UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1980_A1 C54.pdf [ 12.98MB ]
Metadata
JSON: 831-1.0095142.json
JSON-LD: 831-1.0095142-ld.json
RDF/XML (Pretty): 831-1.0095142-rdf.xml
RDF/JSON: 831-1.0095142-rdf.json
Turtle: 831-1.0095142-turtle.txt
N-Triples: 831-1.0095142-rdf-ntriples.txt
Original Record: 831-1.0095142-source.json
Full Text
831-1.0095142-fulltext.txt
Citation
831-1.0095142.ris

Full Text

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 o f 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 in THE FACULTY OF GRADUATE STUDIES (Department o f P h i l o s o p h y )  We a c c e p t t h i s t h e s i s as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA • - J u l y 1979 • 1  @  A r i a d n a Chernavska  In p r e s e n t i n g t h i s t h e s i s  i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r  an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written  permission.  Department o f  PHILOSOPHY  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Abstract The t h e s i s i n v e s t i g a t e s t h e p o s s i b i l i t y o f a c l a s s i c a l 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  semantics i s defined  as a s e t o f mappings each o f w h i c h i s ( i ) b i v a l e n t , i . e . , t h e v a l u e 1 ( t r u e ) o r <p ( f a l s e ) i s a s s i g n e d 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.  In addition, t h i s set  must", be " f u l l " . , i . e . , a n y ' p a i r o f d i s t i n c t p r o p o s i t i o n s i s a s s i g n e d d i f f e r e n t v a l u e s by some mapping i n t h e s e t . When t h e p r o p o s i t i o n s make a s s e r t i o n s about t h e p r o p e r t i e s o f c l a s s i c a l o r o f quantum systems, t h e mappings s h o u l d a l s o be ( i i i . ) " s t a t e - i n d u c e d " , i . e . , v a l u e s a s s i g n e d by t h e semant i c s s h o u l d a c c o r d w i t h v a l u e s a s s i g n e d by. c l a s s i c a l o r by quantum mechanics. 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 , sitions  ( e q u i v a l e n c e c l a s s e s . o f ) propo-  form a B o o l e a n a l g e b r a , and each semantic;(mapping a s s i g n s t h e  v a l u e 1 t o t h e members o f a c e r t a i n subset o f t h 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 t h e members o f t h e d u a l u l t r a i d e a l ,  where  t h e u n i o n o f t h e s e two subsets i s t h e e n t i r e a l g e b r a .  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 a r e 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 p r o v i d e a c l a s s i c a l s e m a n t i c s , w h i c h a l s o s a t i s f i e s (iii).  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 a r e non-Boolean, so  i t i s an open q u e s t i o n whether a semantics  s a t i s f y i n g ( i ) , ( i i ) and  ( i i i ) can be p r o v i d e d . Von Neumann f i r s t proposed (.1932) t h a t t h e a l g e b r a i c s t r u c t u r e s o f t h e subspaces ( o r p r o j e c t o r s ) o f H i l b e r t space be r e g a r d e d as t h e p r o p o s i t i o n a l s t r u c t u r e s P ^ o f quantum mechanics. been f o r m a l i z e d i n two ways: as orthomodular  These s t r u c t u r e s have  l a t t i c e s w h i c h have t h e  b i n a r y o p e r a t i o n s "and", " o r " , d e f i n e d among a l l e l e m e n t s , i  and i n c o m p a t i b l e j£  compatible  ; and as p a r t i a l - B o o l e a n a l g e b r a s w h i c h have t h e ii  "binary o p e r a t i o n s d e f i n e d among o n l y c o m p a t i b l e  elements.  I n the  two b a s i c senses i n w h i c h t h e s e s t r u c t u r e s are non-Boolean are minated.  ) a p p l i c a b l e t o the P  a p p l i c a b l e t o b o t h the  b r a s . Then i t i s shown how  n  l a t t i c e s ; and t r u t h - f u n c t i o -  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 -  t h e l a t t i c e d e f i n i t i o n s o f "and", " o r " , among  i n c o m p a t i b l e s 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 ) f o r any  discri-  And two n o 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 are d i s t i n g u i s h e d : t r u t h -  functionality ( n a l i t y (A)  thesis,  l a t t i c e c o n t a i n i n g i n c o m p a t i b l e , elements.  semantics  In c o n t r a s t , the  Gleason and Kochen-Specker p r o o f s o f 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 f o r quantum mechanics show 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 , t r u t h - f u n c t i o n a l (A)  semantics f o r t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t space  s t r u c t u r e s ; and the presence o f i n c o m p a t i b l e elements i s n e c e s s a r y i s not s u f f i c i e n t t o r u l e out such a  but  semantics.  As f o r ( i i i ) , each quantum s t a t e - i n d u c e d 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 a s s i g n s 1 and'iS' v a l u e s t o t h e elements i n a u l t r a f i l t e r and d u a l u l t r a i d e a l o f P_„.  where, i n g e n e r a l t h e u n i o n  of  an u l t r a f i l t e r and i t s d u a l u l t r a i d e a l i s s m a l l e r t h a n . t h e e n t i r e s t r u c ture.  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 t h e quantum  a n a l o g o f 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 s u b s t r u c t u r e The  of  f i n a l p o r t i o n o f the t h e s i s surveys p r o p o s a l s  f o r the  intro-  d u c t i o n o f h i d d e n v a r i a b l e s i n t o quantum mechanics, p r o o f s o f t h e  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 o f  these  i m p o s s i b i l i t y proofs.  And  arguments i n f a v o u r o f t h e 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 t h a n t h e 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  ." i i i -  reviewed.  TABLE OF CONTENTS  Chapter 0  Introduction  1  Chapter I  Algebraic Preliminaries  8  Chapter I I  A. B. C. D. E. F.  Group and R i n g S t r u c t u r e s The Boolean A l g e b r a and t h e B o o l e a n L a t t i c e Subsets o f a B o o l e a n S t r u c t u r e The Quantum P a r t i a l - B o o l e a n A l g e b r a The Quantum Orthomodular L a t t i c e . . Subsets o f and p QMA QML  G.  Mappings on a S t r u c t u r e  '.'  30  The C l a s s i c a l P r e c e d e n t 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 A. B. C. D.  Chapter I I I  8 10 18 20 24 28  ....  33  The S t a n d a r d 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 The B o o l e a n 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 . . . . . B i v a l e n t Homomorphic Mappings on Any B o o l e a n Structure The A l g e b r a i c Semantics f o r t h e Lindenbaum A l g e b r a .  The C l a s s i c a l P r e c e d e n t f o r a S t a t e - i n d u c e d  Semantics  .  34 36 39 43  Preface A. The S t a t e s o f a C l a s s i c a l System Determine t h e R e a l V a l u e s o f That System's Magnitudes . . . . . . 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 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 , S t a t e - i n d u c e d Semantics f o r t h e B o o l e a n P_ Structures . . . .-' CM 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  43  A. B. C.  57 59  w  Chapter IV  33  D. E. F.  The Fundamental P o s t u l a t e s o f Quantum Mechanics Incompatibility 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 The P a r t i a l - B o o l e a n A l g e b r a and t h e 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 R a m i f i c a t i o n s o f t h e B a s i c D i f f e r e n c e between P.M. and P_MT QMA QML The Two B a s i c Senses i n Which t h 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 A r e Non-Boolean  iv  . .  44 46 48 57  61 66 72 77  Chapter V  The Impossibility of a Bivalent, Truth-functional Semantics f o r the Non-Boolean Propositional Structures Determined by Quantum Mechanics .......  90  Preface 90 A. The Impossibility of a Bivalent, Truth-functional(6,&) Semantics f o r Any P Which Contains Incompatible Elements 91 B. The Kochen-Specker Proof o f the Impossibility o f a Bivalent Homomorphism(o) on a Three Dimensional Hilbert Space P 97 QMA C. Avoiding These Impossibility Proofs 101 D. The Meaning o f the Hidden-Variable Impossibility Proofs f o r the Issue of a C l a s s i c a l Semantics for the Quantum Propositional Structures 103 Summary 109 niuIA  c  Chapter VI  A State-induced Semantics f o r the Non-Boolean Propositional Structures Determined by Quantum Mechanics A. B.  The Quantum State-induced Expectation Functions . The Quantum Expectation Function As an Ultravaluation on an Ultrasubstructure o f P . C. An Example . . . D. A State-induced Semantics f o r the ?.., Structures Summary  Chapter VII  114. 114 .121 . 138 . 147 154  Hidden Variables Reconsidered  162  Preface A. Criticisms of the Hidden-Variable Impossibility Proofs B. Either P.,,. Preservation or Boolean Reconstruction QMA —  162  v  167 199  1  CHAPTER 0  INTRODUCTION  I n 1932, von Neumann p u b l i s h e d t h e f i r s t p r o o f s o f t h e completeness o f quantum mechanics and t h e 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 h i d d e n v a r i a b l e s i n t o quantum t h e o r y . and  T h i r t y y e a r s l a t e r i n 1963, J a u c h  P i r o n published another proof o f t h e 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 t h e y r e g a r d e d as a s t r e n g t h e n i n g result.  o f von Neumann's i m p o s s i b i l i t y  A l t h o u g h von Neumann's p r o o f s were l a t e r c h a l l e n g e d , t h e  completeness o f quantum mechanics and t h e 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 were p r o v e n anew by G l e a s o n i n 1957. And t e n y e a r s l a t e r , Kochen and Specker p u b l i s h e d 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 The p r o o f s by J a u c h - P i r o n  proof.  and Kochen-Specker a r e e s p e c i a l l y  i n t e r e s t i n g because t h e y connect t h e e n t e r p r i s e o f i n t r o d u c i n g h i d d e n v a r i a b l e s i n t o quantum t h e o r y w i t h t h e e n t e r p r i s e o f a s s i g n i n g  0, 1  v a l u e s t o t h e a l g e b r a i c s t r u c t u r e s o f t h e 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 r e g a r d e d as t h e  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 p r o p o s a l s ,  i n 1932 and 1936, t o c o n s i d e r t h e subspaces ( o r  p r o j e c t o r s ) as t h e m a t h e m a t i c a l 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 t o c o n s i d e r t h e o p e r a t i o n s  and r e l a t i o n s among t h e subspaces a s t h e  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 proofs o f the i m p o s s i b i l i t y o f assigning  0, 1  and r e l a t i o n s .  So t h e s e  values t o the algebraic  s t r u c t u r e s o f quantum p r o p o s i t i o n s i n a manner w h i c h p r e s e r v e s t h e l o g i c a l operations  and r e l a t i o n s among t h e p r o p o s i t i o n s a r e p r o o f s — o r  at least  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 o f a c l a s s i c a l , that  b i v a l e n t and t r u t h - f u n c t i o n a l , semantics f o r t h e quantum In Chapter I , the a l g e b r a i c t h e s i s are presented.  n o t i o n s employed  is,a  propositions.  throughout t h i s L  In Chapter I I , the Boolean Lindenbaum a l g e b r a  o f a s e t o f well-formed formulae 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 introduced, L  an We  and t h e n o t i o n  i s defined  o f 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  as a complete c o l l e c t i o n o f a l g e b r a i c  s h a l l see i n Chapter V how  algebraic how  structures  valuations  such a semantics f a i l s t o work f o r t h e  o f quantum p r o p o s i t i o n s .  And we  s h a l l see i n Chapter VI  such a semantics does work f o r c e r t a i n s u b s t r u c t u r e s  propositional  o f t h e quantum  structures. P  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 , l a b e l e d formalized  i n two ways: A  and i n c o m p a t i b l e A, V  defined  V  (and), &  (or)  defined  ; and as p a r t i a l - B o o l e a n  among o n l y  P„„ QML  P«„. QMA  and t h e  T  A l s o , two n o 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 a r e truth-functionality (i)  appropriate  (<i,jfe)  to a  appropriate *  p r o p o s i t i o n a l s t r u c t u r e s may  to a  And two b a s i c  ^Q^A  w  n  ^  c  have  n  In Chapter IV, some formalizations  are  described.  distinguished: P  and t r u t h - f u n c t i o n a l i t y  senses i n which t h e quantum  be s a i d t o be non-Boolean  Then i n Chapter V, i t i s shown how  have been  T  algebras  compatible elements.  d i f f e r e n c e s between the  ,  P.,, which have t h e QML among a l l elements, c o m p a t i b l e i  as orthomodular l a t t i c e s  J  operations  L.  on  are  elaborated.  the l a t t i c e d e f i n i t i o n s o f  A, V  among  i n c o m p a t i b l e s 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 , truth-functional  (O,JD)  i n c o m p a t i b l e elements.  semantics f o r any quantum In c o n t r a s t ,  p r o o f , which s e m a n t i c a l l y bivalent, truth-functional  interpreted (o)  PQ  the Kochen-Specker  M L  containing 1967  impossibility  i s a proof of the i m p o s s i b i l i t y of a  semantics f o r any  three-or-higher  3  dimensional  Hilbert  space  PQJJA  s t r u c t u r e , 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 o f o v e r l a p p i n g maximal Boolean substructures but not  in  » "the presence o f i n c o m p a t i b l e  sufficient  t o r u l e out  quantum p r o p o s i t i o n a l s t r u c t u r e s .  semantics, which i s p a r t l y m o t i v a t e d Chapter I I I , the s t a t e - i n d u c e d P^  the c l a s s i c a l  Each  Exp^,  on a  P^  Exp^,  as  v a l u e s t o the elements i n  P  L.  i n a manner e x a c t l y L truth-functionally  Thus  Exp^.  P  be  even though t h a t s u b s t r u c t u r e may P.,, QM  and  so may  Exp^,  may  W  a  PQ^  l a r g e r than any  L.  PQM W  .  semantics (algebraic)  Thus, when the c l a s s i c a l semantic method i s a p p l i e d  t o a non-Boolean quantum t o a l s o be  Boolean  be a non-Boolean s u b s t r u c t u r e o f  J  i s e x a c t l y l i k e the methodology o f the c l a s s i c a l  semantics f o r an  regarded  r a t h e r than the e n t i r e  In s h o r t , the b a s i c methodology o f the quantum s t a t e - i n d u c e d for  he  i s b i v a l e n t and  P_ QM  substructure of  0, 1  valuation of c l a s s i c a l p r o p o s i t i o n a l  even though the domain where each  and  w i t h the quantum  t r u t h - f u n c t i o n a l l y assigns  a l g e b r a i c v a l u a t i o n s on an  o f the s t a n d a r d  ,  L.  semantic mappings, i s  t r u t h - f u n c t i o n a l i s only a sub-structure of ,  P  semantics f o r a  expectation-functions  as the quantum analog logic,  by the f a c t t h a t , as d e s c r i b e d i n  t o the elements i n a s u b s t r u c t u r e o f  0, 1  state-induced  ( a l g e b r a i c ) semantics f o r a Boolean Lindenbaum a l g e b r a  analogous t o the way assign  f o r the  determined by c l a s s i c a l mechanics i s e x a c t l y analogous t o  M  investigated. values  .  semantics f o r a Boolean p r o p o s i t i o n a l  So the n o t i o n o f a s t a t e - i n d u c e d state-induced  i s considered  T h i s i s the p r o p o s a l o f a  necessary P?,7? QMA  such a c l a s s i c a l semantics f o r a  In Chapter V I , another semantic p r o p o s a l  structure  elements i s  state-induced  , and)  the r e s u l t  i s a semantics (which happens  which i s n o n - c l a s s i c a l i n the sense t h a t  domain o f each semantic mapping  Exp^,  i s a non-Boolean s u b s t r u c t u r e  of  the  4  P„„ . QM  I n c o n t r a s t , i n t h e case o f c l a s s i c a l mechanics, t h e domain o f each  s t a t e - i n d u c e d semantic mapping i s t h e e n t i r e Boolean  P ^ ; and l i k e w i s e ,  i n t h e case 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 , t h e domain o f each a l g e b r a i c v a l u a t i o n i s t h e e n t i r e Boolean  L.  Chapter V I I s u r v e y s h i d d e n v a r i a b l e (HV) p r o p o s a l s , p r o o f s 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 o f t h e s e HV i m p o s s i b i l i t y p r o o f s . Kochen-Specker theories:  p r e s e n t t h e c l e a r e s t n o t i o n o f t h e g o a l o f proposed HV  t o g i v e 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 o f quantum mechanics,  whereby t h e s t a t i s t i c a l r e s u l t s o f quantum mechanics a r e r e p r o d u c e d 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 B o o l e a n s t r u c t u r e  P  of  HV  ( s u b s e t s o f ) a proposed c l a s s i c a l phase space o f h i d d e n v a r i a b l e s . Kochen-Specker  r e q u i r e t h a t 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 o f quantum  mechanics p r e s e r v e t h e f u n c t i o n a l r e l a t i o n s among quantum magnitudes and t h e l o g i c a l o p e r a t i o n s among c o m p a t i b l e quantum p r o p o s i t i o n s ; i n o t h e r words, an HV r e c o n s t r u c t i o n must p r e s e r v e t h e 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 t h e quantum structural condition.  P ^ .  Such a r e q u i r e m e n t may be c a l l e d a  Von Newmann and J a u c h - P i r o n each impose an  a d d i t i o n a l s t r u c t u r a l condition r e q u i r i n g the preservation o f an operation among i n c o m p a t i b l e s . That i s , a c c o r d i n g t o von Newmann and J a u c h - P i r o n , an HV t h e o r y must p r e s e r v e some o f t h e l a t t i c e f e a t u r e s o f t h e quantum PQ^ ; t h i s v i e w i s c r i t i c i z e d i n t h r e e n o t e s a t t h e end o f Chapter V I I . Now Kochen-Specker  show t h a t t h e i r n o t i o n o f an HV r e c o n s t r u c t i o n o f  quantum mechanics  i s p o s s i b l e I F F t h e r e 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 on  P  .  mechanics  ( i ) mappings  I n t h i s way, t h e problem o f h i d d e n v a r i a b l e s f o r quantum i s connected w i t h t h e problem o f a c l a s s i c a l semantics f o r t h e  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 . prove t h a t f o r  ^QJ^  a r e i m p o s s i b l e , and  And  structures, bivalent, truth-functional  (6) • mappings  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 i m p o s s i b l e f o r t h e  quantum mechanics o f t h r e e - o r - h i g h e r HV  as mentioned above, Kochen-Specker  dimensional  H i l b e r t space.  The  other  i m p o s s i b i l i t y p r o o f s s i m i l a r l y i n v o l v e showing t h e i m p o s s i b i l i t y o f  proposed 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 C r i t i c s o f t h e s e HV  HV mappings on t h e  P^  structures.  i m p o s s i b i l i t y p r o o f s argue t h a t t h e  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 t o s a t i s f y t h e v a r i o u s s t r u c t u r a l c o n d i t i o n s imposed by t h e a u t h o r s o f t h e i m p o s s i b i l i t y proofs.  HV  So whether o r not t h e p r o o f s a r e a c c e p t e d depends  upon whether o r not t h e s t r u c t u r a l c o n d i t i o n s a r e a c c e p t e d as imposed r e q u i r e m e n t s .  And  as Bub  quantum mechanics i s i n t e r p r e t e d . dichotomy a r t i c u l a t e d by Bub:  justifiably  makes c l e a r , t h e l a t t e r depends upon  how  I n p a r t i c u l a r , we have the f o l l o w i n g  E i t h e r quantum mechanics i s t a k e n t o be  a  ( p r i n c i p l e ) t h e o r y w h i c h 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 g i v e n by t h e  structure  a b s t r a c t e d f r o m t h e fundamental p o s t u l a t e s o f quantum m e c h a n i c s ; i n t h i s c a s e , the quantum  PQJ^  must be p r e s e r v e d ,  and as shown by Gleason  and  Kochen-Specker, quantum mechanics i s a complete t h e o r y o f quantum phenomena and an HV r e c o n s t r u c t i o n o f quantum mechanics i s i m p o s s i b l e .  Or  the  e n t e r p r i s e o f 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 o f quantum mechanics i s t r e a t e d as paramount, w i t h r e s p e c t t o which t h e quantum be p r e s e r v e d ;  need not PQJ^  i n t h i s c a s e , as proved by Kochen-Specker, t h e quantum  cannot be p r e s e r v e d ,  and as e x e m p l i f i e d by t h e 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 p r e s e r v e P „ . ^ QMA p o s s i b l e and quantum mechanics i s i n c o m p l e t e r e l a t i v e t o such a n HV W  is  6  reconstruction.  Bub argues t h a t b e h i n d each o f t h e s e two p o s i t i o n s t h e r e  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 m o t i v a t e d by t h e  p r e s u p p o s i t i o n t h a t t h e l o g i c a l s t r u c t u r e o f quantum phenomena and quantum t h e o r y must be a B o o l e a n s t r u c t u r e l i k e t h e B o o l e a n J  c l a s s i c a l phenomena and c l a s s i c a l mechanics.  P„„ CM  structure of  The former i s m o t i v a t e d by  a n open a c c e p t a n c e o f t h e non-Boolean c h a r a c t e r o f t h e l o g i c a l s t r u c t u r e o f quantum phenomena and quantum t h e o r y , as m a n i f e s t e d i n t h e non-Boolean ^QMA  s t r u c t u r e ( w h i c h i s a b s t r a c t e d from t h e quantum f o r m a l i s m P„ CM  same way t h a t t h e B o o l e a n of c l a s s i c a l mechanics).  W  by t h e  s t r u c t u r e i s a b s t r a c t e d from t h e f o r m a l i s m  Thus one's views on l o g i c may c o l o u r one's  i n t e r p r e t a t i o n o f quantum mechanics. But r e g a r d l e s s o f t h e above l o g i c a l p o i n t , s i n c e 1967 i t has been c l e a r t h a t 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 o f quantum mechanics which p r e s e r v e s t h e 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 t h e quantum impossible.  P„„ QM  is  And i t i s a r g u a b l e t h a t because t h e c o n t e x t u a l HV t h e o r i e s  do n o t p r e s e r v e t h e quantum  PQJ^ »  such t h e o r i e s a r e n o t r e a l l y  r e c o n s t r u c t i o n s o f quantum mechanics but r a t h e r a r e e n t i r e l y s e p a r a t e t h e o r i e s o f quantum phenomena w h i c h , as Bub p u t s i t , w i l l have t o s t a n d on t h e i r own f e e t .  T h e i r f e e t a r e shaky  s i n c e so f a r , experiments have  f a l s i f i e d t h e d e v i a t i o n s from quantum mechanics p r e d i c t e d by t h e c o n t e x t u a l HV t h e o r i e s .  Thus quantum mechanics, whose s t a t e - i n d u c e d  do p r e s e r v e t h e 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  Exp^  mappings  and do  s u c c e s s f u l l y p r e d i c t t h e r e s u l t s o f e x p e r i m e n t s , marks a r a d i c a l d e p a r t u r e 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 d e p a r t u r e from classical logic .  7  I n t h i s t h e s i s , two t y p e s o f HV t h e o r i e s a r e i n v e s t i g a t e d , namely, what a r e 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 t h e " z e r o t h k i n d " (proved i m p o s s i b l e by von Neumann, J a u c h - P i r o n , G l e a s o n , Kochen-Specker) and HV t h e o r i e s o f t h e " 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 t h e "second k i n d , " t h a t i s , t h e s o - c a l l e d l o c a l HV t h e o r i e s , a r e n o t d i s c u s s e d 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 , t h e c e l e b r a t e d paper by E i n s t e i n , P o d o l s k y , and Rosen, i n w h i c h t h e n o n - l o c a l i t y o f quantum phenomena i s h i g h l i g h t e d , i s n o t d i s c u s s e d i n t h i s t h e s i s . x  B e r n a r d d'Espagnat, i n h i s paper "The Quantum Theory and R e a l i t y " i n a r e c e n t S c i e n t i f i c American ( V o l . 241, No. 5, November, 1979) p r e s e n t s 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 t h e n o n - l o c a l i t y o f quantum phenomena and o f t h e p r o p o s a l o f a l o c a l HV t h e o r y . Though d'Espagnat does n o t say s o , h i s e x p l a n a t i o n o f t h e 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 t h e o r y makes i t c l e a r t h a t t h e 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 . , B o o l e a n , m a n i p u l a t i o n o f t h e 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, 8 3 ) ; he argues t h a t t h e c r u c i a l assumption i n t h e 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 t h e o r y i s n o t t h e assumption o f l o c a l i t y b u t r a t h e r t h e 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 a r e t o be computed as though t h e y 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 . , B o o l e a n , p r o b a b i l i t y space. Thus t h e 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 t h e "second k i n d " may i n f a c t be no d i f f e r e n t from t h e 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 t h e " f i r s t k i n d " w h i c h h i n g e upon attempted B o o l e a n t r e a t m e n t s 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 o f t h e s e 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  S e c t i o n A.  Group and R i n g  Structures  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 elements E={a,b,c,d,e, defined f o r any  from  . . .}  E x E  b , c € E,  to  E  with a binary  (univalent) operation  plus  such t h a t  i s closed with respect  to  b + c € E,  E  and t h e f o l l o w i n g c o n d i t i o n s  + + ; i.e.,  o b t a i n f o r any  elements i n E: (1)  +  i sassociative, i.e.,  (2)  There e x i s t s a d i s t i n g u i s h e d element b + 0 = 0 + b = 0 ,  (3)  F o r any  b € E,  be p r o v e n t h a t inverse of  f o r any  c  c € E  b) and s a t i s f i e s  i s an a d d i t i v e group.  i s a l s o w r i t t e n as <E, +, 0>  F o r example,  i s symmetric d i f f e r e n c e , and I f an a d d i t i v e group +  such t h a t  0  "-b"  I t can  (the a d d i t i v e F o r any  b - c.  s a t i s f y i n g c l o s u r e and ( 1 ) , ( 2 ) , <S, A, 0> i s a s e t - t h e o r e t i c S  i s a s e t o f s u b s e t s o f some s e t ,  i s t h e empty s e t .  <E, t , 0>  i s commutative, i . e . ,  b + c = 0.  b + (-b) = 0 = (-b) + b .  r e a l i z a t i o n o f an a d d i t i v e group, where  (4)  such t h a t  i s u n i q u e ; i t i s d e s i g n a t e d as  The o r d e r e d t r i p l e  A  0 € E  b € E.  there exists a  b, c € E, b + ( - c )  (3)  b + ( c + d) = (b + c ) + d.  i s such t h a t :  b + c = c + b ,  then  <E, +, 0>  i s an  a b e l i a n o r commutative a d d i t i v e group. Now l e t a second b i n a r y  (univalent) operation  dot  •  be d e f i n e d  from any  E x E  to E  b , c € E,  such t h a t  b • c € E  E  i s closed with respect t o  (by c o n v e n t i o n ,  and t h e s e two c o n d i t i o n s  i.e., for  i s also written be),  obtain:  (5)  •  i s associative.  (6)  •  i s d i s t r i b u t i v e with respect to  and  b • c  •,  +,  i.e., b * ( c + d ) = b c + b d  ( c + d) • b = cb + db.  The o r d e r e d q u a d r u p l e  <E, +, •, 0>  and ( l ) - ( 6 ) i s a r i n g .  F o r example,  r e a l i z a t i o n o f a r i n g , where  s a t i s f y i n g t h e two c l o s u r e <S, A, f l , 0>  conditions  i s a set-theoretic  fl i s t h e i n t e r s e c t  operation.  I f a r i n g i s such t h a t : (7)  ••• i s commutative,  t h e n t h e 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 b « l = l « b = b ,  f o r any  The o r d e r e d f i v e - t u p l e  1 € E  such t h a t  b € 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 w h i c h 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 . C o n s i d e r such a r i n g w h i c h a l s o (9)  b • b = b,  f o r any  b 6 E,  satisfies:  t h a t i s , each element i n E  i s idempotent.  2 , (By c o n v e n t i o n ,  b • b  i s also written  b .)  Two c o n d i t i o n s f o l l o w from ( 9 ) : (10)  Each element Proof:  F o r any  ( b + b ) (3), (7)  •  b € E  2  i s i t s own a d d i t i v e  b € E,  (b + b ) = b  = b + b = b + b + b + b .  b = -b.  Q.E.D.  i s commutative.  2  2  inverse.  + b  2  + b  2  + b . 2  And by ( 9 ) ,  So by ( 2 ) , b + b = 0 ,  Thus f o r any  b , c € E,  b + c = b +  and so by (-c) = b -  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.  by ( 3 ) , be = - ( c b ) ,  So by ( 2 ) , be + cb = 0,  and hence by ( 1 0 ) , be - cb.  and so  Q.E.D.  (Halmos, 1963, p. 2) The o r d e r e d 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 . <E, +, •, 0, 1>  And t h e o r d e r e d f i v e - t u p l e  s a t i s f y i n g c l o s u r e , ( 1 ) - ( 1 0 ) i s a Boolean r i n g - w i t h - u n i t .  Or i n o t h e r words, t h e idempotent elements o f a commutative r i n g form a Boolean r i n g , and t h e idempotent elements o f a r i n g - w i t h - u n i t o r 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 . <S, A, f l , 0 , X> where  S  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 Boolean r i n g - w i t h - u n i t ,  i s the set of subsets of a given set  S e c t i o n B.  F o r example,  X.  The Boolean A l g e b r a and t h e Boolean L a t t i c e In a Boolean r i n g - w i t h - u n i t , two b i n a r y o p e r a t i o n s  join '  V  a r e d e f i n e d from  i s d e f i n e d from  follows:  f o r any  b' = 1 - b. algebra.  E  E x E  to  E  i s t h e empty s e t , and  and a unary o p e r a t i o n  b A c = b • c,  The r e s u l t i n g s e x t u p l e  For example,  E  i n terms o f t h e r i n g o p e r a t i o n s  b, c € E,  S  0,11,  b V c  = b +  <E, A, V, ', 0, 1>  <S, R, U, ', 0 , X>  of a Boolean a l g e b r a , where 0  to  meet  A  and  complementation +, •  as  c - ( b * c ) , i s a Boolean  i s a set-theoretic realization  i s the set of a l l subsets of a given set '  are the set operations  X,  i n t e r s e c t , union,  complementation, r e s p e c t i v e l y . From t h e above l i s t o f c o n d i t i o n s ring-with-unit s a t i s f i e s with respect  to  l i s t o f c o n d i t i o n s w h i c h a Boolean a l g e b r a operations.  +  ( 1 ) - ( 1 0 ) which a Boolean and  *,  we can d e r i v e a  s a t i s f i e s with respect  However, i n a d d i t i o n t o t h e c l o s u r e o f  E  w.r.t.  lengthy  to i t s  A, V,  ',  11  and t h e e x i s t e n c e o f t h e d i s t i n g u i s h e d 0 and 1-elements i n E,  the following  f i v e c o n d i t i o n s a r e n e c e s s a r y 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 B o o l e a n algebra:  f o r any  b , c , d € E,  (Bl)  Commutativity:  b A c = c A b  (B2)  Associativity:  (b A c) A d - b A ( c A d)  (b V c ) V d = b V. ( c V d ) , (B3)  Absorption:  and  b V c = c V b,  by ( 4 ) and ( 7 ) .  and  by ( 1 ) , ( 4 ) , ( 5 ) , ( 6 ) , ( 1 0 ) .  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)  Distributivity:  b A ( c V d) = (b A ) V (b A d)  and  c  b V ( c A d) = (b V c ) A (b V d ) ,  by ( 3 ) , ( 6 ) , ( 9 ) .  Among t h e many o t h e r i d e n t i t i e s and c o n d i t i o n s which can be d e r i v e d from ( 1 ) - ( 1 0 ) we note t h e f o l l o w i n g : Idempotence:  b A b = b  D i s t i n g u i s h e d elements:  and  b V b = b.  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 o f complementation:  C h ' ) ' = b,  by ( 1 ) , ( 2 ) , ( 1 0 ) .  Moreover, i n a Boolean a l g e b r a we may d e f i n e a b i n a r y r e l a t i o n i n terms o f t h e meet o r j o i n o p e r a t i o n s b A c - b, every  and  b 5 c I F F b V c = c.  a s : f o r any  b, e € E,  I t follows that  b S c IFF  0 < b 5 1, f o r  b € E. Then by ( 2 ) , ( 3 ) , ( 6 ) , ( 8 ) , and (.1), t h e  s a t i s f i e s the condition: condition together  f o r any  b , c £ E,  '  operation  b 5 c I F F c' 2 b ' .  w i t h (B4) and t h e i n v o l u t i o n c o n d i t i o n d e f i n e  orthocomplementation  x  .  also This ' as  S i n c e 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  for '  i n t h e ordered  <  12  sextuple designation o f a Boolean algebra Any any  x  Boolean algebra a l s o s a t i s f i e s  the following conditions, f o r  b , c 6 E: De M o r g a n ' s l a w s :  (B5),  (b A c )  = b  V c  and  (b A c ) V c = (b V c ) A ( c ^ V c ) , A  1  Modularity  I f b 5 c  c = ( c A b"") V b ,  elements  b < c ,  where  proven as f o l l o w s .  b < c Assume  Then s i n c e  = 0 V (b A c ) = b A c" ", 1  :  b = ( b v c  J  * ) A c  a r e s a i d t o be d i s j o i n t  b 5 c ,  b A c = 0.  b  then  IFF c 5 b .  i.e., b A c = 0  The  b V ( e A c ) = ( b V e ) A c ,  then  since  and  o r orthogonal  Moreover,  b < c  b A c 5 c  A c ,  0 2 e,  f o r every  IFF b A  e € E.  we h a v e  With t h e binary r e l a t i o n  algebra  b 5 c^.  1  Q.E.D. and orthomodularity  <  Consider  c o n d i t i o n s and here  because  i n Sections  d e f i n e d a s above i n a B o o l e a n  conditions (B1)-(B5) t h a t w.r.t.  i s a B o o l e a n l a t t i c e , a s w i l l b e shown b e l o w .  orthocomplemented  Assume  x  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  f o l l o w s from t h e f i v e  = 0,  a n d s o b y (B4-) ,  the 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  it  c  b A 1 - b A ( c V c ) - ( b A c ) V ( b A c"")  compatibility,modularity,  they a r e important  f o r any  f o l l o w s from (B5).  b, c € E  b A c 5 0,  1  then  I f b 5 c  which again  1  IFF  And b y ( B l ) ,  f o l l o w s from (B5).  Orthomodularity:  Any  A n d b y t h e same  Q.E.D.  Modularity: e 6 E.  proven as f o l l o w s .  (b V c ) .  Cc A b"") V b - Cc V b ) A 1 = ( c V b ) .  b V c = c V b.  = b A c .  w h i c h by (B4) a n d 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 conditions,  (b V c )  Cb A c ) V c = Cc A b ) V b ,  Compatibility: By  <E, A, V , , 0, 1>.  and d i s t r i b u t i v e , l a t t i c e  -  D a n d E. algebra,  a Boolean  A -Boolean. i . e . , an  i s defined as f o l l o w s .  an a r b i t r a r y , nonempty c o l l e c t i o n o f e l e m e n t s  13  E = {a,b,c,d,e, . . . }  f o l l o w i n g p r o p e r t i e s , f o r any (5a)  Reflexivity:  ( 2 b)  Anti-symmetry:  (2c)  Transitivity:  The o r d e r e d p a i r  2 c E xE  with a binary r e l a t i o n b,c,d € E:  b 5 b. If b 2 c  and  If b 2 c  <E,2>  c 5 b,  and  c 5 d,  then then  2,  and f o r any  i s t h a t element e € E,  b 2 d.  d e f i n e t h e g r e a t e s t l o w e r bound  and t h e l e a s t upper bound ( l . u . b . ) o f any s u b s e t F  b = c.  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 t o  g.l.b. of  w h i c h has t h e  b € E  i f e 2 f  i s the r e s u l t of interchanging  2  as f o l l o w s .  such t h a t , f o r e v e r y  f o r every  i s defined d u a l l y , i . e . , substitute  F c E  >  f € F  f o r 2. >, A  and  (g.l.b.)  then  f € F,  e 2 b.  The b 2 f,  The l . u . b .  (The d u a l o f any c o n d i t i o n V,  and  and  0  and  1  (Halmos, 1963, pp. 7-8, 2 2 ) . ) The u n i q u e n e s s o f t h e g . l . b . and l . u . b . o f F c E  any  f o l l o w s from  o f whose elements meet o f  b, c,  b,c € E  ( 2 b ) . A lattice  F o r example,  where  c  <S,c,n,U>  b v e  and c a l l e d t h e j o i n o f  i s the s e t - i n c l u s i o n r e l a t i o n .  b € E,  b v c = 1  0,  t h e r e e x i s t s a t l e a s t one (such a  c  and a l . u . b . ,  c € E  w i l l be c a l l e d t h e complement o f  I f a l a t t i c e has a g . l . b . , b € E,  b A b " = 0, 1  0,  (b" ")" " = b, 1  l a t t i c e i s an orthocomplemented  1  and  and a l . u . b . ,  lattice  b 2 c  and i f , f o r  b A c = 0 b  and  and be denoted  <E,2,A,V,',0,1>.  t h e r e e x i s t s a unique orthocomplement b V b~ = 1,  1,  such t h a t  " b " ' ) , t h e n t h e l a t t i c e i s a complemented l a t t i c e  any  and c a l l e d t h e  i s a set-theoretic realization of a l a t t i c e ,  I f a l a t t i c e has a g . l . b . , any  i s a p o s e t any two  b A c  have a g . l . b . , w r i t t e n  and have a l . u . b . , w r i t t e n  b, c.  <E,2,A,V>  1,  b~ € E IFF  <E,2,A,V," ",0,1>. L  and i f , f o r satisfying:  2 b^,  then the  14  I f t h e meet and j o i n o p e r a t i o n s a r e d i s t r i b u t i v e t h e n the l a t t i c e is distributive.  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 , t h e n  complementation i s unique and i s o r t h o c o m p l e m e n t a t i o n ( B i r k h o f f , p. 1 5 2 ) .  An orthocomplemented,  1948,  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  lattice. I t i s easy t o prove t h a 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 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 a l g e b r a as above.  2  d e f i n e d i n a Boolean  F o r ( B l ) , ( B 2 ) , and (B3) ensure t h a t  2  satisfies  ( 2 a ) , ( 2 b ) , and ( 2 c ) . And ( B l ) , ( B 2 ) , (B3) ensure t h a t t h e element i s a l o w e r bound f o r t h e s u b s e t  {b,c}  = b A c,  c A ( c A b) = ( c A c ) A b = c A b,  thus  b A c 2 c. and  b A c 2 b,  And i f d  d A c = d,  d 2 (b A c ) ; t h e element Moreover,  then  thus  i s any l o w e r bound f o r (bAc)  b A c  b V c b A c  and  because  b A (b A c ) = (b A b) A c  {b,c},  A d = b A ( c A d ) = b A d = d ,  i s t h e g r e a t e s t l o w e r bound o f  b V c  a r e each unique because  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 . i s a l a t t i c e w i t h respect t o the as above.  2  thus  i.e., d A b = d and hence  {b,c}.  i s t h e l e a s t upper bound o f t h e s u b s e t and  b A c  Dually,  {b,c}.  A, V  i n a Boolean  Hence, a B o o l e a n a l g e b r a  r e l a t i o n defined i n a Boolean a l g e b r a  And i n o t h e r words, t h e ( B l ) , ( B 2 ) , (B3) c o n d i t i o n s c o m p l e t e l y  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. 1 8 ) . I t f o l l o w s from (B4) t h a t t h e d i s t i n g u i s h e d  0  and  1  elements o f a  Boolean a l g e b r a a r e t h e g r e a t e s t l o w e r bound o f E and t h e l e a s t upper bound o f E, r e s p e c t i v e l y , as shown n e x t . b 21,  and  b € E , b A l = b A ( b v b ) = b, i  b A 0 = b A (b Ab ") = (b Ab) Ab "= b Ab~= 1  i s an upper bound o f e € E  F o r any  such t h a t  E,  1 2 e,  and  0,  so  0 2b; that i s , 1  i s a l o w e r bound o f  E.  I f t h e r e i s an  J  0  so  i . e . , 1 A e = 1,  then e i t h e r  e = 1  o r (B3)  15  i s v i o l a t e d , and d u a l l y , i f t h e r e i s an e V 0 = 0,  then e i t h e r  upper bound o f  E  e = 0  e € E  such t h a t  e 5 0, i . e . ,  o r (B3) i s v i o l a t e d ; t h u s 1 i s t h e l e a s t  and 0 i s t h e g r e a t e s t l o w e r bound o f  Boolean a l g e b r a i s a complemented l a t t i c e .  E.  Q.E.D.  So a  And by ( B 5 ) , a B o o l e a n a l g e b r a  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 a l g e b r a i s a Boolean l a t t i c e w i t h r e s p e c t t o t h e ^ r e l a t i o n d e f i n e d i n a B o o l e a n a l g e b r a as above.  C o n v e r s e l y , i t i s easy  t o prove t h a t a B o o l e a n 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 t h e phrase B o o l e a n s t r u c t u r e and t h e  B = <E,A,V, ",5,0,l>  sextuple  a  Boolean l a t t i c e  t o r e f e r t o b o t h a B o o l e a n a l g e b r a and a  indiscriminately.  The B o o l e a n s t r u c t u r e s d e t e r m i n e d by c l a s s i c a l mechanics, I label  P  and d e s c r i b e i n - C h a p t e r I I I ( B ) , a r e  which  cr-complete and atomic . L  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 , w h i c h I l a b e l  and d e s c r i b e i n Chapter 1 1 ( B ) , a r e complete and atomic i f t h e y a r e 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 a r e d e f i n e d as f o l l o w s . Completeness:  B  i s complete i f e v e r y s u b s e t o f elements i n  g . l . b . and a l . u . b .  B  s u b s e t o f elements i n Atomicity:  B  B  a € B  such t h a t  which c o v e r s t h e  has a g . l . b . and a l . u . b . .  b t c .)  p. 6 9 ) .  B  b > a,  e € B.  i s non-atomic  b € 8,  there i s  where an atom i s a n element  0-element, i . e . , a > 0  not s a t i s f i e d by any  has a  cr-complete i f e v e r y denumerable  i s atomic i f , f o r e v e r y non-zero element  an.atom  and  is  B  ( F o r any  and  b,c € B,  a > e > 0  is  b > c IFF b > c  i f i t has no atoms (Halmos,  1963,  16  I t f o l l o w s t h a t i n an a t o m i c i t dominates  8,  every element i s t h e l . u . b . o f t h e atoms  (Halmos, 1963, p. 7 0 ) .  And i n an atomic  8,  two elements a r e  e q u a l I F F they dominate t h e same atoms ( R u t h e r f o r d , 1965, p. 8 3 ) . any d i s t i n c t elements  b ^ c  such t h a t  a £ c,  a 5 b  but  8  Every f i n i t e  i n an atomic or  a < c  and L  Every f i n i t e  The diagram o f a f i n i t e r e p r e s e n t a t i o n o f an 8.  n t  8  Every f i n i t e  where  n  F o r example:  n  X  8  is  i s t h e number o f  structure  i s i s o m o r p h i c t o t h e power s e t  <E = t h e s e t o f a l l s u b s e t s o f a g i v e n s e t  where t h e number o f elements i n  in  (Z^)  a € 8  a £ b.  i s t h e two-element B o o l e a n  <E = {0,1},A,V,- ,2,0,l>. Boolean s t r u c t u r e  but  t h e r e i s an atom  i s atomic and complete.  isomorphic t o the c a r t e s i a n product atoms i n 8  8,  Thus f o r  X,fl,LI,' ,5,0,X>,  i s t h e same a s t h e number o f atoms i n 8  looks l i k e a two-dimensional  d i m e n s i o n a l cube, where  n  i s t h e number o f atoms  8.  n =  3  In these diagrams, t h e d o t s r e p r e s e n t the elements o f t h e s t r u c t u r e  and  the l i n e s c o n n e c t i n g the dots r e p r e s e n t the o p e r a t i o n s and r e l a t i o n s among the elements,  e.g.,  18  3 '  S e c t i o n C.  Subsets  o f a Boolean S t r u c t u r e  In c o n f o r m i t y w i t h s t a n d a r d mathematical d i s t i n g u i s h between the s t r u c t u r e members o r subsets o f the s e t members o r subsets o f  E  8  and i t s s e t o f elements  to  8  i s a non-empty subset o f  generate a s u b a l g e b r a o r s u b l a t t i c e o f  which  of  8.  in  8  A non-empty 8,  i s said  8.  i s a non-empty ( p r o p e r ) subset  F  of  )  (a)  (  For any  b,c  € F,  For any  b € F  b A c € F.  ) (b)  A (proper) i d e a l i n  B  (a')  c  and f o r any  if  b 2 e  then  e € F.  For any  I  B  of  which s a t i s f i e s :  (a')  b , c € I , b V c € I .  )  (b')  For any  b € I  The d i s t i n g u i s h e d  1-element o f  is itself a f i l t e r  in  and  e € 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 ( p r o p e r ) subset  B  x  8  satisfies:  (  in  A,v,  when c l o s e d w i t h r e s p e c t t o t h e o p e r a t i o n s o f  A (proper) f i l t e r B  So  8.  which i s c l o s e d w i t h r e s p e c t t o the o p e r a t i o n s 8,  E.  a r e a l s o more simply r e f e r r e d t o as  A subalgebra or s u b l a t t i c e of  subset o f  p a r l a n c e , I do not  8.  and f o r any 8  if  b > e  i s a member o f every f i l t e r  D u a l l y , the  i s i t s e l f an i d e a l i n  e € B,  8.  then in  e € I. B  0-element i s a member o f every  and ideal  Moreover, i t f o l l o w s from de Morgan's laws  19  and from one o f t h e c o n d i t i o n s d e f i n i n g t h a t , f o r any f i l t e r an i d e a l i n  B;  F  B,  in  (if b 2 c  t h e s e t o f elements  and d u a l l y , f o r any i d e a l  I  B,  in  then  c  {b € B : b^  i s a f i l t e r i n . B ( S i k o r s k i , 1960, pp. 9, 1 1 ) .  any  I  and d u a l (c) (c*)  And i f b € F  in  For any  b" " € F  b € B,  bjfl,  IFF  1  Thus f o r  b  € I  X  IFF  and i f b € I  b € I. b € F. then  b £ F.  For example, a  f i l t e r and d u a l i d e a l are d e s i g n a t e d i n t h e Boolean s t r u c t u r e below by t r i a n g l e s  is  we have:  b € B,  F o r any then  B  € F}  t h e s e t o f elements  {b• € B : b^ € 1} F  2b)  diagrammed  around t h e elements i n t h e f i l t e r and squares around t h e  elements i n t h e d u a l i d e a l :  l=aVbVcV/d  .0  The u n i o n o f any f i l t e r and i t s d u a l i d e a l i n sublattice of  B  B  i s a subalgebra or a  ( B e l l and Slomson, 1969, p. 1 7 ) .  An u l t r a f i l t e r  UF  in  B  i s a p r o p e r f i l t e r w h i c h i s not t h e  p r o p e r subset o f any p r o p e r f i l t e r i n  B.  An u l t r a i d e a l  UI  in  B  is a  20  B.  p r o p e r i d e a l which i s not t h e p r o p e r subset o f any p r o p e r 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  c o n t a i n e d i n an u l t r a i d e a l ( S i k o r s k i , 1960, pp. 15, 1 7 ) . B  u l t r a f i l t e r and e v e r y u l t r a i d e a l i n any  Moreover,  is  every  i s p r i m e , t h a t i s , f o r any  b,c € B: (d)  If  :  (d»)  b V c € UF,  If  then e i t h e r  b A c € 'Ul,  And e q u i v a l e n t l y , i f b £ UF  then e i t h e r then  ( B e l l and Slomson, 1969, p. 20). Ul  B  i n any  B = UF U U l . or  b £ UF.  Proof:  For any  b f. UF  If  UF  then  b € Ul  b^ € UF,  or  c € UF.  or  c € Ul.  and i f b £ U l  I t f o l l o w s t h a t f o r any  b € B,  and f o r any  b € UF  b 6 UF  or B  i n any  b"" € UF,  b € Ul,  then  UF  and i t s d u a l  and thus  and f o r any  b € B,  b € UF  and so by ( c ) , b € U l .  1  b~ € U l .  Q.E.D. 8  Each u l t r a f i l t e r and i t s d u a l u l t r a i d e a l i n a f i n i t e a t o m i c is a  principal'  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 f i n e d w i t h  r e s p e c t t o a n atom  a € B  as f o l l o w s :  = {b€  UF  B : b > a}  and  3.  = {b € B : b < a }.  B,  t h e r e i s a one-to-one  correspondence between atoms a r d u l t r a f i l t e r s  (and d u a l u l t r a i d e a l s )  UI  And i n a n a t o m i c  3.  (Bell  and Slomson, 1969, p. 2 7 ) . F o r any p a i r o f d i s t i n c t elements an u l t r a f i l t e r  UF  B  in  non-zero element i n a  B  b i- c  B,  i n any  c o n t a i n i n g one but not t h e o t h e r . i s c o n t a i n e d i n some u l t r a f i l t e r  in  there i s A l s o , each B.  ( B e l l and  Slomson, 1969, p. 1 6 ) . S e c t i o n D.  The Quantum P a r t i a l - B o o l e a n A l g e b r a  Kochen and Specker d e f i n e a p a r t i a l - B o o l e a n a l g e b r a by 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 . a f i e l d i s a s e t o f elements  E  first  In s h o r t , a p a r t i a l - a l g e b r a over  w i t h the u s u a l r i n g o p e r a t i o n s  +  and  21  d e f i n e d from relation.  A  to  E,  where  A commutative  namely t h e case where  o S E xE  a l g e b r a i s a s p e c i a l case o f a p a r t i a l - a l g e b r a ,  t = E xE.  form a p a r t i a l - B o o l e a n a l g e b r a operations  AjV,- -,  i s c a l l e d the c o m p a t i b i l i t y  The idempotent elements o f a p a r t i a l - a l g e b r a  <E ,A, A, V,--, 0,1 >  w h i c h has t h e Boolean  1  d e f i n e d i n terms o f t h e r i n g o p e r a t i o n s  1  but t h e b i n a r y o p e r a t i o n s  A, V  +, •,  a r e a g a i n d e f i n e d from o n l y  ^  as u s u a l ,  to  E.  A  Boolean a l g e b r a i s a 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 a l g e b r a , namely t h e c a s e where  ^ = E x E.  (Kochen-Specker, 1965, pp. 180, 183; 1967, pp. 64-65).  U s i n g t h e t e r m i n o l o g y and s t y l e o f S e c t i o n s (A) and ( B ) , t h e s e s t r u c t u r e s a r e d e s c r i b e d as f o l l o w s . A partial-ring-with-unit elements  E = {a,b,c,d,e,  <E,A, + ,»,0,1>  . . . ,} i n c l u d i n g t h e d i s t i n g u i s h e d  with a binary r e l a t i o n of compatibility +  and  (A a)  • d e f i n e d from i  CD t o  b,  c € E,  i . e . , f o r any follow that (Ob) (oc)  E  For every  i f B i c  b,c,d € E,  and  b € E,  b A 1 +  b € E, then  b i b , symmetric, i . e . , c i> b,  i f b o c  and  and n o n - t r a n s i t i v e ,  c od,  i t does n o t  and  and •,  b i 0. i . e . , f o r any  are pairwise compatible then  b,c,d € E, i f  (b+c) i d ,  (b • c ) i d,  etc. f o r a l l combinations. And f o r any s u b s e t  F £ E,  i f a l l t h e elements i n  F  are pairwise  c o m p a t i b l e , t h e n by c l o s u r e t h e y g e n e r a t e a c o m m u t a t i v e - r i n g - w i t h - u n i t . (Kochen-Specker, 1965, p. 180; 1967, p. 6 4 ) . d e f i n e d from o n l y  C3  to  E,  1,  and two b i n a r y o p e r a t i o n s  bod.  c> i s c l o s e d under b, c , d  <!> c E x E  0  such t h a t :  i s r e f l e x i v e , i . e . , f o r any  f o r any  i s a non-empty s e t o f  r a t h e r t h a n from  F i n a l l y , since E xE  to  E,  +, *,  are  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 o r 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 o t h e r c o n d i t i o n s which t h e and  1  elements and t h e  +,  •  o p e r a t i o n s must s a t i s f y .  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  Q  However, s i n c e  i s a commutative-ring-with-unit  and s i n c e any s u b s e t o f m u t u a l l y c o m p a t i b l e 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 c o m m u t a t i v e - r i n g - w i t h - u n i t , t h e + ,•  0  and  1  elements and t h e  o p e r a t i o n s o f 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  t h e c o n d i t i o n s ( l ) - ( 8 ) which d e f i n e t h e  0, 1, +•, *,  all  of a commutative-ring-  with-unit . The idempotent elements o f 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 a l g e b r a A = <E,i,A,v, ",0,l > when t h e J  +  and  1  • as u s u a l and <b , 0, 1  1965, p. 183). elements; b cb 1  AjV,- -  a r e d e f i n e d as above E  i s non-empty;  0, 1  (Kochen-Specker, are the d i s t i n g u i s h e d  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 e v e r y  and  subset  In p a r t i c u l a r ,  o p e r a t i o n s a r e d e f i n e d i n terms o f  b o 0;  F c E,  i>  i s c l o s e d under  i f a l l t h e elements i n  AjV," ";  and f i n a l l y , f o r any  1  F  b € E,  are p a i r w i s e compatible, then  by c l o s u r e t h e y g e n e r a t e a B o o l e a n ( s u b ) s t r u c t u r e .  Moreover, s i n c e a l l t h e  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 a r e idempotent, not o n l y 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) a r e a l l satisfied. terms o f  Thus t h e b i n a r y o p e r a t i o n s A, v +  and  d e f i n e d from  ^> t o  E  in  • as u s u a l s a t i s f y t h e c o m m u t a t i v i t y , 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 w h i c h f o l l o w from t h e c o n d i t i o n s ( l ) - ( 7 ) , ( 9 ) , (10) s a t i s f i e d by from o n l y  to  E,  +  and  r a t h e r than from  • . ExE  And s i n c e to  E,  A, V  are defined  they are i n f a c t  23  partial-operations. terms o f  +  and  The unary o p e r a t i o n  the  (assuming  as u s u a l ) the c o n d i t i o n :  b 5c  and  b 5 c  IFF  A  then  V  or  b V e = c. A  IFF  b k> c.  b A d,  then  follow that  b 5 d.  Thus  in  5  b 5c  bAc  b o c  pQj^  A  and  b V d So  However, i f A  a r e not d e f i n e d i n  5  may 5  b i d  s  t  r  u  c  t  and  u  r  <  =  A  c o d:  = (b A c )  b 5 d.  Ad.  For any  And  IFF  ,  e s i s introduced  (b A c) cb d;  c 5 d  B.  i s not  algebras  determined  and  atomic. € A  b,c,d  if  b 5 c,  such  b d> (c A d)  and  and  algebra c 5 d,  satisfies  (An a d d i t i o n a l c o n d i t i o n on  i n Chapter V I ( D ) n o t h i n g - b e f o r e  t o Kochen-Specker.  but  x  an a t o m i c p a r t i a l - B o o l e a n a l g e b r a  Chapter  VI(D)  .•  notion of a partial-Boolean algebra  f o l l o w i n g c o n s t r u c t i o n due  5  A transitive partial-Boolean b,c,d€A,  is  EJ^ >A,V, ,5,0,1>.  i s a s s o c i a t i v e I F F , f o r any  b i ( c A d)  if  so i t does not  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 ,  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  and  and  not be t r a n s i t i v e , i n w h i c h c a s e  the same a t o m i c i t y c o n d i t i o n as an a t o m i c *QMA  b 5 c  € A, A  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  s a t i s f i e s the c o n d i t i o n :  then  b V c  b,c  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 a l g e b r a s  b A (c A d )  implies  and the j o i n  we can be s u r e t h a t , f o r any  as u s u a l .  V  or  b A c = b,  IFF  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 d e f i n e d i n  A partial-Boolean algebra that  A  and  i s defined i n a partial-Boolean  by quantum mechanics, which s h a l l be l a b e l e d The  in  i s orthocomplementation.  X  as u s u a l , i . e . ,  b ° c,  a p a r t i a l - o r d e r i n g . But considered  E  w h i c h f o l l o w from t h e  X  S i n c e the meet  r e f l e x i v e and a n t i - s y m m e t r i c b jo d,  to  i s d e f i n e d i n terms o f  partial-ordering relation  are d e f i n e d i n b 5 c  5  IFF c^" < b ,  conditions ( l ) - ( 3 ) , (6), (8)-(10).  a l g e b r a i n terms o f  E  1-element as u s u a l s a t i s f i e s the complementation  i n v o l u t i o n c o n d i t i o n s and  The  d e f i n e d from  i s .further> e l u c i d a t e d by t h e  C o n s i d e r a nonempty f a m i l y o f  24 {B.}.„ . such t h a t t h e i n t e r s e c t i o n o f two a l g e b r a s o f 1 i€Index f a m i l y i s i t s e l f an a l g e b r a o f :the f a m i l y ; so - a l l the; B^ share t h e same  Boolean a l g e b r a s the  A n d i f - {e-^, e ,. . . .} a r e elements, o f t h e u n i o n  d i s t i n g u i s h e d 0^ 1 elements.!  -  E = U B. • s u c h t h a t e v e r y r p a i r o*f'theitf l i e in-some'common a l g e b r a = B; i t h e n t h e r e i s a B , k € i n d e x such t h a t  ,.  k  { ^t 2'  * * *} ^ \  union  as f o l l o w s .  e  e  a  E  B. • such t h a t l  that  b A c = d  *  F o r any  b,c,d  € E,  b,c € B. ; b A c = d l in  A  Then a p a r t i a l - B o o l e a n a l g e b r a  B.;  bVc  = d  b i c  in  in  A  A  in  A  i s d e f i n e d on t h e IFF t h e r e e x i s t s  IFF t h e r e e x i s t s a  IFF t h e r e e x i s t s a  B.  x  that b  b V e = d  = c  in  in- B^;  a l l the  8^  .  1  B.; l and  A  b 0  =c  in  in  A  A  IFF t h e r e e x i s t s a  such  B. l  such t h a t  a r e t h e common d i s t i n g u i s h e d elements o f  s t a t e and Hughes proves t h a t e v e r y  A  c o n s t r u c t e d on a f a m i l y o f B o o l e a n a l g e b r a s as  (Kochen-Specker, 1965, pp. 183-184; Hughes, 1978, pp. S e c t i o n E.  such  1  Kochen-Specker  i s o m o r p h i c t o an  B. l  The Quantum Orthomodular  is above.  113-114).  Lattice  Jauch's d e f i n i t i o n o f t h e l a t t i c e s t r u c t u r e s d e t e r m i n e d by quantum mechanics, w h i c h I l a b e l <E,5,A,V, ,0,1> X  E  P  ,  Then Jauch d e f i n e s t h e 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 :  (Let the  l a t t i c e g e n e r a t e d by {SL.}.,_ , I i€Index sublattice  lattice  which i s complete i n t h e u s u a l sense t h a t e v e r y s u b s e t o f  has a g . l . b . and a l . u . b .  the  s t a r t s w i t h an orthocomplemented  SL  F  A subset  F c E  i s a compatible set i f  i s a Boolean s u b l a t t i c e o f t h e o r i g i n a l l a t t i c e .  be t h e f a m i l y o f a l l t h e s u b l a t t i c e s w h i c h c o n t a i n J  Q  = D  SL.  i s t h e l a t t i c e g e n e r a t e d by  F;  F , ( J a u c h , 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  r e f l e x i v e , symmetric, and n o n - t r a n s i t i v e . any  b€E,  b«^0,  b o 1,  and  b i b .  co £ E x E i  s  And i t i s easy t o show t h a t , f o r  25  PQ  In o r d e r t o d e f i n e  »  MTj  Jauch furthermore p o s t u l a t e s t h e  conditions: (P)  If b 5 c  then  b ^> c,  b,c € E.  f o r any  Jauch c a l l s  this  c o n d i t i o n weak m o d u l a r i t y . (Al)  Atomicity (as usual).  (A2)  If a  i s an atom and  e € E  ( J a u c h , 1968, pp. 86-87).  atom, t h e n  a A e = 0,  then  a V e  covers  e,  f o r any  And i t f o l l o w s t h a t i f a  ( a V e) A e"" i s a l s o an atom, f o r any 1  e € E  i s any (Piron,  1976, p. 2 4 ) . P  Thus  = <E,5,A, V,+,<^,0,1>  modular, a t o m i c  i s a c o m p l e t e , orthocomplemented, weakly  lattice.  I t i s easy t o show t h a t i n such a l a t t i c e , f o r any b <^> c (b  (b A c"*") V c = ( c A b ) V b = b V c;  IFF  x  A c ) V (b A c ) x  = b;  and  b <^> c  b d> c  I F F t h e elements  b,c £ E:  IFF  b, b , X  c,-c , X  s a t i s f y t h e d i s t r i b u t i v e l a w f o r any c o m b i n a t i o n ( J a u c h , 1968, p. 87; P i r o n , 1976, p. 2 6 ) . Moreover, s i n c e from weak m o d u l a r i t y t h a t , f o r any b = (b V c ) A c, X  and i f b 5 c  b < c b,c € E ,  then  IFF b A c = b, i f b 5 c  c = ( c A b"*") V b.  i t follows  then This i s the  o r t h o m o d u l a r i t y c o n d i t i o n , a c c o r d i n g t o Rose (1964, p. 331) and a c c o r d i n g to  P i r o n (1976, p. 2 4 ) . The p h r a s e " o r t h o m o d u l a r " subsumes t h e two  c o n d i t i o n s o f o r t h o c o m p l e m e n t a t i o n and weak m o d u l a r i t y ; t h u s c o m p l e t e , a t o m i c , orthomodular  PQ^L  ^  S  A  lattice.  P i r o n develops h i s d e f i n i t i o n o f the complete, atomic, orthomodular lattice  P^„ QML  element  b € PQ^^  the  T  i n a d i f f e r e n t manner w h i c h r e v e a l s t h e f a c t t h a t each may have non-unique complements d e f i n e d i n  unique orthocomplement  besides  b"*". P i r o n s t a r t s w i t h a l a t t i c e w h i c h i s  26  complete i n t h e u s u a l sense.  S i n c e completeness ensures t h a t t h e e n t i r e  l a t t i c e has a g . l . b . which i s t h e d i s t i n g u i s h e d which i s t h e d i s t i n g u i s h e d sextuple  0-element and a l . u . b .  1-element, a complete l a t t i c e i s an. o r d e r e d  <E,5,A,V,0,1>. PQ^  In order t o define  »• P i r o n f u r t h e r m o r e p o s t u l a t e s t h e  l  conditions: ( A l ) and ( A 2 ) , as i n J a u c h . (C)  F o r each element b^ € E,  where  b € E, b, b ^  t h e r e i s a t l e a s t one c o m p a t i b l e complement  a r e complements s a t i s f y i n g t h e u s u a l  complementation c o n d i t i o n  (b A b^ = 0  and  b V b* = 1 ) , and  b, b ^  a r e c o m p a t i b l e i n a sense which P i r o n d e f i n e s i n d e p e n d e n t l y o f t h e A, v,  o p e r a t i o n s and  5  relation.  Most s i m p l y , any  b, c a r e  c o m p a t i b l e i n P i r o n * s sense i f t h e y a r e a s s o c i a t e d w i t h s i m u l t a n e o u s l y measurable quantum p r o p o s i t i o n s . (P)  F o r any c, c^  b,c € E,  i f b 5 c  i sdistributive.  t h e n t h e s u b l a t t i c e g e n e r a t e d by  b , b^,  P i r o n c a l l s t h i s c o n d i t i o n weak m o d u l a r i t y  ( P i r o n , 1976, pp. 21-23). Two r e s u l t s f o l l o w from P i r o n ' s weak m o d u l a r i t y . t h e n by ( P ) , t h e elements  b, b*, c  a r e d i s t r i b u t i v e and so  b V ( b A c ) = (b V b ) A (b V c ) = 1 A (b V c ) = c; l  then  \  similarly, i f b < c  l  c A ( c ^ v b ) = b.  F i r s t , i f b < c,  T h i s r e s u l t w i l l be mentioned a g a i n s h o r t l y .  S e c o n d l y , a c c o r d i n g t o P i r o n , i t f o l l o w s i m m e d i a t e l y 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 t h a t t h e c o m p a t i b l e  complement o f each element i s u n i q u e . b  w i t h i t s unique  b V b^ = 1, pp. 2 3 - 2 4 ) .  (b ) 1  2  1  b* = b,  Thus t h e a s s o c i a t i o n o f a n element  i s orthocomplementation s a t i s f y i n g : and i f b 2 c  then  c  A  5 b  A  b A b ^ = 0,  ( P i r o n , 1976,  27  Substituting  for ^ ,  becomes t h e o r t h o m o d u l a r i t y  t h e f i r s t r e s u l t f o l l o w i n g from ( P )  condition.  And P i r o n p r o v e s t h a t i f t h e  orthocomplement i s i n t e r p r e t e d a s a c o m p a t i b l e complement, t h e n 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, P i r o n ' s weak m o d u l a r i t y c a n be shown t o be e q u i v a l e n t t o Jauch's weak m o d u l a r i t y . i c E xE  Piron l a t e r defines the c o m p a t i b i l i t y r e l a t i o n  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 :  IFF t h e s u b l a t t i c e g e n e r a t e d by  b , b*, c ,  i s distributive.  b o c  With t h i s  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 , Jauch's ( P ) i s e q u i v a l e n t t o P i r o n ' s ( P ) w i t h substituted f o r ^ .  J a u c h a l s o says t h a t h i s weak m o d u l a r i t y i s  e q u i v a l e n t t o t h e p o s t u l a t e t h a t t h e c o m p a t i b l e complement i s u n i q u e , t h a t i s , t h e 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 m o d u l a r i t y  (Jauch,  1968, p. 8 7 ) . So l i k e J a u c h , P i r o n d e f i n e s orthomodular l a t t i c e , . PQ  M L  PQ^L  a  s  a  complete, atomic,  Moreover, P i r o n makes i t c l e a r t h a t an element i n  may have non-unique complements w h i c h s a t i s f y t h e complementation  c o n d i t i o n b u t which a r e n o t c o m p a t i b l e complements and a r e n o t orthocomplements. Thus t h e r e a r i s e s i n P ^  t h e problem o f a complementation w h i c h i s n o t  M L  u n i q u e (and hence i s n o t an o p e r a t i o n ) , a s w i l l be d i s c u s s e d i n Chapter I V ( F ) . The B o o l e a n  L  and  P„„ CM  s t r u c t u r e s and t h e p a r t i a l - B o o l e a n a l g e b r a 43  P.„. QMA  each have o n l y one c o m p l e m e n t a t i o n , namely, t h e o r t h o c o m p l e m e n t a t i o n , w h i c h i s unique. F i n a l l y , as w i t h  PQ  M A  >  o f an o r t h o m o d u l a r l a t t i c e  PQ  M L  >  a  Boolean s t r u c t u r e i s a s p e c i a l c a s e namely, t h e case where  CJ = E x E .  Moreover, any quantum  P ^  can be extended t o an o r t h o m o d u l a r l a t t i c e  PQ  A, V  o p e r a t i o n s among i n c o m p a t i b l e  M L  by d e f i n i n g t h e  elements.  The  28  two s t r u c t u r e s and  P„„. QMA  and  P„ QML  w i l l be f u r t h e r compared i n Chapter I V ( E )  W T  (F).  Section F.  P  Subsets o f  and  ?  The n o t i o n o f 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 , 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 a r e d e f i n e d i n any l a t t i c e , t h e quantum  »  e x a c t l y as t h e y a r e d e f i n e d i n a B o o l e a n  principal e.g., i n  lattice  ( B i r k h o f f , 1967, pp. 25, 2 8 ) . 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 t h e 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  PQJ^  A R E  d e f i n e d as i n a B o o l e a n 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 ) , (b')  (Bub, 1974, p. 1 2 0 ) .  1  m o d i f i e s c o n d i t i o n ( a ) (and d u a l l y , ( a ' ) ) . by t h e f a c t t h a t , f o r any b A c  i s not defined i n  i n a P_ . QMA W  (a ) H  b, c PQJ^  a  i s a non-empty subset r  I f b,c € F ,  J  The m o d i f i c a t i o n i s m o t i v a t e d  i n any f i l t e r •  F c PQ  Hughes' s m o d i f i e d F  then there i s a  of  However, R. Hughes  P^„ QMA  d € F  A  such  M A  i f b & c  definition i s :  , t h a t , f o r any J  such t h a t  d < b  any  then A filter b,c,d € P « „ . • QMA d 5 c.  And Hughes adds as a p r o v i s o t h e c o n d i t i o n : (c„) n  0 £  F.  C o n d i t i o n ( b ) i s l e f t as b e f o r e ; t h a t i s , ( b ) , (a,,), H f i l t e r i n a P_„. . QMA According 8,  the entire  8  and  (c ) H  define a  t o the d e f i n i t i o n o f a f i l t e r i n a Boolean s t r u c t u r e i s a f i l t e r , a l b e i t an improper f i l t e r .  t o Hughes's d e f i n i t i o n o f a f i l t e r i n P Q ^  A  J  "the e n t i r e  But a c c o r d i n g PQ^  a  i s n o t a'  0 € P - „ . but according t o c o n d i t i o n ( c ) , 0 i s not a QMA H member o f any f i l t e r . Conditions ( b ) , ( a ) , ( c ) , a c t u a l l y define a proper H H f i l t e r since  T T  u  29  i n P _ „ . • So we may drop c o n d i t i o n ( c ) and d e f i n e a f i l t e r QMA H  filter ^QMA  a  3  S  N O N  ~  follows.  P ' y subset  E M  The  F  t  PQ^  of  w h i c h s a t i s f i e s (a^) and ( b ) .  a  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 r any  defined i n P„„. QMA where  in a  TT  b A c  b,c € F,  a c c o r d i n g t o ( a ) and assuming t h a t  b i c),  (i<?e.,  t h e element  i s t h e g r e a t e s t l o w e r bound o f  b A c {b,c},  b A c  is  i s a member o f  F,  a s shown i n S e c t i o n  ( D ) ; w h i l e a c c o r d i n g t o (a„) and r e g a r d l e s s o f whether o r n o t b A c  is  H  defined i n P Q ^ » F.  Now i f  a n  Y  o  n  °f "the l o w e r bounds o f  e  i s defined i n P „ . , QMA  b A c  W  b A c € F  i f b,c € F.  member o f  F  {b,c} i s a member o f  t h e n (a„) and ( b ) do ensure t h a t H  F o r by ( a ) , some l o w e r bound o f { b , c } i s a  i f b,c € F,  a n d so by ( b ) , t h e g . l . b .  {b,c}  = b A c  i sa  member o f F i f b,c € F. That i s , though a f i l t e r i n a P Q J ^ defined by c o n d i t i o n ( a ) r a t h e r t h a n ( a ) , n e v e r t h e l e s s a f i l t e r i n . P . „ . does H QMA i  S  IT  s a t i s f y c o n d i t i o n (a) f o r those The  b,c € F  such t h a t  b ch c.  d u a l m o d i f i e d c o n d i t i o n (a') w h i c h , t o g e t h e r w i t h t h e n  unmodified  ( b ) , d e f i n e s an i d e a l  I  1  i na  P^w.  i s , o f c o u r s e , f o r any  QMA  b,c,d ' '  € P „. : QMA  (a') ri  I f b,c € I , t h e n t h e r e i s a  And  N  as above, an. i d e a l i n a  ( a ) f o r those  b,c € I  1  PQ^  d € I  such t h a t  d > b  does s a t i s f y t h e u n m o d i f i e d  such t h a t  PQJ^  A  S  A  proper f i l t e r  P  R O  P  E R  filter  d > c.  condition  b o c.  As i n t h e Boolean c a s e , we d e f i n e an u l t r a f i l t e r a  and  (ultraideal) i n  ( i d e a l ) which i s n o t t h e p r o p e r s u b s e t o f any  (ideal) 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 a r e d e f i n e d w i t h r e s p e c t t o an atom o f  PQ^  A  S  ^  N  S e c t i o n C. Hereafter,  P^  r e f e r s t o both  PQ^  A  N  I  ''QML  indiscriminately.  30  P„ QM  A substructure of  i s a non-empty subset o f elements o f  W  i s c l o s e d w i t h r e s p e c t t o t h e A, y , ^ A, v  operations of  P^^  operations of  P  i s Boolean I F F i t s elements a r e m u t u a l l y  P  Boolean substructure of  P_„ QM  which c o n t a i n s b o t h  P^  M  .  M  A substructure o f  ( i . e . , pairwise)  compatible. generates a  when c l o s e d w i t h r e s p e c t t o t h e o p e r a t i o n s o f i n P.„ , QM  P. # P„ 1 2  J  P^  elements i n P  Any non-empty subset o f m u t u a l l y c o m p a t i b l e  And f o r any  (where t h e  generates a substructure o f  when c l o s e d w i t h r e s p e c t t o t h e o p e r a t i o n s o f  P_„ • QM  M  which  W  a r e p a r t i a l - o p e r a t i o n s , as d e s c r i b e d i n S e c t i o n ( D ) . )  Any non-empty subset o f elements o f  PQ^  P^  P_ QM  t h e r e i s no B o o l e a n s u b s t r u c t u r e i n  P_ , P. . Any element 1 2  P € P. QM W  i s a member o f  some B o o l e a n s u b s t r u c t u r e i n P.,, , a t l e a s t t h e B o o l e a n s u b s t r u c t u r e QM c o n s i s t i n g o f j u s t t h e elements { P ^ P ^ O j l } . A maximal B o o l e a n s u b s t r u c t u r e mBS o f  P_„ QM  i s a B o o l e a n s u b s t r u c t u r e w h i c h i s n o t t h e p r o p e r subset o f P  any o t h e r B o o l e a n s u b s t r u c t u r e o f substructure of  P  i s contained  .  And by Zprn's lemma, any B o o l e a n  i n a maximal one ( V a r a d a r a j a n , 1962,  p. 204). The c e n t r e o f a are compatible  J  the  0, 1  S e c t i o n G.  P. , QM W  P„, QM  which  W  .  T h i s subset  i s i n fact a closed  and moreover, i t i s a B o o l e a n s u b s t r u c t u r e .  contains a t l e a s t the  elements a r e c o m p a t i b l e  0, 1  elements o f  P_„ QM  The since  w i t h every o t h e r element i n P ^  .  3  Mappings on a S t r u c t u r e Let  which have  i s t h e s u b s e t o f elements i n P . QM  w i t h every element i n P ^  substructure of c e n t r e o f any  P... QM  X, y  A, V,--  distinguished  1  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 o p e r a t i o n s d e f i n e d on a s e t o f elements i n c l u d i n g t h e  0-element and  1-element.  Any mapping  m : X  V  from any  31  X  structure Ma  t o any s t r u c t u r e b,c,d € X,  For any  m(b) = m(c)  in  Mb  m(0) = 0  in  m(b)  V.  m(b A c ) = m(d)  /  in  i s unique, that i s , i f b = c  F o r example, i f b A c = d Y  t  i f b  m(l) = 1 If  : X  y  y  h(c).  H2  h(b V c ) = h ( b ) v  h(c).  H3  Hb )  = y,  mapping  in  y.  also assigns:  m : X  {0,1}.  »  then  m  is a  A homomorphic mapping X,  i . e . , f o r any  b,c € X,  =(h(b)) . i  A mapping  m : X -*• y  i f b t c  then  one-to-one i n t o m(X)  m : X -»• V  preserves the operations defined i n  h(b A cO = h ( b ) A  b,c £ X,  then  then  m(b ) = m(c)  i s t h e two-element Boolean s t r u c t u r e  HI  x  then  X  X  i n V.  b i v a l e n t mapping d e s i g n a t e d as h  in X  = c  in  in  /.  Moreover, any n o n - t r i v i a l mapping Mc  a s s i g n s v a l u e s as f o l l o w s :  y.  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  m : X -*• y  A mapping  i . e . , t h e image o f m : X -»• Y  i s s a i d t o be i n j e c t i v e I F F , f o r any  X  under  m  i s s a i d t o be s u r j e c t i v e I F F i s the e n t i r e  y.  An i s o m o r p h i c  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, w h i c h p r e s e r v e s t h e o p e r a t i o n s o f B i r k h o f f , 1948, p. v i i ) .  X  (Lang, 1971, pp. 87, 90, 106;  An imbedding o f one s t r u c t u r e i n t o a n o t h e r i s a  homomorphic mapping w h i c h i s i n j e c t i v e (Bub, 1974, p. 6 8 ) .  Notes 1 ' Hughes d i s c u s s e s t h e problem o f t h e t r a n s i t i v i t y o f 5 i n a p a r t i a l - B o o l e a n a l g e b r a and proves t h a t a quantum p a r t i a l - B o o l e a n a l g e b r a  32  o f subspaces o f 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, p. V I . 1 8 ) .  1978,  2 As d e s c r i b e d i n n o t e 5 o f c h a p t e r I V ( E ) , o r t h o c o m p l e m e n t a t i o n i s d e f i n e d as a t y p e o f mapping, namely, a d u a l automorphism. 3 P i r o n d e f i n e s the centre of a P ; the c e n t r e o f a P- . can be d e f i n e d i n e x a c t l y the same way. ( P i r o n , 1976, p. 29)-. nM  33  CHAPTER I I  THE CLASSICAL PRECEDENT FOR A BIVALENT TRUTH-FUNCTIONAL SEMANTICS  S e c t i o n A.  The Standard Semantics  of C l a s s i c a l Propositional Logic  C l a s s i c a l propositional l o g i c assigns t r u t h values t o a set L =  o f w e l l - f o r m e d formulae by semantic mappings, c a l l e d  v a l u a t i o n s , which a r e b i v a l e n t and t r u t h - f u n c t i o n a l . L  i n i t i a l l y assigns the value  atomic ( s u b ) f o r m u l a e i n L. to  every o t h e r f o r m u l a i n L  0  (false) or  IFF  v(f ) = v(f ) = 1  TF2  v(f V f ) = 1  IFF  v(f) = 1  TF3  v ( f ) =1  values f o r any  1  I F F v ( f ) = 0,  d e s i g n a t e s " o r , " and  0, 1  i n t h e f o l l o w i n g r e c u r s i v e manner:  v(f A f ) = 1  on an  ( t r u e ) t o each o f t h e  And t h e n t h e v a l u a t i o n a s s i g n s  TF1  x  1  A valuation v  or  where  v(f) = 1  "A"  d e s i g n a t e s "and,"  " - L " designates "not."  "v"  T h i s (redundant) l i s t o f  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 t h e 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 t h e valuations. assigned a  The b i v a l e n c y c o n d i t i o n r e q u i r e s t h a t every f o r m u l a i n L 0  or  1  be  value.  A c c o r d i n g t o t h e t r u t h - t a b l e method o f s c h e m a t i z i n g v a l u a t i o n s , t h e r e a r e as many v a l u a t i o n s f o r a s e t L t h e t r u t h - t a b l e f o r L, a different i n i t i a l o c c u r r i n g i n L.  o f formulae as t h e r e a r e rows i n  where each row i n t h e t r u t h - t a b l e i s s p e c i f i e d by  assignment o f  And i f n  0, 1  v a l u e s t o t h e atomic  (sub)formulae  i s t h e number o f atomic ( s u b ) f o r m u l a e i n L,  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 o f  v a l u a t i o n s can be r e g a r d e d 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. T h i s n o t i o n o f 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 S e c t i o n ( D ) . S e c t i o n B.  The B o o l e a n 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  I n t h e 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 start with a set L V, -- o p e r a t i o n s . 1  with respect relation: where  o f formulae w h i c h i s c l o s e d w i t h r e s p e c t Such a c l o s e d  L  t o the  i s p a r t i t i o n e d i n t o equivalence classes  t o the standard, c l a s s i c a l proof t h e o r e t i c equivalence  f o r any -  f . ,f„ € L, f . ~ f ' 1 2 1 2  IFF  n  ^~ i s c l a s s i c a l d e r i v a b i l i t y .  \r f„ o f 1 2  and  n  h f  L = <E = { / f ^ A / f ^ / ,  w h i c h s h a l l be l a b e l e d  equivalence c l a s s containing ±  and  2  € L,  f  /f / A / f / = /f ±  If^l S If^l  2  ±  IFF f ^ Y f  2  .  "/f^/".)  /f / V / f y = /f ±  The  0-element o f  e q u i v a l e n c e c l a s s o f theorems o r t a u t o l o g i e s . i s f i n i t e , then the 2  n  atomic, with exactly  2  atomic ( s u b ) f o r m u l a e i n L  elements and  L  ±  F o r any  V f^;  1-element i s t h e  structure of atoms.  i s i n f i n i t e , then the  / f ^ " ^ 1^1 \  i s the equivalence  When t h e number  L n 2  (The  J_  c l a s s of anti-theorems or c o n t r a d i c t i o n s , while the  (sub)formulae i n L  D f , 2 1  algebra,  . . . },A, V, , 5 , 0 , l > .  i s designated  A f /;  n  The r e s u l t i n g s e t o f e q u i v a l e n c e  c l a s s e s form a B o o l e a n s t r u c t u r e , o f t e n c a l l e d t h e Lindenbaum  f ,f  A,  L  n  o f atomic  i s f i n i t e and  But when t h e number o f L  structure of  L is  i n f i n i t e and a t o m l e s s . For example, t h e c l o s e d s e t L^ 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 e x a c t l y 16 e q u i v a l e n c e c l a s s e s :  R  o f p r o p o s i t i o n a l formulae i n and  S,  i s partitioned into  35  co  Pi  \  <  co  \  0  1  co  co  CO  < < < A A Pi Pi Pi  ^/—N co  \  CO  in  Pi  Pi  \  \  \  \  co  Ml  v '  -<  CO  \  \  >^  A  Pi  >  Pi  -lCO  CO  CO  >  >  > A Pi  Pi  -1  Pi  <  Pi  Pi  \  \  \  \  \  ^  Pi  >  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 e q u i v a l e n c e  c l a s s e s form t h e Lindenbaum a l g e b r a  0=A  A  R  L  diagrammed as  7  N o t i c e t h a t t h e f o u r atoms i n t h i s Lindenbaum a l g e b r a a r e not the e q u i v a l e n c e c l a s s e s o f t h e atomic formulae /R A S/,  /R A S V ,  ' /R"" A S/, 1  R, S, /R  x  but r a t h e r a r e t h e f o l l o w i n g :  A S"""/.  36  Every Lindenbaum a l g e b r a o f ( e q u i v a l e n c e  c l a s s e s o f ) formulae 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 Boolean s t r u c t u r e .  The s i m p l e s t  ( 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 t h e two elements i s often called  ;  i t s h a l l a l s o be l a b e l e d  {0,1}.  0  and  1  and  Any Boolean  s t r u c t u r e c a n be homomorphically mapped onto t h i s s i m p l e s t Boolean s t r u c t u r e , as d e s c r i b e d S e c t i o n C.  next. B i v a l e n t Homompr.phic Mappings on Any Boolean S t r u c t u r e  G i v e n any Boolean s t r u c t u r e h : 8 -+ {0,1}  mappings in B  B  B,  t h e r e e x i s t b i v a l e n t , homomorphic  which can be d e f i n e d w i t h r e s p e c t t o t h e u l t r a f i l t e r s  s i n c e t h e r e i s a one-to-one correspondence between u l t r a f i l t e r s i n  and b i v a l e n t homomorphisms on  homomorphism element  h  b ( 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  w i t h r e s p e c t t o a n u l t r a f i l t e r UF as f o l l o w s : h(b) = 1  i f b € UF  1960, p. 1 6 ) . However, each r e s p e c t t o UF a s : b € UI,  B.  f o r any  h  on a  b € 8,  and B  h(b) = 0  f o r any  i f b j£ UF  (Sikorski,  may be e q u i v a l e n t l y d e f i n e d  h(b) = 1  i f b € UF  where UI i s t h e unique u l t r a i d e a l d u a l t o UF.  and  h(b) = 0 i f  In t h i s t h e s i s , the  l a t t e r i s t a k e n as my u s u a l d e f i n i t i o n o f a b i v a l e n t homomorphism. B,  with  F o r any  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 a r e e q u i v a l e n t because, f o r any  UF and d u a l UI i n B i n Chapter 1 ( C ) .  and f o r any  b € 8,  b f. UF  IFF  b 6 UI,  as shown  But when we c o n s i d e r t h e 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 d e t e r m i n e d by quantum mechanics, i t i s n o t always t h e case t h a t if  b J? UF  then  b € UI.  So t h e 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 V I ( 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 clearly bivalent.  And by  d e f i n i t i o n , a homomorphism s a t i s f i e s t h e c o n d i t i o n s H I , H I , H3 l i s t e d i n  37  Chapter 1 ( G ) , where 1 A 0 = 0 A 1  = 0,  l A i = l v l = l ,  O A O = O v O = 0 ,  l v O = O v l = l , and  1  = 0,  X  0^=1.  I t i s easy  t o show t h a t 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 H I , H2, H3, I F F 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. 9 9 ) . Thus each b i v a l e n t homomorphism on a Lindenbaum a l g e b r a  i s bivalent  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 homomorphism on a  8  c a n be shown as f o l l o w s .  e v e r y u l t r a f i l t e r and d u a l u l t r a i d e a l i n any  o f every b i v a l e n t  As mentioned i n Chapter 1 ( C ) , 8  i s p r i m e , and each  u l t r a f i l t e r t o g e t h e r w i t h i t s d u a l u l t r a i d e a l c o m p l e t e l y exhaust 8 = UF U U I .  Moreover, i t f o l l o w s from t h e e i g h t c o n d i t i o n s  (a')-(d'), listed  truth-functional.  UI  b,b  b  €  2  b„ v b 1 vb  l  b  U3  >2  b b  X  X  8:  IFF  b  l  € UF  and  IFF  b  l  € UI  or  b  2  € UI,  by ( b ) and ( d ' ) .  b  2  € UF,  by ( b ) and ( d ) .  2  6 UF  IFF  b  l  € UF  or  2  € m  IFF  b  l  € UI  and  € UF  IFF  b € UI,  by ( c ) .  6 UI  IFF  b € UF,  by (c»).  So by t h e d e f i n i t i o n o f TF1  TF2  t o UF and UI i s  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 ,  b„ A b € UF 1 2 Ab € UI b  l  U2  1  b  b  h : 8 -»• {0,1}  6 UF,  2  by ( a ) and ( b ) . T  € U I , by ( a ' ) and ( b ' ) .  2  with respect  h(b  A b ) = 1  IFF  h(b ) = h(b ) = 1  h(b  A b ) = 0  IFF  h(b ) = 0  or  h(b ) = 0  v b ) =1  IFF  h(b ) = 1  or  h(b ) = 1  V b ) = 0  IFF  h(b ) = h(b ) = 0  1  h(b h(b  1  1  2  2  (a)-(d),  i n Chapter 1 ( C ) , w h i c h d e f i n e prime UF and prime U I , t h a t  each b i v a l e n t homomorphism d e f i n e d w i t h r e s p e c t  f o r any  8, i . e . ,  1  2  t o UF and U I :  38  TF3  h(b ) = 1 X  IFF . h(b) = 0  h(b ) =0  I F F h ( b ) = 1.  X  Thus each b i v a l e n t homomorphism on a F u r t h e r m o r e , any b^ ^ b  8  8  i s truth-functional.  a d m i t s many b i v a l e n t homomorphisms. I f  a r e any p a i r o f d i s t i n c t elements i n B,  2  Chapter 1 ( C ) , t h e r e i s some u l t r a f i l t e r i n 8 not t h e o t h e r . assigns  8  w h i c h c o n t a i n s one element b u t  Hence t h e r e i s some b i v a l e n t homomorphism on  the value  1  t o one element and  f o r any p a i r o f d i s t i n c t elements h(b^) t h ( b ) ;  that  t h e n as mentioned i n  0  b^ t  to therother.  B  which  I n o t h e r words,  8, , t h e r e i s some  ina  h  such  t h i s has been c a l l e d t h e s e m i - s i m p l i c i t y p r o p e r t y o f  2  (Kochen-Specker, 1967, p. 6 7 ) . And i n p a r t i c u l a r , as Halmos shows, f o r  any nonzero i.e.,  b ^ 0  such t h a t  0-element  ina  h(b) = 1  B,  t h e r e i s some  since every  h  h  such t h a t  assigns  h ( b ) ^ h ( 0 ) = 0,  the value  0  to the  (Halmos, 1963, p. 7 7 ) . The former n o t i o n s h a l l be t a k e n t o d e f i n e  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 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  X,  i s complete I F F , 1  b i c  f o r any d i s t i n c t  i n X,  there  i s an  h  such t h a t  h(b) ± h ( c ) .  C l e a r l y , t h e completeness o f t h e c o l l e c t i o n o f b i v a l e n t homomorphisms on a Boolean s t r u c t u r e When  8  8  i s ensured by t h e s e m i - s i m p l i c i t y p r o p e r t y  i s a t o m i c , t h e n b e s i d e s t h e above-mentioned  of  8.  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 d u a l  u l t r a i d e a l s ) there  ultrafilters  i s a l s o a one-to-one correspondence between  and atoms.  Each atom  8,  UF  namely  a € 8  i s a member o f e x a c t l y one u l t r a f i l t e r i n  = { 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  respect  to  UF and i t s d u a l U l . a a o f b i v a l e n t homomorphisms on an a t o m i c  namely t h e h defined with a I t i s easy t o show t h a t a c o l l e c t i o n 8  B,  i s complete I F F i t i s as l a r g e as  39  the number of atoms i n 8 . Proof:  By d e f i n i t i o n , a complete c o l l e c t i o n i s  large enough so that every atom  a ± 0  l=h(a)^h(0)=0  on  1  value  by some  h  8.  Conversely, consider the c o l l e c t i o n of 8  bivalent homomorphisms on an atomic atoms i n  8.  :  h a  defined ( v i a  there i s an atom  a € 8  If a 5 b  but  h (b) = 1 i h ( c ) . a a b % UF  , and so  a  i s an  h a  on  Section D.  which i s as large as the number of  By d e f i n i t i o n , each bivalent homomorphism in, t h i s c o l l e c t i o n UF  a  and  Now by a theorem due to Rutherford  a fc b.  Since each atom i s assigned the  by exactly one bivalent homomorphism, the complete c o l l e c t i o n i s  as large as the number of atoms.  i s an  i s assigned the value  8  such that  Ul ) with respect to an atom a (Chapter 1(B)), f o r any a 5 b  but  a fc c,  a € 8.  b f c  or  8,  in  a 2 c but  a fc c,  b € UF and c £ UF , and so a a S i m i l a r l y , i f a < c but a fc b, then c € UF a  h (c) = 1 i- h (b). a a such that  Thus f o r any  i n 8»  b / c  and there  h (b) # h ( c ) . Q.E.D. a a  The Algebraic Semantics f o r the Lindenbaum Algebra  These facts about bivalent homomorphisms on a Boolean structure are relevant f o r the concept of a bivalent truth-functional semantics f o r the Lindenbaum algebras of c l a s s i c a l propositional l o g i c . Each u l t r a f i l t e r i n the formulae  L  L  structure of a (closed) set of  i s i t s e l f a subset of (equivalence  classes of) formulae i n L  which i s deductively complete i n the sense that, f o r any UF i n the an  L, If  then  and f o r any formulae I £ UF.  f ^ ' f ^ ^ ^'  ^  And each u l t r a f i l t e r i n L  ^ ^  a n c  ^  ^1 1" ^2 '  i s maximally consistent i n  the sense that the meet of a l l the elements i n any UF i s never the of  L,  L of  i . e . , the conjunction of a l l the (equivalence  0-element  classes) of formulae  in UF i s never a contradiction; but i f any element i n L  which i s outside a  40  g i v e n UF were added t o t h a t UF, then t h e meet o f a l l t h e elements i n UF would be t h e  As d e s c r i b e d  i n t h e p r e v i o u s s e c t i o n , each b i v a l e n t  on a Lindenbaum a l g e b r a any  element  L.  0-element o f  If I € L  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 .  and any  U F c L,  either  If/ € UF L  but not both; hence, no b i v a l e n t homomorphism on t o both 1  If I  and  / f "*"/  1  L.  0-element o f  L.  /fV  or  assigns  h,  € UF  the value  1  the value  And i f a b i v a l e n t homomorphism were  t o any o t h e r element i n  i n t h e u l t r a f i l t e r which d e f i n e s the  Moreover, f o r  s i n c e e v e r y b i v a l e n t homomorphism a s s i g n s  t o a n u l t r a f i l t e r o f elements i n  to assign the value  homomorphism  then  h  L  b e s i d e s those elements  would a s s i g n t h e v a l u e  1  to  So each b i v a l e n t homomorphism c a n be s a i d t o be a L.  maximally c o n s i s t e n t mapping on  Moreover, each b i v a l e n t homomorphism on t h e Lindenbaum a l g e b r a o f an  L  i s t h e a l g e b r a i c v e r s i o n o f one o f t h e s t a n d a r d v a l u a t i o n s  That i s , f o r any g i v e n v a l u a t i o n b i v a l e n t homomorphism formula  f € L,  h^  v^  on t h e  v (f)= h (/f/)  L  on an o f that  L, L  there  i s a corresponding  such t h a t , f o r every  (Bub, 1974, p. 102).  Q  f o r L.  And f i n a l l y , i n  t h i s t h e s i s t h e complete c o l l e c t i o n o f b i v a l e n t homomorphisms on a Lindenbaum a l g e b r a The  i s 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 s e m a n t i c s .  analogy between t h e complete c o l l e c t i o n o f b i v a l e n t  homomorphisms on an  L  L  as f o l l o w s .  may be e l a b o r a t e d  (sub)formulae i n  L  atomic, w i t h e x a c t l y homomorphisms on  L  and t h e t r u t h t a b l e c o l l e c t i o n o f v a l u a t i o n s I f we assume t h a t t h e number  i s f i n i t e , then t h e 2  n  atoms.  contains  2  L  structure of  L  n  f o r an  o f atomic  i s f i n i t e and  Thus t h e complete c o l l e c t i o n o f b i v a l e n t n  b i v a l e n t homomorphisms, j u s t as t h e  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 values t o the n  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  atomic ( s u b ) f o r m u l a e i n L,  homomorphism on t h e  L  assignment o f  values t o the n  L.  formulae i n and  1  to  0, 1  S  structure of  L  and l i k e w i s e each b i v a l e n t  i s s p e c i f i e d by i t s i n i t i a l e q u i v a l e n c e c l a s s e s o f atomic  F o r example, an i n i t i a l assignment o f t h e v a l u e s specifies the valuation  0, 1  v  0  i n the t r u t h table f o r  L  o  given i n Section  (B).  to  to  /R/  and  1  /(R  U F  /  R  J  .  /S/  A  &  /  X  homomorphism formula  h ^  R  x  s p e c i f i e s t h e unique atom  0  /R /A S/ i n t h e A  o f L ^ ; t h i s atom i n t u r n s p e c i f i e s t h e unique =  = S ) / , /R V R / } ; X  Z  S i m i l a r l y , t h e i n i t i a l assignment o f t h e v a l u e s  Lindenbaum a l g e b r a ultrafilter  to R  A  g  ^  {/R  X  A S/, /S/, / R V , /R  and t h i s u l t r a f i l t e r on  , where  ^ x R  A  V  S/, / R  x  V S/, / R  x  V S /, x  s p e c i f i e s a unique b i v a l e n t g  ^(/f/)  =  v  ( ) f  3  f  ° r every  f € L^ • The  concept o f a b i v a l e n t , t r u t h - f u n c t i o n a l s e m a n t i c s f o r a  B o o l e a n Lindenbaum a l g e b r a d e s c r i b e d i n t h i s c h a p t e r w i l l be t r e a t e d  i m this  t h e s i s as a p r e c e d e n t 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 s e m a n t i c s for  t h e Boolean; p r o p o s i t i o n a l s t r u c t u r e s d e t e r m i n e d by c l a s s i c a l mechanics  and  t h e 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 n p a r t i c u l a r , subsequent c h a p t e r s make use o f t h e 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 value  1  P,  a mapping which a s s i g n s t h e  t o an u l t r a f i l t e r UF o f elements i n P  t o t h e d u a l u l t r a i d e a l U l o f elements i n P  and a s s i g n s t h e v a l u e  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 r e s p e c t t o t h e elements i n UF U U l . t r u t h - f u n c t i o n a l mapping d e f i n e d  0  Such a b i v a l e n t ,  w i t h r e s p e c t t o an UF and d u a l U l may be  c a l l e d an u l t r a v a - l u a t i o n because, on a Lindenbaum a l g e b r a 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 , such a mapping i s t h e a l g e b r a i c v e r s i o n o f a s t a n d a r d  42  v a l u a t i o n , which i s r e g a r d e d i n t h i s t h e s i s as t h e paradigm semantic mapping. The  0, 1  v a l u e s a s s i g n e d 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 t h e t r u t h - v a l u e s because, on a Lindenbaum  algebra,  t r u e and f a l s e , a g a i n  an u l t r a v a l u a t i o n i s t h e a l g e b r a i c  version  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 o f t h e 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 t r u t h - f u n c t i o n a l mappings.  P  i s a complete c o l l e c t i o n o f b i v a l e n t ,  So i t i s c l e a r t h a t t h e e x i s t e n c e  o r 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  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  of only  one  does n o t y e t P.  show t h e i m p o s s i b i l i t y o f such a s e m a n t i c s , i t o b v i o u s l y  But i n o r d e r t o s u f f i c e s t o show  t h a t t h e r e 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 T h i s n o t i o n o f 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. I n 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 a l g e b r a o f quantum p r o p o s i t i o n s , labeled ^ » n be imbedded i n t o a B o o l e a n a l g e b r a 8 IFF there exists 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 homomorphisms on P. . Kochen-Specker a l s o d e f i n e a weak imbedding o f a P_ into a 8; such an imbedding e x i s t s IFF t h e r e exists a large enough c o l l e c t i o n o f b i v a l e n t homomorphisms on P so t h a t , f o r e v e r y nonpzero element P i- 0 i n P_„. , t h e r e i s some h : P -> {0,1} such t h a t h(P) t h ( 0 ) , i . e . , h(P) = 1 since every h a s s i g n s t h e v a l u e 0 t o t h e 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 c o m p l e t e . The n o t i o n o f a weakly- complete c o l l e c t i o n o f 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 C h a p t e r s V and V I . c a  n M A  g  43  CHAPTER I I I  THE CLASSICAL PRECEDENT FOR A STATE-INDUCED SEMANTICS  Preface 1 We c o n s i d e r p r o p o s i t i o n s  w h i c h make a s s e r t i o n s about t h e  real-number v a l u e s o f t h e magnitudes, i . e . , measurable p r o p e r t i e s , o f 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 o f a 1 kg s w i n g i n g pendulum i s between 19-20 kg m / s e c . (magnitude) ( system ) (value) As w i l l be d e s c r i b e d operations  i n t h i s chapter,  such p r o p o s i t i o n s and t h e l o g i c a l  "and," " o r , " " n o t " 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  v a r i o u s m a t h e m a t i c a l machinery i n t h e f o r m a l i s m  with  o f c l a s s i c a l mechanics.  These a s s o c i a t i o n s d e t e r m i n e t h e s t r u c t u r e o f a s e t o f 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, a t o m i c B o o l e a n s t r u c t u r e  Moreover, t h e f o r m a l i s m  P  .  o f c l a s s i c a l mechanics i n c l u d e s  b i v a l e n t homomorphisms, o r e q u i v a l e n t l y , s t a t e - i n d u c e d  state-induced  dispersion-free  p r o b a b i l i t y measures, w h i c h can be r e g a r d e d as p e r f o r m i n g t h e s e m a n t i c t a s k o f a s s i g n i n g t r u t h - v a l u e s t o t h e elements o f  P„„ . CM  F o r each b i v a l e n t  homomorphism o r 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 i n d u c e d by t h e s t a t e o f 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 t h e P d e s c r i b i n g t h e system, j u s t as each o f t h e s t a n d a r d  structure of propositions valuations f o r a set L  o f f o r m u l a e o f 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 t h e L.  L  structure of  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  considering the notion of a state-induced  semantics f o r t h e p r o p o s i t i o n a l  s t r u c t u r e s d e t e r m i n e d 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 t h e n o t i o n  44  o f a s t a t e - i n d u c e d s e m a n t i c s f o r t h e p r o p o s i t i o n a l s t r u c t u r e s d e t e r m i n e d by quantum mechanics, as s h a l l be proposed i n Chapter V I . S e c t i o n A.  The S t a t e s o f a C l a s s i c a l System Determine t h e R e a l V a l u e s o f That System's Magnitudes According to the Hamiltonian  f o r m a l i z a t i o n o f c l a s s i c a l mechanics, 2  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 parameterized  which i s  by p o s i t i o n and momentum c o o r d i n a t e s and whose d i m e n s i o n a l i t y  r e f l e c t s t h e degrees o f freedom o f t h e system.  For example, a p h y s i c a l  system w i t h o n l y one degree o f 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 t h e s i m p l e s t phase space w h i c h i s two  dimensional  and has one p o s i t i o n c o o r d i n a t e and one momentum c o o r d i n a t e .  Each p o i n t  w € £!  2,  r e p r e s e n t s a p u r e s t a t e o f t h e system a s s o c i a t e d w i t h  for a  pure s t a t e i s a s p e c i f i c a t i o n o f t h e system's p o s i t i o n and momentum v a l u e s . According  t o c l a s s i c a l mechanics, t h e v a l u e s o f every o t h e r ( m e c h a n i c a l )  magnitude o f t h e system can be c a l c u l a t e d once t h e system's s t a t e i s specified. A  I n p a r t i c u l a r , t h e c l a s s i c a l f o r m a l i s m r e p r e s e n t s each magnitude 3  by a r e a l - v a l u e d , measurable  function  f  R  : 2  on the phase space  a s s o c i a t e d w i t h t h e system s u c h t h a t t h e image o f any p o i n t the f u n c t i o n  f  o f t h e magnitude  i s t h e real-number v a l u e A  A,B, . . .  • • •  f o r any  w € 2,  (f  and  •  + f ) ( w ) = f (w)+f fi  w.  representing the c l a s s i c a l  have t h e r i n g o p e r a t i o n s + and • d e f i n e d among them as  the u s u a l sum and p r o d u c t o f f u n c t i o n s : (f  under  ( t h e real-number l i n e )  when t h e system i s i n t h e s t a t e  The r e a l - v a l u e d f u n c t i o n s magnitudes  a € R  w € 2  (w),  and  •f  f , f^ )(w)  on  = f (w) A  2  and f o r every  "fgCw).  work l i k e t h e a d d i t i o n and m u l t i p l i c a t i o n o f r e a l numbers.)  (Here  +  45  For example, c o n s i d e r as a system a 1 kg pendulum s w i n g i n g so t h a t i t s maximum h e i g h t i s 2 m above i t s minimum h e i g h t .  