a l l p r o j e c t o r s or subspaces o f a three-or-higher dimensional H i l b e r t space; and as s h a l l be described s h o r t l y , t h i s second c o n d i t i o n ensures t h a t ( g e n e r a l i z e d ) d i s p e r s i o n - f r e e HV measures are i n f a c t noncontextual. B e l l c r i t i c i z e s the i m p o s i t i o n of t h i s second c o n d i t i o n upon HV measures because the second c o n d i t i o n r e l a t e s i n a n o n t r i v i a l and u n j u s t i f i e d way the r e s u l t s o f measurements which cannot be performed simultaneously. Although Gleason's proof r e f e r s to an i n f i n i t e set o f subspaces (or p r o j e c t o r s ) o f three-or-higher dimensional H i l b e r t space, i n order to understand B e l l ' s e x p l i c a t i o n and c r i t i q u e o f Gleason*s proof we need only 3 consider the f o l l o w i n g twelve-element fragment of ^ which i n c l u d e s two overlapping mBS's: 182 One maximal Boolean substructure mBS^ i s generated by the three mutually orthogonal ( i . e . , compatible) atoms {P^,P 2,P 3) and the other mBS^ i s 3 generated by {P3,P^,P,-}. A g e n e r a l i z e d measure |_L on P which s a t i s f i e s (Ga) assigns values to these f i v e atoms as f o l l o w s : M-(P1) + ki(P 2) + n(P 3). = p.(P1 V P 2 VP 3) = (i-(l) = 1. and l i ( P 3 ) + n ( P 4 ) + [ii? ) = |j,(P V P 4 v P g ) = n ( l ) = 1. I t f o l l o w s t h a t i f [i (P„) = 1 then (j, (P. ) = [i ( P 0 ) = 0; and s i m i l a r l y , i f u. ( P 0 ) = 1 then w 3 w 1 w 2 J "w 3 [x(P. ) = i x ( P _ ) = 0. (The s u b s c r i p t w i s added because a measure which w 4 w 5 e assigns 0, 1 values i s d i s p e r s i o n - f r e e . ) These two c o n d i t i o n a l s are instances o f the f i r s t c o n d i t i o n which B e l l d e r i v e s from (Ga) and which he l a b e l s (A) ( B e l l , 1966, p. 450). B e l i n f a n t e argues t h a t because B e l l ' s (A) r e f e r s to only one t r i a d of mutually orthogonal atoms at a time, we cannot y et conclude t h a t , i f u. (P ) = 1 then [i. (P. ) = [i (P.) = u. (P„) = u. (P_) = 0' ( B e l i n f a n t e , 1973, W O W l W 2 W 4 w o p. 65). But such a c o n c l u s i o n is_ guaranteed by the second c o n d i t i o n which B e l l derives from (Ga) and which he l a b e l s ( B ) . An instance of (B) i s : I f a w ( P 1 ) = M-W(P2) = 0 then, f o r any other P < P 1 V P 2 , M- (P) = 0. Thus i f [i (P„) = 1, then by ( A ) , u. (P ) = (x (P^) = 0, and then, s i n c e W O W 1 W 2 P 4 ~ P l V P 2 a n d P 5 ~ P l V P 2 ' b y ( B ) M'w ( P4 ) = M'w ( P5 ) = °* These two c o n d i t i o n s (A) and (B) which B e l l d e r i v e s from (Ga) correspond to the two c o n d i t i o n s (KS1) and (KS2) s t a t e d i n Chapter V(B) and to the two c o n d i t i o n s l a b e l e d (61b) and (64) i n B e l i n f a n t e ' s d e s c r i p t i o n of Kochen-Specker's work ( B e l i n f a n t e , 1973, pp. 39, 41). The f i r s t c o n d i t i o n of each p a i r , namely, ( A ) , (KS1), (61b), ensure t h a t the assignment o f 0, 1 values to the atoms i n a given mBS of a P^ M preserve the Boolean operations and r e l a t i o n s , i . e . , the Boolean s t r u c t u r a l f e a t u r e s , o f the mBS. And the second c o n d i t i o n o f each p a i r , namely, ( B ) , (KS2), (64), ensure t h a t 183 the assignment of 0, 1 values to the atoms in any overlapping mBS's in a preserve the overlap patterns among the mBS's. Both sorts of conditions are subsumed by the structural conditions of P^^-preservation, which i t s e l f has two aspects: F i r s t , i t ensures that the Boolean structural features of each mBS in a P are preserved; second, i t ensures that the partial-Boolean structural features of the entire P are pre-served; in particular, i t ensures that the overlap patterns among the mBS's in a P Q J^ a r e preserved. So Bell rightly points to the second condition (B), which he derives from (Ga), as the crucial part of Gleason's impossibility proof. For as described in Chapter V(B), i t is the preservation of the overlap patterns which makes bivalent homomorphisms(<4 ) impossible in Kochen-Specker's Theorem 1 version of Gleason's proof. Bell argues that proposed dispersion-free HV measures need not be required to satisfy (B). For any proposition P which i s less than or equal to P^  V P^ is incompatible with each of P^  , P^ , unless P = P^ or P = P„ . And i f P <J6 P. and P p P„ then a measurement of P cannot be made simultaneously with a measurement of P^  and P^  • Bell also poses the question of how this condition (B), which in fact refers to the values assigned to incompatibles, could follow from condition (Ga) which explicitly refers to only orthogonal elements which are compatible. Bell answers that i t was "tacitly assumed" that a measurement of, say, P must yield the o same value regardless of whether P„ is measured together with P, , P o 1 2 or together with P^ - , P 5 . But since P^  , ? 2 are each incompatible with each of P^ , P g , a measurement of P^  , P , P^  requires an experimental arrangement different from the arrangement by which P , P , P are o H* D measured, so there i s no reason to believe that the result of a measurement of P 3 together with P i > p 2 ' s n°uld be the same as the result of a 184 measurement o f Pg together w i t h P^ , P 5 ( B e l l , 1966, p. 451). An HV theory which allows i t s d i s p e r s i o n - f r e e HV measures to as s i g n d i f f e r e n t 0, 1 values to a given element P € P depending upon which other elements are measured together w i t h P have been c a l l e d c o n t e x t u a l HV t h e o r i e s . And the t a c i t assumption mentioned by B e l l i s the assumption that-an HV theory i s non-contextual; i . e . , i t s d i s p e r s i o n - f r e e HV measures a s s i g n a unique 0 or 1 value to a given element P € P M^ r e g a r d l e s s o f which other elements are measured together w i t h P. Now i n quantum mechanics, the outcome of a measurement of any magnitude A (which i s always one of A's eigenvalues) or of any idempotent magnitude P (which i s always one of P's 0 or 1 eigenvalues) i s determined by the quantum s t a t e ty, though i f ty i s incompatible w i t h any of A's or P's e i g e n s t a t e s , then as described i n the Preface, the quantum formalism a t best determines the p r o b a b i l i t y o f any one of A's or P's eigenvalues being the outcome o f a measurement and determines the average value ( i . e . , expectation-value) of A or P f o r a l a r g e number o f the same measurements of A or P on many quantum systems whose s t a t e i s described by f- In a c o n t e x t u a l HV theory, the outcome of a measurement o f A or P i s determined by the hidden s t a t e and the context o f measurement. A hidden s t a t e , l a b e l e d w above, i s s p e c i f i e d i n a c o n t e x t u a l HV theory by the quantum s t a t e ty together w i t h the hidden v a r i a b l e ( s ) g; so h e r e a f t e r , a hidden s t a t e o f a c o n t e x t u a l HV theory s h a l l be designated by ty, And the context i s taken to be the set of a l l p o s s i b l e outcomes of the measure-ment as s p e c i f i e d by a complete, orthogonal set o f eigens t a t e s o f the measured magnitude. As mentioned i n Chapter IV(A), the eigens t a t e s of any magnitude, as represented by p r o j e c t o r s {P,-},^ T T 1^ a w o n a H i l b e r t space, 185 are orthogonal and s a t i s f y J P. = I . In order t h a t the set o f eigenst a t e s i o f a magnitude be complete, i t s u f f i c e s t h a t each P. i s a one-dimensional p r o j e c t o r on ff; i . e . , an atom i n the P ^ s t r u c t u r e o f 5 H. Thus the context o f a measurement o f a magnitude represented by an operator on an n dimensional H i l b e r t space HU i s s p e c i f i e d by a s e t of n orthogonal one-dimensional p r o j e c t o r s on tfn, i . e . , by a s e t of n mutually orthogonal atoms i n the s t r u c t u r e o f H N . And sin c e a set of n mutually orthogonal atoms i n P ^ .generates a unique maximal Boolean s u b s t r u c t u r e o f P ^ M , ^  the context o f a measurement o f a magnitude represented by an operator on H N can eq u a l l y w e l l be s p e c i f i e d by an mBS i n the P ^ s t r u c t u r e of tfn, as suggested by Gudder (1970, p. 432). In p a r t i c u l a r , when we consider any idempotent magnitude P, which i s * n n represented by the p r o j e c t o r P on H and so i s an element i n the P ^ s t r u c t u r e o f H N , P, qua element o f P ^ , i s i t s e l f a member of any o f the mBS's i n P ^ which s p e c i f y p o s s i b l e contexts o f measurement o f P. A. For P i s i t s e l f a member (or a sum o f members) of any set of n orthogonal, one-dimensional p r o j e c t o r s on Hn r e p r e s e n t i n g a complete, orthogonal s et of eigenstates o f the idempotent magnitude P and so s p e c i f y i n g the context o f a measurement o f P; thus P, qua element of PQ^ , i s i t s e l f a member (or a j o i n of members) of any set o f n mutually orthogonal atoms i n P N ^ s p e c i f y i n g the context of a measurement o f P; and so P, qua element of P N „ , i s i t s e l f a member o f any mBS i n P N „ — — QM J QM s p e c i f y i n g the context o f a measurement o f P. In s h o r t , i n a co n t e x t u a l HV theory, the outcome of a measurement o f any P € P ^ i s determined by the hidden s t a t e y, £ and the context o f measurement, s p e c i f i e d by an mBS i n P . „ w i t h P € mBS. QM The f a c t s t a t e d i n the l a s t sentence can be and has been 186 f o r m a l i z e d i n any number of ways. Most a b s t r a c t l y , s i n c e the outcome o f a measurement o f P i s always one of P's 0 or 1 eigenvalues, we may t a l k o f a c o n t e x t u a l HV theory proposing contextually-dependent 0, 1 value assignments to the elements o f P^ . For example, B e l i n f a n t e t a l k s o f a c o n t e x t u a l HV theory, which he r e f e r s to as a " r e a l i s t i c " HV theory, i n t r o d u c i n g , f o r a given hidden s t a t e ty, £, a b i v a l e n t mapping v whose arguments are quantum p r o p o s i t i o n s and which depend not only upon ty, £, but a l s o upon the context of measurement ( B e l i n f a n t e , 1973, pp. 40-42). Less a b s t r a c t l y , s i n c e i n t h i s chapter and i n Chapters I I I and VI we have described how i n c l a s s i c a l mechanics, quantum mechanics, and proposed HV t h e o r i e s , 0, 1 value assignments to the elements i n P0„„ , Pn., , P„„ *> CM QM HV s t r u c t u r e s are preformed by various kinds of state-induced d i s p e r s i o n - f r e e p r o b a b i l i t y measures, we can i n a s i m i l a r v e i n say t h a t the hidden s t a t e s of a c o n t e x t u a l HV theory induce d i s p e r s i o n - f r e e HV measures which a s s i g n 0, 1 values to elements of P i n a c o n t e x t u a l l y dependent manner. For example, Bub t a l k s i n t h i s way (1974, pp. 146-147; 1973, p. 51). While according to Gudder's way of f o r m a l i z i n g the c o n t e x t u a l HV p r o p o s a l , a hidden s t a t e of a c o n t e x t u a l HV theory induces a d i s p e r s i o n - f r e e HV measure on only an mBS of P so t h a t the c o n t e x t u a l dependence o f the measure i s at l e a s t p a r t l y handled by r e s t r i c t i n g i t s domain to one context, i . e . , one mBS (Gudder, 1970, p. 433). We s h a l l focus upon the n o t i o n o f the hidden s t a t e s o f a c o n t e x t u a l HV theory inducing d i s p e r s i o n - f r e e HV measures which a s s i g n 0, 1 values to the elements of P M^ i n a c o n t e x t u a l l y dependent manner. The contextual-dependence of the d i s p e r s i o n - f r e e measures may be and has been formulated i n two equivalent ways. One way i n v o l v e s c o n t e x t u a l i z i n g proposed g e n e r a l i z e d , d i s p e r s i o n - f r e e HV measures M^ , ^ on P^ by having 187 the domain o f each a, „ be the cross-product o f P and the set o f mBS's i n P„w so that the value which a, „ assigns to an element P € P™ QM y, g QM depends upon which mBS co n t a i n i n g P i s being considered ( i . e . , depends upon the context i n which P i s being measured). Thus a hidden s t a t e y, £ induces a c o n t e x t u a l i z e d , g e n e r a l i z e d , d i s p e r s i o n - f r e e HV measure \H  1 PQM X { m B S i } i €lndex > { 0 , 1 } S U C h t h a t ' f ° r e x a m P l e ' ^ , C ( < P 3 ' m B S l > ) need not equal u, (<P„,mBS, >). According to Bub, the Bohm 1952 HV ^y,g 3 4 proposal i s such a co n t e x t u a l HV theory. However, one would be hard pressed to f i n d anything l i k e t h i s u.. : PAvt x {mBS.