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Systems of quantum logic Hughes, Richard Ieuan Garth 1978

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SYSTEMS OF QUANTUM LOGIC by RICHARD IEUAN GARTH HUGHES B.A.(Hons.), Cambridge University, 1957 M.A., Cambridge University, 1961 THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES The Department of Philosophy We accept t h i s thesis as conforming to the required standard THE ., UNIVERSITY OF BRITISH COLUMBIA July, 1978 i^ c) Richard leuan Garth Hughes 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 t h e 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 t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e 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 u n d e r s t o o d 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 w r i t t e n p e r m i s s i o n . Department o f PHILOSOPHY The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e _ _ 0 C T 1 _ 1 5 J L J £ 7 9 _ i i SYSTEMS OF QUANTUM LOGIC. Dissertation by R.I.G. Hughes Dissertation Supervisor: Dr. Edwin Levy Abstract According to quantum mechanics, the pure states of a micro-system are represented by vectors i n a Hilbert Space. Sentences of the form, "x e L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic i s provided in terms of interpretations of a formal language within an  algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question i s a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. i i i Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this l o g i c . With respect to the given sematics, these c a l c u l i are sound and weakly complete. Strong completeness i s conjectured. Quantum logic deals with the l o g i c a l relations between sentences, and so i s properly called a l o g i c . However, i t i s the log i c appropriate to a limited class of sentences: proposals that i t should replace c l a s s i c a l l o g i c wherever the l a t t e r i s used should be viewed with suspicion. i v TABLE OF CONTENTS Chapter I. Logic and Physical Theory 1.1 Introduction 1.2 A P r i m i t i v e System and i t s States 1.3 The Single P a r t i c l e i n Newtonian x Mechanics 1 1.4 The Systems of Quantum Mechanics 1.5 Summary 2 Chapter I I . Boolean Algebras and L a t t i c e s 2 9 11.1 Introduction o 11.2 Boolean Algebras 11.3 L a t t i c e s 2 11.4 Complemented and D i s t r i b u t i v e L a t t i c e s 2 11.5 Miscellaneous Results 3 11.6 F i l t e r s and U l t r a f i l t e r s 3 Chapter I I I . C l a s s i c a l Sentential Logic 3 111.1 Introduction 3 111.2 A Language, P , for Sentential Logic 4 111.3 Valuations 4 1 111.4 Interpretations within Boolean Algebras 111.5 Proof Theory: the System CN -> 111.6 Proof Theory: the System CL 6 Chapter IV. Newtonian Mechanics and C l a s s i c a l Logic IV.1 Systems and Their States 6 IV.2 The Language Yl 7 IV.3 The Algebra Bn 7 IV. 4 A Language of N-Statements ^ Chapter V. The Formalism of Quantum Mechanics ^ V. l Introduction ^ V.2 Vector Spaces ^ V.3 Linear Operators ^ V.4 Inner Products ^ V.5 Hermitian Operators and Projection Operators 9 V.6 Spectral Decomposition 9 V.7 The S t a t i s t i c a l Algorithm 9 V.8 The Spin-% P a r t i c l e , 1 0 " V Chapter VI. VI. 1 VI. 2 VI. 3 VI. 4 VI. 5 VI.6 Chapter VII. V I I . l VII. 2 VII. 3 Orthomodular L a t t i c e s and P a r t i a l Boolean Algebras The Algebra of Q-Propositions The L a t t i c e of Subspaces The Algebra of Projection Operators P a r t i a l Boolean Algebras F i l t e r s and U l t r a f i l t e r s Two Examples Algebraic Structures and Quantum Logic: Orthomodular L a t t i c e s The System OM(FG) Soundness and Completeness of OM(FG) The System OM(H) 103-132 103- 104 104- 110 110-114 114-124 124-128 128-132 132-150 132-136 136-147 147-150 Chapter VIII. VIII .1 VIII. 2 VIII.3 VIII.4 VIII.5 VIII.6 VIII.7 VIII.8 VIII.9 VIII.10 VIII.11 VIII.12 VIII.13 VIII. 14 Chapter IX. IX. 1 IX.2 IX. 3 Chapter X. X. l X.2 X.3 X.4 X.5 X.6 X.7 Algebraic Structures and Quantum Logic: P a r t i a l Boolean Algebras Introduction The Language Q. The Semantics of Q. Admissible Valuations of Q_ Two Comparisons: Reichenbach's 3-Valued System and Friedman and Glymour's Semantics The System QN The Soundness of QN The Completeness of QN The Deduction, Substitution and Replacement Theorems f or QN The Algebra The System QL The Soundness of QL The Completeness of QL QL as a L o g i s t i c Calculus The L o g i c a l Systems of Kochen and Specker The System LS The Calculus PP1 Conclusions The Nature of Quantum Logic Jauch's Views The Case Against Jauch Operations on the L a t t i c e of Propositions An Inadequate P o s i t i v i s m Theory as Description Quantum "Logic" as Logic The Language M 150-258 150-153 153-155 155-171 172-179 179-184 184-191 192-198 198-209 209-214 215-227 227-237 238-243 244-246 246-258 259-268 259-264 264-267 267-268 269-300 269-273 273-277 277-284 2 84-2 88 288-294 294-298 298-300 v i Chapter XI. The Scope and Limitations of Quantum Logic 301 319 XI.1 P r o b a b i l i t i e s and Truth Values 301-306 XI.2 States and Properties 307-315 XI.3 The Scope of Quantum Logic 315-319 Bibliography 320-324 v i i r Figure 1 Phase Space for 2-Coin System 4 2 Shaded Area = 1101 4 3 Semi-Closed Quadrant 4 4 L a t t i c e of Subareas of the Disc 6 5 Areas of the Phase Space and the Corresponding Statements 8 6 L a t t i c e of Statements about the 2-Coin System 10 7 Phase Space f o r Single P a r t i c l e Moving i n One Dimension. 12 8 Correspondence between Subsets of Phase Space and Theoretical Statements 12 9 Boolean L a t t i c e s with 1, 2 and 3 Atoms 15 10 A Non-Distributive L a t t i c e 15 11 Lines and Planes i n 3-Space 17 12 Subspaces of 3-Dimensional Space 17 13 L a t t i c e of Subspaces of R 3 19 14 106 15 The Algebra B* 121 16 129 17 The L a t t i c e LQ 131 18 The Algebra £ g 131 19 162 20 183 21 Experiment E (schematically) . 280 22 281 1 Chapter I. Logic and Physical Theory 1.1. Introduction. Physics uses the language of mathematics to describe the world. According to a theory l i k e Newtonian mechanics or quantum mechanics, the behaviour of a p h y s i c a l system may be represented by a mathematical model. Conceptually, at l e a s t , we can d i s t i n g u i s h between the mathematical formalism which such a theory employs and i t s p h y s i c a l i n t e r p r e t a t i o n : by in t e r p r e t i n g some of the mathematical expressions appearing within the theory we obtain statements which carry information about the e n t i t i e s or systems with which the theory deals. These expressions display an algebraic structure: they stand i n c o d i f i a b l e r e l a t i o n s one to another. To such algebraic r e l a t i o n s between mathematical expressions there correspond l o g i c a l r e l a t i o n s between the associated statements, that i s , between the sentences which, according to the theory, describe a system. Now the structure which appears within Newtonian physics i s an algebra of sets, and the l o g i c a l r e l a t i o n s which t h i s determines are p r e c i s e l y those of c l a s s i c a l l o g i c . However, the s i t u a t i o n i s d i f f e r e n t i n the case of quantum mechanics, one of the most remarkable and successful theories of modern physics, and one which, f i f t y years a f t e r i t s inception, faces no serious r i v a l s , despite the f a c t that i t s i n t e r p r e t a t i o n s t i l l poses deep p h i l o s o p h i c a l problems. The structures suggested by t h i s theory are not Boolean; correspondingly i t seems that the l o g i c appropriate to the quantum domain i s a deviant, n o n - c l a s s i c a l , l o g i c . Such, i n out l i n e i s the thesis advanced by many advocates of 2 "quantum logic"" 1", and i t i s t h i s thesis which I propose to examine. Among 2 the f i r s t to put i t forward were Birkhoff and von Neumann ; i n 1936 they wrote that t h e i r object was ... to discover what l o g i c a l structure one may hope to f i n d i n p h y s i c a l theories which, l i k e quantum mechanics, do not conform to c l a s s i c a l l o g i c . In t h i s introductory chapter I set out the th e s i s i n more d e t a i l . I take, f i r s t of a l l , a very p r i m i t i v e kind of p h y s i c a l system, and show how the states of that system can be represented mathematically. Though t h i s representation hardly constitutes a p h y s i c a l theory, yet within the mathematics employed we f i n d an algebraic structure which corresponds d i r e c t l y to a l o g i c a l structure. The l o g i c which emerges i s an elementary version of c l a s s i c a l p r o p o s i t i o n a l c a l c u l u s , adequate to deal with a l l the " t h e o r e t i c a l " statements we can make concerning t h i s p r i m i t i v e system. A straightforward extension of these ideas leads us f i r s t to Newtonian mechanics, and then to quantum theory; I in d i c a t e why, on t h i s a n a l y s i s , the l o g i c appropriate to the propositions of Newtonian mechanics i s c l a s s i c a l , But not a l l , and some who would assent to i t i n p r i n c i p l e would take issue with t h i s p a r t i c u l a r formulation of i t . See, f o r example, the discussion of Reichenbach 1s work i n Chapter VIII.5, and that of Jauch's i n Chapter X. Birkhoff and von Neumann (1936), p. 1. The idea of a quantum l o g i c had been suggested e a r l i e r by von Neumann himself, a l b e i t i n a s l i g h t l y tentative manner: i n von Neumann (1932) he writes, "The r e l a t i o n between the properties of a p h y s i c a l system on the one hand, and the projections on the other, makes possible a s o r t of l o g i c a l calculus with these." Another pioneering paper was that of Strauss (see Strauss (1936)). I f i n d no warrant, however, f or Fine's claim (Fine (1972), p. 14 and f n . 17), that Jordan, von Neumann and Wigner (1934) i s an ea r l y paper on quantum l o g i c , although i t treats i n d e t a i l some algebraic structures which a r i s e within the mathematics of quantum theory. 3 1.2. A P r i m i t i v e System and i t s States. Consider a box with a transparent l i d , which contains a penny and a quarter. We can regard t h i s as a p r i m i t i v e p h y s i c a l system; we determine i t s state by looking at the uppermost face of each coin. Since each of the coins can be heads or t a i l s , the system can be i n any one of four s t a t e s : both coins heads, both coins t a i l s , penny heads and quarter t a i l s , or 3 vice versa. We can represent t h i s system and i t s states as follows. To each state of t h i s system there corresponds one quadrant of a d i s c (see F i g . 1). Thus the area above the h o r i z o n t a l l i n e corresponds to the two states i n which penny i s heads, the r i g h t hand h a l f corresponds to the quarter being heads, the upper l e f t quadrant corresponds to the state i n which the penny shows heads and the quarter t a i l s , and so on. These quadrants can be l a b e l l e d 1000, 0100, 0001, 0010 (moving counterclockwise round the d i s c from the top r i g h t quadrant); t h i s allows one to r e f e r to any area A of the d i s c which includes an exact number of quadrants by taking a four place binary number and putting a "1" i n a given place i f the appropriate quadrant i s included i n A. For instance, the number 1101 r e f e r s to the shaded area i n Figure 2. In what follows I want to consider each quadrant as having nothing i n common with the others except the p o i n t at the centre of the d i s c . To achieve t h i s I assume that each quadrant includes only one of the two r a d i i which bound i t . 3 The way the material of t h i s chapter i s organised owes much to Ariadna Chernavska; i n p a r t i c u l a r , she provided the idea of using a system with j u s t four possible states as an example. The two coin system I describe does not d i f f e r i n p r i n c i p l e from the tetrahedral die she used, but o f f e r s c e r t a i n advantages as f a r as exposition i s concerned. 4 5 ( A r b i t r a r i l y I choose, i n each case, the l a s t radius reached as one moves clockwise through the quadrant, as shown i n Figure 3). Let us, for the moment, ignore the system i t s e l f and ju s t look at i t s mathematical representation, that i s , the d i s c divided i n t o four. The d i f f e r e n t areas obtained by combining the quadrants i n d i f f e r e n t ways are re l a t e d to each other, and these r e l a t i o n s h i p s can be displayed with a 4 diagram (see Figure 4). The diagram shows a l a t t i c e of points ; the sixteen points on the l a t t i c e represent areas of the c i r c l e . The lowest point on the l a t t i c e represents the zero area (the area of the poin t at the centre of the c i r c l e ) , and from t h i s l a t t i c e point l i n e s run upwards to the four points which represent the four quadrants. Above these i n turn are points representing l a r g e r areas of the d i s c , and at the top i s the point representing the whole d i s c . We may l a b e l the lowest and highest points on the disc "0" and "1" re s p e c t i v e l y . I f a and b are points on the l a t t i c e , we write a <_ b i f a = b , or i f one can t r a v e l from a to b by moving upwards along the l i n e s of the l a t t i c e . In t h i s l a t t i c e a <_ b whenever the area represented by a i s 'included i n the area represented by b . In other words the r e l a t i o n being displayed here i s that of i n c l u s i o n . Given any two areas of the d i s c (say 1100 and 0110), there i s a l e a s t area which includes both of them (1110) and a greatest area which they have i n common (0100), r e s p e c t i v e l y the union and i n t e r s e c t i o n of the two areas. Now, i f to these two areas there correspond points a and b on the l a t t i c e , there are also points on the l a t t i c e which represent the> union and i n t e r s e c t i o n of the two areas, and which are re f e r r e d to as aVb and aAb 4 The term " l a t t i c e " has a precise meaning f o r mathematicians. Though my use of the term here i s i n accordance with that usage, I postpone a formal account of l a t t i c e s u n t i l l a t e r . (See Ch. II.) 7 r e s p e c t i v e l y . Also f o r any point a , we can f i n d i t s complement a 1 , such that the areas corresponding to a and a' (i) have nothing i n common, and ( i i ) together make up the e n t i r e d i s c . That i s , we have, 1.2.1 aAa' = 0 and aVa' = 1 For example, the point 0100 i s the complement of the point 1011, and v i c e versa. Thus, i f we take the set of points of the l a t t i c e we f i n d that we have two binary operations defined on i t (for any.two points, a, b, we have avb and aAb) and one singulary operation, that of complementation. These operations obey the same laws as t h e i r s e t - t h e o r e t i c counterparts, union, i n t e r s e c t i o n and complementation; i n p a r t i c u l a r the d i s t r i b u t i v e laws hold: fo r any points a, b and c . 1.2.2 av(bAc) = (avb) A (av c) aV(bA C) = (avb) A (av c) . We can describe the r e s u l t i n g mathematical structure equivalently as a complemented d i s t r i b u t i v e l a t t i c e or as a Boolean algebra. To return to the p h y s i c a l system, what so r t of statements can we make describing i t ? The system has a rather impoverished l i s t of properties, since i t j u s t consists of two coins, each of which can be e i t h e r heads or t a i l s . Any d e s c r i p t i o n of these properties of the system i s equivalent to a statement put i n terms of the " t h e o r e t i c a l " representation of the system, i . e . i n terms of areas of the d i s c . Let us symbolize the statements "The penny i s heads" by "p" , "The penny i s t a i l s " by "np" (to be read as "not-p"), "The quarter i s heads" by "q" and "The quarter i s t a i l s " by "nq" . Then symbolizing "and" by "&", we can produce a compound statement in v o l v i n g p and q to correspond to each quadrant of the d i s c , as i s shown i n Figure 5. Figure 5 Areas of the Phase Space and the Corresponding Statements. 9 Notice that i f we replace " I " by "T" and "0" by "F" our previous l a b e l l i n g becomes the column of a truth table appropriate to the statement for that quadrant. S i m i l a r l y each point on the l a t t i c e can be taken to represent a statement inv o l v i n g p and q (see F i g . 6). More s t r i c t l y , i t represents an equivalence class of these statements, such that a l l the members of a given c l a s s have the same truth t a b l e . The maximum point on the l a t t i c e , 1 , represents the class of tautologies and the minimum, 0 , the class of contradictions. Corresponding to the two binary operations on the l a t t i c e are connectives l i n k i n g statements: to the operation v corresponds "v" (read and understood as " o r " ) , to A corresponds while to the singulary operation of complementation corresponds negation. The l a t t i c e now displays r e l a t i o n s h i p s between statements, and the r e l a t i o n <_ which orders the l a t t i c e can be associated e i t h e r with semantic entailment or with d e r i v a b i l i t y . That i s to say, for statements A and B corresponding to points a and b i n the l a t t i c e , a <^  b i f f A \= B i f f A (- B . In f a c t , what we have here i s the Lindenbaum-Tarski algebra for the p r o p o s i t i o n a l calculus with j u s t two atomic sentences.^ C l e a r l y then, i n t h i s simple case, the algebraic structure which appears v i a t h i s representation of the system i s also a l o g i c a l structure. This i s hardly s u r p r i s i n g , since the representation was chosen with p r e c i s e l y that aim i n view. However, we can apply the same ideas very r e a d i l y to something more obviously t h e o r e t i c a l , that i s , to Newtonian mechanics. See Chapter I I I , e s p e c i a l l y Ch. III.5 and III.6. 10 Figure . 6 Lattice of Statements about the 2-Coin System. 11 1.3. The Single P a r t i c l e i n Newtonian Mechanics. The systems dealt with i n c l a s s i c a l - that i s Newtonian - mechanics each consist of a f i n i t e number of p a r t i c l e s . A l l the information which the \ theory y i e l d s about a system at a s p e c i f i e d time may be derived from knowledge of the p o s i t i o n of each p a r t i c l e at that time (where i t i s ) , and i t s momentum (which depends on how f a s t i t i s going and i n what d i r e c t i o n ) . For s i m p l i c i t y I w i l l t a l k about a sin g l e p a r t i c l e system, where the p a r t i c l e i s constrained to move i n one dimension, that i s along a l i n e . ^ Now, whereas the state of the p r i m i t i v e system was characterised by an area ( s p e c i f i c a l l y a quadrant) of a d i s c , here the state of the system i s characterised by the point on the Cartesian plane (q fp) r whose coordinates t e l l us the p o s i t i o n q and the momentum p of the p a r t i c l e (see Figure 7). This plane i s the phase space for t h i s system, as the di s c was the phase space f o r the p r i m i t i v e system. Thus the state of the p r i m i t i v e system was represented by an area of the phase space; the state of the sin g l e p a r t i c l e of Newtonian mechanics i s represented by a point i n the phase space - and i n quantum mechanics, as we s h a l l see, the state of a system w i l l be represented by a vector i n the phase space. So i n one respect things are not what they were. However, i n another respect i t turns out that the Newtonian system and the p r i m i t i v e system are very s i m i l a r , because the relevant mathematical structure i s of the same type i n each case. What i s i s "relevant mathematical structure"? Well, i n the case of the two-coin system, p a r t i c u l a r areas of the phase space (the disc) corresponded to c e r t a i n statements: here we have, not ^This topic receives f u l l e r treatment i n Chapter IV. 12 Figure 7 Phase Space for Single P a r t i c l e Moving i n One Dimension. Figure 8 Correspondence between Subsets of Phase Space and Theoretical Statements. Shaded Area corrsponds to statement, "K.E. > b2/2m V e r t i c a l Line corresponds to statement, "q = a!' 13 necessarily areas, but regions which correspond to p a r t i c u l a r propositions. 2 In Figure 8 the shaded area corresponds to "Kinetic Energy >^  b /2m" (where m i s the mass of the p a r t i c l e ) , and the v e r t i c a l s t r a i g h t l i n e to "Positi o n = a" . Let me amplify t h i s idea of "correspondence". I f I say that to a proposi t i o n A there corresponds a region S , I mean that, when the state in of the system l i e s within S , then A holds. I have talked of "regions"; more mathematically one can t a l k of "Borel subsets of the 7 phase space". Like the set of subareas of the d i s c , the set of Borel subsets of the Cartesian plane has the structure of a l a t t i c e ordered by i n c l u s i o n , and, as before, the binary operations of the l a t t i c e are union and in t e r s e c t i o n , and the complement of a Borel subset of the plane consists of a l l points i n the plane not included i n that subset. Again, we have an isomorphism between t h i s structure and a l o g i c a l structure whose elements are equivalence classes of sentences, and i n which the operations are based on the connectives "v", "&" and " t " ; and, exactly as i n the case with the p r i m i t i v e system, t h i s structure i s a complemented d i s t r i b u t i v e l a t t i c e ( a l t e r n a t i v e l y described as a Boolean algebra), so the r e s u l t i n g l o g i c i s irreproachably c l a s s i c a l . The only difference i s that we are dealing with a somewhat larger l a t t i c e than previously - s p e c i f i c a l l y one with an i n f i n i t e set of equivalence classes of sentences. The extension of these ideas to deal with systems of many p a r t i c l e s i n three-dimensional space makes l i f e more complicated, but i t does not introduce any difference of p r i n c i p l e . For a formal d e f i n i t i o n , see IV.2.2. 14 1.4. The Systems of Quantum Mechanics. To review the argument: i n both of the cases so f a r discussed, the l a t t i c e of (classes of) statements i s isomorphic to the l a t t i c e of subsets of phase space. Since the set-t h e o r e t i c operations of union, i n t e r s e c t i o n and complementation obey Boolean laws, the l o g i c appropriate to these statements i s Boolean, i . e . c l a s s i c a l . Examples of simple Boolean l a t t i c e s appear i n Figures. 4 and 9. Figure 10 however, shows a l a t t i c e of s i x points which i s not Boolean. I t i s non-Boolean because the d i s t r i b u t i v e law does not hold. We have, 1.4.1 x A (uVv) = x A 1 = x but (xAu) v (xAv) = o v 0 = 0 and so x A (uVv) f (xAu) V ( X A V ) Nevertheless t h i s structure conforms to the mathematical d e f i n i t i o n of a l a t t i c e (see Ch. II .3) . This example, of course, merely shows that we can construct non-d i s t r i b u t i v e l a t t i c e s . What i s , f o r us, more important i s that (a) we need to use such a l a t t i c e to display the mathematical structure of a set of subspaces of a given space (as opposed to the set of subsets of that space), and (b) i t i s the subspaces, rather than the subsets, of a space which correspond to the propositions of quantum theory. I f i r s t address myself to poi n t (a). Consider three-dimensional p h y s i c a l space with a designated point, 0 , as o r i g i n . The set of subspaces of the space includes (i) the o r i g i n i t s e l f (the zero subspace), ( i i ) a l l s t r a i g h t l i n e s through 0 , ( i i i ) a l l planes which include 0 , and (iv) the whole of t h i s space. Thus some Figure' 10 A Non-Distributive L a t t i c e . 16 subspaces are included i n others: a l i n e can be included i n a plane, any plane (as well as any li n e ) i s included i n the whole space and so on. But the c r u c i a l point i s that when we form a l a t t i c e of subspaces ordered by i n c l u s i o n , the operations on that l a t t i c e are not the normal set-t h e o r e t i c operations. For example, l e t us take two one-dimensional subspaces at r i g h t angles, S and S (see Figure 11), and ask, "What i s the l e a s t subspace x y which includes both of them?" C l e a r l y the answer i s , "The x-y plane." But t h i s i s not the union of S and S , which j u s t consists of two l i n e s , but x y something larger which can be c a l l e d the span of and , and written (using the l a t t i c e operation symbol) as v . However, the l a r g e s t subspace which S and S have i n common i s t h e i r i n t e r s e c t i o n (the zero x y subspace), and so there the standard s e t - t h e o r e t i c operation i s appropriate. Let us look at the singulary operation, that of complementation. To f i n d the complement of a given subspace, say a plane, S , we cannot j u s t take yz a l l the r e s t of'the space, because we want complementation to be an operation on the set of subspaces, i . e . a function which takes a subspace as argument and y i e l d s a subspace as value. But i f we ask, "What i s the subspace which i s the complement of the subspace S (the y-z plane)?" we f i n d that the answer i s not unique, because f o r any l i n e S^, coming out of t h i s plane we have 1.4.2 S v s , = l S A S , = 0 yz x' yz x 1 We sin g l e out one such complement, , and c a l l i t the orthocomplement of S One subspace i s the orthocomplement of another provided (i) each l i n e through 0 included i n one subspace i s a r i g h t angles to each l i n e through 0 included i n the other, and ( i i ) the span of the two subspaces i s the whole space. A s e l e c t i o n of the subspaces of 3-space i s shown i n Figure 12. We have three 18 axes Cone-dimensional subspaces) x, y and z , mutually at r i g h t angles, and, a d d i t i o n a l l y , i n the x-y plane there are two other subspaces ortho-gonal Cat r i g h t angles) to each other l a b e l l e d u and v . Also shown coming out of the paper are the various planes formed by taking the span of each of x, y, u and v with z . The l a t t i c e of the subspaces formed i n t h i s way i s shown i n 3 Figure 13, i n which every point (except 0 and R ) l i e s immediately above or below, but i s not connected to, i t s orthocomplement. This i s a p e r f e c t l y nice l a t t i c e : i t has various elegant properties, i n c l u d i n g that of g orthomodularity, but i t i s not d i s t r i b u t i v e . Now, according to the thesis I am examining, i t i s t h i s l a t t i c e , or one l i k e i t , which determines the l o g i c a l structure of the set of statements de s c r i p t i v e of a quantum mechanical system. For i f we generalise the properties of vectors i n p h y s i c a l space, i . e . arrows through the o r i g i n whose length and d i r e c t i o n are both s p e c i f i e d , we a r r i v e at the notion of vectors i n a H i l b e r t space, and, according to quantum mechanics, (i) each pure state of a system can be represented by such a vector, and ( i i ) to each proposition about the system there corresponds a subspace of the H i l b e r t space. For instance, consider the proposition that an e l e c t r o n has x-component of spin equal to +% . To t h i s p r o p o s i t i o n corresponds a subspace of the appropriate H i l b e r t space, such that the question, "Does the electron have x-component of spin equal to +% ?" i s the same as the question, " I f v represents the state of the elec t r o n , does v l i e within sx + ? " This i s exactly analogous to the s i t u a t i o n i n c l a s s i c a l mechanics. 2 There we saw that "Does the p a r t i c l e have K i n e t i c Energy >_ b /2m ?" i s 8 As I prove i n Ch. VI; i n that chapter I also define orthomodularity. Figure' 13 Lattice of Subspaces of R3. 20 e f f e c t i v e l y the same question as, " I f w represents the state of the p a r t i c l e , does co l i e within the shaded area of the diagram?" So we are l e d by t h i s chain of reasoning to the conclusion that the l o g i c a l structures appropriate to the propositions of quantum mechanics are not c l a s s i c a l . As we s h a l l see, these structures can be regarded i n two d i f f e r e n t ways, as orthomodular l a t t i c e s or as p a r t i a l Boolean algebras, and the l o g i c s y i e l d e d by these two approaches have quite d i f f e r e n t formal properties. 1.5. Summary. To conclude t h i s preliminary survey, I w i l l bring out the p a r a l l e l s between the three kinds of system I have been discussing with a table. For the p r i m i t i v e system, for a Newtonian system and f o r a quantum mechanical system the table shows (i) the kind of phase space i n which the theory represents the state of the system, ( i i ) the way i n which the state of a system i s represented by the theory, ( i i i ) the regions of the phase space which correspond to the statements, or propositions, about the system, and (iv) the structure which r e s u l t s from an ordering of those regions and which, according to the thesis I am examining, i s to be regarded as the l o g i c a l structure associated with each theory. System Phase space State of system Statement Structure P r i m i t i v e Disc Quadrant Area Complemented d i s t r i b u t i v e l a t t i c e = Boolean algebra Newtonian Quantum mechanical F i n i t e l y dimensional r e a l space Point Borel subset Complemented d i s t r i b u t i v e l a t t i c e = Boolean algebra H i l b e r t space Vector Subspace Orthomodular l a t t i c e P a r t i a l Boolean algebra 22 Chapter I I . Boolean Algebras and L a t t i c e s 1 1 . 1 . Introduction. In t h i s chapter I discuss i n more d e t a i l those mathematical structures mentioned i n Ch. 1 . 2 , Boolean algebras and l a t t i c e s . Standard r e s u l t s are stated without proof, but proofs are given of c e r t a i n r e s u l t s which are important i n what follows. 1 1 . 2 . Boolean Algebras. A Boolean algebra i s an ordered sextuple, 8=<B, v, A , ' , 0 , 1 > , such that (i) B i s a non-empty set; I w i l l ignore the case when B contains j u s t one element (the "degenerate case"); ( i i ) v and A ( c a l l e d " j o i n " and "meet") are binary operations on B ; ("complement") i s a singulary operation on B ; B i s closed under the operations v , A , and ', . ( i i i ) 0 and 1 are two designated elements of B , the zero and the u n i t element respectively; except i n the degenerate case these are d i s t i n c t . Roughly speaking, the operations v, A and '. share the properties of the s e t - t h e o r e t i c a l operations of union, i n t e r s e c t i o n and complementation. (More p r e c i s e l y , the f i e l d of sets <P(A), u , n, , <j>, A> , where P(A) i s the power set of the set A , i s a Boolean algebra.) Various equivalent sets of axioms have been put forward to characterise these operations and to show the properties of the two designated elements. I w i l l use the system preferred by S i k o r s k i , i n which there are f i v e p a i r s of axioms. J This set of axioms i s not a minimal set, as he points out, but i t summarizes the important properties of the operations i n a neat manner, and emphasizes the symmetry of the system"*"^; also, the elements 0 and 1 appear l a t e r i n the development i n a natural way. Note that a l l axioms and theorems containing the l e t t e r s "a", "b", "c", e t c . should be read as though prefaced by the phrase, "For a l l a, b, c e B ..." . Axioms: II.2.1 a v b = b v a a A b = b A a II.2.2 a v (b v c) = (a v b) v c a A (b A C ) = (a A b) A c II.2.3 (a A b) v b = b (a v b) A b = b II.2.4 a A (b v c) = (a A b) v (a A C ) a v (b A c) = (a v b) A (a v II.2.5 (a A a 1) v b = = b (a v a') A b = = b Thus v and A obey commutative, a s s o c i a t i v e and absorptive laws (II.2.1, II.2.2, II.2.3); each i s d i s t r i b u t i v e over the other (II.2.4), and the complement of any element has the properties shown i n II.2.5. From these axioms one can prove the idempotence laws, 11.2.6 a v a = a a A a = a and also show that 11.2.7 a v a' = b v b' a A a 1 = b A b' __ See S i k o r s k i (1964), pp. 3-15. "^ We may derive the p a i r s of axioms II.2.2, II.2.3 from the remainder, and also derive e i t h e r of the d i s t r i b u t i v e laws from the other, together with II.2.1, II.2.2 and II.2.3. However, the complementation laws (II.2.5) are independent. 24 In view of II.2.7, there are elements of B , namely a v a 1 and a A a' , which, although they are obtained from a single element a by the Boolean operations, do not depend on the choice of a . These are the designated elements, 1 and 0 r e s p e c t i v e l y . We have then, by d e f i n i t i o n , 11.2.8 a v a' = 1 a A a' = 0 These elements have the properties that 11.2.9 a V O = a a A l = a 11.2.10 a v l = l a A 0 = 0 Within such an algebra a d u a l i t y p r i n c i p l e obtains, as follows. 11.2.11 I f , i n any theorem, (i) "v" i s substituted f o r every occurrence of "A" , and vice versa, and ( i i ) "1" i s substituted f o r every occurrence of "0", and vice versa, then the r e s u l t i s again a theorem. The truth of t h i s metatheorem i s guaranteed by the symmetrical role s played (i) by v and A , and ( i i ) by 1 and 0 , i n the axioms and d e f i n i t i o n s . Standard theorems of Boolean algebra include the following. 11.2.12 ( a 1 ) ' = a 11.2.13 (a v b)' = a' A b' (a A b)* = a' v b* (de Morgan's laws) 11.2.14 a v b = b i f f a A b =.a Of p a r t i c u l a r importance i n what follows i s the 2-element Boolean algebra, B „ =.< Z 0, v, A, ', 0, 1 > , where Z 9 = {0, 1} . 25 I I . 3. Lattices."*"^" A l a t t i c e i s a set ordered i n a p a r t i c u l a r way; thus before discussing l a t t i c e s one needs to summarize some features of orderings. The ordered p a i r <A, <_> i s a p a r t i a l ordering i f f A i s a non-empty set and 2 < i s a binary r e l a t i o n on A ( i . e . , j< £ A ) such that, f o r a l l a, b, c e A 11.3.1 a <_ a 11.3.2 i f a <_ b and b <_ c , then a <^  c 11.3.3 i f a <_ b and b <_ a , then a = b . In words, <A, _< > i s a p a r t i a l ordering i f f the r e l a t i o n <_ i s r e f l e c t i v e , t r a n s i t i v e and antisymmetric on A . I f , i n addition, f o r a l l a, b e A , 11.3.4 e i t h e r a £ b or b _< a then A i s t o t a l l y ordered by <^  . An obvious example of a t o t a l ordering, <N, <_ > , i s the set of natural numbers ordered by the r e l a t i o n s h i p " i s les s 12 than or equal to". We now introduce the ideas of upper and lower bounds. I f <A, <^ > i s a p a r t i a l ordering, B c A and a e A , then 11.3.5 a i s a <-upper bound f o r B i f f , f o r a l l b e B , b <_ a 11.3.6 a i s a <-lower bound f o r B i f f , f o r a l l b e B , a < b II.3.7 a i s the _<-least upper bound for B i f f a i s a <-bound B and, for a l l c £ A , i f c i s a £-upper bound for B, then a _< c . •^For a comprehensive treatment of l a t t i c e s , see Birkhoff (1967). 