Let  w^  and  w^  be t h e f o l l o w i n g s t a t e s . w^:  At t h e t o p o f i t s swing, t h e pendulum's h e i g h t 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 t h e bottom o f i t s s w i n g , t h e pendulum's h e i g h t 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, J  w^  ,  i s represented  i n the c l a s s i c a l  1 2 f„ = • (momentum) . K 2 • mass  by t h e r e a l - v a l u e d f u n c t i o n pendulum's s t a t e i s  K,  t h e real-number v a l u e o f  formalism  So when t h e  K  2 2 0 kg m /sec .  is  2  And when t h e pendulum's s t a t e i s  w^  ,  the value of  K  i s 19.2 kg m /sec .  So t h e f a c t t h a t t h e real-number v a l u e s o f a c l a s s i c a l magnitudes depend upon t h e system's s t a t e has been f o r m a l i z e d r e p r e s e n t i n g each magnitude f  : °J -* R  A  2  system's  by  by a r e a l - v a l u e d , measurable f u n c t i o n  on a c l a s s i c a l phase space whose p o i n t s r e p r e s e n t  t h e system's  states. A l t e r n a t e l y , each s t a t e mapping from a ( c l o s e d ) s e t  F  w € Q  can: i t s e l f be r e g a r d e d as a  of functions representing  magnitudes t o t h e real-number l i n e , i . e . , point  w € &  and f o r any f u n c t i o n  w : F  : Q -> R,  f  ->• R,  R  w : R  mapping p r e s e r v e s t h e  d e f i n e d among t h e f u n c t i o n s : functions  f" , f A  = w(f ) + w(f ); A  B  g  A  may be c a l l e d t h e s t a t e - i n d u c e d  t h a t each s t a t e - i n d u c e d  on  ffi,  and  w(f  A  +  and  f o r any g i v e n , f i x e d  w(f  + f  ) =  •f ) = (f g  f t  such t h a t , f o r any  w(f.) = f.(w).  A  mapping  (f  classical  + fg)(w)  The  A  mapping. *  operations  w € S2 = f  I t follows  and f o r any  (w)  + f  (w)  • fgXw) = f^w) • f (w) = w(f ) - w C ^ ) . g  A  46  T h i s mathematical machinery o f r e a l - v a l u e d f u n c t i o n s and state-induced  mappings n o t o n l y f o r m a l i z e s t h e procedure by which real-number  values are assigned  t o t h e magnitudes o f a c l a s s i c a l 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  can be a s s i g n e d t o  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 t h e real-number v a l u e s o f 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  S e c t i o n B.  i n Section (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 C l a s s i c a l  When a s e t o f r e a l - v a l u e d , measurable f u n c t i o n s on a closed set with respect t o the  +, •  commutative-ring-with-unit, l a b e l e d 0-element every  i s the constant  w € Si,  and t h e  function  1  p  •f  S i n c e t h e product  p  = f  p  F^  Si t o  p  +, •  0  or  1.  0 to  which  fp  p  p  w € Si,  u n i t , as d e f i n e d i n Chapter 1 ( A ) . A, V,- -,  r = f (w)  I n o t h e r words, each  p  fp  A s e t o f idempotent f u n c t i o n s which i s operations  forms a  or i n o t h e r words, t h e idempotent elements o f  1  f  i t f o l l o w s t h a t t h e real-number v a l u e  {0,1}.  closed with respect t o the  The  w € Si, ( f • f ) ( w ) = f ( w ) .  i s d e f i n e d a s , f o r every  o f a n idempotent f u n c t i o n i s e i t h e r  Boolean o p e r a t i o n s  function  a r e idempotent f u n c t i o n s  i . e . , f o r every  ( f p • f p ) ( w ) = fp(w) * f p ( w ) ,  i s a f u n c t i o n from  t  t o every p o i n t i n Si.  ,  fp • fp  then t h e s e t forms a  f ^ which a s s i g n s t h e real-number  Some o f t h e f u n c t i o n s i n f  Si i s a  F ^ = <{f^,f^, . . .}, + ,• ,0,1>.  1-element i s t h e c o n s t a n t  a s s i g n s t h e real-number  satisfying:  operations,  Mechanics  F  Boolean-ring-with-unit,  form a  Boolean-ring-with-  And_in t h i s B o o l e a n - r i n g - w i t h - u n i t ,  the  and t h e 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  can be d e f i n e d i n terms o f t h e r i n g o p e r a t i o n s  +  a B o o l e a n s t r u c t u r e o f idempotent f u n c t i o n s on a  and Si .  •  5  as u s u a l , y i e l d i n g  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 d e f i n e d w i t h r e s p e c t t o a unique s u b s e t as f o l l o w s : if  w ^ Wp ,  i . e . , w € W^.  2- and  subset o f  w € 2,  f o r any p o i n t  W  p  = f  -1  f (w)=1  Each  Wp  and  f (w)= 0 p  Borel) A.  ({{.1 •})• = {w € 2 : f ( w ) = 1 } ; and each p  t h e idempotent f u n c t i o n s on a 2;  p  p  i s a measurable ( i . e . ,  the s e t - t h e o r e t i c (ortho)complement o f W  the B o r e l s u b s e t s o f  i f w € W  p  c 2  W  2  ,  = 2 - W  i.e., W  p  W  p  .  p  is  Thus  a r e i n a one-to-one correspondence w i t h  each B o r e l s u b s e t u n i q u e l y d e f i n e s a n 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 s u b s e t ( v i a i t s i n v e r s e image subsets o f a 2  form a B o o l e a n - r i n g - w i t h - u n i t  f  p  ('{I})-)'.  (with  The B o r e l  + , *, 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 , t h e empty s e t , and the e n t i r e space  2,  Boolean-ring-with-unit  r e s p e c t i v e l y ) , which i s isomorphic t o the o f idempotent f u n c t i o n s on  with-unit o f B o r e l subsets o f a A, V," ", 5, 1  2  2.  And t h e B o o l e a n - r i n g -  i s a l s o a Boolean s t r u c t u r e ( w i t h  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,  s e t - ( o r t h o C o m p l e m e n t a t i o n , 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 ) , w h i c h i s i s o m o r p h i c t o t h e B o o l e a n s t r u c t u r e o f idempotent f u n c t i o n s on  2 (Bub,  1974, p. 1 0 5 ) . The B o o l e a n 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, o r i s o m o r p h i c a l l y , t h e Boolean s t r u c t u r e o f B o r e l subsets o f t h e phase s p a c e , have each been r e g a r d e d as a p r o p o s i t i o n a l s t r u c t u r e d e t e r m i n e d by classical'mechanics,  labeled  PQ^ •  For in:one-way  o r .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 t h e real-number v a l u e s 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 t h e idempotent f u n c t i o n s on t h e system's phase space o r t h e u n i q u e l y system's phase space.  corresponding B o r e l subsets o f the  F o r example, i n h i s 1932 book, von Neumann argues  48  t h a 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 t h e v a l u e s o f a system's magnitudes can themselves be r e g a r d e d as idempotent magnitudes whose  0, 1  v a l u e s can be i n t e r p r e t e d as t h e " v e r i f i c a t i o n " and t h e n o n - v e r i f i c a t i o n o f the p r o p o s i t i o n s .  M e n t i o n i n g von Neumann's argument,  Kochen-Specker  l i k e w i s e r e g a r d p r o p o s i t i o n s as idempotent magnitudes whose 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 . S e c t i o n ( C ) , why t h e 0, 1  0, 1  values  There i s a b e t t e r r e a s o n , g i v e n i n  v a l u e s e x h i b i t e d by t h e idempotent  may be i n t e r p r e t e d as t h e t r u t h - v a l u e s o f p r o p o s i t i o n s .  magnitudes  Nevertheless, i n  the c l a s s i c a l f o r m a l i s m , idempotent magnitudes a r e r e p r e s e n t e d by t h e abovedescribed  idempotent f u n c t i o n s on a phase space.  On t h e o t h e r hand, i n  t h e i r 1936 p a p e r , 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 t h e s u b s e t s o f t h e 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  the B o r e l s u b s e t s o f t h e system's phase space. P  the Boolean p r o p o s i t i o n a l s t r u c t u r e The elements o f  P ^  Either association yields  = <E = {P'^*^' " ' * ^' ' '~ »->0>-'- * A  associated with propositions.  L  >  f w  The  P  P  s t r u c t u r e o f any  represent or are S2 i s a  cr-complete  P i n a P „ , i s a one-point w CM uniquely corresponding with the singleton Borel  atomic, Boolean s t r u c t u r e .  subset  v  may be thought o f e i t h e r as idempotent f u n c t i o n s o r  as B o r e l s u b s e t s o f t h e phase space; t h e elements o f  idempotent f u n c t i o n  with  And each atom  {w}.  S e c t i o n 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 , f o r t h e Boolean  P_ CM W  S t a t e - I n d u c e d Semantics  Structures  J u s t as t h e real-number v a l u e s o f a system's magnitudes  depend  upon t h e system's s t a t e ( i . e . , upon t h e v a l u e s o f t h e system's p o s i t i o n and momentum), l i k e w i s e t h e t r u t h v a l u e s 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  49  about t h e real-number v a l u e s o f a system's magnitudes depend upon t h a t system's s t a t e .  For example, when t h e pendulum d e s c r i b e d  i n the state  ,  w^  i n Section A i s  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 o f t h e pendulum i s between 19-20 kg m /sec . t h e pendulum i s i n t h e s t a t e true.  w  2  ,  And when  the t r u t h value of that p r o p o s i t i o n i s  The f a c t t h a t a system's s t a t e d e t e r m i n e s t h e t r u e v a l u e s o f  p r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e real-number v a l u e s o f t h e 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 t h e P^ s t r u c t u r e o f t h e s e 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 d e s c r i b e d i n two ways. B o t h ways s h a l l be e l a b o r a t e d , t  1  even though each y i e l d s t h e same n o t i o n o f s t a t e - i n d u c e d a  P^yj .  For one way makes use o f c o n c e p t s i n t r o d u c e d  u l t r a v a l u a t i o n s on i n S e c t i o n A and  t h u s shows t h e c o n t i n u i t y between t h e s t a t e ' s d e t e r m i n i n g t h e real-number v a l u e s o f magnitudes and t h e s t a t e ' s d e t e r m i n i n g t h e t r u t h v a l u e s o f propositions.  And t h e o t h e r way makes use o f the concept o f 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 r e c u r s As d e s c r i b e d preserves  +  and  *,  i n S e c t i o n A, each s t a t e - i n d u c e d i . e . , each s t a t e - i n d u c e d  r e a l - v a l u e d and homomorphic. t h e Boolean s t r u c t u r e and  homomorphic.  and f o r any = 0  f  i n C h a p t e r s V, VI and V I I .  P„„ CM  mapping  mapping on an  i f w € W^,  S,  o f idempotent elements o f an  w(f ) = f (w) =1 p  where  W  p  F  I t f o l l o w s t h a t each s t a t e - i n d u c e d  For by t h e d e f i n i t i o n o f t h e mapping on  w : F  i f w € W  p  U W^=  2.  p  Thus  w : P^  F..„ CM w,  ,  -> R  is mapping on  i s bivalent f o r any  and  -+ {0,1},  w € S2  w(f ) = f (w) p  p  and i n  o t h e r words, each pure s t a t e o f a c l a s s i c a l system i n d u c e 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 mapping ^CM  °^  P^  a s e  s  P  a c e  w : P^  ->- {0,1}  on t h e p r o p o s i t i o n a l s t r u c t u r e  a s s o c i a t e d w i t h t h e system.  ;  50  I n f a c t , as we would e x p e c t , each s t a t e - i n d u c e d i s an u l t r a v a l u a t i o n w h i c h a s s i g n s elements i n  P  elements i n  P ^ ,  and a s s i g n s the v a l u e  atom  i n the  function  f  set  as shown n e x t . PQ  ultrafilter  : P > P } w UF  any p o i n t  w € S2  that i s ,  f • 5 f_  IFF  1  and  .  and ir  P < P"" . w  w  s p e c i f i e s a unique  P_„ , CM  in  P ;  {w}.  And  ;  the  namely the unique  d u a l l y , the  i s the unique u l t r a i d e a l UI ^ w  1  P € UI  w € S2  set  W  "{P € P „ : P < P""} CM w  W  of  namely, t h e o n e - p o i n t idempotent  i s an u l t r a f i l t e r  d e f i n e d by the atom W  J  52,  M  t o the d u a l u l t r a i d e a l o f  o r t h e c o r r e s p o n d i n g s i n g l e t o n B o r e l subset  w  {P € P „ „ CM  w, ' f o r any  0  t o an u l t r a f i l t e r  Each p o i n t  structure of  H  1  the v a l u e  P^  mapping on a  or  f o r any B o r e l subset {w} c W_  r  ,  c S2,  Wp  and a l s o  dual to  w € W  UF  w € Wp IFF  r  P  w  .  Now 5  IFF W  for  < P ,  P,  i.e.,  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 o f t h e mapping  element  P € P„„ CM  ,  wCP)  Hence, each s t a t e - i n d u c e d  =1  if  P € UF  and  w  P_„ CM  mapping on a  w(P)  = 0  if  i s an u l t r a v a l u a t i o n  so i s t h e c l a s s i c a l - m e c h a n i c a l analogue o f a s t a n d a r d v a l u a t i o n  of  5 classical-propositional logic. state-induced  And  0, 1  thus the  P„ CM  u l t r a v a l u a t i o n s t o t h e elements o f  as t h e t r u t h v a l u e s f a l s e and  v a l u e s a s s i g n e d by can be  W  the  interpreted  true.  T h i s n o t i o n o f the s t a t e - i n d u c e d  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 .  A c c o r d i n g t o the m a t h e m a t i c a l f o r m a l i s m o f 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  o f a system can  r e g a r d e d as i n d u c i n g 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 t h e s t r u c t u r e o f t h e system's phase space..  A measure  f u n c t i o n on a Boolean a l g e b r a , e.g.,  ?^  conditions:  on  ,  n  is a  be  P  real-valued  w h i c h s a t i s f i e s the  following  51  (ua)  F o r any c o u n t a b l e s e t { P . } . - , o f d i s j o i n t elements o f P „ 1 i€Index CM M-CV P.) = 1 |i.(P.). T h i s i s t h e 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 °°,  (LIC)  M-(O) = 0.  W  T  It follows that (M-i)  If P  1  H  5 P  2  P € P  f o r every  .  C M  i sisotone, i . e . , ,  then  < n(P ),  \J,(? )  €  f o r any  2  ±  P ^  ( S i k o r s k i , 1960, p. 1 0 ) . 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 e v e r y element u : P „ „ -»• [0,1] , CM  where  the real-number l i n e .  P €?  C  M  ,  0 5 (i(P) 5 1,  that i s ,  [0,1] i s t h e c l o s e d i n t e r v a l from  0  to 1  on  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 condition: (Lid)  n(P ) - ( y ( P ) ) 2  2  = 0,  f o r every  P C  . P  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 Proof:  S i n c e e v e r y element  P € P  i s bivalent. 2  C M  i s idempotent, i . e . ,  P  = P,  2  condition Thus  (|od)  ji(P) = 1  y i e l d s the equation: or  LICP)  = d-i(P)) ,  f o r every  P € P  C M  .  0. Q.E.D. (Bub, 1974, p. 6 0 ) . 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 t h e f o l l o w i n g p r o o f due t o  Gudder (though Gudder does n o t r e f e r t o a Boolean s t r u c t u r e l i k e F i r s t , i t i s easy t o show t h a t , f o r any P > 1  H (P„ V P J = [i. ( P , ) + n C P j - V. CP, A P ). P r o o f : W  l  2  W  l  W  z  W  l  ^  p 2  P  ).  ^ ^CM '  The j o i n  P  V P  J-  ^  of  52  any  € P  P ,P  can be w r i t t e n as t h e j o i n o f t h r e e m u t u a l l y P  e l e m e n t s , e.g., P^ = P  V ?  1  A P^"; and P  2  = P  2  = P  g  g  V P^ V P  A P  .  2  , where  5  +  ^w  =  w  ^w 3  =  ( P  l  (P  V  V  V P  5^  ^w 5  +  ( P  (P„ V P ) = (i (P ) + 1 2 w 1  w  J  V l V  ^w 4 Y  +  (P  )  = ^w  A P^;  ±  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 :  r  +  = ?  g  Then by a d d i t i v i t y ,  [i (P„ V P ) = |J. ( P _ ) + | i (P, ) + | i C P ) . w 1 2 w 3 w 4 w 5  •VV  P  disjoint  C P  J  ^w V ^w V ^w V (  =  V  +  CP ) - (i ( P A P„). 2 "w 1 2  ^w  (  +  C P  l  A  V*  Q.E.D.  (  +  ThuS  With t h i s r e s u l t , i t ' p. : ?^  i s easy t o 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  i s homomorphic, i . e . , f o r any P, P„ ,P„ € ' P _ „ , ' ii (P*") = ( u ( P ) ) 1 2 CM w w pi (P„ V P ) = (j. (.P ). V u. C P J . P r o o f : F o r any P € P „ , w 1 2 w 1 w 2 CM  X  {0,1}  and  M  p, (P V P""") = | i ( 1 ) = 1; W  and by a d d i t i v i t y ,  W  1 = u, ( P ) + n ( P ) ,  Hence  and so  x  W  (J^CP) = 0  W  |i, CP V P ) = u. (P) + u. (P" "). w w w x  u. (P^) = 1 - u. (P) = (u. ( ? ) ) . X  W  W  o r 1,  f o r every  P  V  P  2^  = 1  ,  a  n  d  i  E d i t i o n , f o r Subcase l a , assume  n  Then s i n c e  P^ A P  u. (P„ ) = 1 w 1  and a l s o  2  < P^  and P^ A P  p. ( P . ) = 1. w 2  2  < P  Hence  p, (P. A P . ) = 0. w 1 2 = p. ( P . ) + p. ( P . ) - p, (P. A P . ) , e i t h e r w 1 w 2 w 1 2  For Subcase l b , assume  else  p (P„) = 0 "w 1  Now  W  P € P ^ so i n t h e n e x t p a r t o f t h i s .proof,  t h e r e a r e two c a s e s , one o f w h i c h has two s u b c a s e s . ^w^ l  4  and p ( P j = 1. ^w 2  F o r Case 1, assume (-^(P^ A P ) = 1 . 2  , by c o n d i t i o n (pi.) we have 1  2  p, (P. v P ) = p, (P ) v p, ( P . ) . *w 1 2 *w 1 w 2 Then s i n c e p, (P. ) = 1 w 1  Hence, '  u, (P. v P.) w 1 2 and p ( P ) = 0 o r w 2 0  p (P. v P ) = p (P„) V p ( P _ ) = 1. *w 1 2 *w 1 *w 2  For case 2, assume  u, (P„ V P„) = 0. Then s i 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 o f a c l a s s i c a l system i n d u c e 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 the  p, : P ^ -* {0,1}  which i s a b i v a l e n t homomorphism on  P „ „ s t r u c t u r e o f t h e phase space a s s o c i a t e d w i t h t h e system. CM  53  Moreover, each  u, : P„„ w CM  and i s i n f a c t t h e u l t r a v a l u a t i o n shown n e x t .  2  i s an u l t r a v a l u a t i o n on  w : P  -* {0,1}  2  i s s a i d t o i n d u c e t h e measure. 2  assigns p r o b a b i l i t y  t h e atom  above, as  in  P  subset o f p o i n t s i n  of  i s an a t o m i c measure c o n c e n t r a t e d on a s i n g l e p o i n t  (Bub, 1974, p. 4 7 ) , namely t h e p o i n t  of a  described  P_„ CM  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 a l g e b r a  B o r e l subsets o f a in  {0,1}  )  w  representing  That i s , e a c h 1  u, w  P_„ CM  t o the s i n g l e t o n subset  and a s s i g n s p r o b a b i l i t y  2.  on t h e  0  t h e s t a t e which structure  {w}  (which i s  t o every other s i n g l e t o n  Now s i n c e  u, (P ) = 1 and s i n c e u, preserves ww w t h e ~*~ o p e r a t i o n as shown above, i t f o l l o w s t h a t u. (? ) = 1 - 0. Then ww s i n c e u, i s i s o t o n e , we have, f o r any P € P_ , i f P > P then "w . CM w x  X  r  w  u, (P) = 1, w  and i f P < P  then  X  w  u, CP) = 0. 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 f o r any  P € P„„ , CM  J  H (P) = 1 w  P  i f P € -UF  w  = {P € ?  i f P € U l = {P € "P„, : P 5 P } . w CM w  on  And c l e a r l y , each  M  .  described  x  above; c o n v e r s e l y ,  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 u l t r a v a l u a t i o n , the  0,1  CM  n t t  So each  [i.^ i s t h e v e r y mapping each mapping  state-induced,  a s s i g n s v a l u e s as f o l l o w s :  u, (P) = 0 ^w P^  Thus each  w : P  P ^ .  v a l u e s a s s i g n e d by  : P > P } w (J, w  i s an u l t r a v a l u a t i o n  w : P^  -* {0,1}  and  •+  {0,1}  is a  A l s o , s i n c e each (i w  i s an  t o t h e elements o f  PCM  can be i n t e r p r e t e d as t h e t r u t h v a l u e s 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 d e t e r m i n e s t h e t r u t h v a l u e s o f the p r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e real-number v a l u e s o f t h e system's magnitude i s f o r m a l i z e d v i a t h e n o t i o n o f s t a t e - i n d u c e d on t h e  P  ultravaluations  . s t r u c t u r e o f t h e phase space a s s o c i a t e d w i t h t h e system.  t h i s state-induced  And  p r o c e d u r e o f a s s i g n i n g t r u t h v a l u e s t o t h e elements o f a  propositional structure  P  works e x a c t l y l i k e t h e p r o c e d u r e by which  54  L  t r u t h v a l u e s a r e a s s i g n e d t o t h e elements o f an  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 a n a l o g y between t h e s t a t e - i n d u c e d P  on a  L  and t h e u l t r a v a l u a t i o n s on an  ultravaluations  s u g g e s t s , f o r example, t h a t  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 a s s o c i a t e d phase space the on  we  underlying  s t r u c t u r e diagrammed i n C h a p t e r 11(B) so t h a t each u l t r a v a l u a t i o n I.^ i s i n d u c e d by a s t a t e o f t h e 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 t h e 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' t h e c o n v e n t i o n t h a t we r e a d t h e bottom f a c e o f t h e d i e as t h e outcome o f a throw and t h u s as t h e s t a t e o f t h e d i e .  The phase space a s s o c i a t e d with- t h e d i e c o n s i s t s o f f o u r Si  = {w ,w ,w ,w },  2.  1  U  the d i e . we may  O  each r e p r e s e n t i n g  points  one o f t h e f o u r d i s c r e t e s t a t e s o f  *+  In order that  be t h e p r o p o s i t i o n a l s t r u c t u r e o f t h i s  i n t e r p r e t t h e element  /R/  €  as t h e p r o p o s i t i o n :  ,  "A number l e s s  7  t h a n t h r e e appears (on t h e bottom f a c e o f t h e d i e ) . " a s s o c i a t e d w i t h t h e idempotent f u n c t i o n  f  : °J R  follows:  f o r any  w. 1  f (w.) = 0 R i /S/ €  € S2. ,  f_(.w.) = 1  0  K  i f w. € {w, ,w i X 2  as t h e p r o p o s i t i o n :  W  i  w.  ( S  ^ ^ 2 %^" W  9 V  ,  f (w.) = 1 e  i f w.  0  1  And we may  as  f  € {w ,w }  and  /.  i n t e r p r e t t h e element  "An odd number a p p e a r s . "  i f w.  defined  € {w, ,w }  1  i s a s s o c i a t e d w i t h t h e idempotent f u n c t i o n any  -> {0,1}  U  1  =•' {w ,w }. 3 •H  This p r o p o s i t i o n i s  This  proposition  d e f i n e d as f o l l o w s : and  Each, o f t h e f o u r u l t r a v a l u a t i o n s on  f (w.) = 0 is  for  i f  state-induced  55  because i t i s t h e s t a t e o f t h e d i e w h i c h 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 d u a l 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 o f t h e p o s t u l a t e d system i s t h e c l a s s i c a l - m e c h a n i c a l  analogue o f t h e i n i t i a l assignment o f s p e c i f i e s an atom i n  ,  0, 1  values t o  R  and  S  which  as d e s c r i b e d i n Chapter 11(D).  F i n a l l y , by t h e s e m i - s i m p l i c i t y o f t h e Boolean s t r u c t u r e t h e c o l l e c t i o n o f s t a t e - i n d u c e d u l t r a v a l u a t i o n s on a  P  ,  i s complete.  Thus t h e complete c o l l e c t i o n o f s t a t e - i n d u c e d u l t r a v a l u a t i o n s on a  P  can be r e g a r d e d as a s t a t e - 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 s e m a n t i c s for  P  .  This state-induced  semantics f o r  p r e c e d e n t f o r a proposed s t a t e - i n d u c e d  P^  s h a l l be r e g a r d e d as t h e  s e m a n t i c s f o r t h 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 , as d e v e l o p e d i n Chapter V I .  Notes I u s e t h e 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 " s e n t e n c e " o r " s t a t e m e n t " c o u l d s e r v e as w e l l . 1  way;  2  2  2  As suggested by R. E. R o b i n s o n , t h e u n i t s kg m / s e c , which h e l p make sense o f t h e real-number v a l u e s , may be c o n s i d e r e d t o be p a r t o f 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 t h e 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 r e l ) s u b s e t R c R, t h e s e t W o f a l l p o i n t s w € 2 such t h a t f ^ ^ ^ ^ ^ f a B o r e l s u b s e t o f 2. ( T h i s s e t W i s t h e i n 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 t h e s u b s e t s R £ R and W c 2 r u l e s out s e t s such as t h e s e t o f 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 t h e measure o f t h i s s e t cannot be e x p r e s s e d as a c o u n t a b l e u n i o n o r sum o f t h e measures o f each o f t h e s e t ' s elements. A s i n g l e t o n , o n e - p o i n t s e t i s a B o r e l s e t o f measure 0. w  s  x  t  s  e  l  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 s u b s e t s o f 2 t h a n t h e c l a s s o f B o r e l s u b s e t s , see ( J a u c h , 1968, p. 7 9 ) .  56  Bub d e s c r i b e s t h i s c o n n e c t i o n 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 d e f i n e s a b i v a l e n t homomorphism by t h e S i k o r s k i d e f i n i t i o n , as d i s c u s s e d i n Chapter I l C C ) . J  The domain o f 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 , o r a l g e b r a o f s e t s , i n p a r t i c u l a r , t h e Boolean a l g e b r a o f B o r e l s u b s e t s o f c l a s s i c a l phase space. However, M. S t r a u s s , I . S e g a l , and o t h e r s argue t h a t t h e ( i s o m o r p h i c ) Boolean a l g e b r a o f 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 t h e domain o f t h e measures o f p r o b a b i l i t y t h e o r y ( S t r a u s s , 1973, p.,. 268; S e g a l , 1954, p. 721). S i m i l a r l y , Gleason proposes t h a t we may r e g a r d h i s quantum measures as b e i n g d e f i n e d on t h e s e t o f idempotent o p e r a t o r s on a H i l b e r t space r a t h e r than t h e s e t o f subspaces o f H i l b e r t space ( G l e a s o n , 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 t h e 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 w h i c h makes a s s e r t i o n s about what appears a f t e r a throw o f t h e d i e i s a m o l e c u l a r c o m b i n a t i o n o f R, S. L e t L l a b e l t h e c l o s e d , denumerable s e t o f a l l m o l e c u l a r c o m b i n a t i o n s o f R, S. We have no e q u i v a l e n c e r e l a t i o n w i t h w h i c h t o p a r t i t i o n L i n order t o get t h e e q u i v a l e n c e c l a s s e s w h i c h a r e t h e elements o f L . S t r i c t l y s p e a k i n g , i t i s t h e elements o f L w h i c h 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 S2 . However, l e t e v e r y P € L 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 f p (where fp(w) i s t h e t r u t h v a l u e o f P g i v e n w), and say t h a t , f o r any P, Q € L , P ~ Q IFF f ( w ) = fq(w) f o r e v e r y w € & , where i f fp(w) = fq(w) f o r every w € '&„ , then f p = 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 o t h e r words, P ~ Q IFF P, Q have t h e same t r u t h t a b l e , w h i c h i s t h e semantic c o u n t e r p a r t o f t h e p r o o f t h e o r e t i c e q u i v a l e n c e r e l a t i o n s t a t e d i n Chapter 1 1 ( B ) . 2  2  2  2  0  2  p  2  Q  57 CHAPTER IV THE NON-BOOLEAN PROPOSITIONAL S e c t i o n A.  STRUCTURES DETERMINED BY QUANTUM MECHANICS  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 e x t r e m e l y s i m p l i f i e d e x p o s i t i o n o f some o f t h e m a t h e m a t i c a l f o r m a l i s m o f quantum mechanics.  H whose d i m e n s i o n a l i t y r e f l e c t s  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 t h e degrees o f freedom o f t h e system.  Each magnitude  represented by a s e l f - a d j o i n t operator A  I t i s postulated that a physical  A  A  o n t h e system's  o f t h e system i s H.  The o p e r a t o r  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 t h e c a s e o f a d i s c r e t e s p e c t r u m ) : A = / a.P. , where f o r each i € Index, I i and A|\|A.> = a.|\|f.>. • 1 x i  (I)  T  P. i  Y  = !>!/•. ><\lr. I  1T  The r e a l numbers ^ £ ^ g j a  n (  j  a  r  e  c a  e x  l l e d t h e e i g e n v a l u e s o f A and o f A.  They  a r e t h e real-number v a l u e s and t h e o n l y real-number v a l u e s e x h i b i t e d by t h e 1 magnitude  A.  A pure s t a t e \Jr o f a quantum system i s r e p r e s e n t e d by a v e c t o r  |\|f> i n t h e system's H o r by a d e n s i t y o p e r a t o r P^ = |\|/><\|/| on H.  The o p e r a t o r  P^. i s s e l f - a d j o i n t and i d e m p o t e n t , t h a t i s , P^, i s a p r o j e c t i o n o p e r a t o r w h i c h 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 d e s i g n a t e d P, P^, P^, e t c . Each p r o j e c t o r P on an H c o r r e s p o n d s u n i q u e l y t o a subspace H o f H, where a subspace i s a s e t o f v e c t o r s w h i c h form a c l o s e d l i n e a r m a n i f o l d ( s e e Bub, 1974, pp. 10, 12).  The p r o j e c t o r s  >  a  n  d t  h  vectors { k  e  i 6 I n d e x  i  >  >  i € I n d e x  appearing i n  A.  t h e 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 o p e r a t o r A r e p r e s e n t t h e ( p u r e ) e i g e n s t a t e s o f A and o f A.  The s e t o f e i g e n s t a t e s o f any A a r e m u t u a l l y o r t h o g o n a l ( a s  d e f i n e d 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 i ^ i l | ^ > = l | ^ > f o r every |\|/-> € H.) 1  i d e n t i t y operator which s a t i s f i e s  i s the  58  The s t a t e o f a quantum system d e t e r m i n e s t h e real-number v a l u e s o f the  system's magnitudes v i a t h e f o l l o w i n g f o r m a l i s m .  eigenstate  |YJ>»  f o r some  real-number v a l u e o f eigenstate  A  j € Index,  i n an a r b i t r a r y pure s t a t e  A  A  i s the eigenvalue  then the  a f f i l i a t e d with that  A | Y J > = a,. J.1'Y_.> •  But when a system i s A,  t h e n upon  may e x h i b i t any o f i t s real-number e i g e n v a l u e s ;  quantum f o r m a l i s m does n o t s p e c i f y w h i c h e i g e n v a l u e  However, f o r any pure s t a t e of  a.  \|/ w h i c h i s n o t an e i g e n s t a t e o f  measurement t h e magnitude the  A,  o f t h e magnitude  i s the eigenvalue  | > | r b y the equation  When a system i s i n an  A  will  exhibit.  \|/y t h e p r o b a b i l i t y t h a t t h e real-number v a l u e  a.. ,  f o r some  j € Index,  i s d e t e r m i n e d by t h e  quantum f o r m a l i s m :  P  (II)  ( a t j A  j  )  =  l^j l ^ l  2  = ^ k j ' x t j U>  = oH^.I^ •  T h i s p r o b a b i l i t y i s a real-number i n t h e c l o s e d i n t e r v a l 1  real-number l i n e .  The p r o b a b i l i t y e q u a l s  i n the eigenstate  \|/y a f f i l i a t e d with the eigenvalue  ( a . ) = |<\|f. |\|r.>|2 = 1 .  p'  a.. , 0  t h e system i s i n any one o f t h e o t h e r e i g e n s t a t e s o f  A.  1  :  i.e., (impossibility)  1  The average v a l u e , i . e . ,  the expectation value, o f  system i s i n an a r b i t r a r y pure s t a t e w e i g h t e d sum o f e i g e n v a l u e s o f  (III)  o f the  ( c e r t a i n t y ) I F F t h e system i s  And t h i s p r o b a b i l i t y e q u a l s  %,A  IFF  [0,1]  Y  when t h e  i s d e f i n e d as t h e f o l l o w i n g  A:  Exp, ( A ) = I a.p: ( a . ) = £ ( a . ) o | f l ^ ^ j ^ A  =  A  <Y|I(a )|Y ><^ l|i> i  i ='<Y|A|Y>.  i  1  59  And when t h e system i s i n an e i g e n s t a t e value of  A  i s the eigenvalue  of  A,  then the expectation  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 e x p e c t a t i o n v a l u e o f a p r o j e c t o r a c c o r d i n g t o ( I I I ) , i . e . , f o r any pure A  state  \Jr and f o r any magnitude  A,  A  Ca_.) = E x p ^ C P ^ ) ,  the p r o j e c t o r r e p r e s e n t i n g t h e e i g e n s t a t e a f f i l i a t e d a_. .  In f a c t ,  A  where  P^  is  with the eigenvalue  i n s t e a d o f moving from t h e p r o b a b i l i t y e x p r e s s i o n ( I I ) t o  t h e e x p e c t a t i o n v a l u e e x p r e s s i o n ( I I I ) , t h e f o r m e r e x p r e s s i o n ( I I ) can be d e r i v e d from ( I I I ) , as i s done, f o r example, by M e s s i a h (1966, pp. 176-179). In o t h e r words, ( I ) t o g e t h e r w i t h e i t h e r ( I I ) o r ( I I I ) a r e r e g a r d e d as t h e f o u n d a t i o n a l p o s t u l a t e s o f quantum mechanics.  F o r example, von Neumann  c o n s i d e r s ( I I ) t o be t h e more g e n e r a l p r o b a b i l i t y e x p r e s s i o n b u t he r e g a r d s (III) pp.  t o 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. 200-206).  S e c t i o n B.  Incompatibility 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 t h e s i m u l t a n e o u s m e a s u r a b i l i t y o f any s e t o f magnitudes  ^i^icindex  i s t h a t each magnitude i s e q u a l t o a ( B o r e l ) f u n c t i o n o f some common magnitude, say B;  t h a t i s , f o r each-  Borel function  (Kochen-Specker, 1967, p. 6 4 ) . Now f o r any magnitude  B  g^  and any B o r e l f u n c t i o n  g,  i € Index,  t h e magnitude  magnitude which e x h i b i t s t h e v a l u e  A^ = g^(B) f o r some  g ( B ) i s by d e f i n i t i o n t h a t  g ( b ) when  B  exhibits the value  So when t h e real-number v a l u e o f t h e common magnitude real-number v a l u e o f each  A. = g.(B) i s g . ( b ) . i  of  B  i  B  i s b,  b.  then t h e  Hence a s i n g l e measurement  l  s u f f i c e s t o determine t h e real-number v a l u e s o f a l l t h e magnitudes  60  ^i^i€lndex "  F o r example, a s mentioned i n Chapter I I I ( A ) , e v e r y 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 o f t h e p o s i t i o n and/or momentum magnitudes, and s o a l l c l a s s i c a l magnitudes a r e s i m u l t a n e o u s l y measurable.  But i t i s  not t h e case t h a t e v e r y quantum magnitude i s a f u n c t i o n o f t h e p o s i t i o n and/or momentum magnitudes.  Moreover, t h e quantum p o s i t i o n and momentum  magnitudes a r e t h e m s e l v e s n o t s i m u l t a n e o u s l y measurable.  And i n g e n e r a l ,  t h e s e t o f 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 n o t s i m u l t a n e o u s l y measurable. With r e s p e c t t o t h e ( s e l f - a d j o i n t ) o p e r a t o r r e p r e s e n t a t i o n o f t h e quantum magnitudes, a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e s i m u l t a n e o u s m e a s u r a b i l i t y o f any magnitudes i s t h e c o m m u t a t i v i t y o f t h e i r r e p r e s e n t a t i v e A*  operators.  Any o p e r a t o r s  e i g e n s t a t e s i n common.  A\  A, B  A  A*  commute I F F A, B  have a l l t h e i r  Moreover, any s e t ^ i ^ i g i i e x A  n <  °^ ° P  e r a  "  t o r s  ^  s  m u t u a l l y commutative I F F t h e r e i s an o p e r a t o r B and B o r e l f u n c t i o n s {g.}., , such t h a t A. = g.(B) = g . ( B ) , f o r e v e r y i € Index (von Neumann, x x€Index x x x 6  J  T  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  As  B  has t h e o p e r a t o r  B,  -^^ Vs-  then  g ( B ) has t h e o p e r a t o r  (von Neumann, 1932, p. 204; Fano, 1971, p. 394).  g, i f  As  g(B) = g(B).  Thus i t f o l l o w s t h a t any  quantum magnitudes a r e s i m u l t a n e o u s l y measurable I F t h e i r r e p r e s e n t a t i v e o p e r a t o r s a r e m u t u a l l y commutative; t h e c o n v e r s e i s a l s o shown by von Neumann (1932, pp. 223-228). Commuting o p e r a t o r s and s i m u l t a n e o u s l y measurable magnitudes a r e s a i d t o be c o m p a t i b l e ; such o p e r a t o r s o r magnitudes have a l l t h e i r e i g e n s t a t e s i n common.  O p e r a t o r s w h i c h do n o t commute and magnitudes which a r e n o t  s i m u l t a n e o u s l y measurable .are s a i d t o be i n c o m p a t i b l e ; such o p e r a t o r s o r magnitudes may n e v e r t h e l e s s have one o r s e v e r a l e i g e n s t a t e s i n common so  61  t h a t one o r s e v e r a l o f t h e i r e i g e n v a l u e s  may be s i m u l t a n e o u s l y  determined  by measurement. When, we t a l k o f a p r o p o s i t i o n a l s t r u c t u r e d e t e r m i n e d by quantum mechanics, t h e p r o p o s i t i o n s we c o n s i d e r a r e p r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e real-number e i g e n v a l u e s  o f quantum magnitudes.  P r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e e i g e n v a l u e s magnitudes a r e s a i d t o be c o m p a t i b l e . about t h e e i g e n v a l u e s  o f incompatible  with the f o l l o w i n g exception.  o f 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 magnitudes a r e s a i d ' t o be  I f t h e eigenvalues  incompatible  happen t o be a s s o c i a t e d .  w i t h e i g e n s t a t e s w h i c h a r e s h a r e d i n common by t h e i n c o m p a t i b l e  magnitudes,  t h e n 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 e i g e n v a l u e s o f 2  incompatible  magnitudes a r e s a i d t o be c o m p a t i b l e .  truth-values t o incompatible  The attempt t o a s s i g n  quantum p r o p o s i t i o n s i s a p r o b l e m a t i c  enterprise,  as w i l l be shown i n C h a p t e r 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 t h e c l a s s i c a l c a s e d e s c r i b e d operators  i n Chapter I I I , the s e l f - a d j o i n t  r e p r e s e n t i n g quantum magnitudes have t h e b i n a r y r i n g A  +  and  every  •  d e f i n e d among them a s f o l l o w s :  |\|/> 6 H,  (A+B)|\|/> = &|\|r> + B|\|r>,  f o r any and  operations  A  A, B  on  H  and f o r  (A«B)|\|/> = A(B|\|/>).  The  +  o p e r a t i o n so d e f i n e d i s a s s o c i a t i v e and commutative, as u s u a l .  •  o p e r a t i o n so d e f i n e d 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 r e s p e c t t o  as u s u a l . B(A|\|f>) every  But  •  f o r every |\|r> € H,  i s n o t commutative, i . e . , A,B,|y>.  then  A, B  And t h e +,  A(B|\|/>) need n o t e q u a l  I n p a r t i c u l a r , i f A(B|\|A>) = B(A|\|r>) f o r a r e s a i d t o commute o r t o be c o m p a t i b l e .  s u g g e s t s t h a t a c l o s e d set. o f s e l f - a d j o i n t o p e r a t o r s  This  on. a H i l b e r t space has  62  t h e s t r u c t u r e o f a non-commutative r i n g - w i t h - u n i t whose constant operator  0  satisfying  0,|t  1-element i s the c o n s t a n t o p e r a t o r |^>  € H.  >  I  = 0,  f o r every  satisfying  0-element i s the \ty> € tf, and whose  'l|^> = 1,  f o r every  However, a s e t o f s e l f - a d j o i n t o p e r a t o r s i s not c l o s e d w i t h  respect to  •  unless  • i s r e s t r i c t e d t o commuting, i . e . , c o m p a t i b l e ,  operators.  For a l t h o u g h the sum  o f any two  a s e l f - a d j o i n t o p e r a t o r , t h e product  o f two  s e l f - a d j o i n t operators i s i t s e l f s e l f - a d j o i n t o p e r a t o r s i s not  i t s e l f a s e l f - a d j o i n t o p e r a t o r u n l e s s t h e two commute (von Neumann, p. 98).  1932,  So r a t h e r t h a n a non-commutative r i n g - w i t h - u n i t , a s e t o f  s e l f - a d j o i n t o p e r a t o r s r e p r e s e n t i n g quantum magnitudes which i s c l o s e d w i t h respect to  +  and  •  form a s t r u c t u r e w h i c h may  partial-dot-ring-with-unit +  i s d e f i n e d from  E><E  <E = {A,B, to  E,  and  . . .}, + , *  be c a l l e d a • ,0,1>,  i s d e f i n e d from o n l y  T a k i n g t h i s n o t i o n o f r e s t r i c t i n g the b i n a r y partial-operation  d e f i n e d from o n l y  i  to  where  E  one  •  where  again,  +  and t h e  o c E.xE.  •  o  E*E,  to  E.  operation to a  s t e p f u r t h e r , we  d e f i n e t h e s t r u c t u r e o f t h e s e l f - a d j o i n t o p e r a t o r s t o be a w h i c h has b o t h the  cb c  may  partial-ring-with-unit  o p e r a t i o n s d e f i n e d from o n l y  A>  to  E,  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 a l g e b r a .  And t h e y d e f i n e t h e s t r u c t u r e o f a  quantum system's magnitudes, which a r e r e p r e s e n t e d  by and presumably r e f l e c t  the s t r u c t u r e o f s e l f - a d j o i n t o p e r a t o r s on t h e system's H i l b e r t s p a c e , as a partial-algebra;  i . e . , as a p a r t i a l - r i n g - w i t h - u n i t  But r e g a r d l e s s o f the e x a c t s t r u c t u r i n g  i n my  terminology.  o f the  self-adjoint  operators, 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 o f quantum mechanics i s d i f f e r e n t from t h e s t r u c t u r e o f t h e 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 t h e magnitudes o f c l a s s i c a l mechanics.  63  Thus i t i s r e a s o n a b l e t o expect t h a t t h e s t r u c t u r e o f t h e quantum 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 t h e real-number e i g e n v a l u e s quantum magnitudes i s d i f f e r e n t from t h e Boolean  P  ofthe  structure of c l a s s i c a l  propositions. Nevertheless,  t h e p r o c e d u r e 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 t h e m a t h e m a t i c a l f o r m a l i s m i s e x a c t l y analogous t o t h e procedure by which a from t h e c l a s s i c a l f o r m a l i s m ,  as described  P  o f quantum mechanics structure i s abstracted  i n Chapter I I I ( B ) .  F o r 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 t h e p r o j e c t o r s ( i . e . , idempotent, s e l f - a d j o i n t o p e r a t o r s ) c o r r e s p o n d i n g subspaces o f H.  H  on an  or with the uniquely  And t h e l o g i c a l o p e r a t i o n s  "and," " o r , "  " n o t , " e i t h e r have been i n d i r e c t l y d e f i n e d i n terms o f t h e p r o j e c t o r and  •  operations  intersect  A,  be d e s c r i b e d  +  o r have been d i r e c t l y a s s o c i a t e d w i t h t h e subspace  span  v,  shortly.  and o r t h o c o m p l e m e n t a t i o n  o p e r a t i o n s , as w i l l  _ L  These a s s o c i a t i o n s d e t e r m i n e t h e s t r u c t u r e o f a s e t  o f quantum p r o p o s i t i o n s , o r i n von Neumann's t e r m s , t h e s e a s s o c i a t i o n s determine "a s o r t o f l o g i c a l c a l c u l u s " o r 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. 2 5 3 ) .  In h i s 1932 book, von Neumann d i s c u s s e s c l a s s i c a l and quantum p r o p o s i t i o n s under t h e c a t e g o r i c a l l a b e l : system.  properties of the state ofthe  That i s , von Neumann's p r o p e r t i e s a r e i n f a c t p r o p o s i t i o n s w h i c h  make a s s e r t i o n s about t h e real-number ( e i g e n ) v a l u e s (von Neumann, 1932, p. 249). F o r example: +3gn.  The  s  Pi  Von Neumann argues t h a t each s u e h p r o p o s i t i o n x  o f a system's n  or x  0  a n  electron i s  canf.be a s s o c i a t e d  a magnitude which i s d e f i n e d such, t h a t i t s v a l u e i s 1 i s v e r i f i e d and  "  magnitudes  i f t h e p r o p o s i t i o n i s not v e r i f i e d .  with  i f the proposition I n o t h e r 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 t h e real-number e i g e n v a l u e s  of  64  a quantum system's magnitudes can i t s e l f be r e g a r d e d a s o r a s s o c i a t e d an idempotent magnitude o f t h e system. represented  with  S i n c e an idempotent magnitude i s  by a p r o j e c t o r on t h e 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 c o r r e s p o n d s u n i q u e l y t o a subspace o f t h a t H i l b e r t s p a c e , namely, t o t h e subspace onto w h i c h - t h e p r o j e c t o r p r o j e c t s e v e r y v e c t o r i n H i l b e r t s p a c e , von Neumann c o n c l u d e s t h a t quantum p r o p o s i t i o n s can be a s s 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 o r e q u a l l y w e l l w i t h subspaces o f a H i l b e r t space. For example, c o n s i d e r a p r o p o s i t i o n which a s s e r t s t h a t t h e v a l u e o f t h e magnitude  A  i s i n some B o r e l subset  R  o f t h e real-number l i n e .  Such p r o p o s i t i o n s a r e r e g a r d e d by most a u t h o r s as t h e paradigm quantum ( o r classical), propositions. A  are i t s eigenvalues,  As d e s c r i b e d above, t h e o n l y v a l u e s e x h i b i t e d by and each e i g e n v a l u e  i s uniquely associated  with  a projector  P. = |\^.><^.|. So depending upon how many e i g e n v a l u e s o f A ^ i are i n t h e B o r e l subset R, t h e 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 t h e unique p r o j e c t o r  P. and i t s c o r r e s p o n d i n g subspace H. when ™i n i a. € R, o r t h e unique p r o j e c t o r £ i , corresponding n i=l i V H, when s e v e r a l a ,a ( R, ._, t . 1' n i=l i Y  o n l y one  p  1  subspace  a  n  d  1 T S  V  T  A l l o t h e r a u t h o r s who d i 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 o r a quantum l o g i c a l calculus', a l s o somehow o r o t h e r 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 t h e p r o j e c t o r s on a H i l b e r t space o r t h e subspaces o f a H i l b e r t space.  So t h e s t r u c t u r e o f t h e p r o j e c t o r s on a H i l b e r t s p a c e ,  or i s o m o r p h i c a l l y , t h e s t r u c t u r e o f t h e subspaces o f t h a t H i l b e r t space, i s r e g a r d e d as t h e p r o p o s i t i o n a l s t r u c t u r e d e t e r m i n e d by quantum mechanics, labeled  P  QM  = <E - { P . P ^ P ^ . . . . },A,5,A, v,+,0,l>.  The elements o f  P  QM  may be thought o f e i t h e r a s p r o j e c t o r s o r a s subspaces o f a H i l b e r t space;  65  t h e elements o f P PSP^JP  The P  2  ways.  represent  or are associated with  propositions  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  But b e f o r e d e s c r i b i n g t h e s e two ways i n t h e next s e c t i o n , t h e  f e a t u r e s o f P_  w h i c h a r e common t o b o t h f o r m a l i z a t i o n s a r e f i r s t  w  QM  described, '  as f o l l o w s . P^,«  A  i s an a t o m i c s t r u c t u r e whose atoms, w r i t t e n  QM  sometimes  P  P, o r  '  , a r e t h e o n e - d i m e n s i o n a l p r o j e c t o r s on H,  Y  e.g., P. =  o r t h e c o r r e s p o n d i n g o n e - d i m e n s i o n a l subspaces o f H, which i s t h e range o f P^ . i s then u l l projector distinguished entire  H.  |Y><YI»  y  cl  0  The d i s t i n g u i s h e d  e.g.,  t h e subspace  0-element o f a  P^  M  o r t h e c o r r e s p o n d i n g z e r o - s u b s p a c e o f H; t h e  1-element i s t h e i d e n t i t y p r o j e c t o r  As w i t h t h e L  and t h e P ^  I I and I I I , t h e 0-element o f a  P  I  o r the corresponding  structures described  i n Chapters  i s associated with impossible o r  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 t h e 1-element i s a s s o c i a t e d  with  c e r t a i n o r 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  P„ A P 1 2  F o r any P,,P. € P_„ 1 2 QM J  P  l l  P =  P  = P  2  l  A  ?  2  V Pg ..  2 2  9  P  22  =  P  Kochen-Specker, 1967, written  P„ _L P„ 1 2  f o r any P » ? 1  P cb.P 1  2  IFF P The  e 2  1  P  A  P  l "  ' 3 P  p. 65).  I F F P„ QM  =  '  i  f  J  P  A  P  P 2  2  (  P  a  J  A  U  C  H  >  a  2  r  e  D  I  S  J  O  L  N  T  t  h  e  , 9  ^ ^QM  p  S U G  2  1  ( P i r o n , 1976,  X  i' 2  P  P^ cb P  Any P ^ >  < 1 P  l  P  o p e r a t i o n s as  IFF there e x i s t three mutually  And assuming t h a t  2  . i s r e f l e x i v e , symmetric,  P ^ J 22' 3  d i s j o i n t ( i . e . , o r t h o g o n a l ) elements and  of P  i n terms o f t h e A , v,  and n o n - t r a n s i t i v e , and i s d e f i n e d follows.  A  compatibility relation  6  r  8  e  ^ ^at  i t can be shown t h a t > PP*  i^  P  2 8  »  9 7  J  d i s j o i n t o r orthogonal  p. 29). n  P^ = P ^ V P^  I t follows that, '  p 2  a  n  d  i -P^ •  binary relation  as u s u a l ( i . e . , P  1  < P  2  5  o f P^  IFF P P A P 1  , defined  M  2  = ?  ±  . and  P  ±  i n terms o f A  5 P  2  IFF ?  ±  V P  or V 2  = P ), 2  66  is  a p a r t i a l - o r d e r i n g Ci.e., i t i s r e f l e x i v e , anti-symmetric,  Moreover, the c o m p a t i b i l i t y o f any  p  s 2  ^ ^QM  P  1  i s  a  n  f o r t h e i r b e i n g r e l a t e d t o t h e p a r t i a l - o r d e r i n g 5, then  P  2  And discussed  S e c t i o n D.  ,  f o r any  P ^  6 P  .  