}.._ , * {0,1} i n Bohm's J & Y»C QM l lCI-ndex work or even i n Bub's d e s c r i p t i o n o f Bohm's work (Bub, 1973, p. 51). For again, the above n o t i o n o f an a. measure i s an a b s t r a c t i o n , which helps make sense o f Bub's d e s c r i p t i o n o f Bohm's work and which was suggested to me by B e l i n f a n t e ' s method o f c o n t e x t u a l i z i n g h i s b i v a l e n t v mappings (w i t h R. E. Robinson suggesting the cross-product f o r m u l a t i o n ) . Now the a l t e r n a t i v e way in v o l v e s proposing that a Boolean P s t r u c t u r e be the HV domain o f proposed c l a s s i c a l d i s p e r s i o n - f r e e HV measures u.^  ^ : P^ y -»• {0,1} induced by the hidden s t a t e s , w i t h a c o n t e x t u a l i z e d a s s o c i a t i o n of the elements of P_.w w i t h the elements o f the PTT„ . Thus we have a QM HV co n t e x t u a l i z e d a s s o c i a t i o n % : P^„ x {mBS.}.- T ,—*" Pt7„ such t h a t , f o r QM I i€Index HV example, %(<PQ,mBS >) need not equal %(<P ,mBS >), and so u.. (%(<P ,mBS >)) need not equal (j, (%(<P ,mBS >)). The Bohm-Bub 1966 y>s 3 i ys5 3 " HV proposal i s such a cont e x t u a l HV theory. According to Bub, both ways o f form u l a t i n g the con t e x t u a l HV pro p o s a l , e i t h e r i n terms of c o n t e x t u a l i z e d measures on P.„ or i n terms o f a c o n t e x t u a l i z e d a s s o c i a t i o n o f P.,, w i t h QM QM P^v , are f o r m a l l y equivalent (Bub, 1973, p. 51). C l e a r l y , both have the same e f f e c t , namely, the proposed d i s p e r s i o n - f r e e HV measure induced by a hidden s t a t e i n a con t e x t u a l HV theory does not a s s i g n a unique 0 or 1 188 value to a given element P € P^ M when P i s a member of more than one mBS i n P^^ , i . e . , when P i s a member of two or more overlapping mBS's l n PQM ' Thus the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s of a con t e x t u a l HV theory e s p e c i a l l y break up the overlap patterns among the mBS's of any i n "the manner suggested i n Chapter V(B), namely, by as s i g n i n g d i f f e r e n t values to a s i n g l e element which i s i n more than one mBS of P Q M . So, f o r example, although P g .= Pg i n the twelve-element 3 fragment of P ^ diagrammed above, and although Exp^(Pg) = Exp^(Pg) f o r 3 every quantum Exp^, on P^^ , ne v e r t h e l e s s , i n a c o n t e x t u a l HV theory, Pg may be assigned d i f f e r e n t v a l u e s , as e x e m p l i f i e d i n the previous paragraph. In t h i s sense, the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a c o n t e x t u a l HV theory do not preserve the r e l a t i o n Pg = Pg . That i s , they do not preserve the = r e l a t i o n o f P , and so i t c l e a r l y f o l l o w s t h a t w i t h respect to elements i n overlapping mBS's i n PQJ^ , the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s of a cont e x t u a l HV theory do not preserve any o f the operations and r e l a t i o n s of PQ^ . A c o n t e x t u a l HV theory and i t s d i s p e r s i o n - f r e e HV measures thereby avoid HV i m p o s s i b i l i t y p roofs. Or as Bub puts i t , i n terms o f the second fo r m u l a t i o n o f the co n t e x t u a l HV proposal which i n c l u d e s a Boolean P . and n v c l a s s i c a l d i s p e r s i o n - f r e e HV measures on P , a c o n t e x t u a l HV theory i s a type of Boolean r e c o n s t r u c t i o n o f quantum mechanics (Bub, 1974, p. 146) which avoids the Kochen-Specker i m p o s s i b i l i t y proof by l e t t i n g the a s s o c i a t i o n o f the elements o f P.,, w i t h the elements o f P T T T T be a QM HV c o n t e x t u a l i z e d mapping which breaks up the overlap patterns among the mBS's of P Q ^ 3 r a t h e r than demanding, as Kochen-Specker do, t h a t t h i s a s s o c i a t i o n be an imbedding(i) which preserves P Q J^ » i . e . , preserves a l l the 189 p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P Q ^ » i n c l u d i n g the < r e l a t i o n (and thus the = r e l a t i o n ) and i n c l u d i n g the overlap p a t t e r n s among the mBS's. Now to continue w i t h the t h i r d s o r t o f c r i t i c i s m , l a b e l e d ( i i i ) i n the Preface, o f the i m p o s i t i o n o f the s t r u c t u r a l c o n d i t i o n of Pg^-preservation and o f the development o f the Gleason, Kochen-Specker HV i m p o s s i b i l i t y p r o o f s . I f , as B e l l argues, there i s no reason why the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P , i n p a r t i c u l a r , the overlap patterns among the mBS's of P ^ , must be preserved by non-contextually a s s i g n i n g the same unique 0 or 1 value t o , say, P^ r e g a r d l e s s of whether P i s measured i n the context mBS or i n the context mBS , then the Gleason and Kochen-Specker HV i m p o s s i b i l i t y proofs beg the HV question. For these proofs r e s t upon c o n t r a d i c t i o n s caused by r e q u i r i n g t h a t 0, 1 values be assigned to the elements of a P^ i n a non-contextual, P ^ ^ - p r e s e r v i n g manner which i s not j u s t i f i e d . Moreover, these proofs do not r u l e out a cont e x t u a l HV r e c o n s t r u c t i o n of quantum mechanics, and so they do not r u l e out hidden v a r i a b l e s , as they purport to do. Kochen-Specker's work i s e s p e c i a l l y v u l n e r a b l e to the above c r i t i c i s m because of the f o l l o w i n g ambiguity, pointed out by Bub, i n the manner i n which Kochen-Specker ground the p a r t i a l - B o o l e a n algebra o f quantum p r o p o s i t i o n s which they r e q u i r e an HV theory to preserve: (a) On the one hand, Kochen-Specker regard the quantum p r o p o s i t i o n a l s t r u c t u r e as simply given by the p a r t i a l - B o o l e a n algebra o f p r o j e c t o r s or subspaces o f H i l b e r t space, which has been l a b e l e d P Q J ^ • For according to Kochen-Specker, i t i s a "basic t e n e t " o f quantum mechanics th a t quantum magnitudes are represented by operators on a H i l b e r t space and 190 s i m i l a r l y , quantum p r o p o s i t i o n s , qua idempotent magnitudes, are represented by p r o j e c t o r s on a H i l b e r t space (Kochen-Specker, 1967, p. 65). That i s , the quantum p r o p o s i t i o n a l s t r u c t u r e i s a P^ s t r u c t u r e , i n p a r t i c u l a r , a PQMA s t r u c t u r e , o f p r o j e c t o r s or subspaces of a H i l b e r t space. And f o r example, two p r o p o s i t i o n s are equivalent i n P Q J ^ i f "they are represented by the same p r o j e c t o r (or subspace). (b) But on the other hand, Kochen-Specker d e f i n e a p a r t i a l - B o o l e a n algebra of quantum p r o p o s i t i o n s w i t h respect to a_ set of s t a t e s and measures; the defined s t r u c t u r e s h a l l be l a b e l e d pBA to d i s t i n g u i s h i t from the above P Q J ^ • The d e f i n i t i o n o f pBA may y i e l d , f o r example, t h a t two p r o p o s i t i o n s are equivalent i n pBA i f t h e i r expectation-values are equal f o r a l l quantum Exp^ measures. I f the quantum p r o p o s i t i o n a l s t r u c t u r e which Kochen-Specker r e q u i r e an HV theory to preserve i s such a pBA defined w i t h respect to the quantum measures, then Kochen-Specker's i m p o s s i b i l i t y proof, which r e s t s upon the requirement t h a t the quantum p r o p o s i t i o n a l s t r u c t u r e be preserved, begs the HV question. For there i s no reason why proposed HV measures must preserve such a pBA, and i n p a r t i c u l a r preserve the equivalence = P^ i n pBA, i f P^ = P^ i n pBA only because Exp^(Pg) = Exp^,(P,p f o r a l l quantum Exp^ measures. Moreover, i f d i s p e r s i o n - f r e e HV measures do e x i s t , then the pBA defined w i t h respect to the quantum measures and the HV measures may be d i f f e r e n t from the pBA defined w i t h respect to j u s t the quantum Exp^, measures. These c r i t i c i s m s o f Kochen-Specker's defined pBA are s i m i l a r to the c r i t i c i s m s o f . von Neumann's use of h i s c o n d i t i o n (E) to d e f i n e the sums o f incompatibles. This ambiguity between (a) and ( b ) , and the way i n which (b) leads to a misunderstanding o f Kochen-Specker's work and makes the Kochen-Specker 191 i m p o s s i b i l i t y r e s u l t appear to be e s p e c i a l l y v u l n e r a b l e to B e l l ' s c r i t i c i s m , are described by Bub (1974, pp. 84-88). Bub concludes t h a t Kochen-Specker are best understood r e f e r r i n g to P n M f l r a t h e r than pBA, t h a t i s , Kochen-Specker should have used j u s t the (a) n o t i o n and not discussed the (b) n o t i o n a t a l l . Moreover, as s h a l l be described i n S e c t i o n B ', from Bub's pe r s p e c t i v e on the problem o f h i d d e n - v a r i a b l e s , the ambiguity between (a) and (b) i s not s u b s t a n t i a l l y important, though i t i s confusing and leads to a misunderstanding o f Kochen-Specker's work, and so the ambiguity i s worth c l a r i f y i n g . In the r e s t o f t h i s s e c t i o n , Kochen-Specker's (b) d e f i n i t i o n o f pBA i s elaborated, and a reason why Kochen-Specker may have been motivated to develop t h i s (b) d e f i n i t i o n i s given. According to Kochen-Specker, a p h y s i c a l theory l i k e c l a s s i c a l mechanics or quantum mechanics or a proposed HV theory c o n s i s t s o f a s e t o f magnitudes {A,...