12 I use "N" to denote the set {1,2,3,...} of natural numbers throughout. 26 11.3.8 a i s the <-greatest lower bound f o r B i f f a i s a <_-lower bound f or B and, for a l l c € A , i f c i s a <-lower bound f or B , then c <_ a De f i n i t i o n s II.3.7 and II.3.8 imply that the j<-least upper bound and the 13 _<-greatest lower bound of B are unique, i f they e x i s t . The proof of t h i s i s t r i v i a l , given that i<A, <_> i s a p a r t i a l ordering. The <-least upper bound of B and the ^-greatest lower bound of B are also c a l l e d r e s p e c t i v e l y the <^-supremum of B (sup B) and the <infimum of B (inf B) . If < A, <_ > i s a p a r t i a l ordering and A i t s e l f has an upper bound, then i t i s unique, and i s known as the maximum of A ; s i m i l a r l y , i f A has a lower bound i t i s known as the minimum of A . If < A, £ > i s a p a r t i a l ordering, and every subset of A containing exactly one or two elements has both a <^-least upper bound and a _<-greatest lower bound, then the p a r t i a l ordering forms a l a t t i c e . We write: 11.3.9 sup{a,b} = a v b inf{a,b} = a A b f o r any a, b e A , and provide a formal d e f i n i t i o n of a l a t t i c e i n terms of the operations, v, A, as follows. A l a t t i c e i s a quadruple < A, v, A, <> such that A i s a non-empty set, v and A are binary operations on A , <_ i s a r e l a t i o n which p a r t i a l l y orders A , and for a l l a, b, c e A , 13 There are sets B c_ A for some p a r t i a l orderings A, _< which have no upper bound, e.g. when B = A = N . There are also sets with upper bounds but with no l e a s t upper bound. The same can be sa i d , mutatis mutandis, about lower bounds. 11.3.10 a £ a v b a A b <_ a 11.3.11 b £ a v b a A b £ b 11.3.12 i f a <_ c and b <_ c , then a V b <_ c ; i f c <^  a and c <_ b , then c <_ a A b I t i s simple to prove that, i f L = <A, v, A , <> i s a l a t t i c e , then f o r any element a, b, c of A 11.3.13 a v b = b v a a A b = b A a 11.3.14 a v (b v c) = (a v b) v c a A (b A C) = (a A b) A C 11.3.15 (a A b) v b = b ( a V b ) A b = b which equations may be compared with II.2.1, II.2.2 and II.2.3. The proofs of the idempotence laws, 11.3.16 a v a = a a A a = a are also t r i v i a l . We may note that, i f v and A are operations on a set A , characterized by II.3.13-16, and we write, 11.3.17 a •<_ b i f a v b = b , then < i s a p a r t i a l ordering of A and the structure L = <A, v, A , <^  > i s a l a t t i c e . In other words, II.3.13-16, together with II.3.17, y i e l d a d e f i n i t i o n of a l a t t i c e equivalent to that given above. 28 II.4. Complemented and D i s t r i b u t i v e L a t t i c e s A l a t t i c e may be, but i s not necessarily, complemented or d i s t r i b u t i v e or both. A l a t t i c e <A, v, A , o> i s complemented i f f i t contains a maximum element, 1 , a minimum element, 0, and for each a e A there i s an element b of A such that 11.4.1 a v b = l a A b = 0 b i s known as a complement of a ; i n the general case t h i s complement i s not unique. A l l complemented l a t t i c e s have the following p r o p e r t i e s . For a l l a e A , 11.4.2 a V 0 = a a A l = a 11.4.3 a v 1 = 1 a A 0 = 0 A l a t t i c e i s d i s t r i b u t i v e i f f , f o r a l l a, b, c e A . 11.4.4 a A (b v c) = (a A b) v (a A c) a v (b A C) = ;(a v b) A (a v c) This p a i r of conditions II.4.4 are not independent: f or a l l a, b, c e A , 11.4.5 a A (b V C) = (a A b) v (a A c) i f f a v (b A c) = (a v b) A (a v c) Note further that condition II.4.4 i s equivalent to the apparently weaker condition: II.4.4* a A (b v c) <_ (a A b) v (a A c) To t i e the two notions together, we can show that, i n a l a t t i c e which i s both complemented and d i s t r i b u t i v e , each element has a unique complement. Thus, i f < A, v, A , <_>:•• i s a complemented d i s t r i b u t i v e l a t t i c e , and we denote the unique complement of an a r b i t r a r y a e A by a' , then ' 29 i s a singulary operation on A such that, f o r a l l a £ A , 11.4.6 a v a* = 1 a A a' = 0 I t i s t r i v i a l to prove that 11.4.7 (a A a') v b = b (a v a') A b = b The r e s u l t s produced are s u f f i c i e n t to show that, i f (i) < A, v, A , <_ > i s a complemented d i s t r i b u t i v e l a t t i c e ; ( i i ) 0 and 1 are resp e c t i v e l y the minimum and maximum elements of A ; ( i i i ) f o r any a e A , a' i s the complement of a , then <A, v, A , ' , 0, 1> i s a Boolean algebra. This follows immediately from II.3.13-15, II.4.4 and II.4.7. We now show that every Boolean algebra may be regarded as a complemented d i s t r i b u t i v e l a t t i c e . Let B=<B, v, A , ' , 0, 1 > be a Boolean algebra. We make use of the equivalence II.2.13 to define a r e l a t i o n j< on B as follows. 11.4.8 a <_ b i f f a v b = b i f f . a A b = a We can show that t h i s r e l a t i o n i s r e f l e x i v e , t r a n s i t i v e and antisymmetric: whence < B, <^  > i s a p a r t i a l ordering. As we have noted (see p. 11.6) the commutative, a s s o c i a t i v e , absorptive and idempotence laws (II.2.1-3 and II.2.6) s u f f i c e to show that < B, v, A , _< > i s a l a t t i c e . Further, we know from II.2.10 that a v l = l a A 0 = 0 for a l l a e B whence 30 II.4.9 a £ 1 0<_a for a l l a e B Thus B contains both a maximum and a minimum element (1 and 0 r e s p e c t i v e l y ) . From II.2.8, f o r every a e B there i s an element a' e B such that a v a' = 1 a A a ' = 0 and s o < B , v, A, < > i s a complemented l a t t i c e . D i s t r i b u t i v i t y i s guaranteed both by the f a c t that complementation i s unique and by the d i s t r i b u t i v i t y axiom, II.2.4. Thus, corresponding to any Boolean algebra < B, v, A, ', 0, 1 > there i s a complemented d i s t r i b u t i v e l a t t i c e < B, v, A, <_ > , where a <_ b i f f a v b = b . Such l a t t i c e s w i l l be c a l l e d Boolean l a t t i c e s from now on. II.5. Miscellaneous Results. This section contains a miscellany of d e f i n i t i o n s and theorems for the Boolean algebra <B, v, A, *, 0, 1> upon.which an ordering r e l a t i o n <_ i s defined by II.4.8. For any f i n i t e and non-empty A c B , we may define VA , AA : i f A = {a,,...,a } , then 1 n I I . 5.1 VA = a. v (a. v (.. . (a v a ) ...) dr l 2 n - l n AA = - a A (a A (... (a A a ) ...) dx i 2 n - l n The commutative and as s o c i a t i v e laws guarantee that these elements of B are 14 well defined. In terms of the r e l a t i o n < we have, 14 I t would be more precise to replace II.5.1 by a recursive d e f i n i t i o n . 31 II.5.2 VA = sup A AA = i n f A By induction on the c a r d i n a l i t y of A we can show that II.5.3 II.5.4 VA = 0 i f f A = {0} AA = 1 i f f A = {1} II.5.5 The following theorems hold f o r a l l a, b, c, d e B a < b i f f a A b' = 0 Proof. L e f t to r i g h t : Right to l e f t : assume a <_ b ; then a = a A b a A b" = (a A b) A b' = a A (b A b') = a A 0 = 0 assume a A b' = 0 ; a A b = ( a A b ) v o = ( a A b ) v ( a A b ' ) = a A (b v b') = a V I = a whence a < b 11.5.6(a) I f b <_ c , then a v b <_ a v c Proof: Assume b < c ; then b = b A c and a v b = a v ( b A C ) = (a v b) A (a v c) (II.4.8) (II.3.13) (II.4.6) (II.4.3) (II.2.9) (assumption) (II.2.4) (II.2.8) (II.2.9) (II.4.8) (II.4.8) (II.2.4) 32 whence a v b <_ a v c 11.5.6(b) I f b _< c , then a A b <_ a A c . The proof i s s i m i l a r . II.5.7 I f a £ b v c and a A b _< c , then a <_ c Proof: Assume a <_ b v c and a A b _< c then a = a A (b v c) = ( a A b ) v (a A C ) <_ c v (a A c) = (a A c) v c (II.4.8) (II.4.4) (assumption, II.5.6) (II.3.12) (II.3.14) Thus a < c . 11.5.8(a) I f a A b < d and a A c < d , then a A (b v c) < d Proof: Assume a A b <_ d and a A c <_ d . Then d = d v d = ( ( a A b ) v d) v ((a A c ) v d) = ( ( a A b ) v (a A c)) v d = (a A (b v c ) ) v d Thus a A (b v c) < d (II.3.15) (II.4.8) (II.3.13-3.15) (II.4.4) 11.5.8(b) I f a <_ b v c and a <_ b v d , then a <_ b v (c A d) The proof i s s i m i l a r . II.5.9 a A b < c i f f a < b' v c Proof: Assume a A b < c ; then c = ( a A b ) v c (II.4.8) 33 and b' v c = b 1 v ((a A b) v c) = (b' v (a A b)) v c (II.3.13) = ((b' v a) A (b' v b)) v c (II.4.4) = (b' v a) v c (II..4.1-2) = a v (b' v c) (II.3.12-13) Whence a _< b' v c . The proof of the converse i s s i m i l a r . II.6. F i l t e r s and U l t r a f i l t e r s . Throughout t h i s section, "L" ref e r s to the Boolean l a t t i c e < B, v, A , < > . II.6.1. a i s an atom of L i f f a e B , a ^ 0 , and, for a l l b e B , i f b <_ a , then e i t h e r b = a or b = 0 . 11.6.2 L i s an atomic l a t t i c e i f f , f o r every b e B , e i t h e r b = 0 or there i s an atom a of L such that a _< b . Not a l l Boolean l a t t i c e s contain atoms. 11.6.3 F i s a f i l t e r on L i f f F i s a non-empty subset of B , and, f o r a l l a, b e B , (a) i f a e F and a <_ b , then b e F ; (b) i f a, b e F , then a A b e F ; (c) 0 | F . D e f i n i t i o n s sometimes omit clause (c)., and then define a proper f i l t e r as a f i l t e r which does not contain the zero element. 34 Note that, since f o r a l l a e B we have a _< 1 . 11.6.4 I f f i s a f i l t e r on L , then 1 e F . Also, 11.6.5 I f a e B and a f 0 , then F = {b : b e B and a < b} a dr — and i t follows immediately that, f o r any a e B (a / 0) , 11.6.6 F i s a f i l t e r on L . a F i s the p r i n c i p a l f i l t e r generated by a . a Since, f o r a l l a, b e B , a A b £ a and a A b j< b (II. 3.10), i t follows from II.6.2-3 that 11.6.7 i f F i s a f i l t e r , then a A b e F i f f a e F and b e F . Further, 11.6.8 a <_ b i f f every f i l t e r containing a also contains b . Proof: L e f t to r i g h t : by 11.6.3(a) Right to l e f t : assume that, i f F i s a f i l t e r such that a e F , then b e F . Then b e F , where F i s the p r i n c i p a l f i l t e r generated a a by a ; i . e . , b e {c : c e B and a <_ c} . Whence a _< b . 11.6.9 A f i l t e r on L i s an u l t r a f i l t e r i f f there i s no f i l t e r F* on L such that F c F* . In an atomic l a t t i c e , the u l t r a f i l t e r s are the p r i n c i p a l f i l t e r s generated by the atoms. 11.6.10 Every f i l t e r on L i s contained i n an u l t r a f i l t e r on L . A 35 proof of t h i s theorem, using Zorn's Lemma, appears i n B e l l and Slomson (1969)"^. The c o r o l l a r y below follows immediately from II.6.6 and II.6.10. 11.6.11 Every element of B , except the zero element, i s contained i n at l e a s t one u l t r a f i l t e r . A number of equivalent d e f i n i t i o n s of an u l t r a f i l t e r are a v a i l a b l e , as the next theorem shows. 11.6.12 I f F i s a f i l t e r on L , then-(a) F i s an u l t r a f i l t e r i f f (b) for a l l a> b e B , a v b e F i f f e i t h e r a e F or b e F or both i f f (c) f o r a l l a e B , ei t h e r a e F or a' e F , but not both. P r o o f A s s u m e that F i s a f i l t e r on L . We show f i r s t that (a) implies (b). Since F i s a f i l t e r , and a <_ a' v b , b <_ a v b , we know already that i f a e F or b e F or both, then a v b e F , by 11.6.3(a).. We need only to show that (a) implies the converse of t h i s . The proof i s by contraposition. Assume that a i F, b £ F , but a v b e F . Consider the set G , where G = {c e B and a v c e F> . We show that G i s a f i l t e r which contains F . Since b e B and a V b e F , b e G ; thus G i s not empty, (i) Assume that c e G and c < d . Since c e G , a V c e F . Since 1 5 S e e B e l l and Slomson (1969), pp. 15-16. "*"^ The proof follows that i n van Fraassen (1971), p. 54. 36 c £ d , a v C £ a V d (by II.5.6), whence a v d e F . (II.6.3), and so d e G . ( i i ) Assume that c, d e G ; then a v c e F and a v d e F . Thus by II.6.3, Ca v c) A (a v d) e F , i . e . a v (c A d) e F , and so c A d e G . ( i i i ) By assumption, a i F , and so a v 0 / F . Whence 0 i G . Therefore G i s a f i l t e r . But for any c e F , c £ a V c (II.3.10), whence a v c e F (II.6.3), and so c e G . This shows that F c G ; however, F / G , since b e G , but, by assumption, b e F . I t follows that F c G , i . e . F i s not an u l t r a f i l t e r . We now show that (b) implies (c). Assume (b). Then, since, for a l l a e B , a v a' = 1 e F (by II.6.5), i t follows that e i t h e r a e F or a' e F or both. But both a and a' cannot be members of F , since that would mean that a A a' e F (by II.6.3), i . e . that 0 e F , contrary to 11.6.3(c). F i n a l l y we show that (c) implies (a). Assume that, for a l l a e B , e i t h e r a e F or a' e F . Also assume, per absurdum, that there i s some f i l t e r G such that F c G . Let b e G - F ; then b' e F , and so b' e G . Whence b A b' e G (by II.6.3), i . e . 0 £ G . But then G i s not a f i l t e r , by II.6.3. Thus there i s no f i l t e r G such that F c G , and so F i s an u l t r a f i l t e r . The theorem shows that every u l t r a f i l t e r p a r t i t i o n s B i n t o sets, U, U', such that, f o r a l l a £ B , a £ U i f f a 1 £ U' . Also, from t h i s theorem and the c o r o l l a r y II.6.11, we see that: II.6.13 For any element a £ B (a / 1) , there i s at l e a s t one u l t r a f i l t e r which does not contain a . We can now strengthen II.6.8 as follows. 37 11.6.14 a <_ b i f f every u l t r a f i l t e r on L which contains a also contains b . Proof. The l e f t to r i g h t c o n d i t i o n a l follows immediately from II.6.8. We show the r i g h t to l e f t c o n d i t i o n a l by contraposition. Assume that a j£ b ; then a A b' ^ 0 (II.5.5). Let a A b' = c , and be the p r i n c i p a l f i l t e r generated by c . We have a e F and b 1 e F . This f i l t e r may be extended to an u l t r a f i l t e r , U . c c Then a e U and b 1 € U . By II.6.12, b \ U . Thus there i s an u l t r a f i l t e r on L which contains a but does not contain b . Consider the Boolean algebra B = B, v, A, ', 0, 1 corresponding to the l a t t i c e L . I f U i s an a l t r a f i l t e r on L , then we may define the function h : B -> Z_ , such that U A 11.6.15 h u = 1 i f f a e U ; = 0 i f f a i U Then we can show that 11.6.16 h^ i s a homomorphism of B onto . Proof: By d e f i n i t i o n , h^ i s a function mapping B onto 7,^ . We have also, f o r a l l a, b e B , h (a v b) = .1 i f f a v b € U i f f e i t h e r a e U or b e U or both (II.6.12) i f f h^(a) = 1 or h 0(b) = 1 i f f h ( a ) v h„(b) =1 (II.5.3) 38 h ^ a A b) = 1 i f f a A b e U i f f a e U and b e U (II.6.7) i f f ^ ( a ) = 1 and h (b) = 1 i f f h^(a) A l ^ t b ) = 1 (II..5.4) h u(a') = 1 i f f a' e U (II.6.12) i f f a e U i f f h n(a) = 0 i f f ( h u ( a ) ) ' = 1 Since takes only the values 0, 1, these r e s u l t s s u f f i c e to show that h n ( a v b) = h w(a) v h^b) h D ( a A b) = h D(a) A h y(b) h u W ) = ( h ^ a ) ) ' i . e . , that h^ i s a homomorphism of B onto B^  • We c a l l t h i s the canonical homomorphism associated with the u l t r a f i l t e r U . I f A i s a f i n i t e set of elements of B , and h^ i s a homomorphism of B onto B^  , we define the set h^(A) as follows. II.6.17 hyCA) = ^ h T j ( a ) : a € A ^ Using t h i s notation, and that introduced i n II.5.1, we obtain, f o r a f i n i t e non-empty subset A of B , II.6.18 h y(VA) = vh 0(A) hy(AA) = Ahy(A) 39 Chapter I I I . C l a s s i c a l Sentential Logic 111.1. Introduction. The mathematical structures described i n Chapter II stand i n an intimate r e l a t i o n to c l a s s i c a l s e n t e n t i a l l o g i c (C.S.L.). The purpose of t h i s chapter i s to make t h i s r e l a t i o n s h i p e x p l i c i t , so that when, subsequently, I develop a quantum l o g i c based on p a r t i a l Boolean algebras, the p a r a l l e l s between the two l o g i c s are apparent. A s u i t a b l e language, P , for C.S.L. i s introduced; I use an algebraic approach to provide a semantics f o r P equivalent to the elementary treatment of C.S.L. v i a tr u t h tables: This i s obtained using the concept of a valuation of P , that i s , a mapping of the sentences of P i n t o the algebra ; I then show how a valuation can be regarded as a s p e c i a l case of an i n t e r p r e t a t i o n of P within a Boolean algebra. Two (equivalent) systems of proof are discussed. They are both natural deduction systems, based on the N- and L-systerns of Gehtzen. Again, I emphasize the algebraic aspects of the theory. 111.2. A Language, P , f o r Sentential Logic. In the language, P , there are four categories of symbols: (a) p r o p o s i t i o n a l v a r i a b l e s : "p^", "p 2",... ; the set P = {p^: i £ N} of p r o p o s i t i o n a l v a r i a b l e s i s denumerably i n f i n i t e ; (b) a p r o p o s i t i o n a l constant: "£" ; (c) l o g i c a l connectives: "v", "&", " l " ; 1~7 See "Investigations i n t o L o g i c a l Deduction", i n Gentzen (1969). 40 (d) punctuation symbols: " C", " ) " • Certain f i n i t e sequences of these symbols are c a l l e d formulae. We define a formula r e c u r s i v e l y , as follows: III.2.1 (a) For any i e N , "p " i s a formula of P ; (b) "£" i s a formula of P ; (c) i f A and B are formulae of P , then r(A v B)T and rCA & B)"1 are formulae of P ; (d) i f A i s a formula of P , then i A i s a formula of P (e) Nothing i s a formula of P , except by v i r t u e of (a), (b), (c) or (d) above. The set, F, of formulae of P , i s denumerably i n f i n i t e . From now on, I s h a l l adopt the convention of dropping outer parentheses from formulae, and expressions l i k e r(A v B)"1 , which employ both symbols of P and m e t a l i n g u i s t i c v a r i a b l e s , w i l l be rendered without quasi-quotation marks. Such an expression w i l l accordingly appear thus: A v B . The a d d i t i o n a l connective, "=>", i s used to abbreviate c e r t a i n formulae, as follows: I I I . 2.2 A o B = -IA v B The number of l o g i c a l connectives i n P could.be reduced to two without loss of content. E i t h e r " n " and "&", or " V and "v" could have been chosen, and the t h i r d introduced as an abbreviating connective. For example, we could define "v" i n terms of "&" and "-J" , thus: III.2.3 A v B = -|(1A & ~\B) 41 I have chosen a s l i g h t l y l e s s f r u g a l language to emphasize the correspondence 18 between the language and Boolean algebra: to the three connectives of the language correspond the three operations of Boolean algebra, as we s h a l l see. In t h i s chapter I s h a l l often employ a m e t a l i n g u i s t i c fi g u r e known as a sequent. III.2.4 r -»• A i s a sequent i f f T, A are f i n i t e sets of formulae which are not simultaneously empty. For notational economy, various conventions are used i n wr i t i n g out sequents, and elsewhere. I f r , A are sets of formulae, we write "I" , A" i n place of "V u A"; i f T i s a set of formulae and A i s a formula, we write "T, A" or "A, T" i n place of "T u {A}" . Hereafter, unless otherwise stated, c a p i t a l l e t t e r s from the beginning of the Roman alphabet, A, B, C, etc., w i l l be understood to r e f e r to formulae of P , and Greek l e t t e r s , T, A , to r e f e r to sets of formulae of P . III.3. Valuations. As the recursive d e f i n i t i o n of a formula (p. III.2) shows, to construct a formula we use l o g i c a l connectives to l i n k together p r o p o s i t i o n a l variables or the p r o p o s i t i o n a l constant to form subformulae, which i n turn are link e d together to b u i l d the formula. In the c l a s s i c a l view, the connectives are t r u t h - f u n c t i o n a l : that i s to say, when we use a connective to b u i l d a formula from subformulae, the truth value assigned to the formula depends i n a s p e c i f i e d way on the truth values which we assign to the subformulae. 18 I t i s possible to make do with only one connective; see (e.g.) "New Foundations" i n Quine (1961). 42 These dependencies, or "truth functions", can be shown g r a p h i c a l l y by matrices. v T F T T T F T F T F F F These y i e l d , f o r the defined connective: T F T F T T Thus, i f assign a truth value to each of the p r o p o s i t i o n a l v a r i a b l e s of P and to the p r o p o s i t i o n a l constant, then the t r u t h value of each formula A of P i s uniquely determined. Such an assignment of truth values to each A e F may be c a l l e d a valuation of P . A formal d e f i n i t i o n of valuation equivalent to that sketched above, can be given i n terms of the Boolean algebra 8^ . Consider the Boolean algebra 8 = < Z^, v, A, ', 0 , 1 > . III.3.1 V : F -> Z 2 i s a valuation of P i f f , f o r (a) i f A = e P , then v(A) e { 0 , 1} ; (b) i f A = £, then v(A) = 0 ; (c) i f A = B v C , then v(A) = v(B) v v(C) ; Cd) i f A = B S C , then v(A) = v(B) A v(C) ; te) i f A = B , then v(A) = (v(B))' I f we tr a n s l a t e "v(A) = 1" as "A i s true", and "v(A) = 0 " as "A i s f a l s e " , the equivalence of the two approaches i s e a s i l y seen. Other semantic notions may be defined i n terms of the valuation v . A3 111.3.2 v s a t i s f i e s A i f f v(A) = 1 . 111.3.3 A i s s a t i s f i a b l e i f f there i s some v such that y_(A) = 1 . 111.3.4 A i s C - v a l i d i f f , f o r a l l valuations, v , of P , v(A) = 1 . In t h i s case we write }= A . 111.3.5 v s a t i s f i e s T i f f v(A) = 1 for a l l A e r . 111.3.6 T i s s a t i s f i a b l e i f f there i s some v such that, f o r a l l A e T , v(A) = 1 . 111.3.7 T -> A i s C - v a l i d i f f r -* A i s a sequent, and e i t h e r (a) A i s not empty and, f o r a l l valuations v , i f v s a t i s f i e s T , then there i s some A e A such that v s a t i s f i e s A , or (b) A . i s empty and T i s not s a t i s f i a b l e . Note that, from 111.3.7(a) and III.3.5, we see that, i f T i s empty, then r -> A i s a C - v a l i d sequent i f f f o r every valuation v of P there i s some A eA such that v s a t i s f i e s A . I f the sequent T -> A i s C- v a l i d , we write V H A . I I I . 3.8 T semantically, e n t a i l s A i f f T 1= {A}. In t h i s case we write T r= A . T r i v i a l l y , we have 111.3.9 r r= A i f f T i s f i n i t e and, f o r a l l valuations v , i f v s a t i s f i e s V , then v s a t i s f i e s A , and also 111.3.10 >= A i f f 0 = A . I have defined semantic entailment as a r e l a t i o n between f i n i t e sets of formulae and formulae, and so the notion being used here i s more 44 r e s t r i c t e d than i s customary. We can extend our d e f i n i t i o n as follows: I f T i s any set of formulae, A a formula, then III.3.11 T = A i f f there i s some f i n i t e subset, T' , of T , such that 0) r 1 = A . 19 From the compactness theorem f o r C.S.L., i t follows that III.3.12 r = A i f f , f o r a l l valuations v , i f v s a t i s f i e s T , then v s a t i s f i e s A . However, i n what follows I s h a l l be concerned only with the f i n i t a r y notion defined by III.3.8. 20 III.4. Interpretations within Boolean Algebras. In Chapter VIII I discuss a semantics f o r quantum l o g i c s i m i l a r to 21 that proposed by Kochen and Specker. They employ the notion of v a l i d i t y i n a p a r t i a l Boolean algebra: i n t h i s section I take a p a r a l l e l approach to C.S.L. Let 8 be the Boolean algebra <B, v, A, ', 0, 1 > , where B = {a ,a , ...} (possibly, but not neces s a r i l y , i n f i n i t e ) . To each a = a , a . e B there corresponds a mapping of the formulae of P i l i 2 i n t o B which conforms to c e r t a i n c o n s t r a i n t s . Such functions are c a l l e d i n t e r p r e t a t i o n s of P within 8 . I t i s convenient to use the same symbol 19 See (e.g.) van Fraassen (1971), Ch. I I . 20 In t h i s section the l e t t e r "B" i s sometimes used as a me t a l i n g u i s t i c v a r i a b l e standing f o r a formula of , and sometimes denotes the set of elements of a Boolean algebra. The r i s k of confusion seems n e g l i g i b l e . 21 See Kochen and Specker (1965a) and (1965b). I examine these papers c r i t i c a l l y in.Ch. IX. 45 for the sequence and f o r the associated i n t e r p r e t a t i o n function. Formally, III.4.1 I f a* = a. ,a. , .. . e B N , then a* : F -> B i s an i n t e r p r e t a t i o n of V_ within 8_ i f f (a) a*(p.) = a. for a l l j e N ; (b) a*(£) = 0 (c) a*(AvB) = a* (A) v a*(B) (d) a*(A&B) = a*(A) A a*(B) (e) a*(lA) = (a* (A)) ' I t follows immediately from t h i s d e f i n i t i o n that, i f a* i s an i n t e r p r e t a t i o n of P within B^ ( i . e . , a* e Z 2 N ) ' a * i s a valuation of P , and further, that to each valuation v of P there corresponds a sequence a* = a. ,a. , ... e Z_ N (so that a. e {0, 1} , for a l l j eN) and an i , i 2 I. 1 2 j associated i n t e r p r e t a t i o n function, such that a. = v(p.) , and, for any 1 j 3 formula A e F , v(A) = a*(A) . As before, i f U i s an u l t r a f i l t e r on 8 , we denote by h^ the canonical homomorphism from 8 onto 8^ associated with U . Further, i f a* i s an i n t e r p r e t a t i o n of P within 8 , and U i s an u l t r a f i l t e r on 8 , we denote by "hya*" a function from F onto such that, f o r a l l A e F , 111.4.2 h wa*(A) = h u(a*(A)) . We can now prove the following theorem. 111.4.3 I f a* i s an i n t e r p r e t a t i o n of P' within 8 , and U i s an u l t r a f i l t e r on 8 , then h^a* i s a valuation of P . Proof: Let a* be an i n t e r p r e t a t i o n of P within 8 , U an u l t r a f i l t e r 46 on 8 . Then f o r any A , B e F , (a) l^aMA) e {0, 1} (b) a* U) = 0 i U (c) a*(AvB) e U i f f a*(A) v a*(B) e U i f f (III.4.1c) e i t h e r a*(A) e U or a*(B) e U or both (II.6.12) (d) a*(A&B) e U i f f a*(A) A a*(B) e U i f f (III.4.Id) both a*(A) e U and a*(B) e U (II.6.7) (e) a*(lA) e U i f f (a* (A)) ' e U i f f (III.4.1e) a*(A) i U (II.6.12) and since h ya*(A) = 1 i f a*(A) e U = 0 otherwise i t follows that n y a * i s a valuation of P . 111.4.4 A formula A i s v a l i d within 8 i f f , for a l l a* e B N , a*(A) = 1 From t h i s d e f i n i t i o n i t follows t r i v i a l l y that 111.4.5 t=A i f f A i s v a l i d within 8 2 . We now show that 111.4.6 f=A i f f A i s v a l i d within every Boolean algebra. Proof: L e f t to r i g h t (by contraposition). Assume that there i s some Boolean algebra B , such that A i s not v a l i d within 8 . Then there i s a sequence a* e B N such that a*(A) = b / 1 . But i f b f 1 , we know that there i s an u l t r a f i l t e r U £ B such that b £ U (II.6.13), and so h^a* (A) = ^ ( b ) = 0 . But h ya* i s a valuation of P , i . e . there i s a valuation of P which does not s a t i s f y A . Thus A . i s not C - v a l i d . 47 Right to l e f t . This follows immediately from III.4.5. To deal with sequents we introduce new notation, as follows. III.4.7 a*(T) = {a*(A) : A e T} where T i s any set of formulae. The important theorem below may be regarded as a generalization of III.4.6. I I I . 4.8 T |= A i f f ei t h e r (a) T and A are non-empty, and f o r every sequence a* from every Boolean algebra, Aa*(r) <_ v a * ( A ) or (b) A i s empty, and, for every sequence a* from every Boolean algebra, Aa*(r) = 0 or (c) T i s empty, and, f o r every sequence a* from every Boolean algebra, v a * ( A ) = 1 Proof: L e f t to r i g h t . We d i s t i n g u i s h three cases. (a) F and A are both non-empty. The proof i s by contraposition. Assume that there i s some sequence a* from some Boolean algebra 8 , such that Aa* (T) £ v a * ( A ) . Then Aa* ( F ) A ( v a * ( A ) ) ' i_ 0 (II.5.5). Thus the p r i n c i p a l f i l t e r generated by A a * ( T ) A ( v a * ( A ) ) ' exists--- (II: 6.6) , . and'this-filter^may^be extended to an u l t r a f i l t e r U (II.6.10). Further, since Aa* (T) A ( V a * ' ( A ) ) ' e U , i t follows from II.6.7 that Aa* ( T ) e U and ( V a * ( A ) ) ' e U . Let h y be the canonical homomorphism from 8 onto 8„ associated with U . Then 48 h u(Aa*(r)) = 1 = h u ( v a * ( A ) ) • but h u ( A a * ( D ) = h u(a*(r)) (II.6.18) whence A h u ( a * ( D ) = 1 and so h u ( a * ( D ) = { l } (II.5.4) Also h u ( v a * ( A ) ) ' = (h ( v a * ( A ) ) ) ' (II.6.15) = (vh (a*(A)))' (II.6.18) Thus vh (a*(A)) = 0 and so h u ( a * ( A ) = • Now h ya* i s a valuation of P (III.4.3); w r i t i n g v = h ya* we obtain y_(A) = 1 for a l l A e T v(B) = 0 for a l l B e A That i s , r \f A . (b) A i s empty. We proceed again by contraposition, using a s i m i l a r strategy. Assume that there i s some i n t e r p r e t a t i o n a* within some Boolean algebra 13 , such that A a * ( F ) ^ 0 . Then there i s some f i l t e r F £ B , and hence by (II.6.10) some u l t r a f i l t e r U £ B , such that Aa*(r) e F £ U . Let hy be the associated canonical homomorphism from B onto B^ ; then 1 = h u ( A a * ( D ) = A h 0 ( a * ( D ) (II.6.18) and so h^(a*(r)) = {l} i.e.,there i s a valuation v = b^a* such that v(A) = 1 f o r each formula A e T . Thus T i s s a t i s f i a b l e . 49 (c) T i s empty. Again, i f f o r some i n t e r p r e t a t i o n a* within some Boolean algebra B , v a * ( A ) f 1 , then (by II.6.13) there i s an u l t r a f i l t e r U £ B which does not contain v a * ( A ) , and so a canonical homomorphism h^ from B onto B^ such that 0 = h u ( v a * ( A ) ) = v h u ( a * ( A ) ) (II.6.17) and so h ^ a M A ) ) = { o } (II.5.3) Thus there i s some valuation v = h^a* which does not s a t i s f y any member of A . Right to l e f t . Throughout t h i s p art of the proof we use the f a c t that there i s a 1-1 correspondence between valuations and i n t e r p r e t a t i o n s within 8„ . 2 (a) Assume that T and A are both non-empty, and, f o r any sequence a* from any Boolean algebra, A a * (I") <_ v a * (A) . Let v be an a r b i t r a r y valuation, and a* the corresponding i n t e r p r e t a t i o n within B ^ such that, f o r any formula A , y_(A) = a* (A) . Now, i f y_(A) = 1 f o r a l l A e T , then A a * (T) = 1 . By our assumption, V a * ( A ) = 1 , and so (by II.5.3) y_(B) = a* (B) = 1 for at l e a s t one B e A . (b) Assume that A i s empty, and that, f o r any sequence a* from any Boolean algebra, A a*(T) = 0 . Then, f o r a l l valuations v , y_(A) = 0 for some A e T , and so T i s not s a t i s f i a b l e . 50 (c) Assume that T i s empty, and that, f o r any sequence a* from any Boolean algebra, v a * ( A ) = 1 . Then f o r any valuation v , v(A) = 1 f o r at l e a s t one A e A . Whence, i n each case, from the given assumptions, r (= A . We achieve considerable economy i n the statement of theorems i f we extend the notation of II.5.1 to deal with the empty set. Accordingly, we s t i p u l a t e that, i n any Boolean algebra B , 111.4.9 v0 = 0 A0 = 1 This enables us to restate III.4.8 as follows. III.4.8* T ^  A i f f , for each sequence a*, from any Boolean algebra, Aa* (V) <_ v a * ( A ) (Note that, already, from III.4.7, we know that a*(0) =0 .) As a s p e c i a l case of III.4.8*, we have the following: 111.4.10 T |= A i f f , f o r each sequence a* , from any Boolean algebra Aa* (T) £ a* (A) III.4.6 now appears as a s p e c i a l case of III.4.10. Theorem III.4.10 also gives us an a l t e r n a t i v e d e f i n i t i o n of semantic entailment i n terms of int e r p r e t a t i o n s within a Boolean algebra. A t h i r d a l t e r n a t i v e i s provided by III.4.11, below. 111.4.11 r }= A i f f , f o r each sequence a* from any Boolean algebra, i f a*(B) = 1 f o r a l l B e V , then a*(A) = 1 . Proof: We need only p o i n t out (a) that i f T |= A , and i f , f o r a sequence a* from a Boolean algebra B , a*(B) = 1 f o r a l l B e T , then a*(A) = 1 , from 51 III.4.10, and (b) that i f , f o r any sequence a* from any Boolean algebra, a*(A) ' = 1 whenever a*(B) = 1 f o r a l l B e T , then for any valuation v , i f y_(B) = 1 f o r a l l B e T , then y_CA) = 1 , to show that the theorem holds. The f i n a l theorem of t h i s section r e l a t e s the concept of an i n t e r p r e t a t i o n of P within a Boolean algebra to that of an i n t e r p r e t a t i o n 22 of P through a l o g i c a l matrix. 111.4.12 A couple •-: M = < A , I > i s a l o g i c a l matrix for P i f f A i s an algebra s i m i l a r to the algebra of the set of formulae of P , 23 and I a set of distinguished elements of A . Let X be the c h a r a c t e r i s t i c function of the set I , i . e . the function whose domain i s the set of elements of A , such that . X (a) = 1 i f a e I ; X (a) = 0 i f a i I Then 111.4.13 w is_ an M-valuation of. P i f f M = < A , I >.is a l o g i c a l matrix fo r P , a* an i n t e r p r e t a t i o n of P within A , and w = X .a* I f M = <8, U > , where U i s an u l t r a f i l t e r on a Boolean algebra B , then X i s the canonical homomorphism h„ from 8 onto B „ associated U U 2 with U . In f a c t , from III.4.3 we obtain: 22 See van Fraassen (1971), Chapter III.5; the d e f i n i t i o n which follows comes from Wojcicki (1973), p. 8. 23 An algebra A i s a non-empty set A of elements, together with a set {a), jeJ}, of operations on A, indexed by some set J . A i s s i m i l a r to A i f f A and A are indexed by the same set J , and, i f the operation to. on A i s an n-ary operation, then the operation u>. on A' i s an n-ary 3 operation, f o r a l l j e J . (See Lyndon (1966),"'p. 4). 52 III.4.14 v i s a valuation of P i f f M = < B, U > i s a l o g i c a l matrix for P , U i s an u l t r a f i l t e r on the Boolean algebra 8 , and v i s an M-valuation of P . Consideration of a more general case, when M = < 8 , F > and F i s a f i l t e r on a Boolean algebra 8 , y i e l d s the following theorem. III.4i15 r [( A i f f f o r every sequence a* from any Boolean algebra 8 , a*(A) i s a member of every f i l t e r on 8 which contains each a*(C) i n a*(D i f f f o r every sequence a* from any Boolean algebra B , a*(A) i s a member of every u l t r a f i l t e r on B which contains each a*(C) i n a*(r) . Proof. Let a* be an a r b i t r a r y i n t e r p r e t a t i o n of P within a Boolean algebra 8 , and F be a f i l t e r on 8 . Then, from II.6.7, Aa*(T) e F i f f a*(C) e F f o r each C e T and the r e s u l t follows by III.4.10, II.6.8 and II.6.14. III.5. Proof Theory: the System CN When we move by a l o g i c a l argument from one or more premisses to a conclusion, the conclusion i s sa i d to be derived from the premisses. Proof theory seek to a r t i c u l a t e the procedures of such derivations, and to represent them i n a formal way. In the system CN the derivations are displayed i n tree form. The trees are of f i n i t e length; the top formulae of the trees are the assumptions, and the other formulae each follow from those immediately above 53 them by the a p p l i c a t i o n of one of the s p e c i f i e d rules of inference. Thus a de r i v a t i o n i s broken down in t o steps, each of which i s i t s e l f an 24 elementary l o g i c a l argument with one or more premisses and a conclusion. The rules are shown on p. I I I . 