Q M  f i n a l l y , the operations  The  The  e  s  s  a  y  r  condition  that i s , i f  5  P  2  3  A, V,--  P^  of  1  Partial-Boolean Algebra  M  are defined  and  P„„ QM  and the Orthomodular L a t t i c e Quantum  Structures  s t r u c t u r e has been f o r m a l i z e d i n two ways:  P„„ . QML I r e t a i n the l a b e l  orthomodular l a t t i c e  indiscriminately.  as a  J  PQ  t r a n s i t i v e , atomic, partial-Boolean algebra  (E).  c  transitive).  i n the next s e c t i o n .  Propositional  and  e  and  The  T  a n M A  d  as a c o m p l e t e , a t o m i c ,  These s t r u c t u r e s a r e d e f i n e d i n Chapter P_„ QM  P„__ QMA  to r e f e r to a  b a s i c d i f f e r e n c e between a  t h a t t h e f o r m e r has t h e b i n a r y o p e r a t i o n s c o m p a t i b l e elements w h i l e t h e l a t t e r has The two  A, v A, V  1(D)  P  or a  QML a n  d  a  1 S  d e f i n e d among o n l y d e f i n e d among a l l e l e m e n t s ,  c o m p a t i b l e and  incompatible.  f o r m a l i z a t i o n s do not d i f f e r  r e s p e c t t o any  o f the o t h e r e n t r i e s i n the o r d e r e d o c t u p l e  P  with  .  That t h 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 have been f o r m a l i z e d i n t h e s e two ways i s a t l e a s t p a r t l y due and t h e subspaces o f  H.  t o d i f f e r e n c e s between t h e p r o j e c t o r s  For d e s p i t e t h e one-to-one correspondence between  t h e p r o j e c t o r s and t h e subspaces, t h e a s s o c i a t i o n o f quantum p r o p o s i t i o n s with projectors n a t u r a l l y y i e l d s a  PQ  p r o p o s i t i o n s w i t h subspaces s u g g e s t s a  M A  w h i l e t h e a s s o c i a t i o n o f quantum PQ^  L  J  a s  w i l l be shown i n t h i s  section. I n h i s 1932  book, von Neumann p r o p o s e s a l o g i c a l c a l c u l u s o f  67  quantum p r o p o s i t i o n s w h i c h has "and" and " o r " r e s t r i c t e d t o c o m p a t i b l e propositions.  F i r s t von Neumann d e f i n e s " n o t . "  F o r 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 t h e ' p r o j e c t o r . Pwhose c o r r e s p o n d i n g subspace i s H, t h e p r o p o s i t i o n " n o t p"  c o r r e s p o n d i n g subspace i s E .  A  P  r o  P  o s  i  t :  i-  "Pj_  o n  whose  a  n  P2"  d  ^  s  a s s o c  i ' a  d  t e  with the projector  A  P^ • P^  whose c o r r e s p o n d i n g subspace i s  A  , where  A  among subspaces as t h e s e t - t h e o r e t i c i n t e r s e c t o p e r a t i o n . fp  A J_  S\  I - P = P  N e x t , f o r any c o m p a t i b l e p r o p o s i t i o n s  x  ^1 ' ^2 '  A  i s associated with the projector  or  p^"  i s e q u i v a l e n t t o "not ( ( n o t p^)  von Neumann a s s o c i a t e s  "p^  or p  the p r o j e c t o r  A  subspace i s t h e c l o s e d l i n e a r sum o f  H  A 2  , H  i s i n t e r p r e t e d a s t h e subspace span o p e r a t i o n .  - (P^ * P ) 2  ,  t h a t t h e Boolean o p e r a t i o n s operations  +, •.  operations  A, V , , 1  i.e.,  i s developed and d i s c u s s e d  ,  2  with  whose c o r r e s p o n d i n g V H  2  ,  where  V  Thus von Neumann's 1932 among p r o p o s i t i o n s  among p r o j e c t o r s i n t h e u s u a l way  a r e d e f i n e d i n terms o f t h e r i n g  But t h e b i n a r y "and," " o r " o p e r a t i o n s  only compatible propositions.  p^ , p  A  l o g i c a l c a l c u l u s has t h e "and," " o r , " " n o t " o p e r a t i o n s d e f i n e d i n terms o f t h e +, •  analogously,  2  A  I - (C.I-P ).• (.I-Pj)) = P^ + P  Classically  and ( n o t p ) ) " ;  ," f o r any c o m p a t i b l e  2  AA  A A  i s interpreted  a r e d e f i n e d among  A s i m i l a r c a l c u l u s o f quantum p r o p o s i t i o n s  by S t r a u s s under t h e 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 t h e appellation " p a r t i a l - B o o l e a n  algebra."  A l a t t i c e s t r u c t u r e o f 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. 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  There,  propositions  w i t h t h e s u b s e t s o f a phase space, B i r k h o f f and von Neumann a r e e s p e c i a l l y concerned t o p r e s e r v e t h e r e l a t i o n o f l o g i c a l i m p l i c a t i o n among t h e  68  propositions.  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 , a n t i - s y m m e t r i c , and  t r a n s i t i v e , and so can be r e g a r d e d a s 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 o r quantum m e c h a n i c s , i s a p a r t i a l l y o r d e r e d s e t . t h e n assume 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 has a d i s t i n g u i s h e d  They  0-element,  i n t e r p r e t e d as t h e " i d e n t i c a l l y f a l s e " o r "absurd" p r o p o s i t i o n , and a distinguished  1-element, i n t e r p r e t e d as t h e " i d e n t i c a l l y t r u e " o r  "self-evident" proposition.  Next B i r k h o f f and von Neumann c l a i m t h a t :  " I n any c a l c u l u s o f 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 t h e r e 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 s t r o n g e s t  p r o p o s i t i o n implied by, a  g i v e n 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). I n o t h e r words, w i t h r e s p e c t  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 t h a t any g i v e n p a i r o f 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  and a l . u . b . conjunction  (the j o i n  p^ v P ) > 2  p^ A p ) 2  w h i c h t h e y i n t e r p r e t as l o g i c a l  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  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 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 i m m e d i a t e l y mention t h e p r o b l e m a t i c c h a r a c t e r o f t h e meets and j o i n s o f i n c o m p a t i b l e t h a t t h e meet o r t h e j o i n o f i n c o m p a t i b l e  propositions.  They say  experimental propositions  cannot  i t s e l f be d e f i n e d a s an e x p e r i m e n t a l p r o p o s i t i o n b u t r a t h e r must be e x p r e s s e d as a c l a s s o f l o g i c a l l y e q u i v a l e n t  experimental propositions  they c a l l a p h y s i c a l q u a l i t y .  B i r k h o f f and von Neumann go  Nevertheless,  which  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 t h e subspaces o f a H i l b e r t s p a c e , and t h e y a s s o c i a t e " n o t , " "and," " o r , " among c o m p a t i b l e and  incompatible  69  p r o p o s i t i o n s qua subspaces w i t h t h e subspace  ~h A , V ,  as d e f i n e d by  t  von Neumann i n 1932. x I t i s noteworthy t h a t t h e orthocomplement H  H  o f any subspace  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 t h e s e t - t h e o r e t i c  intersect  H  subspaces  H^ , H^  1  A H^ 2  and t h e c l o s e d l i n e a r sum  H. V H„ 1 2  o f any p a i r o f * J  o f a H i l b e r t space a r e t h e m s e l v e s subspaces.  c l e a r t h a t t h e meets and j o i n s o f i n c o m p a t i b l e  So i t i s  p r o p o s i t i o n s qua subspaces  are a t l e a s t s u r e t o e x i s t , whether a s e x p e r i m e n t a l p r o p o s i t i o n s o r as "physical qualities." B i r k h o f f and von Neumann c o n c l u d e t h a t t h e 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 o f a H i l b e r t space may be r e g a r d e d as t h e l o g i c a l s t r u c t u r e o r 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 , J a u c h shows t h a t t h e subspaces o f an i n f i n i t e d i m e n s i o n a l  H i l b e r t space a r e n o t modular, and so Jauch weakens t h e m o d u l a r i t y on t h e quantum l a t t i c e o f subspaces t o weak m o d u l a r i t y  condition  ( s e e Chapter 1 ( E ) ) .  C o n s e q u e n t l y , a u t h o r s who f a v o u r t h e l a t t i c e f o r m a l i z a t i o n o f 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 c o n s i d e r t h e p r o p o s i t i o n a l s t r u c t u r e o r c a l c u l u s o f quantum mechanics t o be a c o m p l e t e , a t o m i c , o r t h o m o d u l a r ( 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 a r e a s s o c i a t e d w i t h t h e 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 t h a n t h e s u b s p a c e s , t h e n t h e e x i s t e n c e o f t h e meets and j o i n s o f i n c o m p a t i b l e problematic.  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  As mentioned i n S e c t i o n CB), t h e o p e r a t o r s and p r o j e c t o r s on  a H i l b e r t space have d e f i n e d among them.  +  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  But a theorem s t a t e s t h a t t h e p r o d u c t o f 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 I F F t h e two a r e c o m p a t i b l e ; t h e sum o f 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 I F F t h e two a r e o r t h o g o n a l A  1932,  p. 8 1 ) . I n a d d i t i o n , any  (von Neumann,  A A  P  i s a p r o j e c t o r IFF  I-P  i s a projector  (von Neumann, 1932, p. 7 9 ) . So a s e t o f p r o j e c t o r s i s c l o s e d w i t h to  +  and  *  only i f the +  p r o j e c t o r s and t h e  •  operation  operation  1  2  ring-with-unit  where JL c i c  • • • }s-L»i + > • »0,1>, s  JL t o E , and any P_ , P„ , 1 2  i s r e s t r i c t e d t o orthogonal  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  resulting i n a sort of partial-Boolean <E = { P » P »  ExE,  +  i s defined  P  + P  i s a p r o j e c t o r I F F P^_L ? As  t o show t h a t t h e sum l e s s t h e p r o d u c t : i s a p r o j e c t o r IFF P i As  (  P  .  -P P ))  P 2  1  2  =  F o r .p  A  As As  P^ + ?  As As  As +  P  2  2  As  P  As  (I-P )P ))  1  i f i s easy AS  '  °^  a n  Y  - P^g = I - I + P As As  - C(I-P )I -  1  ~ i 2  p  >  2  As As  1  2  P  1  for  any  P  A  i >  p 2  A A  »  I - ((I-P ) • (i-P >) 2  and  (I-Pg)  i s a projector IFF  A, V,- "  So when t h e  p r o j e c t o r s i n terms o f  +  closed with respect  A, V ,  to  1  and x  •  -  $ ? ±  paragraph,  i sa projector;  which i s t h e case I F F  operations  a r e d e f i n e d among  as u s u a l , t h e n a s e t o f p r o j e c t o r s i s  only i f A  and  V  are r e s t r i c t e d t o  c o m p a t i b l e 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 a l g e b r a o 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 . The subspace r e p r e s e n t a t i o n i t s e l f t o a partial-Boolean algebra  2  2  ( ( I - P ^ ) • -(I-P^))  P_ji P  Q.E.D.  2  a r e e a c h p r o j e c t o r s ; and  tI-$ ).<b ( t - P j ) ,  .  P  ^  the l a t t e r i s a p r o j e c t o r I F F 2  i ' 2 '  + P  As As  (I-P^).  AS  A s. , ^ ^ . = i - ((I-P^ -(I-P )).  And by t h e 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 t h e p r e v i o u s A  from  • i s d e f i n e d from i t o E. I w r i t e _L c i because, f o r i f P. X P„ t h e n P„ <t> P„ , but n o t t h e c o n v e r s e . 1 2 1 2  Now a l t h o u g h  A As As = i - ((I-P ) -  respect  o f quantum p r o p o s i t i o n s e a s i l y l e n d s  s t r u c t u r i n g , as f i r s t suggested by  von Neumann i n 1932. M e r e l y r e s t r i c t t h e above d e f i n e d  A, V  operations  among subspaces t o c o m p a t i b l e subspaces ( e . g . , see Kochen-Specker, 1967,  71  p. 6 5 ) . On t h e o t h e r hand, t h e p r o j e c t o r r e p r e s e n t a t i o n  o f 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 a s an o r t h o m o d u l a r l a t t i c e , b u t t h e A , V operations  can be d e f i n e d  multiplication incompatible  •  i n terms o f p r o j e c t o r a d d i t i o n  +  i n t h e u s u a l way among o n l y c o m p a t i b l e s .  p r o p o s i t i o n s qua p r o j e c t o r s , t h e  by J a u c h as f o l l o w s :  P  A P 1 2  = lim(P„ • P „ ) 1 2  n  A, V  and Among  operations  are defined  P„ V P^ = (p^* A P ^ ) ' 1 2 1 2  and  J  n  = I - l i m ( ( I - P ) • (I-P )) A, v  of when  CJauch, 1968, pp. 38, 219).  reduce t o the u s u a l d e f i n i t i o n s o f  P^ A P  2  .  PQ  So an orthomodular l a t t i c e  qua p r o j e c t o r s i s a l s o  A, v  These d e f i n i t i o n s  i n terms o f  +, •  °f quantum p r o p o s i t i o n s  MIj  defined.  Thus r e g a r d l e s s o f whether quantum p r o p o s i t i o n s a r e a s s o c i a t e d w i t h t h e p r o j e c t o r s o r t h e subspaces o f a H i l b e r t s p a c e , b o t h a l t e r n a t i v e s and b o t h have been s t r u c t u r e d a s a P - „ . QMA QML I have d e s c r i b e d how t h e a l t e r n a t i v e s have been f o r m a l i z e d a s a P-„, and QMA as a P i n o r d e r t o h i g h l i g h t t h e p r o b l e m a t i c c h a r a c t e r o f t h e meets and QML  have been s t r u c t u r e d a s a  T  N M T  j o i n s o f incompatibles  defined  i n PQ^ •  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 t h e subspaces o f a H i l b e r t s p a c e , t h e n t h e meets and j o i n s o f incompatibles  a r e a t l e a s t s u r e t o e x i s t and t h e 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 for  example, do n o t r e g a r d  PQ^L •  However, B i r k h o f f and von Neumann,  t h e meets and j o i n s o f i n c o m p a t i b l e  as p r o p o s i t i o n s b u t r a t h e r as " p h y s i c a l q u a l i t i e s . "  propositions  When quantum  p r o p o s i t i o n s a r e a s s o c i a t e d w i t h t h e p r o j e c t o r s on a H i l b e r t space and t h e A,  V,"" 1  operations  multiplication than a P „ . QML W T  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  • as u s u a l , then t h e r e s u l t i n g s t r u c t u r e i s a rather In order t o define a P . o f quantum p r o p o s i t i o n s qua QML — — W T  p r o j e c t o r s , J a u c h must i n t r o d u c e d e f i n i t i o n s o f l i m i t s of i n f i n i t e  products.  A  and  V  which i n v o l v e the  72  S e c t i o n E. — ;  The  P  fact that a PQ^  elements w h i l e a  _ QML  P „, . • QML  compared t o a  N  has  PQ^  l S  i  a  A,V  d e f i n e d among i n c o m p a t i b l e  n  some sense m i s s i n g elements  PQJ^  a n  d  a  PQ^L  c  a  by c l o s i n g t h e i n i t i a l s e t w i t h r e s p e c t t o t h e A , V,-1  d e f i n e d i n each s t r u c t u r e .  A, V  proliferation of lattice  o p e r a t i o n s as  QMA  In c o n t r a s t , the  elements so t h a t t h e P Q J ^ generated  elements does n o t o c c u r i n t h e P « „ QML T  H i l b e r t space.  by c l o s i n g  An example o f t h i s  p r o l i f e r a t i o n o f elements i s g i v e n i n Chapter V I ( C ) .  two-dimensional  he generated  n  among i n c o m p a t i b l e s o f t e n r e s u l t s i n a  a f i n i t e i n i t i a l s e t may be denumerably i n f i n i t e .  lattice  one-dimensional  When t h e i n i t i a l s e t i s f i n i t e , t h e '  by c l o s i n g t h e i n i t i a l s e t i s a l s o f i n i t e .  l a t t i c e definitions of  T  d e f i n e d among i n c o m p a t i b l e  F o r example, g i v e n an i n i t i a l s e t o f  subspaces o f a H i l b e r t s p a c e , both, a  and P _ „ QML  A, v  does n o t have  a  elements may suggest t h a t a  generated  P_„. QMA  R a m i f i c a t i o n s o f t h e B a s i c D i f f e r e n c e between ••  This p r o l i f e r a t i o n o f  s t r u c t u r e s o f subspaces o f  And i t does n o t o c c u r i n h i g h e r  dimensional  H i l b e r t space s t r u c t u r e s when t h e r e a r e c e r t a i n a n g u l a r r e l a t i o n s among t h e subspaces i n t h e i n i t i a l s e t .  An example i s g i v e n i n n o t e 8 below.  In  t h e s e cases when t h e p r o l i f e r a t i o n o f l a t t i c e elements does n o t o c c u r , both the PQML ^ ^QMA generated by c l o s i n g an i n i t i a l s e t have e x a c t l y the same elements. A N <  And  i n any c a s e , i t i s n o t c o r r e c t t o c o n s i d e r a ' J  m i s s i n g elements compared w i t h a ^QML '  ^here  l S  a  PQ ^ • M  P„„ QML T  among t h e s e same elements.  t o be  W  F o r g i v e n any f i n i t e o r i n f i n i t e  corresponding f i n i t e or i n f i n i t e  the same elements as  P„ . QMA  PQ^A  which has e x a c t l y  b u t i s m i s s i n g some o f t h e l a t t i c e P € PQ^  S p e c i f i c a l l y , an element  meet o r j o i n o f two i n c o m p a t i b l e elements i n PQ^L »  relations  e  'g«s  ^  m  =  a  v  he t h e A  ^2 '  73  P„ ^ P , b u t t h e same t h r e e elements P, P„ , P„ , i n t h e P „ . 1 2 ' ' 1 ' 2 QMA which corresponds t o t h a t P„,„. w i l l n o t be so r e l a t e d because A and v with  W  n  QML  PQ^ •  a r e n o t d e f i n e d among i n c o m p a t i b l e s i n a  A  S t r a u s s makes a s i m i l a r p o i n t when he argues t h a t t h e l a t t i c e i n t e r p r e t a t i o n o f an element P^ A P A  2  P  as t h e meet o f two i n c o m p a t i b l e elements  i s a m i s i n t e r p r e t a t i o n because  P i- P  o p e r a t i o n cannot be d e f i n e d i n terms o f t h e  P^ & P  2  .  S t r a u s s concludes  •P •  .  2  I n o t h e r words, t h e  o p e r a t i o n as u s u a l when  t h a t , compared w i t h a (orthomodular)  lattice,  a p a r t i a l - B o o l e a n a l g e b r a does n o t omit any elements b u t r a t h e r p r e v e n t s t h e 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 c o u r s e , a u t h o r s who f a v o u r t h e l a t t i c e s t r u c t u r e c a n argue t h a t t h e i n t e r p r e t a t i o n o f an element  P € P.„  T  as t h e meet o f two i n c o m p a t i b l e elements  QML  P i- P^ • P  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„ A P~ 1 2 >  2  since  Jauch has c r e a t e d t h e 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 o f t h e meet o f two i n c o m p a t i b l e elements i n ^Q^L • Regardless V  o f whether o r n o t t h e l a t t i c e d e f i n i t i o n s o f  A  and  among i n c o m p a t i b l e s 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 , t h e l a t t i c e meets  and j o i n s o f i n c o m p a t i b l e s do cause t r u t h - f u n c t i o n a l i t y problems which a r e p e c u l i a r t o t h e P_,, s t r u c t u r e s b u t a r e a v o i d e d i n t h e P .,. s t r u c t u r e s . R  QML  QMA  T  N  For a t r u t h - f u n c t i o n a l mapping on a o p e r a t i o n and t h e b i n a r y  A, V  t r u t h - f u n c t i o n a l mapping on a and t h e b i n a r y  A, V  a  must p r e s e r v e t h e u n a r y  x  o p e r a t i o n s among o n l y c o m p a t i b l e s ; w h i l e a ^ Q ^ ^ must p r e s e r v e t h e u n a r y  o p e r a t i o n s among c o m p a t i b l e s  Hereafter, l e t truth-functional truth-functional  PQ^  (j>)  x  operation  and i n c o m p a t i b l e s .  r e f e r t o t h e former c o n d i t i o n and l e t  (A,£>) r e f e r t o t h e l a t t e r c o n d i t i o n .  c o n d i t i o n i s n o t a p p l i c a b l e t o a mapping on a  PQ^A  s  i  n c e  The l a t t e r a  ''QI^  B  A  S  N  O  74  o p e r a t i o n s d e f i n e d among i n c o m p a t i b l e s . PQ  a p p l i e d t o a mapping on a P  on a  However, b o t h c o n d i t i o n s can be  » though a t r u t h - f u n c t i o n a l  M L  (d>)  mapping  i g n o r e s t h e l a t t i c e meets and j o i n s o f i n c o m p a t i b l e s and t h u s  A M T  QML  preserves only the partial-Boolean s t r u c t u r a l features of  •  x n  Chapter V ( A ) , i t i s shown how t h e l a t t i c e meets and j o i n s o f i n c o m p a t i b l e s cause t r u t h - f u n c t i o n a l i t y truth-functional  (<^>,&)  problems which- r u l e out a b i v a l e n t , PQ  s e m a n t i c s f o r any  which c o n t a i n s i n c o m p a t i b l e  M L  elements. P.  The f a c t t h a t an o r t h o m o d u l a r l a t t i c e  has  W T  A, V d e f i n e d  QML among i n c o m p a t i b l e s a l s o a f f e c t s t h e n o t i o n o f a complement i n PQJJL • For  as mentioned i n Chapter 1 ( E ) , any element  P  ina  containing  i n c o m p a t i b l e elements may have non-unique, i n c o m p a t i b l e complements. i s , f o r any P  1  are  A ?  = 0  2  P^ € P Q and  P  M L  »  t h e r e may be an element  y P  = 1;  so  P  1  ., P  2  n o t c o m p a t i b l e and a r e n o t orthocomplements.  1 ^ 2 P  )  2  P  l  s  u  c  that  n  1  , P  2  F o r example, c o n s i d e r t h e P^ & P  (and hence  2  :  In t h i s l a t t i c e , P  2 ^  a r e complements, b u t P  orthomodular l a t t i c e diagrammed as f o l l o w s , w i t h P  P  That  P^ ,  . a r e c o m p a t i b l e and a r e orthocomplements; l i k e w i s e ,  , P ^ a r e c o m p a t i b l e and a r e orthocomplements.  c l e a r from t h e diagram: P^ V P ^ = 1.  P  A P  2  = 0,  P  1  V P  2  = 1,  So b e s i d e s i t s unique Orthocomplement  a l s o has two o t h e r complements, namely, t h e element  But moreover, as i s and  P  i  A  P  2  =  °»  P^ , t h e element P  2  and t h e element  P^  75  P  X  j  which a r e n o t c o m p a t i b l e w i t h  and a r e n o t orthocomplements o f  However, when we c o n s i d e r t h e c o r r e s p o n d i n g p a r t i a l - B o o l e a n a l g e b r a which has e x a c t l y t h e same elements as t h e above orthomodular l a t t i c e but does n o t have elements  P^ > P  2  , P^  A, V  d e f i n e d among i n c o m p a t i b l e s ,  a r e n o t r e l a t e d v i a t h e A, V  t h e y a r e n o t complements.  t h e s e same  operations  and so  The o n l y complements i n a p a r t i a l - B o o l e a n  algebra  are t h e orthocomplements which a r e c o m p a t i b l e and u n i q u e , j u s t as t h e o n l y complements i n a Boolean s t r u c t u r e such as t h e c l a s s i c a l orthocomplements w h i c h a r e c o m p a t i b l e and u n i q u e .  functionality  P^  M L  and  are  I n c o n t r a s t , t h e elements  i n an orthomodular l a t t i c e may have o t h e r complements. t h e s e o t h e r complements i n a  L  The presence o f  contributes t o the l a t t i c e  truth-  (<!>,'&). p r o b l e m s , as s h a l l be shown i n Chapter V ( A ) .  presence o f t h e s e o t h e r complements i n a  PQ  ML  And t h e  r a i s e s the question  o f whether  t h e l o g i c a l " n o t " o p e r a t i o n s h o u l d be a s s o c i a t e d w i t h o r t h o c o m p l e m e n t a t i o n o r w i t h complementation.  The f a c t t h a t t h e " n o t " o f c l a s s i c a l l o g i c i s an  o p 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 , p r o v i d e s  a precedent  f o r a s s o c i a t i n g " n o t " with- o r t h o c o m p l e m e n t a t i o n r a t h e r than t h e o t h e r .  • 5  non-unique complementation. PQ  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 a l g e b r a the m a t e r i a l c o n d i t i o n a l terms o f  V  " o r " and  _ L  3  MA  ,  o f ( c l a s s i c a l ) f o r m a l l o g i c can be d e f i n e d i n  " n o t " as u s u a l ; moreover, as so d e f i n e d , t h e  m a t e r i a l c o n d i t i o n a l i n P .„. QMA RT  i s t r a n s i t i v e as u s u a l .  But i n a  P  , QML  t h e m a t e r i a l c o n d i t i o n a l cannot be d e f i n e d as u s u a l , which r a i s e s t h e question  o f how t o d e f i n e ^ i n PQ^L * In c l a s s i c a l l o g i c , t h e m a t e r i a l c o n d i t i o n a l i s defined a s , f o r  76  any  formulae  jr  f . , f . € L, 1' 2  f. ? f = 1 2  df. f. v f . 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 and  h f  3 f  |= f  3 f  ,  in the L  ,  then  then  J= r* 3 f  structure o f  namely t h e element r  t~ f , ? f  5,  i f H f  f ^ , f 2 , f g •€ L,  ., o r e q u i v a l e n t l y , i f \= f  If^l => ff^l = If^ ? f^l  / f ^ / V If^l.  3 f^  3 f  ... A l g e b r a i c a l l y , f o r any elements  L,  and  If /, /f„/  i s an element i n  L,  And t h e r e l a t i o n s o f l o g i c a l i m p l i c a t i o n  X  1= a r e i n t e r p r e t e d as t h e p a r t i a l - o r d e r i n g  o r semantic e n t a i l m e n t  relation  And t h e m a t e r i a l  where f o r any  /fl € 1,  < / f / I F F IfI = t h e  1-element.  Then t h e 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 follows: /f / 2  X  F o r any  v / f / = 1,  /f / ,  /f / , /f /  € L,  g  then  i n terms o f  V P  X 2  g  = 1,  then  v ' / y  = 1 and  / f ^ v / f / = 1. 3  and  c o n d i t i o n , i . e . , f o r any P  1  J.  PQJ^ »  With r e s p e c t t o a quantum defined  i f / y  P 1  P^ V P  > g  V P  conditional  a s above does s a t i s f y t h i s  ' 3 P  2  "the m a t e r i a l  .= 1.  P  €  QMA '  i f  P  l" 2 V  P  =  1  a  n  transitivity d  But w i t h r e s p e c t t o a quantum  the m a t e r i a l c o n d i t i o n a l i s d e f i n e d  i n terms o f ^~  and  v  > i f  as u s u a l , t h e n  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 I F F t h e l a t t i c e i s B o o l e a n , as shown by Fay (1967, p. 2 6 7 ) . A c c o r d i n g t o J a u c h and o t h e r s who worry about how t o d e f i n e t h e 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 . these l a t t i c e t h e o r e t i c i a n s conclude that of  x  , V  as u s u a l i n a quantum  ?Q^  L  => cannot be d e f i n e d  (Jauch-Piron,  And so  i n terms  1970, p. 174).  So t h e  c o r r e c t d e f i n i t i o n o f t h e m a t e r i a l c o n d i t i o n a l and even t h e p o s s i b i l i t y o f a rule like  modus ponens have been c o n t r o v e r s i a l i s s u e s among  lattice-theoreticians. Yet a n o t h e r r a m i f i c a t i o n o f t h e b a s i c d i f f e r e n c e between  PQJ^  77  and  P„„, QML  i s described  S e c t i o n F.  i n the next s e c t i o n .  The Two B a s i c Senses i n Which t h e Quantum P r o p o s i t i o n a l Structures  A r e Non-Boolean  I n c o n t r a s t t o t h e Boolean p r o p o s i t i o n a l o r l o g i c a l  structures  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 , t h 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 a r e s a i d t o be non-Boolean. PQ^L  an orthomodular l a t t i c e non-Boolean i n v a r i o u s  a  n  senses.  d  P  a  a r t  i i-Boolean  However, b o t h  algebra  a  c  a  n  ^  e  I n t h i s s e c t i o n f o u r senses a r e d e s c r i b e d ,  three o f which are e q u i v a l e n t . The most c e l e b r a t e d an a l g e b r a  sense i s t h e f a i l u r e o f d i s t r i b u t i v i t y .  distributive.  So i f t h e A , v  o p e r a t i o n s i n an a l g e b r a  d i s t r i b u t i v e , t h e n t h e s t r u c t u r e i s non-Boolean. P«., QML T  A, V  o r l a t t i c e i s Boolean, then i t s binary  which c o n t a i n s  1  e 2  distributive  ^QML  '  or l a t t i c e are not  In particular,  any quantum  F o r a s mentioned i n Chapter 1 ( E ) , f o r JL  _1_  P »l  operations are  i n c o m p a t i b l e elements e x h i b i t s a t l e a s t one i n s t a n c e  of the f a i l u r e o f d i s t r i b u t i v i t y .  any  If  t  identity  h  e  f  o  u  r  e  l  e  m  e  n  t  s  P  i»  P  i •» 2 ' 2 ' P  s a t i s f y the  P  f o r any c o m b i n a t i o n s o f t h e s e elements I F F P^<b  •  It follows that d i s t r i b u t i v i t y f a i l s i n PQ if" j_ ^ 2 ' P.. »P~ 6 P „ , „ • Most a u t h o r s who f a v o u r t h e l a t t i c e f o r m a l i z a t i o n o f t h e 1 2 QML P  P  f  o  r  a  n  y  ML  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. 8 3 1 ) , Jauch (1963, p. 8 3 1 ) , Putnam (1969, p. 2 2 6 ) , Friedman and Glymour (1972, pp. 18, 2 0 ) , f o c u s upon t h e f a i l u r e o f d i s t r i b u t i v i t y as t h e p e c u l i a r l y non-Boolean f e a t u r e o f t h 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  which  d i s t i n g u i s h e s them from t h e 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 t h a t a l a t t i c e i s d i s t r i b u t i v e  78  IFF e v e r y p a i r o f elements i n i t i s c o m p a t i b l e ( J a u c h - P i r o n , 1963, p. 8 3 1 ) . P^„  It follows that a  T  i s non-Boolean i n t h e f a i l u r e o f d i s t r i b u t i v i t y  QML sense I F F i t c o n t a i n s i n c o m p a t i b l e e l e m e n t s .  And hence, we can be s u r e 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 t h e  meets and j o i n s o f i n c o m p a t i b l e s . Since a  ^  a s  A  >  v  operations are d i s t r i b u t i v e i n a  d e f i n e d among o n l y c o m p a t i b l e s t h e s e PQ^A •  can n e i t h e r c a p t u r e t h e sense i n which a distinguish a  ^Q^  a  Thus t h e f a i l u r e o f d i s t r i b u t i v i t y ^Q^  a  i s non-Boolean n o r  from t h e B o o l e a n p r o p o s i t i o n a l s t r u c t u r e s d e t e r m i n e d  by c l a s s i c a l mechanics. However, P i r o n d e f i n e s a n o t h e r sense o f non-Boolean f o r t h e P_,„  QML s t r u c t u r e s w h i c h i s e q u i v a l e n t t o t h 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 and which can a l s o be a p p l i e d t o t h e  structures.  Piron defines the  c e n t r e o f a l a t t i c e as s t a t 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 I F F i t s c e n t r e i s t h e e n t i r e l a t t i c e ( P i r o n , 1976, p. 2 9 ) . So i f t h e c e n t r e o f a l a t t i c e i s s m a l l e r t h a n t h e e n t i r e l a t t i c e , i . e . , i f t h e r e i s an element i n t h e l a t t i c e which i s n o t c o m p a t i b l e w i t h a l l o t h e r e l e m e n t s , t h e n t h e l a t t i c e i s non-Boolean.  Any quantum  i n c o m p a t i b l e elements i s non-Boolean i n t h i s sense.  containing  And P i r o n t a k e s t h i s  f a c t t o be t h e p e c u l i a r l y non-Boolean f e a t u r e o f t h e quantum structures.  By t h e d e f i n i t i o n o f t h e c e n t r e , a  PQ^L  i  s  PQJ^  non-Boolean i n t h e  P i r o n sense I F F i t c o n t a i n s i n c o m p a t i b l e e l e m e n t s . So we expect t h a t a P - . „ i s non-Boolean i n t h e P i r o n sense I F F i t i s non-Boolean i n t h e f a i l u r e  QML  o f d i s t r i b u t i v i t y s e n s e , as i t i s easy t o show. a  P w rt  T  9  I fdistributivity fails i n  t h e n as mentioned above, n o t a l l p a i r s o f elements i n P-....  QML  compatible.  are  ylXlL  And so by t h e d e f i n i t i o n o f c e n t r e , t h e c e n t r e o f  smaller than the e n t i r e  PQ  ML  .  Conversely, i f the centre o f  is P^  M L  is  79  smaller than the entire  PQ L » "that i s , i f there i s an element M  P Q ^ > then that  which i s not i n the centre of  at least one other element i n P Q P^.T QML  a r  m l  •  P  P € PQ^  L  i s incompatible with  Hence not a l l pairs of elements i n  e compatible, and so d i s t r i b u t i v i t y f a i l s i n P_ . Q.E.D. QML • WT  J  But unlike the f a i l u r e of d i s t r i b u t i v i t y 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 QMA '  Pp  ML  P_,,. QMA  defined exactly as the centre of a J  . And as defined i n Chapter 1(D), a partial-Boolean algebra i s i n  fact a Boolean algebra IFF i t s elements are a l l mutually compatible, i . e . , P_„. i s non-Boolean i n the QMA Piron sense i f i t s centre i s smaller than the entire P»„. . And as before, QMA IFF i t s centre i s the entire algebra.  Thus a  6  PQJ^  a  i s non-Boolean i n t h i s Piron sense IFF i t contains incompatible  elements. or a  P„ „ QMA W  S i m i l a r l y , the mere presence of incompatible elements i n a PQJ/^ i s a necessary and s u f f i c i e n t condition f o r the u l t r a f i l t e r s J  (and dual u l t r i d e a l s ) i n P Q  O  R  ML  with a t h i r d sense of non-Boolean.  'QMA  t C >  ^  e n  0  t  P ^ » t h i s provides us r  m e  For as mentioned i n Chapter 1(C), the  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 ) i n a Boolean structure are a l l prime. So i f the u l t r a f i l t e r s i n a  ? .. QML n  T  or a  structure can be said to be non-Boolean. PQ in  m  P.... QMA  are not a l l prime, then that * '  As shown i n Chapter VI(B), i f a  contains incompatible elements, then there i s at least one u l t r a f i l t e r PQ  M  which i s not prime, where a prime u l t r a f i l t e r s a t i s f i e s the  condition (d) stated i n Chapter 1(C). Hence the presence of incompatible elements i n a  P„, QM  i s a s u f f i c i e n t condition f o r P_„ QM  i n the sense that i t s u l t r a f i l t e r s are not a l l prime.  to be non-Boolean Moreover, t h i s condition  -80  i s also necessary.  F o r i f a l l t h e elements o f a  compatible, then that are  a l l prime.  So a  are  n o t a l l prime I F F  P^  P^  are mutually  i s i n f a c t a B o o l e a n s t r u c t u r e whose u l t r a f i l t e r s  P  i s non-Boolean i n t h e sense t h a t i t s u l t r a f i l t e r s P  c o n t a i n s i n c o m p a t i b l e elements. PQ^L  In summary, w i t h r e s p e c t t o a  »  failure of  t n e  d i s t r i b u t i v i t y s e n s e , t h e P i r o n s e n s e , and t h e n o t - p r i m e u l t r a f i l t e r sense of non-Boolean a r e a l l e q u i v a l e n t .  And w i t h r e s p e c t t o a  » *  n e  Piron  sense and t h e n o t - p r i m e u l t r a f i l t e r sense o f non-Boolean a r e e q u i v a l e n t . For  t h e s e senses o f non-Boolean a r e 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 i n c o m p a t i b l e elements i n a  ^Q^L  O R  ^QMA  A  *  However, t h e r e 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 n o t b i c o n d i t i o n a l l y connected w i t h t h e mere p r e s e n c e o f i n c o m p a t i b l e elements. "to  P~„. QMA  T h i s sense i s suggested by Kochen-Specker, who r e f e r structures although t h e i r r e s u l t s also apply t o * &  J  A c c o r d i n g t o Kochen-Specker, a  i  s  specifically  P_ QML  structures.  WT  d i s t i n g u i s h e d from t h e Boolean PQJ^  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 t h e cannot be imbedded i n t o a B o o l e a n a l g e b r a . Kochen-Specker prove t h a t a 8 on  PQ^  c a n a  ^  e  And i n t h e i r Theorem 0,  imbedded i n t o a Boolean a l g e b r a  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 o f b i v a l e n t PQJ^  "that, f o r any p a i r o f d i s t i n c t elements  s o  t h e r e i s a t l e a s t one b i v a l e n t homomorphism h ( P ^ ) i- h ( P ) . 2  H i l b e r t space * homomorphisms. H i l b e r t space  h :  P^ i P^ {0,1}  homomorphisms  in  ^Q^A »  such t h a t  Next Kochen-Specker produce a f i n i t e , t h r e e - d i m e n s i o n a l P.,,. QMA  w h i c h i s shown i n t h e i r Theorem 1 t o admit no b i v a l e n t  Kochen-Specker c o n c l u d e t h a t t h e t h r e e - o r - h i g h e r d i m e n s i o n a l PQ^A  s t r u c t u r e s o f 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, t h e s e 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 d i s c u s s e d i n Chapter V; t h e  Kochen-Specker p r o o f o f Theorem 0 i s d i s c u s s e d h e r e . The " o n l y i f " h a l f o f t h e b i c o n d i t i o n a l statement o f Theorem 0 f o l l o w s i m m e d i a t e l y from t h e s e m i - s i m p l i c i t y p r o p e r t y o f Boolean s t r u c t u r e s . Let  %  i.e.,  be t h e proposed imbedding. in  % : PQJ^  t h a t t h e imbedding 8  4-  f o r any elements  An imbedding i s by d e f i n i t i o n  8  t %(? )«  J  h(%(P )) 4 h(%(P )) 1  f o r any •?  2  h o% : P^ •+ {0,1} QMA which s e p a r a t e s  P  1  4 P  2  ,  Kochen-Specker's  d  assuming  ±  4 Y 2  h : 8 -*• {0,1}  in P  .  f o r any •P »P 1  2  € P^  M A  such  Thus t h e c o m p o s i t i o n  i s t h e d e s i r e d homomorphism o f P.... QMA  W A  n  e x i s t s , thesemi-simplicity property o f  guarantees t h a t t h e r e i s a b i v a l e n t homomorphism  that  A  2  injective,  onto  {0,1}  .  p r o o f o f t h e c o n v e r s e 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 s u g g e s t s how t o c o n s t r u c t a C a r t e s i a n p r o d u c t Boolean s t r u c t u r e i n t o which, a of b i v a l e n t homomorphisms on let  {h.}.,, , I lCIndex  ?-.,. c a n be imbedded i f t h e r e q u i s i t e s e t QMA — e x i s t . Assume t h a t t h i s s e t e x i s t s :  be t h e s e t and l e t  s  be t h e c a r d i n a l i t y o f t h i s s e t . J  Then an imbedding o f P Q J ^ i n t o t h e C a r t e s i a n p r o d u c t Boolean s t r u c t u r e (Z > 2  s  ,  i . e . , % : PQJ^  element that for  P € P^  M A  z 2  s ) »  i s g i v e n by t h e a s s o c i a t i o n o f each  with- t h e f u n c t i o n  Sp(h^) = l u ( P ) every  (  i € Index.  f o r every  g  : { i}iei n  p  i € Index,  n d e x  f * } d e f i n e d so 0  1  where o f c o u r s e  So f o r example, t h e image o f any g i v e n  under t h e imbedding i s % ( P ) = ^ ( P ) ,h ( P ) , . . . >\CP)>  h ^ ( P ) € {0,1} P €PQ^  * (Z )  A  S  2  (Kochen-Specker, 1967, p. 6 7 ) . T h i s 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 a g a i n shortly. The Kochen-Specker imbeddings and homomorphisms p r e s e r v e t h e o p e r a t i o n s and r e l a t i o n s o f a  PQJ^ structure.  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 a l g e b r a s  = (h(b))"\ h ( l ) = 1  t e r m i n o l o g y , an y  t  h  an i n j e c t i v e  and when  y  s a t i s f i e s , f o r any  b A c € X : h ( b ) i h ( c ) , h(b A c ) = h(b) A h ( c ) ,  c o m p a t i b l e elements h(b~)  X, y,  (Kochen-Specker, 1967, pp. 66-67).  I n my  s a t i s f y i n g t h e above i s a homomorphism(A) from h  X  s a t i s f y i n g t h e above i s an imbedding(A) o f X  i s {0,1},  i . e . , when  above i s t r u t h - f u n c t i o n a l ( A ) .  h  i s b i v a l e n t , an  Thus Kochen-Specker's  h  to  into  y,  s a t i s f y i n g the  Theorem 0 b i c o n d i t i o n a l l y  c o n n e c t s t h e p o s s i b i l i t y o f imbedding(o) a . P Q ^ i n t o a Boolean s t r u c t u r e w i t h t h e e x i s t e n c e o f 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^^  ,  o r i n o t h e r 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 ( 6 ) s e m a n t i c s f o r PQJ^ • imbedding(o)  And i t i s t h e i m p o s s i b i l i t y o f  i n t o a Boolean s t r u c t u r e , o r e q u i v a l e n t l y , t h e  i m p o s s i b i l i t y o f 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 which Kochen-Specker  f o c u s upon as t h e d i s t i n g u i s h i n g non-Boolean  PQJ^ structures.  o f t h e quantum  P„,„ QML  feature  Of c o u r s e , t h i s sense o f non-Boolean  can a l s o be a p p l i e d t o t h e quantum of a  J  P^  M L  s t r u c t u r e s , a l t h o u g h an imbedding(A)  i n t o a Boolean s t r u c t u r e 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 ( c - ) P_ QML  semantics f o r a  M T  i g n o r e t h e l a t t i c e meets and j o i n s o f i n c o m p a t i b l e s  and p r e s e r v e o n l y t h e 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 t h e o t h e r senses o f non-Boolean d e s c r i b e d above, t h e presence o f i n c o m p a t i b l e elements i n a  P^  M A  or  P^  M L  i s a necessary condition f o r  t h a t s t r u c t u r e t o be non-Boolean i n t h e Kochen-Specker  sense.  mentioned i n Chapter 1(D) and ( E ) , i f a l l t h e elements o f a ^QML  3  r  e  m  u  t  u  a  l  l  y  compatible, then that  d e f i n e d i n Chapter 1 ( B ) .  P^  M  F o r as PQ  M A  or a  i s a Boolean s t r u c t u r e as  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 o f 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 a n o t h e r Boolean s t r u c t u r e .  (The s u f f i x  (i) is  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 a r e m u t u a l l y compatible.)  So i f a  P  i s non-Boolean i n t h e Kochen-Specker s e n s e ,  t h e n t h e elements o f t h a t s t r u c t u r e cannot be m u t u a l l y c o m p a t i b l e , t h a t i s , the s t r u c t u r e must c o n t a i n i n c o m p a t i b l e elements.  However, u n l i k e t h e  o t h e r senses o f non-Boolean d e s c r i b e d above, t h e mere presence o f i n c o m p a t i b l e 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 b e i n g non-Boolean i n t h e Kochen-Specker sense. r e g a r d l e s s o f t h e presence o f i n c o m p a t i b l e e l e m e n t s , a  P  In p a r t i c u l a r , structure of  t w o - d i m e n s i o n a l 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 space can be imbedded (A)  PQ^  structure o f two-dimensional H i l b e r t  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 o f t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t space R  PL  QM W  s t r u c t u r e s w h i c h makes them non-Boolean i n t h e  Kochen-Specker sense i s t h e presence o f o v e r l a p p i n g maximal Boolean substructures. namely, i t s e l f .  Any Boolean s t r u c t u r e has o n l y one maximal Boolean s u b s t r u c t u r e , 2 And t h e t w o - d i m e n s i o n a l H i l b e r t space  P  structure  diagrammed below has many maximal B o o l e a n s u b s t r u c t u r e s , but t h e y do n o t overlap:  0  84  Except f o r t h e t r i v i a l Boolean s u b s t r u c t u r e c o n t a i n i n g j u s t t h e 0, 1 2 elements o f P_„ , QM f o u r elements  e v e r y o t h e r Boolean s u b s t r u c t u r e o f  ,  , 0, 1,  Boolean s u b s t r u c t u r e mBS. . l  f o r some  2 P_ QM w  i = 1,2, . . . ,  The mBS's o f  contains the and i s a maximal  P , do share t h e 0, 1 QM ' rtl  but do n o t share any o t h e r elements and so a r e n o n - o v e r l a p p i n g . n e x t , any t w o - d i m e n s i o n a l H i l b e r t space  P^  o f t h e s e t o f mBS's i n P  and  2  As shown  can be imbedded(eb) i n t o t h e  2  C a r t e s i a n p r o d u c t Boolean s t r u c t u r e ( Z A ' )  elements  2T  where  r  i s the c a r d i n a l i t y  i s the dimensionality of the Hilbert  space. 2 Each mBS.. i n P  i s i s o m o r p h i c t o t h e C a r t e s i a n p r o d u c t 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, t h e h ^ , h^ on mBS,  1  and t h e h  c  P  , h, d  l  P  on  l  mBS_  2  P  2  l i s t e d i nthe following table:  P  2  0  1  \  1  0  0  1  \  0  1  0  1  h  1  0  0  1  0  1  0  1  c h  d  S i n c e t h e mBS's  of  P^  M  do n o t o v e r l a p , i t i s p o s s i b l e t o d e f i n e a t l e a s t 2  2 »r  b i v a l e n t homomorphisms(i)  on t h e e n t i r e  P^  M  by s i m p l y a d d i n g  t o g e t h e r t h e b i v a l e n t homomorphisms on each mBS^ . F o r example, assume t h a t r i s j u s t 2, i . e . , c o n s i d e r t h e s i x - e l e m e n t fragment o f P' QM consisting of just  mBS^  and mBS^  together.  The above f o u r b i v a l e n t  85  homomorphisms  » ^  h a  added t o g e t h e r  »  n  9  h  c  o n  P  + h  a  c  h  b  +  h  d  =  h  b  +  h  c  =  h  a  +  h  d  =  = h, 1 H  h  2  3 \  '  2  M B S  '  r e s p e c t i v e l y , can be  b i v a l e n t homomorphisms(o) on  ,2 QM  ?  l  l  P  P  2  2  P  0  1  1  0  1  0  0  1  0  1  0  1  0  1  0  1  1  0  0  1  1  0.  0  1  0  1  PQ  S i m i l a r l y , f o r any 2 • r  i  2 •2  as f o l l o w s t o y i e l d  t h i s s i x - e l e m e n t fragment o f  h  m B S  d  with  M  b i v a l e n t homomorphisms Co)  r  mBS's,  on t h e e n t i r e  i t i s possible t o define  P„„ .  And t h u s , as 2  P € P„, ,  Kochen-Specker show i n t h e i r p r o o f o f Theorem 0, f o r each element  QM  the mapping of  2  P. QM W  %CP) = <h ( P ) , h ( P j , . . . , h 2  2  QM  d e f i n e s an imbedding(o)  i n t o t h e C a r t e s i a n p r o d u c t Boolean s t r u c t u r e r  i s also written: P~„  2 > r  CP)>  nT_^CZ )? '  Por  2  consisting of just  mBS„ 1  '  exam  and  Cartesian product Boolean.structure Chapter 1(B) as f o l l o w s : % ( P j ) = <0,1,1,0>;  2 «r  (Z_) 2  .  The l a t t e r  P - ' - » "the s i x - e l e m e n t fragment o f e  mBS.  2  /  can be imbedded(c>) i n t o t h e 2•2  (Z ) 2  4  = (Z ) 2  diagrammed i n  % C ) = ^ ( P ^ h CP )..h CP ),h CP )> = <1,0,0,1>; p  1  % ( P ) = <1,0,1,0>;  % ( P ) = <0,1,0,1>; X  % ( 1 ) = <1,1,1,1>:  %(0) = <0,0,0,0>. I f t h e maximal Boolean s u b s t r u c t u r e s d i m e n s i o n a l H i l b e r t space  o f any  three-or-higher  s t r u c t u r e d i d n o t o v e r l a p , t h e n 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 t h e C a r t e s i a n \n *r Boolean s t r u c t u r e of  mBS's  (^ ) 2  ,  where a g a i n  i n t h e s t r u c t u r e and  n  r  product  i s the c a r d i n a l i t y of the set  i s the dimensionality of the H i l b e r t  86  space.  F o r each  structure  (Z )  homomorphisms.  mBS^ n  a n c  ^  b  y  i na  P  n>3  s t r u c t u r e i s isomorphic t o t h e Boolean  s e m i - s i m p l i c i t y has e x a c t l y of P  So i f t h e mBS's  n  bivalent  d i d n o t o v e r l a p , t h e n i t would  be p o s s i b l e t o s i m p l y add t o g e t h e r t h e s e b i v a l e n t homomorphisms on each mBS.  yielding at least  structure.  n«r  b i v a l e n t homomorphisms (A)  on t h e e n t i r e  And t h u s by t h e Kochen-Specker Theorem 0, t h e  c o u l d be ( Z _ )n«r  imbedded(i) i n t o t h e C a r t e s i a n p r o d u c t Boolean s t r u c t u r e T*  Tl  r  1  which i s a l s o w r i t t e n : n. . (Z„). . i=l 2 f1 a t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t However, t h e mBS's o n>3 space  P  PQ~ .  C o n s e q u e n t l y , t h e attempt t o d e f i n e b i v a l e n t homomorphisms(A) on a  may o v e r l a p and do o v e r l a p i n quantum m e c h a n i c a l l y r e l e v a n t  n>3 P^jyj , by s i m p l y a d d i n g t o g e t h e r t h e s e p a r a t e b i v a l e n t e x i s t i n g on each  mBS  homomorphisms  o f P^~ , i s p r o b l e m a t i c and i n f a c t i s i m p o s s i b l e .  C h a p t e r V ( B ) . prove An example ity i s Kochen-Specker this o i mfp a o s ts ri ib vi il ai lt y ;e xtcheepitri opnr t o ooft hiiss d ii sm cp uo ss ss ei db iil n n>: g i v e n i n t h e note below; such e x c e p t i o n a l ? ~ s t r u c t u r e s a r e n o t quantum QM g 3  mechanically relevant. In  summary, t h e r e a r e two b a s i c senses i n which t h 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 may be s a i d t o be non-Boolean and may be d i s t i n g u i s h e d from t h e B o o l e a n 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 t h e  f a i l u r e o f d i s t r i b u t i v i t y , t h e P i r o n , and t h e n o t - p r i m e u l t r a f i l t e r o f non-Boolean; t h e presence o f i n c o m p a t i b l e elements i n a  P^  M  senses  i s necessary  and s u f f i c i e n t f o r t h e 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 o t h e r b a s i c sense i s suggested by Kochen-Specker's p r e s e n c e o f i n c o m p a t i b l e elements i n a  P^  M  work; t h e mere  i s necessary but i s not s u f f i c i e n t  87  f o r t h e 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 ^ T h i s f a c t i s a c t u a l l y d e r i v e d from one o r t h e o t h e r o f t h e fundamental p o s t u l a t e s ( I I ) o r ( I I I ) which d e f i n e p . and Exp,(A) ( M e s s i a h , 1966, pp. 178, 297). ' Y  2 A c c o r d i n g t o t h e t e r m i n o l o g y o f 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 s i m u l t a n e o u s l y d e c i d a b l e . Von Neumann's n o t i o n o f t h e simultaneous d e c i d a b i l i t y o f p r o p o s i t i o n s i s a refinement o f h i s notion o f the s i m u l t a n e o u s m e a s u r a b i l i t y o f magnitudes. The l a t t e r r e q u i r e s t h a t t h e s e l f - a d j o i n t o p e r a t o r s r e p r e s e n t i n g t h e magnitudes commute. The f o r m e r r e q u i r e s t h a t o n l y t h e p r o j e c t o r s r e p r e s e n t i n g t h e p r o p o s i t i o n s commute, but t h e magnitudes mentioned i n t h e p r o p o s i t i o n s need n o t be s i m u l t a n e o u s l y measurable, i . e . , t h e i r o p e r a t o r s need n o t commute. So w h i l e t h e o p e r a t o r s r e p r e s e n t i n g s i m u l t a n e o u s l y measurable magnitudes share a l l t h e i r e i g e n s t a t e s , the o p e r a t o r s r e p r e s e n t i n g t h e magnitudes mentioned i n s i m u l t a n e o u s l y d e c i d a b l e p r o p o s i t i o n s need share o n l y t h e e i g e n s t a t e ( s ) s p e c i f i e d by t h e p r o p o s i t i o n s . Von Neumann has h i s own u n u s u a l use o f t h e terms c o m p a t i b l e and i n c o m p a t i b l e . Nevertheless, simultaneously decidable propositions are c o m p a t i b l e i n t h e 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 r e s p e c t t o an orthomodular l a t t i c e P^„. , t h i s condition i s weak m o d u l a r i t y , which c h a r a c t e r i z e s t h e quantum ^ P structures. With r e s p e c t t o a p a r t i a l - B o o l e a n a l g e b r a PQ^A ' t h i s condition h o l d s because by d e f i n i t i o n , P. < P„ I F F P. A P = P. , b u t P A P„ i s defined i n P IFF P . N M  1  Q  M  2  4.  