}, a set o f s t a t e s . .. },• and a set o f ( c l a s s i c a l ) p r o b a b i l i t y measures {p ,...} on the real-number l i n e R, or more ty,A e x a c t l y , on the Boolean s t r u c t u r e B^ o f B o r e l subsets o f R. For any B o r e l subset R c R, f o r any magnitude A, and f o r any s t a t e ty, Pty € [0,1] i s the p r o b a b i l i t y t h a t the r e a l - v a l u e o f A i s a member of R. These p measures on B D are r e l a t e d to the more f a m i l i a r ty , A K e x p e c t a t i o n - f u n c t i o n s Exp and are r e l a t e d to the HV measures p, of a Kochen-Specker type o f HV r e c o n s t r u c t i o n o f quantum mechanics, by equations given below. Now Kochen-Specker argue t h a t the magnitudes of a p h y s i c a l theory are not independent o f each other but r a t h e r are f u n c t i o n a l l y r e l a t e d , e.g., 2 2 the magnitude A i s c l e a r l y a f u n c t i o n of A. And the f u n c t i o n A o f the magnitude A can be measured by simply measuring A and squaring the r e s u l t i n g value. That i s , the r e a l value o f any ( B o r e l ) f u n c t i o n g(A) of 192 any magnitude A i s c a l c u l a t e d by simply a p p l y i n g t h a t f u n c t i o n g to the r e a l value o f A. The l a s t sentence i s a statement o f what may be regarded as an uncontentious general p r i n c i p l e which a p p l i e s to the magnitudes o f any p h y s i c a l theory. Kochen-Specker a l s o assume that the magnitudes o f a p h y s i c a l theory are determined by the p^ A measures i n the f o l l o w i n g sense: (*) For any magnitudes A, B, i f p^ A ( R ) = p^ g ( R ) f o r every s t a t e ty and any B o r e l subset R c R, then A = B. With (*), the above general p r i n c i p l e suggests the f o l l o w i n g d e f i n i t i o n , which Kochen-Specker l a b e l ( 3 ) , f o r a f u n c t i o n g(A) of any magnitude A: (3) For any A and any B o r e l f u n c t i o n g, Pty g ( A ) ^ = Pty A^ S f ° r a n y s t a t e Y a n d a n v R - R (Kochen-Specker, 1967, pp. 61, 63). In f a c t , (3) can be regarded as a restatement of the uncontentious general p r i n c i p l e . For i f the r e a l value o f A i s a member of some B o r e l subset R £ R, then by the general p r i n c i p l e , the r e a l value o f g(A) i s a member of the B o r e l subset g(R) £ R. Likewise, i f the r e a l value of g(A) i s a member o f some R c R 5 then by the general p r i n c i p l e , the r e a l value o f A i s a member of the B o r e l subset g (R) c R. So assuming t h a t Pty g ( A ) ^ R ^ 1 S t n e P r°b ability t h a t the r e a l value of g(A) i s i n R, and assuming t h a t the p^. A measures determine the magnitude o f a p h y s i c a l theory i n the above (*) sense, then by the general p r i n c i p l e we can be sure t h a t P V , g ( A ) ( R ) = P Y i A ( s " 1 ( R ) ) -Moreover, w i t h respect to a Kochen-Specker type o f HV r e c o n s t r u c t i o n o f quantum mechanics, i n which each quantum magnitude A i s 193 represented by a f u n c t i o n f : S2 -»• R on the HV phase space S2 and the r e a l value of A f o r any hidden s t a t e w 6 8 i s ^ A( w)» t n e general p r i n c i p l e y i e l d s the i d e n t i t y : f o r any w € S2, ^ g ( A ) ^ W ^ = 2^ A^ W'^ * ^° i n a Kochen-Specker type of HV r e c o n s t r u c t i o n , the f u n c t i o n s { f ^ , . . . } re p r e s e n t i n g the quantum magnitudes i n the proposed HV theory must s a t i s f y the f o l l o w i n g s t r u c t u r a l c o n d i t i o n l a b e l e d (4) by Kochen-Specker: (4) For any quantum magnitude A and any B o r e l f u n c t i o n g, f g ( A ) = * ( f A > ' Kochen-Specker aim t o show that an HV r e c o n s t r u c t i o n o f quantum mechanics which s a t i s f i e s (4) i s impossible. But f i r s t Kochen-Specker r e p l a c e (4) by a more t r a c t a b l e s t r u c t u r a l c o n d i t i o n as f o l l o w s . Using (*) and ( 3 ) , Kochen-Specker d e f i n e the r e l a t i o n o f commeasurability, i . e . , c o m p a t i b i l i t y , among the magnitudes of a p h y s i c a l theory as s t a t e d i n Chapter IV(B). Then u s i n g (*) and (3) again, Kochen-Specker d e f i n e the r i n g operations + and • among commeasurable magnitudes as f o l l o w s : For any magnitudes A^ , A^ , i f , A^ are commeasurable, then f o r some magnitude B and B o r e l f u n c t i o n s g^ , g^, k± = g 1(B) and A 2 = g 2 ( B ) , and then (5) A 1 + A 2 = ( g l + g 2 ) ( B ) , A 1 • A 2 = ( g l • g 2)(B). With + and • so defined among compatible magnitudes, the set of magnitudes of a p h y s i c a l theory acquires the s t r u c t u r e o f a p a r t i a l - a l g e b r a , or i n the terminology o f Chapter 1(D), a p a r t i a l - r i n g - w i t h - u n i t . And thus the set o f p r o p o s i t i o n s o f a p h y s i c a l theory, qua idempotent magnitudes, i . e . , qua idempotent elements o f a p a r t i a l - r i n g - w i t h - u n i t , has the s t r u c t u r e of a p a r t i a l - B o o l e a n algebra. In p a r t i c u l a r , by (*), ( 3 ) , and ( 5 ) , the 194 mutually compatible magnitudes o f c l a s s i c a l mechanics form a commutative-r i n g - w i t h - u n i t , which i s a s p e c i a l case of a p a r t i a l - r i n g - w i t h - u n i t , namely, the case where a l l elements are mutually compatible, as described i n Chapters I I I ( B ) and 1(D). And the p r o p o s i t i o n s o f c l a s s i c a l mechanics form a Boolean a l g e b r a , which again i s the s p e c i a l case o f a p a r t i a l - B o o l e a n algebra where a l l elements are mutually compatible. Li k e w i s e , the magnitudes of a proposed Kochen-Specker type o f HV theory form a commutative-ring-with-u n i t , and the p r o p o s i t i o n s o f such an HV theory form a Boolean algebra. And f i n a l l y , by (*), ( 3 ) , and (5) the magnitudes o f quantum mechanics form a p a r t i a l - r i n g - w i t h - u n i t , and the p r o p o s i t i o n s of quantum mechanics form a p a r t i a l - B o o l e a n a l g e b r a , l a b e l e d pBA. This completes the Kochen-Specker d e f i n i t i o n o f a pBA of quantum p r o p o s i t i o n s , which s h a l l be f u r t h e r discussed s h o r t l y . Kochen-Specker then note t h a t t h e i r c o n d i t i o n (4) i m p l i e s t h a t the p a r t i a l - o p e r a t i o n s + and •, which are def i n e d among ( j u s t ) compatible quantum p r o p o s i t i o n s { p i > p 2 > ' a n d a m o n § "'-he compatible HV re p r e s e n t a t i v e s {f , f ,•••} o f quantum p r o p o s i t i o n s by the c o n d i t i o n 1 2 ( 5 ) , are preserved by the mapping % which a s s o c i a t e s the quantum p r o p o s i t i o n s w i t h t h e i r HV r e p r e s e n t a t i v e s , i n t h i s case % : pBA P^v . For example, as elaborated by Bub, f o r any compatible , P 0 » which are by the d e f i n i t i o n o f c o m p a t i b i l i t y B o r e l f u n c t i o n s o f some common P, say P 1 = g l ( P ) and P 2 = g 2 ( P ) , we have: f ^ = f ^ p ^ p j = (by (5)) f(g1+g2)(P) = ( b y C 4 ) ) ( g l + g 2 ) ( f P ) = ( b y C 5 ) ) g l ( f P ) + g 2 ( V = ( b y M ) f , , + f , . = f + f (Bub, 1974, p. 87). So, f o r example, i f % ( P 1 ) = f p and % ( P 2 ) = f p and % ( P 1 + P 2 ) = f p + p , then by (4) and 195 (5) we have: % ( P 1 + P ) = %(P ) + % ( P 2 ) . Thus the mapping % which a s s o c i a t e s quantum p r o p o s i t i o n s w i t h t h e i r HV r e p r e s e n t a t i v e s preserves the p a r t i a l - o p e r a t i o n + among compatible quantum p r o p o s i t i o n s . S i m i l a r l y , i t can be shown t h a t , by (4-) and ( 5 ) , % preserves the p a r t i a l - o p e r a t i o n among compatible quantum p r o p o s i t i o n s . And so w i t h the operation and the p a r t i a l - o p e r a t i o n s A, V defined i n terms of +, • as u s u a l , the mapping % : pBA -»• P m r preserves these A, V, operations s i n c e i t HV preserves the +, • opera t i o n s . In other words, % i s an imbedding(i>) (Kochen-Specker,. 1967, pp. 63-66). However, as pointed out by Bub, i t i s c l e a r t h a t i n t h i s (b) d e f i n i t i o n o f pBA, Kochen-Specker r e l y upon the ^ measures to d e f i n e , b y ' ( * ) , the equivalence o f quantum p r o p o s i t i o n s , and to d e f i n e , w i t h (*) and ( 3 ) , the f u n c t i o n a l r e l a t i o n s and the c o m p a t i b i l i t y r e l a t i o n s among the quantum p r o p o s i t i o n s . That i s , the pBA s t r u c t u r e o f quantum p r o p o s i t i o n s which Kochen-Specker r e q u i r e an HV r e c o n s t r u c t i o n to preserve i s defined w i t h respect to the ^ measures. These measures on 8^ are r e l a t e d to ex p e c t a t i o n - f u n c t i o n s Exp by the equation: For any magnitude A and any s t a t e v, Exp.(A) = f r dp, . ( { r } ) , r 6 P. And the p. . measures are R Y J - ° ° ty,A y,A r e l a t e d to the p measures o f a Kochen-Specker type of HV r e c o n s t r u c t i o n by the equation: For any magnitude A, any s t a t e y, and any B o r e l subset R c R, p^ a(R) = [^(f" A (R)) (Kochen-Specker, 1967, p. 61). Now so f a r , A and Y designate any magnitude and any s t a t e i n any p h y s i c a l theory. So w i t h respect to the issu e o f a proposed HV r e c o n s t r u c t i o n o f quantum mechanics, i t i s not c l e a r whether the set of \|/ s t a t e s , which v i a the p measures y, A d e f i n e s pBA, i n c l u d e s j u s t the quantum s t a t e s , which are u s u a l l y designated by Y» °r in c l u d e s both the quantum s t a t e s and the hidden s t a t e s proposed 196 by an HV r e c o n s t r u c t i o n . And as suggested above, the pBA defined w i t h respect to j u s t the quantum s t a t e s may be d i f f e r e n t from the pBA defined w i t h respect to both the quantum and the hidden s t a t e s ; i n p a r t i c u l a r , w h i l e the former i s isomorphic to P Q ^ A » t n e l a t t e r might not be. I f Kochen-Specker mean the set of s t a t e s which d e f i n e , v i a the ^ measures, t h e i r pBA to i n c l u d e both quantum and hidden s t a t e s , then they are presuming t h a t hidden s t a t e s e x i s t and they thus beg the HV question i n a t r i v i a l way. I f i t does not matter whether the set of s t a t e s i n c l u d e s j u s t the quantum s t a t e s or i n c l u d e s both quantum and hidden s t a t e s , then Kochen-Specker beg the HV question i n the sense t h a t they presume t h a t the pBA defined w i t h respect to the quantum y s t a t e s i s the same as the pBA defined w i t h respect to both quantum y and hidden w s t a t e s ; i n p a r t i c u l a r , they presume th a t \i (P.) = u. (P„) j u s t as Exp (P ) = Exp (P ). w o w o y o y d But i n a c o n t e x t u a l HV theory, an element P 3 which i s a member of two or more overlapping mBS's i n pBA i s not assigned a unique value f o r a given hidden s t a t e w s p e c i f i e d by y, £, e.g., u\. (<P ,mBS >) may not equal y , c , o 1 [i.