15-16 i n the form of schemata. I f a formula appears written i n brackets above one of the premisses of a r u l e , t h i s shows that assumptions of that form, although they may be required i n the de r i v a t i o n of the premisses, are discharged by the a p p l i c a t i o n of that r u l e , i . e . that the conclusion does not depend on that assumption. More formally, we may define the notion of a d e r i v a t i o n as follows. (Unless otherwise stated, the term "formula" i s used i n t h i s section i n the sense of "formula token".) An inference f i g u r e i s an array of formulae of the form B., B B 1 2 . n where B,, B . B , A are formulae of s p e c i f i e d types. When we write 1 2 n down such a f i g u r e we apply an inference r u l e (or rule of inf e r e n c e ) . The formulae B n, ..., B are the upper formulae of the fig u r e ; A i s the 1 n lower formula of the f i g u r e . A d e r i v a t i o n tree i s e i t h e r a si n g l e formula, or a f i n i t e array of formulae, each of which occurs i n an inference f i g u r e , and exactly one of which i s not the upper formula of an inference f i g u r e . Those formulae of a der i v a t i o n tree which are not lower formulae of any inference f i g u r e are the i n i t i a l formulae of the tree; that formula which i s not an upper formula of any inference f i g u r e i s the end formula of the tree. A formula B stands above a formula A i n a d e r i v a t i o n tree i f f 24 I t i s not the l i m i t i n g case of a d e r i v a t i o n : as we s h a l l see, the (somewhat degenerate) tree A shows that A (-^A . e i t h e r Ca) B i s an upper formula of an inference fi g u r e of which A i s the lower formula, or (b) there i s a formula C such that B stands above C and C stands above A . We indicate that B i s an i n i t i a l formula standing above A by w r i t i n g B A If A i s a formula i n a deri v a t i o n tree, the limb branching from A i s that part of the tree which includes only A and a l l formulae standing above A . Thus an i n i t i a l formula i s a limb of the tree and the limb branching from the end formula comprises the tree i t s e l f . Note also that any limb written out on i t s own i s a deri v a t i o n tree. The length of a limb i s defined r e c u r s i v e l y . Let denote the length of the limb branching from the formula A . Then (a) i f A i s an i n i t i a l formula, L = 1 ; (b) i f A i s the lower formula of an inference fig u r e whose upper formula are B ,. . . ,B , then L = Max {L ,...,L } +•1 i n A B B 1 n The length of a deri v a t i o n tree = L^ , where A i s the end formula of the tree. Certain inference rules may perform a discharging function; that i s , when the figu r e corresponding to such a rule appears within a tree, a l l i n i t i a l formulae of a s p e c i f i e d type which stand above c e r t a i n upper formulae of the fi g u r e are sa i d to be discharged by the a p p l i c a t i o n of that r u l e ; i n the case when the upper formula of the inference fi g u r e i s i t s e l f an i n i t i a l formula of the s p e c i f i e d type, i t i s discharged by the a p p l i c a t i o n of the r u l e . 55 A d e r i v a t i o n tree constitutes a d e r i v a t i o n of a formula A from a ( f i n i t e ) set of formulae, T , i n the system CN i f f (a) A i s the end formula of the tree; (b) the set of undischarged i n i t i a l formulae of the tree i s a subset of T ; (c) each inference f i g u r e appearing i n the tree i s an instance of one of the following schemata. [A] [B] ( v i ) A B (VE) '. '. A v B C C A V B A V B (&I) A B (SE) A & B A & B A & B A B [A] hA] (£1) A TA (£E) TA Rules, vE, E have a discharging function: the i n i t i a l formulae discharged i n each case are of the type shown i n square brackets standing above one of the upper formulae of the f i g u r e i n question. The length of a d e r i v a t i o n i n CN i s defined as the length of the corresponding d e r i v a t i o n tree. I f a d e r i v a t i o n of A from Y e x i s t s , we write Y H^ A . In the case when T = 0 , we write |- A . The rules f o r "=>" appear as derived r u l e s . [A] ( = 1 ) . (=>E) A A o B B A B B 56 The following theorems are t r i v i a l . For a l l A , B , C, V, I I I . 5 . 1 T , A | - A I I I . 5 . 2 r , 1 I~ N A I I I . 5 . 3 r , A & 1 Af- & I I I . 5 . 4 h A V I A 1 N 1 1 1 . 5 . 5 T , A | - N B and T, B \-^C imply T, A |- C The system CN i s sound, that i s , f o r a l l A , T, 1 1 1 . 5 . 6 T h A implies T = A N The proof i s straightforward: we use I I I . 3 . 9 as an e f f e c t i v e d e f i n i t i o n of semantic entailment, and proceed by induction over the length of a d e r i v a t i o n tree. CN i s also strongly complete, that i s , for a l l A , T, III . 5 . 7 T h A implies T |—^A To show t h i s , I f i r s t develop a T-algebra f o r P . This i s a 2 5 g e n e r a l i z a t i o n of a version of the Lindenbaum-Tarski algebra for P . In a l l that follows, T i s assumed to be f i n i t e . We define the r e l a t i o n ~ r on F as follows. For formulae, A, B, I I I . 5 . 8 A ~ r B i f f T, A |-NB and T, B |~NA I I I . 5 . 1 , I I I . 5 . 5 and I I I . 5 . 8 guarantee that ~ r i s an equivalence r e l a t i o n on F , that i s , i s r e f l e x i v e , t r a n s i t i v e and symmetric. F i s p a r t i t i o n e d 2 5 The Lindenbaum-Tarski algebra i s the T-algebra obtained when T = 0 . For an a l t e r n a t i v e version, constructed by reference to the semantics of P , see Lyndon ( 1 9 6 6 ) pp. 2 8 - 3 1 . i n t o equivalence classes by t h i s r e l a t i o n : we denote the cla s s containing the formula A by [A] . III.5.9 [A] =.{B : B.e F and B ~ F A} Since f o r a l l A, B e V , we have T, A j ~ ^B , T, B |~NA , i t follows that a l l members of T belong to the same equivalence c l a s s , i . e . III.5.10 if A, B e r then [ A ] r = [ B ] r We define I I I . 5 . i i [r] = {A : r | N A } We now define an ordering r e l a t i o n on F/~^ (the set of equivalence c l a s s e s ) . 111.5.12 [ A ] p < [ B ] r i f f T, A |-NB III.5.5 and III.5.8 guarantee that the choice of representative element of [A]j, , [B]j, i s i r r e l e v a n t . C l e a r l y , <_ i s r e f l e x i v e (by III.5.1), t r a n s i t i v e (by III.5.5) and antisymmetric (by III.5.8). I f r h N A , then, f o r a l l B , V, B |-NA , i . e . 111.5.13 For a l l B , [ B ] r <_ [T]^ . Also, from III.5.2, f o r a l l A III.5.14 [ d r < [ A ] r that i s , F/~^ , contains a maximum element, a n o ^ a minimum element, [ l]^ We define, f o r a l l A, B III.5.15 [ A ] r V [ B ] r = [A v B ] f [ A ] f A [ B ] r = [A & B ] r . We can show straightforwardly that the choice of representative element of each cla s s i s i r r e l e v a n t , i . e . that v and A are we l l defined operations on F/~ r . Further, i t i s t r i v i a l to prove that, with respect to the ordering £ , 1 1 1 . 5 . 1 6 [ A ] R V [ B ] R = sup{[A] r, [ B ] R [ A ] J , A [ B ] r = i n f { [ A ] R , [B]^ Thus < F/~ , v, A , <_ > forms a l a t t i c e . From I I I . 5 . 3 - 4 , together with I I I . 5 . 1 2 - 1 4 , we know 1 1 1 . 5 . 1 7 [ A ] r V [ i A ] r = [ r ] r [ A ] R A h A ] r = [ A ] R f o r a l l A . And so the l a t t i c e i s complemented: f o r any A we have 1 1 1 . 5 . 1 8 [ A ] ' = [TA] We can e a s i l y show that, f o r a l l A, B, C, 1 1 1 . 5 . 1 9 [A&(BvC)] r = [(ASB)v(A&C)] r [Av(B&C)] r = [(AvB)&(AvC)] p and i t follows immediately (using I I I . 5 . 1 5 ) that the l a t t i c e i s d i s t r i b u t i v e . Thus C = <F/~ , v, A , •, [£], [T] > i s a Boolean algebra: we c a l l C the T-algebra f o r P . We now use t h i s algebra i n the proof of the Completeness theorem: I I I . 5 . 7 r I=A implies V h~ A N Proof of I I I . 5 . 7 (by contraposition): Consider the i n t e r p r e t a t i o n p* of P within C , such that each formula B of P i s mapped onto the equivalence cl a s s which contains i t , 2 6 „ The structure of these proofs may be i n f e r r e d from Chapter V I I I . 8 . There I go through ( s l i g h t l y more complicated) proofs f o r the p a r a l l e l constructions i n Quantum Logic. i . e . such that p*(B) = [BJ r . I t remains to show that t h i s i s an i n t e r p r e t a t i o n of the kind defined by III.4.1. However, l e t P* = [ p ^ / I P 2 ] and f o r any p r o p o s i t i o n a l v a r i a b l e , set p*(p^) = [p^lp ' a n ( ^ w r i t e P * ( £ ) = U ] „ , i n accordance with III.4.1a-b. Now consider the algebra of formulae, F = < F , d, c, n > , with operations, d, c, n, corresponding to the connectives v, &, ~l , so that d(A,B) = A v B , c(A,B) = A & B, n (A) = -jA Now from III.4.1c-e we see that (a) p* i s a homomorphism from F i n t o C • and from III.5.15, III.5.16 we may regard (b) [ J as a homomorphism from F i n t o C . But i t i s c l e a r that F i s a free algebra with the set of a l l p r o p o s i t i o n a l 27 variables and constants as b a s i s . Whence any mapping of the set of pr o p o s i t i o n a l v a r i a b l e s and constants into F/~ extends uniquely to a homomorphism of F i n t o ^ , and, since f o r any p r o p o s i t i o n a l constant or var i a b l e , p , P*(P) = [ p ] r i t follows that f o r any formula A , p*(A) = [ A ] r . Now assume that T |— ^ . Then A / [ r ]^ but, f o r a l l B e Y , B e [ Y ] T . Whence p* (A) f i r ] , but Ap * (D = [ T ] r ; and so Ap * (D | p * ( A ) 27 See Lyndon (1966), p. 10. 60 since [Y]^ i s the maximum element of C . Thus, from III.4.11, r 1 A III.6. Proof theory: the System CL In the system CL derivations are again displayed i n tree form, but the tree have sequents, rather than i n d i v i d u a l formulae, as t h e i r basic constituents. To d i s t i n g u i s h the two kinds of tree, I s h a l l r e f e r to those used i n CL as sequent trees rather than as d e r i v a t i o n trees. However, the formal d e s c r i p t i o n of a tree supplied i n III.5 may be ca r r i e d over, i f we replace the word "formula" wherever i t occurs i n that d e s c r i p t i o n by the word "sequent". Thus we obtain d e f i n i t i o n s appropriate to CL of inference f i g u r e , upper sequent, lower sequent, sequent tree, i n i t i a l  sequent, end sequent, stands above, limb, length of a limb, a p p l i c a t i o n of a. rul e of inference. A sequent tree constitutes a der i v a t i o n of a sequent T -> A i n the system CL i f f (a) T -> A i s the end sequent of the tree; 28 (b) a l l i n i t i a l sequents are of the form A -> A or & ; (c) each inference f i g u r e i s an instance of one of the schemata displayed on p. 61 I f a d e r i v a t i o n of the sequent T -> A e x i s t s , then we write T J- A ; i f T I- A and r = 0 , we write j-A ; i f T (—A and A = 0 , we te r hL . Again, the r u l e s f o r " 3 " appear as derived r u l e s (see p. I I I . 22). CL i s sound and complete, i n the following senses. wri Gentzen's o r i g i n a l L-system d i d not use the absurd sentence. I introduce the axiom to unify the presentation of CL and CN and to make the same language appropriate to both. 61 III.6.1 T f- A implies T t= A XJ III.6.2 Tf= A implies r f- A Inference f i g u r e s . (I) S t r u c t u r a l D i l u t i o n r -> A r A Cut r -> A , A r , A -> A r , A -> A r -> A, A r -> A (II) Operational Introduction E l i m i n a t i o n r -> A, A r -> B, A r , A + A r , B -> A r -* A&B , A r , A&B •> A r , A&B -> A r -> A, A r ->• B, A r , A •» A r , B -> A r -+ AVB, A r -> AVB, A r , AVB -> A r , A -» A r ->- A, A r -> T A , A r T A -»- A (III) Derived r , A -> B, A r -> A, A r , B -> A r -> A 3 B, A r , A ^ B A The proof of III.6.1 i s by strong induction on the length of a d e r i v a t i o n . The strategy of the proof of III.6.2 i s s i m i l a r to that used i n proving III.5.7. We p a r t i t i o n F using the equivalence r e l a t i o n ~ 111.6.3 A ~ B i f f A f- B and B f- A XJ L and write 111.6.4 [A] = {B : B e F and A ~ B} We define the ordering r e l a t i o n <_ on the set F/~ of such equivalence cl a s s as follows: I I I . 6.5 I A] <_ [B] i f f A J-LB <_ i s a p a r t i a l ordering on F/~ ; and i f , as before, we define 111.6.6 [A] V [B] = [AvB] [A] A [B] = [A&B] [ A ] ' = h A ] and a d d i t i o n a l l y , write 111.6.7 [AvHA] =1 U] = 0 then we f i n d that 111.6.8 T = < F/~, v, A , *, 0, 1 > i s a Boolean algebra. To prove the Completeness theorem we need the following Lemmata and c o r o l l a r y . 111.6.9 h TB i f f Av A(-_B XJ XJ Proof: L e f t to r i g h t , by the d i l u t i o n r u l e . Right to l e f t by the sequent tree below. 63 A A -> A, T A ->• AvT A Av A +B •+ B, A VIA -> B III.6.10 I f A&B j-CvD , then A, B f- C,D L L Proof: By the sequent tree below. A -> A B -»- B A,B -> A A,B B A/B -> A&B A&B -> CvD C -> C D D A/B -> A&B, CvD A,B,A&B •> CvD C -> C,D D -* C,D A,B -> CvD CvD -> C,D A,B -> CvD,CD A,B,CVD -> C,D A,B -> C,D III.6.11 I f A.,&(Afc. . . (A .&A)...) B v ( B v . . . ( B vB ) . . .) then 1 2 n - l n L 1 2 n - l m T K A ' where r = {A , A , ... A } , A = {B , B , ... B } . L 1 2 n 1 2 m This i s an obvious extension of III.6.10. We may now prove III.6.2. Assume T |= A . Then f o r a l l i n t e r p r e t a t i o n s a* within a l l Boolean algebras, Aa * ( r ) £ v a * ( A ) (III.4.8*) Thus f o r the i n t e r p r e t a t i o n p* such that f or a l l A , p*(A) = [A] , Ap * ( r ) < vp * ( A ) Case (a) T, A are not empty. Assume r = {A.,...,A } 1 n A = { B B } 1 m A p * ( D = A { [ A ] : A e D = [A &(A &...(A &A )...)] 1 I n - l n vp * ( A ) = v{[B]: B e A} = [B v(B v...(B vB )...)] 1 2 m-1 m And so [ A . & ( A . & . . . ( A _ & A ) . . . ) ] < [B 1v(B v...(B ,vB ) 1 z n - l n 1 2 m-1 m whence, by the d e f i n i t i o n of 5 (III.6.5), A . & ( A _ & . . . ( A .&A )...) K B v(B v...(B ,vB).. 1 2 n - l n L 1 2 m-1 m and, by III.6.11, V |- A Case (ti) A i s empty, T = {A^,...,An} v p * ( A ) = 0 , and so Ap * ( r ) = 0 Thus A { [ A ] : A e T} = 0 = [A,&(A„&...(A , & A ) . . . ) ] 1 2 n - l n and so A,&(A„&...(A , & A ) . . . ) k I 1 Z n - l n L Using the A-axiom and the cut rule , we obtain A1&(A.&...(A & A )...) K 1 Z n—I n L And, from III.6.10, r PL Case (b) T ± s empty, A = {B^...^} Since Ap*(T) = 1, i t follows that v p * ( A ) = 1 Thus v{[B]: B e F} = 1 = [B.v(B„v...(B .vB )...)] ± z m-1 m Whence K B v(B„v...(B ,vB )...) (by III.6.9) * L 1 I m-1 m and so |- A Thus, i n each case, r \- A XJ Since from III.5.4-5 and III.6.1-2 we have III.6.12 r |- A i f f r |=A i f f r k A 1 N 1 L we see that the two systems of proof, CN and CL, may be regarded equivalent. Chapter IV. Newtonian Mechanics and C l a s s i c a l Logic 66 IV.1 Systems and Their States. In Chapter 1.3 I b r i e f l y discussed the theory of c l a s s i c a l (Newtonian) mechanics, and sketched an argument to show that the "appropriate" l o g i c for this physical theory was C.S.L. In this chapter I give a f u l l e r description of the theory and emplify the argument, using some of the ideas developed i n chapter I I I . A physical theory such as Newtonian physics or quantum mecha-nics provides a mathematical model which represents the behaviour of phy-s i c a l systems. We descripe a system as f u l l y as the theory permits when we specify ( i ) i t s state, and ( i i ) the way i n which this state w i l l change this time. I w i l l i l l u s t r a t e this with an example from Newtonian physics. A p a r t i c l e constrained to perform simple harmonic motion along a straight l i n e (e.g. by o s c i l l a t i n g at the end of a spring) constitutes a simple system. There are certain constants which affect the behaviour of the p a r t i c l e , such as i t s mass m, and the e l a s t i c i t y of the spring (given by the spring konstant, k). Given these constants, the state of that p a r t i c l e at any instant i s determined when we know i t s position, q, (measured from some suitable point on the l i n e , the midpoint of i t s o s c i l l a t i o n s , say), and i t s instantaneous momemtum, p. The evolution of the state of the system through time i s given by the equations, IV. 1.1. dq = 9H IV. 1.2. -dp = 9tf dt 8p dt 8q IV.1.3. H = p 2 k q 2 2m 2 I f we specify that, when t=o, q = q Q and p = 0 ( i . e . that the p a r t i c l e i s instantaneously at rest at a distance q Q from the midpoint of i t s o s c i l l a t i o n ) these equations y i e l d IV, 1.4 q •= q c o s ( / _ \ t ) P = - v^ cm.q si n ( / _ ^ . t ) m m and the theory predicts that the par t i c l e w i l l execute sinusoidal o s c i l l a t i o n s of period 2V k The systems deals with by Newtonian mechanics comprise aggre-gates of p a r t i c l e s , and the example above may be generalised as follows. The instantaneous state of a system i s specified once we know, for each p a r t i c l e i t s three coordinates of position, q^, p^, q^, and three co-ordinates of momentum, p^, p^, p^ . For any p a r t i c l e of the system, the time evolution equations are: IV.1.5 dq. = 9H IV.1.6 -dp. = 8H i r i  dt 3p. dt 3q. l l where H i s a function of the position and momentum coordinates of a l l the par t i c l e s of the system. This function depends on the constants for the system; i t i s known as the Hamiltonian of the system. In general, the solution of sets of equations of the kind shown above i s (at least) not 29 straightforward. In what follows we w i l l not be concerned with the evo-l u t i o n of a system through time, but with the information contained i n the spe c i f i c a t i o n of i t s instantaneous state. See (e.g.) Goldstein (1960), Chapter 7. 68 I f we know the state of a system, we may assign values to physical magnitudes for that system; thus we specify the properties which the theory declares that system to possess. In the f i r s t example the p a r t i c l e has such magnitudes as v e l o c i t y , k i n e t i c energy and potential energy, and these are known, within a constant factor which r e f l e c t s our choise of units, once we know the p a r t i c l e ' s position and momentum. 2 2 , m p p , and kq .) Notice, that for a c l a s s i c a l (They are, respectively, ^ > 2 m ~2 system, a l l these magnitudes can simultaneously be assigned values. More formally we may say that the (pure) states of a physical system are represented i n the theory by points i n a phase space, ft. In the case of the single p a r t i c l e constrained to move i n one dimension, each state i s represented by a point, (q,p), i n a 2-dimensional space; for' a system of n p a r t i c l e s moving i n 3-dimensional r e a l space, the state space i s a space of 6n dimensions. With each physical magnitude the theory associates a mapping from points i n the phase space to points on the r e a l l i n e . In Newtonian mechanics these mappings are real-valued functions, taking as arguments coordinates of points i n phase space. Now, for a given system X, l e t us consider a physical magnitude M, and the associated function m : ft -> R. For each r e a l number r e R, ' . x ' ^{ rj i - s a region of phase space such that the state of X i s repre-sented by a point w within that region i f and only i f the statement "M has value r" i s true of X. More generally, for the system X i n state o>, 69 IV.1.7 val(M) e S c R i f f u e 1(S) Since'val(M) = {r} i f f val(M) e r , the particular case i s subsumed in this general statement.' We callsentences of the form "val(M) e S" N-statements. Clearly, the N-statements descriptive of any single system are not simantically independent: the truth-value of a particular N-statement i s ties in with the truth-values of others. To put this another way, Newtonian theory imposes constraints on the mappings of the set S^ of N-statements into {0j1} which are admissible valuations of S„. Now the state of a system N determines which N-statements hold for that system; further, given any two distinct states, x and y, of a system, there w i l l be some N-statements which hold when the system is in state x but not when i t i s in state y. Thus to each (type of) system, and to each point i n the phase space for that system, we can associate an admissible valuation of S„T and vice N versa. For conveince we use the same symbol, w , for this valuation as 30 for the corresponding point in the phase space , and write: IV.1.8 w : S^ -»- {0,1} is an admissible valuation for S^ i f f w e f t for some system X, and a (val(M) e S) = 1 i f f oi e _ 1 ( S ) CJ (val(M) e S) = 0 otherwise. _ A similar approach is found in (e.g.) Bub and Demopoulos (1974). They c a l l N-statements "theoretical propositions" and use phrase "basic (theoretical) propositions" for N-statements of the form "val(M) = r" (which,-for van Fraassen (1974), are the "elementary statements" of the theory). What I have termed a "state of a system, Bub and Demopoulos there c a l l an "atomic event" (in the history of a system): for them the " s t a t i s t i c a l state" of the system is the assoc-iated function from S^ onto {0,1}. So that the approach may be gene-ralised to quantum mechanics, Bub and Demopoulos regard this function as a probability measure on S . 70 This double use of the symbol co enables us to think of the phase space ft for a given system as a subset of the valuation space (the 31 ' set of admissible valuations) for the set of N-statements. For a system X, and an N-statement A, we denote by cr (A) that subset of ft such that, X for a l l a) e ft, IV. 1.9 w(A) = 1 i f f OJ e a V(A) While we r e s t r i c t discussion to the system X, a (A) i s the truth set of A. Let M be a physical magnitude, and m^ : ft-:- -*• R be the associated function for the system X. Then IV.1.7-9 y i e l d , IV.1.10 a x ( v a l (M)e S) = _ 1 ( S ) T r i v i a l l y , IV.1.11 a (val(M) e R) = ft a„(val(M) i R) = 0 We see that, for a given system X, to each N-statement A there corresponds a subset a (A) of the phase space. Conversely, to each subset x Z of the phase space of a system X there corresponds a class [£] x of N-statements. This class contains a l l those N-statements which are true of X i f any only i f the state of X l i e s within that subset I of ft. For any E _c ft, IV. 1.12 [Zl = {A: A i s an N-statement and cr (A) = l] X df 1 X  J I t follows immediately that, for a l l w e f t , Z _£ ft, IV.1.13. we Z i f f w(A) = 1 for a l l A e [ Z ] v Cf. van Frassen (1971). 71 Thus to assert of a given system that w e E i s to assert that a l l the N-statements i n the class [E ] hold. Accordingly, we term a sentence of the form "For system X, OJ e E" (E £ ft) an N-proposition, and denote i t by "[E] ". Newtonian theory determines the function f from the set of ordered pairs <X,A> (where X i s a system, A an N-statement) to the set of N-propositions, such that f(<X,A>) = [a (A)] ; that i s to say, X X the theory determines the way i n which the instantaneous properties of a system are related to i t s state. IV.2 The Language N. In this section I r e s t r i c t discussion to one p a r t i c u l a r class of N-propositions. A l l the N-propositions i n this class are descriptive of one system X; however, not a l l N-propositions descriptive of X are Included i n i t . Let ft be the phase space for X. We have seen that to each subset E £ ft there corresponds the N-proposition [E] . We x now take a p a r t i c u l a r set of such subsets, namely the set of a l l Borel subsets of ft, and ( i ) construct a Boolean algebra -B on that set, and ( i i ) consider the language *v whose atomic sentences are the N-proposi-tions corresponding to i t s members. In the following section we construct the algebra <fe of N-propositions isomorphic to and then look at the tx> ft interpretations within -4$ of p, the language described i n Chapter I I I . We f i r s t examine the set of a l l Borel subsets of ft. To do t h i s we need to introduce the notion of a ring of sets. I f x i s an arbitrary space, then a ring of sets i s a family A of subsets of x 72 which includes the empty set, and which i s closed under the operations of union and complementation with respect to x« I t follows that i t i s also closed under the operation of intersection, and that i t includes the set X i t s e l f . I f , a d d i t i o n a l l y , A i s closed under the operation of count-able union ( i . e . provided that, i f A^ e A for i e N, then .-5]A.^ e A), then A i s a i - r i n g . I f A i s a Spring, i t i s also closed under the ope-rati o n of countable intersection. IV.2.1 32 We can show that, for any set C of subsets of x> there i s a unique minimal {T-ring B of subsets of x> such that C c B . As we have seen, for a system of n p a r t i c l e s ft i s a r e a l space of 6n dimensions. Thus ft = R^n, and, for any w e f t , w = < < u , , w „ , w , > , where w . e R for 1 < i < 6n. Consider the sub-1 2 on l set of ft Xco : w e ft and a. < w . < ^ , 1 < i< 6nV 1 1 1 ' ° Such a set i s a semiclosed i n t e r v a l of ft. We may form the set of a l l semiclosed intervals of ft, and apply theorem IV.2.1 to j u s t i f y the follow-ing d e f i n i t i o n . IV.2.2 B^ i s the minimal 5"-ring of subsets of ft which contains a l l the semiclosed intervals of ft. Bfi i s the Borel ring of subsets of ft: i t s members are the borel subsets of ft. In addition to the semiclosed i n t e r v a l s of ft, B^ contains a l l the open and closed subsets of ft, and the sets which contain a single point w e f t . By r e s t r i c t i n g ourselves to B^ c S(ft) (where S (ft) i s the 3 2See P i t t (1963), p.3 73 power set of ft) we have not discarded any subset of ft which i s physically s i g n i f i c a n t . Let •£ = <B , u , n , ' - , 0, ft>, where u , n , receive their customary set-theoretic interpretations. The -B i s a Boolean algebra. Note that, although B^ i s closed under the operations of count-able union and countable intersection, we do not consider these opera-tions i n the algebra -B . -B i s a non-denumerable atomic l a t t i c e , which i s ordered by inclusion. The atoms are the sets containing just one point i n ft, i.e . the sets -jw^  fSr w e f t . Thus to each w e f t there corresponds an u l t r a f i l t e r U on -B , namely the p r i n c i p a l f i l t e r W it generated by w (see the remark following II.6.9 on p.II.12). We now consider just those N-propositions which correspond to the members of B n: we write, Since, i n what follows I confine attention to a single system X, I s h a l l omit reference to i t as redundant. By "an N-proposition" I s h a l l mean a member of B . N-propositions, and with connectives "v", " A " , Following (e.g.) 33 van Fraassen , we may define the language ^ as an ordered p a i r , <S , V > , where S. i s the set of sentences of 11 , and V. the set of admissible valuations. S- i s defined recursively as follows. TV/ ~33 See van Fraassen (1971), p.31 IV.2.3 We now construct a language, «L , whose atomic sentences are 74 IV.2.4(a) I f E e B , then "w e E" i s a sentence of \,; <b) i f A and B are sentences of ti, , then *(A v B)* and *(A A B)5 are sentences of Tt ; (c) i f A i s a sentence of *h, , then ~A i s a sentence of \,; (d) nothing i s a sentence of . , except by virtue of (a), (b) or (c) above. Writers of H _ adopt,the convention of omitting outermost parentheses from sentences; I s h a l l also omit quasi-quotation marks when discussing 1fL , and use "[E]" to denote "u e E", as i n Section 1. The admissible valuations of /h, map the set of sentences of into {0,1} : we have . We make each admissible valuation cor-respond to a point w e ft , by wri t i n g : IV.2.5 v^ i s an admissible.valuation for i f f there i s a point w e f t , and v i s the function vw :S^ -»--Z„ , such that w TL 2 (a) v [E] = 1 i f f w e E; w and, for a l l A, B e S^ and E, IT e Bfi, i f v,(A) = y ([E]) and w w v (B) = v ( [ n ] ) , then W W (b) v^(A v B ) = v,([E] v [n]) = y ([E u n ] ) v^(A A B) = v^ ([E] A [ n ] ) = v ([E n n ]) (c) v w ( ~ A ) - v u ( ~ [ E ] ) = v w <[!]) . The admissible valuations of It are simply related to those of SN ( t h e s e t o f N-statements): from IV.1.9 and clause (a) above we see that, for any w e f t and any N-statement A, IV.2.6 w(A) = v ([cr (A)]) W A 75 Clause (a) also ensures that the N-proposition "co e Z " carries i t s usual meaning: for any atomic sentence [ E ] of Ou , IV.2.7 v [ E ] = >u ( E ) to- ' &) .34 where i s the Dirac Measure' on B n associated with the point co. CO " Thus the admissible valuations of 4u behave in a way of which Tarski would 35 have approved , for we may rewrite IV.2.5(a) (or IV.2.7) to obtain, IV. 2.8 "co € Z " i s true i f f co e Z . Hence ^ i s a formalised language whose atomic sentences receive the same interpretation as their counterparts in mathematical English. In what follows, by using " [ Z ] " to denote a sentence of 0/|_ , and taking "co e Z " to be a sentence of (mathematical) English, I avoid, on the one hand, a plethora of quotation marks, and, on the other, a confusion between the language Ofl and the metalanguage. Definition IV.2.5 also dictates the interpretation which the connectives of <TL receive. We have: IV.2.8 v ( [ Z ] v [n]) = i f f v [ Z u n] = 1 CO CO i f f co e Z u n i f f co e Z or co e I I i f f v [ Z ] = 1 or v [ I I ] = 1 CO CO 3 4See Fano (1971), p.207 3 5See Tarski (1956), pp.155-6 76 IV.2.9 v •([!] A [II]) = 1 i f f v JZ nn] = 1 OJ • OJ i f f OJ c :E fvil i f f we E and OJ e II i f f v [E] = I and v [II] = 1 OJ OJ IV.2.10 v (~[E]) = 1 i f f v [E] = 1 OJ OJ i f f OJ e E i f f OJ i E i f f v [E] = 0 OJ I t follows that the admissible valuations of 1^ characterise the semantic behaviour of the connectives "v", " A " , of /)\, i n exactly the same way that the (admissible) valuations of ^> characterise the seman-t i c behaviour of the connectives "v", "&", " - j " of c l a s s i c a l l o g i c . We can express this by saying that the l o g i c of i s c l a s s i c a l . IV. 3 The Algebra *^ . We may define operations, V Q , A q , - q, on the set of N-propositions, as follows. IV.3.1 [ E] V q [ii] = d f [E u n] [E] A q [ ] = d f [E n II] - o M =df & These operations correspond i n an obvious way to the connectives of fa . Now consider the a l g e b r a - ^ = < B^, V Q , A Q J - Q , [ 0 ], [ ft] >. We may regard [ ] as a one-to-one mapping of B^ into B^, which, by IV.3.1, i s a homomorphism of into ^  thus >B i s isomorphic to and s o - ^ ^ i s a Boolean akgevra, -g i s the algebra of N-propositions. 77 The algebra 4^ may be used to obtain interpretations of the language -fc discussed i n Chapter I I I . From Ch.III.4, we know that, for any Boolean algebra , there i s a class of interpretations of p within 3. Thus there e x i s t interpretations of p within B . Further, since ft the elements of Btf are the atomic sentences of the language Tl,, i n this case the interpretations are not merely interpretations i n the technical 36 sense , that i s , functions defined by sequences of members of a Boolean algebra, but they are also interpretations i n the more general sense of mappings of the sentences ( i n this case, formulae) of one language, p , into the sentences of another, Moreover, as we have seen, each atomic sentence "[£]" of1T- can be read as the sentence "OJ e E" whose symbols have their customary meaning. Thus, v i a the interpretation ( i n the tech-n i c a l sense) of p within IT- , the formulae of p receive an interpretation (in the general sense) i n mathematical English. The question arises of what truth values the formulae of p receive on such interpretations. Consider f i r s t the algebra of Borel subsets of ft. As we have noted (p. IV.7) 3^  i s atomic, and the u l t r a -f i l t e r s on 3 . The atoms are the sets {OJ}; where OJ e ft, and since 3 16 ft i s ordered by in c l u s i o n , to each w e f t there corresponds the u l t r a f i l t e r U = {E : { w } c E e B„} OJ I t follows that, i f h i s the canonical homomorphism associated with U , OJ OJ since, for any E e B^ , IV.3.1 h (E) = 1 i f f E e U OJS OJ we have IV.3.2 h j E ) = ^ ( E ) 36 Some writers use "semi-interpretation" for t h i s technical sense of i n -terpretation. See the discussion of Friedman and Glymour (1972) i n Ch.VIII.5. 78 where ju is the Dirac measure on ft associated with w. to Now i s isomorphic to 3 ^ a n d s o t o t n e homomorphism h^ B„ Z 0 there corresponds a homomorphism h r n:B- -> Z„, such that, for a l l U I [tO J T \ L Z-Bft' IV.3.3 h [ £ j ] [Z] = hJZ) From IV.2.7 and IV.3.2-3, we see that for a l l w e f t , IV.3.4 v j Z ] = h [ w ] [Z] for a l l Z e ; and also that, for a l l Z, II e B , IV.3.5 v ([I] v [n]) = v ([Z u n]) (IV.2.5) w to = h [ f i j ] [E u n] (IV.3.5) = h [ f t j ] ([E] v Q [n]) (IV. 3.1) IV.3.6 v w([Z] A [II]) = h[u] ([Z] A Q [n]) (similarly) IV.3.7 v w(~[Z]) = h [ < u ] (-Q[Z]) (similarly) Thus each canonical homorphism h ^ r B ^ -> Z£ on 46^ extends uniquely to an admissible valuation v :S Z0 of and, conversely, w H z that to each such admissible valuation there corresponds a unique canonical homomorphism on 4^ . Now consider the valuations of p obtained by f i r s t interpreting p in ^  (equivalently, by providing an interpretation of p within ), and then providing an admissible valuation of It is clear that these are valuations of p of a kind we have already encountered in Ch.III.4, be-ing of the form h r [Z] * , where h r , i s the canonical homorphism asso-Ito]- [to J ciated with the u l t r a f i l t e r U ^ , and [Z] * , is a sequence of members of 79 We may summarize what has been said. According to Newtonian mechanics, to any physical system there corresponds a phase.space, ft , and a corresponding algebra 4 L of N-propositions of a language % . There i s a unique admissible valuation of f L associated with each state of to of the system: we may say that the admissible valuations of 'Hi are 3.7 state-induced . For any state co e ft we can effect a p a r t i t i o n of into two sets: the f i r s t , the u l t r a f i l t e r U on generated by [{w}], contains those N-propositions which are true when the system i s i n state co; the other contains those which are fa l s e . The admissible valua-tion of determined by co- , r e s t r i c t e d to the set B^ of N-propositions, i s the characteristic function on BL of the u l t r a f i l t e r U r -,, that i s , \ [co]' ' the function xTT such that, for a l l Z e B , U[co] " IV.3.8 X u [ z ] =,1 ± f [ z ] € u M = Q [co] Note that, i f AA. i s the Dirac measure on B,_ associated with co, then, co ft for a l l E e Bfi , IV.3.9 X u [Z] = VU^E) [ C J ] Since <$L and are both Boolean, and U r n i s an u l t r a f i l t e r , both ti, ft [w] ' y and /U^ are homomorphic mappings (of and >Bfi respectively) onto Now the algebra provides a class of interpretations of p (alte r n a t i v e l y , within/fi ). 37 I owe this to E. Levy and A. Chernavska. 