Von Neumann r e s t r i c t s "and" and " o r " t o what he c a l l s s i m u l t a n e o u s l y d e c i d a b l e 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 a r e c o m p a t i b l e i n t h e 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. 5  T h i s p o i n t was suggested by Dr. R. E. Robinson. In h e r d i s c u s s i o n o f 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 c l a i m s t h a t an element i n such a s t r u c t u r e may have more t h a n one orthocomplement (Haack, 1974, p. 1 6 1 ) . Though i t i s t r u e t h a t an element may have more t h a n one complement, t h e orthocomplement o f an element i s by d e f i n i t i o n u n i q u e . F o r a c c o r d i n g t o B i r k h o f f , t h e a s s o c i a t i o n o f an element b w i t h i t s orthocomplement b i s a type o f mapping (namely, a d u a l automorphism h : X ->• X which i s an isomorphism o f 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 e v e r y b,c € X, b < c I F F 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 C h a p t e r 1 ( G ) , c o n d i t i o n Ma, t h e image o f any element b € X under any mapping h : X -* S i s u n i q u e , e.g., h ( b ) = i s unique.  88  L i k e w i s e t h e f a i l u r e o f b i v a l e n c e sense o f non-Boolean proposed by van F r a a s s e n i s b i c o n d i t i o n a l l y connected w i t h t h e mere presence o f i n c o m p a t i b l e 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 F r a a s s e n , 1973, p. 8 9 ) . 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 n o t o v e r l a p and P!?,,^ s t r u c t u r e s QMA b i v a l e n t homomorphisms(i) even though t h e i r QM mBS's do which admit o v e r l a p , may be g e n e r a t e d by c l o s i n g c e r t a i n l i m i t e d s e t s o f o n e - d i m e n s i o n a l ,n>3 w i t h r e s p e c t t o t h e A, v , operations subspaces ( o r p r o j e c t o r s ) o f H of P F o r an example o f t h e l a t t e r , c o n s i d e r t h e f o l l o w i n g o r QMA QM s t r u c t u r e g e n e r a t e d by c l o s i n g an i n i t i a l s e t o f twelve-element QM f i v e o n e - d i m e n s i o n a l subspaces o f H w i t h r e s p e c t t o t h e A, v, x o p e r a t i o n s o f P^ , where {P^,?^?^} a r e m u t u a l l y c o m p a t i b l e and l i k e w i s e x  t  M  {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 o f b i v a l e n t homomorphisms(6) on t h i s t w e l v e - e l e m e n t P. : P  h  l  h  2  h  3  \ h  5  l  p  l  P  2  P  2  P  3  P  3  P  4  P  4  P  5  P  5  collection  0  1  1  0  0  1  0  1  1  0  0  1  0  1  0  1  1  0  0  1  0  1  1  0  0  1  0  0  1  1  0  0  1  0  1  0  1  0  1 ft 1  1  0  0  1  1  0  0  1  0  1  1  0  0  1  0  1  0  1  1  0  0  1  ( J u s t t h e f i r s t t h r e e b i v a l e n t , homomorphisms(A) a weakly complete c o l l e c t i o n . )  h^ ,  T h i s twelve-element  P^  , h^ ,  constitute  i s a l s o an example  o f a phenomenon mentioned i n S e c t i o n ( E ) above, namely, an example o f how t h e p r o l i f e r a t i o n o f l a t t i c e elements due t o t h e l a t t i c e d e f i n i t i o n s o f  89  A , V among i n c o m p a t i b l e s does n o t o c c u r i n PQ^L s t r u c t u r e s generated byc l o s i n g a f i n i t e i n i t i a l s e t o f o n e - d i m e n s i o n a l subspaces o f f/ ~ when t h e r e a r e c e r t a i n a n g u l a r r e l a t i o n s among t h e subspaces i n t h e i n i t i a l s e t ; most s i m p l y , i n t h i s c a s e , P , P , P , P a r e a l l i n t h e same 3 3 t w o - d i m e n s i o n a l subspace P„ . And i n t h i s c a s e , t h e P „ „ and t h e P _ „ . •. . 3 QML QMA generated by c l o s i n g t h e i n i t i a l s e t o f f i v e o n e - d i m e n s i o n a l suspaces o f H w i t h r e s p e c t t o t h e A, V , - - o p e r a t i o n s o f P Q ^ d PQ r e s p e c t i v e l y , each have e x a c t l y t h e same t w e l v e e l e m e n t s , as diagrammed above. n  3  T  1  a  l  n  M A  N  >3  N e v e r t h e l e s s , as e x e m p l i f i e d by Kochen-Specker, f o r H , the s e t s o f o n e - d i m e n s i o n a l 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 d e s c r i b e a c t u a l quantum m e c h a n i c a l systems and s i t u a t i o n s y i e l d , upon closure,  N  >3  P_„  s t r u c t u r e s whose mBS's do o v e r l a p and o v e r l a p i n such a n—3 way t h a t b i v a l e n t homomorphisms(£>) on P a r e r u l e d out. (Kochen-Specker, ^ n—3 1 9 6 7 , S e c t i o n 4 ) . I n o t h e r words, quantum m e c h a n i c a l l y r e l e v a n t P Q s t r u c t u r e s have o v e r l a p p i n g mBS's which r u l e o u t b i v a l e n t homomorphisms(6). M  90  CHAPTER V  THE IMPOSSIBILITY OF A BIVALENT, TRUTH-FUNCTIONAL SEMANTICS FOR THE NON-BOOLEAN PROPOSITIONAL STRUCTURES DETERMINED BY QUANTUM MECHANICS  Preface As d e s c r i b e d  i n Chapter I V ( F ) , t h e r e a r e two b a s i c senses i n  which t h 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 may be s a i d t o be non-Boolean. And  as mentioned i n Chapter 1 1 ( 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 o f 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 t h e Boolean s t r u c t u r e i s a l o g i c a l o r propositional structure.  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 o r n o t i t admits s u c h a s e m a n t i c s i s an open question.  With r e s p e c t t o t h e non-Boolean quantum p r o p o s i t i o n a l  structures,  answers t o t h i s q u e s t i o n have a l r e a d y been g i v e n o r a t l e a s t suggested by von Neumann, J a u c h - P i r o n , Gleason and Kochen-Specker.in t h e i r p r o o f s and arguments a g a i n s t  t h e 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 i n quantum mechanics.  For as s h a l l be d e s c r i b e d  i n Section  ( D ) , when i n t e r p r e t e d  semantically  Gleason's i m p o s s i b i l i t y p r o o f and Kochen-Specker's Theorem 1 show t h e 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(i>) semantics f o r d i m e n s i o n a l H i l b e r t space interpreted semantically, dispersion-free  s t r u c t u r e s , whether  PQ^A  O  R  three-or-higher ^QML. *  A N (  ^  von Neumann's p r o o f o f t h e i m p o s s i b i l i t y o f  quantum ensembles and J a u c h - P i r o n ' s C o r o l l a r y 1 suggest t h e  i m p o s s i b i l i t y o f 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 ,&)  s e m a n t i c s f o r two-or-fhigher  d i m e n s i o n a l H i l b e r t space  •OJ2  P^T  QML  structures.  The p r o o f s by von Neumann and  J a u c h - P i r o n must be i n t e r p r e t e d a s r e f e r r i n g t o t h e orthomodular s t r u c t u r e s and t h e t r u t h - f u n c t i o n a l i t y ( o , & )  lattice  c o n d i t i o n because v o n Neumann  and J a u c h - P i r o n do d e f i n e o p e r a t i o n s among i n c o m p a t i b l e s and do r e q u i r e the p r e s e r v a t i o n o f these operations.  I n t h e n e x t s e c t i o n , von Neumann,  J a u c h - P i r o n s u g g e s t i o n 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>,&) f o r any PQJ^ W h  Semantics  i c n  Contains Incompatible  Elements  2  C o n s i d e r t h e fragment  o f the PQ  s t r u c t u r e o f two-dimensional  M  H i l b e r t space diagrammed below, with- P^ H> P^ :  2  As mentioned i n Chapter I V ( E ) , In b o t h s t r u c t u r e s , t h e 0, 1  PQ  2 M L  and P  Q  have t h e same e l e m e n t s .  M  elements a r e e q u a l t o t h e f o l l o w i n g meets  and j o i n s o f c o m p a t i b l e s : P^ A P^" = 0, P A P^ = 0, P V ? = 1, P v P = 1. I n a d d i t i o n , t h e 0, 1 elements a r e e q u a l t o t h e f o l l o w i n g x  2  ±  X  2  meets and j o i n s o f i n c o m p a t i b l e s i n P Q P  1  V P  2  = 1, . P  1  V P  2  M L  :  P^  A  p  = 2  °»  P  l  A  P  2  =  ->-  °*  = 1.  Any n o n - t r i v i a l mapping  m  on P  Q M L  assigns the value  each meet which e q u a l s t h e 0-element, e.g., m ^  0 to  A P > = m(0) = 0. 2  92  Likewise,  m  assigns the value  1  t o each j o i n which equals the 1-element.  1  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 Proof:  Any bivalent  m  diagrammed above i s impossible. 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 m(P^ A P ) = m(0) = 0, so by truth-functionality(i,&>),  We have m(P  1  m(P  A P ) = m(P ) A m(P ) = 0. 2  1  ) = 0.  Hence  2  m(P^ A F^) = m(0) = 0,  m(P„ A P ) = m(P„) A m(P ) = 0. 1 2 1 2 A  Hence  x  Case 2: Assume  mCP^) = 0.  V P ) = m(l) = 1 2  element  2  So by truth-functionality(A,<!S)  Also we have  m(P^  m(P ) t 1  P  x  .  1 A 1 = 1,  m(P ) = (m(P )) " = 0 J  2  2  m(P ) = 0. 2 x  = 1.  So we have a contradiction.  Then as i n Case 1, mCP^ V P ) = m(l) = 1 and 2  y i e l d contradictory assignments of values to the  Q.E.D.  2 QML OVIT  i s impossible.  any two-or-higher dimensional Hilbert space incompatible elements.  semantics f o r t h i s  This proof can be generalized to include n  >2  PQ  M L  structure which contains  (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 a  a  thus  so by truth- f u n c t i o n a l i t y (A ,<J6),  Hence a bivalent, truth-functional(A,^>) fragment of P  since  0-element and  1-element which are compatible.) The generalization makes use of the following lemmas: Lemma A. For any atom P^ i n any P^ and for any element P €P , , P A P QM a nr  Assume  IFF P < P a  P 6 P. a  or P < P"". a 1  By d e f i n i t i o n of A , there exist three J  mutually d i s j o i n t elements  P ^ P ^ P g € •?  such that  ?  a  = P  1  V P  g  93  and  P = P„ V P 2.  .  o  Since  and t h e r e i s no element Since P  ?  = P H  ±  V ?>  -P  3  ., o r P  P  a  V P  1  P € P„„ x QM  and -P V ?  3  i s an atom,' P. V P  o  1  such t h a t  = P  3.  1  .  3  > 0  P_ v P„ > P > 0. 1 3 x  > Pg ., e i t h e r  = G. ' and P  O  O  = P  P  = 0 and  I f the former, then  P = P  V P > P . I f t h e l a t t e r , t h e n P = P V 0 = P ; and . Z a. cL A z. s i n c e P. , P_ a r e d i s j o i n t , P = P„ and P = P„ a r e d i s j o i n t . 1 2 a 1 2 Assume P < P, t h e n P i P. (See n o t e 3 o f Chapter I V . ) J  a  a  Likewise, i f P 5 P , ' a  then  X  A P.  P  P A P"", where a '  P A P"" I F F a  1  1  Q.E.D.  a Lemma B.  F o r any atom  P £ P „ , QML A  i f P £ P a  T  By a s s u m p t i o n , J  P P  P > P a x  i n any  a  then  P > 0 a  c  such t h a t  P  > 0.  o r P AP=0. a  a  = P A P a  I F F P 5 P, a  Lemma C.  E v e r y element  it  P. •  1  Y  o r an  element  P A P = 0. a € P.,„ QML  P x  So e i t h e r  The former i s r u l e d o u t because  which c o n t r a d i c t s t h e a n t e c e d e n t o f  P A P = 0. a  P. l  Q.E.D.  P i 0  i n PQ  be any atom i n P..,_ QML J  1 M L  S  T  h e j o i n o f t h e atoms  such t h a t  P. 5 P, I  and l e t  be t h e ( 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  P .„ QML  j o i n e x i s t s because  RT  J  We want t o show t h a t so  ^  dominates. Let  V  d  But P > P A P > 0. a a  = P AP a  Hence  n  and t h e r e i s no element  a  Lemma B.  a  P  A P.  P = P^ .  Clearly,  Now i f P A P = 0. 1  X  And l e t P = V P. . x ^ I  i s complete.) ^  then  P = 1 A P = (P  X X  and t h u s  X  P.. = P.  X  V P ) A P X  X  = (P A P ) V ( P A P ) = (P A P) V 0 = P X  P^ < P, and  A P,  X  i.e., P < P  X  Assume on t h e c o n t r a r y t h a t  X  PJ" A P / 0.  Then  94  since  P QML  is atomic, there is an atom  rtWT  Pa SxP ^ A P , for  some  so  Pa'x 5 P  i , and so  1  P <P.  and  in  P a  a  Since  P„„ QML T  such that  P 5 P , aP =xP . ,  a  P < P , i . e . , P A P = P . And since a x X a x a = 0 , a contradiction. Q.E.D. x x x  Pa S P x , Pa = Pa A P = P A P X  The join of a l l the atoms in any  Lemma D. the  P Q J ^ is equal to  1-element. P . be any atom in l  Let  P. and let QML WT  V  P . be the ^ l  (finite or infinite) join of a l l the atoms i n P. QML WT  Assume on the contrary that  V . P . j- 1. Then i  ( V P . ) " t 0, i L  1  and so  ( VP.) i  X  > P . ,  1  I t follows that  AMT  P . . Clearly, D  3  0= ( V P l  which is impossible. Lemma E. any P QML in P-„ . QML  for some atom JA C V P.) l  1  X  V P .> P .. i •3 1  > (V P . ) A P . = P . , l J  Hence  ( V P.) = 1. Q.E.D. i Any proposed bivalent, truth-functional(<i,jb) mapping on must assign the value 1 to at least one of the atoms  T  A g a i n , let &  P . . be any atom in x J  join of a l l the atoms in J  P_,. QML  T  and let  V P . be the ^ l  P.„ . I assume that the truthQML T  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 ,^) mapping m 1  V  #  X  on  PQ  M L  assigns the value  m ( P . ) = m ( V P.) = m ( l ) = 1. And for every 1  *  1  X  P ., m(P.)= 0 1  3L  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 P „ x QML x A  T  95  1  i s assigned the value  by  2  m.  Q.E.D.  Besides these lemmas, the generalization makes use of the d i s t i n c t i o n between i r r e d u c i b l e and reducible  P „ QML RT  structures, defined '  T  0  at least the  1  and  0, 1 '  contains just the J  P-„. • A T ., whose centre QML QML elements i s i r r e d u c i b l e . A P_,, whose centre QML elements of  T  n  T  T  0, 1  contains other elements besides the PQ^  described i n Chapter IV(F), any  elements i s reducible.  o  n  "  t  a  i  n  s  incompatible' elements.  P Q J ^ containing  C l e a r l y , any  incompatible elements i s either i r r e d u c i b l e or reducible. c  As  contains incompatible elements IFF  l  i t s centre i s less than the entire structure.  ^QML  P Q J ^ contains  As defined i n Chapter 1(F), the centre of any  as follows.  And any i r r e d u c i b l e  PQJ^ may  A-reducible  have a i l i t s elements  l  mutually-compatible, but such a reducible P Q ^ i s i n fact, a Boolean structure. P.... QML  I f the centre of a reducible  contains any atoms of J  P.„_ QML  ,  then the structure does admit some bivalent, truth-functional(<_!>,&) mappings, as s h a l l be described'-in a-brief'-digression. Each central'.atom P of such a reducible P_,„ s p e c i f i e s an u l t r a f i l t e r UF and dual u l t r a i d e a l Ul QML c c c  by the usual d e f i n i t i o n s : UF J  Ul  c  {P  =  € P«.„  :  QML  c  = { P € P ., QML N  P 5 cP" "}. 1  And  compatible with every other element i n P € P Q ^ »  for every element P > P  or  c  either of  UF  P c  P*"> >  P  c  P or c  .  T  : P > P } c  since each c e n t r a l atom i s by d e f i n i t i o n P Q ^ > i t follows by Lemma A that,  and for any given c e n t r a l atom  Since by d e f i n i t i o n of  P 5 P  or a member of  1  c  .  Ul ; c  and  ,  So every element i n P  thus  O V T T  QML  = UF  c  P  X  > P  P«., QML  U Ul  IFF  C  ,  either  P < P  1  c  ,  i s either a member  T  .  c  c  P  Then as w i l l be  shown i n Chapter VI(B), i t follows by the conditions defining an u l t r a f i l t e r and dual u l t r a i d e a l that the bivalent homomorphism defined with respect to  UF  and  Ul  h  c  :  PQ  M L  •+  {0,1},  as usual, truth-functionally(i,&)  96  assigns  v a l u e s t o e v e r y element i n PQJVJL •  0, 1  assigns the value value P P  5 P c  p a r t i c u l a r , each  n  t o i t s a f f i l i a t e d c e n t r a l atom P i P a c  t o e v e r y o t h e r atom J  i s compatible w i t h  a a  0  1  x  P c  i n P_„ . QML T  and so by Lemma A,  i s r u l e d out since  P c  P  c  h  c  and a s s i g n s t h e  F o r e v e r y o t h e r atom J  P 2 P"^ ( t h e a l t e r n a t i v e a c  i s an atom): t h u s  P € UI . a c  There a r e  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 t h e r e a r e c e n t r a l atoms i n P_ . T h i s ends t h e d i g r e s s i o n . QML Now t h e p r e v i o u s Theorem 0 i s g e n e r a l i z e d as f o l l o w s . WT  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  i m p o s s i b l e f o r any ( t w o - o r - h i g h e r d i m e n s i o n a l H i l b e r t space) F*QML Case 1:  which contains incompatible  Irreducible PQ  M L  •  B  elements.  Lemma E, any proposed b i v a l e n t ,  v  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  t o a t l e a s t one atom i n P_„„ , s a y m(P ) = 1. S i n c e P i s not i n QML a a t h e c e n t r e o f t h e i r r e d u c i b l e P„„ , t h e r e 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, 1  T  J  T  3.  A P = 0  P  cl  and P  cL  A P  X  = 0.  cL  t h e case 1 o f Theorem 0 t h a t P . X  ^QML  m  a s s i g n s c o n t r a d i c t o r y v a l u e s t o t h e element  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 l  S  i P m  o  s  s  an i r r e d u c i b l e Case 2: P^wr QML  cL  Then i t f o l l o w s by t h e r e a s o n i n g g i v e n i n  ikl * e  P„,„ QML  Reducible '•'  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 i s impossible. P QML A M T  ( c o n t a i n i n g incompatible elements).  c o n t a i n s a t l e a s t one n o n - c e n t r a l atom.  Any r e d u c i b l e  F o r i f e v e r y atom i n P.., QML  T  J  were c e n t r a l , t h e n s i n c e t h e c e n t r e i s a s u b l a t t i c e c l o s e d w i t h r e s p e c t t o t h e j o i n o p e r a t i o n i t f o l l o w s by Lemma C t h a t e v e r y element i n be c e n t r a l , i . e . , t h e r e would be no i n c o m p a t i b l e elements i n P Q  w  M L  . A  o  u  l  d  97  non-central atom i n  x  S  c l e a r l y d i s t i n c t from the  0-element, and so  a complete c o l l e c t i o n 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 i n case 1 of t h i s proof, any proposed bivalent, t r u t h - f u n c t i o n a l C o m a p p i n g non-central atom i n element i n PQJ^  which assigns the value  to a  w i l l assign contradictory values to some other  which i s incompatible with that atom.  l  1  So although a  reducible P Q J ^ admit some bivalent, truth-functional(c}),j6) mappings, as shown i n the digression above, a reducible P does not admit enough m  a  v  O V T T  QML  such mappings to constitute a bivalent, truth-functional(o,&) semantics f o r P  QML  '  Q  -  E  '  D  -  3  One way of avoiding the contradictions which, y i e l d t h i s i m p o s s i b i l i t y proof i s to weaken the truth-functionality(o,<iS) condition to just t r u t h - f u n c t i o n a l i t y ( o ) , thus allowing the semantic mappings on a  P QML  to ignore the l a t t i c e meets and joins of incompatibles.  Such bivalent,  truth-functional (A) mappings which preserve the partial-Boolean s t r u c t u r a l features of a  PQ^  O  R  A  L  ^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 i n Chapter IV(F), two-dimensional  Hilbert space  P  structures do admit a complete c o l l e c t i o n o f bivalent homomorphisms (A), i . e . , they do admit a bivalent, truth-functional(A) semantics, i n spite of the fact that they contain incompatible elements. dimensional Hilbert space  F^Z? QM  But three-or-higher  structures do not admit bivalent  98  homomorphisms(o).  The peculiar s t r u c t u r a l feature of P ^  which rules  A  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 i s a necessary condition). In t h e i r Theorem 1, Kochen-Specker consider a p a r t i c u l a r f i n i t e 3 P«„« QMA  and show that t h i s structure does not admit even a single bivalent &  homomorphism(o). 3  on  PQJ^  s a t i s f i e s , f o r any three mutually orthogonal atoms  h(P ) V h ( P ) 2  1  By d e f i n i t i o n , a proposed bivalent homomorphism(<b) ,h  V h(P ). = h ( P v P 3  1  2  vP )  f o r 1 < i i j 5 3.  3 one of every three mutually orthogonal atoms i n PQ^A ^ 3 value  1  p. 67).  by a bivalent homomorphism(A>) on  (KS1)  P" QM  For any n h(P„) 1 and  V  2  s  Thus exactly  assigned the  on any n  dimensional  satisfies:  mutually orthogonal atoms  h(P ) v . . .  h(P.) A  h ( P . )  V  h(P ) = n  = h(P. A  P . )  h(P,  1  P  v  ^> 2' " * " ' n ^ ^QM ' P  P  2 0  = h(0) = 0  P  V  . . . v  P  n  = h ( l ) = 1,  )  for 1 < i  i  j < n.  By the Lemma A of the previous section, any two atoms i n a  P^ A  are orthogonal IFF they are compatible.  By closure with respect to the A ,  V , ^  mutually orthogonal ( i . e . ,  operations of P ^ , a set of n A  compatible) atoms i n a  P^  generates a Boolean substructure o f P^  p a r t i c u l a r , such a set generates a maximal Boolean substructure of since the maximum number of mutually compatible atoms i n a  P^  M  . In  P^  structure  14.  of  n  >  PQ^A (Kochen-Specker, 1967,  More generally, a bivalent homomorphism(cb) h  Hilbert space  3  € PQ^  = h ( l ) = 1, and  G  h(P.) A h.(Pj) = hCP. A P j ) = h(0) = 0  P^JP^JP^  dimensional Hilbert space i s n.  Thus condition (KS1) refers to  the (maximal) Boolean substructures of a P Q  M  and ensures that t h e i r Boolean  99  s t r u c t u r a l f e a t u r e s a r e p r e s e r v e d by a b i v a l e n t homomorphismCi).  And j u s t  t h e u s u a l d e f i n i t i o n o f 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 t h e 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 n o t f o c u s a t t e n t i o n upon t h a t p e c u l i a r l y non-Boolean structural feature o f  s t r u c t u r e , namely, t h e presence o f o v e r l a p p i n g  mBS's i n P ^ . 3  QM  B i v a l e n t homomorphisms(i) on a Boolean s t r u c t u r a l f e a t u r e s o f e v e r y  PQ~ p r e s e r v e n o t o n l y t h e  (maximal) Boolean s u b s t r u c t u r e b u t 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 o f t h e e n t i r e n—3  PQ^»  i  n  particular,  t h e o v e r l a p p a t t e r n s o f t h e mBS's i n PQ^ . One way i n which t h e o v e r l a p p a t t e r n s c a n 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 v a l u e s t o be a s s i g n e d t o a g i v e n element  P  w h i c h i s a member o f two o r more o v e r l a p p i n g mBS's;  the value assigned t o P  i n t h e c o n t e x t o f one mBS may be d i f f e r e n t from  the value assigned t o P  i n t h e c o n t e x t o f a n o t h e r mBS.  Such proposed  v i o l a t i o n s o f t h e o v e r l a p p a t t e r n s a r e f u r t h e r d i s c u s s e d i n Chapter V I I . b i v a l e n t homomorphism(i>) on a i n t h i s way ( o r i n any way). i.e.,  A  does n o t v i o l a t e t h e o v e r l a p p a t t e r n s F o r a b i v a l e n t homomorphism(o) i s a mapping,  h ( P ) i s u n i q u e , as s t a t e d i n Chapter 1 ( G ) , and i n . p a r t i c u l a r , h ( P )  does n o t depend upon w h i c h mBS i s b e i n g c o n s i d e r e d .  Kochen-Specker do n o t  e x p l i c i t l y s t a t e t h i s aspect o f a b i v a l e n t homomorphism(o); here and i n Chapter V I I i t s h a l l be r e f e r r e d t o a s : (KS2)  The v a l u e s a s s i g n e d by a b i v a l e n t homomorphism(i)  h : • ^QM'  a r e unique and do n o t v a r y w i t h o r depend upon t h e mBS's o f Pq^ .  T h i s aspect o f t h e n o t i o n o f 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  P ^  s h a l l r e s t a t e Kochen-Specker's p r o o f o f 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  A  i n a manner which e l u c i d a t e s t h e e f f e c t o f t h e o v e r l a p 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). By ( K S 1 ) , any b i v a l e n t homomorphism(cb) the value  1  t o e x a c t l y one atom i n t h e mBS  Kochen-Specker l a b e l h(p ) = 1 Q  to  p^ * q  and t h u s  A  ,r  n  .  t h e atoms i n mBS  n  o f c o u r s e determines n  .  mBS- . 0  h  F o r example, c o n s i d e r an mBS  two atoms i n e v e r y mmBS h. to  the values assigned t o a l l the  Q  containing  h ( p ) = 1. Q  A  mBS  which overlaps w i t h p  and two o t h e r  0  And so by ( K S 1 ) , t h e o t h e r  a r e each a s s i g n e d t h e v a l u e  These v a l u e assignments i n t u r n determine t h e atoms i n e v e r y  mBS  which c o n t a i n s  p  initial  p l a c e s r e s t r i c t i o n s upon t h e  t o t h e atoms i n e v e r y  atoms; by assumption and by ( K S 2 ) ,  s p e c i f i e d by t h e t h r e e atoms  But i n a d d i t i o n , t h i s  assignment o f v a l u e s t o t h e atoms i n mBS v a l u e s a s s i g n e d by  must a s s i g n  QMA  T h i s i n i t i a l assignment o f v a l u e s  n  o t h e r non-atomic elements o f mBS  on P  L e t us i n i t i a l l y assume t h a t  h ( q ) = h ( r ) = 0. n  Q  h  0 by  t h e v a l u e s a s s i g n e d by h  w h i c h o v e r l a p s w i t h any o f t h e  mBS  which 3  overlap with  mBSg .  PQJ^ s t r u c t u r e  And t h i s p r o c e s s c o n t i n u e s t h r o u g h t h e  u n t i l we g e t h(qg) = 1, which c o n t r a d i c t s t h e statement  that  h(q )  (which f o l l o w s by (KS1) from t h e i n i t i a l assumption t h a t  h(p ) = 1). A 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 i n s t e a d i n i t i a l l y assume t h a t  n  =  0  101  h(q ) Q  = 1  and  h ( p ) = h.(.r ) = 0. Q  Q  we i n i t i a l l y assume t h a t  h-Cr-p). = 1  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 and  any b i v a l e n t homomorphism(c>) on t h i s P one o f t h e t h r e e atoms  h ( p ) = M q ^ ) = 0. Q  Thus,  must a s s i g n t h e v a l u e  PQ , q ^ , r ^ i n mBS  since 1 to  y e t a l l three attempts lead  Q  PQ  t o 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  c o n s i d e r e d by  M A  Kochen-Specker i s i m p o s s i b l e . In o t h e r 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_,„. QMA this  i s i m 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 3 PQJ^  l S  impossible.  Kochen-Specker a l s o c o n s i d e r a much s m a l l e r 3 27 atoms and 16 o v e r l a p p i n g  mBS's.  This  3 PQ^  a  which  contains  does admit some b i v a l e n t  homomorphisms (i>), b u t as Kochen-Specker p o i n t o u t , t h e r e a r e 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) a s s i g n s d i f f e r e n t v a l u e s t o t h e s e two d i s t i n c t elements.  That i s , t h e c o l l e c t i o n  o f b i v a l e n t homomorphisms(A) which do e x i s t on t h i s So l i k e t h e r e d u c i b l e orthomodular l a t t i c e  the d i g r e s s i o n i n t h e p r e v i o u s s e c t i o n , t h i s  P_„_ QML 3 PQ^  PQ^  1 S  no* complete.  structures discussed i n  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, b u t i t does n o t 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. 6 7 ) .  The Kochen-Specker r e s u l t i s f u r t h e r d i s c u s s e d i n Chapter V I I . S e c t i o n C. A v o i d i n g  i n Section  (D) and  These I m p o s s i b i l i t y P r o o f s  There a r e a t l e a s t two ways o f a v o i d i n g n o t o n l y Theorem A b u t a l s o t h e Kochen-Specker i m p o s s i b i l i t y p r o o f .  One way i s t o f u r t h e r weaken  the t r u t h - f u n c t i o n a l i t y ( < M c o n d i t i o n ; a n o t h e r way i s t o r e s t r i c t t h e domains  102  o f proposed semantic mappings on  P ^  to c e r t a i n substructures of  N  P^  .  With r e g a r d t o t h e l a t t e r , f o r example, i f t h e domain o f t h e mapping  m  P  on t h e  2 N M T  QMLJ  r e s t r i c t e d such t h a t  d i s c u s s e d i n t h e b e g i n n i n g o f S e c t i o n (A) were  mCP^),  mCP^)  a r e n o t d e f i n e d when  mCP^)  is  7 d e f i n e d , then  m  would n o t a s s i g n c o n t r a d i c t o r y v a l u e s .  i f t h e domain o f each proposed semantic mapping on a t o an  mBS  avoided.  of  P ^  ,  N  ' I n o t h e r words,  P ^ A  were r e s t r i c t e d  t h e n b o t h i m p o s s i b i l i t y p r o o f s would c l e a r l y be  More i n t e r e s t i n g l y , semantic mappings which a v o i d b o t h i m p o s s i b i l i t y  p r o o f s y e t whose domains a r e s u b s t r u c t u r e s o f  P  which i n c l u d e o v e r l a p p i n g  mBS's a r e d e s c r i b e d i n Chapter V I ; t h e y - a r e t h e quantum  state-induced  expectation-functions. With r e g a r d t o t h e f i r s t mentioned way o f a v o i d i n g b o t h i m p o s s i b i l i t y p r o o f s , Friedman and Glymour p r o p o s e , f o r quantum  PQJ^  s t r u c t u r e s , semantic mappings which a r e r e q u i r e d t o p r e s e r v e t h e -~~ J  o p e r a t i o n and t h e  5  relation of  PQ  M L  but a r e a l l o w e d t o i g n o r e t h e meets  and j o i n s o f b o t h c o m p a t i b l e s and i n c o m p a t i b l e s (Friedman-Glymour, 1972). However, as shown i n Chapter V I ( B ) , t h e Friedman-Glymour semantic mappings a r e 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 s u b s t r u c t u r e s o f which i n c l u d e o v e r l a p p i n g  mBS's; i n t h i s r e s p e c t , t h e Friedman-Glymour  mappings a r e e x a c t l y l i k e t h e quantum s t a t e - i n d u c e d mentioned above. to just  x  , 5  P.... QML  expectation-functions  So a weakening o f t h e 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  p r e s e r v a t i o n n e v e r t h e l e s s ensures t h e p r e s e r v a t i o n o f t h e  meets and j o i n s o f c o m p a t i b l e substructures o f  P-„ QML T  and i n c o m p a t i b l e elements i n c e r t a i n  .  More e x t r e m e l y ,  the s o - c a l l e d contextual 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 a r e not r e q u i r e d t o p r e s e r v e  even  103  the  o p e r a t i o n and t h e <  x  impossibility proofs.  S e c t i o n D.  relation of  P  and w h i c h a v o i d both  T h i s proposal i s d i s c u s s e d i n Chapter V I I .  The 'Meaning o f t h e H i d d e n - V a r i a b l e  I m p o s s i b i l i t y Proofs f o r the  I s s u e o f a C l a s s i c a l Semantics f o r t h e Quantum P r o p o s i t i o n a l Structures In h i s 1957 p r o o f o f t h e completeness o f quantum m e c h a n i c s , Gleason r e f e r s t o t h e i n f i n i t e s e t o f a l l subspaces ( o r p r o j e c t o r s ) o f a t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t s p a c e , b u t Gleason does n o t e x p l i c i t l y s t a t e whether t h e s t r u c t u r e o f such a s e t i s an o r t h o m o d u l a r l a t t i c e o r a partial-Boolean algebra. ^QML  a  n  d  P  c o r r e s  o n  ding  elements, but the  PQ  As mentioned i n Chapter I V ( E ) , such 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 t h e same  infinite M L  has i t s  A, V PQ  and i n c o m p a t i b l e elements w h i l e t h e among o n l y c o m p a t i b l e elements.  o p e r a t i o n s d e f i n e d among c o m p a t i b l e M A  has i t s  A , V  operations defined  N e v e r t h e l e s s , 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 a l g e b r a s t r u c t u r e s because t h e mapping  ^ LI  w h i c h he d e f i n e s on t h e 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)  F o r any denumerable.' c o l l e c t i o n orthogonal subspaces,  P.) = £ |x(P.);  u.(v •  LL(P  1  V  p 2  ) =  LL(P ) 1  +  n(P ) 2  ^i^ieindex  1  X  •  1  °  f  m  u  t  u  a  l  l  y  f o r example,  X  ( G l e a s o n , 1957, p. 885).  T h i s 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 a r e d i s p e r s i o n - f r e e , i . e . , b i v a l e n t , t h e n t h e mappings p r e s e r v e t h e u n a r y  8 o p e r a t i o n and b i n a r y do n o t p r e s e r v e t h e  A ,V A ,V  o p e r a t i o n s among c o m p a t i b l e s .  But t h e mappings  o p e r a t i o n s among i n c o m p a t i b l e s .  I n o t h e r words,  d i s p e r s i o n - f r e e h i d d e n - v a r i a b l e mappings w h i c h s a t i s f y (Ga) a r e b i v a l e n t  104  homomorphisms(o), and v i c e v e r s a .  Viewed s e m a n t i c a l l y , such mappings a r e  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). PQJ^ •  Such mappings p r e s e r v e t h e o p e r a t i o n s and r e l a t i o n s o f a But such mappings on a  P  R T M T  i g n o r e t h e l a t t i c e meets and j o i n s o f  QML  incompatibles.  PQ  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  of subspaces, a l t h o u g h h i s r e s u l t s a l s o do a p p l y t o  M a  structures  PQ^L structures.  C l e a r l y , s i n c e d i s p e r s i o n - f r e e Gleason mappings i g n o r e the meets and j o i n s of  i n c o m p a t i b l e s , t h e y do not r u n i n t o t h e t r u t h - f u n c t i o n a l i t y ( o , & )  problems w h i c h a r e t h e b a s i s o f Theorem A. i n t o t r u t h - f u n c t i o n a l i t y ( o ) problems.  However, t h e mappings do r u n  F o r a c o r o l l a r y t o Gleason's  completeness p r o o f shows t h a t proposed d i s p e r s i o n - f r e e h i d d e n - v a r i a b l e mappings s a t i s f y i n g (Ga) a r e i m p o s s i b l e on t h e i n f i n i t e s e t o f a l l subspaces of a t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t space.  T h i s c o r o l l a r y i s known a s  Gleason's p r o o f o f t h e 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 . The Kochen-Specker  1967 Theorem 1, d e s c r i b e d i n S e c t i o n B i s a  f i n i t e v e r s i o n o f Gleason's i m p o s s i b i l i t y p r o o f which makes e x p l i c i t t h e f a c t t h a t Gleason's p r o o f c o n s i d e r s b i v a l e n t homomorphisms(o) on s t r u c t u r e s ( a l t h o u g h Gleason's r e s u l t a l s o a p p l i e s t o  N  M  A  structures).  Moreover, w i t h t h e i r o r t h o h e l i u m example, Kochen-Specker 3 quantum m e c h a n i c a l r e a l i z a t i o n o f t h e i r f i n i t e  P  provide a concrete,  (1967, pp. 71-74).  Thus Gleason's p r o o f , which r e f e r s t o a l l subspaces o r p r o j e c t o r s o f a H i l b e r t s p a c e , i s p r o t e c t e d from c r i t i c s who argue t h a t o n l y some f i n i t e set  o f o p e r a t o r s i n f a c t r e p r e s e n t quantum magnitudes o r argue t h a t o n l y  some " e s s e n t i a l " magnitudes need be a s s i g n e d v a l u e s by proposed d i s p e r s i o n - f r e e h i d d e n - v a r i a b l e 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 , p. 376).  1970,  105 I n c o n t r a s t t o t h e G l e a s o n , Kochen-Specker p r o o f s , b o t h t h e von Neumann and t h e J a u c h - P i r o n  i m p o s s i b i l i t y p r o o f s c o n s i d e r mappings which  a r e r e q u i r e d t o p r e s e r v e an o p e r a t i o n among i n c o m p a t i b l e s , and b o t h p r o o f s i n c l u d e t h e case o f t w o - d i m e n s i o n a l H i l b e r t space. In h i s 1 9 3 2 p r o o f s o f t h e completeness o f quantum mechanics and t h e i m p o s s i b i l i t y o f d i s p e r s i o n - f r e e h i d d e n - v a r i a b l e ensembles i n quantum mechanics, von Neumann does n o t 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 r o j e c t o r s o f H i l b e r t space.  PQ^  O  ^ Q M L  R  A  s t r u c t u r e s o f subspaces o r  R a t h e r , von Neumann c o n s i d e r s  expectation-  f u n c t i o n s whose domain i s t h e ( i n f i n i t e ) s e t o f quantum magnitudes represented  by s e l f - a d j o i n t o p e r a t o r s on a H i l b e r t space o f any d i m e n s i o n ,  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 p r e s e r v e t h e + d e f i n e d among t h e magnitudes r e p r e s e n t e d  by o p e r a t o r s .  operation  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 r e q u i r e m e n t s can be shown t o be 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 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  o n l y t h e idempotent quantum magnitudes r e p r e s e n t e d 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 expectation-function.  d  e  f i  n  e  d  i  n  he a d i s p e r s i o n - f r e e  w  As d e s c r i b e d i n Chapter I V ( D ) , t h e s t r u c t u r e o f t h e  p r o j e c t o r s on a H i l b e r t space i s a ^QM  Exp  P^  M  ,  terms o f t h e r i n g o p e r a t i o n s  w i t h t h e A, v, * -1  +, *,  operations of  as u s u a l f o r  compatible  p r o j e c t o r s and by means o f Jauch's d e f i n i t i o n s f o r i n c o m p a t i b l e p r o j e c t o r s . Moreover, w i t h r e s p e c t t o t h e idempotent magnitudes r e p r e s e n t e d by p r o j e c t o r s , t h e 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  Exp  w  mappings ensures t h a t t h e mappings a r e b i v a l e n t , as  shown by a s i m p l e p r o o f .  Thus von Neumann's  E*P  w  mappings a r e b i v a l e n t  106  mappings on  P^  structures.  n  expectation-function  Exp  F u r t h e r m o r e , von Neumann r e q u i r e s any  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>)  F o r any c o m p a t i b l e magnitudes  A,B, . . . ,  Exp(A + B + . . .) = Exp(A) + Exp(B) + . . . (vN&)  F o r any i n c o m p a t i b l e  magnitudes  A,B, . . . 10  Exp(A + B + . . .) = Exp(A) + Exp(B) + . . . In p a r t i c u l a r , an and  incompatible  Exp  must p r e s e r v e t h e +  operation  among c o m p a t i b l e  idempotent magnitudes r e p r e s e n t e d 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), t h e c o n d i t i o n (.vN<t) ensures t h a t t h e b i v a l e n t mappings p r e s e r v e t h e unary  operation  among c o m p a t i b l e p r o j e c t o r s . ' ' ' 1  and t h e b i n a r y  Thus t h e von Neumann  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 ) mappings on G l e a s o n , Kochen-Specker mappings. required t o preserve the +  P^  A, V  Exp  Exp  w  operations  mappings a r e  s t r u c t u r e s , as a r e t h e  A  However, von Neumann's mappings a r e a l s o  operation  among i n c o m p a t i b l e s .  So c o n s i d e r i n g  j u s t t h e idempotent magnitudes r e p r e s e n t e d 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 his  Exp w  operation  P^,„ s t r u c t u r e a s t h e domain o f QML mappings, because he r e q u i r e s h i s mappings t o p r e s e r v e a b i n a r y  among  incompatibles.  In t h e i r 1963 p r o o f o f t h e 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 i n quantum mechanics, J a u c h - P i r o n 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  structures.  The mappings a r e r e q u i r e d t o  satisfy certain conditions, especially:  (JPo)  F o r any elements w(P ) 1  P »P  + w(P ) = w(P  1  ^ Q M L'  € 2  V P ) + wCP 2  i f  1  P  l ^ 2 P  A P ). 2  t  h  e  n  107  (JPiS)  F o r any subset J  w(P.)  = 1  {P.}.,,. , i i€Index  f o r every  o f elements i n a  i € Index,  then  w(A  P„ , i f QML WT  P.) = 1; i  f o r example, i f wCP^  = w ( P ) = 1, 2  then  w(P^ A P > = 1 2  ( J a u c h - P i r o n , 1963, p. 8 3 3 ) . L i k e (Ga) and (vNo), t h e c o n d i t i o n ( J P i ) ensures t h a t t h e b i v a l e n t mappings p r e s e r v e t h e unary -1  o p e r a t i o n and t h e b i n a r y  A, v  o p e r a t i o n s among  12 compatibles.  So t h e J a u c h - P i r o n mappings a r e b i v a l e n t and  truth-functional(cS).  But t h e mappings a r e i n a d d i t i o n r e q u i r e d t o 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 t h e i n c o m p a t i b l e elements o f a  •  A  o p e r a t i o n among c o m p a t i b l e and  So G l e a s o n , Kochen-Specker, von Neumann,  and J a u c h - P i r o n a l l r e q u i r e t h e i r proposed h i d d e n - v a r i a b l e mappings t o be t r u t h - f u n c t i o n a l ( < o ) , b u t i n a d d i t i o n , von Neumann and J a u c h - P i r o n t h e i r mappings t o p r e s e r v e an o p e r a t i o n among i n c o m p a t i b l e s .  require  And i t i s  p r e c i s e l y t h e s e 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 t h e von Neumann and t h e J a u c h - P i r o n p r o o f s t o work a t a l l and which a l l o w t h e i r p r o o f s t o i n c l u d e t h e t w o - d i m e n s i o n a l H i l b e r t space case which i s e x c l u d e d from t h e G l e a s o n , Kochen-Specker p r o o f s . S p e c i f i c a l l y , u s i n g t h e t r a c e - f o r m a l i s m developed i n h i s completeness p r o o f , von Neumann shows 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 s a t i s f y h i s c o n d i t i o n s a r e i m p o s s i b l e on t h e ( i n f i n i t e ) s e t o f o p e r a t o r s on a H i l b e r t space o f any d i m e n s i o n (von Neumann, 1932, pp. 320-321).  And  J a u c h - P i r o n 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 r e s p e c t t o 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 a r e i m p o s s i b l e . S e m a n t i c a l l y i n t e r p r e t e d , s i n c e t h e truth-functionality(<J>,cH)  c o n d i t i o n i s even s t r o n g e r t h a n  t h e c o n d i t i o n s imposed by e i t h e r von Neumann  or J a u c h - P i r o 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 suggest t h a t i n g e n e r a l and  108  i n c l u d i n g t h e t w o - d i m e n s i o n a l H i l b e r t space c a s e , quantum  structures  do n o t admit b i v a l e n t , t r u t h - f u n c t i o n a l ( c - , & ) mappings and hence do n o t 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 > , ^ ) s e m a n t i c s ; t h i s i s proven i n S e c t i o n A a s Theorem A. There i s an i m p o s s i b i l i t y p r o o f by Z i e r l e r and S c h l e s s i n g e r i n v o l v i n g a c o n d i t i o n which i s as s t r o n g as my t r u t h - f u n c t i o n a l i t y ( c b , & ) 14 condition.  I n 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 t h a t i f t h e r e  i s a s t r o n g l y a d d i t i v e embedding set  P  when  o f an orthomodular p a r t i a l l y o r d e r e d  i n t o a Boolean a l g e b r a , t h e n t h e j o i n  P^  commutes w i t h  P  P^  (i.e.,  P^  P^ V P^  ) '  p  only  s t r o n g l y a d d i t i v e embedding  A  2  exists i n P  5, " , V , and moreover i s monomorphic, i . e . , i f m(P^) < nKP^)  preserves then  m  J  < ?  ( Z i e r l e r - S c h l e s s i n g e r , 1964, pp. 254-255, 260).  2  I t i s easy t o prove t h a t a monomorphic mapping preserves  <  i s i n j e c t i v e , i . e . , f o r any P^ t P^  m : P -»• 8  i n P,  which  m(P^) ^ mtP^).  Proof: Assume on t h e c o n t r a r y t h a t m(P^) = m ( P ) i n 8. Then s i 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 „ ) . S i n c e m i s 1 2 2 1  <  2  monomorphic, i t f o l l o w s t h a t contradicts  P^ f ?  2  .  P^ 5 P  Q.E.D.  and  2  - i '  p  P  2  t  n  u  s  P  i  =  p  w n 2  i s i n f a c t an imbedding  Theorem 3.1 says:  (A,<&)  of P  of P  P. ,P„ € P,,_ , 1 2 QML r t  P „_ , QML rt  into a  So t h e c o n t r a p o s i t i v e o f and P„ & P„ , t h e n 1 2 Or i n o t h e r words,  P_„ , which has V d e f i n e d f o r QML Theorem 3.1 y i e l d s : I f a P.,,. c o n t a i n s i n c o m p a t i b l e QML  elements, t h a t i s , i f t h e j o i n in  B.  P„ V P„ e x i s t s i n P 1 2 into a 8 i s impossible.  w i t h r e s p e c t t o an orthomodular l a t t i c e any  into  I f the join  an imbedding  n  And s i 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 o f P  8  i°  t h e n an imbedding  P^ v P  2  T  o f incompatible  (o,&) o f P_„ QML T  into a  P^ , P B  2  exists  i s impossible.  109  Then assuming t h a t t h e r e 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 i m b e d d i n g ( i , & ) o f a ^QML  ^  n t o  e x i s t s on  ^  a  QML  P„.„  e x i s t s I F F a complete c o l l e c t i o n o f b i v a l e n t homomorphisms(<^,&) , t h e above r e s t a t e m e n t  of the c o n t r a p o s i t i v e of  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 e q u i v a l e n t t o my Theorem A: ^QML  C O T V t a i n s  semantics  If a  i n c o m p a t i b l e elements, t h e n 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 = , & )  f o r P.  QML W T  i s impossible.  Summary  The g e n e r a l f a c t o f t h e i m p o s s i b i l i t y o f 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 t h e 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 s h o u l d be more s u b t l y demarcated a c c o r d i n g t o whether the s t r u c t u r e s a r e t a k e n t o be orthomodular l a t t i c e s  P.„  T  or  QML partial-Boolean algebras  P Q ^ ' a c c o r d i n g t o whether t h e semantic mappings a  a r e r e q u i r e d t o be truth-functional(i>,<&) o r t r u t h - f u n c t i o n a l ( i > ) ; and a c c o r d i n g t o whether t w o - o r - h i g h e r  d i m e n s i o n a l H i l b e r t space  o r t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t space  P^  P  structures  s t r u c t u r e s are being  considered. ^ X  I f t h e quantum  P  s t r u c t u r e s a r e t a k e n t o be o r t h o m o d u l a r  l a t t i c e s , then b i v a l e n t mappings which p r e s e r v e t h e o p e r a t i o n s and r e l a t i o n s of a  P  A  must be t r u t h - f u n c t i o n a l ( o , < i $ ) .  Then as suggested  by von Neumann  QML  and J a u c h - P i r o n and as proven i n S e c t i o n A, t h e mere presence o f i n c o m p a t i b l e elements i n a P ^ „ i s s u f f i c i e n t t o r u l e out any s e m a n t i c a l o r h i d d e n -  QML  T  v a r i a b l e p r o p o s a l which imposes t h i s s t r o n g c o n d i t i o n , f o r any  ,  d i m e n s i o n a l H i l b e r t space  2  P ~_  Q sMt r uLc t u r e .  two-or-higher  Thus from t h e orthomodular  l a t t i c e p e r s p e c t i v e , t h e 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 t h e p e c u l i a r l y non-Boolean f e a t u r e o f t h 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 i s t h e e x i s t e n c e o f i n c o m p a t i b l e magnitudes and p r o p o s i t i o n s . However, t h e 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 t h e semantic o r h i d d e n - v a r i a b l e mappings on t h e  ^Q^L  s t r u c t u r e s , a l t h o u g h s u c h mappings i g n o r e t h e l a t t i c e meets and j o i n s o f i n c o m p a t i b l e s and p r e s e r v e o n l y t h e p a r t i a l - B o o l e a n a l g e b r a f e a t u r e s of the  ^Q^L  structures.  structural  0r a l t e r n a t i v e l y , t h 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 can be t a k e n t o be p a r t i a l - B o o l e a n a l g e b r a s , where b i v a l e n t mappings w h i c h p r e s e r v e t h e o p e r a t i o n s and r e l a t i o n s o f a need o n l y 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  p r o o f s show t h a t any s e m a n t i c a l o r h i d d e n v a r i a b l e p r o p o s a l w h i c h 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 i m p o s s i b l e f o r any d i m e n s i o n a l H i l b e r t space R  P?