^ ^ (<P3,mBS|+>). And f i n a l l y , i f Kochen-Specker mean the s e t o f s t a t e s which d e f i n e t h e i r pBA to in c l u d e j u s t the quantum s t a t e s , then they beg the HV question i n the manner described on page 190. For then by quantum s t a t e s determine the i d e n t i t y of the quantum magnitudes and quantum p r o p o s i t i o n s ; i . e . , f o r any quantum p r o p o s i t i o n s P ^ ' P 2 ' P l = P 2 ^ ^y P ~ ^y P e v e r y 9. u a n t u m s " t a t e V a n d any B o r e l subset R c R. Or i n other words, by the above equation connecting Exp^ w i t h p^ A we have: For any quantum p r o p o s i t i o n s P^ , P^ , p ^ = p 2 i f Exp^,(P^) = Exp^CP^) f o r every quantum s t a t e . But there i s no reason why proposed d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a 197 proposed HV theory must preserve t h i s equivalence which i s defined with respect to the quantum states and measures. Thus contextual HV measures which do not preserve the equivalences i n pBA may be proposed e s p e c i a l l y i n order to avoid the Kochen-Specker HV i m p o s s i b i l i t y proof. So I f Kochen-Specker had only the (b) d e f i n i t i o n of pBA, then t h e i r c r u c i a l imbedding(A) condition would u n j u s t i f i a b l y demand the preservation of a structure defined with respect to maybe just the quantum states and measures. However, Kochen-Specker have not only the defined pBA but also the (a) PQJ^ given by the basic tenets of quantum mechanics. And as Bub argues, both the Kochen-Specker HV i m p o s s i b i l i t y proof and the contextual HV counter-proposal are best understood i f we give Kochen-Specker the benefit of the doubt and resolve t h e i r ambiguity between (a) and (b) i n favour of the (a) PQ^a • The very f a c t that Kochen-Specker require that the quantum p r o p o s i t i o n a l structure be preserved i n an HV reconstruction suggests that they regard i t as something more than a merely s t a t i s t i c a l structure defined with respect to the dispersive quantum states and measures. Moreover, Kochen-Specker s p e c i f i c a l l y declare t h e i r Theorem 1 to be a f i n i t e version of Gleason's i m p o s s i b i l i t y proof, which r e f e r s to the projectors or subspaces of H i l b e r t space. Thus Kochen-Specker*s f i n i t e version of Gleason's proof may likewise be understood as r e f e r r i n g to the PQJ^ structure of projectors or subspaces of H i l b e r t space rather than r e f e r r i n g to the pBA structure. Kochen-Specker may have been motivated to develop t h e i r (b) d e f i n i t i o n of pBA i n order that t h e i r contentious imbedding(A) condition should f o l l o w from the uncontentious general p r i n c i p l e as described above. But then Kochen-Specker should have used the general p r i n c i p l e only to 198 support, via their imbedding(c>) condition rather than to help define, via (*)» (3), (4), (5), a pBA of quantum propositions. For example, we may take the quantum propositional structure to be a PQJ^ °f projectors or subspaces of Hilbert space; so = i f p^ = P2 ' a n d t h e partial-operations +, • are defined among compatible propositions as projector addition and multiplication. Then we may s t i l l argue that in a proposed HV reconstruction of quantum mechanics, where any quantum proposition P is represented by an idempotent function f p : Si -> {0,1} on the HV phase space Si and any Borel function g(P) is correspondingly represented by the idempotent function ^g(p) » t n e uncontentious general principle requires that the 0, 1 values issued by ^g(p) m u s " t be g-functions of the 0, 1 values issued, by f p . And the fulfillment of this requirement is best ensured by making fg(p) = §(fp)» f ° r a nY p a n d any Borel function g. Thus we have condition (4), from which the imbedding (A) condition follows as described above. In other words, the crucial Kochen-Specker imbedding(A) condition, which requires that PQJJA be preserved in any proposed HV reconstruction of quantum mechanics, _is_ supported by the uncontentious general principle which i t seems no c r i t i c of Kochen-Specker's HV impossibility proof could reasonably object to. However, without realizing or disregarding the above elaborated connection between the general principle and the imbedding(i) condition, c r i t i c s of the Kochen-Specker proof may argue that even i f Kochen-Specker are understood as referring to PQJ^ rather than pBA, their proof begs the HV question because their imbedding(A) condition, which requires pQ^-preservation and which rules out hidden-variables, is not j u s t i f i e d . In other words, c r i t i c s may argue that there i s no reason why a proposed HV 199 r e c o n s t r u c t i o n must preserve even t h i s PQ^A given by the fundamental p o s t u l a t e s o f quantum mechanics. In f a c t , B e l l must be understood as making t h i s f u r t h e r argument, f o r he addresses h i m s e l f to the Gleason i m p o s s i b i l i t y proof and thus to an (a) type of s t r u c t u r e r a t h e r than a (b) type of s t r u c t u r e . Bub rescues the Gleason, Kochen-Specker proofs from t h i s c r i t i c i s m , as described i n the next s e c t i o n . S e c t i o n B. E i t h e r P ^ ^ - p r e s e r v a t i o n or Boolean Reconstruction Bub argues t h a t the concept of an HV r e c o n s t r u c t i o n of quantum mechanics does not make sense unless the quantum p r o p o s i t i o n a l s t r u c t u r e i s preserved. For according to Bub, quantum mechanics i s a p r i n c i p l e theory r a t h e r than a c o n s t r u c t i v e theory. The d i s t i n c t i o n i s due to E i n s t e i n and i s d escribed by Bub as f o l l o w s . C o n s t r u c t i v e t h e o r i e s "aim to reduce a wide c l a s s of d i v e r s e systems to component systems of a p a r t i c u l a r k i n d (e.g., the molecular hypothesis o f the k i n e t i c theory of gases)." In c o n t r a s t , p r i n c i p l e t h e o r i e s "introduce a b s t r a c t s t r u c t u r a l c o n s t r a i n t s t h a t events are h e l d to s a t i s f y , " e.g., s p e c i a l and general r e l a t i v i t y can be viewed as p r i n c i p l e t h e o r i e s of space-time s t r u c t u r e (Bub, 1974, pp. v i i , 14-2). Bub regards quantum mechanics and c l a s s i c a l mechanics as p r i n c i p l e t h e o r i e s of l o g i c a l s t r u c t u r e because, . . . they introduce c o n s t r a i n t s on the way i n which the p r o p e r t i e s of a p h y s i c a l system are s t r u c t u r e d . The l o g i c a l s t r u c t u r e of a p h y s i c a l system i s understood as imposing the most general k i n d of c o n s t r a i n t on the occurrence and non-occurrence of events. (Bub, 1974, p. 149) The l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of a p h y s i c a l system i s given by the p r o p o s i t i o n a l s t r u c t u r e as determined by the mathematical formalism of the 2 0 0 p h y s i c a l theory d e s c r i b i n g the system, namely, the c l a s s i c a l P and the quantum P . So at the very core o f quantum mechanics i s the non-Boolean P s t r u c t u r e , which Bub and Kochen-Specker e x p l i c i t l y and Gleason i m p l i c i t l y t a k e ( s ) to be a P n M. • And according to Bub, (1) the question of the completeness o f quantum mechanics must be posed w i t h respect to PQMA ' ^ e q u a n ' t u m p r o b a b i l i t y measures are defined on P Q J ^ a n d the s t a t i s t i c a l r e s u l t s of quantum mechanics make sense w i t h respect t o P Q J ^ » and ( i i i ) any HV r e c o n s t r u c t i o n o r extension o f quantum mechanics must preserve the quantum P Q J ^ • Now as shown by Kochen-Specker and by Gudder, a Boolean HV r e c o n s t r u c t i o n of quantum mechanics which does not preserve the P Q ^ A s t r u c t u r e i s always p o s s i b l e . By a t r i v i a l c o n s t r u c t i o n , Kochen-Specker show that i s i s always p o s s i b l e to introduce a c l a s s i c a l measure space <!)2,P , u> = X which reproduces the quantum s t a t i s t i c s but does not preserve nv PQMA ( K o c n e n ~ S p e c k e r , 1 9 6 7 , p. 6 3 ) . And Gudder proves t h a t i t i s always p o s s i b l e t o introduce a cont e x t u a l HV Boolean r e c o n s t r u c t i o n which reproduces the quantum s t a t i s t i c s and preserves the Boolean s t r u c t u r a l f eatures of the mBS's o f P Q ^ a but which breaks up the overlap patterns among the mBS's and so does not preserve P Q ^ (Gudder, 1 9 7 0 , pp. 4 - 3 4 — 4 3 6 ) . However, as Bub argues: The c o n t r i b u t i o n o f Kochen-Specker l i e s i n showing that the problem o f hidden v a r i a b l e s i s not th a t o f f i t t i n g a t h e o r y — i . e . , a c l a s s o f event s t r u c t u r e s — t o a s t a t i s t i c s . This can. always be done i n an i n f i n i t e number of ways; i n p a r t i c u l a r , a Boolean r e p r e s e n t a t i o n i s always p o s s i b l e . Rather, the problem concerns the ki n d o f s t a t i s t i c s d e f i n a b l e on a given c l a s s o f event s t r u c t u r e s . (Bub, 1 9 7 4 , p. 8 8 ) . The event s t r u c t u r e s given by the fundamental p o s t u l a t e s of quantum mechanics 201 are the non-Boolean P s t r u c t u r e s , i n p a r t i c u l a r , the s t r u c t u r e s . So the problem o f the completeness o f quantum mechanics and the concordant problem of hidden v a r i a b l e s i s c o r r e c t l y addressed w i t h respect to the quantum PQJ^ » a s done by Gleason and Kochen-Specker. In Bub's view, Gleason's completeness proof shows t h a t the quantum formalism generates a l l _n>3 p o s s i b l e ( g e n e r a l i z e d ) p r o b a b i l i t y measures on the s t r u c t u r e s o f three-or-higher dimensional H i l b e r t space. That i s , w i t h respect to P Q ^ » the quantum mechanics o f three-or-higher dimensional H i l b e r t space i s complete. And i t f o l l o w s as a c o r o l l a r y t h a t , f o r PQ^ a » d i s p e r s i o n - f r e e ( g e n e r a l i z e d ) p r o b a b i l i t y measures which preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of PQJ^ a r >e impossible. And so by Kochen-Specker's Theorem 0, an imbedding(i) o f P N M F L i n t o a Boolean s t r u c t u r e i s impossible. Thus a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics which preserves P Q ^ i s impossible. That i s , w i t h respect to PQ^A > a n HV r e c o n s t r u c t i o n o f the quantum mechanics o f three-or-higher dimensional H i l b e r t space i s impossible. The above i n t e r p r e t a t i o n o f Gleason and Kochen-Specker's work a c t u a l l y depends upon our acknowledging the p r i o r i t y o f the PQ^a s t r u c t u r e as the core, or at l e a s t p a r t o f the core, of quantum mechanics which must  be preserved. For d i s p e r s i o n - f r e e HV measures and a Boolean HV r e c o n s t r u c t i o n which do not preserve PQ^a a r e always p o s s i b l e . So i f F*QMA were not r e q u i r e d to be preserved, then i n s p i t e o f Gleason's completeness proof, the f a c t t h a t a l l the measures generated by the quantum formalism are d i s p e r s i v e would s i g n a l the incompleteness o f quantum mechanics r e l a t i v e to a p o s s i b l e Boolean HV r e c o n s t r u c t i o n which i n c l u d e d d i s p e r s i o n - f r e e HV measures. 