80 The composition of the function x T T with such an interpretation of p yields a valuation of p . Valuations of this kind we may c a l l N-Valuations. From what was said above, these N-valuations, l i k e the admissible valuations of *K. , are c l a s s i c a l : i n t e r a l i a they are both bivalent and truth-func-t i o n a l . At this point a very b r i e f anticipation of the comparable featu-res of quantum l o g i c , as i t i s applied to quantum mechanics, i s i n order. I t we rewrite the summary which appears above i n terms of the states of quantum mechanical systems, we find that a p a r a l l e l analysis goes through almost without change. The c r u c i a l difference i s that, i n the quantum case, the relevant algebras are p a r t i a l Boolean algebras: the characte-r i s t i c functions on the u l t r a f i l t e r s of these p a r t i a l algebras again act as admissible valuations for a language (the language 'HI), but these are no longer homomorphic mappings onto $ . Hence the valuations of quantum mechanics are not truth-functional. A f u l l discussion of the language TR. appears i n Chapter X.6 81 IV.4 A Language of N-statements. For a givex? system, each N-proposition [Z] i s an equivalence class of N-statements (see p.IV.5), that i s , statements ascribing physi-c a l properties to the system. Thus, given a physical system, we could construct a language 9\, , whose atomic sentences were N-statements, and define connectives i n such a way. that / B l k , regarded as an algebra of ft equivalence classes of N-statements, was.again the algebra of this lan-guage. As i n the language •71, i t would be found that these connectives were just l i k e their c l a s s i c a l counterparts, and that the appropriate semantics for 'K was bivalent and truth functional. The admissible s valuations of IL would be homomorphic extensions of the admissible va-luations of S^, and, l i k e the admissible valuations of I t , they would behave c l a s s i c a l l y . They would also be state-induced, r e f l e c t i o n the fact that on Newtonian theory, to specify the (pure) state of a system i s to specify what properties a system i n that state has. 82 Chapter V. The Formalism of Quantum Mechanics. V . l Introduction The notion of a physical system, capable of being in.various states, appears i n quantum mechanics as w e l l as i n Newtonian mechanics. However, as I pointed out i n Chapter 1.4, the state of a quantum mecha-n i c a l system i s represented, not by a point i n a f i n i t e l y dimensional phase space, but by a vector i n a (possibly i n f i n i t e l y dimensional) H i l -bert space. This i n i t s e l f does not mark a r e d i c a l difference between 38 the theories: as Glymour has pointed out , every point i n a phase space can be regarded as a vector i n that space, and conversely; the phase spaces of Newtonian mechanics are r e a l spaces, whereas those of quantum mechanics are complex spaces (see V.2, below), but this on i t s own would be consistent with a view of quantum mechanics as merely an extension of Newtonian mechanics. A more profound difference l i e s i n the way the two theories assign values to the physical magnitudes for a system. Whereas Newtonian mechanics assigns to each magnitude a function from the phase space to the r e a l numbers, according to quantum mechanics each magnitude i s to be associated with a Hermitian operator on the H i l b e r t space ( i . e . with a function which maps the space onto i t s e l f ) . Further, i n quantum mechanics, unlike Newtonian mechanics, i t i s not always possible to assign a de f i n i t e value to every magnitude for,, a system, even when the state of that system i s precisely known; rather, the theory only t e l l s us the pro-b a b i l i t y that a given value for a magnitude w i l l be yielded by experiment. 38 See Glymour (1976), p.171. 83 There is a recipe (the s t a t i s t i c a l algorithm of quantum mechanics) for determining these probabilities given the state of a system and the ope-rator associated with each magnitude. A precise statement.of this algo-rithm, and an account of i t s significance for quantum logic, required a 39 brief outline of the mathematical theory of Hilbert spaces 40 V.2 Vector Spaces We generalise the properties of physical (3-dimensional) space to obtain the notion of a vector space. A vector space is defined over a f i e l d : typical examples of a f i e l d are (a) the set of real numbers, and (b) the set of complex numbers, on which the operations of addition and multiplication are defined in the usual manner; I w i l l not offer a for-mal definition of a f i e l d . According to the f i e l d over which i t i s de-fined, a vector space is known as a real vector space of a complex vector space. Elements of the f i e l d w i l l be referred to as scalars. The usual sumbols "+" and w i l l be used for the operations of addition and multiplication of scalars. Let ^ = < F,+,.,-,0,1 > be a f i e l d . A vector space over £ i s a quintuple, 1/" = < V , + * , 0 * > , satisfying the following conditions. 39 Another treatment can be found in Bub (1974), Chapter 1; like Bub, I use the orthodox approach f i r s t developed by von Naumann. (See von Neumann (1932)). 40 Proofs of the resulta quoted here may be found in Fano (1971) or Jordan (1969). 84 (a) V i s a non-empty set (whose members we c a l l "vectors"); (b) 0* i s a designated vector i n V (called the "zero vector"); (c) +* i s a binary operation on V, -* i s a unary operation on V, and .* maps F X V into V, such that, for a l l x, y, z e V and for a l l a, b e F, V,2.1 X+*y = y+*X V.2.2 X+*(y+*z) = (x+*y) +*Z V.2.3 X+*0* = X V.2.4 X+*-*X = 0* V.2.5 (a+b).*X = a.*X +* b.*X V.2.6 a.*(X+*y) = a.*X +* a.*y V.2.7 l.*X = X V.2.8 (a.b).*X = a.*(b.*X) From these axioms we obtain, for a l l X e V and a e F, V.2.9 (-a).*X = -*(a.*X) V.2.10 0.*X = 0* V.2.11 a.*0* = 0* These equations show that no problems w i l l arise i f "+*", ".*", "-*" and "0*" are written as."+", ".", "-" and "0" respectively; further, we may omit ".", for notational economy. I w i l l continue to denote vectors by l e t t e r s from the end of the,alphabet and scalars by l e t t e r s from the beginning of the alphabet (with or without subscripts). The following are examples of vector spaces. (i) The set C n of a l l n-tubles of (complex) numbers, with addition of two vectors X = <x^, x^, . . . j X ^ > and y = < y^, y^, ...,y > defined by x+y = < x,+y„, x„+y , ...,x +y > and m u l t i p l i c a t i o n of x by a scalar 1 1 1 1 n n 85 3 defined by ax = < ax^jax^, ...ax^ >. (Note that the space R finds a model i n physical space, since any point i n physical space can be asso-ciated with a t r i p l e of r e a l numbers < x, y, z> . 2 ( i i ) Set ^ he 1 of a l l i n f i n i t e sequences of numbers 2 < x,, x~, .. .,x , ... > such that Z. |x.| i s f i n i t e , with addition and 1 2 n x i subtraction defined as above. 2 ( i i i ) The set L of a l l functions of a r e a l variable, T(a), for which the Lebesgue i n t e g r a l J*|Y(a)|2da i s f i n i t e , with addition of two vectors ¥ and <f> defined by (f+ cp) (a) = ¥(a)+<j)(a) , and mu l t i p l i c a t i o n by a scalar c defined by (c T) (a) = c ¥(a). At the r i s k of anachronism, one may say that Heisenberg chose ( i i ) as the phase space for his early version of quantum mechanics (some-times called "matric mechanics") , while Schrodiriger chose ( i i i ) i n deve-loping wave mechanics. Matrix mechanics and wave mechanics were shown to be mathematically equivalent by Schrbdinger i n 1926 4\ r e f l e c t i o n the fact 2 2 42 that L and 1 are isomorphic The vectors x^, x^, .. . ,• x are l i n e a r l y dependent i f there are scalars, a^, a^, a^, not a l l zero, such that a,x. + a„x„ + ... + a x =0 1 1 2 2 n n If this equation entails a. = a„ = ... = a =0, then the vectors are 1 2 n l i n e a r l y independent. I f a l l f i n i t e subsets of an i n f i n i t e set of vectors are l i n e a r l y independent, then the i n f i n i t e set of vectors i s l i n e a r l y independent. 41 See Jammer (1961) pp. 271-6, and Bub (1974) pp. 3-8 4 2See Fano (1971), pp. 269-270. 86 I f a set of vectors i s closed tinder the operations of vector addition and mu l t i p l i c a t i o n by a scalar (+* and . * ) , then i t constitutes a linear manifold. A vector space or linea r manifold i s said to be n-dimensional i f i t contains a set of n l i n e a r l y independent vectors but no set of more than n l i n e a r l y independent vectors. I f i t contains an i n f i n i t e set of independent vectors, i t i s i n f i n i t e l y dimensional. Every closed lin e a r manifold i s called a subspace,.that i s , a linear manifold L i s a subspace i f f every convergent sequence of vectors <x^ > i n L converges to a vector x i n L. Clearly this d e f i n i t i o n assu-mes that we have defined a topoloty on the vector space, so that the notion of convergence i s to hand. A suitable topoloty i s introduced i n Chapter V . 4 ; at present we may note that every f i n i t e l y dimensional line a r manifold i s a subspace. I f every vector x within a subspace i s expressible as a linear combination of the vectors x^, x^ , i . e . there e x i s t scalars a^, a^ such that x = ^ -^ a-^ x >^ then the vectors x^, span the subspace. A set of l i n e a r l y independent vectors which span a given subspace i s a basis for that subspace. We may show that a li n e a r manifold i s n-dimensional i f f i t has a basis of n vectors. V.3 Linear.Operators. 87 An operator A on the vector space V i s a function whose domain and range (respectively "I>A"> " R A") a r e b o t h subsets of V. V.3 . 1 A = B i f f D. = D„ and Ax = Bx for a l l x e D . A B A An operator i s a li n e a r operator, i f f , for a l l x,y e D^ , and for a l l c e F, V.3.2 A(x+y) = Ax + Ay V.3.3 A(cx) = c(Ax) The sum and product of two linea r operators, A and B are defined v i a the equations: V.3.4 DA1T, = DA n D„ (A+B)x = Ax + Bx for a l l x e DA._ A+B A B A+B V.3.5 = {x: x I" T>^ and Bx e D^ } (AB)x = A(B x) for a l l X £ DAB-In general AB ^ BA ; i f AB = BA , then A and B commute. 2 I f A = AA = A , then A i s an idempbtent operator. We denote by I the id e n t i t y operator, and by 0 the zero operator, defined as follows. For a l l x e V, V.3.6 Ix = x Ox = 0 I f , for some operator A, non-zero vector x and scalar a, we have V. 3. 7 Ax = ax then x i s an eigenvector of A, and a i s the eigenvalue of A corresponding to the eigenvector x. I w i l l express this more succinctly by saying that <x,a> i s an eigenpair of A. I f there are eigenpairs, <x,a> , <y,a> , of A such that x f cy for any c e F, then we have degeneracy, and any vector 88 of the form c^x + c^y (where and are scalars) is also an eigenvec-tor of A with eigenvalue a. Note that two vectors, A and B commute i f f they have a l l their eigenvectors in common. V.4 Inner Products. We may assign to each pair of vectors, x, y, a scalar, denoted by "(x,y)" with properties as follows. (Note that the notation "a*" is here used to denote the complex conjugate of a.) For a l l x,y,z e V, c e F, V.4.1 (x,x) > 0 (x,x) =0 i f f x = 0 V.4.2 (x,y) = (y,x)* V.4.3 (x,cy) = c(x,y) V.4.4 (x,y+z) = (x,y) + (x,z) In doing so we define an inner product (sometimes called "sca-lar product") on the vector space V* . Inner products may be defined on the vector spaces described in Section 2. In examples (i) and ( i i ) we . may write (x,y) = and in example ( i i i ) (¥,<f>) = fV(a)* <f>(a)da. 2 2 The conditions on the vectors of 1 and L (examples ( i i ) and ( i i i ) ) guarantee that the inner products defined on those spaces are f i n i t e , and i t is not hard to show that in each case the conditions V.4.1-4 are satis-fied. 89 A vector space on which an inner product i s defined i s known as a pre-Hilbert, or Euclidean, space. The conditions on the inner product y i e l d the following r e l a -t i o n , known as Schwarz's Inequality. For a l l x,y e V, V.4.5 (x,y) (y,x) < (x,x) (y,y) (x,y) (y,x) = (x,x) (y,y) i f f x and y are l i n e a r l y dependent. I f we write |x| = ^^/(x,x) , then (using Schwarz's Inequality to get V.4.8) we obtain, for a l l c e F, x e V, V.4.6 |x| > 0 |x| = 0 i f f x = 0 V.4.7 |cx| = c|x| V.4.8 |x+y| < |x| +|y| Thus |x| has the properties of a norm , and may be regarded as the 3 length of the vector x. Note that, i f V = R and v = < x,y,z > e V, then 2 2 2 2 J v 1 = x + y + z , i n accordance with Pythagoras' Theorem. Any vector X |^ |. has norm one, and i s said to be normalised. We may now define a metric, and hence a topology, on any Pre-Hil b e r t space i n terms of this norm: we denote the distance between any two vectors, x and y by "d(x,y) M, and write V.4.9 d(x,y) = d f |x-y| I t i s the resulting topology which i s referred to i m p l i c i t l y i n the f o l -lowing d e f i n i t i o n s . See. Fano (1971) p. 173. 90 V.4.10 A pre-Hilbert s p a c e ^ is complete i f f every Cauchy sequence of points in V converges to a point in V. A complete pre-Hilbert space i s known as a Hilbert space. Two vectors x and y such that (x,y) = 0 are known as orthogonal 3 vectors. (In R such vectors are perpendicular, one to the other.) i f x^ , X 2 » ...,x_., ... are mutually orthogonal non-zero vectors (i.e. x_^  4 0 but x^ .) =0 for a l l i , j , i ^ j ) , then they are linearly inde-pendent. The converse i s not true; however, given a set of linearly independent vectors we can produce an equipollent set of mutually ortho-gonal vectors, each of.unit length, which span the same subspace. The procedure for doing so is called the "Qram-Schmidt Orthonormalisation 44 Process" . Mutually orthogonal vectors of unit length are known as orthonormal vectors. If v,, v , ... are orthonormal vectors which 2 n span a subspace L, then v^, V n, ... is an orthonormal basis for L. In this case, for a l l x e L, there are scalars a.., a such that 1 n V.4.11 x = E^a^v^ and = (v^x) for a l l i If there i s a denumerable basis for a Hilbert space, then the space i s separable; in what follows, only separable Hilbert spaces defined over the f i e l d of complex numbers w i l l be considered. V.5 Hermitian Operators and Projection Operators. With every vector x we can sssociate a linear functional f on x V (i.e. a linear function f :V -> C), such that, for a l l y e V, 44 See Jordan (1969), p.7. 91 V.5.1 f (y) = (x,y) 45 f can be shown to be continuous. Conversely, Riesz's Theorem states x that, If f is a continuous linear functional on V, then there exists a unique vector x such that, for a l l y e V, V.5.2 f(y) = (x,y) Thus there exists a 1-1 correspondence between the vectors of V and the continuous linear functionals on V. 46 Now for a given vector x, bounded linear operator A, there is a continuous linear functional f such that, for a l l y e V, V.5.3 f(y) = (x,Ay) and so, by Riesz's Theorem, there exists a vector z such that, for a l l Y e v, V.5.4 (z,y) =,(x,Ay) and since, for a given A, z i s uniquely determined by x, we can write V.5.5 z = A +x Where A*" Is an operator on V. It follows that* for a l l vectors x,y in V, V.5.6 (A+x,y) = (x,Ay) A + is the adjoint of A, and can be shown to be a bounded linear operator. A bounded linear operator A is self-adjoint or Hermitian i f f A + = A, i.e., for a l l x, y e V V.5.7 (x,Ay) = (Ax,y) = (y,Ax)* 4 5See Fano (1971), p.260. 46 A linear operator A i s bounded i f f there is a non-negative real num-ber, c, such that for a l l x eV, |Ax| <c|x| 92 Clearly, i f A i s Hermitian, then, for a l l x e V, (x,Ax) i s r e a l ; the converse also holds. I t i s easy to show that every eigenvalue of a Hermitian operator i s a r e a l number. We may note at this point that i t i s the Hermitian operators on V which are taken to correspond to the physical magnitudes of a quantum mechanical system. I f L i s a subspace of a H i l b e r t space H, then we say that a vector x i s orthogonal to L (x j _ L) i f f x i s orthogonal to a l l vectors i n L, i . e . , for a l l y e L, (x,y) = 0. The set {x: x._]_ L } can be shown to be a subspace. I t i s the orthogonal complement of L, denoted by L—. Now for any vector x e H and subspace L c H there i s a unique vector x^ e L such that ( x - x ^ _j_ L. ^ i s the projection of x on L. for each vector x e H we have: V.5.8 x = Xj+x^J. where x^_j_ = x - x j X • We define the projection operator P onto the subspace L by the equation, Li V.5.9 P Lx = ^ for a l l x e H . P i s easily shown to be a bounded li n e a r operator. The operators I and 0 are projection operators onto the whole space and the n u l l space respec-t i v e l y . I f P i s the projection operator onto the subspace L, then I -Li P^ i s the projection operator onto the subspace L. Note that, i f { v., ...,v , ...} -forms an orthonormal basis for I n the subspace L, then V.5.10 P Tx = E.(v.,x)v. for a l l x e H . L i l l 93 From V.5.10 and V.4.1-4 i t follows that, for arbitrary sub-spaces L, M, V.5.11 L £ M i f f (x,P Tx) < (x,P x) for a l l x e H . L L We can express V.5.11 i n terms of an ordering r e l a t i o n among operators. We write, for operators A, B on H, V.5.12 A < B i f f (x,Ax) < (x,Bx) for a l l x e H and obtain, for subspaces L, M, V.5.13 L c M i f f P < P„ — L M We also have: V.5.14 L c M i f f P P U = PT = P P ~ L M L ML The following important result appears as a theorem on this presentation, but may be used as a d e f i n i t i o n of a projection operator. V.5.15 A linea r operator P i s a projection operator i f f P i s both 2 + idempotent and Hermitian i f f P = P = P V.6 Spectral Decomposition. For any Hermitian operator on a f i n i t e l y dimensional space, and for the important class of compact Hermitian operators on a separable H i l -47 bert space ' the following results hold. I f A i s such an operator on a space H, i t admits a set of eigen-pairs {<v^, a^ >} . I f there i s no degeneracy, V.6.1 the set of eigenvectors, {v^}, of A forms an orthonormal basis for H. 47 For a d e f i n i t i o n of a compact operator see Fano (1971) p. 189. 94 I f there i s degeneracy, then we may s t i l l form an orthonormal basis for H from a set of eigenvectors; however, this basis w i l l not be uniquely defined, since i f <v.,a.> and <v, ,a.> are both eigenpairs, and v. and v, 3 J k 3 3 K are l i n e a r l y independent, then <v^,a^> i s also eigenpair, provided that v 1 l i e s i n the subspace spanned by v., v, . In this case we may select 1 J k the members of any orthonormal basis for this subspace to be members of the basis for H. In what follows we s h a l l assume that such a judicious selection has been made wherever i t was needed, to y i e l d the set of eigen-vectors {v^} as an orthonormal basis for H. From V.6.1 We see that, i f I i s the projection operator on to the one-dimensional subspace spanned by the eigenvector v^, then V.6.2 I . I . = 6..I. (where <S. . i s the Kronecker <5-f unction) , I j IJ I x i and also V.6.3 A = Z.a.I. x i x Proof of V.6.3: From V.6.1, we may express any x e H i n the form: V.6.4 x = E.c.v. 3 3 3 We then obtain: Ax = A E.c.v. = E.c.Av. = E.c a v. ; 3 J 3 3 3 3 ] ] ] ] but since, I.v. = 6..v. , we also have E.a.I.x = E.a.I.E.c.v. = E.a.c.v. x x x i i i J J J J J J J Since ( i ) the eigenvalues of a Hermitian operator are a l l r e a l , and ( i i ) a greatest eigenvalue, a , can be shown to exis t for a l l the max operators of the kind we are considering, we may l a b e l the eigenvalues so that 95 V.6.5 a, < a„< ... < a 1 2 max Now consider the operators V.6.6 P = I. for a e R a a. < a I x P i s the projection operator onto the subspace of H spanned by eigen-cl vectors with eigenvalues a^ such that a^a. Thus (by V.5.13-14), V.6.7 a < b i f f -P < P, i f f P P, = P = P.P a b a b a b a We have: V.6.8 P = 0 for a < a. a 1 V.6.9 P = I for a > a a max We may think of P as a function of a whose values are projection opera-tors, and which only changes i n value at eigenvalues of A. At each eigenvalue a^ the value of P "increases" by 1^, and the space spanned by P increases i n dimension (by one dimension i f there i s no degeneracy) 3. We have, V.6.10 I. = P - P where 0 < e < 'a. - a. .. l a. a.-e l l - l l l P i s known as the spectral measure corresponding to A. We 3. now generalise this to the case of an arbitrary s e l f - a d j o i n t operator. A spectral measure i s a family of projection operators, {P } , depending on a r e a l parameter a, with the following properties. V.6.11 I f a < b , then P £-P.; a b V.6.12 I f e > 0 , then P , x -> P x a s e + 0, for a l l a e R and a l l a+e a x e H; V.6.13 P x -> 0 as a->--», a ' P x - > x as a + * for any x e H . a J 96 We can express V.6.12 and V.6.13 by saying that P is continuous from the EL right, and that i t s strong limit, is 0 as a tends to - v , and i s I as a tends to + <o . The Spectral Decomposition Theorem states that: Corresponding to each self-adjoint operator A there is a unique (A) spectral measure, {P } , such that, for any x e H, 3. V.6.14 (x,Ax) = /ad(x,P ( A )x) - KO a We express this by writing, V.6.15 A = / a dP ( A ) a which may be compared with V.6.3. From V.6.11^12 we know that, for a (A) given x e H, (x,P x) i s a monotonically increasing function of a which 3. i s continuous from the right: thus i t may be used to define a Stieltjes measure on the real line (see V.6.16, below), and to form the Stieltjes integral in V.6.14. (A) Set set of points {a.} at which, for some x e H, (x,P x) is l a an increasing function of a is known as the spectrum of A. In the case (A) when A admits a set of eigenvalues {a.}, (x,P x) i s discontinuous 1 a(A) exactly at the set of points {a.} . If P is continuous from the l e f t X 9. (from V.6.12 i t is always continuous from the right), then no such discon-tinuities occur and A has a continuous Spectrum. For a given (normalised) vector x e H, we may define a measure on the real line in terms of the spectral measure of A. On the semiring consisting of the semiclosed intervals of the real line, (a,b] = S, we define the Stieltjes measure, 97 V.6.16 A A x ( S ) = (x,P b ( A )x) - (x,P a ( A )x) .= (x,P s ( A )x) where V.6.17 P < A ) - P b ( A ) - P a ( A ) • S b a If we extend A to be measure on the Borel a-ring of R, B(R), then the A X measure obtained satisfies the following conditions: V.6.18 ( 0 ) = 0 > A x (R) = 1 V.6.19 0 < (S) < 1 for a l l S B(R) ; V.6.20 A (S u T) = A (S) + A (T) for a l l S,TeB(R) such that A X A X A X S n T = 0 But these are the defining conditions for a probability measu-48 A, re ; thus ™ is a probability measure on the Borel a-ring of R. A X Alternatively we can generalise the notation introduced in (A} V.6.17 so that P v 1 i s defined for a l l S e B(R). We can then extend A ^ to a function on B(R) by writing V.6.16* ^ A x ( S ) = (x,P s ( A )x) for a l l S e B(R) and we then find that this function is just the probability measure on 49 B(R) obtained before (A) Now i t can be shown that P g is a projection operator on H, (A) that i s , that P i s idempotent and self-adjoint. Whence V.6.21 (x,P s ( A )x) = ( P s ( A ) x , P s ( A ) x ) = |P s ( A )x| 2 for any x e H, S e B(R), Hermitian operator A. This in turn gives: 48 See P i t t (1963), p.4-49 I am grateful to Dr. T. Gardner of the University of Toronto, who pro-ved this result for me. 98 V.6.22 A A x ( S ) = |P^ A )x| 2 V.7 The S t a t i s t i c a l Algorithm. The mathematical theory of Hi l b e r t spaces i s connected to the physical theory of quantum mechanics by the s t a t i s t i c a l algorithm: this relates the probability that a measurement of a physical magnitude w i l l y i e l d a re s u l t within a pa r t i c u l a r Borel subset of the r e a l l i n e to the state of the system being examined. Here I discuss only the s i t u a t i o n when the system i s i n a pure  state. In such a case the state of the system i s characterised by a (normalised) vector x, i n a Hi l b e r t space H. The algorithm can be ex-tended to deal with systems i n mixed states"*^: such states are charact-erised by weighted sets of such vectors, {w,x,. w„x„, ...w x }, with . 1 1 2 2 n n n > 2, 0 < w. for 1 < i< n, and § w. =1. 1-1 1 With each physical magnitude M A we associate a Hermitian ope-rator A on H. For conciseness we write "val(M ) e S" (where s e B(R)) A for "The value of the physical magnitude M for the system l i e s within the Borel subset S of the r e a l s , " and "Px(val(M A) e S)" for the probabi-l i t y that this i s the case for a given system i n state x. We now state the s t a t i s t i c a l algorithm for pure state as follows. V.7.1 p x(val(M A) 6 S) = |P^ A )x| 2 We have already seen (V.6.22) that, for a given vector x, self-adjoint operator A, |Pg A^x| 2 i s a probability measure on B(R). 5 0See Bub (1974) pp. 24-28. 99 case Now consider the case when A i s a self-adjoint operator with a discrete ( i . e . non-continuous) spectrum, and S = {a}, a e R. In,this (A) P r i = 0 when a i s not an eigenvalue of A; i f a i s an eigenvalue ta} (A) of A, then P, , = I , where I i s the projection operator onto the i a J- a a subspace spanned by the eigenvectors with eigenvalue a. Thus, i f a i s not an eigenvalue of A, then p (val(M.) = a) = 0 for a l l x e H. I f (A) P x(val(M^) = a) = 1, then ( x j P ^ a j x) = 1 and so (i ) a i s an eigenvalue (A) of A, and ( i i ) x i s a vector i n the subspace onto which P^J projects. This follows from the fact that* i n the notation of Section 6, V.7.2 P | | A ) = l i m ( P ^ A ) - P ^ ) (=1 when a i s an eigenvalue of A) The converse of the previous result i s easily obtained, and so we have: V.7.3 p (val(M.) = a) = 1 i f f the state x of the system i s an eigenstate of A, with eigenvalue a. Now, i f A and B are non-commuting operators, there are eigen-states of A which are not eigenstates of B, and so i f the system i s i n an eigenstate x of A, there may be no eigenvalue b of B such that P x(val(Mg) = b) = 1. In such a.case, experiments on w i l l y i e l d the value a, but those to determine w i l l y i e l d no one value with certainty. This i s i n contrast with the situ a t i o n i n c l a s s i c a l physics, where the state of a system determines uniquely the values which a l l physical mag-nitudes possess. For an arbitrary Hermitian operator A ( i . e . one not necessarily possessing a discrete spectrum), we may define the expectation value of a physical magnitude M for a system i n state x as follows. 100 V.7.4 Exp x(M A) = / ; a d .(A^ ({a}) I n t u i t i v e l y , each possible value a of i s here weighted by the proba-b i l i t y of i t s occurrence. Then we have, (A) V.7.5 Exp (M ) = fZ a d(x,P x) (from V.6.16) = (x,Ax) (V.6.14) As we might expect from V.7.3 above, i f < x,a> i s an eigenpair of A, then V.7.6. Exp (M.) = (x,Ax) = (x,ax) = a(x,x) = a . X A 1 V.8 The Spin-2 P a r t i c l e The summary of the formalism of quantum mechanics I have just given i s both abstract and compressed. I w i l l i l l u s t r a t e this account with the case of the spin-^ p a r t i c l e , or fermion. This i s a category of p a r t i c l e which includes both protons and electrons. We may regard a fermion as a simple system; a f u l l s p e c i f i c a t i o n of the state of such a system includes a s p e c i f i c a t i o n of the p a r t i c l e ' s spin: further, the treatment of i t s spin can be detached from discussion of any additional s p e c i f i c a t i o n of i t s state ( i n terms of i t s momentum, say). The spin state of a fermion can be represented by a vector i n 2 the 2-dimensional H i l b e r t space• C. Amont the relevant magnitudes (or observables, as they are sometimes known) are the three components of spin, with operators S^, S^, S z given by the P a u l i matrices: 101 V.8.1 S = 1 X 2 i \ s = I / o - I S y -2 1 0 / {1 Oj These operators do not commute: i n fact we have V.8.2 S S - S S = iS and so on, c y c l i c a l l y , x y y x z The eigenpairs of S are < x , -~ > and .< x , -4~ > , and, on X ™~ z_ "T* Z. this representation"'"'" V.8.3 x_ = , x + = ^ f , ^ ) Note that x_ and x + are normalised, and also that V.8.4 (x +,x_) = 0 2 Thus the eigenvectors of S form an orthogonal basis for C . X The projection operators onto the subspaces spanned by the e i -genvectors x_ and x + are, respectively, V.8.5 P = 1 / 1 - l \ X" 2 / V.-1 1 J (S ) Let {P x } be the spectral measure associated with S . Then a x V.8.6. P ( S x ) = 0 for -• < a < « a I = P for -\ < a< \ x- 2 2 =P + P , = I for ^  < a < + » x- x+ 2 and, i n accordance with V.6.3, V.8.7 S = -h + h. x 2 x- 2 x+ Note also that ( S x } n _ „ (S„) V.8.8 P — P r 1-, i — j . r i . , x- {--} x+ {1/2} "'"'"Note that i n V.8.3 I use the notation 11 ( , )" i n order to specify a vector i n C , rather than to indicate an inner product, as i n V.8.4. 102 Now consider a fermion i n the state x +. For a given S e B(R), (S ) l e t L g be the subspace which P g x projects onto. Then we find that V.8.9 x + e L g i f f | e S ; x + e Lg i f f | i S and so V.8.10 p (val(S ) e S) = |P^ Sx ) x,| 2 = 1 i f \ e S X j X O "T* = 0 otherwise 1 52 In other words, a measurement of w i l l y i e l d the value ^ with certainty (Here, and i n what follows, I use the same symbol to represent the physical magnitude and the operator corresponding to i t ) . The only possible values for the y-components of spin are l i k e -wise +^ and We have, for the system i n state x +, V.8.11 P x + ( v a l ( S y ) - f) = l P { 1 / ^ y ) x + | 2 P x + ( v a l ( S y ) - - f > - | P { _ l | V x + | 2 But we have: V.8.12. P { 1 / < S y > = l A - A whence V.8.13 |P { 1 /<S y) x + | 2 = 1 = |P {_ 1 / 2^y> x + | 2 And so the two possible outcomes of an experiment to determine S are y equiprobable, i f the system i s i n an eigenstate of S^. As we might expect V.8.14 Exp (S ) = ( S j,S x.) = 0 = Exp (S ) x + y +' y + x_ y This comparatively simple example of a quantum mechanical system and the associated observables w i l l be useful to us i n l a t e r chapter. 52T In natural units. 103 Chapter VI. Orthomodular Lattices and P a r t i a l Boolean Algebras. VI.1 The Algebra of Q-propositions. In the discussion of Newtonian mechanics i n Chapter IV, sen-tences of the form "Val(M) e S" (where M was some physical magnitude) were called "N-statements"; to each N-statement A and system x there corresponded a subspace a (A) of the phase space ft, and we saw that, for A a system i n state w, A i s true i f f u e. a v(A) (IV. 1.9) A Analogously, we may c a l l sentences of the form "Val(M) e S" Q-statements when they are applied to the systems deals with i n quantum mechanics. In Chapter V we saw that P x(val(M A) £ S) = |P^ A )x| 2 (V.7.1) and from this i t follows that (A) ( A ) VI.1.1 p x(val(M A) e S) = 1 i f f (Pg x , P g x) = 1 i f f x e Lg , where L g i s the subspace• (A) of H., onto which Pg projects. (The second equivalence derives from the fact that x i s normalised and (A) P i s a projection operator). Thus, i f H i s the Hilbe r t space for a quantum mechanical system X, then to each Q-statement "Val(M A) e S" there corresponds a projection (A) operator Pg on H, and hence a subspace of H onto which this operator projects; i f , accordingly, we denote the Q-statement by A and the cor-104 responding subspace by p (A) , then we have, for a system X i n a pure A state x, (A) VI.1.2 A i s certain i f f x e p A This may be compared with IV.1.9, above. Again, given a system X, by analogy with the c l a s s i c a l case; we may c a l l sentences of the form "x e L" Q-propositions (where L i s a sub-space of H), and we can construct an algebra of Q-propositions isomorphic to the algebra of subspaces of H. However, the algebra that results i s not Boolean. In this chapter I w i l l present two alternative ways to con-struct this algebra, y i e l d i n g on the one hand an orthomodular l a t t i c e , and on the other a p a r t i a l Boolean algebra. VI.2 The La t t i c e of Subspaces. The set, Sp(H)., of a l l subspaces of a H i l b e r t space H i s p a r t i -a l l y ordered by the r e l a t i o n of inclusion. For subspaces L, M we WBite L < M i f f L c M. Note that this r e l a t i o n i s defined only between subspaces and not between subsets of H. We use the fact that the i n t e r -section of two subspaces i s i t s e l f a subspace to define operations as follows. For any L, M < H, VI.2.1 L V M = n {N: N < H & L £ N & M < N} dt VI.2.2 L A M = , c L n M dx I t i s t r i v i a l to show that these are, respectively, sup {L,M} and.inf {L,M} with respect to the r e l a t i o n < . They are, i n t u i t i v e l y , the smal-l e s t subspace which includes both L and M, and the largest subspace which 105 they both include. We have, for subspaces L , M , VI.2.3 L v M =-{x: x = y + z for some y e L , z £ M } VI.2.4 L A M = {x: x e . L & x £ M } Both these sets of vectors are closed under the operations of vector addi-tion and scalar m u l t i p l i c a t i o n . Since, for any pair of subspaces, L , M , s u p { L , M } and i n f { L , M } both e x i s t , <Sp(H), v, A , <> i s a l a t t i c e (see Ch. II . 3 ) . This l a t t i c e i s atomic, the atoms being the one-dimensional subspaces of H. Elements within the l a t t i c e are not uniquely complemented: for instance, as the 2 diagram shows, i n R we may have VI.2.5 L V M = R 2 = L V M ' VI.2.6 L A M = 0 = L A M ' but see Fig. 14. VI.2.7 M ^ M ' (Note that i n the diagram the vectors labelled L , M , M ' do duty for the one-dimensional subspaces spanned by these vectors). Because complementa-tion i s not unique, the l a t t i c e i s non-distributive: i n the example above we see that VI.2.8 L A ( M v M ' ) = L A R 2 = L + 0 = 0 v 0 = ( L A M ) v ( L A M ' ) This confirms our previous remarks, that, i n general, the l a t t i c e X of rl 5 3 subspaces of Hilbe r t space i s not Boolean . However, X i s an example H 54 of an orthomodular l a t t i c e 53 Note, however, that when H = R , then f i s isomorphic to # „ . 54 H 2 For an extended discussion of orthomodular l a t t i c e s , see Holland (1970) 106 L - S i -Figure 14. 107 X = < B, v, A , < > i s an orthomodular l a t t i c e i f f VI.2.9(a) 3L i s a complemented l a t t i c e ; (b) for each a e B, there exists a unique element a^ - e B such that (i) a-L i s a complement of a; ( i i ) ( a V - = a ; ( i i i ) i f a < b , then br- < a-- ; (c) for a l l a, b e B, i f a < b, then b = a v (b A a —) [Note that, since i n any l a t t i c e a < b implies a v (b A a-^-) < b, we could replace (c) by the apparently weaker condition: (c*) for a l l a, b e B, i f a < B, i f a < b, then b < a v (b A a^ -) ]. <£. i s an orthocomplemented l a t t i c e i f f VI.2.9(a) and (b) hold. In an orthocomplemented l a t t i c e de Morgan's laws hold: i f a, b e B, then VI.2.10 (a v b ) - = aX A b- (a A b)-^ = sX v b-1 Proof: a < a v b b < a v b (II.3.10) (av'b)X<J- ( a v b ) i < b l ( v i . 2 . 9 ( b ) ) ( i ) (a v b ) - < a 1 A b-L (II.3.11) a A b < a a A b < b ( I I . 3.10) sX < (a A b ) ^ b^ < (a A b) 1- (VI.2.9(b)) ( i i ) a 1 v b-L < (a A b) ~ (II.3.11) ( i i i ) a A b < ( a 1 v t r 1 ) 1 (VI.2.9(b)) From ( i i i ) , using VI.2.9(b), we obtain, (iv) ar- A b 1 < (a v b)-*-From ( i ) and (iv) , (a v b) 1- = a 1- A b-1-The other law i s proved i n sim i l a r fashion. 