~«  or  QMA  P^r  structures.  three-or-higher  But such s e m a n t i c a l  QML  o r h i d d e n - v a r i a b l e p r o p o s a l s a r e p o s s i b l e f o r any t w o - d i m e n s i o n a l 2  Hilbert  2  space P Q or PQ s t r u c t u r e s , i n s p i t e o f the presence of i n c o m p a t i b l e s , and i n s p i t e o f t h e f a c t t h a t t h e s e s t r u c t u r e s a r e non-Boolean i n t h e P i r o n 16 sense and i n the n o t - p r i m e u l t r a f i l t e r sense. M A  MIj  Notes 1 I t i s worth n o t i n g t h a t t h e s e v a l u e a s s i g n m e n t s would be a c c e p t a b l e t o t h e l a t t i c e t h e o r e t i c i a n s J a u c h (1968, p. 7 6 ) , Putnam (1969, p. 222), van F r a a s s e n (1973, p. 9 0 ) , Friedman and Glymour (1972, p. 1 8 ) . For t h e s e a u t h o r s do a s s o c i a t e t h e 0 element o f a P with contradictory p r o p o s i t i o n s and t h e 1 element w i t h t a u t o l o g i c a l propositions. So even though some o f t h e s e a u t h o r s do n o t d i s c u s s semantic p r o p o s a l s f o r P , a l l would a c c e p t t h e v a l u e a s s i g n m e n t s m(P A P ) = m(0) = 0 and 1  QML  m(P  V P ) = m ( l ) = 1, 1  2  2  f o r any proposed semantic mapping  m  on a  P  . QML  For example, Putnam e x p l i c i t l y d i s c u s s e s t h e c o n j u n c t i o n o f 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 i n c o m p a t i b l e , o n e - d i m e n s i o n a l subspaces  Ill  whose i n t e r s e c t i s t h e 0 subspace, e.g., our A = 0 and P^ A P^ = 0. Such a c o n j u n c t i o n 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 t o t h e v a l u e assignments m(P A P ) = 0 and m(P A p^) = 0.  2 Thanks t o Edwin Levy, L. P e t e r B e l l u c e , and R i c h a r d E. Robinson f o r a u d i t i n g these p r o o f s . 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, a d d i n g t h a t i t i s a s t a n d a r d p r o o f i n Boolean l a t t i c e theory. Dr.. Robinson suggested a more economical r e s t a t e m e n t o f the p r o o f s . 3  T h i s i m p o s s i b i l i t y h o l d s whether a semantics f o r a P i s taken t o be a complete c o l l e c t i o n o r a weakly 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 ( i ) mappings. The n o t i o n o f a weakly complete c o l l e c t i o n i s d e f i n e d i n n o t e 1 o f Chapter I I . The d i m e n s i o n o f a H i l b e r t space H i s t h e maximum number o f l i n e a r l y independent v e c t o r s i n t h e H i l b e r t space ( J a u c h , 1968, p. 20), and i s d e s i g n a t e d by t h e s u p e r s c r i p t n = 1,2 So any tf has  n  0  l i n e a r l y independent v e c t o r s ; t h i s s e t o f v e c t o r s a r e a b a s i s  f o r tf (Lande, 1972, p. 47). A b a s i s may be o r t h o g o n a l i z e d (by the Gram-Schmidt p r o c e s s ) and n o r m a l i z e d (by d i v i d i n g each v e c t o r by i t s l e n g t h ) y i e l d i n g an o r t h o n o r m a l b a s i s . Thus t h e maximum number o f m u t u a l l y n  o r t h o g o n a l v e c t o r s i n any H i s n. S i n c e each v e c t o r |y> corresponds u n i q u e l y w i t h t h e o n e - d i m e n s i o n a l p r o j e c t o r P^ = | Y Y | and t h e o n e - d i m e n s i o n a l subspace H^ w h i c h i s the range o f P^ , t h e maximum n  >  number o f m u t u a l l y o r t h o g o n a l , o n e - d i m e n s i o n a l any the a of And  H P  n  <  p r o j e c t o r s o r subspaces o f  i s n. .Each o n e - d i m e n s i o n a l p r o j e c t o r o r subspace i s an atom i n s t r u c t u r e o f t h e H i l b e r t space, and by Lemma A, any two atoms i n o r t h o g o n a l IFF t h e y a r e c o m p a t i b l e . Thus the maximum number a  r  e  m u t u a l l y c o m p a t i b l e atoms i n t h e P*V s t r u c t u r e o f any f f i s n. . • 0M so I c l a i m w i t h o u t p r o o f t h a t when a set of n mutually compatible n  or o r t h o g o n a l atoms i n a P ^ i s c l o s e d w i t h r e s p e c t t o the operations of  5  P_„ we o b t a i n an QM  A , v, "" L  mBS o f P*\, . QM  3  The t h r e e o r t h o g o n a l atoms i n each mBS o f t h e P which Kochen-Specker c o n s i d e r i n t h e i r Theorem 1 a r e r e p r e s e n t e d by a triangle i n t h e c o m p l e t i o n o f 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 s u b p o r t i o n o f t h i s g r a p h , which Kochen-Specker draw s e p a r a t e l y as I" , c o n t a i n s 13 p o i n t s (atoms) and e i g h t o v e r l a p p i n g t r i a n g l e s (mBS's) upon completion. There a r e 15 such s u b p o r t i o n s i n r , so t h e c o m p l e t i o n o f r c o n t a i n s 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 t h e p o i n t s p = a , nMk  Q  q  = b, and r = c , so t h a t t h r e e p o i n t s and two t r i a n g l e s a r e r e d u n d a n t , o 2 o Thus t h e c o n s i d e r e d by Kochen-Specker c o n t a i n s 192 atoms and 118 mBS's.  112  T h i s i m p o s s i b i l i t y h o l d s whether a s e m a n t i c s f o r a t a k e n t o be a complete o r 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.  P  is  7 I n p r e c i s e l y t h i s manner, t h e semantic mappings proposed by I . Hacking f o r t h e quantum P s t r u c t u r e s , namely, t h e e v a l u a t i o n s , s i d e - s t e p t h e Theorem A i m p o s s i b i l i t y proof. T h i s p r o p o s a l was made i n an u n p u b l i s h e d , 1974 paper w h i c h has s i n c e been r e s c i n d e d . N M T  g  A d i s p e r s i o n - f r e e G l e a s o n mapping i s b i v a l e n t , as mentioned by Gudder; a p r o o f i s g i v e n by Bub and r e s t a t e d i n note 9 below. And Gudder p r o v e s 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 a r e b i v a l e n t homomorphisms(6) (Gudder, 1970, pp. 433-434). A v e r s i o n o f t h i s p r o o f 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 I F F , f o r any quantum m e c h a n i c a l magnitude A, E x p ( A ) = ( E x p ( A ) ) . So w i t h r e s p e c t t o 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) = ( E x p ( P ) ) . That i s , Exp(P) = 0 o r 1. So an Exp on a P d i s p e r s i o n - f r e e I F F , f o r any element P € P , Exp(P) = 0 o r 1 ^ (Bub, 1974, p. 6 0 ) . 2  Q  2  M  ^ C o n d i t i o n (vNo) a l o n e 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#) t o g e t h e r a r e 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, 3 1 1 ) . 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 w h i c h p r e s e r v e t h e + o p e r a t i o n among c o m p a t i b l e o p e r a t o r s o r p r o j e c t o r s a l s o p r e s e r v e t h e • o p e r a t i o n among c o m p a t i b l e s (Kochen-Specker, 1967, p. 8 1 ) . S i n c e t h e A, v,operations o f a P s t r u c t u r e c a n be d e f i n e d i n terms o f t h e r i n g o p e r a t i o n s +, •, a s u s u a l among c o m p a t i b l e p r o j e c t o r s , mappings on a P w h i c h p r e s e r v e t h e +, • o p e r a t i o n s among c o m p a t i b l e s a l s o p r e s e r v e t h e A, V , o p e r a t i o n s among c o m p a t i b l e s . X  X  1-  x  12 The p r o o f 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) a s w e l l as w i t h c o n d i t i o n ( G a ) . 13 J a u c h - P i r o n ' s C o r o l l a r y 1 speaks o f c o h e r e n t 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 o r t h o m o d u l a r l a t t i c e ( J a u c h - P i r o n , 1963, pp. 831, 8 3 4 ) . 14 Z i e r l e r and S c h l e s s i n g e r ' s work was c a l l e d t o my a t t e n t i o n by P r o f . W. Demopoulos. 15 Some a u t h o r s do n o t make t h e s e d i s t i n c t i o n s . F o r example, M. Gardner c l a i m s t h a t , "Kochen and Specker have proven t h a t t h e r e i s no homomorphism o f P [ i . e . , P ^ ^ ] into !' (Gardner, 1971, p. 519)«. Gardner does c l a r i f y t h a t t h e homomorphisms c o n s i d e r e d by Kochen-Specker a r e homomorphisms(i). But Gardner does n o t m e n t i o n t h a t t w o - d i m e n s i o n a l ' ' 2 H i l b e r t space PQ^ s t r u c t u r e s a r e exempt from Kochen-Specker's p r o o f ; a  113  t h a t Kochen-Specker g i v e an example o f a homomorphisms ( i ) i n t o  Z^  Pq^  w h i c h does admit some  ( i . e . , b i v a l e n t homomorphisms(o)) b u t does n o t  admit a complete c o l l e c t i o n o f s u c h mappings; and t h a t t h e Kochen-Specker proof also applies to P s t r u c t u r e s , as e x p l a i n e d i n S e c t i o n s B and D. QML  16 Most o f t h i s c h a p t e r i s t o be p u 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 s e m a n t i c s f o r t h 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 , " i n a •forthcoming i s s u e o f P h i l o s o p h i a .  114  CHAPTER V I  A STATE-INDUCED SEMANTICS FOR THE NON-BOOLEAN PROPOSITIONAL STRUCTURES DETERMINED BY QUANTUM MECHANICS  S e c t i o n A.  The Quantum S t a t e - i n d u c e d  Expectation-Functions  As d e s c r i b e d i n Chapter I V ( A ) , t h e quantum f o r m a l i s m 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,  system by a o n e - d i m e n s i o n a l p r o j e c t o r on A  r e p r e s e n t s each pure s t a t e \|r o f t h e H,  and r e p r e s e n t s each magnitude H.  o f t h e system by a s e l f - a d j o i n t o p e r a t o r on  o f each o f t h e system's magnitudes ^  A,B, , . .  o f t h e system a c c o r d i n g t o t h e e x p r e s s i o n  one o f t h e real-number e i g e n v a l u e s t h e system i s one o f t h e e i g e n s t a t e s Exp. (A) = a . . j  The e x p e c t a t i o n  value  i s determined by t h e s t a t e  Exp^.(A) = < ^ | A | \ ) / > ,  ^ ' i ^ i ^ index  °^  ^j^i^index  °^  A  w  A  '  n  e  the state o f  n  e  which i s  *S'>  Thus, when t h e s t a t e o f a system i s an e i g e n s t a t e o f any o f  :  t h e system's magnitudes, t h e n t h e s t a t e o f t h e system d e t e r m i n e s t h e e x a c t v a l u e s o f t h o s e magnitudes v i a t h e e x p e c t a t i o n - f u n c t i o n . o f t h e system i s n o t an e i g e n s t a t e o f a g i v e n magnitude d e t e r m i n e s t h e p r o b a b i l i t i e s o f t h a t magnitude i t s eigenvalues  according to the expression  A  When t h e s t a t e \|f A,  then the s t a t e  e x h i b i t i n g any one o f  p. ( a . ) = Exp(P. ) , where &  Y»A  1  •  P. i s the projector representing the eigenstate associated with the i . eigenvalue a ^ . So f o r any o f t h e system's magnitudes, each pure s t a t e ^ o f t h e system d e t e r m i n e s , v i a t h e 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 t h e v  1  115  e x a c t real-number v a l u e o f t h e magnitude o r t h e p r o b a b i l i t i e s o f t h e magnitude e x h i b i t i n g any one o f i t s e x a c t ( e i g e n ) v a l u e s . state Exp^,  the expectation-function  Exp^,  unambiguously d e f i n e s t h e s t a t e  \|/  I n f a c t , f o r any pure s t a t e  i s unique t o  \|/,  and  conversely,  (Fano, 1971, p. 399). the expectation-function  a c t s as a mapping from t h e s e t o f magnitudes r e p r e s e n t e d t h e real-number l i n e , i . e . , Exp^. : {A,B, (Ea)  And f o r any pure  . . .}  ->• R,  Exp^.  by o p e r a t o r s t o  which s a t i s f i e s :  Exp^(A + B + . . .) = Exp^.(A) + Exp^(B) + . . . A>6  (Eb)  If  (Ec)  E x p ^ ( I ) = 1.  For any pure s t a t e Exp^, : {A,B,  Exp^(AY>0.  then  \|/,  . . .} -»• R  (Fano, 1971, p. 398; von Neumann, 1932, p.  a s s o c i a t e d mapping  .,", T  t h e u n i q u e l y a s s o c i a t e d mapping may be c a l l e d t h e quantum s t a t e - i n d u c e d  j u s t a s , f o r any pure s t a t e  w  308)  mapping,  o f a c l a s s i c a l system, t h e u n i q u e l y  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 ) , ( E b ) , ( E c ) , ensure t h a t , w i t h r e s p e c t t o t h e idempotent elements o f form a  P^  Exp^. : PQ^  s t r u c t u r e , each [0,1].  s t a t e - i n d u c e d mapping p r o b a b i l i t y measure  Exp^  {A,B,  . . .}  which  i s a p r o b a b i l i t y measure  C l a s s i c a l l y , t h e analogous r e s u l t i s t h a t each c l a s s i c a l w :{f , f , . . . } - » • R u.^ : P ^ -»• {0,1}  o f idempotent elements o f  i s a dispersion-free  w i t h respect to the  P^  M  structure  ^ '^B' * ' A  Moreover, as i n t h e c l a s s i c a l case d e s c r i b e d i n Chapter I I I ( C ) , t h i s m a t h e m a t i c a l machinery o f quantum s t a t e - i n d u c e d mappings on t h e s e t o f  116  o p e r a t o r s on a H i l b e r t space n o t o n l y f o r m a l i z e s t h e p r o c e d u r e by w h i c h real-number v a l u e s and p r o b a b i l i t i e s a r e a s s i g n e d t o t h e magnitudes o f a quantum system, b u t 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 p r o c e d u r e by w h i c h 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 a s s i g n e d t o t h e p r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e real-number v a l u e s o f a quantum system's magnitudes, as s h a l l be shown i n t h i s  chapter.  As d e s c r i b e d i n Chapter I V ( C ) , p r o p o s i t i o n s w h i c h make a s s e r t i o n s about t h e v a l u e s 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 o r subspaces o f t h e system's H i l b e r t space, and t h e l o g i c a l o p e r a t i o n s among t h e 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 o r d e f i n e d i n terms o f o p e r a t i o n s among t h e p r o j e c t o r s o r subspaces, y i e l d i n g a propositional structure mappings  P.,. . I n o r d e r t o d e s c r i b e how t h e s t a t e - i n d u c e d QM Exp^, a c t w i t h r e s p e c t t o a P^ , we f o c u s t e m p o r a r i l y upon t h e M  elements o f  P^  a s p r o j e c t o r s , w h i c h a r e by d e f i n i t i o n idempotent,  n  s e l f - a d j o i n t , bounded o p e r a t o r s whose o n l y e i g e n v a l u e s 0  and  1.  With respect t o a  state-induced  Exp, on *Y  p r o b a b i l i t y measure  P  QM  (ua)  P_„ QM  P^ n]  o f p r o p o s i t i o n s qua p r o j e c t o r s , each  s a t i s f i e s t h e f i v e c o n d i t i o n s which d e f i n e a  U-, a s l i s t e d i n Chapter I I I ( C ) .  € P  and f o r any  QM  2  V P 2  = P. + f 1 2  P^ , P*  - P 1  • P 2  0  and  P. A P = 6 1 2 P € P_„). QM  (since  A  1  P  1  So i f P. , P 1 2  Exp^, on a  P  then  J  P^ , P 2  2  1  = P • P . And i f 2 1 2  are d i s j o i n t , then  + P ) = Exp^CP^) + E x p ^ ( P > . 2  then  P 6 P  P. A P < P„ A P ^ = fl and 1 2 2 2  Thus by ( E a ) , f o r any d i s j o i n t = Exp^(P  P.  a r e d i s j o i n t , i . e . , P^ < P ^ ,  2  F o r any  :  As s t a t e d i n Chapter IV(D'), i f P^ 6 P* , 1  a r e t h e real-numbers  ,  P  and  P > 0  f o r every J  V P = P + P . 1 2 1 2  Exp^P^ v P > 2  And f o r any c o u n t a b l e s e t  117  {P. }." , 1 i€Index Exp,(V  P.) = 2  Thus ( LQ) i s s a t i s f i e d ,  Exp. (P.).  i (Mb)  P_„ , QM  o f p a i r w i s e d i s j o i n t elements o f  i P € P  Every element  i s by d e f i n i t i o n idempotent, i . e . ,  **2  A  P = P ;  i t f o l l o w s t h a t e v e r y element i s n o n n e g a t i v e , i . e . ,  P > 0  ( v o n Neumann, 1932, p. 3 0 8 ) . Hence by ( E b ) , f o r e v e r y  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 o p e r a t o r (Fano, 1971, p. 2 8 8 ) , Exp^(P) 5 °°, A  f o r every  P € P^  (Fano, 1971, p. 3 9 6 ) .  0 5 Exp^(P) < «»,  So we have:  P € PQ  f o r e v e r y element  .  M  Thus (ub) i s  satisfied. (uc)  By ( E b ) , i f P = 0 Cue)  ([il)  then  Exp^(£) = 0,  i . e . , Exp^(0) = 0.  i s satisfied.  I n Halmos's d i s c u s s i o n o f t h e p r o b a b i l i t y measure that the isotone character of \i  character of Exp^.  P  on a  i s shown as f o l l o w s .  1  1  P € P^  M  ,  A P )  T h i s r e s u l t f o r any  F o r any  i n particular, for P = P  ^»  P  because  A  2  U-, he says  u. f o l l o w s from t h e m o n - r i e g a t i v e  (Halmos, 1950, p. 3 7 ) .  Exp^(P ) < Exp^(P ) + E x p ^ ( P any  Thus  2  ^ PQM '  Exp^(P) > 0 f o r  A P^ .  And as s t a t e d  X  2  P  i n Chapter I V ( D ) , i f P < P , t h e n $ o P (where P A IFF P d> £ ) and P. A P = P ; and so by t h e m u t u a l 1 2 1 2 1 ±  2  P  2  2  x  0  A  compatibility of ^  P  2  = P  = P  A  2  V (P  A  2  and  P  2  2  A P^  = ExP^C^) +  2  A  (P  A  A P^).  = Exp^CP^ V ( ?  A  V P  1  A  , P  2  A  I = P  A  A  P^ , P  A  I  ) = (P  1  So i f P  < P  ±  A P^)).  and by t h e d e f i n i t i o n o f  2  A  2  A  ,  A  P) 1  Moreover, s i n c e  P^(P  A 2  P  1  X  )  *  H  E  N  C  E  '  F O R  A  N  V  2  P p  A  Exp^(P ) 2  I> A P  a r e d i s j o i n t , and so by ( u a ) : E x  A I  V (P  then  I,  X  2  <  p ,  P^  1  Exp^(P^ V ( P €  PQM '  2  L F  A P^ ))  118  V  5 P  ±  So (M-n)  then  2  E x p ^ ) 5 E x p ^ ) + Exp^(P  2  A P^) = E x p ^ ) .  E x p ^ i s an i s o t o n e mapping, t h a t i s , (|ii) i s s a t i s f i e d .  By ( E c ) , P € P  Q M  E x p ^ ( I ) = 1, ,  thus ( u i )  i ssatisfied.  that i s , Exp^ : P  0 < Exp^CP) < 1,  So c o n d i t i o n s ( E a ) , ( E b ) , ( E c ) , ensure t h a t an  And so f o r every  Q M  Exp. on a y  -  [0,1].  P.„ QM  satisfies  the f i v e c o n d i t i o n s w h i c h 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 Boolean s t r u c t u r e , on a non-Boolean a  P ^  e.g., on a P ^  P ^ , N  structure.  N  u. i s d e f i n e d on a  w h i l e t h e quantum  Exp^ i s defined  So t h e quantum e x p e c t a t i o n - f u n c t i o n s on  c a n be r e g a r d e d a s g e n e r a l i z e d p r o b a b i l i t y measures w h i c h s a t i s f y  A  a l l t h e u s u a l 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 b u t w h i c h a r e d e f i n e d on a non-Boolean  P.„ QM  s t r u c t u r e r a t h e r t h a n on a B o o l e a n  structure. 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  d e f i n e d 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 d e f i n e d by J a u c h - P i r o n  p. 8 3 3 ) . Bub and J a u c h - P i r o n  (Bub, 1974, p. 89; J a u c h - P i r o n , 1963,  agree t h a t t h e 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  ML  ,  *QML '  PQ^ »  i n s u c h a way t h a t on e v e r y (maximal) B o o l e a n s u b s t r u c t u r e o f t  h  e  §  e n e r a l : J  -  p r o b a b i l i t y measure non-Boolean  z e d  A  P  A M A  p r o b a b i l i t y measure r e d u c e s t o t h e c l a s s i c a l  p.. I n a d d i t i o n ,  P.„. , P.„ , QMA QML T  with respect t o the entire  b o t h Bub and J a u c h - P i r o n  require that a ^  g e n 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 r e s p e c t t o o r t h o g o n a l 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 n o t s t a t e t h i s r e q u i r e m e n t e x p l i c i t l y , b u t i t i s c l e a r t h a t 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 ( G a ) .  I n 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 t o 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 w h i c h amounts t o ( G a ) , namely, t h e c o n d i t i o n ( J P o ) s t a t e d i n Chapter V ( D ) .  And e l s e w h e r e , Jauch e x p l i c i t l y  imposes (Ga) r a t h e r t h a n ( J P o ) (1976, p. 1 3 5 ) . PQ  M  does s a t i s f y ( G a ) .  Any quantum  F o r as shown above, any  Exp^  Exp^, P^  on a  on a satisfies  M  (pa) w h i c h i s e q u i v a l e n t t o (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 a r e e q u i v a l e n t n o t i o n s , as s t a t e d i n Chapter I V ( 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  c o n d i t i o n (JP&) s t a t e d i n Chapter V ( D ) , and J a u c h - P i r o n quantum  further discussed  s a t i S :  fy t  n  e  claim that the  Exp^. mappings do s a t i s f y (JPi*) (1963, p. 8 3 3 ) .  impose t h i s c o n d i t i o n .  t o  Bub does n o t  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  i n Chapter V I I ; i n p a r t i c u l a r , J a u c h - P i r o n ' s  imposition  o f (JP{6) as p a r t o f t h e 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 s t a t e - i n d u c e d  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 ( a s d e f i n e d by e i t h e r Bub o r J a u c h - P i r o n ) on  PQ  M  ,  j u s t as each c l a s s i c a l s t a t e - i n d u c e d  mapping  w :  -*• {0,1}  u. : P „ „ -*• {0,1}, as d i s c u s s e d i n w CM Chapter I I I ( C ) . But t h e c l a s s i c a l measures [i a r e 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 t h e c o n d i t i o n (u.d) w h i c h e n s u r e s i s a c l a s s i c a l p r o b a b i l i t y measure  b i v a l e n c y , w h i l e t h e quantum measures v a l u e s between  0  and  1  Exp^  assign dispersive, p r o b a b i l i t y  t o some elements o f  P ., . QM n  Moreover, u n l i k e t h e  c l a s s i c a l measures w h i c h a r e t r u t h - f u n c t i o n a l mappings on Exp^  measures a r e n o t t r u t h - f u n c t i o n a l  Conditions  1  M  ,  t h e quantum  (o,&)) mappings on  P^  .  M  (Ea) and ( E c ) do ensure t h a t t h e quantum measures p r e s e r v e t h e -  operation of ExpCP ) = ,  ( ( i ) or  P^  1  P^  ,  i . e . , f o r any  Exp^  by s u b s t i t u t i o n , E x p ( l - P )  on a = ,  P^  and f o r any  P € P^  M  ,  by ( E a ) , E x p ( I ) - Exp(P) = ,  120  by ( E c ) 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  But t h e quantum measures do n o t always p r e s e r v e t h e A, v  PQM . F o r example, c o n s i d e r t h e p r o j e c t o r s P  =  2  |^><-^ j such 2  state  Y P  2  and  2  •  The  P^  Y  2  such t h a t  E x p ^ i n d u c e d by t h i s s t a t e  0  P* = 0;  A  Y  Y °  E X P  N3  E X P  Exp^(P) = 0. P  {  V '  E  X  Rewritten: P  Y  ><:  Y V (  t h e n by ( n i ) ,  P > P^ ,  |Y Y1  P^ =  In a d d i t i o n , f o r any element if  (  P  ) =1  <0  QM  respect t o the subset  2  =  ,  F o r any  X  '  ^  }  i f P & P^ ,  then  and i f P < P , X  P  E x p ^ on a  \|r which  =  ±  -  E  X  h  i  %  )  =  t h e n by ( u i ) ,  i f P > P, i f P  Y < Pf" Y  i f P fa P,  {P € P^  E x p ^ : P ^ -»• [0,1] A  : P > P^  or  And each quantum  i sbivalent with  P < P^ } ; by Lemma A o f which a r e c o m p a t i b l e  Exp^, a s s i g n s p r o b a b i l i t y v a l u e s between  t o a l l o t h e r elements i n P_„  ,  i . e . , t o a l l elements i n  QM which a r e incompatible w i t h  P^ ,  ° '  Exp^(P) € ( 0 , 1 ) :  and f o r any element  AM  A  1  H,  s t r u c t u r e o f t h e system's  nM  Chapter V ( A ) , t h i s i s t h e subset o f elements i n P ^  and  -*• [0,1]  r e p r e s e n t s t h e pure s t a t e  Exp^(P) = 1;  So each quantum e x p e c t a t i o n - f u n c t i o n  P^ .  E x p ^ : P^  w h i c h a r e c o m p a t i b l e w i t h t h e atom  I'^'i  =  P € P  £(0,1)  with  because, f o r  I n p a r t i c u l a r , a c c o r d i n g t o t h e quantum f o r m a l i s m , f o r any  QM '  P  and  \|r o f a quantum system i s b i v a l e n t w i t h r e s p e c t  namely, t h e subset o f elements i n P^  Exp^ .  P^, 9 P^  t 0.  2  t o a c e r t a i n s u b s e t o f elements o f t h e P  induced  d  a s s i g n s v a l u e s as f o l l o w s :  However, each quantum e x p e c t a t i o n - f u n c t i o n  P^, w h i c h qua p r o j e c t o r  n  and c o n s i d e r t h e pure  Y  i = 1,2, Exp^(p\) = j l ^ x ^ l l '  i n d u c e d by t h e pure s t a t e  a  b u t Exp,(p\) A E x p , ( P ) t 0 \|f 1 2  = 0,  (Exp(P)) .  operations o f  |y^><Y^I  P^ =  P^  r e p r e s e n t e d by t h e p r o j e c t o r  Exp. (P. A P ) = Exp, (6) f 1 2 each  P^ cV ?  that  2  ,  t h e atom w h i c h qua p r o j e c t o r  P_„  Q M P^  0  121  r e p r e s e n t s t h e s t a t e ty w h i c h induced  Exp^, . P. € P^„ , ty QM i s a closed substructure o f  In t h e n e x t s e c t i o n , I s h a l l show t h a t , f o r any atom J  {P € P  t h e subset  : P > P.  QM ^QM '  a  n  d  '  t  i e  ^P  6 0  ^ ^ 3  o r P 5 P \" }  ty 1 0 1 1 1 -  ((A)  ^  ^  1 1 1 1 0 1 1  ty 0 1 1  E x  s n  o  t  o n l  y b i v a l e n t but also  P^„ . QM The Quantum E x p e c t a t i o n - F u n c t i o n As an U l t r a v a l u a t i o n on an  truth-functional S e c t i o n B.  o r (o,&))  P ^ -*-  on t h a t s u b s t r u c t u r e o f  U l t r a s u b s t r u c t u r e o f P_„ QM As d e s c r i b e d i n Chapter 1 ( F ) , t h e n o t i o n s o f a f i l t e r and d u a l ideal are defined i n a  ?..„ QML  by t h e c o n d i t i o n s ( a ) , ( b ) , and t h e d u a l  c o 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 t h e s e n o t i o n s i n P Q J ^ w h i c h has t h e  a  A, V  o p e r a t i o n s d e f i n e d among o n l y  compatibles,  c o n d i t i o n s ( a ) and ( a ' ) a r e m o d i f i e d t o t h e c o n d i t i o n s (a„) and ( a ' ) g i v e n 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 satisfies the u n m o d i f i e d 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 QMA J  (a') and ( b ' ) .  As i n t h e c a s e o f a B o o l e a n s t r u c t u r e , an u l t r a f i l t e r P  ( u l t r a i d e a l ) i n a quantum  i s a proper f i l t e r ( i d e a l ) which i s not t h e  p r o p e r s u b s e t o f any p r o p e r f i l t e r ( i d e a l ) i n P^ . Making use o f Lemma B o f Chapter V ( A ) , i t i s easy t o p r o v e t h a t the subset o f elements  {P € P^^  i s an u l t r a f i l t e r  in  UF^  P  '• P - P^}, .  Q M L  a r e members o f t h e s e t S = {P € P  2  > P^ ,  then  satisfies (a). if thus  P  ±  - P S  2  >  P  1  A P  2  > P^ A P  F o r any P then s i n c e  € P ^  1  P  s a t i s f i e s ( b ) . So  2  F o r any  = P^  proper f i l t e r , t h a t i s , S 4 P  i.e., P  and f o r any P P  ±  ,  Q M L  and so P^ A P  > P^ we have S  € P  P ^  : P > P^}, 2  P^ € P ^  f o r any g i v e n atom  € S  2  > P^ ,  2  ±  i f P  M L  , ?  ±  2  > P^, and  € S; (i.e.,  thus P  2  S >P^),  and so P_ 6 S; L  i sa f i l t e r i n P„„ . Moreover, S i s a QML , e.g., 0 J? S s i n c e 0 Z P^ . And T  Q M L  122  finally,  i s n o t t h e p r o p e r s u b s e t o f any p r o p e r f i l t e r i n P Q ^ •  S  assume on t h e c o n t r a r y t h a t t h e r e i s a p r o p e r f i l t e r S  F.  c  P € P „ such t h a t QML  Then t h e r e i s a n element  i.e.,  P £ P^ .  so by ( a ) , Thus,  Since  P^ € S c F,  P A P^ € F.  0 € F,  RT  But since  P^ , P  O  R  such t h a t  b u t P A S,  a r e members o f F and  P t P^ , by Lemma B,  F = P  and so by ( b ) ,  P €F  T  both  i n PQ^L  F  P  P A P^ = 0.  , w h i c h c o n t r a d i c t s t h e assumption  A M T  QML  that  i s a p r o p e r f i l t e r i n P_„_ . Q.E.D. QML The p r o o f t h a t t h e s u b s e t o f elements  F  U I ^ i n PQ proceeds d u a l l y . S i m i l a r l y , t h e subset o f elements  ultraideal  P, € P_.,. , y QMA  F o r any Pj_>P2 ^ QMA '  ^  P  l' 2  P  P  a  S = {P € P „ „ . : P > P.}, QMA ty  i.e., P  d €S  such t h a t  and d 5  (a ).  F o r any P  l ~ 2 ' ?  t  h  e  n  s  i  1  d <  n  c  € P  1  e  s a t i s f i e s ( b ) . So  P  Q  r  e  m  e  h  m  > P.  P  w  e  n  a  v  e  P  e  r  €S  ?  a  d = P^;  f i l t e r , that i s , S 4 P Q  ,  M  (i.e., n  d  s  o  P l  P  €  since  t h e r e i s an element Since is a  P^, € S c F, d €F  0 5 P^ by ( b ) ,  P € P^  m  both  such t h a t  such t h a t  P^, , d 5 P^  and P^ 5 P^ . However,  P  in P P € F  S ;  P^, 2 P,  Q  M  . Q.E.D.  h  S  u  s  S  i s a proper  b u t P £ S,  Since  S  r  S c F.  P^  Then  i . e . , P Z P^ .  and so by ( a > , H  there  i s an atom, o n l y  and so by ( a ) , H  F = PQJ^ » w h i c h c o n t r a d i c t s t h e assumption t h a t  filter in P  t  . such t h a t  a r e members o f F  and d 5 P.  satisfies  • P ° assume on t h e  a  F  S  0 Z• P' . And f i n a l l y ,  i s n o t t h e p r o p e r s u b s e t o f any p r o p e r f i l t e r i n PQJJ contrary that there i s a proper f i l t e r  v  > P^), i f  2  A  e.g., 0 £ S  n  as shown n e x t .  thus  i s a f i l t e r i n P .. • Moreover, QMA n  a  then there i s a  ty  namely,  l ~ ty '  o r  °f t h e s e t  s  2  2  ^  P  and P„ > P, ,  ty  ,  P - ^>  UF. i n P„... , \|r QMA '  and f o r any P  M  2 ~ \|/  S  {P € PQJ^  i s an u l t r a f i l t e r  A  s  P  T  ML  g i v e n atom  [ {  {P € P . „ : P 5 p f } i s an QML ty  0 € F. F  But then  i s a proper  123  {P € P  The p r o o f t h a t t h e subset o f elements UI^. i n P  an u l t r a i d e a l  : P < P^"}  is  proceeds d u a l l y .  A M A  Any s u c h u l t r a f i l t e r r e s p e c t t o an atom o f  N M A  P ^  UF.  and d u a l u l t r a i d e a l  UI,  defined with  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  A  u l t r a i d e a l , r e s p e c t i v e l y , a s mentioned i n Chapter 1(C) and ( F ) . o f an i n f i n i t e d i m e n s i o n a l H i l b e r t  space  P^  ,  I n t h e case  n o t every u l t r a f i l t e r and  2 not every d u a l u l t r a i d e a l i s p r i n c i p l e .  N e v e r t h e l e s s , s i n c e a quantum pure  s t a t e , as r e p r e s e n t e d by a v e c t o r i n H i l b e r t structure of Hilbert  P ^  s p a c e , i s an atom i n t h e  A  space, each ( p u r e ) s t a t e - i n d u c e d mapping i s _ d e f i n e d  w i t h r e s p e c t t o a p r i n c i p l e u l t r a f i l t e r and d u a l p r i n c i p l e u l t r a i d e a l i n ^QM  " ^°  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 ,  principle ultraideals, labeled semantics  UF^ ,  labeled  UF^. , and  i n t h i s discussion o f a state-induced  P.„ . QM As mentioned above, any f i l t e r i n a for a  ( a ) , ( b ) , and any i d e a l i n a filter ina  T  by d e f i n i t i o n  satisfies  by d e f i n i t i o n s a t i s f i e s ( a ' ) , ( b ' ) .  Any  P Q J ^ 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 ( b ) and a l s o s a t i s f i e s ( a ' ) . UF^  PQ^A  ^y d e f i n i t i o n s a t i s f i e s  Moreover, i t i s easy t o show t h a t any  1  ultrafilter  P_„ QML  and d u a l u l t r a i d e a l  UI^ i n a  PQ  M  s a t i s f y the conditions  ( c ) and ( c ) s t a t e d i n Chapter 1(C): 1  (c)  F o r any  P € P  (c')  F o r any  P € P^  Proof:  F o r any  And f o r any  Q M  P € P^  P € P  Q M  ,  , p  X  ,  ? S UF  ,  p € X  IFF P € U I ^ .  U I ^ I F F P € UF^ .  X  p €  +  UF^ I F F P"*" > P^ I F F P ^ - P I F F P € U I ^ .  € UI^ IFF P^ < P ^  I F F P^.< P I F F 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 U I ^ = {P € . P , : P >-P o r P <-P } X  Q|{  +  124  A, v , o p e r a t i o n s o f  is closed with respect to the ? *V  For any elements P P P  A P  1  2  €  l 2 % ' VP  P  2  * Q M'  €  UF^ , by ( b ) P  € U I ^ and  1  1  €  P  by  € UI^ .  X 2  (b  '  v P  1  }  P  i f  o  t  h  I f both  € U I  l' 2  P  € UF^  2  l 2 t AP  b  ?  €  \)/ '  U F  (since P ^  ( s i n c e  i  P  ?  t  h  n  b  ^  y  v P >,  and by ( c ' )  2  t h e n by ( a ' )  ~V'  2  e  ±  € UI^ , A P  n  < P  ±  3 a s shown n e x t .  P^ ,  and by  ( c )  € UF, and P € UF; . I f P„ € UF, and P € UI; , t h e n by ( b ) 1 Y 2 y i f Y P v P € UF^ , by (b') P A P 6 U I ^ , by ( c ' ) P ^ € U I ^ , and by ( c ) X  X  n  2  1  P  2  1  2  € UF, . S i n c e a f i l t e r 2 y nonempty and s i n c e , f o r any  F  X  and an i d e a l  P € F,  i t f o l l o w s by ( b ) t h a t t h e 1  o t h e r words, l e t t i n g  US,  °  P 5 1,  element o f  J i t f o l l o w s by (b') t h a t t h e 0  I  element o f  l a b e l the union  a r e each by d e f i n i t i o n  and f o r any P_„  P € I , P > 0,  i s a member o f  QM P_ Q Mi s a  y UI, . I n  \|/  member o f  w  UF. U U I . ,  Y  UF, , and  we have  Y  0 € US, ,  Y  Y  1 € US^ , and f o r any elements P ^ P g Q M ' l' 2 y ' P„ A P„ € US, , P. V P € US, , and P € US, . Thus, f o r any atom 1 2 1 2 \|/ 1 2 \|c P, € P ™ > "the s u b s e t US, = UF , U UI, i s a c l o s e d s u b s t r u c t u r e o f P.„ €  Y  Q M  Y  P  i f  Y  Y  -  \(r  w h i c h 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 . subalgebra  P  P  €  U  S  t  h  6  n  X  of  P.„. ,  and  QMA  US,  Y  i s analogous t o t h e r e s u l t :  4  Specifically,  i sa sublattice  of Q  US^  M L . P^„  This  T  8  I n any B o o l e a n s t r u c t u r e  i sa result  (algebra or  l a t t i c e ) , t h e u n i o n o f a f i l t e r and d u a l i d e a l form a s u b s t r u c t u r e (subalgebra  or sublattice) of  8 ( B e l l and Slomson, 1969, p. 1 7 ) .  However, i t i s i m p o r t a n t US^  = UF^ U U I ^ w i t h r e s p e c t t o t h e A, V, "~  operations of  1  f o r any elements P, € US;  or  1 \|r P. € US . 2  t o note t h a t t h i s c l o s u r e o f  \|/  P  0  P^Pg € P € US. ,  2 F o r any  QM  ,  neither that i f  € US^  then  1 2 \|/ 1 such meets and j o i n s w h i c h a r e  0  € US,  then  P. € US,  or  Y  P.„ ,  Q M  US.  J  b u t whose c o n s t i t u e n t elements  Y Y  2  i na  ^  b o t h members o f US,  v P  P. A P  \j/  t h e m s e l v e s members o f  1  guarantees  n  nor that i f  US,  •  P  P^  are hereafter called  P„ , P„ 1  are not  2  U S - e x t r a meets and j o i n s .  —  Y  125  I t i s a l s o w o r t h n o t i n g t h a t , f o r any atom  i na  P ^ , the  US. i s t h e u n i o n o f a l l t h e B o o l e a n s u b s t r u c t u r e s i n P.,, ty QM P^. , and i n p a r t i c u l a r , US^, i s t h e u n i o n o f a l l t h e  ultrastructure which c o n t a i n  _Q>3  overlapping  i n P^^  mBS's  which c o n t a i n {P € P^  p r e v i o u s s e c t i o n , by Lemma A,  P^, . As mentioned i n t h e  : P > P^  M  or  P 5  }  = {P € P Q : P A P^}, t h a t i s , US^, i s t h e ( u n i q u e ) subset o f a l l elements i n P.., w h i c h a r e c o m p a t i b l e w i t h P. . L e t mBS. . be any QM * ty ty,i M  mBS  J  in  P^„ w h i c h c o n t a i n s  QM  P. ,  and l e t  U  ty  mBS. .'s i n P_„ . Y,i QM  mBS, . be t h e u n i o n o f a l l such  .  tyyl  I t i s easy t o show t h a t , f o r any g i v e n atom  '  J  J  °  ty  P. € P.,, QM  P € P „ „ , P € US. I F F P € U mBS, . . I f P € US,, QM ty ty,i ty and so t h e s e t o f elements {P-,P ",P^,P^ ,0,1} form a B o o l e a n  and f o r every element  9  then  p i  i  J  s u b s t r u c t u r e i n P.,, QM  which contains  P. ty  X  and w h i c h , by Zorri's lemma, i s J  i t s e l f c o n t a i n e d i n some maximal B o o l e a n s u b s t r u c t u r e P. ; Y  thus  P € U mBS. . . i  and so  P € US^, .  mBS^ ^  containing  elements i n an  Q.E.D.  Conversely,  i f P f U  So f o r any g i v e n atom  P^ , mBS^ ^ £ US^ c P ^ .  mBS^. ^  a r e compatible  with  .  mBS. . w h i c h c o n t a i n s  mBS, . , t h e n ty,i  P^, (! P ^  M  ?6  P. , ty  and f o r every  i np a r t i c u l a r , a l l the  P^, and a r e a l s o m u t u a l l y  c o m p a t i b l e , w h i l e a l l t h e elements i n US^ a r e c o m p a t i b l e w i t h P^ b u t 5 need n o t be m u t u a l l y c o m p a t i b l e . S i n c e , as d e s c r i b e d i n Chapter I V ( F ) , 2 the mBS's i n a t w o - d i m e n s i o n a l H i l b e r t space ^ do n o t o v e r l a p , e.g., 2 2 any atom P^ € P ^ i s a member o f o n l y one mBS i n P ^ , M  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 B o o l e a n s u b s t r u c t u r e o f P^„ . B u t s i n c e t h e mBS's i n a QM three- o r higher-dimensional  H i l b e r t space  P^  M  may o v e r l a p , e.g., any  P, € P ^ may be a member o f many mBS's i n P^„ , US, = U mBS, . ty QM 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 atom  3  J  126  V  may c o n t a i n i n c o m p a t i b l e elements and t h u s may i n some sense be a P  non-Boolean s u b s t r u c t u r e o f  n>3 QM '  As s t a t e d i n t h e p r e v i o u s s e c t i o n , any v a l u e s as f o l l o w s :  P € P ^ ,  F o r any  A  P^  Exp^, on a  i f P > P;  1  Exp^,(P) =  assigns  M  i f P < P. Y 6(0,1) i f P ^ - P . Y, V P < P , }, and Y 0  Since  UF  +  = {P 6 P  Q M  : P > P^},  US, = {P 6 P - „ : P i P , } , Y QM y v a l u e s as f o l l o w s :  UI^  i t f o l l o w s t h a t any  F o r any  P ^'PQ > M  x  {P € PQM  Exp^(P) =  Exp. Y  on a  1  i f P 6 UF  0  i f P € UI  J(0,1) T h i s r e s u l t s u g g e s t s t h a t each ultrasubstructure be a f f i l i a t e d . )  US^ .  truth-functional Thus an US^,  ( H e r e a f t e r , an  Of c o u r s e , an  elements i n US', ^ .  Exp^_ on a  Exp. Y  P^  P_„  i f P^US^-UF^'UUI^  i s an u l t r a v a l u a t i o n on t h e  Exp^, and i t s US^  may be s a i d t o  i s b i v a l e n t with respect to the  Moreover, i t s h a l l be shown below t h a t an  ((A) o r (A,&))  Exp^, i s  w i t h r e s p e c t t o t h e elements i n US^ .  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) o r (A,<&))  d e f i n e d w i t h r e s p e c t t o t h e u l t r a f i l t e r UF^, and t h e d u a l  UI^, ,  t h a t i s , an  Exp^  assigns  mapping on ultraideal  i s an u l t r a v a l u a t i o n on t h e a f f i l i a t e d  ultrasubstructure  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 d u a l  conditionals. and f o r any  UI  UI^. i n a F o r any P ^  € P  P UF^  Q M  y i e l d t h e f o l l o w i n g b i c o n d i t i o n a l s and  N M  and d u a l  (qua  p^),  P. A P. € UF, I F F P. € UF, Y Y  UI^  f o r any  and  P^  ina  P„  F £ ±  € UF  M  ,  f o r any  ?  , P  2  Q M  P € PQ (qua  by ( a ) and ( b ) ;  P  ,  M  Q  M  )  127  P„ A P 1 2  € U l . I F P, € U l , y 1 y  or P  € U l , by ( b ' ) ; y  2  P. V P , ( UF, I F P. € UF. o r P <E UF. , by ( b ) ; 1 2 y 1 y 2 y ' -' ' P V P € U I ^ I F F P^^ € U I ^ and P € U I ^ , by ( a ' ) and ( b ' ) ;  U2  0  1  U3  2  2  P-- € UF^ I F F P € U I ^ , by ( c ) ; 1  P"" € U I ^ I F F P 6 UF^ ,  by C c ' ) .  1  I t c l e a r l y f o l l o w s t h a t t h e Exp^, on t h e elements i n UF. y  P^  which a s s i g n s t h e value  and a s s i g n s t h e v a l u e  0  1 to  t o t h e elements i n U l , y  s a t i s f i e s a l l o f t h e c o n d i t i o n s TF1, TF2, TF3, w h i c h d e f i n e a t r u t h - f u n c t i o n a l mapping and a r e l i s t e d i n Chapter 1 1 ( C ) , e x c e p t t h e f o l l o w i n g two: if  Exp^(P  I f Exp^(P  A P ) = 0,  1  V P ) = 1,  1  then  2  then  2  Exp^CP^ = 0  Exp^,(P ) = 1  or  1  or  E x p ^ ( P ) = 0.  Exp^,(P ) = 1; 2  These two  2  c o n d i t i o n a l s a r e m i s s i n g from t h e 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 E x p ^ because t h e f o l l o w i n g two c o n d i t i o n a l s a r e m i s s i n g from t h e 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 P  l  €  U  F  y  o  r  P  2  €  U  F  y '  I f  l  P  A  2  P  €  U  UF^ ,  y '  I  t  h  e  n  UI^, : l  P  €  U  I  If P y  °  r  V P  1  P  2  €  U  I  € UF^ ,  2  then  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 d i s c u s s e d n e x t . Using the d e f i n i t i o n ultrafilter  (d)  UF^  If P  I f we t a k e  P  1  i na  V P  2  P  i s prime I F F , f o r any  QM  € UF^ ,  t o be a  QM  P  s t a t e d i n Chapter 1 ( C ) , we s h a l l say t h a t an  then  P  ±  € UF^  and i f P If ? 1  Q M  or  2  ,  P.^  P  2  € P  ,  QM  € UF^ .  then  P^^ v P  2  d e f i n e d and so t r i v i a l l y , t h e a n t e c e d e n t o f ( d ) does n o t o b t a i n , 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  QM  * P  »  P  i s  P  r i m e  I F F  >  f  o  r  a n  y  p  !'  p 2  Q M  i s not So no  UI^ i na  128  (d')  I f P A P € Ul, , 1 2 Y Every u l t r a f i l t e r  then  P. € U l . 1 Y  or  P. € U l . . 2 Y  ( 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. P^  But as s t a t e d w i t h o u t p r o o f i n Chapter I V ( F ) , i f a  i n P Q ^ which i s n o t  i n c o m p a t i b l e elements, t h e n t h e r e i s some u l t r a f i l t e r P ^  prime; i . e . , i f a u l t r a f i l t e r i n P^  c o n t a i n s i n c o m p a t i b l e elements, t h e n n o t every  i s prime.  M  • r  contains  M  T h i s c l a i m s h a l l be proven w i t h t h e h e l p 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:  ,  Q M  i f UF^  P € UF,  Y  U  or  ina  P^  ,  M  i f UF^  I n o t h e r words, f o r any  (where  Y  UF. Y  P € P  s a t i s f i e s ( d ) , t h e n , f o r any  P € Ul.  i s prime, ina ,  Q M  either  U l . i s t h e u l t r a i d e a l dual",to Y  V  F o r any  UF^  P € UF^. o r P £ UF  X  P^  ina  ,  M  P £ UF^ .  Assuming t h a t For  1  P € UF^  or  either  1  P € UF  or  P r o p o s i t i o n B:  and so by  so i f P £ UF  P € Ul; .  then  P € UI^ .  So f o r any  Q.E.D.  V  I f a l l t h e atoms i n a  c o m p a t i b l e , t h e n every element  satisfies (d),  implies  Y  QM  either  1  P " € UF^ ; 1  UF^  ,  M  P V P"" = 1 € UF^ ,  € UF^ . And by ( c ) , P"" € UF^  P € P_„ ,  P € P^  and f o r any  P"" t UF.  implies  (d), either P  UF^  UF. U U l , = P „ , . Y Y QM  then P  F o r any  P 4 0  PQJ^ in  a  r  PQ^A  e  mutually 1  S  T  N  E  J °  I  N  °^  t h e atoms i t dominates. Let VP. I  P. l I  be any atom i n P _ „ . QMA J  such t h a t  P. < P, l  and l e t  be t h e ( 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 s u c h atoms.  (This  j o i n i s d e f i n e d because by a s s u m p t i o n , a l l t h e atoms i n P Q J ^  A  R  E  129  mutually compatible.)  The r e s t o f t h e p r o o f proceeds e x a c t l y as PQ  t h e p r o o f o f Lemma C i n C h a p t e r V ( A ) , w i t h  M A  substituted f o r  P  QML " Now t h e c l a i m s t a t e d above may be p r o v e n as f o l l o w s . Theorem B:  P.„ QM  I fa  every u l t r a f i l t e r Proof: and  UF. Y  J  i n P^  i n P^  UF^  i s prime.  M  P ^  contains incompatible  N  i s prime.  M  0  Y  M  i s c o m p a t i b l e w i t h every element i n P ^  M  i s c o m p a t i b l e w i t h every o t h e r atom i n P ^ , N  are mutually compatible. generates  P^  ,  M  t h a t i s , t h e atoms i n P ^ A  M  f o r as s t a t e d i n Chapter 1 ( D ) , ( E ) , ( F ) , any s e t o f P  elements i n a  g e n e r a t e a B o o l e a n s u b s t r u c t u r e when P ^ . Moreover, f o r P  by Lemma C o f Chapter V ( A ) , every element  j o i n o f t h e atoms i t dominates. P r o p o s i t i o n B, e v e r y element  every element  P  i n P^  P t 0  in  i sf a  r  e  n  e  in P M  qua i s the  QML P  qua  A M A  mutually compatible.  w i t h r e s p e c t t o t h e A, v,  P ^ . And so a l l elements i n P ^ N  which c o n t r a d i c t s t h e assumption t h a t  P ^  » by  j o i n o f t h e atoms i t Thus  i s a member o f t h e B o o l e a n s u b s t r u c t u r e  by c l o s i n g t h e s e t o f atoms i n P ^ operations o f  P ^ 0  And s i m i l a r l y , f o r P ^  d o m i n a t e s , where a l l t h e atoms i n P Q J ^  Q.E.D.  i n p a r t i c u l a r , each atom  I t f o l l o w s t h a t t h e s e t o f atoms i n P ^  closed with respect t o the operations o f »  P. i n i  1  mutually compatible  QML  ;  Thus each atom  a B o o l e a n s u b s t r u c t u r e when c l o s e d w i t h r e s p e c t t o t h e A, V, --  operations o f  PnMT  UF. U UI, = US, Y V  P. € P.„ . Q  f o r some atom  elements  Then by P r o p o s i t i o n A, f o r  i n P.„ , UF, U UI, = P.„ , where Q M Y Y Q M Y  = {P 6 P „ : P A P,} QM \|/ P  UF^  Assume on t h e c o n t r a r y t h a t  every u l t r a f i l t e r  every  contains incompatible elements, then n o t  are mutually  generated  J _  compatible,  c o n t a i n s i n c o m p a t i b l e elements.  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 with " u l t r a f i l t e r  UF^. ,"  contains incompatible  can a l s o be p r o v e n .  UI^,"  I n s h o r t , any  interchanged P^^  which  elements c o n t a i n s an u l t r a f i l t e r w h i c h does not  s a t i s f y (d) and c o n t a i n s an u l t r a i d e a l w h i c h does not s a t i s f y ( d ' ) , and thus c o n t a i n s an u l t r a s u b s t r u c t u r e P  QM  US^ = UF^ U UI^,  which i s a proper subset o f  ' US^, c p ^  However, f o r any P.,, QM  t h e elements o f  which are i n  ,  i f we r e s t r i c t o u r a t t e n t i o n t o  US. , y  t h e n we do have, f o r any J  P,,P„ € US. = UF, U U l , : 1 2 Y Y  Y  (d)  If  P  (d')  If  P, A P  V P  1  1 Proof: UF^ p  p  €  P  U  I  P. " A P 1 2 1  P  ,  then  P  € UF^  or  P  € Ul, ,  then  P.  € Ul,  or  P„ € U l , .  2  ±  Y  x  l ^ 2  Y *  S o  and  b y  = (P, v P ) 1 2  X  a n d  s  x t  f  o  l  l  o  P  '  ( c )  0  '  P  * UF^  1  w  p  * UF^  2  i"' 2 P  € UF. h  a  t  L  .  Y  t  Y  1  Assume on t h e c o n t r a r y t h a t  but  i' 2  € UF^  2  p  P^  . 6  U  v  p  (If  Y '  i S  P  s o  P. V P , 1 2  i»  P  2  ±  V P^  A (P  ±  v P ) 2  € UF^  c o n t r a d i c t s t h e assumption t h a t filter, in  P^  M  .  .  L  a r e  m  UF^  u  t  M  ,  u  a  l  l  y  M  ,  contrary that,  UF  y  M  ,  U UI^ = P  then Q M  ,  Q M  ,  which  i s an u l t r a f i l t e r , which i s a p r o p e r  i . e . , US^, = P ^  i f UF^ U U I ^ = P Q  then  Then by (a) a g a i n ,  US^,  Q.E.D.  t o be an  r a t h e r than  improper  US^, c p ^  above works as a p r o o f o f t h e c o n v e r s e o f P r o p o s i t i o n A: PQ  ,  QMA  compatible  UF^ = P  The p r o o f o f (d') proceeds d u a l l y .  P^  '  PQJ^  And so by ( b ) ,  I t i s noteworthy t h a t i f we t a k e substructure of  ( a )  ultrafilter  € US^ = UF^ U U I ^ ,  2  i s defined i n  ' 2' ' P  P  l f  b y  and so t h e i r meets and j o i n s a r e a l l d e f i n e d i n 0 = (P  member o f t h e  a  2  a n d  Y  2  Then s i n c e F  € UF^ .  2  UF^,  i s prime.  and f o r any  P ^  F o r any  Proof: * P  ,  Q M  then the UF^,  in a  Assume on t h e ,  P  1  V P  2  € UF^  131  but  P. * UF.  P . £ UF. . 2 y  and  I t y  P  Then s i n c e ty  = UF. U UI. , ty  A M  P,,P„ € U I . 1 2 ty  The r e s t o f t h e p r o o f c o n t i n u e s as above t o t h e end o f t h e p e n u l t i m a t e sentence.  Q.E.D.  IFF UF^ U U I ^ = P  Thus we have: .  Q M  F o r any  UF^, i n a  And e q u i v a l e n t l y , f o r any  i s n o t prime I F F UF^ U UI^, + P  Q M  ,  P^  UF^  ,  UF^, i s prime P  ina  i . e . , I F F UF^ U U I ^  ,  Q M  UF^  i s a proper  P.„ . QM  substructure of  Nevertheless, the p o i n t o f the proof given i n the paragraph p r e c e d i n g t h e p r e v i o u s p a r a g r a p h i s t o show t h a t , even when i . e . , even when  U I ^ a r e each n o t prime i n P ^  UF^, and  UF^, U U I ^ c p ^ ,  with respect  US. = UF. U UI. c P  t o t h e elements i n t h e u l t r a s u b s t r u c t u r e  ,  UF.  Y  ty ty ty QM does s a t i s f y ( d ) and UI^, does s a t i s f y ( d ) , and so UF^, and UI^, may each be s a i d t o be prime w i t h r e s p e c t t o US^ . C o n c o r d a n t l y , f o r any atom P'I € P~,. » even when t h e s t a t e - i n d u c e d e x p e c t a t i o n f u n c t i o n Exp. on P . ty QM ^ *ty QM T  W  does n o t s a t i s f y a l l o f t h e c o n d i t i o n s l i s t e d as TF1, TF2, TF3, n e v e r t h e l e s w i t h r e s p e c t t o t h e u l t r a s u b s t r u c t u r e US^ the c o n d i t i o n s :  F o r any  p  P^ d> P^ 6 US  ) QML '  TF1  f o r any  >  QM  ( q u a  V P ) = 1  IFF  Exp^(P  1  V P ) = 0  IFF  Exp^CP^  1  IFF  E X  A P ) = 0  IFF  >  ( P  l  Exp (P y  2  2  A  1  P  2  )  2  =  E  E  *PY V (  X  P  1  ( P Y  }  PY V (  P  E x p ^ does s a t i s f y a l l P - ^ P j € US^ c ? ^  ):  t  2  - 0  or  Exp^(P ) = 0  = 1  or  Exp^(P ) = 1  2  2  = Exp (P ) = 0  Exp^(P) = 0  Exp^(P ) = 0  IFF  Exp^(P) = 1  t  2  E x p ^ i s an u l t r a v a l u a t i o n on t h e u l t r a s u b s t r u c t u r e  Exp^, ,  as d e f i n e d w i t h r e s p e c t t o  UF^_  (qua  = Exp (P ) = 1  IFF  1  ,  M  QMA  Exp^P ") = 1 i  Thus  P  P^  f o r any  1  E x t  TF3  c  ,  Exp (P vk  TF2  P € US c P^  c  and t h e d u a l  US^, ;  UI^, ,  that i s ,  i s a bivalent  132  ((i)  truth-functional  (o,&))  or  e x a c t l y , each quantum s t a t e - i n d u c e d (A)  assigns  0, 1  i s a subalgebra P  of  PQ^ *  A  n  d  e  a  c  (&,&)  US. , \j/  The t r u t h - f u n c t i o n a l m a  Y  (<}>,#)  0, 1  More  "truth-functionally  quantum s t a t e - i n d u c e d  n  assigns  i t s a f f i l i a t e d ultrasubstructure  M L  PQJ^  Exp^ on a  v a l u e s t o t h e elements i n i t s a f f i l i a t e d  truth-functionally  US^ c P Q  US^ = UF^ U UI^. .  