202 Now as Bub mentions, the completeness of quantum mechanics with respect to PQJ^ » i . e . , the f a c t that the quantum formalism generates a l l possible (generalized) p r o b a b i l i t y measures on any 9 guarantees that the PQJ^ structure given by the fundamental quantum postulates and the pBA structure defined with respect to the quantum measures are isomorphic (Bub, 1974, p. 45). So the ambiguity, described i n Section KAJ, i n Kochen-Specker's notion of the quantum pr o p o s i t i o n a l structure as a (a) given PQJ^ a n d a (b) defined pBA i s not harmful but merely confusing. In p a r t i c u l a r , we can be sure that i f Exp^P^) = Exp^(P 2) f o r a l l quantum Exp. , then P, = P. i n P.„. . * f 1 2 QMA Moreover, i f we acknowledge the p r i o r i t y of the PQJJa structure i n quantum mechanics, then the s t r u c t u r a l condition of PQ^a preservation and the Gleason, Kochen-Specker HV i m p o s s i b i l i t y proofs emerge unscathed by the three sorts of c r i t i c i s m s described i n the previous section. In p a r t i c u l a r , P^^-preservation must s t i l l be required of a proposed HV theory i n s p i t e of the f a c t that t h i s condition leads to contradictions which make the HV theory impossible and of the zeroth kind, i n Belinfante's terminology. And P^^-preservation must be required of the proposed dispersion-free measures of an HV theory i n s p i t e of the considerations of measurement i n t e r a c t i o n which B e l l r a i s e s i n order to dissuade our imposing t h i s condition. And f i n a l l y , the Gleason, Kochen-Specker proofs cannot be charged with begging the HV question because they impose the Pg^-preservation condition, f o r the question of an HV reconstruction of quantum mechanics does not even make sense except with respect to the quantum PQJ^ structure, which must be preserved. In contrast, an HV advocate may choose to regard to quantum P^ 203 structure, whether P Q ^ A O R ^ Q M L ' a s n°"t worthy of preservation when considered with respect to the l a r g e r enterprise of providing a c l a s s i c a l , Boolean reconstruction or r e - i n t e r p r e t a t i o n of quantum mechanics, e s p e c i a l l y because such a reconstruction i s possible i f the quantum P N ^ i s not preserved. So rather than affirming the p r i o r i t y of the quantum structure i n the i n t e r p r e t a t i o n of quantum mechanics, an HV advocate may instead a f f i r m that ( i ' ) the problem of the completeness of any phys i c a l theory only makes sense when posed or framed with respect to a Boolean logical-property-event structure, ( i i ' ) the p r o b a b i l i t y measures of any s t a t i s t i c a l theory l i k e quantum mechanics are to be defined on a Boolean structure, and ( i i i 1 ) a Boolean HV reconstruction of quantum mechanics need not preserve the quantum P N ^ structure. As described by Bub, i f we acknowledge the p r i o r i t y of a Boolean HV reconstruction of quantum mechanics by aff i r m i n g these three primed conditions, then quantum mechanics i s incomplete and an HV reconstruction i s possible and completes quantum mechanics. Most simply, a Boolean structure always admits dispersion-free measures., yet quantum mechanics lacks dispersion-free measures. So with respect to a Boolean logical-property-event structure, quantum mechanics i s incomplete; and quantum mechanics i s completed when reconstructed as a Boolean HV theory which includes dispersion-free measures. Moreover, i f we acknowledge the p r i o r i t y of a Boolean HV reconstruction of quantum mechanics, then the ambiguity i n the Kochen-Specker notion of the quantum p r o p o s i t i o n a l structure i s again not harmful but merely confusing; f o r neither the (a) given P nor the (b) defined pBA need be preserved. I t also follows from the above acknowledgement that the s t r u c t u r a l condition of P „ „ .-preservation succumbs QMA 204 to the three s o r t s o f c r i t i c i s m s described i n the previous s e c t i o n , as do the s t r u c t u r a l c o n d i t i o n s (vn£>) and (JP&). In p a r t i c u l a r , s i n c e there i s no reason why an HV r e c o n s t r u c t i o n must s a t i s f y any of these s t r u c t u r a l c o n d i t i o n s , the von Neumann, the Jauch-Piron, the Gleason and the Kochen-Specker i m p o s s i b i l i t y proofs do beg the HV question s i n c e each r e s t s upon c o n t r a d i c t i o n s caused by the i m p o s i t i o n of an u n j u s t i f i e d c o n d i t i o n . B e l l ' s c o n s i d e r a t i o n s o f measurement i n t e r a c t i o n lend f u r t h e r support to the r e j e c t i o n o f the s t r u c t u r a l c o n d i t i o n s as u n j u s t i f i e d . And s i n c e the s t r u c t u r a l c o n d i t i o n s l e a d to c o n t r a d i c t i o n s , i n other words, s i n c e HV t h e o r i e s which i n c l u d e these s t r u c t u r a l c o n d i t i o n s are of the zeroth k i n d and are imp o s s i b l e , we can be sure t h a t the s t r u c t u r a l c o n d i t i o n s are p r e c i s e l y what a proposed HV r e c o n s t r u c t i o n o f quantum mechanics must not be r e q u i r e d t o s a t i s f y . So there are these two ways of i n t e r p r e t i n g quantum mechanics: E i t h e r the s t r u c t u r e i s regarded as the core o f quantum mechanics which must be preserved, i n which case quantum mechanics i s complete (as proved by Gleason) and a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics i s impossible (as proved by Kochen-Specker). Or the p o s s i b i l i t y o f a Boolean r e c o n s t r u c t i o n o f quantum mechanics i s regarded as the most important consi&eratd'onrintthe interpretarion'-.ofc quantum:.mechanics,-' in-'which case a co n t e x t u a l Boolean HV r e c o n s t r u c t i o n which does not preserve P_,,. i s c QMA p o s s i b l e and quantum mechanics i s incomplete r e l a t i v e to t h i s r e c o n s t r u c t i o n . The a r t i c u l a t i o n o f t h i s dichotomy i s Bub's d e c i s i v e c o n t r i b u t i o n to the i n t e r p r e t a t i o n o f quantum mechanics and the problem of hi d d e n - v a r i a b l e s (see, e.g., Bub, 1973, p. 48). And n o t i c e t h a t t h i s dichotomy undercuts the three s o r t s o f arguments described i n Sectio n '.A . For re g a r d l e s s o f 205 the i n c o n s i s t e n c y and question begging c l a i m s , and re g a r d l e s s of B e l l ' s c o n s i d e r a t i o n s o f measurement i n t e r a c t i o n , the s t r u c t u r a l c o n d i t i o n s and the HV i m p o s s i b i l i t y proofs e i t h e r stand or f a l l depending upon which s i d e of the dichotomy one favours. In f a c t , which s i d e of the dichotomy one favours a l s o determines whether the i n c o n s i s t e n c y and question begging claims stand or f a l l . In the r e s t o f t h i s s e c t i o n , some arguments i n favour o f the P ^ ^ - p r e s e r v a t i o n s i d e o f t h i s dichotomy are described. One might a l s o consider regarding the orthomodular l a t t i c e P A M T r a t h e r than the QML p a r t i a l - B o o l e a n algebras P N M » as the core o f quantum mechanics which must QMA be preserved; some arguments ag a i n s t r e g a r d i n g p . T as the core o f quantum QML 7 mechanics are suggested by various p o i n t s made throughout t h i s t h e s i s . Both sides o f the dichotomy imply the i m p o s i t i o n o f s t r u c t u r a l c o n d i t i o n s on a proposed Boolean HV r e c o n s t r u c t i o n o f quantum mechanics. C l e a r l y , on the P ^ ^ - p r e s e r v a t i o n s i d e , the Boolean s t r u c t u r a l f e a t u r e s of each mBS i n a P^ M and the overlap p a t t e r n s among the mBS's i n a P^~ must be preserved. And on the Boolean r e c o n s t r u c t i o n s i d e , the Boolean s t r u c t u r a l f e a t u r e s o f each mBS i n a P^ may be preserved but, by v i r t u e o f the Gleason, Kochen-Specker r e s u l t s , the overlap p a t t e r n s among the mBS's i n a PQJ^3 cannot be preserved. So i t i s not the case t h a t one s i d e o f the dichotomy imposes s t r i n g e n t s t r u c t u r a l c o n d i t i o n s w h i l e the other s i d e does not. Rather, both sides impose e q u a l l y s t r i n g e n t c o n d i t i o n s : e i t h e r the overlap p a t t e r n s among the mBS's must be preserved, or the overlap patterns must be v i o l a t e d . The simple proposal that i n a proposed Boolean r e c o n s t r u c t i o n , the operations and r e l a t i o n s among compatibles ought to be preserved w h i l e 206 the operations and r e l a t i o n s among incompatibles ought to be ignored, does not help decide between the two sides of the dichotomy. A l l the elements i n an mBS of a P are mutually compatible; and f o r any non-overlapping mBS. , mBS. of P.,, , every element P. € mBS. (except the distinguished 0, 1 elements) i s incompatible with every element P^ € mBS_. (except the distinguished 0, 1 elements). But the elements i n any overlapping mBS's n>3 of P n^ are i n e x t r i c a b l y compatible and incompatible with each other i n the following sense. On the one hand, i f the operations and r e l a t i o n s among compatibles are preserved, then the overlap patterns are preserved, and then i t follows, as B e l l r i g h t l y argues, that some r e l a t i o n s among incompatibles are also preserved. On the other hand, i f the overlap patterns are not preserved, then these r e l a t i o n s among incompatibles are not preserved, but also some r e l a t i o n s among compatibles are not preserved. For example, consider the r e l a t i o n P^ < P^ V P,_ among the elements P^ , P^ , P i n the two overlapping mBS's of the twelve-element Pn^ diagrammed i n Section ''A.. I f the overlap pattern between mBS^ and mBS^ i s preserved, then t h i s r e l a t i o n i s preserved even though P. <J5 P, and P. c$ P , as B e l l 1 4 1 5 c r i t i c i z e s . But i f the overlap pattern i s not preserved, then even though P^ A> P^ V P , t h i s r e l a t i o n i s not preserved i n the sense that, f o r example, i n a contextual HV theory, f o r a given hidden state f , £, P^ may be assigned a value which i s not less-than-or-equal-to the value assigned to P^ v P 5 , i . e . , (<Pl,tnBS1>) £ g ( < I V v p 5 » m B S 4 > ^ I n short, r e l a t i o n s among compatible elements i n overlapping mBS's cannot be preserved without also preserving r e l a t i o n s among incompatibles, and r e l a t i o n s among incompatible elements i n overlapping mBS's cannot be ignored without also ignoring r e l a t i o n s among compatibles. 