108 We now show that, i f ^  i s an orthocomplemented l a t t i c e , then VI. 2.11 ( i ) a < b implies b < a v (b A a-^ ) for a l l a, b e B i f f ( i i ) a < b implies b A (b~ v a) < a for a l l a, b e B Proof: Assume (i ) and that c < d. Then, by VI.2.9(b), d- < cr-, and so cr- < d~- v (cr- A d) ( i ) (d^ v ( ( J - A d))- 1 < c (VI.2.9(b)) d A (d-L v c) < c (VI.2.9(b) ,10) But c and d were a r b i t r a r i l y chosen; whence (i ) implies ( i i ) . A simi-l a r proof shows that ( i i ) implies ( i ) I t follows that i - i s an orthomodular l a t t i c e i f f wt. i s an ortho-complemented l a t t i c e and VII. 2.9(c**) for a l l a, b e B, i f a < b then b A (bX V a) < a We now show that VI.2.10 <£. „ i s an orthomodular l a t t i c e ti Proof: (a) We have seen that i s a l a t t i c e with l a t t i c e operations £1 defined by VI.2.1-2. Also, for a l l L e Sp(H), VI.-2.11 0 < L L < H (0 i s here the zero subspace, spanned by the zero vector) And, for any subspace L, there i s a subspace such that, for a l l x e H, x = P Tx + P i x and P x J _ P . x (from V.5.8-9) , whence Li Li L Li VI.2.12 L v L- = H L U - = 0 I t follows that V i s complemented. 109 (b) We now show, that, for any L, lA- i s the orthocomplement of L. (i) We have already shown that Ir- i s a complement of L. ( i i ) For any x e H, L e Sp(H), x = P Lx + P L x x = P ^ x + P ( L < l ) 1 x whence P Lx = P ^ L X ) X X and, since x was a r b i t r a r i l y chosen, i t follows that P^ = P(LJ-)J-» a n d s o VI.2.13 L = ( I A ) X ( i i i ) Again, for any x e H, L, M e Sp(H), from V.5.8-9, V.4.4, (x,x) = (x, P Lx + PLJ.X ) = (x,P Lx) + (X,PLJX) = (X .PJ JX) + (x.P^x) Now, i f L < M , then (x,P Lx) < (x.P^x), (V.5.11) and so (x,P jx) < (x,P j x ) , i . e . M X < lA . Whence, r l Li VI.2.14 i f L < M , then M X < L A . (c) Assume L < M . I f x e M , then x = P Lx +'Pjj,x , and (i ) P^x e L; also ( i i ) P ^ x e L"*", and ( i i i ) P ^ x e M . (We have ( i i i ) , since, from (i ) P^x e M , by assumption x e M , also Pj4,x = x - P^x . and M i s closed under vector addition and scalar m u l t i p l i c a t i o n ) . From VI.2.4, using ( i i ) and ( i i i ) , we see that (iv) P j l x e M A L x . Thus, from VI.2.3, using ( i ) and ( i v ) , x e L v (M A L4*) . I f x e L v ( M A L^), then (from VI.2.3) there are y, z such that x = y + z, and y e L, z e M A l A . But since, by assumption, L < M , i t follows that y £ M ; also z e M (by VI.2.4) and so x = y + z e M . Thus, VI.2.15 i f L < M , then M = L v ( M A l A ) This concludes the proof. 110 To each subspace L of H there corresponds a projection opera-tor P on H. Thus, isomorphic to the l a t t i c e of subspaces we can con-struct a l a t t i c e of projection operators, i n which the ordering r e l a -tion and the operations of j o i n and meet are defined as follows. VI.2.16 PT < P, i f f L < M (note that this i s consonant with L M V.5.13) VI.2.17 P T v P = P L M df LVM VI.2.18 PT A P>(r = ,f P_ L M df LAM The r e l a t i o n between the operations v and A, thus defined on the set P(H) of projection operators on H, and the operations +, on P(H), defined by V.3.1-2, merits some scrutiny. VI.3 The Algebra of Projection Operators. The general definitions given i n Chapter V.3 of addition and m u l t i p l i c a t i o n on the set of linear operators on H may be s i m p l i f i e d i f we consider only those operators whose domain coincides with H. Let L(H) = {A: A i s a lin e a r operator and D(A) = H}. We have, for A, B e L(H), VI.3.1 (A + B)x = Ax + Bx VI.3.2 (A.B)x = A(Bx) We can also write (-A)x = ,,. -(Ax) , and then straightforwardly we can dr show that Ar = < L(H) ,+ ,.,-,0,1 > i s a ring with identity"'"'. The ring i s non-commutative since, i n general, AB 4 BA. 55 For a definition-of- a-ring, see Monk (1969), p.124. B r i e f l y , we re-quire that + arid . are both associative; <•+ i s commutative; . i s d i -s t r i b u t i v e over.+ from both the l e f t and the right; 0 i s the addi-tiv e i d e n t i t y ; I i s the m u l t i p l i c a t i v e i d e n t i t y and commuted with a l l A e L(H); - i s the operation of forming an additive i n v e r s e c l e a r l y s a t i s f i e d a l l these conditions. I l l The projection operators on H are a l l members of L(H), but we cannot think of + and as operations on P(H) (the set of a l l projec-tion operators on H) , since P(H) i s not closed under these operations. 2 However, for P , P M e P(H), i f P L and P M commute, then (P L- P M) = PT.P„.PT.P„ = P T 2.P M 2 = P T.P M , and(P T.PJ + = P M +.P T + = PM.PT = L M L M L M L M L M M L M L " P T.P w , and so PT.P„ is a projection operator. In this case, since L M L M for any x e H, (a) Y .V^-e L and P M.P Lx e M , (b) i f x e L n M, then P^.P^x = x i t follows that P^ '^ M "*"S t*ie P r oJ e c ti° n operator onto L n M, in fact, VI.3.3 i f P T, P„ commute, then PT.P.. = P T A P „ . L M L M L M whence VI.3.4 i f PT.P„ = P„.P, and L A M = 0, then (a) PT.P„ = 0 and (b) L M M L L M L 1 M to show (b), we have, for any x e L, y e M, (x,y) = (P Lx,P My) = (x,PL.PMy) = (x,0) = 0 Also, VI. 3.5 i f PT ,P„ = PXjr.PT and L n M = 0, then PT + P„ = PT w v r . L M M L L M LVM Proof: Assume PT .Pw = P.,.PT and L n M = 0 . We show f i r s t that L M M L Pj^ +P^  is a projection operator. ( P L + V 2 " P L 2 + PL' PM + PM'PL + PM 2 = PL + PM ( u S l n g *I.3.4(a)) ( P L + P M ) + " P L + + PM + - PL + PM Let P^ +P^ . project onto subspace N. Assume x e N; then x = (PT+P„,)x = P x + P„pc , i.e. there is a y = P Tx e L and a z = P„x e M, L M L M L M such that x = y+z. Whence x e L v M, and so (i) N £ L v M. Now assume, x a L v M. Then there are y,z, such that y e L , z £ M, and x = y+z. 112 Also, since P L.P M =0 L I M (by VI.3.4(b)), and so Y^z = 0 = P My. Thus V = ( P L + V X = ( PL + PM } ( y + 2 ) " V + P L Z + V + PM Z = ^ z = x It follows that ( i i ) L v M c N. From (i) and ( i i ) , N = L v M . We now write, for subspaces, L, M VI.3.6 L - M =,„ L n M-^-df from which i t follows that L-M 1 M, and we can now prove the following theorems. VI.3.7 If P T.P„ = Pljr.PT and K = L n M, then L-K 1 M-K L M M L VI.3.8 If PT.P. = P^.PT , then L M M L ( a ) P L V PM = PT + P M - PT .PM (b) PT A P = P .P L M L M L M L M Proof of VI.3.7: Assume PT .P„, = P„.PT and write K = L n M. For L M M L any u e L, v e M, by VI.3.6, P Tu = u = P Tu + PT Tru ; P>Tv = v = PTrv + P^ . t.v and so L K L-K M K M-K P = P - P • P = P - P L-K L K ' M-K M K If follows that P L_ K.P M_ K= (P L-P R) (P M-P K) 2 = P P _ P p _ p p +p^-L M K M L' K K = PT .P^ . - ?v - Yv.+ P„ (from V.5.14, since L M K K K K c L , K c M) = P p _ p L M rK = p p _ p M L rK = P ,,.PT .„ by parallel reasoning. M—K L—K From VI.3.4, since (L-K) n (M-K) = 0 , L-K 1 M-K Proof of VI.3.8: We have already shown (b) (see VI.3.3, above). To show (a), assume PT .P,, =-P„.PT . Then L M M L (PT + P M - PT . P J 2 = P 2 + PT .PM - P T 2 . P M + L M L M L L M L M PM.PT + P 2 - P M.P T.P M -M L M M L M 113 PT-PM.PT - P T.P M 2 + P T.P M.P T.P M L M L L M L M L M = P + P„ - P, .P„„ (by commutation and L M L M idempotence) <PL + PM " W + - P L + + PM + " ( PL' PM ) + = PT + P w - PT .Pw (since PT .P.. i s also a L M L M L M projection operator). Therefore PT + P w - PT.P„ is a projection operator. L M L M v J v We have (P_ + P„ - PT .P„.) = P„T for some subspace N. Assume L M L M N x e N. Then (PT + P„ - PT .P..)x = x . We know from VI. 3.3 that L M L M PT ,P„ = PTr , where K = L A M, and so x = P Tx + (P„ - PT.)x. But K c M, L M K L M K -and so P^x e M and (P^ - P R)x = z e M ; also P^x = y e L , and so there are y,z such that y e L , z e M and x = y+z . Thus x e L v M , and so (i) N c L v M. Now, for any y e L , P^y = y. , and so (P + P M ~ PL' PM ) y = ( PL + PM " PM' PL ) y = y + PM y " V = J ' L £ N ' Similarly M £ N , and so ( i i ) L v M c N. From (i) and ( i i ) , N = L v M. There is a relation between subspaces which determines whether or not.the corresponding projectors commute. We write: VI.3.9 Subspaces L, M of H are compatible i f f L-K 1 M-K where K = L n M . That i s to say, L and M are compatible i f f they are orthogonal except for an overlap. From this definition we obtain, for L, M e Sp(H), VI.3.10 L and M are compatible i f f PT .P.. = PlT.PT . L M M L Proof: Right to l e f t by VI.3.7. Left to right: assume L and M compatible; then 114 P L ' P M " (°L-K + V (PM-K + V " PL-K'PM-K + VK^K + VM-K + \ ' But we know, by assumption, that L-K 1 M-K , and, by d e f i n i t i o n , that L-K IK 1 M-K ; i t follows that P L_ K«P M_ K = 0 = P K-P M_ K = PL_K'PK ' Thus PT .P„ = PT;r = P„.PT (by p a r a l l e l reasoning) . L M K M L VI.4 P a r t i a l Boolean Algebras. In section VI.2 i t was shown that, i n general, the l a t t i c e sL^ of subspaces of a Hilbe r t space H i s not Boolean. However, within the l a t t i c e there are d i s t r i b u t i v e sublattices. For instance, i f we n. choose an orthonormal basis {v.} for H, then we can take the set Sp, , i [v] (h) of subspaces of H'generated from the one-dimensional subspaces de-fined by these basis vectors by the operations v, A and-1 ; the sub-spaces of Spj-v^(H) are a l l mutually compatible, i n the sense defined by VI . 3 . 9 , and i t i s easy to show that r , (H) = <Spr ,(H) , v, A , <> i s [vj r [ v ] a l a t t i c e . Further the orthocomplement L X of any given L e S p ^ O l ) i s the only complement of L which l i e s w i t h i n , S p ^ ( H ) . Thus, within ^-j-v-|(H) complementation i f uniquely defined, and so this l a t t i c e i s Boolean. Considerations of th i s kind prompt us to consider an a l t e r -native algebraic structure on the set Sp(H). In this algebra the ope-rations v and A are not defined for a l l pairs of members of Sp(H),, but only for those pairs of members which are compatible. Correspondingly, not a l l expressions of the k i n d " L v M " , L A M" are w e l l formed: how-ever, wherever the requisite compatibility conditions are met the equations of Boolean Algebra hold. We c a l l such a structure a p a r t i a l Boolean 115 algebra, and define i t formally as follows. VI.4.1 'By= <B, k, v, A , ', 0, 1> is a partial Boolean algebra i f f (a) B i s a set containing at least two elements; (b) 0 and 1 are the designated elements of B; (c) $ i s a relation on B (the compatibility relation); $ is symmetric and reflexive; (d) for a l l a e B, <a,l> e $ and <a,0> e $: we write a$l , a$0; (e) v and A are partial operations on B; each of them is defined for <a,b> i f f <a,b> € $; (f) ' i s a unary operation on.B; (g) i f {<a,b> ,<b,c>i,<c,a>,} £ $ , then (avh) $c, ( a A h ) $c, (a')$b; (h) i f {<a,b>, <b,c>, <c,a>,} £ $,, then the Boolean polynomials in a, b, c form a Boolean algebra with minimum 0 and maximum 1 , i.e. a, b and c generate a Boolean algebra under the operations of v, A ,,and ' . Every partial Boolean algebra is isomorphic to an algebra $ ' 56 constructed as follows Let ( i e I) be a non-empty family of non-degenerate Boo-lean algebras, such that "g^ = <Bi, v, A , ' , 0^  1> and This theorem (proved here as VI.4.4) is stated but not proved by Kochen and Specker (see their (1965a) p. 270). 116 VI. 4.2 (a) for any B., B. there exists a k e I such that B. n B. = B, ; • ' I ' J 1 3 k (from which we see that a l l algebras i n {{^} have a common zero element and a common unit element) '•ii J. (b) for any a, b, c £ . u B., i f a, b £ B. , b, c £ B. , c,a e B. l e i x x 1 x 2 x 3 for some i^,i2»i2 e I> then there i s a k e I such that a,b,c e B^. (c) v andA are functions, v : D -> . u T B . , A : D . u T B . , such that v xel x A xel 1 2 D = >D c ( n B.) and <a,b> e D i f f there i s a B. such that — i s l 1 V X V A a e B. and b £ B^ ; ' i s a function with domain and range . u _ B . . X £ l i We now from the required algebra: VI.4.3 % * = <B', $,V,A,',0,1>, where (a') B' = . u T B . i s the set of elementa of fa ' '> X£l x (b') 0 an 1 are the designated elements common to a l l the alge-bras 13 ' ; (c') a$b for a,b £ B' i f f there exists an i £ I such that a,b £ B ±; (e') a v b = c i n 13' i f f there exists an i £ I such that a v b = c i n "8^; a A b = c i n $ ' i f f there exists an i e I such that a A b =•c i n 0 ^ ; ( f ) a' = b i n D ' i f f there exists an i e I such that a' = b i n Q . x We may c a l l a family } of Boolean algebras conforming to VI.4.2 a pasted family of Boolean algebras 5 7, and of the algebra 5 7See Bub (1974), pp. 67-68. 117 $ ' formed i n accordance with VI.4.3 we may say that i t i s constructed  from {'QL} . Where no ambiguity i s l i k e l y to r e s u l t , the expression 'g' and {B^} m a v -° duty for each other. VI.4.4(a) Every p a r t i a l Boolean algebra can be,constructed from a pas-ted family of Boolean Algebras.• In this proof, and i n that of the converse, the bracketed l e t t e r s " ( a ) " , "(h)" refer to the clauses of VI.4.1, and " ( a r ) " , " ( f * ) " to the clauses of VI.4.3. Proof. Let "B be the p a r t i a l Boolean algebra <B, $, v, A , ', 0, 1> . Consider any set A^ cB, such that the members of A_^  are p a i r -wise mutually compatible elements of B. From (g) we see that the mem-bers of the set C(A^) = B^ of elements generated from A^ by the opera-tions of v, A and ' w i l l be pairwise compatible; from (h) i t follows that B^ w i l l form a Boolean a l g e b r a ^ under these operations. Let {•Q^} be the family of Boolean algebras thus formed. We now show that {$^} i s a pasted family of Boolean algebras from which T) may be con-structed. (i) The members of the intersection of two such algebras, ^^» e {$^} w i l l be pairwise compatible. Further, B^ n B_. w i l l be closed under the operations V, A and this follows.immediately from the fact that, i f a, b e B. n B. then avb e B. and avb e B. , whence avb e B . n B. 1 3 1 3 1 3 (and s i m i l a r l y for v, ' ) . Thus, from (h), we may form the Boolean alge— b r a * 1 3 v e .} , where B , = B . n B . . k 1 k 1 3 118 ( i i ) Now, i f for arbitrary a,b e B we have a,b e B^ for some i e l , then, since :/& ^ i s a Boolean algebra, a v b and a A b are defined, and so (from (e)) a$b. I t follows that i f there.are Boolean algebras /JB. , » within i3, and a,b € B. , b,c e B. , c,a e B. , then Xl X2 3 3 Xl X 2 X 3 a, b and c are pairwise compatible. Thus, from (h) , there i s a Boolean algebra "fi ^  £ s u c n that a,b,c e B . k T r i v i a l l y , v, A , and ' are functionf of the required kind. (i) and ( i i ) together show that {$_^ } i s a pasted family of Boolean algebras. Since $ i s a re f l e x i v e r e l a t i o n , from (h) we see that each a e B l i e s i n some Boolean algebra e further B^ c B for a l l i e l , and so B = .U TB. . Since, for a l l a, aVa' = 1, and a A a' = 0, i t also follows that a l l members of ("3^ } contain common zero and unit elements. Thus'Tj conforms to (a') and ( b f ) . The s a t i s f a c -tion of the remaining clauses of V I . 4 . 3 i s guaranteed by the d e f i n i t i o n of {£.} . Hence ^3 i s constructed from • VI.4 . 4(b) Every algebra constructed from a pasted family of Boolean algebras i s a p a r t i a l Boolean algebra. Proof: Let $ = <B, $, v, A , * , 0, 1> be the algebra constructed from the pasted family } . We see that 1 (a) Since Each algebra i s non-degenerate, B = ^U^B^ has at least two elements. 119 (b) From ( b f ) , there are minimum and maximum elements of B, namely 0 and 1. (c) The r e l a t i o n $ defined by (c') i s clea r l y reflexive and symmetric, and we also have (d) a$l, a$0 , for a l l a e B, (e) v and A are p a r t i a l operations on B; each of them i s de-fined for <a,b> i f f there i s an i e I such that a,b e B.^  i f f a$b. (f) From ( f ) , together with VI. 4.2 (a), we see that ' i s a unary operation on B; ( f ) alone does not make complement-ation unique. However, i f there are i , j such that a e B., a e B., then a e B.nB., and so a' e B.nB. since i J 1 3 ±3 B.nB. = B, and i s a Boolean algebra, l j k k (g) I f {<a,b> , <b,c> , <c,a>} c $, then, from VI.4.2(b), there i s a k e I such that a,b,c e B^; whence (avb) e B^, (aAb) eB f c, a' e B k, and so (avb) $c, (aAb) $c, a* $b. (h) From (c') and (g) (above), i f {<a,b> , <b,c> , <c,a>} £ $, then the Boolean polynomials i n a, b, c form a Boolean alge-bra. From (a) - (h), above, we see that 'ft i s a p a r t i a l Boolean algebra. I f U i s a p a r t i a l Boolean algebra, then we may define a rela -t i o n < on B as follows. For any a,b e B, 120 VI.4.5 a < b i f f avb = b i f f a A b = a We may compare II.4.8; here, however, a necessary condition for a < b is that a$b. If a and b are not compatible, then avb and aAb are not defined. This means that, although < i s reflexive and symmetric, i t is not always transitive. That i s , in a partial Boolean algebra $ we may have, for some a,b,c e B, a < b and b < c but a £c. An example of such a part i a l Boolean algebra is shown graphi-cally on the next page. The algebra 13 is constructed from the pasted family {jj^, % , <%J , where ^  and 4*2 are two 16-member Boolean alge-bras, whose members are represented in the diagram by circles and dia-monds respectively, and B 3 = B 1 n B 2 = {0,l,b,b*}. Reading the dia-gram in the way described in Chapter 1.2, we have a,b e B 1 and a < b, b,c e B 2 and b < c , but there is no %± € * 2 ' * 3 } S U ° h t h a t *' ° e B± . Thus a £ c. 58 We define a transitive partial Boolean algebra as follows ' • VI.4.6 T) = <B, $, V , . A , 0, 1> is a transitive partial Boolean alge-bra i f f ^  i s a partial boolean algebra, and, for a l l a,b,ceB, i f a < b and b < c, then a < c. Another significant class of p. B. a's is the class of 59 associative p. B. a.'s discussed by Gudder. 5 8 Such a structure is discussed bri e f l y by Kochen and Specker (see their (1965a), p. 265). 59 See Gudder (1971). He shows that each associative p. B. a. can be regarded as an orthomodular poset of a particular kind, and conversely. o Figure 15. The Algebra B* . Note that, although a < b and b ^ c , yet a £ c 122 VI.4.7 The partial Boolean algebra is associative i f f , for a l l a,b,c, e B such that a$b, b$c, (a) a $ ( b A C ) i f f ( a A b)$c and (b) a$(bAc) implies a A(bA c ) = ( a A b ) A C. We can readily show that every associative p. B. a. is transitive, as follows. Assume $ is an associative p.B.a. and that for some a,b,c e B we have a < b, b ^  c . Then a$b and b$c , and b = b A c . Thus a $ b A c , and, from the associativity condition, a A b$c . But a A b = a , whence a$c , and (again using the associativity condition) a A c = ( a A b ) A c = a A ( b A c ) = a A b = a . Thus a < c , as required. It i s an open question whether every transitive p.B.a. is associative. We now construct a partial Boolean algebra of subspaces of a Hilbert space, H. We f i r s t define a relation $ on Sp(h): for arbitrary L,M e Sp(H). VI.4.8 L$M i f f L i s compatible with M i f f PT,P„ = P„.PT L M M L Partial operations v, A^ are defined on Sp(H) by writing, for L,M e Sp(H) VI.4.9 (a) L v c M = L v M provided L$M; (b) L A^ M = L A M provided L$M; (c) where <L,M> k $. I v c M and L M are undefined. We can now show that VI.4.10 B(H) = Sp(H), $, v c, A 0, 1> is an associative partial Boolean algebra. 123 Proof: Clauses (a) - (f) of VI.4.1 are cle a r l y s a t i s f i e d , given VI.4.8-9, above. To show (g), assume that subspaces L,M,N are pair-wise compatible, i . e . that each pair of the corresponding projection operators commutes. Also, given this assumption, P = P + p _ p p P = P P LvM L M L" M LAM L' M whence P T X R T , . P , , T = P T . P „ .+ P „ . P „ ~ P T . P „ ' P X T LVM N L N M N L M N _ p p +p p _ p p p = p p N L N M N L M V LVM and P T » M ' P W = P T - P ™ - I \ T = P M - P T - P M = P M - P T A M LAM N L M N N L M N LAM and P _ l .0M = (I - P T). p„ = P - P T .P M = P„ - p M . p T = p M. p Tl L M L M M L M M M L M L from which we see that (LVJtf) $N , (LA^) $N , L"'" SM . To show (h) we note f i r s t that, i f L,M,N are mutually com-patible subspaces of H, then by (g), above, a l l the subspaces which may be generated from L,M,N by the operations of v, A and ^ are pair-wise compatible: c a l l the set of these subspaces C {L,M,N}. Now, since <Sp(H), v, A , <> i s a complemented l a t t i c e , equations II.2.1-3 and II.2.5 a l l hold (by II.3.12-14, II.4.1-2) 6 0 for the operations v, A , X on Sp(H); i f L,M and N are mutually compatible subspaces, these equations also hold for the operations v^, A , since on the set C {L,M,N } v coincides with v and A with A . To show the d i s t r i — c c b u t i v i t y laws (II.2.4) we use the fact that i f two projection operators are equal, then they both project onto the same subspace. Assume L,M,N to be mutually compatible subspaces. Then we have 6 0Reading, of course, "L", "M", "N" for "a", "b", 11 c". 124 P^ CMA^ ) = P L + P M.P N - P L.P M,P N (VI.3.8) P(LV .M)A (L N) " (PL + PM " W ' (?L + ?N " W c c ° 2 2 = p / + P .P„ - P_ ZP M + L L N L N PM'PL = PM'PN " PM'PL'PN 7 PT.P„.P_ - PT.PM.P„ + P T - P M - P T - P M LML LMN LMLN = PT + PT .P.. - P T .P„ + P M.P T + P„.P-T - PT . L LN LN ML M N L PM'PN " PL'PM " EL ' V PN + PL,PM*PN L M N L M.P N . I t follows that L v (M A N) = (L v M) A (L V M) and a simi l a r c c c c c argument shows that L AC (M VC N) = (L A^  M) (L AC M) . But II.2.1-5 are the defining equations of a Boolean algebra, and so (h) follows. We have shown that -ft (H) i s a p a r t i a l Boolean algebra. To see that ^  (H) i s associative, consider subspaces L, M, N of H such that L$M and M$N. Assume L$ (MAN) . Then PT .P., = P„,.PT , P„.P„ = L M ML M N PN'PM A N D PL'(PM-V = (PM-V-PL ' Thus (P L-P M).P N = V P r p L = p N - p M . p L - P N - ( P L - V • ™ S S u f f l C e S to show both (a) and (b) of VI.4.7. A F o r t i o r i , (H) i s a tr a n s i t i v e p a r t i a l Boolean algebra. VI.5 F i l t e r s and U l t r a f i l t e r s . In Chapter II.6 we discussed f i l t e r s and u l t r a f i l t e r s on a Boolean algebra. We now deal with the p a r a l l e l concepts on a tran-125 s i t i v e p a r t i a l Boolean algebra"""". Consider the t r a n s i t i v e p.B.a. ^ = <B, $, v, A , 0, 1>. A f i l t e r on "£) i s a non-empty subset F of B such that, for a l l a,be B, VI.5.1(a) i f a e F and a <b, then b e F ; (b) i f a,b e F, then there i s a c e F such that c < a and c ^ b; (c) 0 j F .. The p r i n c i p a l f i l t e r F generated by any non-zero element a.of B i s defined exactly as i n the case of a Boolean algebra. We write VI.5.2 F = {b: b e B and a < b} . a F a i s cle a r l y a f i l t e r . VI.5.3 A f i l t e r F on ^  i s an u l t r a f i l t e r i f f for any f i l t e r G on , i f F cG, then F = G. Now for any f i l t e r F, consider the set of those f i l t e r s G such that F c G. I t i s clear that this set i s p a r t i a l l y ordered by inclusi o n : further, that any chain {G^, G n, ...}. of such f i l t e r s has an upper 62 bound i n that set, namely the union of the chain . Thus, by Zorn's Lemma, the set contains a maximal element U such that F cu. Thus every f i l t e r on can be extended to an u l t e r f l i t e r on . We may compare the following with II.6.7 and II.6.12. For any f i l t e r F on $ , VI.5.4(a) i f a A b e F, then a e F and b e F ; (b) i f a e F and b e F , then a A b e F i f f a$b ; 61The d e f i n i t i o n given here i s not the same as that provided by Bub, who takes over the d e f i n i t i o n of a f i l t e r i n a Boolean algebra without modi-f i c a t i o n . The two def i n i t i o n s are not equivalent, unless / i s i t s e l f a Boolean algebra (see VI.6). See Bub (1974) . 62 {G^, . .. , G , ...} • i s a chain i f f G,' <= G0' c ... c G c . . . 1 n 1 - 2 - - n -126 VI.5.5(a) i f either a e F or b e F , then a v b e F i f f a$b ; (b) there i s an u l t r a f i l t e r U on a transitive p.B.a 13 > and elements a,b e B such that a v b e U , but a j U and b I U ; VI.5.6(a) i f a e F , then a' 4 F ; (b) there i s an u l t r a f i l t e r U on a transitive p.B.a $ , and an element a e B such that a | U and a' £ U . Proof: VI.5.4(a) and VI.5.5(a) follow from VI.5.1(a). To show VI.5.4 (b) , assume that a e . F , b e F and a$b . Then there i s an element a A b € B, and also an element c e F such that c < a and c < b . But in this case c ^ a A b and so a A b e F, asrequired. Assume, contra VI.5.6(a), that for some a e B, both a e F and a' e F ; then since a$a' , from VI.5.4(b) we have a A a' = 0 e F , contradicting VI.5.1(c). This proves VI.5.6(a). VI,5.5(b) and VI.5.6(b) are illustrated i n Sec-tion VI.6. Note that VI.5.5(b) follows from VI.5.6(b), since a v a' = 1 and so a v a' i s a member of every f i l t e r . For a l l a,b e B we have VI.5.7 a < b i f f every f i l t e r on ^ containing a also contains b . The proof i s t r i v i a l , given VI.5.1(a), and the fact that F is a f i l t e r . a An atom of a p.B.a. i s defined in exactly the same way as an atom of a Boolean algebra (see II.6.1), and the u l t r a f i l t e r s on an ato-mic and transitive p.B.a are the principal f i l t e r s generated by the atoms, as before. If ^  i s such a p.B.a, then, for a l l b,c ,e B, 127 VI.5.8 Every u l t r a f i l t e r on •$ containing b also contains c i f f a <c for each atom a of fo such that a ^ b . This follows d i r e c t l y from VI.5.7, above. We now show that, for any p.B.a. *£j which i s isomorphic to a p.B.a. <£j (H) of subspaces of Hilb e r t space, VI.5.9 a < b i f f every u l t r a f i l t e r on ^ containing a also contains b . And conjecture that this result can be'generalised to a l l t r a n s i t i v e par-t i a l Boolean algebras. The (generalised) l e f t to right conditional, of course, follows immediately from VI.5.7; we need only show i t s converse. Proof: /fy (H) i s an atomic p.B.a.; the atoms of $ ( H ) are the one-dimensional subspaces of H. Now assume that for L,M e Sp(H), N < M for each one-r dimensional, subspace N of H such that N ^ L. Let {v.} be an orthonormal basis for L, and 1^ be the projection operator onto the sub-space N. spanned by v.. Then P = Z.I. . Also, since, by assumption, X Li X 1 for each i , N. < M , i t follows that, for each i , I.P„ =1. (by V.t.14). Thus P L.P M = ( S ± I ± ) . P M = = S ± I ± = P L , whence L V M . This together with VI.5.8, proves VI.5.9. In general, u l t r a f i l t e r s on a p a r t i a l Boolean algebra are not prime: that i s , i n accordance with VI.5.6.(b) we may have, for the u l t r a -f i l t e r U, a v b e U , but neither a e U nor h e U . For this reason we cannot associate a canonical homomorphism with each u l t r a f i l t e r on a p a r t i a l Boolean algebra $ as we could i f $ were a Boolean algebra. In fact, we know from an important theorem of Kochen and Specker that there i s i n general no homomorphism of a p a r t i a l Boolean algebra onto the two-128 _ 63 element Boolean algebra 'Q 9 . VI.6 Two Examples As examples, we can construct (i) an orthomodular lattice and 3 ( l i ) a partial Boolean algebra, using the set S of subspaces of R f i r s t examined in Chapter I and illustrated below. S is the set of a l l sub-3 spaces spanned by the vectors u,v, x,y, z o f R , where u, v, x and y a l l l i e in the plane w; z i s at right angles to this plane; u ]_ v; x J_ y; and u is at 45° to x (see Figure 16). We use mx" to refer to the one-dimensional subspace spanned by the vector x, etc., and we label the four planes at right angles to w as follows: x v z = p (i.e. p is the x-z plane), y V z = (J, u V z = r, v v z = s . 3 Then S = {0,x,y,u,v,z,w,p,q,r,s,1} , where 1 = R . = <S, v, A , <> now forms an atomic orthomodular lattice. The atoms of JL g are u, v, x, y and z. We see that w = x vy = x Vu = x vv = y Vu = y vv = u vv etc. and w Vz = 1 = pvy = pvu = pVv etc. Note that the operations v and A are defined for a l l pairs of members of S. We also have 0^- = 1, x^=q,y^ = p,z^- = w , u ^ - = s , v - ^ = r . The lattice is represented in Figure VI.3 (on p. VI.23), in which each element lie s above or below i t s orthocomplement. 63 See Kochen and Specker (1967). They prove that no such 2-valued homo-morphism is possible from a p.B.a. of subspaces of a Hilbert space of 3 or more dimensions. See also Bub (1974), pp. 69-71. Figure 16 130 We may a l t e r n a t i v e l y , form the (transitive) p a r t i a l Boolean a l -gebra fl g = < § $, v, A , ' , 0,1> . As i n the algebra jQR elements of S are compatible i f f they are orthogonal except for an overlap. Note that this r e l a t i o n i s r e f l e x i v e and symmetric but not t r a n s i t i v e : we have, for example, x$z, z$u , but not x$u . The operations v and A are only defined for compatible ele-ments. Thus xvy = w = uVv but xvu ^ w ^ xVv because xvu and xVv are not defined. In the algebra complemen-tation i s defined as was orthocomplementation i n the l a t t i c e • The algebra i s shown diagrammatically i n Fugure VI.4. I t may be regarded as a pasted family of Boolean algebras: we have $ s = ^, $ 2' $ 3} where ^ ^ i s the algebra generated from the atoms x, y and z, whose elements are represented by c i r c l e s i n the dia-gram, ^ 2 i s t h e algebra generated by u, v and z, with members represen-ted by squared, and £ 3 i s the intersection of ^ and $ , so that B 3 = {0,z,w,l} . Examples of f i l t e r s on 1tt _ are the sets F = {p,l} , F = S p x {x,p,w,l} and F^ = {x,p,q,r,s,1} . These are, respectively, the p r i n c i -pal f i l t e r s generated by p, x and z. Since x and z are atoms, F and F are u l t r a f i l t e r s . Note that (i) uVv = w e F , but u k F and v i F , z x ' x corrobating VI.5.5(b) and that ( i i ) u' = s , but u | F and s i F , x x corroborating VI.5.6(b). Also, although p e F and r e F , there i s 132 no element p A r e F^ , since p and r are not compatible, and so p A r i s not defined. This illustrates the distinction between the definition of a f i l t e r on a partial Boolean algebra (VI.5.1) and that of a f i l t e r on a Boolean algebra (II.6.2-4): although p A r i s not de-fined in 4^ , there is nevertheless an element of S which l i e s beneath both p and r, namely z. These examples il l u s t r a t e salient differences between an ortho-modular lattice and the corresponding p.B.a., and between the properties of f i l t e r s on Boolean algebras and partial Boolean algebras. We may observe that is.^ is not a typical lattice of subspaces of a vector space, since i t has very few elements. If such a lattice is constructed with incompatible atoms which are not a l l coplanar, then the requirement that the set of elements be closed under v and A usually means that the l a t -tice is too large to furnish a readily visualised example. Chapter VII. Algebraic Structures and Quantum Logic: Orthomodular Lattices. VII.l The System OM(FG) I have shown that the set of subspaces of Hilbert space can be viewed as an orthomodular lattice or, alternatively, as a partial Boo-lean algebra. Each of these structures has been advocated as the app-ropriate structure for an algebra of Q-propositions. Birkhoff and von 64 Neumann, in their pioneering paper of 1936 , considered a lattice of '64 See Birkhoff and von Neumann (1936). 133 propositions. More recently the lattice approach has been favoured by writers like Jauch and Putnam, with widely divergent views about the 65 role and significance of quantum logic . Putnam, in the course of ma-king some sweeping claims for quantum logic, suggested that we should simply "read off" our logic from the lattice . Reading rather more carefully than he did, Friedman and Glymour later proposed the follow-6 7 ing "Gentzen-style" set of rules of inference for quantum logic Note that the syntax of their calculus resembles the syntax of f\ but contains "0" in place of as the absurd sentence, and contains the additional propositional constant "1". VII.1.1 We have the following schemata. (a) S e l f - d e r i v a b i l i t y Axiom A.J-~A (b) T r a n s i t i v i t y (Cut) Rule A h B B h C A |-C (c) &-Elimination Axiom A&B \- A A&B \- B See Jauch (1968) and Putnam (1969). I consider Jauch's views in detail in Chapter IX, and those of Putnam in Chapter X. 'Putnam (1969) , p. 222. See Friedman and Glymour (1972). As they point out (p.20), Putnam's proposed rule of inference, A,B f*A&B is not valid lattice-theoretic rule. The description of the rules VII.l.l(a)-(j) as "Gentzen—style" is theirs (p.25). 134 (d) &-Introduction Rule (e) v-Elimination Rule (f) v-Introduction Axioms (g) 1-Rule (h) 0,1 Axioms (1) (j) Orthomodularity Rule A f- B A |- C A H B&C A K B t- c AvB h C A f- AvB A H nB B (- AvB B M A 0 > A 1 V Av"»A B \- A A&(1AVB) |-B A h i A&TA h 0 I c a l l this system "OM(FG)"; as they present i s , the system 6 8 i s a hybrid of an N-calculus and an L-calculus . Regarded as an L-calculus, OM(FG) d i f f e r s from a Gentzen system i n two related respects. F i r s t , none of the sequents involved contains more than a single fo r -mula on either side of the t u r n s t i l e ; secondly we have a curious ad-mixture of axioms and rules of inference. Viewed as an N-calculus, the system contains a number of redundancies, notably the self - d e r i v a -b i l i t y axiom and the t r a n s i t i v i t y rule. However, to analyse the system i n this way i s to miss the point. Friedman and Glymour set out to provide a proof system whose Lindenbaum-Tarski algebra i s an orthomodu-l a r l a t t i c e , and thei r rules were chosen with that only i n mind. A 68 Compare the c a l c u l i CN and CL i n Chapter I I I . 135 comparison of rules (a) - (f) with the defining equations for l a t t i c e s i n general, and of rules (g) - (j) with those for orthomocular l a t t i c e s i n p a r t i c u l a r , shows that the resulting l a t t i c e coincides with the or-thomodular l a t t i c e i n the following sense. Consider the language f ' , l i k e the language ^ of chapter III.2 except that, (i ) the propositional constant "0" replaces the constant of f; ( i i ) f 1 includes the propositional constant "1" ( i i i ) "0" and "1" are formulae off 1 : i s not a formula off' . We may now define an interpretation a* of_Q% within the  orthomodular l a t t i c e jL, = <B, v, A , <> exactly as, by III.4.1, we de-fined an interpretation of 'f3 within a Boolean algebra, except that I I I . 4.1(b) i s replaced by I I I . 4. Mb1) a*(0) = 0 a*(l) = 1 and that 111.4.1(e) i s replaced by III.4.1(e') a* (7 A) = (a*(A))- L The function a* i s the function f used by Friedman and Glymour, and called by them a semi-interpretation. We now find that, for any formulae A, B off ' , VII. 1.2 A f-B i f f a*(A) < a*(B) for every interpretation a* of within any orthomodular l a t t i c e ; 1 |- A i f f a*(A) = 1 for every interpretation a* off* 1 136 within any orthomodular l a t t i c e ; A ^ 0 i f f a*(A) =0 for every interpretation a* o f f ' within any orthomodular l a t t i c e . The second and t h i r d of these results are obvious c o r o l l a r i e s of the f i r s t , whose proof i s given i n Section 2. VII.2 Soundness and Completeness of OM(FG). A calculus for which theorem VII.1.2 holds may be called sound  and complete with respect to interpretations within ah orthomodular l a t - t i c e . To prove that OM(FG) i s such a calculus, we f i r s t show that to any formulae, A, B of p1 there correspond l a t t i c e polynomials,/.,^ , such that, i f B i s the set of members of the l a t t i c e , £ , VII.2.1(1) for a l l a1,...,a e B, d < 3 i f f , for a l l interpreta-tions a* o f f 3 ' within , a*(A) < a*(B) . Secondly, we show that VII.2.2 A h B i f f , for a l l a.,...,a e B, a 3 . 