mapping on  US^ ,  which  Exp^, on a  v a l u e s t o t h e elements i n  which i s a s u b l a t t i c e o f  character of  P„„ . QML T  Exp^ on t h e domain  seem s u r p r i s i n g i n t h e l i g h t o f t h e Chapter V(A) d e s c r i p t i o n (o,&)  of the t r u t h - f u n c t i o n a l i t y  o f i n c o m p a t i b l e elements i n P ^ f o r any elements  P^ , P^  V  PQ^  operations o f  problems caused by t h e meets and j o i n s .  M L  Yet  Exp^  i n US^ c P Q ^ .  among any c o m p a t i b l e  US^ ; i n o t h e r words,  Exp^  satisfies  Thus  TF1, TF2, TF3,  Exp^, p r e s e r v e s t h e  A,  and i n c o m p a t i b l e elements i n  i s truth-functional  (A,&)  on  US^ c P Q  M L  .  As mentioned i n Chapter V ( C ) , Friedman and Glymour p r o p o s e , f o r t h e quantum  P  structures,  A M T  QML  preserve the -  s e m a n t i c mappings w h i c h a r e r e q u i r e d t o  o p e r a t i o n and t h e S  L  r e q u i r e d t o p r e s e r v e t h e A, v  Friedman-Glymour  PQ^  but a r e not  o p e r a t i o n s among e i t h e r c o m p a t i b l e o r  PQ^ •  i n c o m p a t i b l e elements o f  relation of  However, i t i s easy t o show t h a t a  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  P  of  ,  j u s t l i k e t h e quantum s t a t e - i n d u c e d  QML  Exp^,  mapping.  The Friedman-Glymour v : PQ ^  S3-valuations conditions: (i) (ii)  {0,1} and need o n l y s a t i s f y t h e f o l l o w i n g two  M  F o r any  p 1  s e m a n t i c mappings a r e c a l l e d  »  ^ QML '  p  P  2  v ( P ) = 1 IFF v t P ^ ) = 0 1  I f v(P )= 1 1  and  P  1  < ?  2  ,  then  v ( P ) = 1. 2  I t f o l l o w s from ( i ) , ( i i ) , t h a t , f o r any S 3 - v a l u a t i o n  v  on a  P^„ QML T  and  133  f o r any g i v e n element v(P) = 1 P > P  Q  i fP > P  Q  P  € P^  Q  Q  and v ( P ) = 0  i f P < P^".  , t h e n by ( i i ) , v ( P ) = 1. And s i n c e  v(P ) = 1  and P < P  t h e n by ( i ) ,  x  ,  then  P  2 P  X  P  i s a n atom  Q  ultraideal by  U I , i n P„ Y  WT  QML  {P € P  and t h e s e t  T  < P , i f X  Q  L  x  UF. i n P.„ ty QML  ,  F o r i f v ( P ) = 1 and  P < P ^ IFF P  t h e n as shown i n t h i s s e c t i o n , t h e s e t {P € P g ^ ultrafilter  M L  and so by ( i i ) , v(P" ) = 1, and  v ( P ) = v ( ( P ) ) = 0. When x  P € PQ  , i f v ( P ) = 1, t h e n f o r any  M L  :  AMT  QML  in P  P^  P - P^)  is  : P 5 P\f~}  a  QML  ,  n  i sthe dual  Y  . And i t f o l l o w s from t h e c o n d i t i o n s s a t i s f i e d  UF^, and U I ^ t h a t a mapping l i k e t h e S 3 - v a l u a t i o n w h i c h a s s i g n s t h e  value  1  t o t h e elements i n UF^ and a s s i g n s t h e v a l u e  0  t o the  elements i n UI^, i s n o t o n l y b i v a l e n t b u t a l s o t r u t h - f u n c t i o n a l ( i , & ) o f P.„  t h e u l t r a s u b s t r u c t u r e UF, U UI, Y  5  Y  Q  T  . So b e s i d e s b e i n g b i v a l e n t and  M L  P Q ^ » f^e S3^valuations a r e  preserving with respect t o the entire  a l s o b i v a l e n t , truth-functional(«b,^S) 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  expectation-functions. Of c o u r s e , f o r any atom US, = UF, U UI, ty  r  e  u l t r a v a l u a t i o n s on t h e f h e quantum s t a t e - i n d u c e d  P^, € P^  ,  i fthe ultrasubstructure  i s a n improper s u b s t r u c t u r e o f P_„ , ty ty QM  then t h e quantum e x p e c t a t i o n - f u n c t i o n  i . e . , i f US, = P „ . , ty QM  Exp^, » w h i c h i s induced by t h e pure  s t a t e r e p r e s e n t e d 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) u l t r a v a l u a t i o n on t h e e n t i r e  P  on  structure.  o r (<!>,&))  In p a r t i c u l a r , as described  i n t h e d i g r e s s i o n p r i o r t o t h e p r o o f 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 c e n t r e w h i c h i n c l u d e s a n atom d i g r e s s i o n ) o f P^  , so t h i s  P, ( l a b e l e d ty P^ i s c o m p a t i b l e w i t h e v e r y  P i n the c P € P^ ,  t h e n t h e 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 t h e 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 t h e value  elements i n UF^, and a s s i g n s t h e v a l u e  0  1  to the  t o t h e elements i n U I ^ ,  134  namely, the state-induced assigns  0, 1  Exp^ ,  t r u t h - f u n c t i o n a l l y ((<b)  values to every element i n  However, i f  P^  UF^  and so by the converse of Proposition A, .  Pq^ = US^ = UF^ U UI^ .  contains incompatible elements, then as shown  by Theorem B, there i s some u l t r a f i l t e r  UF^. U UI^. c  (<J>,i4))  or  in  P^  which i s not prime,  UF^ U UI^ 4 Pq^ , i . e . ,  i t i s p r e c i s e l y because every quantum  P^  containing  incompatible elements has at l e a s t one ultrasubstructure which i s smaller than the entire  P„„ QM  that I have chosen to assign  0, 1  the elements of any propositional or l o g i c a l structure  if  rather than according to Sikorski's d e f i n i t i o n of a bivalent  if  P £ UF.  B,  e.g.,  P € P,  v(P) =1  P € UF  if  and  P € UF  v(P) = 0  and  v(P) = 0  With respect to a Boolean propositional or l o g i c a l structure L  UF U Ul = 8 Ul)  f o r any element  if  according to the  For any element  homomorphism:  v(P) = 1  P  definition: P € Ul,  P € P,  truth-values to  or  P  ,  the two d e f i n i t i o n s are equivalent because  f o r every  UF  and dual  Ul  i n a Boolean structure i s prime.  in  8,  since every  UF  (and dual  So each may be regarded as 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 Boolean propositional or l o g i c a l structure  8.  mapping on a  That i s , each d e f i n i t i o n defines a bivalent, truth-functional B  with respect to an  UF  and dual  Ul  in  8;  such a mapping  i s c a l l e d an u l t r a v a l u a t i o n because, with respect to the Lindenbaum algebra L  of c l a s s i c a l propositional l o g i c , such a mapping i s the algebraic version  of a standard valuation. whenever  UF U Ul  c  P.  But the two d e f i n i t i o n s are not equivalent  In p a r t i c u l a r , the two d e f i n i t i o n s are not equivalent  with respect to a quantum  Pq^  which contains incompatible elements and  thus contains at least one ultrasubstructure According to both d e f i n i t i o n s , any  UF, U Ul, P„ . ty ty QM P € P^ such that c  w  135  P^€  UF^, U U I ^  value  0  i s assigned the v a l u e  i f P 6 UI^ ,  P € U I ^ IFF P £ UF^, . UF^, U UI^,  PQ  c  ,  M  P„„ QM  P € P_„, QM  and i s a s s i g n e d  P € UF^  both d e f i n i t i o n s are equivalent.  P^„ • QM  elements o f  P € UF^  because f o r any s u c h  0, 1  U UI^ ,  In p a r t i c u l a r , a  ((A)  (o,&))  or  on t h e u l t r a s u b s t r u c t u r e  But t h e two d e f i n i t i o n s d i f f e r w i t h r e s p e c t t o t h e which are outside o f a given  such t h a t  P £ UF, II U l , c p y y QM  However, t h e assignment o f t h e v a l u e  UF, U U l , y y  0  P  P £ UF. U U l . , y y '  i . e . , P £ UF. y  and  i.e.,  or  If  P  £ UF^  X  P^ € U I ^ .  t o every  which c o n t r a d i c t s the assumption Cc'),  P € UF^. ,  P £ UF^ U U I ^ , P"* £ UF^ 1  But  ,  P v P  P  X  € UF^  P £ UI^, .  £ UF^. U U I ^ .  X  ((^»)  = 1 € UF^  P £ UF^ U U I ^ c P  ,  ,  q m  and so (P V P ) X  V  or  UF^  .  according (o,<iO)  P € P^^  ,  i f  , And  P^ £ UF. U U l , , y y '  and •  t h e n by ( c ) , i f P  X  v(P V P ) x  P € UI^ ,  € UI^, ,  P £ UF^, .  t h e n by Thus i f  In p a r t i c u l a r , both  and so a c c o r d i n g t o t h e S i k o r s k i d e f i n i t i o n , x  according  F o r assume on t h e c o n t r a r y t h a t  which c o n t r a d i c t s t h e a s s u m p t i o n then a l s o  0  P £ UF^ U U I ^  For any  P £ Ul. , y ' P  Every  i s not a member o f  assignment, as shown by t h e f o l l o w i n g example. P^ £ UF^. U U I ^ .  P.„ . QM  &  t o t h e 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  then  c  i s assigned the value  t o t h e S i k o r s k i d e f i n i t i o n s i n c e e v e r y such  P £ UF^ II U I ^  the  values according to 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 c  if  So w i t h r e s p e c t t o a g i v e n u l t r a s u b s t r u c t u r e  mapping w h i c h a s s i g n s  UF', U U l , y y  1  = 1.  P,  v ( P ) = viP ')  = 0.  3  Hence, f o r any  = 1 j* 0 = v ( P ) V v ( P ) . x  So a l t h o u g h a  mapping w h i c h a s s i g n s v a l u e s a c c o r d i n g t o t h e 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 t h e e n t i r e on t h e 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&))  I n c o n t r a s t , t h e o t h e r d e f i n i t i o n which uses the  condition " i f P 6 U l " t h e q u e s t i o n s o f how  P.,, QM  r a t h e r than t h e c o n d i t i o n " i f P £ UF"  l e a v e s open  and what v a l u e s a r e t o be a s s i g n e d t o such elements  136  P f. UF, U UI, y Y  c  P„„ . QM  So the o n l y d i f f e r e n c e between the two  definitions  i s t h a t one l e a v e s t h e s e q u e s t i o n s open w h i l e t h e o t h e r a s s i g n s t h e v a l u e 0  t o t h e elements o u t s i d e a g i v e n u l t r a s u b s t r u c t u r e . ((A)  assignments a r e not t r u t h - f u n c t i o n a l  (A,&)),  or  p a r t o f 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 beyond s a t i s f y i n g i n a t r i v i a l way  Since these  P^  a c t u a l l y adds l i t t l e  M  t a k e n t h e d e f i n i t i o n w h i c h uses t h e c o n d i t i o n " i f P € U I " P_„ QM  value  i n c l u d i n g them as  the b i v a l e n c y desideratum.  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  0  Thus we have  as t h e  .  As d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s , a s t a t e - i n d u c e d ultravaluation outside the A, V  5  Exp^.  UF^, U UI^.  c  a s s i g n s v a l u e s between P^  r e l a t i o n of operations of  .  P_„ QM P^  And  Exp^,  0  and  1  t o t h e elements  does p r e s e r v e the  ^~  o p e r a t i o n and  as i t a s s i g n s t h e s e i n t e r m e d i a t e v a l u e s , but a r e not p r e s e r v e d .  b i v a l e n t nor t r u t h - f u n c t i o n a l  So an  ((A) o r (A,&))  Exp^,  the  i s neither  on the e n t i r e  P.,,  .  QM Friedman and Glymour propose t h a t t h e i r S 3 - v a l u a t i o n s on a  PQ  ML  '  w h i c h have been shown t o 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 o f ^nuT  >  also assign  ultrasubstructures.  0, 1  v a l u e s t o t h e elements o u t s i d e t h e i r  Most s i m p l y , t h e v a l u e  atom ( o n e - d i m e n s i o n a l  0  subspace) and t h e v a l u e  orthocomplement o f every atom ( t w o - d i m e n s i o n a l  may 1  be a s s i g n e d t o every  may  be a s s i g n e d t o t h e  subspace) o u t s i d e a g i v e n 3  P.,, QML  u l t r a s u b s t r u c t u r e o f a t h r e e - d i m e n s i o n a l H i l b e r t space Glymour, 1972,  p. 2 7 ) .  A g a i n , t h e -1  F*QML a r e p r e s e r v e d by such an u l t r a s u b s t r u c t u r e .  0  T  o p e r a t i o n and the  5  (Friedmanrelation of  v a l u e assignments t o t h e elements o u t s i d e  And t h i s p r o p o s a l a v o i d s a t l e a s t some o f the  truth-functionality(A,&) value  0, 1  affiliated  problems o f t h e more s i m p l e p r o p o s a l t h a t t h e  be a s s i g n e d t o every element 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 .  Friedman-Glymour do not d e s c r i b e how  0, 1  v a l u e s may  But  be a s s i g n e d f o r , s a y ,  137  4 PQ  a f o u r - d i m e n s i o n a l H i l b e r t space two-dimensional  M L  w h i c h has n o t o n l y one- and  subspaces b u t a l s o t h r e e - d i m e n s i o n a l subspaces o u t s i d e a n y 4  g i v e n u l t r a s u b s t r u c t u r e o f PQ^  l  • And o f c o u r s e , t h i s Friedman-Glymour  p r o p o s a l , and any o t h e r p r o p o s a l o f a b i v a l e n t semantics P _. 0  T  f o r t h e quantum  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>,iS) 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 )  p r o b l e m s , as  shown by Kochen-Specker. While addressing the issue 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  t y p e o f quantum p r o p o s i t i o n a l l o g i c , Levy p r o p o s e s  QMA  t h a t , b e s i d e s t h e 0, 1  v a l u e s a s s i g n e d by a s t a t e - i n d u c e d u l t r a v a l u a t i o n  t o t h e elements i n an u l t r a s u b s t r u c t u r e o f inappropriate, labeled  N,  »  a  t h i r d truth value,  be a s s i g n e d t o t h e elements o u t s i d e a g i v e n  u l t r a s u b s t r u c t u r e , . Such a t h r e e - v a l u e d s e m a n t i c s f o r a quantum P„,, I s , o f c o u r s e , n o t b i v a l e n t and i s a l s o n o t t r u t h - f u n c t i o n a l QML  o  (A) o r  T  (<b,&)),  as Levy m e n t i o n s .  truth-functionality  ((i)  r  An example o f a v i o l a t i o n o f '.• or  (A,#))  i s g i v e n a t t h e end o f t h e n e x t  section. T h i s Levy p r o p o s a l o f t h r e e - v a l u e d 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 p r e v i o u s p r o p o s a l s o f a t h r e e - v a l u e d f o r quantum p r o p o s i t i o n s . value  I  semantics  F o r example, Reichenbach a s s i g n s h i s t h i r d  truth  ( I n d e t e r m i n a t e ) t o quantum p r o p o s i t i o n s w h i c h a r e m e a n i n g l e s s  a c c o r d i n g t o t h e B o h r - H e i s e n b e r g i n t e r p r e t a t i o n o f quantum mechanics. particular, i f  P^  t h e n a t most one o f P^ , P^  the other i s meaningless,  and a l s o  m e a n i n g l e s s (Reichenback,  1965, pp. 143-145).  P^ & P^ 9  P^ A P^  i s meaningful  and P^ V P^  In while  a r e each  However, even though  t h e y may b o t h be t o g e t h e r i n some u l t r a s u b s t r u c t u r e o f P^  M  , in  w h i c h case b o t h o f them, and t h e i r meet and t h e i r j o i n a r e a l l a s s i g n e d t h e  138  usual  0, 1  t r u t h v a l u e s by t h e s t a t e - i n d u c e d  ultravaluation affiliated  with that ultrasubstructure. I n s h o r t , a l t h o u g h s e m a n t i c mappings on a v a l u e s between  0  and  1  which a s s i g n  o r 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  to  P.„ are not b i v a l e n t QM UF, U U I , c P^„ , nevertheless y \|A Q  t h e elements o u t s i d e a g i v e n u l t r a s u b s t r u c t u r e o f semantic mappings on t h e e n t i r e  P„„ QM  such mappings a r e t r u t h - f u n c t i o n a l b i v a l e n t , namely, on mappings f o r P ^ of  PQ  M  M  UF^ U U I ^ .  when  M  ((A) o r  (o,ciS))  wherever t h e y a r e  Thus t h e p r o p o s a l o f such semantic  has t h e v i r t u e o f c l e a r l y d e m a r c a t i n g t h e s u b s t r u c t u r e s  w i t h respect t o 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)  v a l u e assignments a r e p o s s i b l e , namely, t h e u l t r a s u b s t r u c t u r e s f o r any atom S e c t i o n C.  or  (<!>,&))  UF^ U U I ^ ,  P. € P - „ . Y QM An Example P^  C o n s i d e r t h e fragment o f t h e p r o j e c t o r s ) o f three-dimensional  M  s t r u c t u r e o f subspaces ( o r  H i l b e r t space diagrammed below:  1  0  139  T h i s fragment c o n t a i n s f o u r maximal B o o l e a n s u b s t r u c t u r e s : by t h e atoms  {P.,P ,P }, 0  generated 7  g  generated  o  {P^P^P,^,  and  mBS  mBS's  of these nine one-dimensional  subspaces o f  algebra generated  S  PQ^A  of  ^  s  t n  «  PQJJ^  operations of  6  by t h e atoms  h^,  , P  A, v, ^~  operations  by c l o s i n g  However,  S .with respect to the  among i n c o m p a t i b l e elements, as mentioned i n Chapter I V ( E ) . P  which i s compatible w i t h  g  t h e i n c o m p a t i b l e elements  P ty P_  So the j o i n P„ V P o  P  0  V P  0  0  P„ V P  ^  »  their join  ' 4  p  P  V P  c  b  ,  A, V  n  o  /  But now  P  .  V  P  V P^ = P ^ ,  3  o p e r a t i o n d e f i n e d among t h e s e two  =  P  u  a n d  S  °  P  3  V  P  = K  8  and so  P_ • A P 3 u  P, V P 3 u  = ( P A P-*-)^ = P -. 3 u v  P A y  P  g  P_.  And t h e  /  join  element i s i n t r o d u c e d  P ty P  and t h e meet o f t h e i r orthocomplements  8  r  A  b  ;  their  and a l s o  X  and  P <A 3  v  A ?  u  •  Thus  P  8  ^  X  P . u  X  1  P  A  1  Clearly, {P ,P g  ?  =  u  P^~  *  Let  P 3  P  > P  .P^}  X  v  C l e a r l  A P^ u  X  3  0  join  o  A pj~ 0  a r e each n o t  0  e q u a l t o any o f t h e twenty elements i n t h e above diagram. P  c  P  incompatible elements.  o  0  A  4  and  o  element.  so no new  c o n s i d e r t h e two i n c o m p a t i b l e elements  P„ V P  "  P.f'  4  S i m i l a r l y , c o n s i d e r t h e i n c o m p a t i b l e elements  Again t h e i r j o i n  by h a v i n g t h e  3  This  1-  i s an example o f what S t r a u s s would c a l l t h e l a t t i c e m i s i n t e r p r e t a -  b  Pg ty Py .  P  focus  Consider  = P." .  o P„ A P  where  '  P  P. V P 3  L e t us  ' 5  P  2  0  and t o  c  p  does not i n t r o d u c e any new  t i o n o f t h e element  0  ,  0  0  i s a l s o equal t o  A,.  denumerably i n f i n i t e and so e x e m p l i f i e s the  l S  p r o l i f e r a t i o n o f l a t t i c e elements due t o t h e l a t t i c e d e f i n i t i o n s o f  on the element  9^  t h e n the p a r t i a l - B o o l e a n  o f 20 elements diagrammed above.  t h e o r t h o m o d u l a r l a t t i c e generated V,  mBS,.  S = {P^*^' ' ' '  w i t h r e s p e c t t o the  f i n i t e fragment  4  o v e r l a p s i n c e t h e y share atoms.  I f we had s t a r t e d w i t h t h e i n i t i a l s e t  by c l o s i n g  c  5  0  generated  g  generated  {P ,P, j P } , 3  C l e a r l y , these f o u r  g  by t h e atoms  o  by t h e atoms  {P ,P ,P }.  mBS  0  z  1  mBS^  and  y'  P  = P P  Let  X  v  3  and  so  > P , v u  are three mutually  and  so  compatible  140  atoms w h i c h generate a n o t h e r 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 t h e s e elements a r e diagrammed below; f o r c l a r i t y , a l l t h e elements o f t h e f i r s t diagram have been o m i t t e d except t h e P  , PJ*", P 3  P , 0, 1  i_  But b e s i d e s  P, 5 P , 3 u  we a l s o have: X  X  D  8  elements:  X  Q  P A P 8 u  3  = P w  P > P , w u  P  5 P , u  and so  X  8 n  X 0  3  A P 8  P. <^ P 8 u  X. n  = P u  and so  and a l s o  P. v P = ( P A P )"* = P . 8 u 8 u w  and so  X  P i w  and so  X  when we l e t P  P 8 n  and  P i w  1  -  P . u  X  P, V P = P , 3 8 u  P "d>p ' . L e t 8 u A  J  r  Clearly, '  X  P > w X  J  Thus  {P„,P ,P } 8 u w  F 8 n  and  are three  J  m u t u a l l y c o m p a t i b l e atoms w h i c h g e n e r a t e y e t a n o t h e r maximal B o o l e a n s u b s t r u c t u r e , s a y mBS J  = ( P .VP ) A ( P V P 8 8 8 u X  D  (P  8 n  A P  Q  x  w  0  w 1  .  Moreover:  P. V P = P. V ( P A P ' ) 8 w 8 8 u X  ) = 1 A P = P . u u X  X  ) V (P A P ) = P„ v 1 = P„ , w w 8 8 X  So  L  P A P " = (P V P ) A P u w 8 w w X  and t h u s  1  X  ,  141  P  = (P AP ) = P VP A l l t h e s e r e l a t i o n s a r e diagrammed below; f o r 8 u w u w c l a r i t y , a l l t h e elements o f t h e f i r s t diagram have been o m i t t e d e x c e p t t h e x  JP  3  ' 3 P  ' 8  ' 8  P  ' °'  P  1  e  l  e  m  e  n  t  s  :  0 P  S i m i l a r l y , c o n s i d e r t h e two i n c o m p a t i b l e elements P  their join  3  V P  <iS P 3  y  and t h e meet o f t h e i r orthocomplements  P  X  3  A  ; 9 9  are  each not e q u a l t o any o f t h e 26 elements i n t h e above (combined) diagrams. Let  P^ A P 3 9  X  = P x  and so  P  0  3  vP  9  = (P 3  0  X  A P ^ ^ P . 9 x 1  Thus two more  elements have been i n t r o d u c e d , and as d e s c r i b e d above, by c l o s u r e , f o u r more elements  P  = P  AP ., P  x.  = P  0  •3  VP \, P x '  z  = P .AP i , and.^ P 9 x n  =. P  n  9  VP  x  will b  ,P } and {P ,P ,P } w i l l be two s e t s o f m u t u a l l y x y y x z c o m p a t i b l e atoms, each g e n e r a t i n g two more mBS's, mBS and mBS , in y z i n t r o d u c e d , where  {P„,P  y  \j  Q  142  the  3  P QML  generated by c l o s i n g t h e i n i t i a l  V  operations o f  P  <^J P z  o  PQ^ .  P  ty  g  ,  may be j o i n e d and meeted t o i n t r o d u c e even more elements.  incompatible with  the j o i n s o f  P^  P^ ,  P  element b e s i d e s  , Pg , P  g  , P  ?  Q  , P  A,  w i t h r e s p e c t t o the  Likewise, the incompatible p a i r s  c o u r s e , when we f o c u s upon another is  set S  P  ,  g  , P  g  w i t h each o f t h e s e elements w i l l  u  say  , P  P^ ,  , P  v  And, o f which  , P  w  x  , P , %  i n t r o d u c e even more  3  elements, S  etc.  P generated by c l o s i n g t h e i n i t i a l f i n i t e s e t QML A, V , o p e r a t i o n s o f PQ L w i l l contain a  Thus t h e  with respect to the  N M T  M  3  denumerable i n f i n i t y o f elements. corresponding elements,  3  PQJ^ ° ^  infinite  Nevertheless, the i n f i n i t e ^  a  s  u  b P S  A  C  E  S  of  H  3  WT  and t h e  each c o n t a i n t h e same  so i t i s n o t c o r r e c t t o c o n s i d e r a p a r t i a l - B o o l e a n a l g e b r a o f  subspaces t o be m i s s i n g elements compared w i t h an orthomodular subspaces.  lattice of  The p o i n t o f t h e above example i s t o show how, when an  orthomodular  lattice  o f subspaces i s generated  subspaces by c l o s i n g t h e i n i t i a l of  P_ QML  >  the l a t t i c e  from an i n i t i a l A, v,  set with respect to the  definitions of  A, V  set of operations  among i n c o m p a t i b l e s may r e s u l t  i n a p r o l i f e r a t i o n o f elements which does n o t o c c u r when a p a r t i a l - B o o l e a n a l g e b r a o f subspaces i s generated set  with respect to the Let  A , v, --  from t h e same i n i t i a l PQ^ '  operations o f  1  s e t by c l o s i n g t h e  us assume t h a t t h e quantum system, which i s a s s o c i a t e d w i t h  3 H  and which  pure s t a t e  y  P.,P , . . . ,P 0  o  Q  ,  r e p r e s e n t p r o p o s i t i o n s about, i s i n the  r e p r e s e n t e d by t h e p r o j e c t o r  P  3  which i s t h e atom  P 3  in  the (combined) diagram, which i s a fragment o f t h e system's p r o p o s i t i o n a l  3 structure  P  .  So we f o c u s on t h e s t a t e - i n d u c e d e x p e c t a t i o n - f u n c t i o n  Exp  3 and to  its affiliated  u l t r a s u b s t r u c t u r e US  t h e twenty element  3  PQ^  O  U Ul  = UF O  o  generated by t h e i n i t i a l  e P  .  With r e s p e c t  v^JXl  s e t S,  we have:  g  143  3  U F  =  { 1  ' 3' 3 P  P  1  V P  4>  =  3  P  V P  2  =  ?  1  • 3 P  4  V P  *S ' 3  =  ?  V P  5  K  =  }  a  n  d  3 UIg = { O J P ^ ' 2 ' l P  P  5' 4^*  , P  W i t h r e s p e c t t o t h e denumerably  P  g e n e r a t e d by t h e i n i t i a l s e t S, 3  U F  ^  =  P  3' 3 P  = 3 P  V P  6  i  vP  =  3  P  =  "' 3  P  P  2  V P  7  P„ V P ~ = P = 3 9 x  » 3 P  V P  V P  8  =  P  M L  we have:  *t>  2 =  P^  infinite  3  P  u = 3 P  P. V P , P„ V P 3 y ' 3 x  V  VP  4  *£ ' V  =  v » 3  P  P  = P" ", y  V  P  u  =  P  P  5  =  P  ^ =  v >  e t c . , . . . a denumerable  1  3 i n f i n i t y o f t w o - d i m e n s i o n a l subspaces o f H , UI  = { 0 , P , P ,P ,P ,P ,P ,P ,P ,P ,  e t c . , . . . a denumerable  X  o  ^  o  X  j  M  "  U  v  x  y  3 o f o n e - d i m e n s i o n a l subspaces o f H ,  And w i t h r e s p e c t t o t h e i n f i n i t e UF  3 PQ  each c o n t a i n i n g  a  3 PQ^L  °^  element,  P  d  n  MA  i n both structures includes the 1  each c o n t a i n e d  s u  ,  P3^*  infinity J-  i n P^ }.  3 hspaces of H ,  and t h e 3  nondenumerable i n f i n i t y o f a l l t w o - d i m e n s i o n a l subspaces o f H P  .  And  UI  i n both structures includes the 0  element,  o  o  P  and t h e  o  nondenumerable i n f i n i t y o f a l l o n e - d i m e n s i o n a l subspaces o f H in  containing  j_  Pg .  3  contained 3 PQJ^  H e r e a f t e r , l e t us j u s t f o c u s on t h e twenty element 3 g e n e r a t e d by S and t h e denumerably i n f i n i t e PQ g e n e r a t e d by S. C l e a r l y , US = UF U UI i s l a r g e r t h a n t h e two maximal B o o l e a n Ml  O  O  o  substructures mBS_ and mBS which c o n t a i n P i n t h e twenty element 3 P^.,. • And l i k e w i s e US„ i s l a r g e r t h a n any o f t h e maximal B o o l e a n QMA 3 substructures the denumerably  mBS- , mBS_ , 2 D  mBS  i n f i n i t e fragment  v P  , mBS 3 N M T  •  ,  y  e t c . , which c o n t a i n  P„ i n o  Moreover, by i n s p e c t i o n i t i s  QML  clear that i n the f i n i t e  3 P_„. , QMA  US. = mBS U mBS.. ; and by i n s p e c t i o n i t 3 2 5 3 i s c l e a r t h a t i n t h e denumerable PQ^ » c o n s i d e r i n g j u s t t h e 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 t h e 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 n  l  P  0  3  ,  t h e l i s t e d elements i n US„ = mBS„ U mBS,. U mBS U mBS . 3 2 5 v y  That i s ,  144  US3  e q u a l s t h e u n i o n o f a l l t h e mBS's  containing  P ., as proven i n 3  S e c t i o n B. I t i s a l s o w o r t h 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 conditions  P  (a„), ( b ) , t o d e f i n e a f i l t e r i n a n  , then the yMA  3 S' = UF„ U {P„} would be a p r o p e r f i l t e r i n t h e twenty element P . 3 7 (jMA diagrammed above. U s i n g ( a ) , i t i s easy t o show t h a t S' i s n o t a f i l t e r set  N M  n  P?„  in this  .  A  I f S'  i s a f i l t e r , then since  P , p i " € S',  by (a„),  X  QMA  2  n  1  t h e r e i s an element d € S' such t h a t d < P and d s P . I n t h e twenty element ?\ ,P < , 0 < P , 2 P , P3 < P , P 5 P^ , 0 5 P \ 1  X  X  1  X  X  X  m  < P , P„ 5 P , 7 8 7  P, < P , 4 7 6 X  X  0 £ S',  But  ?  and P„ < P 9 7  X  and so  S'  X  i s not a f i l t e r .  ;  0 < P  so o n l y J  Q.E.D.  ?  and 0 < P . 7  X  X  2  But using ( a ) , i t turns 3  out t h a t  S'  PQ^ '  i s a p r o p e r f i l t e r i n t h e twenty element  a f i l t e r , then since  P^jPy € ' »  by ( a ) , t h e meet o f P  s  X  , P  I f S' i s i sa  X  3 member o f S', b u t t h i s meet i s n o t d e f i n e d i n t h e twenty element P Q ^ since P P^ • ( I f t h e meet were d e f i n e d , as i n a p containing P 2 7 QML 2 3  and  P^",  t h e n t h e meet  be a f i l t e r . ) S'  element i n S' X  = P  X  € S'.  Thus  I I  X ?  € S'.  A P  = 0;  X  P  X  thus s i n c e 1  0 £ S',  S'  S'  would n o t  element, e v e r y o t h e r element i n  and so t h e meets o f P  X  a r e n o t d e f i n e d i n t h e twenty element s a t i s f i e s ( a ) . Also  w i t h every other 3 PQ^ *  A  So  S'  D  X  1 > P  X  I  , Pj" > P ,  S'  and  X  I  i s a f i l t e r i n t h e twenty element  0 £ S',  Moreover, s i n c e  s a t i s f i e s ( b ) , and o n l y  N  S' = UF. U { P } O  s a t i s f i e s ( b ) ; f o r UF O 1, P  X  Moreover, except f o r t h e  i s incompatible with  1 A P  P  X  P  I  I  . Q.E.D. 3 PQ^ * ^3  i s a proper f i l t e r i n t h i s 3 i s t h e p r o p e r s u b s e t o f a p r o p e r f i l t e r i n t h i s P Q ^ • Thus U F i s not 3 an u l t r a f i l t e r i n t h i s P Q ^ ' undesirable r e s u l t o f using (a) r a t h e r t h a n ( a ) t o d e f i n e a f i l t e r i n a P^„„ . H QMA R e t u r n i n g t o t h e s t a t e - i n d u c e d Exp,, , w h i c h a s s i g n s t h e v a l u e 1 S  M  a  T T  v  e  r  v  g  O  145  t o elements i n UF„ and a s s i g n s t h e v a l u e  0  t o elements i n UI  o  easy t o f i n d examples o f how Exp  6 ' 7 * QMA P  P  Pg,p  X  :  P  6  7  A  ?  =  4  P  3 '  t 1).  3  °  S  E X P  3  ( P  6  A  *7 >  °"  =  x  B  u  t  ( a n d t 1) and E x p ^ P  3  E x p . ( P ) A Exp ( P ) t 0 = E x p ( P . A P  Thus  mapping on  • C o n s i d e r t h e c o m p a t i b l e elements  A  € U I  i s not a t r u t h - f u n c t i o n a l ( o )  £ U F U U I , so E x p ( P ^ ) t 0 3  i ti s  o  3 PQJJ  t h e e n t i r e twenty element P  ,  o  x  X  c  ).  x  0  ) t 0 (and  1  Similarly, i t i s  easy t o f i n d examples o f how E x p 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 t h e e n t i r e denumerable P^,„ . C o n s i d e r t h e i n c o m p a t i b l e elements QML 3  P  3' 8 P  so  € P  QML  Exp (P Q  :  P  3  A  v  P  B  *  ) = 0, and  X 0  a  1  O  3  3  P  = 1  0  €U I  3  o  3  V P  X  X  b  /  n  o  t UF„ U U I o  , t h e meet  0  and j o i n s i n S e c t i o n B, where h e r e ,  US^, i s U S  j o i n s w h i c h cause t r u t h - f u n c t i o n a l i t y  3  V Pg).  P^ A P  o  b  a r e examples o f what were c a l l e d  x  1  g  p ,P ,P  3 »  (and t 0 ) .  0  o  ^  But  so E x p ( P ) t 1  O  g  S i n c e t h e elements and t h e j o i n  ,  V '  V E x p ( P ) = 0 V E x p ( P ) = E x p ^ P g ) ji 1 = E x p ^ P  X  3  V  S  0  O  Exp (P )  ' °  3  P. t U F U UI  O O  Thus  6 U F  X  /  U S ^ - e x t r a meets  . These a r e the meets and  ( ( b ) o r (o,^A))  problems f o r  Exp  3  .  Whether they a r e t h e meets and j o i n s o f c o m p a t i b l e elements o r o f i n c o m p a t i b l e elements i s i r r e l e v e n t . Exp P  QM  What makes s u c h meets and j o i n s p r o b l e m a t i c f o r  i s t h a t one o r t h e o t h e r o r b o t h o f t h e i r subformulae a r e elements o f  o w n  ^  c n  a  r  e  rcof members o f U S  truth-functionality  ((A>)  . Moreover, e v e r y v i o l a t i o n o f  o r (i>,#>))  U S ^ - e x t r a meets and j o i n s . is truth-functional  3  P^  by an Exp^, on a  M  i n v o l v e s such  F o r as has been shown i n S e c t i o n B, any  ((<A) o r ( A , & ) on t h e domain  Exp^,  US^, , t h a t i s , Exp^,  does p r e s e r v e t h e meets and j o i n s o f t h e elements o f P QM W  w h i c h a r e members  J  of  US . +  As mentioned i n S e c t i o n B, t h e 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 t h e l i g h t o f t h e Chapter V  (A) d e s c r i p t i o n o f t h e t r u t h - f u n c t i o n a l i t y ( i , < ) 6 )  problems caused by t h e  146 meets and j o i n s o f i n c o m p a t i b l e elements i n P Q ^ L • However, we can f i n d many examples o f t h e truth-functionality(h,k>) ultrasubstructure  US.  of  Exp  P.„„  o f t h e denumerable  on t h e  3  considered i n t h i s section.  T  Q M L  3 Consider the incompatible p a i r s  P  jk P  X  ,  g  P ty>P  , P ty ?^ ,  X  y  f o l l o w i n g meets and j o i n s o f t h e s e i n c o m p a t i b l e p a i r s : P A P  .  Clearly,  P  € UF  X  , P  € Ul  ,  and  and t h e  ±  P  P A P  X  A P^ , P^ V P^ ,  = 0 Z Ul  X  ,  thus  E x p ( P ) A E x p ( P ) = 1 A 0 = 0 = E x p ( P A P ) . C l e a r l y , P <E U F , P € UI„ , and P V P = 1 6 UF„ ; thus Exv>AP ) V Exp„(P ) = 1 V 0 y 3 u y 3 3 u 3 y = i = Exp,(P V P ) . C l e a r l y , P € U l , P € U l , and P A P X  A  3  3  5  X  3  5  3  X  X  X  o  i  = .0 € U l - ;  i  y  x  o  x  o  x  Exp (P ) A Exp„(P ) = 0 A 0 = 0 = Exp (P  thus  O  1  O  X  \j  X  1  o  x  A P ).  I t i s a l s o easy t o f i n d examples o f v i o l a t i o n s o f truth-functionality^,^)  by a semantic mapping  v a l u e s t o t h e elements' i n elements o u t s i d e o f J i n S e c t i o n B. x, x Pg,Pg,P^ £ U S  ,  QML  which a s s i g n s  and i n a d d i t i o n a s s i g n s  3  and so  values t o the  = v(P ) X  P  X  = 1  elements  PQML Q • A c c o r d i n g t o t h e Friedman-Glymour  i n t h e denumerable  va^A  0, 1  0, 1  a c c o r d i n g t o t h e Friedman-Glymour p r o p o s a l mentioned  g  C o n s i d e r t h e t h r e e .mutually c o m p a t i b l e  ^ )  P  US  US  v  M L  and  v ( P ) = 0.  However,  ?  ) = v(P ) = 0 / l A l =  v ( P  ?  g  L )  A  P A P = P X  Y  X  ^ 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 US  o  v  which a s s i g n s  and i n a d d i t i o n a s s i g n s t h e v a l u e  0, 1 N  v a l u e s t o t h e elements i n  t o t h e elements o u t s i d e o f  US  o  a c c o r d i n g t o t h e Levy p r o p o s a l mentioned i n S e c t i o n B, c o n s i d e r t h e s e two j o i n s o f c o m p a t i b l e elements i n t h e twenty element  V  V  S l n C e  P  Similarly, since P  V P = X  c  b  b  6' 6' 7 * 3 QMA ' P  P  c  b  1 € UF„ , o  truth-functional(A),  P  V P_ = P /  so  U S  X  o  £ US  0  o  C P  V  P  V ( P  =  V P 6 6  1  6 " J  and )= V ( P  7 '  , v ( P _ v ? ) = N. But  v ( P - V P ) = 1. 6 6 X  V (  P : QMA  n  b  y  I n o r d e r t o show t h a t  v  i s not  assume on t h e c o n 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 ) .  ) =N  147  Then  1 = v(P  = N,  i . e . , 1 = N,  S e c t i o n D.  VP ) 1  6  = v ( P ) V v(P^) g  = N V N = v(P  ) V v(P ) = v(P ?  which c o n t r a d i c t s t h e p r e s u p p o s i t i o n t h a t  A State-induced  P_„ QM  Semantics f o r t h e  N t  V P )  g  ?  1.  Structures  As d e s c r i b e d 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 s e m a n t i c s f o r a Lindenbaum B o o l e a n a l g e b r a 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 t h e Lindenbaum a l g e b r a .  And as  d e s c r i b e d i n Chapter I I I , a s t a t e - 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 s t a t e - i n d u c e d u l t r a v a l u a t i o n s on t h e  P_„ CM  .  With these c l a s s i c a l  p r e c e d e n t s i n mind, i n o r d e r t o f u l l y e l a b o r a t e t h e n o t i o n o f a s t a t e - i n d u c e d s e m a n t i c s f o r a quantum  P^  M  ,  i t remains t o be shown t h a t the c o l l e c t i o n P_„ QM  o f s t a t e - i n d u c e d u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f a  is  complete. We can e s t a b l i s h completeness i n t h e r e q u i r e d sense i f we can show t h a t , f o r any g i v e n p a i r o f d i s t i n c t elements the. s e t o f atoms dominated by dominated by P^. < P^ , mapping  Pj •  P^  Exp^,  P^ ,  i s an atom dominated by i . e . P^ £ P  by d e f i n i t i o n a s s i g n s t h e v a l u e s 7  as p o i n t e d out by van F r a a s s e n ,  in a  2  ,  PQ^ ,  P  2  .  P^  P^ , i . e . ,  Exp(P^) = 1 ^ E x p ( P j ) .  i f t h e elements o f a  t h e s e t o f atoms dominated by  dominated by  ,  then the state-induced  P^  M  Now  a r e r e g a r d e d as  subspaces o f 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 a  P  i s n o t t h e same as t h e s e t o f atoms  For c l e a r l y , i f P^  b u t n o t dominated by  P^ ? P^  P^ i- P j  in  d i f f e r s from t h e s e t o f atoms  F o r as s t a t e d i n C h a p t e r I V ( A ) , a subspace o f a H i l b e r t  space i s a s e t o f v e c t o r s ( w h i c h 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 o f a H i l b e r t space a r e d i s t i n c t IFF t h e 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 v e c t o r i n a H i l b e r t space i s  148  P^  u n i q u e l y a s s o c i a t e d w i t h an atom i n t h e  s t r u c t u r e o f the H i l b e r t space.  M  However, we may a l s o c o n s i d e r s u p p o r t i n g t h e completeness r e s u l t by an a l g e b r a i c p r o o f w h i c h does n o t i n v o k e t h e subspace c h a r a c t e r o f t h e P . QM  elements o f a  A  P , an a l g e b r a i c p r o o f o f t h e QML  F o r t h e case o f a  completeness r e s u l t c a n e a s i l y be shown t o f o l l o w from Lemma C o f Chapter V ( A ) . An a l g e b r a i c p r o o f o f t h e completeness r e s u l t f o r a  x s  m  o  r  e  difficult.  N e v e r t h e l e s s , t h e weak completeness o f t h e c o l l e c t i o n o f s t a t e - i n d u c e d u l t r a v a l u a t i o n s on a  P_.,„ QMA  P r o p o s i t i o n C:  or a  P ., QML n  i s e a s i l y proved as f o l l o w s :  T  F o r any P ^ ,  the c o l l e c t i o n o f state-induced  N  u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f P c o m p l e t e , i . e . , f o r any element Exp^  By t h e a t o m i c i t y o f P ^ , N  A  such t h a t  in P  f o r any P f 0  N  P^. € P ^  i n P ^ , t h e r e i s an  P f 0  Exp^(P) 4 Exp^(O).  such t h a t  atom  P^ 5 P,  Y  Exp^ which assigns the value  t h e members o f UF^. and a s s i g n s t h e v a l u e  0  Exp^(P) = 1 t 0 = Exp^(O).  satisfies:  t h e r e i s an  P, w h i l e t h e d u a l u l t r a i d e a l ' 0 s i n c e 0 5 P f~. Thus t h e  nxt  state-induced u l t r a v a l u a t i o n  N M  and so t h e u l t r a f i l t e r  UF. = {P € P . „ : P > P,} c o n t a i n s Y QM \|r U l , = {P € P : P < P-f"} c o n t a i n s Y QM y  UI^  i s weakly  A M  1 to  t o t h e members o f  Q.E.D.  P.„_ , t h e completeness r e s u l t i s an immediate QML consequence o f t h e f o l l o w i n g P r o p o s i t i o n D w h i c h f o l l o w s from Lemma C. For t h e case o f a  P r o p o s i t i o n D: P;P_  ,  1 2  P  Y  5  p  i  a n d  F o r any P . „ QML J  and f o r any P. ,P„ € P^,, 12 QML  T  J  P. € P _ „ Y QML  t h e n t h e r e i s an atom P  Y*  P  2  '  O  R  Assume on t h e c o n t r a r y t h a t  P  Y  -  P  a n d  P^ i ?  2  P  Y*  ,i f  such t h a t e i t h e r  T  2  T  P  i '  a n d . f o r e v e r y atom  € PQML »  149  P. < P„ I F F P, < P . 1 2  Y  n  Let  {P.}.„ l i£Endex  by  P„ 1  J  and l e t  {P,}, „ , k k €Index by  be t h e s e t o f atoms o f  P  V P . . l i  be  P^„ QML  w h i c h a r e dominated  T  be t h e j o i n o f a l l such atoms. L e t t h e s e t o f atoms  P.., QML  w h i c h a r e dominated  T  and l e t  V P be t h e j o i n o f a l l such atoms. k a s s u m p t i o n , f o r every atom P^ € P , 6 {Pi} 2  Q M L  Y  P  V  €  {P  k>k€Index •  t  h  u  ^i>i€Index  S  P. = V P, . B u t by Lemma C, I . k k  .  l  thus  By  k  P  i € l n d e x  k>k*Index '  =  {P  l  = V P. . l  a  n  d  ^F  °  S  and P = V P. ; i 2 . k k 0  P^ = P j , w h i c h c o n t r a d i c t s t h e a s s u m p t i o n  P^ ^ P j . Q.E.D.  Now t h e 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 t o P r o p o s i t i o n D: state-induced  F o r any  > fhe collection o f  u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f  i s c o m p l e t e , i . e . , f o r any P^ f P j i n such t h a t If P  Y  P  and  P  ±  QML P  f  S  U  C  ^ P  t h e n by P r o p o s i t i o n D,  h  t  h  •  1  a  t  Similarly, U F  l *  Y'  e  Q M L  t  h  Y  e  r  P  S P  y "  l  l  P  a  n  n  d  S  °  E x p  v  r  1  d  =1 B U T  \|r^ 2^ P  Y* 2 '  P  : P > P^} b u t P  E X P  2  ,  ±  Y^ 1^ P  °  P  thus  i f P^ < P j and P^ % ? A  t h e r e i s an atom  and P^ $ P j ,  b u t E x p ^ ( P j ) t 1;  E x p ( P j ) * Exp j (P ). y  l  If P  6 UF = {P € P  Exp^P^ = 1  P  » t h e r e i s an Exp^.  E x p ^ P ^ / Exp^(Pj).  P^ ^ P j , €  P.... QML  Y  RP  5  P  2  then  % UF^ . And so E x p ^ P ^ t Exp^(Pj). then ^  P j € UF  1 ;  t  h  u  y  but  S  Q.E.D.  For t h e case o f P„„„. , we now assume t h a t QMA  P„„. QMA  structures are  not o n l y 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 a l g e b r a s ( a s d e f i n e d i n Chapter 1 ( D ) ) , b u t a l s o s a t i s f y t h e f o l l o w i n g  150  C o n d i t i o n A:  PQ^  Every maximal Boolean s u b s t r u c t u r e o f a  ^  S  atomic. And we make use o f t h e f o l l o w i n g two lemmas proved by Edwin Levy: Lemma F: are  F o r any  , ?  1  P^  i na  2  ,  2  elements  or  2 ~  P  l '  P  P^.?^ € P  If P A P  ,  2  °  r  P  1" " 2 ' L  = P  of  P  V P , P  then  P  (2) I f P thus P  l l  P  P  =  P  3  2  =  = 0  2  and  such t h a t  Q M  P  < P °»  .  1  t  h  e  = P  P  l  P  P  2  °  =  P  n  e  < P  1 1  r  e  a  r  e  n  o  and  1  n  _  ?  z  e  r  disjoint  o  < P  2 2  2 <  P  n'  = P  ± ±  ( a n d  < P ).  S  °  = P  P  2  ?  then  ^  V'  5  ( 5 )  (6) I f P ^ = P  1  ;  1  s  u  c  "that  n  = P  = P  p  =  P  3  = 0,  g  2  ;  I f  °'  =  .  2  V P  1 1  •  P 3  =  P^^ < P  V Pg = P  2  22  (1) I f P ^ = 0  thus  = P^  ±  I f  2 2  element.  V ^  2 2  and  3  = 0, l  P  P  2 2  3  2 2  2 ' 3 ^ *QM  P  We have e i g h t c a s e s depending upon which  = P  2  = 0 V P  2  (3) I f ?  (and so  V P .  and  g  then  n  r t  a r e o r a r e not equal t o the 0 = P  3  = 0,  2 2  then there  t h e n by t h e 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 ( C h a p t e r I V ( C ) )  P  , Pg  2 2  = 0 V P  1  ,  2  1  °  P  t h e r e e x i s t t h r e e m u t u a l l y o r t h o g o n a l elements P  i f P d> ?  ,  these four 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^ < ?  Proof:  P  t  then  h  e  ?  n  = ?  ±  2  = 0.  ( C l e a r l y , t h e r e s u l t s o f c a s e s ( 3 ) , ( 4 ) , ( 5 ) , ( 6 ) , a r e subsumed by t h e r e s u l t s o f c a s e s ( 1 ) and ( 2 ) . )  V  thuS P  2  =  22  P  V  °  =  P  22  ( 7 ) I f P„ = 0, 3  ;  P  2  S  l  n  c  e  P  then  P. = P.. v O = P . , and 1 11 11  n - i -22 * P  ( 8 )  I f  l l  P  *  0  P.. 4 0 and P 4 0, t h e n 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 o u t as f o l l o w s , l e a v i n g j u s t P ^ < P^ . S i n c e P 4 0,  and P  P  ll 3  0  =  5  P  P  ll  l  I F F  '  assumption Q.E.D.  P  W  3 ~ ll  '  P  e  h  P  a  g  V  e  P  A  n  d  s  3 ~ ll  4 0.  P  Thus  i  n  c  °  e  b  y  n l y  P X  assumption i f  P  3  =  °'  P^  g  B  u  t  f  o  r  t  h  l  s  s c  a  i.e., s  e  P ^ < P^ . M u t a t i s mutandis f o r  b  P  2 2  y  <  p 2  •  151  Lemma G: P_„ QM  a r e a l s o atoms o f  Proof: substructure atom o f  A l l t h e atoms o f a maximal B o o l e a n s u b s t r u c t u r e o f a  Assume on t h e c o n t r a r y t h a t t h e r e i s a maximal B o o l e a n  mBS„ G  rnBSg  P_„ . QM  i n P.., QM  b u t PQ  and an element  i s n o t an atom o f  s t r u c t u r e , t h e r e i s an atom  P € 'P„„ a QM  € P QM  0  rt  P  . Since  such t h a t  P^  P < P a 0 A  i s an  i s an atomic  (P t P a 0  since  n  assumption  is_ an atom o f mBSg .) Now s i n c e a l l elements i n a maximal  Boolean s u b s t r u c t u r e a r e m u t u a l l y P A PQ . P t 0  M  ; and P  P„ 0  P  Q  P^  such t h a t  niur  by a s s u m p t i o n , PQ  i s n o t an atom o f  P  compatible,  j£ mBS  &  f o r e v e r y element  I t f o l l o w s by Lemma F t h a t , f o r e v e r y element  and P * P  ,  Q  e i t h e r (1)  P < P  Q  s i n c e by  Q  , o r (2)  P  Q  P € mBSg ,  P € m^S^ < P,  s  u  c  o r (3)  that  n  P 1 P, Q  o r ( 4 ) t h e r e a r e nonzero elements  P',P' d mBS such t h a t P' < P and 0 0 P' < P . S i n c e by a s s u m p t i o n , 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 o u t . S i m i l a r l y , s i n c e by a s s u m p t i o n , 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 o u t . p o s s i b i l i t y ( 2 ) , i f P < P, 0 r  pip.  P < P. a G  f o r e v e r y element a  o P.  And f o r  we have  P €  Moreover, f o r P = P. , 0  every element  P < P^", and so a such t h a t  P = 0,  Q  P i P.  mBSg  P ^ 0 P A 0 a  and P c > P  v  EL  P < P, a  P k P^, a  IIIBSQ  plus  hence  P i a  P.  we l i k e w i s e have  .  P i P. a  P i a  P.  That i s , f o r  So t h e s e t o f m u t u a l l y  P^  P^  M  (and perhaps o t h e r s ) . P^  ,  which  as a maximal B o o l e a n s u b s t r u c t u r e . 0  So  and P ^ PQ , we have  i s the proper subset o f a Boolean s u b s t r u c t u r e o f  c o n t r a d i c t s t h e d e f i n i t i o n o f mBS  and so  i . e . , P- < P ,  U {P^} g e n e r a t e a B o o l e a n s u b s t r u c t u r e o f  w h i c h c o n t a i n s a l l t h e elements o f  Q.E.D.  we have  we l i k e w i s e have  n  U  mBS  since  P ^ P a 0  since  P € mBS- ,  c o m p a t i b l e elements  Thus  P < P. a 0  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,  then s i n c e  P  then s i n c e  Now c o n s i d e r i n g  152  Furthermore, given the conjecture  t h a t every  mBS  ofa  P^ A  i s atomic, i t  i s a t r i v i a l p o i n t t h a t a l l atoms o f P_„ w h i c h a r e i n an mBS o f P.,, QM QM a r e a l s o atoms o f t h e mBS. F o r t h e o n l y way a n atom of P which i s AM  i n a n mBS P € mBS  of P „ QM  c o u l d n o t be an atom o f mBS  rt  i s i f some o t h e r  element  were between  P and t h e 0-element i n mBS b u t n o t i n P.„ . a QM But s i n c e mBS i s a s u b s t r u c t u r e o f P.,, , i f some P € mBS were such t h a t QM 0 < P < P_ i n mBS t h e n a l s o 0 < P 5 P_ i n P_, , and so P would n o t QM be an atom o f P_„ . QM We a l s o make use o f t h e f o l l o w i n g r e s u l t s .  As mentioned i n  C h a p t e r 1 ( D ) , Hughes has p r o v e n t h a t any p a r t i a l - B o o l e a n a l g e b r a i s i s o m o r p h i c t o a p a r t i a l - B o o l e a n a l g e b r a c o n s t r u c t e d on a f a m i l y o f B o o l e a n ^i^i€lndex  '  3  ^  S  e s c r >  i b e d by Kochen-Specker. A  constructed partial-Boolean algebra b,c,d  €A, b V c = d  algebras  Among o t h e r c o n d i t i o n s , t h e  s a t i s f i e s , f o r any elements  i n A IFF there i s a  8.  such t h a t  b V c = d in  l  8^ . Now as p a r t o f h i s p r o o f , Hughes shows t h a t any p a r t i a l - B o o l e a n can be c o n s t r u c t e d on t h e f a m i l y o f i t s own B o o l e a n s u b a l g e b r a s . p a r t i c u l a r , any subalgebras.  c  a  n b  So i n  c o n s t r u c t e d on t h e f a m i l y o f i t s own B o o l e a n  e  Thus we have t h e f o l l o w i n g  P r o p o s i t i o n E: P  algebra  =  P  l 2 V  P-„. QMA  P  i  F o r any P  ^QMA  n  such t h a t  I  F  F  t  n  e  r  e  and f o r any P . P ^ P j € P  Q M  i  s  a  P = P, v P„ 1 2  B  o  o  l  e  a  substructure  n  Q M  ,  BS o f  i n BS.  We a l s o make use o f t h e s e two lemmas.  Lemma H: QMA '  P  i , e  F o r any P , P € P 1  ''  i  f  P  l^ 2 ' P  bound o f { ,P } i n P Pl  2  ,  2  t  h  Q M  e  n  P  .  l  V  i fP P  2  1  S  t  v P  J  h  e  l  e  a  i s defined i n  2  s  t  upper  153  Proof:  Clearly, '  upper bound o f P v P  {P^Pj}.  and P. v P 1 2  And f o r any  (and P A P ^ ) , and  = P  1  PQMA  P. v P „ > P. 1 2 1  J  > P. ; 2  P € PQ  P > P ',  P = P V P  = P  2  P = P VP  2  V P  = (P V P ) V P  2  P A ( P VP)).  (i.e., 1  Halmos' Lemma:  1  since  vP  P , i P and 1 and moreover,  2  i.e.,P > P  2  1  i.e.,  t h e n because  P A (P v P )  So  2  = P V ( P vP ),  2  ,  1  satisfies:  (P. V P j o P 1 2  we have  0  0  i s an  n  i . e . , P v P j = P,  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 a l g e b r a w h i c h  P, A ( P V P) I F F (P„ v P )<A P, 1 2 1 2  P. v P 1 2  i f P > P  ,  MA  thus  2  .  Q.E.D.  I n an a t o m i c B o o l e a n a l g e b r a , e v e r y element i s t h e  j o i n ( l e a s t upper bound) o f t h e atoms i t dominates (Halmos, 1963, p. 7 0 ) . Now we may prove t h e f o l l o w i n g Theorem C f o r  P  . ,  which  QMA  c o r r e s p o n d s t o t h e above P r o p o s i t i o n D f o r P ^ „ . QML Levy, R o b i n s o n , Chernavska.)  (The p r o o f i s due t o  T  Theorem C:  P  F o r any  Q  and f o r any  M  P .P ±  2  P -;";^'P ., t h e n t h e s e t o f atoms dominated by the  s e t o f atoms dominated by  P, € P„„„ ty QMA  L e t A„ 1  i.e., P  QMA  c n  a  r  A^ = A  2  = 0  A^ = A  2  t 0.  e  dominated by  Since  P^  P  ( t h e empty s e t ) , Since  c o m p a t i b l e atoms o f (1)  and  A^ = A PQ  MA  2  J  2  .  Assume  then i- 0,  P.,,. QMA  , i f  i s not equal t o  P, £ P. , ty 2  o r P. < P. ty 2  w h i c h a r e dominated by J  and l e t A„ 2  n  i  M  ( i . e . , t h e r e i s an atom  2  P. < P. 1  be t h e s e t o f atoms o f  A. = {P. € P ... : P. 5 P.}; 1 ty QMA ty 1 w n  P^  Q  P^P,).  and Proof:  such t h a t e i t h e r ty  P  € P  be t h e s e t o f atoms o f  A^ = A  P^ = P  2  P. , 1  = 0.  2  .  Clearly, i f  Assume t h e n t h a t  t h e r e i s a nonempty s e t Ai_ o f m u t u a l l y  each o f w h i c h i s dominated by  dominates each member o f A^ ,  P^  P^  and by  P  2  i s c o m p a t i b l e w i t h each  .  154  member o f  .  Thus,  A^ U {P- }  i s a s e t o f m u t u a l l y c o m p a t i b l e elements  PQJJ^ • Hence t h e r e i s a B o o l e a n s u b a l g e b r a o f  of  and a l s o c o n t a i n i n g a l l members o f  PQJ^  containing  P^  A^ ; and t h i s B o o l e a n s u b a l g e b r a i s  P „ « • By C o n d i t i o n A, QMA i s a t o m i c , and by Lemma G, a l l o f i t s atoms a r e atoms o f P • Let  c o n t a i n e d i n a maximal B o o l e a n s u b a l g e b r a mBS' mBS'  of  W  N M F L  A' = {Pi, } i ^ i j be t h e s e t o f a l l atoms o f mBS' w h i c h a r e dominated by i P^ ; c l e a r l y , A' c A^ . Now by Halmos's Lemma, P i s t h e l e a s t upper n c  bound o f P  l  =  V  P  A'  i in  so  P^  A'  ^  e x  i n mBS',  i n  P  OMA '  A  n  i •  N  o  w  p 2  i.e., P d  Lemma H,  A',  P! i * i V P '  i n mBS*.  namely,  A',  and hence  P^  P^  t P  Q.E.D.  1  = P  2  .  So i f P  2  ,  And as i n t h e PQ^L  dominates t h e l e a s t  P^ .  (2) By a s i m i l a r argument i t c a n be shown t h e t P  Then by P r o p o s i t i o n E  i s t h e l e a s t upper bound o f i i dominates e v e r y member o f A^ = , and A' c A^ , b y  dominates e v e r y member o f  upper bound o f  = V  then  h^t  .  dominates  P^ •  Thus,  c a s e , t h e 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 t o Theorem C:  F o r any  PQJ^ »  t n  e collection of  s t a t e - i n d u c e d u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f i s complete, Exp^  i . e . , f o r any  such t h a t  P^ 4 P  i n PQ^ >  2  P .. QMA N  t h e r e i s an  E x p ^ C P ^ 4 Exp^,(P ). 2  The p r o o f o f t h e c o r o l l a r y proceeds as i n t h e  PQJJL  case.  Summary As d e s c r i b e d i n C h a p t e r s 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 B o o l e a n a l g e b r a  L  of classical propositional  A  155  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  and a s t a t e - i n d u c e d ,  P_„ CM  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  of c l a s s i c a l P ^ .  mechanics i s a complete c o l l e c t i o n o f s t a t e - i n d u c e d u l t r a v a l u a t i o n s on In b o t h c a s e s , an u l t r a v a l u a t i o n i s a mapping w h i c h a s s i g n s t h e v a l u e the elements i n an u l t r a f i l t e r  UF  i n the dual u l t r a i d e a l  Ul;  w i t h r e s p e c t t o an  and d u a l  UF  b i v a l e n t mapping on a  0  UF  or a  1  and d u a l  Ul  t o t h e elements defined  UF U U l  i s assigned  s a t i s f i e d by any  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 UF U U l . UF  UF U U l = P„, CM  and L,  to  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  i . e . , e v e r y element i n  a r e B o o l e a n , f o r any  UF U U l = L the e n t i r e  Ul.  And i t f o l l o w s from t h e c o n d i t i o n s  t r u t h - f u n c t i o n a l mapping on structures  0  t h u s an u l t r a v a l u a t i o n i s s a i d t o be  UF U U l ,  value.  and a s s i g n s t h e v a l u e  1  P ^  ,  structure.  u l t r a v a l u a t i o n s on an  L  or a  Moreover, because t h e  and d u a l  Ul  i n an  L  L,  P^  or a  P^  n  ,  n  t h u s t h e domain o f each u l t r a v a l u a t i o n i s  And t h e completeness o f t h e c o l l e c t i o n o f P  property o f Boolean s t r u c t u r e s .  CM  i s ensured by t h e  semi-simplicity  F u r t h e r m o r e , f o r t h e case 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 , each u l t r a v a l u a t i o n on t h e c l a s s e s of well-formed formulae i n a (closed) s e t o f one o f t h e s t a n d a r d v a l u a t i o n s  for  L,  L  structure of equivalence L  i s an a l g e b r a i c  version  which i s p a r t o f t h e r e a s o n  u l t r a v a l u a t i o n s a r e so c a l l e d and i s t h e main r e a s o n why u l t r a v a l u a t i o n s  on  any o t h e r 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 a r e r e g a r d e d i n t h i s t h e s i s as semantic mappings.  And f o r t h e case o f c l a s s i c a l mechanics,  ultravaluations  a r e s a i d t o be s t a t e - i n d u c e d because i n f a c t i t i s t h e s t a t e s o f a c l a s s i c a l m e c h a n i c a l system w h i c h i n d u c e mappings, namely, 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, each o f w h i c h UF  w  U UI^ = P  nM  uw  structure of propositions  v a l u e s o f t h e system's  magnitudes.  classical  i s an u l t r a v a l u a t i o n on t h e w h i c h make a s s e r t i o n s  about t h e  156  P.„ QM  When we c o n s i d e r a assertions  structure of propositions  w h i c h make  about t h e v a l u e s o f a quantum m e c h a n i c a l system's magnitudes, t h e  s t a t e s o f t h e system s i m i l a r l y i n d u c e mappings, namely, d i s p e r s i v e  generalized  p r o b a b i l i t y measures, each o f w h i c h P^ is u l t r a v a l u a t i o n on UF, U UI, c P . And as i n t h e c l a s s i c a l c a s e s , each s t a t e - i n d u c e d ty y ~ QM E x  ultravaluation  a  n  UF^ U U I ^ ;  E x p ^ i s a b i v a l e n t mapping on  from t h e c o n d i t i o n s  s a t i s f i e d by any J  UF,  and d u a l ty  J  UI, ty  each s t a t e - i n d u c e d 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 I ^ .  UF U UI = L  and i t f o l l o w s  ina  P_„ QM  ((A) o r  that (<t>,&))  But u n l i k e t h e c l a s s i c a l c a s e s i n w h i c h  UF U UI = P „ „ f o r e v e r y u l t r a f i l t e r and d u a l u l t r a i d e a l w w CM in L, P^y , f o r t h e quantum c a s e , i f P contains incompatible elements, t h e n n o t e v e r y u l t r a f i l t e r and d u a l u l t r a i d e a l i n P . „ i s such t h a t QM UF. U U I . = P.., , r a t h e r , f o r some UF, and d u a l U I , , UF, U UI, c P . ty ty QM ty ty ty ty QM When UF, U UI, i s l e s s t h a n t h e e n t i r e P^„ , we c a n a t l e a s t be s u r e t h a t ty ty QM and  J  N M  UF, U U I . ty  i s a closed  ultrasubstructure. Exp^,  P_„ , QM  However, t h e a f f i l i a t e d ((A)  i sa bivalent, truth-functional  ultrasubstructure and  substructure o f ty  US^  of  P^^ .  w h i c h may be c a l l e d an J  state-induced or  (o,<J6))  ultravaluation mapping on j u s t  that  Thus w h i l e e v e r y u l t r a v a l u a t i o n on an  every s t a t e - i n d u c e d u l t r a v a l u a t i o n on a J  P„„ CM  i sa bivalent,  L  truth-  f u n c t i o n a l mapping on t h e e n t i r e s t r u c t u r e , a t l e a s t some o f t h e s t a t e - i n d u c e d ultravaluations truth-functional PQ^  rather  on a ((o)  P^  M  or  containing (A,&))  t h a n on t h e e n t i r e  i n c o m p a t i b l e elements a r e b i v a l e n t ,  mappings on j u s t u l t r a s u b s t r u c t u r e s  P^  M  .  Moreover, t h e completeness o f t h e  c o l l e c t i o n o f state-induced ultravaluations PQ  M  must be p r o v e n , as done i n S e c t i o n However, t h e f a c t t h a t  on t h e u l t r a s u b s t r u c t u r e s  of a  D.  UF, U UI, c p ty  of  ty  QM  f o r some  UF, ty  and d u a l  157  UI^,  in a  P^  c o n t a i n i n g i n c o m p a t i b l e elements need not be a  A  problematic  f e a t u r e and _is_ not t h e o n l y p r o b l e m a t i c f e a t u r e o f the quantum structures.  As d e s c r i b e d i n Chapter V ( B ) , i f we  P^ n  i g n o r e the l a t t i c e meets and  j o i n s o f i n c o m p a t i b l e s and c o n s i d e r t h e p r o p o s a l o f a b i v a l e n t , truth-functional(<!?) elements i n  P^ n  semantics  P  ,  AM  the presence o f  incompatible  i s n e c e s s a r y but not s u f f i c i e n t t o 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) space  f o r a.  P „ . For a two-dimensional H i l b e r t QM 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  P^ n  semantics  for  A  o f t h e p r e s e n c e o f i n c o m p a t i b l e 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  o f t h r e e - o r - h i g h e r d i m e n s i o n a l H i l b e r t space 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 )  _n—3 V  s t r u c t u r e s w h i c h does  semantics  i s t h e presence o f  _n>3 o v e r l a p p i n g maximal B o o l e a n s u b s t r u c t u r e s i n o f i n c o m p a t i b l e elements i s a n e c e s s a r y  y  ,  f o r w h i c h the p r e s e n c e  ( b u t 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 a p p l y h e r e i n the Chapter VI d i s c u s s i o n o f the p r o p o s a l o f a semantics  for  P^ n  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  ( s t a t e - i n d u c e d ) 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„„ : QM  I f we i g n o r e t h e l a t t i c e meets and j o i n s o f i n c o m p a t i b l e s consider the proposal o f a b i v a l e n t , truth-functional(<b)  semantics  and for  P  nM  UF, U U l , c P„„ (rather y y QM t h a n UF. U U l , = P ) f o r some UF, and d u a l U l , i n any p . „ . which y y QM y y QM c o n t a i n s i n c o m p a t i b l e elements would by i t s e l f be h a r m l e s s l y u n p r o b l e m a t i c 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 — c o n s i s t i n g of u l t r a v a l u a t i o n s , then the f a c t that nM  J  2  Pq  For example, b o t h t h e s e " i f ' s " o b t a i n i n a P 2 and as d e s c r i b e d i n Chapter I V ( F ) , a does admit a complete c o l l e c t i o n M  were n o n - o v e r l a p p i n g .  of bivalent,  truth-functional(o)  nM  mappings, where by i n s p e c t i o n i t i s c l e a r  t h a t 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 ( i ) mappings on  the  2 six-element  P  e x p l i c i t l y c o n s i d e r e d i n t h a t Chapter I V ( F ) i s i n f a c t a  sum  158  o f two u l t r a v a l u a t i o n s , each d e f i n e d on one o f t h e two u l t r a s u b s t r u c t u r e s o f P;L . That i s , UF, = {P € P ? : P > P. } = {P.,.1}. QM 1 QM 1 1  the s i x - e l e m e n t U I  1  UF  =  { P  €  P  QM  :  M  ~ l  P  P  = {P ,l},  2  ^  mBS's o f  < 1' >'  =  P  0  U I = {P ,0},  2  e q u a l s an  }  2  W  I  2 P.„ QM  do n o t o v e r l a p .  a r e u l t r a v a l u a t i o n s on  u l t r a v a l u a t i o n s on mappings  h^ , h  2  US  2  U S  1  US  1  =  U F  U  U I  2  = UF  2  1  =  ;  U U I = mBS 2  2 P_„ QM  equals t h e union  Moreover, t h e two mappings  2  US^ ,  US.  e a r l i e r i n that section, the  (And t h e s i x - e l e m e n t US^ U US -)  . So each  2  t h e two mappings  h  h  a  ,  , h^ a r e  c  and each o f t h e 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 , h^ , h ^ , on t h e e n t i r e s i x - e l e m e n t P^^ i s t h e sum o f  an u l t r a v a l u a t i o n on six-element  d  and a s d e s c r i b e d  o f t h e two u l t r a s u b s t r u c t u r e s ,  n  and  2  2 i n P. , QM  mBS.  a  ,  2  ,  US^  p l u s a n u l t r a v a l u a t i o n on  and more g e n e r a l l y , any  2  P  US  .  2  Thus, t h e  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 t h e e n t i r e P , each o f w h i c h 2 i s a sum o f u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f P . „ . So t h e f a c t QM t h a t UF. U UI, c P f o r some UF, and d u a l U I , i n any P . „ c o n t a i n i n g Y Y QM ty ty QM J  incompatible  elements need n o t be a p r o b l e m a t i c  feature. P  u l t r a v a l u a t i o n s on t h e u l t r a s u b s t r u c t u r e s o f a  2  In p a r t i c u l a r , the  containing  incompatible  elements may be added t o g e t h e r truth-functional(<!>) truth-functional(<i>)  t o y i e l d a complete c o l l e c t i o n o f b i v a l e n t , 2 mappings on t h e e n t i r e P ^ , and t h u s a b i v a l e n t , 2 semantics f o r P^ , i n s p i t e o f t h a t f a c t . M  However, n e i t h e r o f t h e above, u n d e r l i n e d  " i f s " obtain i n a  _n>3 three-or-higher P^~^  d i m e n s i o n a l H i l b e r t space  PQ  .  M  That i s , t h e mBS's  may o v e r l a p , and t h e u l t r a s u b s t r u c t u r e s i n a  m  a  v  of a  be l a r g e r t h a n  -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  of a l l the overlapping  mBS's  US^  i na  in  So t h e f a c t t h a t  i s equal t o the union  w h i c h share t h e atom P^ , as shown >3 UF, U UI, c P f o r some UF, and d u a l Y ty QM ty n  i n S e c t i o n B.  P ^  159  TJI—3  UI^.  i n any  particular,  P Q  M  containing incompatible elements i s problematic.  the best we can do f o r such a ((6)  truth-functional  or  In  i s to define bivalent,  (<!>,&)) mappings on i t s ultrasubstructures.  We  cannot add together these ultravaluations on the ultrasubstructures of a PQ " to get bivalent, truth-functional(A) mappings on the entire P Q J ^ ' But the fact that UF. U U l . Pl„ i s not the only reason why we cannot add \|/ \|/ Q M the ultravaluations i n the suggested manner; the other reason i s that an ultrasubstructure UF^ U UI^, i n a P ^ i s a union of overlapping mBS's 1  3  c  J  i n  pn>3 QM ' Other sorts of semantic mappings may be and have been proposed f o r  the quantum  P  structures.  But i n t h i s t h e s i s , only two semantic  proposals have been seriously considered: truth-functional semantics f o r P ^ A  semantics f o r P , . QM nx  the proposal of a bivalent,  and the proposal of a state-induced  The former i s motivated by the success and usefulness J  of such a semantics f o r c l a s s i c a l l o g i c a l and propositional structures such as  L,  P  n M  .  semantics f o r a  The l a t t e r i s motivated by the fact that the state-induced P , CM  consisting of state-induced  u.  W  mappings already  present i n the formalism of c l a s s i c a l mechanics, works exactly l i k e the algebraic version of the standard, b i v a l e n t , truth-functional semantics of c l a s s i c a l propositional l o g i c . for  And the proposal of a state-induced semantics  i s motivated by the fact that the quantum formalism, l i k e the  c l a s s i c a l 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 f o r a  P ^ , n  i t i s worth considering  the notion of a state-induced semantics consisting of the state-induced Exp^,  mappings already present i n the quantum formalism.  semantic mappings on an  L,  Like the c l a s s i c a l  l i k e the c l a s s i c a l state-induced  u, w  mappings  160  P„„ , CM  on a PQML  ,  and l i k e t h e Friedman-Glymour S 3 - v a l u a t i o n s proposed f o r a  the state-induced  P „ . So t h e b a s i c semantic method i n a l l t h e s e QM  ultrasubstructures of i s t h e same.  A  incompatible elements, equal t o the e n t i r e ' P  3  cases  The c r u c i a l d i f f e r e n c e between t h e c l a s s i c a l and t h e quantum  c a s e s i s t h a t , f o r some  in a P^~ QM  E x p ^ mappings a r e u l t r a v a l u a t i o n s on t h e  UF^  and d u a l  P  UF^ U U I ^ i s s m a l l e r t h a n ,  A M  P  U I ^ i n any  and moreover,  containing  A M  A M  r a t h e r than  being  UF^ U U I ^ i s l a r g e r than any  PQ because UF^, U U I ^ i s t h e u n i o n o f a l l o v e r l a p p i n g w h i c h c o n t a i n t h e atom P, . y  mBS  mBS's i n  M  Notes: 1  Though o t h e r c o n d i t i o n s a r e sometimes t a k e n as d e f i n i n g t h e o r t h o g o n a l i t y r e l a t i o n , e.g., P , P are orthogonal IFF ^ ' °' these c o n d i t i o n s a r e s a t i s f i e d by any p  1  P  1' 2 P  * QM P  I  F  F  P  l  5  P  2 '  i , e  P i r o n t a k e s t h e P^ < P ^  ''  I  F  F  p  2  i»  condition  P  =  2  2  a  r  e  d  i  J  s  o  i  n  t  '  A n d  »  f  o  r  example,  as d e f i n i n g t h e 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  T h i s was p o i n t e d o u t t o me i n d e p e n d e n t l y by Dr. L. P. B e l l u c e and Dr. J . V. W h i t t a k e r . 3 The f a c t t h a t P A P and P^^ V P are not defined i n P when P^ ty P does n o t mean t h a t t h e union UF U U l i s i n any way n o t c l o s e d w i t h r e s p e c t t o t h e A , V o p e r a t i o n s o f QMA ' * ' °Perations o f P < E , C 3 , < , A , V , , 0 , 1 > a r e d e f i n e d from 1  2  2  2  P  T h  e  A  V  =  X  N M A  i c E x E t o E r a t h e r than from E x E t o 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 o r 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 r e s p e c t t o t h e A , V o p e r a t i o n s o f P _.. , I mean c l o s u r e w i t h r e s p e c t t o t h e s e o p e r a t i o n s qua p a r t i a l - o p e r a t i o n s . A  H Thanks t o Dr. Edwin Levy f o r s u g g e s t i n g t h e " u l t r a "  terminology.  5 I n a n e a r l i e r d r a f t , I c l a i m e d 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 E x p ^ on a P _ 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 r e s p e c t t o a Boolean s u b s t r u c t u r e o f m u t u a l l y c o m p a t i b l e elements i n P . Thanks t o 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 , t h e subset o f elements i n a P which a r e assigned 0 , 1 v a l u e s by an E x p ^ on P may i n c l u d e i n c o m p a t i b l e elements and so may be l a r g e r than M  A M  A M  161 any Boolean substructure of  P-,„ . QM Page 1 of a manuscript by Edwin Levy c i r c u l a t e d i n December, 1977.  7 As the external examiner, van Fraassen pointed out t h i s alternate proof of the completeness r e s u l t i n h i s 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 for  any magnitude.  w  s p e c i f i e s an e x a c t v a l u e  But i n quantum mechanics, a pure s t a t e  Y  specifies  an e x a c t ( e i g e n ) v a l u e f o r o n l y t h o s e magnitudes whose e i g e n s t a t e s a r e compatible w i t h with  Y  m a  Y * Any magnitude  A  whose e i g e n s t a t e s a r e i n c o m p a t i b l e  y > upon measurement, e x h i b i t any o f i t s ( e i g e n ) v a l u e s .  quantum f o r m a l i s m , f o r t h e g i v e n s t a t e determined by  Exp^(A) =  aJlY^yll  Y >  "the average v a l u e o f  [|Y~.><Y1|^, 3 Yj  with the (eigen)value  a_. .  which e x a c t ( e i g e n ) v a l u e  A  A  and t h e p r o b a b i l i t y t h a t  e x h i b i t any one o f i t s ( e i g e n ) v a l u e s , say Exp,(P, ) = Y Yj  In the  where  a_. , P,  is A  will  i s d e t e r m i n e d by represents the eigenstate of  But t h e quantum f o r m a l i s m does n o t d e t e r m i n e w i l l exhibit.  c h a r a c t e r i z e d by t h e same quantum s t a t e  Y  d i f f e r e n t v a l u e s f o r t h e same magnitude  A,  I n o t h e r words, quantum systems e x h i b i t , upon measurement, y e t t h e quantum f o r m a l i s m does  not d e t e r m i n e w h i c h o f t h e d i f f e r e n t v a l u e s o f  A  w i l l be e x h i b i t e d . F o r  t h i s r e a s o n , i t has been argued t h a t quantum mechanics i s i n c o m p l e t e and s h o u l d be supplemented by a h i d d e n v a r i a b l e t h e o r y which r e f l e c t s t h e d i f f e r e n t p o s s i b l e outcomes o f a measurement o f  A  f o r a given Y -  In terms o f t h e 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 t h e f a c t , d e s c r i b e d i n Chapter V I , t h a t any  P^  which c o n t a i n s  i n c o m p a t i b l e elements has a t 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 U I ^  A  asso  163  which i s s m a l l e r than t h e e n t i r e  P-  w  ,  and each element o f  P.„  QM  outside  UEy  i s a s s i g n e d a v a l u e between  state-induced by  Exp^, .  Exp^  0  and  1  by t h e a f f i l i a t e d  r a t h e r than b e i n g a s s i g n e d an e x a c t  A t t h e v e r y l e a s t , such a v a l u e between P jE US^ ,  as t h e p r o b a b i l i t y t h a t an element  P,  0  0 and  or 1  1  value  i s interpreted  qua idempotent magnitude,  w i l l upon measurement e x h i b i t i t s ( e i g e n ) v a l u e probability that  which i s  QM  1,  t h a t i s , as t h e  qua p r o p o s i t i o n , i s t r u e o f a system o r ensemble o f  systems whose s t a t e i s ty. So a g a i n , quantum systems c h a r a c t e r i z e d by t h e same quantum s t a t e ty e x h i b i t upon measurement, sometimes t h e t r u t h - v a l u e 0  and sometimes t h e t r u t h - v a l u e  1  f o r t h e same p r o p o s i t i o n  P A US^ ,  but which o f t h e s e t r u t h - v a l u e s w i l l be t h e outcome o f a measurement i s n o t determined  by t h e quantum f o r m a l i s m . Now i f we presume t h a t t h e p h y s i c a l t h e o r y o f quantum phenomena  s h o u l d i n c l u d e a f o r m a l i s m w h i c h does d e t e r m i n e , g i v e n t h e s t a t e o f a quantum system, e x a c t l y whether any  P € P  i s t r u e o r f a l s e , then quantum  mechanics i s indeed an i n c o m p l e t e t h e o r y and we must seek a supplementary formalism.  The p r o p o s a l s o f such supplementary f o r m a l i s m s have been c a l l e d  hidden-variable theories.  H i d d e n - v a r i a b l e (HV) t h e o r i e s a r e e x t e n s i o n s o r  r e c o n s t r u c t i o n s o f quantum mechanics w h i c h i n t r o d u c e f u r t h e r s p e c i f i c a t i o n s o f t h e s t a t e o f a quantum system so t h a t t h e s o - c a l l e d hidden s t a t e determines  t h e e x a c t v a l u e s o f magnitudes and p r o p o s i t i o n s w h i c h a r e a s s i g n e d  d i s p e r s i v e v a l u e s by a quantum s t a t e - i n d u c e d state  ty  Exp^, .  i n d u c e s t h e g e n e r a l i z e d p r o b a b i l i t y measure  which i s d i s p e r s i v e w i t h r e s p e c t t o every  P £ US^, ,  So w h i l e a quantum Exp^. : P^^ ->• [0,1] a hidden s t a t e induces  o r 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 w h i c h somehow a s s i g n s an e x a c t  0  or  1  v a l u e t o 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, t h e presumed i n c o m p l e t e n e s s o f quantum mechanics i s r e f l e c t e d by t h e f a c t t h a t t h e s e t o f quantum  Exp^, measures  i s a p r o p e r s u b s e t o f a l a r g e r s e t o f measures which i n c l u d e s t h e 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 t h e o r y 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 t h e HV t h e o r y , o r t h e y may be some 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 measures d e f i n e d on t h e quantum and  P ^ A  ( o r on s u b s t r u c t u r e s  of  PQ^)* Von Neumann, J a u c h - P i r o n  Gleason-Kochen-Specker prove t h e i m p o s s i b i l i t y o f t h r e e k i n d s o f  generalized described  i n Chapter V(D);  as s h a l l be e l a b o r a t e d impossible  P ^  d i s p e r s i o n - f r e e measures on t h e quantum  A  s t r u c t u r e s , as  t h e y t h u s r u l e o u t t h r e e k i n d s o f HV t h e o r i e s ,  below.  But b e s i d e s t h e s e t h r e e proposed b u t  k i n d s o f 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 a v o i d t h e above i m p o s s i b i l i t y p r o o f s have a l s o been proposed. a l l , f o u r c a s e s , each quantum  In  Exp^ measure i s r e p r e s e n t e d i n t h e proposed  HV t h e o r y a s a m i x t u r e o r complex, e.g., a convex sum o r w e i g h t e d i n t e g r a l , o f d i s p e r s i o n - f r e e HV measures.  And a l l f o u r k i n d s o f HV p r o p o s a l s 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 t h e complexes w h i c h r e p r e s e n t t h e quantum  Exp^, measures i n t h e HV t h e o r y must y i e l d s t a t i s t i c a l r e s u l t s w h i c h  r e p r o d u c e t h e r e s u l t s g i v e n by t h e quantum  Exp^, measures (and so f a r  o b s e r v e d by e x p e r i m e n t ) (Kochen-Specker, 1967, p. 59; B e l i n f a n t e , 1973, p. 9 ) . However, a s Kochen-Specker a r g u e , t h e i m p o s i t i o n o f t h i s s t a t i s t i c a l c o n d i t i o n a l o n e does n o t y e t t a k e i n t o c o n s i d e r a t i o n t h e 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 t h e quantum magnitudes (and propositions).  These r e l a t i o n s a r e embodied i n t h e a l g e b r a i c s t r u c t u r e o f  the quantum magnitudes, and c o n c o r d a n t l y ,  i n the  P  A M  structure o f the  165  quantum p r o p o s i t i o n s .  Von Neumann, J a u c h - P i r o n ,  G l e a s o n , and Kochen-Specker  do t a k e 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 o r a l l o f the o p e r a t i o n s  and r e l a t i o n s o f  P  must be p r e s e r v e d i n an HV  r e c o n s t r u c t i o n o f quantum mechanics. structural conditions.  Such r e q u i r e m e n t s may be c a l l e d  As shown a t l e n g t h i n Chapter V(D),  each o f t h e s e  a u t h o r s imposes a s t r u c t u r a l c o n d i t i o n w h i c h b o i l s down t o t h e r e q u i r e m e n t t h a t 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 t h e quantum of  PQ^  P  (i.e.,  ,  must p r e s e r v e t h e 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  P^^-preservation),  o r i n o t h e r 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  .  I n a d d i t i o n , von Neumann and J a u c h - P i r o n each impose a s t r u c t u r a l c o n d i t i o n , labeled  (vNJ6)  and  (JP&) i n Chapter V(D),  w h i c h 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 p r e s e r v e an o p e r a t i o n among i n c o m p a t i b l e s .  So  von Neumann's n o t i o n and J a u c h - P i r o n ' 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 o f t h e s e s t r u c t u r a l c o n d i t i o n s a r e  s a t i s f i e d by t h e quantum  E x p ^ measures on  P^  .  The c o n t e n t i o u s  issue  i s whether o r n o t t h e proposed d i s p e r s i o n - f r e e HV measures i n t r o d u c e d  by a  proposed HV e x t e n s i o n 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 required t o 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  t h r e e k i n d s o f HV p r o p o s a l s w h i c h 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 t o s a t i s f y t h e t h r e e d i f f e r e n t s e t s o f s t r u c t u r a l c o n d i t i o n s imposed by von Neumann, J a u c h - P i r o n ,  and Gleason-Kochen-Specker have been  shown by t h e s e a u t h o r s t o be i m p o s s i b l e ; measures a r e t h e m s e l v e s i m p o s s i b l e  e i t h e r t h e d i s p e r s i o n - f r e e HV  o r e l s e complexes o f t h e d i s p e r s i o n - f r e e  HV measures cannot r e p r o d u c e t h e s t a t i s t i c a l r e s u l t s o f t h e 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 . above HV  However, c r i t i c s o f  i m p o s s i b i l i t y p r o o f s and a d v o c a t e s o f the c o n t e x t u a l HV  the  proposals  have brought f o r t h the f o l l o w i n g t h r e e s o r t s o f arguments a g a i n s t  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 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  c o n d i t i o n s which a r e imposed upon t h e proposed HV measures, and s t r u c t u r a l c o n d i t i o n s i m m e d i a t e l y r u l e out a HV t h e o r y . concluding  t h a t an HV t h e o r y  conditions,  ( i i ) The  i s i m p o s s i b l e , we  other  so  the  But r a t h e r than  should r e j e c t the  structural  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 of  different  measurements i n ways w h i c h a r e not j u s t i f i e d i f we t a k e i n t o a c c o u n t t h e i n t e r a c t i o n between measuring i n s t r u m e n t s and  quantum phenomena.  i m p o s i t i o n o f t h e s t r u c t u r a l c o n d i t i o n s begs t h e q u e s t i o n , and c o n d i t i o n s s h o u l d be r e j e c t e d . c o n d i t i o n s and question and  ( i i i ) The  i m p o s i t i o n of the  these  structural  t h e development o f t h e i m p o s s i b i l i t y p r o o f s beg t h e  i n o t h e r ways.  Thus the  HV  So t h e s t r u c t u r a l c o n d i t i o n s s h o u l d be r e j e c t e d  the von Neumann, J a u c h - P i r o n ,  G l e a s o n , and  Kochen-Specker p r o o f s do  not  i n f a c t show t h e 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. In S e c t i o n A, t h e s e c r i t i c i s m s a r e d e s c r i b e d  in detail.  Then i n  S e c t i o n B, a n o t h e r p e r s p e c t i v e on quantum mechanics and t h e problem o f hidden-variables (vN^)  and  (JP&)  i s introduced, according  to which the s t r u c t u r a l  (and t h u s t h e von Neumann and  the Jauch-Piron  t o 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 o f (and t h u s the G l e a s o n and criticisms.  conditions  proofs)  succumb  P ^-preservation n  Kochen-Specker p r o o f s ) are r e s c u e d from t h e s e  167  S e c t i o n 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 t h e q u e s t i o n o f whether t h e d i s p e r s i v e ensembles o f quantum systems can be r e s o l v e d i n t o sub-ensembles which a r e 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 v i e w , an HV r e c o n s t r u c t i o n o f 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 a r e 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 , whether t h e d i s p e r s i v e quantum  and so t h e q u e s t i o n i s  E x p ^ f u n c t i o n s can be r e p r e s e n t e d  m i x t u r e s o r weighted sums o f 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^  as functions  (von Neumann, 1932, pp. 305-307, 324). Von Neumann d e f i n e s an e x p e c t a t i o n - f u n c t i o n by a l i s t o f c o n d i t i o n s , one o f which subsumes t h e two c o n d i t i o n s l a b e l e d (vNo) and (vN#>) i n Chapter V ( D ) . expectation-function operators  The domain o f an  i s t h e s e t o f quantum magnitudes a s r e p r e s e n t e d  on a H i l b e r t space.  by  And t h e f u n c t i o n a l r e l a t i o n s among t h e  magnitudes a r e g i v e n by t h e f u n c t i o n a l r e l a t i o n s among t h e o p e r a t o r s , i s , by t h e a l g e b r a i c s t r u c t u r e o f t h e o p e r a t o r s .  that  So i n terms o f t h e quantum  p r o p o s i t i o n s qua idempotent m a g n i t u d e s , a n e c e s s a r y c o n d i t i o n f o r a n 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 v i e w , t h e e x i s t e n c e • 1  of dispersion-free  E x p ^ f u n c t i o n s on t h e quantum  structures.  As mentioned i n Chapter V ( D ) , u s i n g h i s t r a c e - f o r m a l i s m , p r o v e s t h a t no such d i s p e r s i o n - f r e e as von Neumann's i m p o s s i b i l i t y p r o o f . homogeneous e x p e c t a t i o n - f u n c t i o n s  Exp  w  exist; I referred to this result  I n a d d i t i o n , von Neumann p r o v e s t h a t  do e x i s t and i n f a c t c o r r e s p o n d t o t h e  quantum  E x p ^ f u n c t i o n s induced by t h e pure quantum ty s t a t e s .  quantum  Exp^, cannot be r e p r e s e n t e d  f i r s t because t h e quantum d e f i n i t i o n a homogeneous  von Neumann  So t h e  as mixtures o f d i s p e r s i o n - f r e e  Exp^ ,  Exp^, a r e themselves homogeneous (where by Exp  cannot be r e p r e s e n t e d  as a w e i g h t e d sum o f  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 t h e d i s p e r s i o n - f r e e  E x  P  d  o  w  not e x i s t (von Neumann, 1932, p. 3 2 4 ) . I t i s t h u s t h a t an HV r e c o n s t r u c t i o n o f quantum mechanics i s i m p o s s i b l e , a c c o r d i n g  t o 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 p r o o f by a r g u i n g t h a t i t r e s t s upon an i n c o n s i s t e n c y between t h e r e q u i r e m e n t t h a t HV expectation-functions  s a t i s f y (vNjS) and t h e r e q u i r e m e n t t h a t HV  expectation-functions  be d i s p e r s i o n - f r e e .  of the expectation values o f incompatible  F o r (vN&) r e q u i r e s t h e a d d i t i v i t y magnitudes and  incompatible  p r o p o s i t i o n s qua idempotent magnitudes, and t h e d i s p e r s i o n - f r e e v a l u e o f any magnitude o r p r o p o s i t i o n i s an e i g e n v a l u e proposition.  But s i n c e t h e e i g e n v a l u e s  expectation  o f t h e magnitude o r  o f incompatible  magnitudes o r  p r o p o s i t i o n s a r e n o t 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 w h i c h 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 i m p o s s i b l e  ( B e l l , 1966, p. 4 4 9 ) . The  Kochen-Specker v e r s i o n o f von Neumann's i m p o s s i b i l i t y p r o o f shows c l e a r l y how (vN#) i s t h e c u l p r i t i n t h e p r o o f 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 p r o p o s a l s  whose imposed  c o n d i t i o n s a r e i n c o n s i s t e n t w i t h each o t h e r a r e c a l l e d HV t h e o r i e s o f t h e z e r o - t h 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 n o t s u r p r i s i n g . B e l l a l s o a p p e a l s t o t h e problem o f measurement i n t e r a c t i o n i n o r d e r t o argue t h a t HV measures ( o r e x p e c t a t i o n - f u n c t i o n s ) (vn^).  The r e s u l t o f a measurement o f t h e sum  + P^  o f two  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 s i m p l y a d d i n g t o g e t h e r s e p a r a t e measurements f o r ^ > 2 ' P  P  F  °  (1932, p. 310), a measurement o f a sum an e x p e r i m e n t a l by w h i c h  P^  r3  S  e  x  e  m  Plifi  P^ + P^  e  d  need n o t s a t i s f y incompatible  the r e s u l t s of  by von Neumann  o f incompatibles  involves  arrangement which i s e n t i r e l y d i f f e r e n t f r o m t h e arrangements  and  P  Q  a r e each measured s e p a r a t e l y .  Now a l t h o u g h t h e  169  expectation-value e q u a l t h e sum P^  , Pj  assigned  to  + P^  by any quantum  o f the e x p e c t a t i o n - v a l u e s  a s s i g n e d by  Exp^.  Exp^,  always does  t o each o f  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 o r n e c e s s a r y f e a t u r e o f  the  quantum  Exp^  measures.  the  quantum  Exp^,  measures, e s p e c i a l l y when, as B e l l s u g g e s t s ,  Rather, i t i s a very p e c u l i a r feature of  w i t h Bohr "the i m p o s s i b i l i t y o f any  one remembers  sharp d i s t i n c t i o n between t h e  o f atomic o b j e c t s and t h e i n t e r a c t i o n w i t h measuring i n s t r u m e n t s serve t o d e f i n e t h e c o n d i t i o n s under w h i c h t h e (Bohr quoted by B e l l , 1966,  p. 447).  the  behavior which  [quantum] phenomena appear"  B e l l c o n c l u d e s t h a t t h e r e i s no  r e a s o n t o demand t h a t 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 r e s p e c t t o i n c o m p a t i b l e magnitudes and p r o p o s i t i o n s , as (vN&) So when von Neumann imposes h i s c o n d i t i o n  requires.  and t h e n p r o v e s t h a t  d i s p e r s i o n - f r e e HV measures a r e i m p o s s i b l e and t h u s p r o v e s t h a t an  HV  r e c o n s t r u c t i o n o f quantum mechanics i s i m p o s s i b l e , he i s open t o t h e charge o f begging t h e HV q u e s t i o n s i n c e (vNi$) i s u n j u s t i f i e d . F u r t h e r m o r e , von Neumann's i m p o s i t i o n o f (vNsK) on proposed d i s p e r s i o n - f r e e HV measures begs t h e HV q u e s t i o n i n a n o t h e r way.  One  t h e c o n d i t i o n s which von Neumann i n c o r p o r a t e s as p a r t o f h i s l i s t  of  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  of  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)  If  A,B,...  a r e a r b i t r a r y magnitudes, t h e n t h e r e i s an  a d d i t i o n a l magnitude  A + B + •••  (which does not depend on  t h e c h o i c e o f t h e e x p e c t a t i o n - f u n c t i o n ) , such t h a t Exp(A + B + •••) (von Neumann, pp.  309,  = Exp(A) + Exp(B) +  311).  •••  W i t h 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 t h e sum o f i n c o m p a t i b l e magnitudes, e.g., t h e sum o f  A, B  i s t h a t magnitude w h i c h s a t i s f i e s ( E ) f o r a l l  expectation-functions.  Von Neumann m o t i v a t e s A  The  sum o f t h e o p e r a t o r s  t h i s d e f i n i t i o n by two f a c t s :  A  A, B  ( r e p r e s e n t i n g t h e magnitudes  A, B) i s  i t s e l f a s e l f - a d j o i n t o p e r a t o r which can r e p r e s e n t 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 ,  Now i f we assume t h a t d i s p e r s i o n - f r e e HV e x i s t , then t h e sum o f Exp  w  A, B  E x  P  Exp^(A + B) = Exp^(A) + E x p ^ ( B ) . e x p e c t a t i o n - f u n c t i o n s do  w  a s d e f i n e d by a l l t h e quantum  may be d i f f e r e n t from t h e sum o f  A, B  as d e f i n e d by j u s t a l l t h e A  quantum  Exp^ .  represent  And f o r example, a l t h o u g h  quantum  Exp^, ,  the operator  Exp  w  A  A  A  does  E x  P  w  »  ± n  t h e sum o f  A, B  which case  c o n t r a r y t o von Neumann's (vN^) c o n d i t i o n ,  i f t h e d i s p e r s i o n - f r e e HV  sums a r e t h e same.  A + B  as d e f i n e d by a l l t h e  may n o t r e p r e s e n t  Exp^, and HV  (A + B) i Exp (A) + Exp ( B ) , w w  Of c o u r s e ,  A, B  A  A  A + B  as d e f i n e d by a l l t h e quantum A  the operator  t h e magnitude which i s t h e sum o f A  Exp^ and HV  E P X  a W  r  e  i m p o s s i b l e , t h e n t h e two  However, von Neumann imposes (vN#) w h i c h presumes t h a t  the two sums a r e t h e same (and so presumes t h a t d i s p e r s i o n - f r e e HV  E x  P  d o w  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  (A + B) = Exp (A) + Exp ( B ) , W  W  Exp to satisfy w and t h e n von Neumann p r o v e s t h a t t h e  W  proposed d i s p e r s i o n - f r e e HV  E x  P  a  r  impossible.  e  w  Thus von Neumann i s  b e g g i n g t h e HV q u e s t i o n because t h e 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 b e i n g p r o v e d , namely, t h e i m p o s s i b i l i t y o r n o n - e x i s t e n c e  of  2  d i s p e r s i o n - f r e e HV  E x  P  w  functions.  As mentioned i n Chapter V(D), u s i n g t h e 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 t h a t d i s p e r s i o n - f r e e measures a r e  i m p o s s i b l e 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  171  von Neumann's o l d r e s u l t , i . e . , von Neumann's p r o o f o f t h e i m p o s s i b i l i t y o f d i s p e r s i o n - f r e e measures, p r o v e n w i t h o u t t h e c o n t e n t i o u s c o n d i t i o n ( v N ^ ) . However, J a u c h - P i r o n argue t h a t t h e 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 PQ  the quantum orthomodular l a t t i c e  structures are not i r r e d u c i b l e but  M L  So C o r o l l a r y 1  rather are reducible l a t t i c e s with n o n - t r i v i a l centres.  PQ^ •  does n o t r u l e o u t d i s p e r s i o n - f r e e measures on t h e quantum Now a c c o r d i n g t o J a u c h - P i r o n , a quantum  P  w h i c h does admit  QML  h i d d e n - v a r i a b l e s i s c h a r a c t e r i z e d by t h e f o l l o w i n g p r o p e r t y : on a  PQML  w h i c h a d m i t s h i d d e n - v a r i a b l e s c a n be r e p r e s e n t e d as a w e i g h t e d  sum o f d i s p e r s i o n - f r e e measures on Exp^  E v e r y measure  PQ^  measure on  c  a  n  l  »  ^  n  p a r t i c u l a r , every quantum Then i n t h e i r C o r o l l a r y 3  be so r e p r e s e n t e d .  and a g a i n i n t h e i r Theorem 2 , J a u c h - P i r o n p r o v e t h a t an orthomodular l a t t i c e a d m i t s h i d d e n - v a r i a b l e s o n l y i f a l l i t s elements a r e m u t u a l l y i . e . , o n l y i f t h e l a t t i c e i s Boolean.  So any quantum  compatible,  P Q ^ which contains  i n c o m p a t i b l e elements does n o t admit h i d d e n - v a r i a b l e s , and hence h i d d e n - v a r i a b l e s a r e i m p o s s i b l e i n quantum mechanics ( J a u c h - P i r o n , 1 9 6 3 , 835-837).  pp.  Bub's e l u c i d a t i o n o f J a u c h - P i r o n ' s work shows c l e a r l y how c o n d i t i o n (JP&) i s t h e 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 ) . quantum  F o r Bub shows how t h e  P Q J ^ cannot be r e p r e s e n t e d a s w e i g h t e d sums  Exp^, measures on a  o f d i s p e r s i o n - f r e e measures on  Pq^  L  when t h e d i s p e r s i o n - f r e e HV measures  a r e r e q u i r e d t o s a t i s f y (JP£>) (Bub, 1 9 7 4 - , pp.  61-62).  F o r example, c o n s i d e r  a quantum  Exp^. which a s s i g n s v a l u e s t o two i n c o m p a t i b l e atoms  °  3  F  3  since  P  QML  3  f  o  l  l  o  w  s  P^ A p^ = 0 ,  :  E x  P ( y  p Y  > =  1»  E x  P V y  (  =  h*M  E x p ^ ( P ^ A P^) = E x p ^ ( 0 ) = 0 .  €  (0  »» 1)  P^ , P^ and  According t o the  Jauch-Piron c h a r a c t e r i z a t i o n o f a hidden-variables proposal, i f ?  n  QML  admits  172  h i d d e n - v a r i a b l e s then t h i s weighted 43  sum  2  \.w. ,  Exp^, measure on  where  . 1 1  2  I  (HV)  dispersion-free assignment  2  w^  each  l l  = 0,  Y  Y  f o r every  w^  •  ML  N O W  w^  M  L  c a n be r e p r e s e n t e d a s a  ±  n  w.  is a  i  o r d e r t o reproduce t h e  must a s s i g n t h e v a l u e  i n t h e sum r e p r e s e n t i n g  X.w.(P.) = 1 = Exp, ( P . ) . ^  PQ  A  and each  i  measure on  Exp^(P^,) = 1,  i . e . , f o r every  \. = 1  . l  P  And s i n c e  Exp^, ,  P. A P  V  V  ^( ^.^  w  p  = 0,  =  t o P^. , ^  s o  that  w.(P,» A P ) = w.(0)  cp  i n t h e sum r e p r e s e n t i n g  1  l  Y  cp  i  E x p ^ . Moreover, none o f t h e  w^ c a n a s s i g n t h e v a l u e 1 t o P^ because by ( J P ^ ) , . ^ Y ^ ^ w.(P ) = 1 y i e l d s w.(P, A P ) = 1, w h i c h c o n t r a d i c t s w.(P, A P ) = 0; P  w  icp  so  1  J  w^P^  =  ^  f°  re v e r  Y  Y  9 w  £  nonzero v a l u e a s s i g n e d by sum w h i c h reproduces weighted reproduce  1  i n t h e sum r e p r e s e n t i n g Exp^, t o P^  V  Exp^,(P^) = 1.  N  D  <P  by any weighted  That i s , a  measures s a t i s f y i n g  t h e v a l u e assignments o f t h i s quantum  A  E x p ^ . Thus t h e  cannot be reproduced  t h e v a l u e assignment  sum o f d i s p e r s i o n - f r e e (HV)  =  (JPii)  cannot  Exp^, measure.  So we c a n v i e w t h e i m p o s s i b i l i t y o f a J a u c h - P i r o n t y p e o f HV p r o p o s a l as b e i n g due t o an i n c o n s i s t e n c y between t h r e e c o n d i t i o n s imposed on proposed  HV  measures:  the structural condition  (JP&),  the dispersion-  f r e e c o n d i t i o n , and t h e s t a t i s t i c a l c o n d i t i o n , w h i c h r e q u i r e s t h a t t h e v a l u e assignments o f t h e quantum weighted  Exp^, measures be reproduced  sum o f d i s p e r s i o n - f r e e HV  measures.  by, e.g., a  Thus, as B e l i n f a n t e s a y s ,  r a t h e r t h a n p r o v i n g t h e 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 , J a u c h - P i r o n have merely shown t h a t t h e i r t y p e o f HV  proposal i s o f the zero-th kind  ( B e l i n f a n t e , 1973, p. 5 9 ) . B e l l ' s objection to the s t r u c t u r a l condition to h i s o b j e c t i o n to o f t h e i r meet  (vN^).  P^, A P^  When  P^ , P^  (JP#) i s s i m i l a r  a r e i n c o m p a t i b l e , a measurement  i n v o l v e s an e x p e r i m e n t a l arrangement w h i c h d i f f e r s  from t h e arrangements by w h i c h  P^. and P^  a r e each measured s e p a r a t e l y .  173  yet  (JP#>)  value  1  r e q u i r e s a proposed d i s p e r s i o n - f r e e  to  P, A P Y  separately. Exp^  i f i t a s s i g n s t h e value.- 1  cp  In s p i t e o f the d i f f e r e n t experimental  measures do s a t i s f y  (JP&).  Exp^, measures i n an  measure t o a s s i g n t h e t o each  P. , P  Y  arrangements, t h e quantum  HV  HV  measures w h i c h r e p r e s e n t  (JP$>),  measurement i n t e r a c t i o n . condition  (JP<!0  the  HV  So when J a u c h - P i r o n  HV  question since t h e i r imposition of  q u e s t i o n i n t h e f o l l o w i n g manner.  b i l i t y o f r e p r e s e n t i n g t h e quantum of  dispersion-free  measure  impose t h e i r s t r u c t u r a l HV  measures and t h e n show  a r e i m p o s s i b l e , t h e y a r e open t o t h e charge o f b e g g i n g (JP&)  Bub a l s o argues t h a t t h e J a u c h - P i r o n the  HV  (JPv*).  e s p e c i a l l y when we r e c a l l t h e problem o f  on proposed d i s p e r s i o n - f r e e  that hidden-variables  the  reconstruction likewise satisfy  But i t i s n o t r e a s o n a b l e t o 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 must i t s e l f s a t i s f y  ,  <P  And so i t i s r e a s o n a b l e t o r e q u i r e t h a t  the w e i g h t e d sums o f d i s p e r s i o n - f r e e quantum  HV  HV  Exp^  measures on  P_„  i s not j u s t i f i e d .  i m p o s s i b i l i t y p r o o f ( s ) beg  Jauch-Piron measures on a  T  .  prove the impossiPQ^  That i s , t h e  a s l  HV  mixtures measures  QML  considered  by J a u c h - P i r o n  d e f i n e d on t h e quantum  a r e 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  PQ^  l  •  But t h e n t h e J a u c h - P i r o n  out t h e 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 t h e quantum mixtures o f d i s p e r s i o n - f r e e  HV  p r o o f does n o t r u l e Exp^, measures as  measures w h i c h a r e c l a s s i c a l p r o b a b i l i t y  measures d e f i n e d on a B o o l e a n s t r u c t u r e (Bub, 1974, p. 6 3 ) . The same c r i t i c i s m can be d i r e c t e d a g a i n s t t h e p r o o f s and arguments by w h i c h von Neumann p u r p o r t s hidden-variables.  F o r von Neumann r e f e r s t o d i s p e r s i o n - f r e e oHV  expectation-functions  d e f i n e d on t h e s e t o f quantum p r o p o s i t i o n s , qua  idempotent magnitudes r e p r e s e n t e d quantum  P  Q M  .  t o show t h e i m p o s s i b i l i t y o f  by p r o j e c t o r s , whose s t r u c t u r e i s a  S i m i l a r l y , Gleason's i m p o s s i b i l i t y p r o o f and t h e  174  Kochen-Specker Theorem 1 v e r s i o n o f G l e a s o n ' s p r o o f a r e a l s o s u b j e c t t o t h i s criticism.  F o r G l e a s o n ' s p r o o f shows t h a t h i s s o r t 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 HV measures ( w h i c h s a t i s f y  (Ga) and thus a r e  P^^-preserving)  n^3 are impossible  on t h e quantum  P ~ N  v e r s i o n , b i v a l e n t homomorphisms(i)  s t r u c t u r e s , t h a t i s , i n Kochen-Specker's a r e i m p o s s i b l e on  PQ^'  But Gleason&