207 The c o n t e x t u a l HV proposals are the se r i o u s contenders on the Boolean r e c o n s t r u c t i o n s i d e o f the dichotomy. As Gudder makes c l e a r , c o n t e x t u a l HV t h e o r i e s preserve the Boolean s t r u c t u r a l f e a t u r e s of the mBS's i n a P s t r u c t u r e (Gudder, 1970, p. 435). But as described i n S e c t i o n A , the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a cont e x t u a l HV theory v i o l a t e the overlap patterns among the mBS's by as s i g n i n g d i f f e r e n t 0 or 1 values to a given element P € P^ when P i s a member of overlapping mBS's. That i s , the value assigned to P when considered i n the context o f one mBS may be d i f f e r e n t from the value assigned to P when considered i n the context o f another mBS. In t h i s sense, the i d e n t i t y P = P i s v i o l a t e d ; so c l e a r l y , any other o p e r a t i o n or r e l a t i o n among elements i n overlapping mBS's may be v i o l a t e d . And, f o r example, the d i s p e r s i o n - f r e e HV measures of a con t e x t u a l HV theory cannot even preserve j u s t the -1- o p e r a t i o n and the S. r e l a t i o n of P because, as shown i n Chapter V(C), j u s t x , 2 p r e s e r v a t i o n i s s u f f i c i e n t to ensure that a l l operations and r e l a t i o n s among compatible and incompatible elements i n the u l t r a s u b s t r u c t u r e s o f P.,, are preserved, where an u l t r a s u b s t r u c t u r e QM n>3 i n a PQ^ i s a union of overlapping mBS's; thus the overlap patterns among mBS's i n the u l t r a s u b s t r u c t u r e s o f PQ^ 3 a r e preserved i f •J~ , < are preserved. Quantum mechanics i t s e l f i s the se r i o u s contender on the Pq^-preservation s i d e o f the dichotomy; HV proposals which preserve PQJ^ are impossible. The quantum Exp^, measures on a P^ do preserve the 5, = r e l a t i o n s and the ^ op e r a t i o n o f , and they do preserve the A, v operations among any compatible p a i r s o f elements i n P^ (even though Exp^ may not be b i v a l e n t w i t h respect to every element i n PQ M)J thus the Exp^ measures do preserve a n d do preserve the overlap 208 p a t t e r n s among the mBS's. And i n p a r t i c u l a r , w i t h respect to the domain US^ where each Exp^ i s b i v a l e n t , Exp^ preserves a l l the operations and r e l a t i o n s among a l l (compatible and incompatible) elements i n the overlapping mBS's i n US, , as shown i n Chapter V I ( B ) . t Tutsch gives an example o f how the quantum mechanical o r d e r i n g o f p r o p o s i t i o n s , i . e . , the 5 r e l a t i o n of and thus the = r e l a t i o n , i s not preserved i n the Bohm-Bub co n t e x t u a l HV theory. In t h i s theory, once the hidden s t a t e \|/, £ and the context o f measurement are s p e c i f i e d , the outcome of a measurement o f a magnitude A (which i s an eigenvalue o f A) i s determined by the s o - c a l l e d polychotomic a l g o r i t h m , h e r e a f t e r c a l l e d the HV alg o r i t h m . Tutsch's example shows how according to the HV a l g o r i t h m , f o r a given hidden s t a t e y, g» "the outcome o f a measurement o f the magnitude /S /, the absolute value of s p i n - 1 i n the z d i r e c t i o n , i s the z eigenvalue 0 , w h i l e the outcome o f a measurement of the magnitude S , z s p i n - 1 i n the z d i r e c t i o n , i s the eigenvalue - 1 . C l e a r l y , /S / i s a Z f u n c t i o n o f S ; each magnitude i s represented by an operator on z three-dimensional H i l b e r t space; and both magnitudes share the eigenstate y n a s s o c i a t e d w i t h t h e i r 0 eigenvalues and represented by the one-dimensional p r o j e c t o r P Q = I Y Q > < Y Q I « l"n both quantum mechanics and a co n t e x t u a l HV theory, the outcome 0 f o r a measurement o f /S / or f o r a z measurement o f S^ i s connected w i t h the assignment o f the value 1 to the element P n € which qua p r o j e c t o r represents the eigenstate YQ • That i s , i n quantum mechanics, f o r a given quantum s t a t e Y» "the outcome o f a measurement o f /S / i s the eigenvalue 0 ( i . e . , Exp (/S /) = 0) IFF z z Exp i(P.) = 1 ; and l i k e w i s e , the outcome of a measurement o f S i s the Y u z eigenvalue 0 ( i . e . , Exp^.(Sz) = 0) IFF Exp^,(P Q) = 1 . S i m i l a r l y , i n a 209 con t e x t u a l HV theory, f o r a given hidden s t a t e y, and f o r any context mBS, the outcome o f a measurement o f /S / i s the eigenvalue 0 IFF Z LL, >,(<Prt,mBS>) = 1: and l i k e w i s e , the outcome o f a measurement of S i s *Y>£ 0' z the eigenvalue 0 IFF |j, (<P ,mBS>) = 1. Furthermore, i n the quantum 3 p r o p o s i t i o n a l s t r u c t u r e P of three-dimensional H i l b e r t space, the element P^ , which qua p r o j e c t o r represents the eigenstate YQ , represents both o f the f o l l o w i n g p r o p o s i t i o n s : "The eigenvalue o f /S / z i s 0." "The eigenvalue of S i s 0." Since both p r o p o s i t i o n s are z 3 represented by the same element P^ = P^ i n P , each p r o p o s i t i o n i m p l i e s the other i n the sense t h a t P^ < P^ and P^ > P^ (where the 2 of i s i n t e r p r e t e d as l o g i c a l i m p l i c a t i o n ) . And s i n c e , f o r every quantum s t a t e w, Exp,(/S /) = 0 IFF Exp,(P ) = 1 IFF Exp,(S ) = 0, T Y z Y 0 Y Z each p r o p o s i t i o n i m p l i e s the other i n the sense t h a t , f o r any quantum s t a t e Y» i f the outcome o f a measurement o f /S / i s the eigenvalue 0, then z the outcome of a measurement o f S i s the eigenvalue 0, and z 8 conversely. But i n s p i t e of the f a c t t h a t quantum mechanically, each o f the above p r o p o s i t i o n s i m p l i e s the other ( i n both senses o f i m p l i e s ) , Tutsch gives an example of how i n a co n t e x t u a l HV theory, the p r o p o s i t i o n "The eigenvalue o f /S / i s 0." need not imply ( i n e i t h e r sense) the z p r o p o s i t i o n "The eigenvalue o f S i s 0." z According to the d e s c r i p t i o n of c o n t e x t u a l HV t h e o r i e s given i n Section ;A , such a d e v i a t i o n from quantum mechanics i s p o s s i b l e because PQ i s a member of ( a t l e a s t ) two over l a p p i n g mBS's s p e c i f y i n g p o s s i b l e contexts of measurements o f /S /, S , described as f o l l o w s . Besides the z z 0 eigenvalue-j the magnitude S has two other eigenvalues, 1, -1, each z ass o c i a t e d w i t h an eigenstate represented by a one-dimensional p r o j e c t o r 210 P. = l Y i ^ V i I a n d P -I = Iv i ^ V I j but the magnitude /S / has only J. J L J L "~" J. ~~ J. —• J- Z one other eigenvalue besides 0, namely, the eigenvalue 1 a s s o c i a t e d w i t h an eigenstate represented by a two-dimensional p r o j e c t o r , say _ i = P l v P _ i (i«e., the eigenvalue 1 o f /S^/ i s degenerate). Though P = P V P , i t i s e q u a l l y t r u e t h a t P = P V P f o r J L j J L JL JL 1 )~i 3. J J any orthogonal P ,P, which s a t i s f y P v P. = P. V P . . So the set a b a b 1 -1 {P n,P^,P_^} ( i . e . , the mBS^ generated by th a t set) may be the context o f a measurement o f /S / as w e l l as the set ( i . e . , the mBS Z U cl JD 3. generated by that s e t ) ; but only the set {FQ,F^,F_^} ( i . e . , mBS^) may be the context of a measurement o f S z . And c l e a r l y , mBS^ overlaps w i t h mBS s i n c e both share the element P„ . Now as described i n Section A , a 0 ' f o r a unique hidden s t a t e y»5» i t i s p o s s i b l e t h a t y, (<P ,mBS >) y ,5 0 a 4 u, „(<P„,mBS.>).L So given the connection between the outcome 0 f o r a y,g 0 1 measurement o f /S / or S and the assignment o f the value 1 to the z z element P. described above, t h i s p o s s i b i l i t y : u,, (<P„,mBS >) 0 Y>£ 0 a f- u. „(<P_.,mBS. >) means t h a t i t i s p o s s i b l e that i f /S / i s measured i n y,£ o i z the context mBS and S i s measured i n the context mBS„ , then f o r a a z 1 unique hidden s t a t e y,£, the outcome of the measurement of /S / i s the z eigenvalue 0 (which occurs IFF u, (<P„,mBS >) = 1 ) , w h i l e the outcome y,£ 0 a of the measurement of S i s one o f S 's other eigenvalues not equal to z z 0 (which occurs IFF (i (<P ,mBS >) ^ 1 ) . In h i s example, Tutsch gives a y s s " i hidden s t a t e which, according to the HV a l g o r i t h m , assigns values which exemplify t h i s p o s s i b i l i t y . In p a r t i c u l a r , h i s hidden s t a t e y i e l d s the outcome 0 f o r /S / but the outcome -1 f o r S . And although Tutsch z z does not e x p l i c i t l y s t a t e t h a t i n h i s example, /S / i s measured i n a z context d i f f e r e n t from the context i n which S i s measured, Tutsch does z ' 211 conclude t h a t h i s example could mean t h a t the two p r o p o s i t i o n s : "The eigenvalue o f /S / i s 0." and "The eigenvalue o f S i s 0." r e f e r to . z z " p r o p e r t i e s o f the system pl u s apparatus and hence, d i f f e r e n t apparatus may produce d i f f e r e n t r e s u l t s . " This c o n c l u s i o n suggests that i n h i s example, /S / i s measured i n a context d i f f e r e n t from the context i n which S i s z z measured (Tutsch, 1969, pp. 1118-1119). B e l i n f a n t e speaks o f Tutsch's example as an example o f a paradox 9 which i s derived from the HV al g o r i t h m and which i s r e l a t e d to but i n f a c t worse than the Kochen-Specker t r o u b l e s (which motivate the c o n t e x t u a l HV proposals) i n th a t such paradoxes are "much l e s s ( i f a t a l l ) j u s t i f i a b l e as a ' r e s u l t o f the i n f l u e n c e of the measuring arangement'" ( B e l i n f a n t e , 1973, p. 135). Assuming that the above a n a l y s i s of Tutsch's example i s c o r r e c t , the example i s not an example o f a paradox. For /S / and S are z z measured i n d i f f e r e n t contexts ( i n v o l v i n g d i f f e r e n t experimental arrange-ments), and as Bub makes c l e a r , we must expect the d i s p e r s i o n - f r e e measures induced by the hidden s t a t e s of a c o n t e x t u a l HV theory to a s s i g n d i f f e r e n t values even to the same magnitude when measured i n d i f f e r e n t contexts. Gudder, who was i n contact w i t h Tutsch at the time o f the p u b l i c a t i o n o f each o f t h e i r papers, l i k e w i s e understands Tutsch's example as i n v o l v i n g measurements i n d i f f e r e n t contexts (Gudder, 1970, p. 436). Moreover, the "paradoxes" e x e m p l i f i e d by Tutsch's example are r e l a t e d to the Kochen-Specker t r o u b l e s only i n the sense that such "paradoxes" are feat u r e s o f a cont e x t u a l HV theory which are necessary i n order to avoid the Kochen-Specker HV i m p o s s i b i l i t y proof. For i f the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a contextual HV theory d i d not a s s i g n d i f f e r e n t values to the same magnitude when measured i n d i f f e r e n t contexts and ins t e a d 212 assigned unique values to an element even though P Q i s a member of overlapping mBS's, then such measures would be r u l e d out by the Kochen-Specker proof. Gudder suggests that the s o r t of c o n t e x t u a l HV d e v i a t i o n s from quantum mechanics which are e x e m p l i f i e d by Tutsch's example may be candidates f o r experimental v e r i f i c a t i o n or f a l s i f i c a t i o n . In a d d i t i o n to Tutsch's s o r t o f d e v i a t i o n s , there i s a l s o the