1 n The required re s u l t then•follows immediately. Let i» = <B, v, A , <> be an orthomodular l a t t i c e , with ortho-complementation denoted by "1". Clearly, a recursive d e f i n i t i o n of a l a t t i c e polynomial of £ could be supplied analogous to the recursive d e f i n i t i o n of a formula o f ^ ' . Then, i f a and . 3 are l a t t i c e polynomials i n a1,...,a , we c a l l each statement of the form (I) or (II) which holds £ a l a t t i c e theorem. 137 (I) for a l l a, ,. . . ,a e B, <r < 6 1 n (II) for a l l a 1 (...,a e B, a = g I n Note that a.,...,a play the role of variables i n such statements, and 1 n that a and g may also contain the constants 0 and 1. Now consider an assignment of members of B to a^,...-,an which assigns b^ e B to a^, ... b e B to a . We may extend t h i s to an assignment function <j> :V->-, n n B, where V i s a denumerable set of l a t t i c e variables {a-,...a ,...} . ' I n To each such assignment function <f> there corresponds an interpretation a* o f f ' w i t h i n i . , such that a*(p^) = b^ = <|>(a^ ), and conversely. Now from the l a t t i c e polynomial a we may obtain the formula A of f ' by systematically replacing each occurence of each variable a^ i n a by the propositional variable p_^ , and each occurence of v, A J i n a by v, & and*v , respectively. 0 and 1, where they occur i n a, may be l e f t un-changed. S i m i l a r l y , we may transform a formula of 1**' into a l a t t i c e polynomial, and we see that a one-to-one correspondence exists between, l a t t i c e polynomials and formulae o f f * 1 , . I f A i s the formula obtained from the polynomial a according to the recipe above, we write A = T (a) where V i s a 1-1 function. We now show that, I f A, B are formulae of P ' , and a and 6 are the l a t t i c e poly-nomials such that ^(a) = A , I|J(3) = B , then VII.2.1(1) for a l l a.,...,a e B, a < g i f f for a l l interpreta-1 n tions a* o f f ' i n *, a* (A) < a*(B) . (II) For a l l a.,...,a e B, a = g i f f for a l l interpre-1 n tations a* of P' i n £ , a*(A) = a* (B) . 138 Proof. Let <f> be the assignment function such that <j> (a_^ ) = b_^  ; then cp extends uniquely to a homomorphism c\>* of the set of l a t t i c e polynomials into B. We know also that i s a homomorphism of the set of l a t t i c e polynomials onto the set of formulae of P' . But the interpretation a* corresponding to cp i s a homomorphism of the set of formulae o f f ' into B; whence a*.ijj i s a homomorphism of the set of l a t t i c e polynomials into B. Further, for a l l a^, a* 0 ( a ± ) ) = a*( P i) = b ± = tpiaj and since the extension of <f> to cp* i s unique, cp* = a*.ip Since the set of assignment functions i s i n one-to-one correspondence with the set of interpretations, the results follow. We now show that, i f , for the l a t t i c e polynomials a, 3 , i n a. ,...,a , A = I/J (a) and B = i H 3 ) > then 1 n VII.2.2 A f-B i f f for a l l a,,...,a e B, a < 3 1 n To prove this we f i r s t formalise the notion of a proof of a l a t t i c e theorem; we then show that each such proof of a theorem of form (I) can be transformed to a proof of a de d u c i b i l i t y relationship i n OM(FG), and conversely. The d e f i n i t i o n of an orthomodular l a t t i c e e f f e c t i v e l y provides a set of axioms and rules of inference to be used i n deriving l a t t i c e theorems. Additionally, i n informal proofs we use a replacement rule (a rule of inference j u s t i f i e d by the p r i n c i p l e of i n d i s c e r n a b i l i t y of identicals) and a substitution rule which brings into play the (usually t a c i t ) quantification over a l l members of the l a t t i c e which prefaces 139 each l a t t i c e theorem. The system LT which appears below i s a formali-sation of these derivations; I take the claim that LT i s adequate to generate a l l l a t t i c e theorems to be uncontroversial. In the proposed axioms and rules, a, 3, y, 6 are arbitrary l a t t i c e polynomials i n a^,...,an. The polynomial Y A i s a polynomial which contains a as a suhpolynomial, and y^ i s a polynomial obtained from Y A by substituting 3 for some or a l l of the occurrences of the polynomial ct i n y^. (y/a^)a i s the polynomial obtained from a by re-placing every occurrence of the l a t t i c e variable a^ i n a by the poly-nomial y (and s i m i l a r l y for (y/a^g ). Each statement "a < 3", "a = 3" i s to be read as though prefaced by "For a l l a^,...,an e B". VII.2.3(a') a < a (II.3.1) (b') a < 3 3 < Y (II.3.2) a < Y (c*) a < 3 3 * a (II.3.3) (II.3.10) (d') a < a v 3 3 ^ a v 3 (e') a v 3 < a a v 3 < 3 (f*) a < y 3 ^  Y (II.3.11) a v 3 < Y (g') y < a y < 3 (II.3.11) Y < a v 3 (h') 0 < a a< 1 (II.4.2-3, II.3.16) 140 ( i ' ) 1 = a v a^ - 0 = a A a-'-I | Y (VI.2.9) (j') ( a 1 ) 1 = a (k'> a < g g-L < q^ (1') g < a (VI.2.9(c**)) a A (a^- v g) < g (m') a=g j < 8 a=g 6 < y _a - _a_ Yg - 6 6 - Yg (Replacement) a = g Y = 6 a « - T f (n') a < g (Substitution) (Y/a ±)a < (Y/a±)g The following are theorems or derived rules of LT. (o') a < g-*- (by Tree 2.1, g < below) (p') 1 < a v a-*" a A a ^ < 0 (Trees 2.2, 2.3) (q') a = g a = g (Trees 2.4, 2.5) a < g g < a (r') a = g (Tree 2.6) Ya " Yg We prove these results using the derivation trees below. (€>•) ( j ' ) a < g1 1 1 I I ( k , ) (gV = g : (g1)1 < al (m') (Tree 2.1) g < a± 141 (p') d') (a') l = a v a - L : l < l , , s (m') 1 < a v or1 Tree 2.2 (q*) (a') a = B a < a , , (m ) a < B Tree 2.4 (r') (a') a a (m ) Y a = Y B (!') (a') 1 0 = a A A X 0 < 0 a A a X < 0 Tree 2.3 (a') a = (m') a < a , , . B < a Tree 2.5 Tree 2.6 To correspond to various rules and axioms of LT, we need a num-ber of results for OM(FG). I f i r s t prove a Substitution Theorem and a Replacement Theorem. Replacement Theorem. Let A, B, C be any formulae of 4*' , and be a formula obtai-ned by replacing some or a l l of the occurrences of p_^  and C by the for-mula A, and C be the formula obtained by replacing by B ju s t those B occurrences of p. and C which were replaced to form C.. (We do not i A rule out that p^ not occur i n C) . Then, for any formula A,B,C o f f ' and an a r b i t r a r i l y chosen propositional variable p^, VII .2.4 I f A hB and B H A, then C. h C_ and C |-CA A B B A Proof. The theorem i s t r i v i a l i n the case when C does not contain p.. as a subformula. We prove the more general case by induction on the 142 length L of the formula C, defined as follows. (a) I f C i s the propositional variable p^, then L(C) = 1 ; (b) I f C i s the propositional constant 0 or the propositio-n al constant 1, then L(C) = 1 ; (c) I f C = AvB , or C = A&B , where A and B are formulae, then L(C) = max (L(A), L(B)} +1 (d) I f C = "»B , for some formula B, then (C) = L(B)+1 I f L(C) = 1 , then either C = 0 = C„ , or C = 1 = C„ , or i 4 j and A B A B C. = p. = C„ , or C = p, and C, = A and CL, = B. Obviously, i n a l l four A r j B x A B J cases, VII.2.4 holds. Now assume that VII.2.4 holds for each formula D such that L(D) < n. Consider a formula C of length n+1; there are three cases. (i) There are formulae D,E and C = DvE | ( i i ) There are formulae D,E, and C = D&E ; ( i i i ) There i s a formula D and C = ~»D Case ( i ) Assume A f* B and B f- A . Then from the induction hypo-thesis, VII.2.4 holds for D and E, i. e . we have °A ^°B 5 DB ^*DA > EA V \ > EB ^ A ' Consider the derivation tree 2.7 , below. D A [ - D B DB h V E A EA V \ \ V °B V EB DA h V E A BA f V EA °B ^ V E B E A h V E B V E A V D B V E A D B V E A k D B V E B Tree 2.7 D vE (-D„vE„ A A B B 143 This shows that, on our assumptions, C. V C , and a si m i l a r proof A ii shows that CL h C. D A Case ( i i ) We make the same assumptions as i n case ( i ) . Then the tree 2.8 shows that C A Y and a simi l a r proof that Gg hC^ D A & E A V °A °A ^ DB D B & E A ^ EA E A M B D A & E A ^ D B D A & E A ^ E A D B & E A ^ DB D B & E A ^ B D A &E A M B & E A D B &E A HD B &E B Tree 2.8 D. &EA Y D &E,, A A B B Case ( i i i ) From the same assumptions as before we know that D (- D A B and conversely. The tree below now shows that Cg h C A and a si m i l a r proof shows that C A \- C^ DA ^ D B DB K l l D B DA t "°B Tree 2.9 f TD A Thus VII.2.4 holds for any formula of length n+1. This concludes the proof. Substitution Theorem. For any formula A o f f ' , l e t (C/p^)A be the formula which results from replacing every occurrence of the propositional variable p\ i n A by the formula C. 144 VII.2.5 I f A f-B , then (C/p )A H (C/p ±)B . Proof. Assume A }-B. Then a derivation exists i n GM(FG) of B from A. We construct a derivation of (C/pI)B from (C/p^A by taking the derivation of B from A and replacing the propositional variable p^ wher-ever i t occurs by the formula C. Since a l l the axioms and rules of OM(FG) are i n the forma of schemata the resulting array of formulae i s a derivation of (D/p^B from (C/p^A i n OM(FG) . whence (C/p±)A h (C/p±)B. We may introduce the metalinguistic r e l a t i o n 4 p on the set of formulae o f f ' , so that "A *| |>B" abbreviates "A h B and B \- A" : VII.2.6(k) A f-B B |-A A -f h B (1) A-ll-B A H |- B A KB B I-A Theorems VII.2.4-5 now appear as the following meta-rules of OM(FG). VII.2.4(m) A i CA-*HCB VII.2.5(n) A|- B (C/p.)A h (/p.)B Sundry .theorems and derived rules of OM(FG) which we s h a l l need are s t a — ted below. VII. 2.7 (o) H h A v T A O-ff-A&lA (p) l l A + r - A 145 (q) A f B IB h 1A (r) A ^ M C A h D A 4 f- B D A -t K B C A "i ^  D 69 Proofs. (o) follows immediately by (k) from (h) and ( i ) . (p) i s interesting i n that i t s proof requires the orthomodularity rule ( j ) . For reasons of space, I show the two parts of the proof separately (trees 2.10, 2.11); the theorem follows from these by (k). (q) i s shown by 2.12; the f i r s t two cases of (r) are shown by trees 2.13, 2.14, and the t h i r s i s obtained from these by using (k) and (1). (a) 1 A ^ U («) A h 17A Tree 2.10 (a) Tree 2.11 ^ A * " A (g) <f> 1A f VIA TI1A h 777AvA (b) 69 Here and i n what follows, l e t t e r s i n parentheses (e.g., (j)) refer to axioms, rules or theorems of OM(FG) (see VII.2.2, VII.2.4-7); primed l e t t e r s (j') refer to those of LT (see VII.2.3). 146 (h) ( i ) (f) "PA f* 1 1 f» AvIA ... A (-TllAvA 1A f- l l l A v A , . (a) (b; (.e; ( a ) l l A f-AvlA AvlA V T l l A v A ( b ) 1 A h 1 A ( g ) TJA |- l l A VIA j- IVtAvA ( d ) A\- H A ( j ) l l A f" TlA&(WlAvA) TIA&CJTJAvA) |~A ( b ) IT A |- A (o) B H h TIB ( 1 ) A h B B \-VB ( b ) Tree 2.12 A \- TIB ^ (m) (m) Tree 2.13 Tree 2.14 r J U r CA ^  h CB (1) V P LB (1) CB ^ C A CA h D (b) D f " C A CA h C B (b) C B hD D h C B I t i s now a straightforward matter to transform a proof of OM(FG) into a proof of LT, and vice versa. To transform a l a t t i c e theoretic proof to a proof i n OM(FG) we replace every occurrence of each l a t t i c e polynomial a by the formula ijj(a) , and each occurrence of "<" and "=" by " K" and 'Ml-", respectively. Then the resu l t i n g array constitutes a proof i n OM(FG), since 147 (1) every axiom of LT i s transformed to an axiom or a theo-rem of OM(FG) ; (2) every rule of inference i n LT i s transformed either into a rule of inference of OM(FG) or into a derived rule of OM(FG). Ad (1). The axioms (a'), ( d ' ) 5 (e') and (h') are transformed to (a), (f) , (c) and (h), respectively; ( i ' ) and (j') are transformed to theo-rems (o) and (p). Ad (2). Rules (b') 5 ( c ' ) , ( f ) , (g') and (1') are transformed d i r e c t l y into rules (b), (k), (e), (d) and ( j ) , respectively. The transforma-tions of (k')» (mT) and (n') appear as the derived rules (q), (r) and (n) i n OM(FG). Similarly a proof i n OM(FG) may eas i l y be transformed to a proof i n LT. The only axioms and rules of OM(FG) which to not transform d i r e c t l y into axioms or proofs rules of LT are (g), (i) and (1), which appear as (o') , (p 1) and (q'), respectively. That concludes the proof of VII.2.2. VII.1.2 follows immedia-tely. I t i s a t r i v i a l consequence of this theorem that OM(FG) i s sound with respect to the semantics S3 which Friedman and Glymour provide; I postpone a discussion of this semantics u n t i l Chapter VIII.5. VII.3 The System OM(H) A more elegant proof theory than OM(FG) for the same purpose was 148 suggested by Hacking 7^. He proposed an L-calculus i n which the rules d i f f e r from those of CL i n only two respects, one of them t r i v i a l , (1) The axiom A -»• A i s r e s t r i c t e d to atomic sentences A. This modification (which i n fact brings Hacking nearer than CL to Gent-zen's o r i g i n a l formulation) i s not s i g n i f i c a n t , since, by induction on the length of a formula we can show that i n Hacking's system A -> A i s a derivable sequent for any formula A. (2) The thinning (delution) rule of CL i s replaced by a res t r i c t e d version of i t : i n the orthomodular sequent calculus we have r -> A [-'->- A, A ' r ' ,A A ' r -> A r,A -*• A r -+ A,A where T' £ T , A - ' £ A A -> -> A A •+ B B -y A Hacking shows that this modification rules out the derivation tree of CL which we would normally use to show the d i s t r i b u t i v e law: A &(BvC) -> (A&B)v(B&C) and this sequent i s said not to be derivable i n this system. In fact the resultant calculus, OM(H), i s stated to be equiva-lentxto OM(FG). However, the nature of this equivalence i s problematic. We can show f a i r l y e a s i l y that, i f we regard each figure A |*B i n OM(FG) as a sequent, then a l l the axioms of OM(FG) are derivable sequents of OM(H), and that a l l rules of inference of OM(FG) appear as derived rules i n OM(H). (I expect those containing "0" and "1", which are eliminable 7 0See Hacking (1977). 149 from OM(FG), with no resulting loss of content). Thus OM(FG) i s con-tained i n OM(H) . But while each figure A t~ B may be regarded as a sequent, the converse i s not true. Further, although i n c l a s s i c a l logic any given sequent A^,A2, ... , A^ B^,B2 y . • • »B^ i s deductively equivalent to the sequent A1&(A2& . .. (A n - 1&A n) ...)-> B 1v(B 2v . .. ( B ^ v B ^ . . . ) the r e s t r i c t i o n s on the d i l u t i o n rule seem to block the proof of such an equivalence i n OM(H). Consider the sequent A,B -> C,D From this we may derive the sequent A&B -»• CvD i n OM(H) exactly as i n c l a s s i c a l l o g i c (see tree 3.1). However, there seems to be no way to derive the converse. Tree 3.2 i s a derivation tree of CL but not of OM(H), since the applications of the c l a s s i c a l delution rule at * i n this tree cannot be j u s t i f i e d i n OM(H). I t i s d i f f i c u l t to see what legitimate alternative i s available, and this i t i s not clear i n what sense OM(H) i s contained i n OM(FG). A,B + C,D A,B -> CvD,D A,B -»• CvD Tree 3.1 A&B CvD C +C * D-D * A+A^ B + B A C ->C,D D-C,D A,B - A A,B -B A&B - CvD CvD - C,D 150 A,B -> A&B A&B + C,D Tree 3.2 A,B -> C,D Chapter VIII. Algebraic Structures and Quantum Logic: P a r t i a l Boolean Algebras. VIII. 1 Introduction. Systems of quantum log i c based on p a r t i a l Boolean algebras, while prefigured i n the work of Strauss, were f i r s t systematically de-veloped by Kochen and Specker, and have since been investigated by Bub and Demopoulos^"'". The development here p a r a l l e l s the approach to classocal l o g i c outlined i n Chapter III.4-6. In the same way that i n c l a s s i c a l l o g i c we may use a sequence of members of a Boolean algebra to provide an interpretation of a formal language, i n quantum l o g i c we may use a sequence of members of a p a r t i a l Boolean algebra to do so; we may t a l k , a l b e i t loosely, of a sequence interpreting the language. In Both cases a sequence of elements of an algebra i s used to map formu-lae of the language onto elements of the algebra; however, i f the a l -gebra i s a p a r t i a l Boolean algebra, not a l l formulae w i l l f a l l i n the domain of ( i . e . w i l l be mapped by) a given interpretation. I f a sequence from the p a r t i a l Boolean algebra i s to provide an interpre-tation of the complex formula A&B (or of the formula AvB) within , See Strauss (1936) and (1937-8), Kochen and Specker, (1965a), (1965b) and (1967), Bub and Demopoulos (1974) and Bub (1974). 71 151 then (1) this sequence must provide an interpretation of the two formu-lae A and B within ^  - that i s , the interpretation provided must map both of these sentences onto elements of the algebra 4$ - and (2) these elements must be compatible. Semantic notions l i k e v a l i d i t y and semantic entailment may be defined i n terms of such interpretations. I also furnish two systems of proof which are sound and (weakly) complete with respect to this semantics. These are natural deduction systems, a calculus QN which resembles CN, and a calculus QL resembling CL. These c a l c u l i make e x p l i c i t the points i n derivations where compatibility requirements have to be met, i f a step si m i l a r to one i n a c l a s s i c a l derivation i s to be made. For instance, i n QN, to derive A&B i t i s not s u f f i c i e n t that we should be able to derive both A and B: we must also know that they can be meaningfully conjoined. This l a t t e r r e l a t i o n between A and B i s expressed by a proviso f(A,B) , so that i n place of the c l a s s i c a l rule A B A&B we have the rule f(A,B) A B A&B A l l the necessary modifications to c l a s s i c a l rules can be made by using such provisos, which are either of the form 'f(A)' or of the form 1(A,B)* (where A and B are formulae). The system also contains 152 rules for manipulating such provisos, e.g. a commutation r u l e , f(A,B) f(B,A) Such rules are paralleled i n the system QL. The semantics provides interpretations of these provisos, as well as interpretations of formulae. The proviso 'f(A)' i s to be under-stood as saying "A i s meaningful", and *f(A,B)'' as saying "A and B can be meaningfully connected". Note that although the interpretation of rf(A,B^ i s closely t i e d i n with questions of compatibility, i t i s impre-cise to regard M:(A,B) as saying that A and B are compatible. The rel a t i o n of compatibility i s defined on a p a r t i a l Boolean algebra, not on the set of (uninterpreted) sentences of the language; however, i f a sequence interprets A and B we may talk of the interpretations of A and B as compatible, since these interpretations are members of a par-t i a l Boolean algebra. The proviso 'f (A,B)* i s interpreted just when this compatibility relationship holds; likewise the proviso 5(A^P i s interpreted by just those sequences which interpret A. whenever a proviso i s interpreted by a sequence from a p a r t i a l Boolean algebra, i t i s mapped onto the unit element of that algebra. In the sections which follow, these ideas are developed more f u l l y and more formally. After a section i n which the syntax of a language Q i s given, sections 3—5 are devoted to semantics: Section 3 includes a d e f i n i t i o n of an interpretation of Q within a p a r t i a l Boolean algebra, and a discussion of semantic entailment and v a l i d i t y ; i n Section 4 I provide a d e f i n i -t i o n of a valuation of Q, and i n Section 5 this semantic approach i s com-pared with those of Reichenback and of Friedman and Glymour. The two 153 systems of proof, QN and QL, are treated i n Sections 6—10 and 11-14 respectively. VIII.2 The Language Q. The language Q contains the following symbols: propositional variables: "q-^"* "q 2"• ••• ( t n e s e t o f such variab-les i s denumerably i n f i n i t e ) ; a propositional constant: l o g i c a l connectives: "v" , "&", 11V ; a proviso—forming symbol: " f " ; punctuation symbols: " ( " , " ) " , "," . A formula of Q i s defined recursively, as follows: VIII.2.1(a) for any i e N, "q^ 1 i s a formula of Q ; (b) " A" i s a formula of Q ; (c) i f A i s a formula of Q, then I#JA i s a formula of Q ; (d) i f A and B are both formulae of Q, then '(AvB)1 and are both formulae of Q : (e) nothing i s a formula of Q except by virtue of (a), (b), (c) or (d), above. A proviso of Q i s defined as follows: VIII.2.2(a) i f A i s a formula of Q, then If(A) 1 i s a proviso of Q ; (b) i f A and B are formulae of Q, then f(A,B)' i s a proviso of Q ; (c) nothing i s a proviso of Q, except by virtue of (a) or (b) above. 154 In what follows, outermost parentheses w i l l be omitted from formulae, and I w i l l leave out quasi-quotation marks. The defined connective "=>" may be introduced. For any f o r — mulae A and B, VIII.2.3 A = B = lAvB d t For this d e f i n i t i o n , A = B i s a formula of Q. Appreviations: I f f = {A.,,A„, ... ,A } , where A_ , ... ,A are formulae of Q, 1 2 n 1 n and B i s a formula of Q, then VIII.2.4 f ( D = d f { f ( A ± , A j ) : A±,A_. e ["} VIII.2.5 f ( B , D = J 4 r {f(B,A.): A. e D d r l i I f T = 0 , and B i s a formula of Q, then VIII.2.6 f ( D = d f 0 f ( B , D = d f (f(B)} From VIII. 2.4 we see that, i f f = {A} , then f ( D = {f (A,A) }. Note also that f ( D I f(B,f) unless f = {B} or T = 0. The figure used i n QL i s the sequent; this i s a m e t a l i n g u i — s t i c figure s i m i l a r to that used i n CL. VIII. 2.7 $, T -+ A i s a sequent i f f $ i s a f i n i t e (and possibly empty) set of provisos, T and A are f i n i t e sets of formu-lae, and T and A are not simultaneously empty. For notational economy, various conventions are used i n writing out sequents, as before. I f $, ^ are sets of provisos, then we write 155 " instead of "$ u i p " ; i f P i s a proviso and $ i s a set of pro-visos, then we write "P,$" instead of "{P} u $"; i f A i s a formula and T i s a set of formulae, we write " A , r " or " I " , A " i n place of " { A } u [ " ; i f T , A are sets of formulae then we write " I " , A " instead of "T u A " . Thus {f ( A , B ) , f(C)} U *, T U {A} -»• {C , A & B } U ( A u E ) appears as f (A.,B) ,f ( C ) , $ , T , A -> C , A & B , A , Z In what follows, A , B , C , D w i l l be taken to be a r b i t r a r i l y chosen formulae, P and Q to be a r b i t r a r i l y chosen provisos,F and A to be f i n i t e and possibly empty sets of formulae, and $ and ip to be f i n i t e and possibly empty sets of provisos. R e c a l l , however, that when T and A appear i n the context of the sequent $ , T — A , they cannot both be emp ty. VIII.3 The Semantics of Q. Let $ be the p a r t i a l Boolean algebra < B , $ , V , A , ' , 0 , l > . n 72 We denote by a* the sequence <a. ,a. , ...> e B To each such se-1 1 12 quence a*, there corresponds a function which maps the set of formulae and provisos of Q into B . Such functions are called interpretations of Q within ffi ; for convenience we use the same symbol for both the se-quence and the associated interpretation. Formally, l e t a* = N <a. ,a. ,...> e B ; then the interpretation a* i s the function whose _ Now, as before, we may regard the r i s k of confusion engendered by using " B " to denote both the set of elements of an algebra and a sen-tence of Q as negl i g i b l e . 156 domain and values are defined as follows. (We denote the demain of a* VIII. 3.1(a) I f A = q , then ( i ) A e for a l l a*, and ( i i ) a* (A) = a i ; •(b) i f A = A , then ( i ) A-e D ^ for a l l a*, and ( i i ) a* (A) = 0 ; (c) i f A = -|B , then ( i ) A e D ^ i f f B e and ( i i ) a*(A) = (a*(B))' when A e D .; a* (d) i f A = BvC , then (i) A e D ^ i f f B e and C e and a*(B)$a*(C) , and ( i i ) a*(A) =...a*(B)v a*(C) when; A e Da,; (e) i f A = B&C , then ( i ) A e D . i f f B e D . and C e D . ' a * a* a* and a*(B)$a*(C) , and ( i i ) a*(A) = a*(B) A a*(C) when A e V ; (f) ( i ) f(A) e i f f A e D^; (II) a*(f(A)) = 1 when A 6 °a*; (g) ( i ) f(A,B)e D a A i f f A e D ^ and B e D ^ and a*(A)$a*(B); ( i i ) a*(f(A,B)) = 1 when f(A,B)e D a* We can eas i l y show that (h) i f A = B3C , then ( i ) A e D . i f f B e D . and C e D . a* a* a* and a*(A)$a*(B) , and ( i i ) a*(A) = (a*(B))' v a*(V) when A e D a*' Note also that VIII.3.2 A e D . i f f a*(f(A)) = 1 a* 157 VIII. 3 .3 f(B,C) eD ^  i f f BvC e i f f B&C e i f f ' B 3 C 6 D a * i f f f ( C ' B ) e D a * VIII.3 . 4 I f a* i s an interpretation, then a* interprets A i f f A e Da * > J*. Interprets $ u f i f f for each P e $ and for each A e\, P e D . and A e D .; a* interprets $ uT a* a* c  compatibly i f f a* interprets $ u F and, for a l l A,B e f~, a*(A)$a*(B). Note that, for any P e $ , A e T, we have a*(P)$a*(A) provided a* i n t e r -prets $ u r. I f a* interprets $ u T, we write VIII. 3.5 a*(0 u D = £ (a*(M): M e $ u { } at I f T = {A^, ...>An}, then provided a* interprets compatibly, we write VIII.3 . 6 va* ( D = d f (...(a*(A 1)v a*(A2>)v ...)v a*(A n) Aa* ( D = d f (...(a*(A1)A a*(A 2))A . . . ) A a*(A n) Also, for any interpretation a*, we set VIII.3 . 7 v a* ( 0 ) = 0 Aa* ( 0 ) = d f 1 C l a s s i c a l l y there are various ways to define the notion of v a l i d i t y and the r e l a t i o n of semantic entailment. From Chapter I I I we have, for c l a s s i c a l l o g i c : (a) A i f f , for a l l valuations v of P , v(A) = 1 (III. 3 . 4 ) (g) f: A i f f , for a l l interpretations a* within # 2, a*(A) = 1. (Y) t" A i f f , for a l l interpretations a* within every Boolean algebra a*(A) = 1 (III. 4 . 6 ) ( 6 ) r |s A i f f , for a l l valuations v o f p , i f v(B) = 1 for a l l B e T, then v(A) = 1. 158 (e) r frA i f f , for a l l interpretations a* within , i f a*(B) = 1 for a l l B e f, then a* (A) = 1. ( T ) n :)sA i f f , for any interpretation a* within any Boolean algebra, i f a*(B) = 1 for a l l B e Y, then a*(A) = 1 (III.4.11) (n) r i f f , for any interpretation a* within any Boolean algebra, A a* (r) < a*(A) (III.4.1o). (9) T | i A i f f , for any interpretation a* within any Boolean Algebra $ , a*(A) i s a member of each f i l t e r F on such that a*(r) £ F (III.3.14). (i) r flA i f f , for any interpretation a* within any Boolean algebra $ , a*(A) i s a member of each u l t r a f i l t e r U on $ such that a*(r) £ u (III.4.13). Among these altenatives, (a) and (6) are i n terms of (admissible) valuations of ^  (mappings of the set of formulae o f f into {0,1}), while the remainder are i n terms of interpretations of "P within a Boolean a l -gebra. In Chapter I I I we started the discussion of semantics by d e f i -ning a valuation of ^  , and both v a l i d i t y and semantic entailment were then defined i n terms of these valuations; thus (a) and (6) were used as d e f i n i t i o n s , and were easily seen to be equivalent to (6) and (e) respectively, while (y), (<5) , (n), (0) and (l) appeared as theorems. The l o g i c we are now developing takes as i t s s t a r t i n g point the fact that the set of Q-propositions for a quantum mechanical system forms a par-t i a l Boolean algebra. Accordingly, the reverse of the c l a s s i c a l pro-cedure i s appropriate: we define v a l i d i t y and semantic fefttailment i n 159 p, terms of interpretations within p a r t i a l Boolean algebras, and consider subsequently (in Section 4) what a suitable d e f i n i t i o n of an admissible valuation might be. In this discussion I only consider interpretations within tran-s i t i v e p.B.a.'s; the reasons for r e s t r i c t i n g the semantics i n th i s way are given at the end of the Section. From a formal point of view i t might be preferable to take a more general approach and tocconsider i n -terpretations within a l l p.B.a.'s; i n f a c t , much of the discussion which follows would be applicable to a general semantics of this kind. We may note, however, that t r a n s i t i v i t y holds for a l l those p.B.a.'s i n which we are p a r t i c u l a r l y interested, namely the p.B.a.'s of subspaces of a Hilbe r t space. We use the analogue of (y) to provide a d e f i n i -tion of Q-validity. VIII.3.8 A i s Q-valid i f f , for any interpretation a* within any tr a n s i t i v e p.B.a. $ , i f A e D ., then a*(A) = 1. a* Since, as we observed i n Chapter VI.5, there are, i n general, no homomorphic mappings of a p.B.a. onto the two element Boolean algebra, equivalent definitions analogous to (a) and (g) are not r e a l l y available. S i m i l a r l y , we look towards (x), (n), (0) or ( i ) , rather than (<5) or (e) when we set out to define semantic entailment, and, again, we have no assurance that the equivalences which hold for Boolean alge-bras also hold for p a r t i a l Boolean algebras. Which of these we choose, and how we modify i t for use i n quantum l o g i c , depends i n part on how strong a condition of compatibility we f e e l should be met by the i n t e r -160 •pretations of the sentences involved. The strength of this condition i s a metter of taste: as a result there are a number of possible d e f i -nitions of semantic entailment available to us. The l i s t of nine possible definitions below i s not exhaustive. VIII.3.9(a) $, T (r A i f f , for any sequence a* from any t r a n s i t i v e a p a r t i a l Boolean algebra, i f a* interprets $ u r, and i f a*(B) = 1 for a l l B e T, then a* interprets A and a*(A) = 1. (b) r |^ A i f f , any interpretation a* within any tran-s i t i v e p a r t i a l Boolean algebra which interprets $ u T compatibly interprets A, and i f , for each P e $ and each B e T, a*(P) = a*(B) = 1, then a*(A) = 1. (c) $, T feA i f f , for any interpretation a* within any t r a n s i t i v e p a r t i a l Boolean algebra, i f a* interprets $ u T compatibly, then a* interprets A and A a*(T) < a*(A). (d) $, r f^A i f f , for any interpretation a* within any t r a n s i t i v e p a r t i a l Boolean algebra, i f a* interprets $ u T compatibly, then a* interprets {A} u $ u T compatibly, and Aa*(r) < a*(A). (e) $, T |^ A i f f , for any interpretation a* within any t r a n s i t i v e p a r t i a l Boolean algebra, i f a* interprets $ u r compatibly, then a* interprets {A} u $ u V compatibly, and i f , for a l l P e $, B e T, a*(P) = a*(B) = 1, then a*(A) = 1. 161 (f) $, r ^A i f f , for any interpretation a* from any tr a n s i t i v e p a r t i a l Boolean algebra, i f a* interprets $ u T, then a* i n t e r -prets A, and i f , for a l l P e $, B e V, a*(P) = a*(B) = 1, then a*(A) = 1. (g) $, T |sA i f f , for any interpretation a* within any tr a n s i t i v e p a r t i a l Boolean algebra ^ 3 » i f a* interprets $ u V compatibly, then a* interprets A, and a*(A) i s member of each f i l t e r F on ^  such that a* (O £ F. (h) T f^ A i f f , for any interpretation a* within any tr a n s i t i v e p a r t i a l Boolean algebra ^ , i f a* interprets $ u T compatibly, then a* interprets A, and a*(A) i s a member of each u l t r a f i l t e r O on ^  such that a*(r) £ U. (j) Like (h), but with the word "compatibly" omitted. We abbreviate the statement"*, r j: A" to "E ( a ) n , and use simi-Si l a r abbreviations for the other cases. The r e l a t i v e strengths of the definitions are i l l u s t r a t e d i n Figure 19; the interpretation of this diagrem i s straightforward: we have E(d) implies E(g), E(g) implies E(h), and so on. Most of these implications are obvious from the definitions above. The equivalence of de f i n i t i o n s (c) and (g) appears 162 163 from VI.5.4(b) and VI.5.7. The l i s t of omplications shown by the diagram may not be complete: i t i s an open question whether (b) i s equivalent to ( c ) , and (d) to (e); i f the conjectured generalisation of VI.5.9 holds, then (h) i s equivalent to (g), and hence to (c). However, the following examples show that not a l l the proposed d e f i n i -tions are equivalent: we have VIII.3.10 A ^ a q l V q 2 b u t - ~ ^ b q l V q 2 V q l M 2 b u t q l ' , q l / d q2 V q 2 £d ql & q2 b u t q l ' q 2 ^ f q l & q 2 q 2 ' q l & 1 q l M l b u t V ^ ^ l ^ l whence, i n general, E(a) -j- E(b) , E(c) + E(d) , E(d) + E(f) and E(f) + E(d). Three considerations influence our selection of a d e f i n i t i o n of semantic entailment. (i) We want our d e f i n i t i o n to be i n keeping with the algebraic approach we are taking. On this approach the primary semantic notion i s that of an interpretation within a p a r t i a l Boolean algebra, the struc-ture of this algebra being a formal r e f l e c t i o n of two fac t s , f i r s t that we are considering interpretations under which not a l l sentences can be regarded as meaningful, and secondly that within these interpretations questions of meaningfulness hang together i n a systematic way. ( i i ) More s p e c i f i c a l l y , we may consider those interpretations which we have chiefly i n mind when we construct bur l o g i c . These are i n t e r -pretations of Q within the p a r t i a l Boolean algebras of Q-propositions for quantum mechanical systems (more b r i e f l y , "quantum mechanical i n t e r -164 pretations"). Our i n t u i t i o n s about the concept of entailment as i t applies to Q-propositions should, as far as possible, be realised by our chosen d e f i n i t i o n . ( i i i ) F i n a l l y , we are influenced by questions of s i m p l i c i t y : we do not want our d e f i n i t i o n to be one which makes our l o g i c unduly d i f f i c u l t to work with. For instance, we want other semantic concepts l i k e v a l i -dity to be straightforwardly related to semantic entailment, and we want to be able to produce a workable system of proof which i s sound with respect to the d e f i n i t i o n we choose. These considerations p u l l i n d i f f e r e n t directions. For instance while (a) i s arguably the simplest of the suggested d e f i n i t i o n s , consi-deration (i) t e l l s against i t s adoption i n the following way. We have (vacuously) " (r aq 1&q 2 > but on de f i n i t i o n s (b)-(j) a si m i l a r statement of entailment i s disallowed. E f f e c t i v e l y these l a t t e r d efinitions im-pose a condition of meaningfulness on the formula to the right of the t u r n s t i l e . We may f e e l that without such a condition we should not do 73 ju s t i v e to the ideas we are trying to capture i n the l o g i c . On the other hand, the condition imposed by (d) i s arguably too stringent: i t requires not only that the formula A to the right of the t u r n s t i l e be meaningfully interpreted, and that a*(A) l i e above Aa*(T), where r i s the set of premisses, but also that a*(A) be compatible with each member 73 Note that such a meaningfulness condition i s not the same as a r e l e - vance condition, as the term i s technically used (see, e.g. Meyer (1971)), though i t does mean that there are formulae which are not entailed by a contradiction. 