f o l l o w i n g s o r t of c o n t e x t u a l HV d e v i a t i o n s from quantum mechanics which has been the subject of experimental t e s t . In a c o n t e x t u a l HV theory, a pure quantum s t a t e ty ^ s t r e a t e d as a mixed s t a t e w i t h respect to the p o s s i b l e hidden s t a t e s (each represented by Y together w i t h some g ) , and the mixed s t a t e ty describes an ensemble of hidden s t a t e s w i t h a s o - c a l l e d e q u i l i b r i u m d i s t r i b u t i o n o f hidden v a r i a b l e s . In order t h a t the s t a t i s t i c a l c o n d i t i o n , mentioned i n the Preface to t h i s chapter, be s a t i s f i e d , t h i s e q u i l i b r i u m d i s t r i b u t i o n of hidden v a r i a b l e s together w i t h ty must reproduce, v i a the HV a l g o r i t h m , the s t a t i s t i c a l r e s u l t s of quantum mechanics given by the Exp^, measures ( B e l i n f a n t e , 1973, p. 136); or i n other words, the s t a t i s t i c a l r e s u l t s of quantum mechanics are derived from the HV a l g o r i t h m by assuming t h a t the hidden v a r i a b l e s are i n , an e q u i l i b r i u m d i s t r i b u t i o n . For example, as described by B e l i n f a n t e , consider a l a r g e number o f quantum systems whose quantum s t a t e i s ty on which we perform measurements of a magnitude A and, whenever the outcome i s a p a r t i c u l a r eigenvalue a.. , we f o l l o w up by measuring a d i f f e r e n t magnitude B. In quantum mechanics, the average value of B i s determined not by Exp,(B) but r a t h e r by Exp , ( B ) , where ty. i 3 i s the eigenstate of A a s s o c i a t e d w i t h the eigenvalue a.. ; t h a t i s , the f i r s t measurement of A i s assumed to have reduced the i n i t i a l s t a t e t 213 Y to the eigenstate • °^ A - In a co n t e x t u a l HV theory, i n order t h a t the HV al g o r i t h m reproduce t h i s quantum mechanical r e s u l t Exp, ( B ) , besides the r e d u c t i o n o f Y t° Yj > ^ m u s t a l s o be assumed t h a t the hidden v a r i a b l e s , which together w i t h y_. describe!, the hidden s t a t e s of the ensemble o f quantum systems a f t e r the measurement o f A, are i n an e q u i l i b r i u m d i s t r i b u t i o n before the measurement o f B occurs ( B e l i n f a n t e , 1973, pp. 139-140). However, a f t e r the measurement o f A, the HV al g o r i t h m c a l c u l a t i o n s y i e l d a non-equilibrium or biased d i s t r i b u t i o n o f hidden-v a r i a b l e s . I t i s assumed th a t such biased d i s t r i b u t i o n s of hid d e n - v a r i a b l e s very r a p i d l y r e l a x to the e q u i l i b r i u m d i s t r i b u t i o n which reproduces the Exp (B) r e s u l t . But i f the measurement o f B i s performed before the Yj biased d i s t r i b u t i o n r e s u l t i n g from the measurement o f A has r e l a x e d t o the e q u i l i b r i u m d i s t r i b u t i o n , then the biased d i s t r i b u t i o n p r e d i c t s v i a the HV algor i t h m s t a t i s t i c a l r e s u l t s f o r B which d i f f e r from what quantum mechanics p r e d i c t s v i a i t s Exp (B) formalism ( B e l i n f a n t e , 1973, p. 163). Yj These s o r t s o f d e v i a t i o n s from quantum mechanics which are connected w i t h non-equilibrium d i s t r i b u t i o n s o f hidden v a r i a b l e s a f t e r measurement are d i f f e r e n t from the Tutsch s o r t o f d e v i a t i o n s which are connected w i t h d i f f e r e n t contexts o f measurement. For as described by Bohm-Bub (1966, p. 466), the non-equilibrium s o r t o f d e v i a t i o n s occur f o r measurements o f magnitudes represented by operators on two-dimensional H i l b e r t space. But c l e a r l y , the c o n t e x t u a l s o r t of d e v i a t i o n s can occur f o r measurements only o f magnitudes represented by operators on a three-or-higher dimensional H i l b e r t space s i n c e only P Q ^ s t r u c t u r e s have overlapping mBS's. The existence o f these d e v i a t i o n s make i t a t l e a s t i n p r i n c i p l e p o s s i b l e to experimentally v e r i f y o r f a l s i f y the p r e d i c t i o n s o f the HV 214 a l g o r i t h m and thus to decide between quantum mechanics and the proposed c o n t e x t u a l , Boolean HV r e c o n s t r u c t i o n s of quantum mechanics. Experiments t e s t i n g f o r the non-equilibrium d i s t r i b u t i o n s o r t of d e v i a t i o n s w i t h a time -13 i n t e r v a l of l e s s than 10 seconds between the measurements of d i f f e r e n t magnitudes ( l i k e the measurements o f A and B described above) have so f a r found no d e v i a t i o n s from the p r e d i c t i o n s of quantum mechanics and have thus f a l s i f i e d the p r e d i c t i o n s of the HV a l g o r i t h m . However, HV advocates may argue t h a t the time i t takes a b i a s e d d i s t r i b u t i o n of hidden v a r i a b l e s -13 to r e l a x to the e q u i l i b r i u m d i s t r i b u t i o n i s l e s s than the 10 second i n t e r v a l between the measurements o f the experiments mentioned above. For as described by B e l i n f a n t e , " i t has not yet been e s t a b l i s h e d how f a s t one may t h e o r e t i c a l l y expect biased h i d d e n - v a r i a b l e d i s t r i b u t i o n s to r e l a x . . . .' So even i f the HV a l g o r i t h m i s f a l s i f i e d at an even s h o r t e r time i n t e r v a l i n some f u t u r e experiment, HV advocates may nevertheless continue to argue t h a t the s h o r t e r time i n t e r v a l i s not yet short enough to capture the b i a s e d d i s t r i b u t i o n of h i d d e n - v a r i a b l e s before i t r e l a x e s to the e q u i l i b r i u m d i s t r i b u t i o n which reproduces the quantum mechanical p r e d i c t i o n s . Thus w h i l e experiments have so f a r f a l s i f i e d and may continue to f a l s i f y the HV a l g o r i t h m , i t may be that no experiment w i l l ever c o n c l u s i v e l y decide between quantum mechanics and the proposed c o n t e x t u a l HV t h e o r i e s ( B e l i n f a n t e , 1973, pp. 88, 100). Nevertheless, quantum mechanics i s so f a r supported by.experimental evidence. And as pointed out by B e l i n f a n t e , the formalism of quantum mechanics i s simpler than the formalism of the c o n t e x t u a l HV t h e o r i e s . So by the usual c r i t e r i a o f experimental evidence and formal s i m p l i c i t y , quantum mechanics i s a b e t t e r theory of quantum phenomena than i s a c o n t e x t u a l HV 215 theory. So why i s quantum mechanics s t i l l challenged by the c o n t e x t u a l HV proposals? Four reasons are d e s c r i b e d , to the end o f t h i s s e c t i o n . 1. One reason quantum mechanics i s v u l n e r a b l e to a con t e x t u a l HV proposal i s because even i f i t i s granted t h a t the c l a s s i c a l n o t i o n o f a p r o b a b i l i t y measure defined on a Boolean s t r u c t u r e may be g e n e r a l i z e d so as to be defined on the non-Boolean quantum P N ^ s t r u c t u r e s , the notion o f a ge n e r a l i z e d measure on P N ^ i s open w i t h regard to the i s s u e o f which operations and r e l a t i o n s o f P A ^ ought to be r e q u i r e d to be preserved by the g e n e r a l i z e d measures. As described i n Chapter IV(A), Bub and Jauch-Piron each d e f i n e two d i f f e r e n t s o r t s o f g e n e r a l i z e d measures on P 0 ^ . The cont e x t u a l HV measures can be regarded as a t h i r d s o r t o f g e n e r a l i z e d p r o b a b i l i t y measure on P N ^ (even though the domain o f a c o n t e x t u a l HV measure i s P_„ x {mBS.}.- _ , ). A l l three s o r t s 'of g e n e r a l i z e d measures QM 1 i J i € I n d e x 6 preserve the Boolean s t r u c t u r a l f e a t u r e s of the (maximal) Boolean substructures o f P A M . In a d d i t i o n , Bub, Gleason, Kochen-Specker, and Jauch-Piron r e q u i r e t h a t a g e n e r a l i z e d p r o b a b i l i t y measure s a t i s f y Gleason's a d d i t i v i t y c o n d i t i o n (Ga) which ensures t h a t d i s p e r s i o n - f r e e g e n e r a l i z e d p r o b a b i l i t y measures preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P N ^ , i n p a r t i c u l a r , preserve the overlap patterns among the mBS's of P Q ^ -Jauch-Piron f u r t h e r r e q u i r e t h e i r g e n e r a l i z e d measures t o s a t i s f y (JP&). An argument ag a i n s t the i n c l u s i o n of (JP^) as pa r t o f the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure i s given i n the note below. X^ Here we consider whether or not (Ga), which e n t a i l s P n ^ A ~ p r e s e r v a t i o n , ought to be included. The d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a c o n t e x t u a l HV theory do not and cannot s a t i s f y (Ga) because together w i t h 216 the d i s p e r s i o n - f r e e c o n d i t i o n , (Ga) y i e l d s P ^ ^ - p r e s e r v a t i o n and (Ga) a l s o y i e l d s the c o n d i t i o n l a b e l e d (B) by B e l l , n e i t h e r o f which i s s a t i s f i e d by co n t e x t u a l HV d i s p e r s i o n - f r e e measures. However, the i n c l u s i o n of at l e a s t Gleason's a d d i t i v i t y c o n d i t i o n as par t of the c o n d i t i o n s d e f i n i n g a gen e r a l i z e d p r o b a b i l i t y measure on a PQ^ i s s t r o n g l y supported by the precedent t h a t i n c l a s s i c a l p r o b a b i l i t y theory, c o n d i t i o n (ua) i s inc l u d e d among the c o n d i t i o n s d e f i n i n g a c l a s s i c a l p r o b a b i l i t y measure on a Boolean s t r u c t u r e , as s t a t e d i n Chapter I I I ( C ) . (see f o r example, Kolmogorov, 1933, p. 2). Since elements i n a P^^ are d i s j o i n t IFF they are orthogonal, or i n other words, o r t h o g o n a l i t y i s the quantum analogue o f d i s j o i n t e d n e s s , c o n d i t i o n (Ga), which r e q u i r e s t h a t a g e n e r a l i z e d p r o b a b i l i t y measure on a PQ^ be a d d i t i v e w i t h respect to orthogonal elements o f P^ M , i s the quantum analogue o f c o n d i t i o n ( p a ) , which r e q u i r e s t h a t a c l a s s i c a l p r o b a b i l i t y measure on a Boolean s t r u c t u r e be a d d i t i v e w i t h respect to d i s j o i n t elements. Or i n other words, (Ga) i s simply the c o n d i t i o n (p,a) as a p p l i e d to the quantum P^ M s t r u c t u r e s . So i t i s arguable that because (pa) i s one of the c o n d i t i o n s d e f i n i n g a c l a s s i c a l p r o b a b i l i t y measure, (Ga) ought to be one o f the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure. Moreover, as elaborated at the end of S e c t i o n A , the c o n d i t i o n of p Q ^ - p r e s e r v a t i o n which f o l l o w s from (Ga) i s independently supported by the uncontentious general p r i n c i p l e according to which the r e a l value of say B o r e l f u n c t i o n o f any magnitude i n any p h y s i c a l theory i s c a l c u l a t e d or determined by simply applying t h a t B o r e l f u n c t i o n to the r e a l value o f the magnitude. Since any magnitude i s compatible w i t h any B o r e l f u n c t i o n o f i t s e l f , the general p r i n c i p l e r e f e r s to the p r e s e r v a t i o n o f f u n c t i o n a l r e l a t i o n s among compatible magnitudes ( o r p r o p o s i t i o n s ) . So i n a c o n t e x t u a l 217 HV theory, w h i l e the f u n c t i o n a l r e l a t i o n s among compatible elements i n any mBS o f P ^ are preserved, the f u n c t i o n a l r e l a t i o n s among compatible elements i n overlapping mBS's o f P^ M are not preserved s i n c e the P Q M A s t r u c t u r e i s not preserved and thus the general p r i n c i p l e which e n t a i l s p Q ^ - p r e s e r v a t i o n i s not s a t i s f i e d i n a c o n t e x t u a l HV theory. For example, as suggested by Gudder, i f one considered two d i f f e r e n t mBS's c o n t a i n i n g P and g(P) r e s p e c t i v e l y , then one would get independence of the re p r e s e n t i n g f u n c t i o n s f„ , f , r a t h e r than the f u n c t i o n a l r e l a t i o n f / T, N = g ( f _ ) . P g(P) g(P) P And the excuse given by c o n t e x t u a l HV advocates f o r t h i s v i o l a t i o n of the general p r i n c i p l e i s that the c o n s i d e r a t i o n of two d i f f e r e n t mBS's i n v o l v e s two separate measurements w i t h d i f f e r e n t experimental arrangements, and so i n such cases one would expect to o b t a i n independent r e s u l t s f o r P and g(P) (Gudder, 1970, p. 4-35). This excuse ignores o r makes l i g h t o f the f a c t t h a t , as determined by quantum mechanics and as (so f a r ) experimentally observed, the r e s u l t s o f any measurements o f P, g ( P ) , are not independent but r a t h e r are f u n c t i o n a l l y r e l a t e d i n accordance w i t h the general p r i n c i p l e . 2. As suggested again i n the previous paragraph, quantum theory i s v u l n e r a b l e to the co n t e x t u a l HV proposals i f measurement i n t e r a c t i o n o r measurement disturbance i s regarded as the cause or b a s i s o f the non-c l a s s i c a l p e c u l i a r i t i e s o f quantum mechanics and as ( a t l e a s t p a r t o f) the reason why the von Neumann, Jauch-Piron, and Kochen-Specker type of HV proposals are impossible. For example, according to Heisenberg's v e r s i o n of the Copenhagen i n t e r p r e t a t i o n of quantum mechanics, one reason why quantum ensembles cannot be r e s o l v e d i n t o subensembles which are d i s p e r s i o n - f r e e (as re q u i r e d i n von Neumann's HV proposal) i s because quantum systems are dis t u r b e d by measurement. And f o r an example o f how measurement c o n s i d e r a t i o n s 218 support c o n t e x t u a l HV proposals, we have of course B e l l ' s argument, from the pe r s p e c t i v e o f Bohr's v e r s i o n of the Copenhagen i n t e r p r e t a t i o n , t h a t s t r u c t u r a l c o n d i t i o n s l i k e P n M A ~ p r e s e r v a t i o n which r e f e r even i n d i r e c t l y to measurements of incompatible magnitudes must not be imposed upon the proposed d i s p e r s i o n - f r e e measures of an HV theory because of the i n e x t r i c a b l e wholeness o f quantum phenomena and measuring devices. Now the outcome of a measurement at best determines an assignment of 0, 1 values to a maximal Boolean subst r u c t u r e o f elements i n a P A ^ . For a measurement o f any magnitude A can at best be a measurement of what i s c a l l e d a complete s e t of commuting magnitudes i n c l u d i n g A (and i n c l u d i n g j u s t A i f none of A's eigenvalues are degenerate) whose eigens t a t e s ^ i ^ i € I n d e x ' a s r e P r e s e i r t e d by (n) orthogonal atoms ^ i ^ ^ I n d e x i n a P Q ^ s t r u c t u r e which generate a unique maximal Boolean subst r u c t u r e mBS^ of P Q ^ » s p e c i f y the context o f the measurement o f A. And the outcome o f the measurement, which i s an eigenvalue a^ of A a s s o c i a t e d w i t h an eigenstate \|/\ i n the set ^ i ^ i ' g i n d e x ' determines v i a E x p ^ i n quantum mechanics and v i a u. i n a c o n t e x t u a l HV theory, an assignment o f 0, 1 values to the elements i n th a t mBS. . The c o n t e x t u a l HV measure does no A more without changing i t s 0, 1 value assignments to the members o f mBS . However, without changing i t s value assignments to the members of mBSA , the quantum measure Exp : US ->• {0,1} i n a d d i t i o n assigns 0, 1 values ^ j * j , to every element i n the u l t r a s u b s t r u c t u r e US^ = {P € P Q M : P o P.} 2 m B S A > where (unless US happens to equal mBS.) US i s a union o f ¥j A Yj overlapping mBS's i n c l u d i n g mBS . A These a d d i t i o n a l 0, 1 value assignments by the quantum measure Expr" mean the f o l l o w i n g . Let B be any magnitude which shares the 219 eigenstate t y w i t h A even though B k> A ( i . e . , B and A do not share a l l t h e i r e i g e n s t a t e s ) . E i t h e r alone ( i f none o f B's eigenvalues are degenerate) o r as p a r t o f a complete set o f commuting magnitudes, B s p e c i f i e s a unique maximal Boolean substructure mBS^ of P n^ which c l e a r l y overlaps w i t h mBSA s i n c e the atom P , which qua p r o j e c t o r represents the eigenstate ty. , i s a member of both mBS and mBS . 3 A And s i n c e every element i n mBS^ i s compatible w i t h P^ , c l e a r l y mBSg c US, . Thus Exp, assigns 0, 1 values to every element i n mBSg . And these value assignments by Exp mean tha t i f B i s measured a f t e r Yj A i s measured, or i f B, in s t e a d o f A, had been measured w i t h the outcome b^ , then the outcome o f the measurement o f B, namely, the eigenvalue b_. a s s o c i a t e d w i t h the eigenstate , determines an assignment o f 0, 1 values to the elements i n mBS such that the values assigned to the common elements i n mBS fl mBS match the value assignments A 15 determined by the outcome o f the ( f i r s t ) measurement of A. Thus the 0, 1 value assignments by the quantum measure Exp to every element i n both mBS. c US, and mBS,, c US, are determined by J A ty. B ty. J the outcome o f one measurement y e t r e f e r to the outcomes o f more than one measurement. For A and B cannot be measured simultaneously, i . e . , A & B. (And i f A i B, then mBS^ = mBSg i n P^ .) In other words, the f a c t t h a t a quantum Exp^ measure assigns 0, 1 values to overlapping mBS's of elements i n a manner which preserves the overlap p a t t e r n s says something about d i f f e r e n t measurements of incompatible magnitudes. S i m i l a r l y , i f proposed d i s p e r s i o n - f r e e HV measures are r e q u i r e d to a s s i g n 0, 1 values to o v erlapping mBS's of elements i n a manner which preserves the overlap p a t t e r n s , then t h i s requirement does r e f e r to d i f f e r e n t measurements of 2 2 0 incompatible magnitudes, as the c o n t e x t u a l HV advocates argue. For example, 3 the 1 9 2 atoms contained i n 1 1 8 overlapping mBS's i n the P Q ^ a considered by Kochen-Specker i n the Theorem 1 p a r t o f t h e i r HV i m p o s s i b i l i t y proof represent the eigenst a t e s of 1 1 8 incompatible magnitudes which cannot a l l be measured together, y e t Kochen-Specker r e q u i r e proposed d i s p e r s i o n - f r e e HV measures to preserve the overlap p a t t e r n s among the eigenstates o f these magnitudes. This requirement, which i s p a r t o f the P ^ ^ - p r e s e r v a t i o n c o n d i t i o n , i s very hard t o motivate i f measurement i n t e r a c t i o n , as des c r i b e d by Bohr and B e l l , o r measurement d i s t u r b a n c e , as described by Heisenberg w i t h h i s Uncertainty P r i n c i p l e , are t r e a t e d as c e n t r a l i n the i n t e r p r e t a t i o n o f quantum mechanics, as the cause o f the n o n - c l a s s i c a l p e c u l i a r i t i e s o f quantum mechanics, and as the reason why hidden-v a r i a b l e s are e i t h e r impossible or e l s e dependent upon the context o f measurement. In c o n t r a s t , i f the non-Boolean P_„. s t r u c t u r e a b s t r a c t e d from QMA the fundamental p o s t u l a t e s o f quantum mechanics i s t r e a t e d as c e n t r a l i n the i n t e r p r e t a t i o n o f quantum mechanics, then the n o n - c l a s s i c a l p e c u l i a r i t i e s of quantum mechanics are regarded as due to the non-Boolean character o f the P „ „ . s t r u c t u r e r a t h e r than due to measurement i n t e r a c t i o n o r QMA d i s t u r b a n c e . X x And as Kochen-Specker and Bub make c l e a r , c o n s i d e r a t i o n o f measurement i n t e r a c t i o n or disturbance are beside the p o i n t i f the problem of hidd e n - v a r i a b l e s i s c o r r e c t l y understood as posing the question o f whether the s t a t i s t i c a l r e s u l t s o f <H,P„„ B,Exp,> can be r e c o n s t r u c t e d i n QMA terms o f a c l a s s i c a l measure space <®,P „,u> i n a manner which preserves HV the core s t r u c t u r e of quantum mechanics. For example, i n s p i t e o f Heisenberg's U n c e r t a i n t y P r i n c i p l e , the s t a t i s t i c a l r e s u l t s o f 2 2 <W , P^ A,Exp^> can be c l a s s i c a l l y r e c o n s t r u c t e d , as Kochen-Specker 221 demonstrate by producing an HV theory f o r t h a t p a r t of quantum mechanics which i n v o l v e s j u s t two-dimensional H i l b e r t space (Kochen-Specker, 1967, pp. 75-80, 86). 3. Metaphysical p r e j u d i c e s , l i k e the " r e l i g i o u s b e l i e f t h a t 'nature must be d e t e r m i n i s t i c ' . . ." mentioned by B e l i n f a n t e (1973, p. 18) make quantum mechanics e s p e c i a l l y v u l n e r a b l e to the c o n t e x t u a l HV proposals. As described by Bub, the main reason why quantum mechanics i s v u l n e r a b l e to c o n t e x t u a l HV proposals i s because o f the p r e s u p p o s i t i o n t h a t the l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of r e a l i t y and of any p h y s i c a l theory about any p o r t i o n of r e a l i t y is_ and can only be a Boolean s t r u c t u r e . Bub argues that behind the a f f i r m a t i o n of the three primed c o n d i t i o n s (i')» ( i i ' ) , ( i i i ' ) d e s c r i b i n g the Boolean r e c o n s t r u c t i o n s i d e o f the dichotomy i n the i n t e r p r e t a t i o n o f quantum mechanics i s the (metaphysical) p r e s u p p o s i t i o n t h a t the l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of quantum phenomena must be a Boolean s t r u c t u r e , l i k e the Boolean l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of c l a s s i c a l phenomena and c l a s s i c a l mechanics. In c o n t r a s t , behind the a f f i r m a t i o n of the three un-primed c o n d i t i o n s ( i ) , ( i i ) , ( i i i ) d e s c r i b i n g the P ^ ^ - p r e s e r v a t i o n s i d e o f the dichotomy, there i s an open acceptance of the n o t i o n of a non-Boolean l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of quantum phenomena and quantum mechanics (Bub, 1973, p. 54; 19