165 of a * ( r ) . Further, this condition acts i n an apparently arbitrary way, since i t means that V I I I.3.11 q 1& 1q 1 {= dq 2 b u t q i ' i q i f d q2 ' w h e r e a s ' f o r instance, both q-jS"^ £ cq2 a n d li'^i ]f cq2 ' However, (d) shares with ( c ) , (g), (h) and (j) a generality of approach absent from those alternatives which focus attention on the interpretations assigning the unit element to each member of T. And i t may be argued that this generality i s desirable, given the algebraic approach we are taking. This argument has pa r t i c u l a r force when we consider quantum mechanical interpretations. When we have these i n t e r -pretations i n mind, i f we enquire what i s entailed by a set V of formu-lae, we are not very interested i n the special case when a*(B) = 1 for every B e l : i n this case each interpreted sentence of T merely t e l l s us that the state vector for our system l i e s i n the appropriate Hilbert space. We are more interested i n the general case, when some constraints are l a i d on the system's state. Considerations ( i ) and ( i i ) , then, prompt us to eliminate (a), (b), (e) and ( f ) , given the a v a i l a b i l i t y of other alternatives. I f we take account only of consideration ( i i ) , then (h) or (j) seems most appropriate. These are the formulations which ask us to consider the u l t r a f i l t e r s of a p.B.a. In the p.B.a. (H) of subspaces of a Hilbert space for a given quantum mechanical system, the u l t r a f i l t e r s have a special significance. Such a p.B.a. i s atomic; each atom, or 166 one-dimensional subspace of H, corresponds to a pure state of the system. Further, the u l t r a f i l t e r s on ^ ( H) are the p r i n c i p a l f i l t e r s generated by these atoms. Thus, i f a* i s a quantum mechanical interpretation, and U x an u l t r a f i l t e r corresponding to a pure state x, to say that a*(A) e U x i s e f f e c t i v e l y t o say that, on this interpretation, A holds whenever 74 the system i s i n state x . We see that, I f a* i s a quantum mechanical interpretation for some system, and $ £ D^, then $, T J^ A implies that whenever the state of the system i s such thatj on this interpretation, the formulae i n T a l l hold, then A also holds. D e f i n i t i o n (h) i n t r o -duces a compatibility condition: i f a* i s a quantum mechanical i n t e r -pretation for some system, and $ £ ^a*» then $, T j^A implies that, provided the formulae i n T a l l receive mutually compatible interpreta-tions, whenever the state of the system i s such that, on this interpre-t a t i o n , a l l those formulae hold, then A also holds. These two d e f i n i -tions, (j) i n p a r t i c u l a r , seem closest to the concept of semantic en-tailment we would want to employ when dealing with quantum mechanical systems. In addition as talk of a formula "holding" implies, d e f i n i -t i o n (j) may be d i r e c t l y t i e d to the d e f i n i t i o n of an admissible valua- tion of Q (see Section 4), which w i l l make the notion of a formula "holding" more precise. However, to adopt (j) leads to certain d i f f i c u l t i e s . Consider a set T of formulae and an arbitrary u l t r a f i l t e r U on a p a r t i a l Boolean algebra 4$ • Since not a l l members of an u l t r a f i l t e r need be mutually Compare the discussion of Newtonian systems on p. IV.13. 167 compatible, we may have an interpretation a* of Q within $ on which a*(T) £ u, but Aa * ( r ) i s undefined. For this reason, (l 1.' q2 £ j q l & q 2 " This may be seen as the kind of oddity we should be prepared to accept i n a quantum l o g i c ; however, the fact that, on this d e f i n i t i o n , we do not know that we can interpret a set T of assumptions conjunctively means that there can be no equivalence between (j) and another d e f i n i t i o n expressed ( l i k e ( c ) ) i n terms of the ordering r e l a t i o n < on ^  . For-f e i t i n g such an equivalence makes the task of producing a proof theory corresponding to the semantics much more d i f f i c u l t , i f not unmanageable. Of the remaining d e f i n i t i o n s , now consider d e f i n i t i o n (c). This i s a general algebraic d e f i n i t i o n which introduces meaningfulness conditions of an appropriate strength. Thus considerations of the f i r s t kind favour i t s adoption. I t i s equivalent to (g), and we conjecture that i t i s equivalent to (h), which i s one of the alternatives picked out on the basis of consideration ( i i ) . We w i l l f i n d that i t i s eas i l y related to the concepts of Q-validity of formulae and of sequents, and that the l i n k with the notion of an admissible valuation of Q, while not as direct as that we would obtain by adopting d e f i n i t i o n ( j ) , i s nevertheless straightforward. Add i t i o n a l l y , we may produce a natural system of proof which corresponds to i t . For these reasons, we adopt d e f i n i t i o n (c) as a. d e f i n i t i o n of  semantic entailment, despite the fact that there i s one serious associa-ted with this choice. This problem i s common to definitions ( c ) , (g) 168 and (h). On these d e f i n i t i o n s , the entailment r e l a t i o n between pairs of sentences i s t r a n s i t i v e : for example, 0, A (r^ B and B |r^ C toge-ther imply $ u f, A jr^C. However, the resulting l o g i c i s i n a related way non-normal. From ( i ' ) $, T (=A and ( i i ) <S>,F , A fiB we cannot infer ( i i i ' ) $, F ^rB. From ( i ' ) , A i s interpreted whenever $ LT T i s interpreted compatibly; and whenever $ u r u {A} i s interpreted compa-t i b l y , then B i s interpreted, by ( i i ' ) . However, i t does not follow that B i s interpreted whenever $ u r i s interpreted compatibly, as ( i i i ' ) requires: there may well be an interpretation a * which interprets $ u n compatibly, and thus, i n accordance with ( i ' ) interprets A, but on which a*(A) i s not compatible with a l l members of a*(F); ( i i ' ) now t e l l s us nothing about whether or not B i s interpreted by such an i n t e r -pretation, nor i s the required information given by the relations Aa*(F) < a*(A), A a * ( r ) A a*(A) < a*(B). In formal terms: l e t "C^^ be the semantic consequence opera-tion on the power set of the set of formulae of Q, that i s , the opera-tion t ( k ) such that $ A eC^ (D i f f $, F |»A (k = a, b, ... j ) I t we use d e f i n i t i o n (c), (g) or (h), this i s not a closure operation, for we do not have t(J> < c ) < r » = ( D . We may note that the semantic consequence operations associa-ted with definitions (j) and (d) are both closure operations. Both escape the problem I have outlined above, for opposite reasons. On 169 the one hand,when we write $, T .^A we do not consider only those interpretations which interpret $ u T compatibly; on the other, i f $, T J; A then we know that a*(A) i s compatible with a*(T) whenever $ u T i s compatibly interpreted. Thus i n each case the argument which showed that "Cf^  was not a closure operation f a i l s against and This discussion shows clea r l y that there i s no unequivocally best choice among the alternative d e f i n i t i o n s . In adopting (c) we do violence to certain metalogical principles i n order better to accomodate other considerations. We write, V I I I . 3.12 $, T (?A i f f $ , I f^ A and, for a sequent, VIII. 3.13 $, r j=J A i f f $, r + A i s a Q-valid sequent i f f , for any interpretation a* from any t r a n s i t i v e p.B.a., i f a* interprets $ u V compatibly, then a* interprets A compa-t i b l y and Aa*(r) < v a*(A). De f i n i t i o n VIII.3.13, of a Q-valid sequent, appears as the natural ana-logue i n quantum l o g i c of the c l a s s i c a l theorem III.4.8*: (k) r -> A i s a C-valid sequent i f f for every interpretation a* within every Boolean algebra, Aa*(r) < va*(A). These definitions y i e l d , unsurprisingly, VIII. 3.14 *, E |SA i f f $, r [=• {A} 170 For any formula A of Q we see that VIII.3.15 A i s Q — v a l i d i f f f (A) {» A By our d e f i n i t i o n , only formulae can be semantically entailed by a given set $ of provisos and a given set Y of formulae. We may, however, talk of a proviso being guaranteed by $ u V, and for conveince use the same symbol to express this r e l a t i o n . Bearing i n mind that, i f P e D . , then a*(P) = 1, we write, a* VIII. 3.16 $, r |rP i f f for any sequence a* from any t r a n s i t i v e p a r t i a l Boolean algebra, i f a* interprets $ u T compatibly, then Pe D ^ . Note that VIII. 3.17 $, r j=A implies $, Y ^ f (A) $, T JrAvB implies $, Y )= f(A,B) etc. F i n a l l y , I return to the question of t r a n s i t i v i t y . From our de-f i n i t i o n VIII.3.1, a sequence of members of any p a r t i a l Boolean algebra 4S> can supply an interpretation of Q within "fa ; however, to investigate questions of v a l i d i t y and semantic entailment we look only at interpreta-tions within t r a n s i t i v e p a r t i a l Boolean algebras. The decision to re-s t r i c t discussion i n this way i s linked with the choice of (c) , rather than (a) as the d e f i n i t i o n of semantic entailment. With a d e f i n i t i o n l i k e (a) problems of t r a n s i t i v i t y do not arise: consider a d e f i n i t i o n ( a + ) , l i k e (a) except that i t considers interpretations i n any p.B.a.; we find that, i f $, A f=a+B and $, B (c^C , then $, A js^C , 171 i n accordance with our i n t u i t i o n s about semantic entailment. However, were we to adjust d e f i n i t i o n (c) i n a si m i l a r way, to obtain ( c + ) , then prima facie we could encounter the following s i t u a t i o n . For formulae A, B, C and a set $ of provisos, we could have $, A and $, B , and, for some interpretations which interpreted $ u {A} , a*(A) < a*(B) and a*(B) < a*(C) (as required), but a*(A) incompatible with a*(C). In that case a*(A) i a*(C), and so $, A ^ C . We have (reluctantly) accepted that "C i s not a closure opera-tion for our l o g i c , but the abandonment of t r a n s i t i v i t y of entailment between pairs of formulae seems too high a price to pay i n the interests of generality. Moreover, i t i s not a price that i s exacted by our decision to employ a quantum lo g i c . For, as we have seen, the algebra % (H) of sub spaces of a Hilbert space i s a t r a n s i t i v e p.B.a., and so, i f a* i s an interpretation oa Q within $ (H), we can never have a*(A) < a*(B) and a*(B) < a*(C), but a*(A) i a*(C). Thus a minimal s a c r i f i c e of generality allows us to adhere to a basic metalogical p r i n c i p l e . We consider only t r a n s i t i v e p.B.a.'s (of which the p.B.a.'s of subspaces of Hilbert space form a proper subset) i n our discussion of semantic entailment; we do likewise when we define v a l i d i t y , so that the two concepts may be yoked together i n a natural way. Note f i n a l l y , that u l t r a f i l t e r s , whose properties we use in the next section, can only be usefully defined on t r a n s i t i v e p.B.a.'s as they were i n Chapter VI.5. 172 VIII.4. Admissible Valuations of Q The words "truth" and " f a l s i t y " have not yet entered the discussion of the semantics of Q. To put this another way, there has been no discussion of the admissible valuations of Q. As we have noted, this i s i n contrast with the approach used for c l a s s i c a l l o g i c : i n that case a valuation for P was defined before the general idea of an interpretation of P within a Boolean algebra was introduced. Subsequently these valuations were i d e n t i f i e d with the M-valuations for P corresponding to the l o g i c a l matrices of the form <B,U> (with U an u l t r a f i l t e r on the Boolean algebra B). Here, however, the development moves the other way. We move from a d e f i n i t i o n of a Q-;ogocal matrix, and of an interpretation of Q within a p.B.a., to that of an M-valuation, and arrive f i n a l l y at the d e f i n i t i o n of an admissible valuation of Q. Unlike c l a s s i c a l l o g i c , quantum log i c offers no equivalent d e f i n i t i o n of an admissible valuation independent of the idea of an interpretation of Q within a p.B.a. VIII.4.1. M=.<B,I> i s a Q-logical matrix i f f B = < B , $ , V , A , 1 , 0 , 1 > i s a t r a n s i t i v e p a r t i a l Boolean algebra and I c B. In passing we may explore the sense i n which any p a r t i a l Boolean algebra i s " s i m i l a r " to the algebra of the set of formulae of Q, and thus the extent to which this d e f i n i t i o n accords with the standard d e f i n i t i o n of a l o g i c a l matrix (III.4.12). E f f e c t i v e l y , the proviso-forming symbol " f " acts as the analogue i n Q of the r e l a t i o n $ on B. However, the correspondence i s not 173 exact. F i r s t , a proviso, as defined by VIII.2.2, does not always involve two formulae: f(A) i s a proviso as we l l as f(A,B) . We can meet this problem by regarding f(A) as an abbreviation for f(A,A) ; within the l o g i c being developed f(A) and f(A,A) are equivalent both semantically and deductively (see Sections 3 and 6). But, more fun-damentally, i f A and B are any two formulae of Q , then A&B and AvB are also both formulae, independently of any considerations involving f(A,B). Thus the algebra of the set of formulae of Q i s only a p a r t i a l Boolean algebra i n the Pickwickian sense that, since f(A,B) i s a proviso for a l l formulae A and B , we may therefore regard any two formulae of 75 Q as compatible. I do not take considerations of this kind to v i t i a t e the approach being used; rather, I draw attention to them to emphasise the divergences which exist between other logics and this version of quantum l o g i c . We define an M-valuation of Q i n the same way that we defined an M-valuation of P . For any Q-logical matrix, M = <B,I> , l e t X-|-:B ->{1,0} be the characteristic function of I. Then VIII.4.2 w i s an M-valuation of Q i f f M = <B,I> i s a Q-logical matrix, a* i s an interpretation of Q within B and w = x-j-* 3* Thus to every l o g i c a l matrix, M = <B,I> , there corresponds a set of M-valuations of Q. The M-valuation X^ , a* ^ s a function from a subset of the set of formulae and provisos of Q into {0,1} , whose domain coincides with D^ . 75. Of course, i n general i t i s misleading to think that the notion of compatibility should be applied to formulae . 174 We can now define an admissible valuation of Q . VIII.4.3. v i s an admissible valuation of Q i f f there i s a Q-logical matrix M = <B,U> , where U i s an u l t r a f i l t e r on the t r a n s i t i v e p.B.a. B , and v i s an M-valuation of Q. A s i m i l a r statement of equivalence appears as a metatheorem of, c l a s s i c a l l o g i c . (See III.4.1) In c l a s s i c a l l o g i c , as we saw, valuations are bivalent and truth functional. Here, however, although any admissible valuation of Q can y i e l d only 0 or 1 as a value, i n nearly every case some formula w i l l l i e outside i t s domain; indeed, this may w e l l happen to a complex formula even when both i t s immediate subformulae are evaluated. In addition, the characteristic function X J J of an u l t r a f i l t e r i s not i n general a homomorphism of the p.B.A. onto ~^2' Forboth these reasons, the admissible valuations of Q are not truth functional. Nevertheless, we can display some systematic features of the way i n which the value assigned to a complex formula l i k e A&B depends on the evaluations, not only of i t s components A and B , but also of the proviso f(A,B) . These dependencies are shown i n Tables VIII.1 and VIII.2, overleaf. A B f(A,B) A&B AvI 1 1 1 1 1 1 1 N N N 1 0 1 0 1 1 0 N N N 1 N 1 X 1 N N N N 0 1 1 0 1 0 1 N N N 0 0 1 0 1 0 0 0 N N 0 N 1 X 0 N N N N N 1 1 X N 1 N N N N 0 1 X N 0 N N N N N 1 X N N N N N Table VIII.1 A ~A 1 0 0 ? N N Table VIII.2 176 In these tables "N" stands for "No evaluation": since a valua-tion v i s a composite function X y a * > i f a formula l i e s outside the domain of a* , then i t l i e s outside the domain of v and thus receives no evalution.' Clearly, i f a formula A i s i n the domain of a* , then either a*(A) e U and v(A) = 1 , or a*(A) I U and v(A) = 0 Given any proviso P , i f P e for some a* , then a*(P) = 1. Since the maximum element of any p.B.a. i s a member of every u l t r a f i l t e r , i t follows that we can never have v(P) = 0 : i f a proviso i s evaluated, i t i s assigned the value 1. The values (or lack of them) assigned to A ,,B and f(A,B) are not altogether independent. I f A or B i s not assigned a value, then f(A,B) cannot be assigned a value. Correspondingly an "X" appears on certain lines of Table VIII.1, showing that the assignments given to A , B and f(A,B) on those lines are mutually inconsistent. The remaining lines are j u s t i f i e d by VIII.3.1 and VI.5.4-6. At one place i n each table, "?" appears. This indicates that (i) given v(A) = 0 for some admissible valuation v and formula A , we cannot deduce the value of v(~A) , and ( i i ) although we may know that v(A) = 0 = v(B) , we cannot thereby claim that v(AvB) = 0 . In neither of these cases i s the value of the complex sentence undefined. Any admissible valuation which assigns 0 to A assigns some value to ~A , and any admissible valuation whic assigns values to A , B and f(A,B) also assigns a value to AvB . Rather, i n each case the value assigned to the complex sentence i s not uniquely determined by the values assigned to i t s components. This follows from VI.5.5-6. ' To find out the value 177 of v(~A) and v^(AvB) i n these cases we need to look at the p a r t i c u l a r Q-logical matrix and interpretation a* involved i n the admissible valuation v. Note that "N" appears i n the tables as a result of the r e s t r i c -tions on the domain of an admissible valuation, and that "?" appears because, i n general, the characteristic function of an u l t r a f i l t e r on a p.B.a. B f a i l s to map B homomorphically onto B^ Now consider an admissible valuation obtained v i a a quantum mechanical.interpretation of Q . As we saw i n Section 3, i f B(H) i s the p.B.a. of subspaces of the Hilbert space for a quantum mechanical system, each u l t r a f i l t e r U on B(H) corresponds to a pure state of the system. I t follows that the admissible valuations associated with these u l t r a f i l t e r s are state-induced (see Chapter IV.3). That i s , corresponding to each valuation of Q associated with B(H) there i s a pure state x together with a one-dimensional subspace L " £ H (whe-e x £ L ) and a X X corresponding u l t r a f i l t e r U x on B(H) , such that, for some interpretation a* of Q within B(H) , v = x^j « a* > and, for any formula A of Q, x VIII.4.4. v(A) = 1 i f f x e a*(A) Conversely, i f the (pure) state of the system i s represented by a vector x e H , and a* i s any interpretation of Q within B(H) , then there i s an admissible valuation v of Q , such that, f° r any formula A of Q, VIII.4.4. v(A) = 1 i f f x £ a*(A) as before As we would hope, the question of semantic entailment i s connected to that of assigning truth values to sentences. We have, 178 VIII.4.5 $, r j r A implies that, for any admissible valuation v , i f ( i ) v(P) = 1 for a l l P e $ ; ( i i ) v(B) = 1 for a l l B e T ; ( i i i ) v(f(B,C)) = 1 for a l l B,C e r , then v(A) = 1 Proof. Assume that' $, T |* A and that v i s an admissible valuation for which ( i ) , ( i i ) and ( i i i ) a l l hold. Then there i s a t r a n s i t i v e p.B.a. B , an u l t r a f i l t e r U on B, and an interpretation a* of Q within B such that ( i ' ) a*(P) =1 for a l l . P e $ ; ( i i ' ) a*(B) e U for a l l B e T ; ( i i i ' ) a* interprets r compatibly. From ( i i i ' ) we know that Aa*(T) i s w e l l defined, and from our assumption, Aa * ( r ) < a*(A) (by VIII.3.9(c)). From the generalised l e f t to right conditional of VI.5.9 i t also folfflws that every u l t r a f i l t e r on B containing Aa * ( r ) also contains a*(A) . But, from ( i i ' ) and VI.5.4, Aa*(T) e U , and so a*(A) £ U . Whence y_(A) = 1 as required. I f , as I have conjectured, VI.5.9 holds for a l l t r a n s i t i v e p.B.a.'s then the converse of VIII.4.5 also holds, and we have an alternative (and recognisably orthodox) d e f i n i t i o n of semantic entailment available, couched i n terms of admissible valuations and equivalent to the d e f i n i t i o n adopted i n Section 3. Note that the condition expressed by clause ( i i i ) of VIII.4.5 i s needed, since our d e f i n i t i o n of semantic entailment only asks us to consSfer those interpretations of Q within a t r a n s i t i v e p.B.a. which interpret $ u V compatibly. Had we adopted VIII.3.9(j) as our d e f i n i t i o n of semantic entailment, we could have eliminated this clause, thus r e l a t i n g the concepts of admissible valuation and semantic entailment i n an even more t r a d i t i o n a l way. 179 From now on, I w i l l use 11 Qx to refer to the couple <FQ,V^> , where F^ is the set of formulae and provisos of Q , and V^ is the set of admissible valuations of Q. VIII.5 Two Comparisons: Reichenbach's 3-valued System, and Friedman and Glymour's Semantics. At this point i t is instructive to compare briefly the formal aspects of Q and of two other systems of quantum logic. I deal f i r s t with the system suggested by Reichenbach.76 Reichenbach believed that the use of a three-valued logic would result in the suppression of the "causal anomalies" which bedevil the interpretation of quantum mechanics, Unlike the other systems I have -described, his system did not derive directly from an analysis of the algebraic structure of Hilbert space. Accordingly, he employed no function analogous to the function a* which maps F^ into a partial Boolean algebra. In Reichenbach's system, the valuations of a formal language map the set of a l l formulas of teh language onto a set of 3 truth values, True, Indeterminate and False. The set of connectives is functionally adequate (in fact, highly redundant), and what results is a trivalent truth functional logic of the kind investigated by Post.^ The connectives include "~", "v", 'V and whose truth func-tional behaviour is shown by the matrices overleaf. 76. 77. See Reichenbach (1944), §§29-37, particularly §33. See Post (1921). 180 v T I F -> T I F T I T T T T T T F F T F I F I T I I I T T T I I F T F T I F F T T T F T Cyclic Diametrical Negation Negation VIII.5.1 A!B = Av~A -> —B dr 7 8 Thus 11!" is a defined connective with matrix ! T I F T F T F I T T T F F T F Reichenbach claimed that the formula Av~A •+ ~~B expresses the fact that A and B are complementary: i t can be raid as, "If A i s true or false, then B is indeterminate." Its (diametric) negation can be presumably be interpreted as saying that A and B are compatible. Thus a relation of compatibility appears between sentences of the language, which can be stated within the language using just the available sentential connectives. Even on this brief survey, the formal differences between Reichenbach's system and Q are obvious. Within Q a relation of simultaneous meaningfulness i s not expressible by means of a formula of Q but only.by a.proviso. I t can be argued that the semantics of Q are trivalent (even though each admissible valuation i s only bivalent), but the valuations are certainly not truth functional. This, in turn, derives from their algebraic origin, to which nothing in Reichenbach's system corresponds. These origins are, however, shared by the Friedman-Glymour system, to which I now return. I outlined the syntax and proof theory of this system in Chapter VII, but.at that stage I omitted a discussion 78. This way of describing Reichenbach's procedure appears in van Fraassen (1974) , p.585... _ - . . .. 181 of their semantics. As with Q , evaluation of a formula is a two-stage process. The set of formulae is f i r s t mapped onto the set of elements of an orthomodular lattice £ , and this set is in turn mapped onto {0,1} Their nomenclature is different from mine: what I have called (in Chapter VII) an "interpretation of P* within JC" they term a "semi-interpretation", and they use the phrase "admissible valuation" to refer to the mapping from X. into {0,1}. I restrict my discussion to their semantics S3, that i s , to the only semantics which to them "seems 79 mathematically and philosophically interesting." I w i l l adjust their terminology to bring i t into line with my own. Thus the mappings from X into {0,1} which they c a l l "admissible valuations" I w i l l c a l l 11 A-functions", while an admissible valuation of P' w i l l be a function which maps the set F of formulae of P' into {0.1} . If A is the set of elements of an orthomodular lattice •£ , then VIII.5.2. A:A -> {0,1} is a A-function for £, i f f (i) for a l l a e A , A(a) =1 i f f A(a X) =0 ; ( i i ) for a l l a,b e A , i f A(a) = 1 and a < b , then A(b) = 1 VIII.5.3. v:F {0,1} is an admissible valuation of P' • i f f a* is an interpretation of P' within an orthomod-ular lattice JC, A a A-function for JC, and v = A. a* The methodological similarity between the Friedman-Glymour approach and that taken in Section 4 is evident. We may also compare VIII.5.4 and VIII.5.5, below. 79. Friedman and Glymour (1972), p.20. 182 Let a* be an interpretation of p' within an orthomodular l a t t i c e C ,, A a A-function for £ , and v the admissible valuation for P* such that v = A.a* . Then, for any formulae, A, B of P', VIII.5.A v(A) = 1 i f f v(-»A) = 0 ; i f v(A) = 1 and a*(A) < a*(B) , then v(B) = 1 Let a* be an interpretation of Q within a t r a n s i t i v e p.B.a. B ,. U an u l t r a f i l t e r on B , and v the admissible valuation of Q such that v = Xjj»a* • Then, for any formuale A , B of Q, VIII.5.4. I f v(A) = 1 then y_ClA) = 0 ; i f v(iA) = 1 then v(A) = 0 ; i f v(A) = 1 and a*(A) < a*(B) , then v(B) = 1 . Thus i n both logics an ordering r e l a t i o n on an algebraic structure mediates the r e l a t i o n of semantic entailment within the l o g i c . The admissible valuations of P' are not truth functional, as an elegant theorem by 80 Chernavska shows. For consider the orthomodular l a t t i c e of subspaces of R2. I f A i s a A-function for R 2 , then we cannot have A(0) = 1 , since that would y i e l d both A(R 2) = 0 (by VIII.5.2(i)) and A(R 2) = 1 (by V I I I . 5 . 2 ( i i ) ) . Now consider two pairs of one -dimensional subspaces, such that L x L , L ± L , and L and L are oblique to each other, x y u . v x u By V I I I . 5 . 2 ( i ) , exactly one of each pair i s mapped onto 1 by A . Without loss of generality we can assume that ^( i - x ) = 1 = A(L^) . In this case A(L^ A L y) = A(L^) = 1 , but A(L v A Ltt) = A(0) = 0 . Thus 80. Chernavska (1978). Figure 20. 1 8 4 a A-funeta on is not, i n general, a homomorphism, and : the admissible val-uations are not truth functional. Also, as we have just seen, there is more than one atom in R2 which is assigned the value 1 by a A-function. It follows that an admissible valuation for P T obtained via the orthomodular lattice for a 81 quantum mechanical system is not state-induced. In sum, while the admissible valuations of both P' and Q are not truth functional, they differ i n that those of P' have as domain the set of a l l formulas of the language, and so are s t r i c t l y bivalent, rather than bivalent wherever defined. In each case, valuations may be obtained via the algebra of subspaces of Hilbert space for a quantum mechanical system; however, those of Q are state-induced, whereas those of P' are not. On the question of compatibility, there are no expressions in P' corresponding to the provisos of Q , but neither does i t seem possible to express this notion by means of a formula of P' . VIII.6 The System QN. The system of proof QN i s a modification of the system CN. Like those of CN, the derivations of QN take the form of trees; we may easily modify the description of a tree which appeared in Chapter III.5 to make i t appropriate to QN by adding the words "or proviso(s)" after 81. Friedman and Glymour show that i t is possible, in the case of a 3-dimensional space, to construct a A-function such that exactly one atom of the corresponding l a t t i c e i s assigned the value 1 . However, this does not yield a state-induced valuation of the kind just discussed in Section 4, since each plane is also assigned 1 , rather than just those planes which contain the privileged atom. 185 each occurence of the word "formula(e)" i n that description. Thereby we obtain suitable definitions of an inference figure, a. rule of inference, an upper formula (or proviso), end formula (or proviso), stands above, the limb branching from A (or P), the length of a_ limb, and discharged. We also need to supplement the discussion of the discharging function of certain rules, since i n QN certain provisos as self-discharging, that i s , whenever they appear as i n i t i a l provisos i n a tree they are discharged, and no inference rule need be applied to effect t h i s . VIII.6.1 A derivation tree constitutes a derivation i n QN of a formula A (or proviso P) from a f i n i t e set $ of .provisos and a f i n i t e set T. of formulae, i f f (a) A (P) i s the end formula (proviso) of the tree; (b) the set of undischarged i n i t i a l provisos of the tree i s a subset of $ ; (c) the set of undischarged i n i t i a l formulae of the tree i s a subset of T ; (d) each inference figure occurring i n the tree i s an i n -stance of one of the following schemata. &+ f(A,B) A B &- A&B A&B ' A&B A B [A] [B] v+ f(A,B) A f(A,B) B v- f(A,T) f(B,T) AvB C C AvB AvB C where r i s the set of undischarged i n i t i a l formulae which stand above the occur-rences of C as an upper formula of the figure 186 [A] h A ] £ + A I A z_ f ( A , r ) a f ( A , r ) i A A where T i s the set of undischarged i n i t i a l formulae standing above £ Introduction f(A) Assumption B f(A,B) provided no undischarged i n i t i a l formulae stand above the upper formulae of this inference figure. ^-compatibility Commutation f(A) Selection f(A,£) f(A,B) f(B,A) Analysis Construction f(A&B) f(A,B) f(A,B) f(A) f (AvB) f(A,B) f(A,B) f(B,C) f(C,A) f (A&B.C) f(lA,B) f(A,B) f(A,B) f(B,C) f(C,A) f(AvB,C) f(A,B) f(TA,B) Replacement' [A] [C] f(B,C) C A f(A,B) provided no undischarged i n i t i a l formulae stand above the occurrences of B and C i n this figure, except for (i) occurrences of A above C , ( i i ) occurrences of C above A , and ( i i i ) occurrences of formulae D , E as upper formulae of an assumption rule. 187 i T r a n s i t i v i t y ' [A] [B] provided no undischarged i n i t i a l formulae stand above the occurren-f(A) B C ces of B and C i n this inference f(A,C) figure except for (i) occurrences of A above B, ( l i ) occurrences of B above C, ( i i i ) occurrences of formulae D,E as upper formulae of an assumption rule. In both the rules marked"!" the existence of the derivations shown above the inference figure i s mandatory. f(q^) i s a self-discharging proviso. In the schemata given, assumptions which are discharged are shown i n square brackets. We write $, T \-A (or $, T \"~P) i f f a derivation i n QN of A (or P) from $ u r e x i s t s . We have the following derived rules. VIII. 6.2 [A] i n the =>+ and rules, i s the set of un-discharged i n i t i a l formulae which stand =>+- f(A,T) f(A,B) B above B and above A respectively A B 3- f(A,T) f ( B , D A A 3 B B Re f l e x i v i t y f(A) D e d u c i b i l i t y 0 [A] f(A,A) f(A) F(A,B) Provided no undischarged i n i t i a l formulae stand above the occurrence of B i n this figure, except for (i) occurrences of A, ( i i ) occurrences of formulae D,E as upper formulae of an assumption rule. 188 Synthesis f(A,B) f (A,B) f(A,B) f (A&B) f (AvB) f (A=>B (1+) f(A) (1-) fC»A) fOA) f(A) Construction f ( A , B ) f (B,C) f(C,A) f(AoB,C) f ( A ) i s a self-discharging proviso. Trees 6.1 and 6.2 below j u s t i f y the 3+ and the =>- rules; since we can show that f(A) h Av"»A (see trees 8.7 and 8.8). The r e f l e x i v i t y and deducibility rules are special cases of the t r a n s i t i v i t y rule when A = B = C and when B = C respectively. Note that the derivation of B from A i s mandatory i n the dedu c i b i l i t y rule. The synthesis rules are a l l obtainable from trees l i k e tree 6.3. below. The T+ and T-rules are ea s i l y obtained by using the rules for r e f l e x i v i t y , 1-construc-tion or 1-analysis, and selection. The derivation of the =>-construc-tion rule i s t r i v i a l , given the v-construction r u l e . Tree 6.4 shows that f(") i s self-discharging. In both tree 6.1 and 6.2 r i s the set of undischarged i n i t i a l formulae standing above B . [A] f(A) '. Tree 6.1 . f(1A ,B) B f ( l A , B ) [1A] f(A,r) . f(nA,r) f(A,r,) A V J A I A V B ? A V B "»AvB 189 1 AvB  f CAvB) Tree 6.2 fQA,B) A [IA] f(A,B)  f(B,A) f ( B , D f(nA,T) f (B,T) HAvB B [B] Tree 6.3 f(A,B) Tree 6.4 [ f ( q 3 ) ] f(A,B) f(A) f ( q 3 , A ) f(A,B) f(B,A) f(A,A) f^'^3) f (AvB ,A) f ( ~ ) f(AvB) By induction on the length of a formula, using the analysis, commutation and s e l e c t i o n r u l e s , we obtain the Sub formula Theorem: VIII.6.3 I f A i s a subformula of B, then f(B) f-f (A) , f(B,C)f-f(A) and f(C,B) {- f ( A ) , for any formula C. I t i s i n t e r e s t i n g to note how l i t t l e d i f f e r e n t the rules for the connectives are from t h e i r counterparts i n the c l a s s i c a l system. Provisos are required i n the rules for the introduction of binary connectives, and also i n any rules which have a discharging function. We can r e l a t e the notions of d e r i v a b i l i t y i n the two systems CN and QN. I f i r s t s t ate, without proof, a metatheorem of CN. 190 VIII.6.4 I f T i s a set of formulae of p, A a formula of p such that T hjgA , then a derivation tree exists constituting a de-ri v a t i o n of A from Y i n CN, such that, for each formula B occuring i n the tree, either (a) B = A , or (b) B = ", or (c) B e Y, or (d) B i s a subformula of A , or (e) B i s a subformula of a member of Y, or (f) B i s constructed from formulae of type (a), (b), (c), (d) or (e) by using the sentential connectives. 82 We may c a l l a derivation tree of this kind an economical derivation tree Now for any formula A of Q, l e t A° be the formula of p which results from replacing each occurrence of the propositional variable q. i n A by the propositional variable p. of p; i f T = {A.,...,A } where 1 x I n A, A n are a l l formulae of Q, l e t r° = {A^,...,An°}. Then, for any formula A of Q, and set Y of formulae of Q, VIII.6.5 r° (-NA° implies f(A) , Y f-A (where A° i s the set of formulae occuring i n an economical derivation of A° from r ° ) . Proof. Assume r°f-NA° ; then a tree exists , constituting a derivation of A 0 from r° i n CN, whose end formula i s A° and whose undischarged i n i t i a l formulae are a l l members of r ° . We now replace each formula B° occurring i n this tree by the corresponding formula B of Q, and add 82 Prawitz (1965), Corollary 1 to Theorem 3 p.42, provides a stronger version of this theorem; I t may seem surprising that we have to include clause ( f ) . As an example of a derivation tree i n which a formula of type (f) occurs, consider a tree to show *7(A&B) /• lAvlB. This w i l l include an occurrence of 7(iAvlB) as a discharged assumption. 191 provisos as required so that the resulting tree constitutes a d e r i -vation of A from $ U-Y i n QN, where $ i s some set of provisos. By inspection of the rules for QN, we see that each proviso thus added w i l l be of the form f(C,D) , where C° , D° are formulae appearing i n the derivation of A° from T°. I t follows that each proviso thus added i s a member of f ( A ) , and hence that the res u l t i n g tree consti-tutes a derivation of A from f(A) u r i n QN. Also, for any set $ of provisos, set T of formulae of Q, and formula A of Q, we have VIII.6.6 $, T )-A implies T° \-^A° The proof i s obvious. Now consider the case when A° i s a theorem of CN. In this case an economical derivation of A° from the empty set of formulae e x i s t s , and each formula occurring i n the tree i s either or i s con-structed from A and the subformulae of A° by using the sentential con-nectives of p . Let A be the set of subformulae of A (the formula of Q corresponding to A°). Then by induction on the length of a formula of Q, using the construction and commutation rules, we see that f(A) f-f(B,C), where B, C are any two formulae constructed from the sub-formulae of A using the sentential connectives of Q; further, for any such formula B we then have f(A) [~f(B, A), by the selection and "-compatibility rules. Hence, for any formulae D°, E° occurring i n an economical derivation of the theorem A° of CN, we have f(A) |-f(D,E). BY VIII.6.5 i t follows that 192 VIII. 6.7 (-^ jA0 implies f(A) f- A where A i s the set of sub-formulae of A. VIII.6.8 A i s a theorem of QN i f f f(A) f A. VIII.7 The Soundness of QN. We show that QN i s sound, i . e . that, i n QN, VIII.7.1 $, T f*A implies $, V (: A ; $, Y \r, P implies $, A Is P • Proof. We proceed by induction on the length of a derivation tree. We say the system i s n-sound i f f , whenever a derivation of A' from $'* u r' exists constituted by a tree of length 1^, < n, we have $', r' ^-A', and whenever a derivation of P' from $' u I" exists con-st i t u t e d by a tree of length Lp?- n> w ^ have $' , I" |:P! T r i v i a l l y , the system i f 1-sound, since A |t.A and P \s P for any formula A and proviso P of Q and for any propositional variable of Q we have M ( q ± ) . Now assume that the system i s n-sound,and that a derivation of A from $ u I exists constituted by a tree of length n+1. Then A i s the lower formula of an inference figure which i s an instance of one of these schemata: v+, v-, &+, &-, A+, We consider each case sepa-rately; i n what follows, "interpretation" i s understood to mean an interpretation within a t r a n s i t i v e p a r t i a l Boolean algebra. 193 Case (v+). A = BvC and i s the lower figure of a v+ inference figure. Then there are formulae B, C such that $, Y |-f(B,C) and either $, Y |-B or$, r \-C. We consider only the former case; the other follows ob-viously. In this case, from our assumptions ( i ) $, Y )s B and ( i i ) $, T |=f(B,C). From ( i i ) , each interpretation a* which interprets 3> u Y 'compatibly interprets B and C compatibly. Thus a* interprets BvC, and a*(B)v a*(C) ex i s t s . From (i) we obtain • A A * ( T ) $ a*(B)< a*'(B)y..a*(C) = a*(BvC) By t r a n s i t i v i t y , A a * ( T ) < a*(BvC) and so $, Y ^ A . Case (v-) A i s the lower formula of a v- inference figure. Then there are formulae B, C and a Set A of formulae of Q such that A £ Y, and $, T f-f(B,D) and $, T f f(C,D) for each D e A , <J , Y [-BvC, $ , A , B |-A and $, A , C |-A . From our assumptions, (i) $, Y ^ f (B,D) and <J>, r [? f (C,D) for each De A ; ( i i ) $, r |» Bvc; ( i i i ) $ , A , B A and $, A , C j» A. Consider an assumption a* which interprets $ u Y compatibly. From ( i i ) , a* interprets B and C compatibly, and so, using ( i ) , a* interprets $u i u {B,C} compatibly. From ( i i i ) , A a * ( A ) A a*(B)< a*(A) and A a * ( A ) A a*(C)< a*(A) . Let b = A a * ( A ) A a*(B) ; c = A s * ( A ) A a*(C). Then b$a*(A) and c$a*(A); also, since a* interprets A u {B,C} compatible, b$c. Thus we may apply II.3.11 to obtain (iv) b v c < a*(A) . But, by II.2.4, again using the fact that a* interprets A u {B,C} compatibly, b v c = A a * ( A ) A (a*(B) v a*(C)) = A a * ( A ) A a*(BvC) (v) . Now, from ( i i ) , a*(BvC)$ A a * ( D ; also A a *(r) $ A a * ( A ) , since A £ Y , and we 194 already know that Aa*(A)$a*(bvC) . Thus, since A a * ( r ) ^ A a*(A) , we have (vi) A a * ( r ) A a*(BvC) < Aa*(A) Aa*(BvC) (from II.5.6). But from ( i i ) , v A a * ( r ) < a*(BvC) , and so we can use ( i v ) , (v) and (vi) to obtain Aa*(r)< A a * ( r ) Aa*(BvC)< a * ( A ) Aa*(BvC)< a * ( A ) By t r a n s i t i v i t y , Aa*(T)< a * ( A ) and so 0, T J; A . Case (&+) A i s the lower formula of an &+ inference figure. Then ethere are formulae B,C, such that A = B&C and $, T |-f(B,C0 , $ , r | i B and T («-c . By our assumptions ( i ) every interpretation a* which interprets $ u r compatibly interprets B and C compatibly, and, for each such interpretation ( i i ) Aa*(T)< a*(B) and ( i i i ) Aa*(T)< a*(C). Let a* be such an assignment. Then, from ( i ) , a* interprets B&C, and a*(B)A a*(C) e x i s t s . Further, since A a * ( F ) , a*(B) and a*(C) are pair-wise compatible, from ( i i ) and ( i i i ) we obtain Aa*(r)< a*(B)A a*(C) = a*(B&C) = a * ( A ) whence $, T )s A . Case (&-) A i s the lower formula of an &- inference figure. We deal with the case when the upper formula of the figure i s of the form A&B; the other case i s dealt with s i m i l a r l y . In this case there i s a formula B such that $, T f* A&B ; whence, from our assumptions, $ , r ^ A&B , and every interpretation a* which interprets $ u T compa-t i b l y interprets A&B. Further for such an interpretation, Aa*(r)< a*(A&B). But every interpretation which interprets A&B interprets both A and B, and so a * ( A&B) \= a * ( A ) A a*(B)< a * ( A ) . By t r a n s i t i v i t y , we see that $ r | = A . 195 Case ("+) A = A and i s the lower formula of a "+ inference figure. Then there are formulae, B -»B, such that both $, V \- B and $, T \- 1B ; hence, by our assumptions, ( i ) $, V ^ B and $ , T (- IB, Let a* be an interpretation which interprets $ u ' T compatibly. Then A a * ( T ) < a*(B) and A a * ( T ) < a*OB) = (a*(B))' . We see that a*(B)$ A a*(r) and (a*(B))'$ A a*(r) , and we know that a*(B)$(a*(B)) 1 Whence, by Boolean algebra, a*(T)< a*(B)A (a*(B))' = 0 = a*C) I t follows that $, T |= A Case ("-) We deal only with the f i r s t case; the other i s si m i l a r . A = IB and i s the lower formula of a inference figure. Then there i s a set A of formula such that A c r and $, r \- f (B,C) for each C e A . By our assumption, every interpretation a* which interprets $ u T compatibly interprets $ u A u {B} compatibly. Further, on each such interpretation, Aa * ( A)A a*(B) = 0 , whence, using II.5.5, A s * ( A ) < (a*(B))' Thus Aa*(T)< A a * ( A ) < (a*(B))' = a*(*iB) by t r a n s i t i v i t y , T (sA Now, retaining the assumption that the system i s n-sound, assume that a derivation of a proviso P from $ u T e x i s t s , constituted by a tree of length n+1. Then P i s the lower proviso of one of the proviso ru-les. We deal separately with the assumption, replacement and t r a n s i -t i v i t y rules; the proof i n the other cases i s straightforward. 196 Since, for any proviso P and interpretation a*, a*(P) = 1 provided a* interprets P, i t i s s u f f i c i e n t to point out, i n each of these cases, that whenever the upper formula or provisos of the inference figure are interpreted, so i s the lower proviso. This follows t r i v i a l l y from the defi n i t i o n s of an interpretation (VIII.3.1) and a p a r t i a l Boolean a l g e — bra (VI. 4.1). As sumption Rule: P — f(A,B) and i s the lower proviso of an i n f e -rence fugure of the form A B f(A,B) From the r e s t r i c t i o n s on the r u l e , we know that there are no undischarged i n i t i a l formulae standing above A or B. Thus we have Y j* A , ¥ f- B , for some (possibly empty) set of provisos ¥. Whence, by the induction hypothesis, i f a* interprets ¥, a*(A) = 1 = a*(B). Thus a*(A)$a*(B), and so a*(f(A,B)) = 1. I t follows that V (s f (A,B) Replacement Rule: P = f(A,B) and i s the lower proviso of an i n f e -rence figure of the form [A] [C] f(B,C) C A f(A,B) Then we know from the r e s t r i c t i o n s on the rule that there i s a set of provisos such that ( i ) $, r |-Q for a l l Q e Y , and also ( i i ) ¥, A (- C and ( i i i ) ¥, C (-A . In addition we have (iv) $, T \- f(B,C) . 197 These results may be obtained by derivations of length less than or equal to n. Now l e t a* be an interpretation which interprets $ u T compatibly. Then, from (i) a* interprets each Q e and, from (iv) a* interprets B and C compatibly. Then, from ( i i i ) (since by the hypothesis ¥, C |s A) , a* interprets A, and ( i i ) and ( i i i ) now y i e l d a*(A)^ a*(C) and a*(C)^ a*(A) ; whence a*(A) = a*(C). Now a*(B)$a*(C) and so a*(A)$a*(B) , i . e . a*(P) = 1. I t follows that <S>, T |= P T r a n s i t i v i t y Rule: P = f(A,C) and i s the lower proviso of an i n f e -rence figure of the form [A] [B] f(A) B C f(A,C) Then, from the r e s t r i c t i o n s on the r u l e , there i s a set of provisos such that ( i ) $ , r (-Q for each 0 e and also ( i i ) ¥, A (- B and ( i i i ) ¥, B |-C . Additionally, we have (iv) $, T |~ f (A) . A l l these results may be obtained by derivations of length less than or equal to n; hence the induction hypothesis may be applied. Let a* be an i n t e r -pretation which interprets $ u T compatibly. Then, from ( i ) , a* i n -terprets each Q e and, from ( i v ) , a* interprets A; whence, by ( i i ) a* interprets B, and so, by ( i i i ) , a* interprets C . Also, since a*(Q) = 1 for each Q e ¥, a*(A)< a*(B) and a*(B)< a*(C). Thus, since a* i s an interpretation within a t r a n s i t i v e p.B.a., a*(A)$a(C). 198 I t follows that a*(f(A,C)) = 1 , and so $, Y f? P . In every case, i f A i s derived from $ u T using a tree of length n+1, then $, V (s A , and i f P i s derived from $ u r using a tree of length n+1, then <J>, r fr P . . Thus, i f the system i s n-sound, then i t i s n+l-sound. By induction, QN i s n-sound for any positive integer n. Since $ , r |»A ( $, T f-P ) only i f there i s a tree of f i n i t e length which constitutes a derivation of A (P) from $ u r , i t follows that QN i s sound. VIII.8 The completeness of QN We now show that QN i s weakly complete, i . e . that f(A) | ; A implies f(A) f- A . Given a set of provisos 0, we may form the set Z of formulae such that VIII. 8.1 A e E i f f $ (- f (A) Note that * e Z and q. £ X for a l l i e N; also B e Z i f f(B) e $ or f(BC,) e $. We now form the $-algebra of Q as follows. Consider a re l a t i o n - on Z such that, for A, B Z VIII. 8.2. A = B i f f both $,A|-B and $, B f-A . Clearly, this r e l a t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e . Let Z/~ be the p a r t i t i o n of Z effected by -, and l e t [A] be the member of E/~ which contains the formula A e Z. Let $ be a rel a t i o n on I./- such that VIII.8.3. [ A ] $ [ B ] i f f $ h f ( A > B > 199 The tree 8.1 below shows that this r e l a t i o n between classes i f indepen-dent of the choice of representative element, i . e . that, i f A - C and B - D, then [A]$[B] i f f [C]$[D]. In this tree and i n other trees displayed i n this analysis, the provisos i n $ are presumed to stand above the formulae or provisos shown. Note also, that i n this case we know from the d e f i n i t i o n of = that the r e s t r i c t i o n on the replacement rule i s observed. [D] [B] f(A,B) [C] [A] f(D,A) Tree 8.1 f(C,D) Since $ |- f (A) and f(A) f (A,A) , i t follows that 0 |- f (A,A) , for a l l A e E. Hence $ i s r e f l e x i v e , and, from the irrelevance of the choice of representative element we see that, i f [A] = [B], then $ |-f(A,B) . The commutation rule shows that $ i s symmetric. Note also that, i f A e E and $, A hB , then, by the deducibility r u l e , $, f(A) f-f(A,B) ; i t follows that B e E, since we also have f(A,B) (- f (B) , and that [A] $ [B]. We set V I I I . 8.4 o = d fn 1 = d f [ 1 ^ I f B i s a formula such that $ |- B , then [B] - [.-)*] , since we also have, from tree 8.2, $ [• V • 200 [ f C ) ] f e n r] Tree 8.2 *V The (derived) synthesis rules show that, i f A, B e T, and [A]$[B], then AvB, A&B are members of Z, while the analysis rules show that the converse also holds. We define operations v and A between compatible elements of E/~ : VIII.8.5 [A] v [B] =,„[AvB] ., , r. l t ! r i l l 1 J L J df L provided [A]$[B]: otherwise [A] A [B] = ..[A&B] , A , , df v and A are undefined. Trees 8.3 and 8.4 suf f i c e to show that, i f A = C, B =D and [A]$[B] , then [A] v [B] = [C] v [D] and [A] A [B] = [C] A [D]. Thus, i n the d e f i n i t i o n above, the choice of representative elements of [A] and [B] i s irrelevant. [A] [B] AvB f(AvB f(C,D) G f(C,D) D f(A,B) AvB CvD CvD _ 0 083 Tree 8.3 CvD A&B A&B f(C,D) Tree 8.4 C&D 83 I omit the derivation of f(B,A) from AvB. 201 We know, by that i f A e Z, then ">A e Z , and, by r e f l e x i v i t y 84 and 1-construction, that [A]$pA]. We write VIII. 8.6 [A]' = d fHA] and show (tree 8.5, below) that our choice of representative element i s irre l e v a n t , i.e. that i f A = B , then ""A - TB . [B] f(A,B) f( nA,B) A "»A f(B,"»A) * Tree 8.5 IB We now prove that VIII. 8.7 ^ 0 = < E y /~ ' v» A» '» °' 1 > l s a t r a n s i t i v e p a r t i a l Boolean algebra. We have already s a t i s f i e d clauses (b), (c), (e) and (f) of the de f i n i t i o n of a p a r t i a l Boolean algebra (VI.4.1); i t remains to show that (a), (d), (g), and (h) also hold, and that fa i s t r a n s i t i v e . (a) Since q^ e Z and e Z , and for any set $ »f provisos, $ 5 f* " (since QN i s sound), we know that E / ~ contains at least two elements. (d) We have, by "-cpmpatibility, [ A]$[A], for a l l A e Z . I t follows from the 1-construction rule that ["]"]$ [A]. (g) Now assume that [A]$[B] , [B]$[C] , [C]$[A]. I t follows from the construction rules that [AvB]$[C], [A&B]$[C] and [*JA]$[B]; whence "84" Here and elsewhere we also use the fact that $ i s symmetric. 202 ([A] v [B])$[C] , ([A] A [B])$[C] and ([A])'$[B]. (h) We define an ordering r e l a t i o n < on E/~, such that VIII. 8.8 [A] < [B] i f f $, A f-B I t i s t r i v i a l to show that the choice of representative element of [A] and [B] i n this d e f i n i t i o n i s i r r e l e v a n t , and that < i s r e f l e x i v e , a n t i -symmetric and t r a n s i t i v e . Thus E/~ i s p a r t i a l l y ordered by <. Further, for any A e E, tree 8.6 shows that [-*-] < [A]; since we know that $, A |- V (by tree 8.5), we also have [A] < ["]"]. I t follows that 0 and 1 are the minimum and maximum elements ofE/~ with respect to ^ . f(A) f(A,~) p ] Tree 8.6 A Now consider formulae A, B, C such that [A], [B] and [C] are pairwise compatible. From (g), by induction on the length of a for-mulae, we can show that a l l Boolean polynomials i n [A], [B] and [C] are pairwise compatible. We now furnish trees, where necessary, to show that for such formulae A, B and C i n E; (1) [A] < [AvB] [B] < [AvB] (2) [A&B] < [A] [A&B] < [B] (3) i f [A] < [C] and . [B] < [C] , then [AvB] < [C] (4) i f [C] < [A] and [C] < [B] , then [C] < [A&B] (5) [AvIA] = [ T ] (6) [A&nA] = [-] (7) [A&(BvC)] < [(A&B0v(A&C)] 203 and claim that these relations together show that the Boolean polynomials i n [A], [B] and [C] generate a Boolean l a t t i c e , and hence a Boolean a l -gebra, under the operations ofv, A and '. (See II.3.10-11, II.4.6, II.4.4*.) Since, by assumption, $ |- f(A,B) , (1) follows d i r e c t l y from the v+ ru l e , and (2) d i r e c t l y from the &- rule. (3) and (4) are obtainable, using the v- and the &+ rules together with the fact that $ f- f(A,B) and $ j- f(B,A). To show (5) we prove that $ [• AvIA. Note f i r s t that tree 8.7, below, shows that (a) f(A)pf(A,1A) (3) f(A ) K f(A , n(Av3A)) (y) f (A) |-f (l(AvlA)) f(A) f(A,A) f(A) fOA,A) f(A,A) f(A) (a) f(A,1A) f(1A,A) f (A,A) f (Av">A,A) f (1(A vlA) ,A) (3) .... f(A , 1(AvlA)) f(T(AvlA),A) Tree 8.7 (y) .... fO(AvlA)) Tree 8.8 now displays the required r e s u l t . 204 f(A) (a) f(A,lA) [A] Tree 8.8 f(A) AvlA [-J(AvlA) ] ( 3 ) f(A) f (A,1(AvlA)) (a) f(A,lA) IA f(A) AvlA [1(Av7A)] (Y) : fO(AvlA)) AvlA Now consider tree 8.9. since ["] = 0, this confirms (6). A&"»A A&1A IA Tree 8.9 The required d i s t r i b u t i o n law i s shown, using tree 8.10. The assump-tions ( 6 ) , ( e ) and (T) are j u s t i f i e d , since we know that $ h f(A,B) , $ j-f(BC,) and $ f" f(C,A) , and we can construct trees to show the following, using only the construction rules and the commutation rule. (6) f (A,B) , f(B,C), f(C,A) \- f(A&B,A&C) (e) f(A,B), f(B,C), f(C,A) L f(B,A&(BvC)) ( x ) f(A B) , f(B,C), f(C,A) [» f(C,A&(BvC)) A&(BvC) A&(BvC) (6) A [B ] j. (6) A [C] ( O (T ) f(B,A&(BvC)) f(C,A&(BvC)) BvC (A&B)v(A&C) (A&B)v(A&C) A&(BvC) f(A&B,A&C) A&B f(A&B,A&C) A&C (A&B)v(A&C) 20 5 Tree 8.10 Note that, at the steps marked the proviso f(A,B) and the proviso f(A,C) have been omitted. We have shown that $ i s a p a r t i a l Boolean algebra of equi-valence classes of the formulae i n £: we c a l l i t the $-algebra of Q. To show that $ ^ i s t r a n s i t i v e , we prove f i r s t that VIII.8.9 [A] < [B] i f f [A&B] = [A] i f f [AvB] = [B] . We show [A] < [B] -> [A&B] = [A] -> [AvB] = [B] [A] < [B]. (1) Assume [A] < [B]. As was noted i n the discussion follow-ing VIII.8.3, this implies $ \~ f (A,B) ; hence [A]$[B] and [A&B] exis t s . T r i v i a l l y , $, A&B f- A . From tree 8.11, $, A |-A&B , and so A [A&B] = [A] f(A,B) A B A&B Tree 8.11 (2) Assume [A&B] = [A]. Then by the analysis rule we obtain $ \- f(A,B). Application of the v+ rule now yields $, A f-AvB, and tree 8.12 shows that $, AvB \- A. Whence [AvB] = [A]. [A] Tree 8.12 8 5 A&B f(A,B) AvB B [B] B In this tree, and i n some of those which follows, I have omitted some provisos which are obtainable t r i v i a l l y from those shown: here f(B,A) i s not displayed. 206 (3) Assume [AvB] = [B]. Then by the analysis r u l e , $ |- f(A,B). Further, since f (A,B) , A f* AvB and (by assumption) $, AvB \- B, i t follows that $, A |-B. Thus [A] < [B ]. Since ^ i s t r a n s i t i v e , we conclude that ^ i s a t r a n t i t i v e p a r t i a l Boolaen algebra. The weak completeness of QN follows straightforwardly; we have VIII.8.10 f ( A ) ^ A implies f(A) f"A. Proof. Consider the formula A. Let $ = (f(A)}, and T.^ be the set of formulae such that B e E ^ i f f $ [- f (B) . Then we may form the $-algebra of Q. Clearly, A e E ^ , and we have [A] = 1 i f f f(A),A (• V and f(A) , 1 ~ j-A i f f f ( A ) , 1 " (- A i f f f(A) |-A. Let be the set of provisos such that P e i f f f(A) f-P. Note that B e E . i f f f(B) e 4\. A A We define a function a* from u E , onto E , / ~ such that A A A a*(B) = [B] for a l l B e E ^ ; a*(P) = 1 for a l l ? e \ Now e E ^ and q± e E ^ for each i e N. Thus, by the construction of the algebra a* i s an interpretation of 0 within $ ^ such that D = V u Z. . a* A A Assume f (A) (i A. Then a*(A) = 1 on every interpretation a* within any Boolean algebra $ , such that A e D .. Thus on the i n t e r -a5* pretation a* within $ defined above, [A] = a*(A) = 1, and so f (A) f* A. 207 Unfortunately, this proof does not extend readily to a strong completeness theorem. We can, however, derive the following. VIII. 8.11 I f $, T (-f(A), then $, A (s A implies $, Y |-A 86 Proof. Let Y be the set of sentences {B,,—,B } . (We assume 1 n that i f ±4j, then B^ 4 B..) We construct the sentence CON(T) as follows. Consider each possible conjunction C.. such that C. = B. & (B. &( ...&B. ) ...) (1 < j < n!) " I l l 1 J 1 2 n By s t i p u l a t i n g some prescribed alphabetical order of the symbols of the language, we can arrange that the numerical subscripts j on the formulae C_. correspond to a systematic ordering of the set {Cj: 1 ^  j ^  n!}. We now write CON(T) = C.&(C_&( ... &C ,) ...) dt 1 z n! We have ( i ) r (*C0N(r) ; ( i i ) CON(T) f- B for a l l B e Y. To derive ( i ) , observe that the requisite provisos for the applications of the &+ rule may be obtained by the assumption rule together with the &-con-struction rule. Also, by the assumption r u l e , ( i i i ) T |- f ^ ^ B ^ ) for a l l B.,B. e r. Now assume that $, r f- f(A). Set ¥ = $ uf ( r ) and consider the set of sentences ~ such that D e - i f f $, f (r) f- f(D) A e E , by assumption, and, by the &—construction r u l e , C0N(r)e £. We form the y-algebra of Q, as before. In this algebra, since [Bi]$[B_.] for a l l B i > B j e r (by ( i i i ) , above), we have (iv) [CON(r)] = A {[B]: B e r } . As before, l e t a* be the interpretation of Q within ^ , such that a*(q ) = [ q ± ] , a*(^) = 0. 86 Recall from Section VIII.2 that we use r to denote a f i n i t e set of formulae. 208 Then $, Y (~ A i f f $, f (r) , r f- A (using ( i i i ) ) i f f $, f ( T ) ,CON(T) (~ A (using ( i i ) ) i f f [CON(r)] < [A] i f f A {[B]: B e Y] < [A] (from (iv)) i f f Aa*(r) < a*(A) Thus, i f for every interpretation a* into any t r a n s i t i v e p a r t i a l Boo-lean algebra which interprets $ u Incompatibly, Aa*(r) < a*(A) , then $ , T L A. The result follows. Thus, given VIII.8.11, i f we show that (*) $, T |=A implies $, Y f- f(A) we w i l l have shown that QN i s strongly complete, i . e . that $, T ^A implies*, r |- A. So f a r (*) has resisted proof, and so the strong com-pleteness of ON remains conjectural. However, we do have a version of the strong completeness result analogous to the weak completeness theorem: we have shown that VIII. 8.12 f ( A ) , $ , r (rA implies f ( A ) , $ r |-A. This may be a l l that can be proved. One obvious move, i n the search for a more general r e s u l t , i s to take the method by which we construc-ted the $-algebra for Q, and to construct a $-Talgebra instead. A simi l a r move was used i n Chapter I I I to prove the strong completeness of CN. However, i f we try t h i s , we soon run foul of the r e s t r i c t i o n s on the t r a n s i t i v i t y and replacement rules, and any adjustments we make to avoid these d i f f i c u l t i e s lead i n turn to further problems l a t e r on. 209 Other strategies prove equally unsuccessful, but by trying them we arrive at a number of interesting theorems about QN. These I discuss i n the next section. VIII.9 The Deduction, Substitution and Replacement Theorems for QN, One obvious strategy, when we seek to move from a weak comple-teness result to a strong completeness theorem, i s to use a deduction theorem. As can be seen from the derived rule for , the deduction theorem i n QN taken the following form: VIII. 9.1 f(A,B),A |- B i f f f (A,B) f- A = B i f f F(A=>B)|-A=> B. We also have VIII.9.2 f (A,B) , A M i f f f (A 3 B) |s A 3 B. To prove the l a t t e r r e s u l t , r e c a l l that f(A,B) and f(A => B) are both interpreted by just those interpretations a* which interpret A and B compatibly. Thus, i f a* i s an interpretation within a p.B.a. which interprets f(A,B), then a* (A) and a*(B) both l i e i n a Boolean subalgebra of fa. Thus, from I I . 5.5, a*(A) < a*(B) i f f (a*(A))' v a*(B) = 1, as required. Now these r e s u l t s , together with the weak completeness theo-rem (VIII.8.10) and the soundness theorem (VIII.7.1), y i e l d VIII.9.3 f (A,B)A (-B i f f f (A,B)A (c B or more generally, VIII.9.4 f(A,B), $, A f B i f f f(A,B ) , < 5 , A (*B. 210 I f the sentence CON(T) i s defined as i n Section 8, then VIII. 9.5 $, T f-B i f f $,CON(r) |~ B $, T \: B i f f $,CON(T) y B And, applying VIII.9.4, we obtain VIII.9.6 f(C0N(r),B), <S>, T |~B i f f f(C0N(r),B), <S>, T f-B which, i n turn, yields V I I I . 9 . 7 f ( B , r ) , $ , r | - B i f f f ( B , r ) , $ , r ( - B . However, this result i s weaker than those already obtained i n Section 7 and 8 (see VIII.7.1 and VIII.8.12). Another route to a strong completeness theorem for a sentential c a l -— 87 cuius i s suggested by the work of Wojcicki ; Wojcicki approaches the topic of completeness v i a a discussion of l o g i c a l matrices (see Ch.III. 4). Given any l o g i c a l matrix M = <A>, I>for p, he can define a corres-ponding "matrix consequence" operator C^ on the power set of the set of formulae of p; this can be related to the operation of deductive con-sequence on this power set, that i s , the operation such that, for any set of formulae of p, A e C (r) i f f r \- A . In f a c t , the proof consists i n showing that, given that certain constraints are met, i f the algebra & i s the algebra of the set of formulae of the language and I i s appropriately chosen, then D,, = C . M D However, for a number of reasons Wojcicki's results cannot be applied to the calculus QN. In the f i r s t instance, as I pointed out —— I owe this suggestion to Bas van Fraassen. I give the barest precis of Wojcicki's work: a f u l l discussion would take us too far a f i e l d . See Wojcicki (1973). 211 i n Section 4 , i t i s doubtful whether we can regard the set of formulae of Q, together with the set of connectives of Q as forming a l algebra of the appropriate kind. Secondly, the constraints, to be met include the requirement that a substitution theorem hold for ON, and, t h i r d l y , Wojcicki makes the related s t i p u l a t i o n that i n a sentential calculus C^(0) must be invariant under substitution, i . e . , that f- A imply |- (B/q_^)A (where (B/q^)A i s the formula obtained from A by replacing each occurrence of q^ i n A by the formula B) . This t h i r d condition i s not met by QN: we have, for any ato-mic formula q^, |. q ^ v l q ^ > b u t not, i n general, f- BvlB. Instead we have f(B)f- BvTB , r e f l e c t i n g the form taken by the substitution theorem for QN, which we now state. Let (B/q^) $, (B/q ^ r be the set of provisos and the set of formulae obtained by replacing each occurrence of q^ i n every proviso i n $ and i n every formula i n T by the formula B. Then VIII.9.8 $ , r ^A implies f(B),(B/q )* , (B/q )T f- ( f i / q ) k -Proof. Assume that $, V f- A. Take the derivation tree T which shows that$, r f-A, and systematically replace each occurrence of q^ i n this derivation tree by the formula B. Then the r e s u l t , T', i s s t i l l a derivation tree i n QN; however, i f the self-discharging proviso f(q.) appears as an i n i t i a l proviso i n T, then the proviso f(B) which appears i n the corresponding place i n T.' w i l l not be discharged. Thus the set of undischarged i n i t i a l provisos and formulae of T' w i l l be a subset of {f(B)} u (B/q i) $ u (B/q i ) r , and so the tree shows that 212 f(B) ,(B/q.)$, (B/q.) T |- (B/q.)A. We also have a Replacement Theorem for QN. VIII.9.9 I f A, B and C are any formulae of Q, q^ any propositional variable, then f (B) ,A \r B and f (A) ,B f- A i f f both (i). f(B),(A/q i)C r (B/q ±)C and f(A),(B/ q i)C H (A/ q i)C and(ii) f(B),f((A/q ±)C) |- f((B/q ±)C) and f (A) ,F( (B/q ±) C) |-f((A/q.)C Proof. The right to l e f t conditional follows immediately, i f we consider the case when C = q.. l Left to r i g h t . Assume f(B),A (- B and f(A),B f- A. We show (i ) by induction on the length of a formula. The length of a formula of Q i s defined as was the length of a formula of 1s" i n the proof of VII.1.1, with "q/' replacing "p , replacing "0" and reference to "1" e l i m i -nated. For economy, we write (A/q.)C = C . In the proof we w i l l 1 A not consider the t r i v i a l case qhen q^ i s not a sub formula of C. In that case ( i ) and ( i i ) hold, obviously. I f C i s a formula of length 1, then C. = A, C = B and ( i ) A B holds from our assumption. Now assume that, for a l l formulae C of length L < n, ( i ) holds. Let C be a formula of length n _ l . Then either (a) there i s a formula D such that L(D) = n and C = or there are formulae D,E, each of length less than or equal to n, and (b) C = DvE or (c) C = D&E. 213 Case (a) : C = By the induction hypothesis i £ " V : Tree 9.1 f(D A) B f (B) ,DA (- DB and f (A) ,Dg h ^ f(D A,D B) I B" By the subformula Theorem f("1D ,D ) D "ID A IJ A A (VIII.6.3), 1D A |-f(A) . f(D»,lDA) A...' 1DB These rel a t i o n s , together with tree 9.1, show that f(B),TD A |- IDg Simi l a r l y f (A) ,1D f Case (b): C = DvE. Note f i r s t , that by the introduction rule and the subformula theorem, D vE f- f(A). Then, from the indue-tion hypothesis, together with tree 9.2, we see that f (B) >D^vE^ f" f ( D B ' V W [Eg] [E A] F (W E A C : B ] [ : A 1 Tree 9.2. f ( E B ' V ' '•; ° A °B f (DB E ) J B From tree 9.2, also, ^ V E A L f ( D A ' E ^ ) > a n d from the induction hypothe-s i s f (B) ,DA J- Dg ; f(B),E j- Eg. These r e s u l t s , together with tree 9.3, show that f(B),C J- Cg, and by a p a r a l l e l route we may arrive at f ( A ) , C B L C A . 214 Tree 9.3 f \] [ EA ] £ ( W D B f(D B,E B) E B £ (W V EA V ! B V E B Case (c): C = D&E A tree s i m i l a r to tree 9.2 shows that f ( B ) , D A & E A f- f(Dg,Eg), and that, together with the induction hypothesis, allows to conclude from tree 9.4 that, i n this case, f ( B ) ,C. |- G„. In A D p a r a l l e l fashion we obtain f(A),D L C . B > A D &E D.&E, A A A A Tree 9.4 D, E. A A F ( D B ' V D B E B D B & E B Tree 9.5, below, shows that ( i ) implies ( i i ) [ CA ] f ( CA> S Tree 9.5 F ( C A ' C B ) f ( CB' CA> f(cB) That concludes the proof of VIII.9.9. 215 VIII.10 The Algebra ^  . I now construct a t r a n s i t i v e p a r t i a l Boolean algebra of equi-valence classes of formulae of Q. The algebra i s more general than the $-algebra constructed i n Section 8, since each formula of Q i s a member of some equivalence class. I t i s the analogue for Q of the Lindenbaum-Tarski algebra for p; even so, i t does not y i e l d a stron-ger completeness theorem than the one we have. A set of formulae of Q i s A-derivable i f f A e T^, We f i r s t introduce the comcept of an A-derivable set of for-mula of Q. VIII.10.1 and, for a l l B e r ^ , ( i ) {f (D) : D e T }, A (-B and ( i i ) {f (D) : D e T^}, B |~ A The set {A} i s A-derivable. Consider an A-derivable set T . A From the ded u c i b i l i t y rule we see d i r e c t l y that, i f B e r , then ( i i i ) {f (D) : D e r } f- f(A,B) , and i t follows from tree 10.1 below that, i f B, C e 1^, then (iv) {f (D) : D e r } f- f(B,C) . Note that the r e s t r i c t i o n s on the replacement rule are observed. In this and subsequent trees, the provi-sos from the set {f(D): D e T^} are assumed but not displayed wherever • the proof requires them. f(A,C) f(C,A) [B] [A] f(B,C) Tree 10.1 216 For any formula A of Q, l e t {T } be the class of A-derivable sets. We define VIII.10.2 [A] = u {T } df A We see immediately that [A] e {T^} . 88 We introduce notarion as follows : VIII.10.3 f[A] = - (f(C): C e [A]} at For every B e [A], since there i s some A-derivable set T £ [A] such that 89 B e T , we have VIII.10.4 f[A],B f- A ; f[A],A f- B ; f[A] (~ f(A,B) This may seem, at f i r s t sight, confusing: we s h a l l need to distinguish between f ( A ) , f [ A ] , and f( [ A ] ) . (Recall that f([A]) = {f(B,C): B,C e [A]} .) However, the dis t i n c t i o n s involved are straightforward, and the notational economies achieved are considerable. Note that f(A) 4 f[A] 4 f([A]) ± f ( A ) , but f(A) 6 f[A]; f(A,A) e f( [ A ] ) . 89 VIII.10.4 and s i m i l a r statements of d e r i v a b i l i t y involve a s l i g h t abuse of notation: the r e l a t i o n $ V |» A was defined between f i n i t e sets of provisos and formulae, $ and r , and a formula A. However, the set f[A] of provisos w i l l be i n f i n i t e . This need not worry us: i n this context alone ( i . e . , wherever the expression " f [ A ] M occurs i n a de r i -v a b i l i t y relation) we may consider the re l a t i o n to be extended to the i n f i n i t e case, so that 0, T [• A i f f there i s some f i n i t e subset ¥ of $ such that A i s derivable from Y u V. Elsewhere I hew to a U n i -tary version of d e r i v a b i l i t y . 217 We can define an equivalence r e l a t i o n = on the set of formulae of Q as follows. VIII.10.5 A and B are Q-equivalent formulae of Q (A = B) i f f (i) f [A],f [B],A \- B and ( i i ) f [A], f [B ], B \. A. To show that = i s an equivalence r e l a t i o n , we prove f i r s t that VIII.10.6 I f A and B are Q-equivalent, then [A] = [B]. Proof. Assume A = B. Then ( i ) f [ A ] , f [ B ] , A/-B and ( i i ) f [ A ] , f [ B ] , B Y A. From ( i ) and ( i i ) we obtain, for an arbitrary D e [B], ( i i i ) f [A], f [B ], A h D and (iv) f [ A ] , f [ B ] , D L A . I t follows that [A] u [B] £ [A]. A p a r a l l e l argument shows that [A] u [B] £ B , and hence that [A] = [B]. It now follows immediately from VIII.10.7 below, that = i s an equivalence r e l a t i o n . VIII.10.7 The family 11 of sets [A], where A i s a formula of Q, pa r t i t i o n s the set of formulae of Q. Proof. T r i v i a l l y , every formula of Q belongs to at least one such set, and each such set i s non-empty. We can also prove that, i f C e [a] and C e [B] , then [A] u [B] c [A], as follows. Assume C e [A] and C e [B]. Let D by any member of [A] u [B]. Then either D e [A] or D e [B]. I f D e [B], we have f[A],A |-C 5 f[B],C h B ; f[B],B hD , and so (i) f[A] u f[B],A KD. S i m i l a r l y , ( i i ) u f[A] u f[B],D Y A. Results ( i ) and ( i i ) are t r i v i a l i f D 6 [A]. Hence we conclude that 218 [A] u [B] e {T }. Whence [A] u [B] c [A], as required. Thus 11 i s a p a r t i t i o n of the set of formulae of Q. (b) ["] and [ T ] are designated elements of Q. We define V I I I .10.8 o = d f r ] ; i = d f n A i We know that |- "1 *, witness tree 10.2. I t follows that, i f A i s a theorem of QN ( i . e , we have f(A) (- (A), then A e [*!"]. [f(A)] Tree 10.2 f(A/0 [A] 1 A (c) We now define a compatibility r e l a t i o n $ on H. VIII.10.9 [A]$[B] i f f f[A],f[B] f- f(A,B) . I t i s easy to show that the choice of representative element i s i r r e -levant. I f C e [A] and D e [B] , then we have f[A],C | A ; f[A],A K C ; f[B],B h D. From the further assumption that f[A],f[B] )-f(A,B), we can see by tree 10.3 that f[A],f[B] |- f(C,D) • D B : : c A f ( A , B ) B D . . Tree 10.3 f(D,A) A f(C,D) $ i s symmetric (obviously), and, since f[A] y f(B,C) for a l l B,C e [A], $ i s also r e f l e x i v e . (e) P a r t i a l operations v, A on J are defined as follows. VIII.10.10 [A] v [B] = f [AvB] provided [A]$[B] ; [A] A [B] = JJT[A&B] otherwise these operations dt are undefined. 219 To see that the choice of representative elements of [A] and [B] i s i r r e l e v a n t , consider the set A of formulae, such that A = {CvD: C e [A] , D e [B]} Let $ = {f(EvF): EvF e A} . We assume that [A]$[B] , and show that, for any CvD e A, (i) $, AvB (-CvD ; ( i i ) $, CvD f- AvB For any G e [A] u [B] , ( a ) $ hf(G) (by v-analysis (commutation) and selection). I t follows, by the assumption of compatibility, that (3) $ f-f(A,B) > $ h f(C,D). Then (i ) follows by tree 10.4, and ( i i ) may be obtained i n p a r a l l e l fashion. ( a ) ( a ) Tree 10.4 [A] [B] (3) ! (3) '. f(C,D) C f(C,D) D (3) f(A,B) AvB CvD CvD CvD From ( i ) and ( i i ) , A c [AvB] , and this i n turn shows that our choice of representative element was i r r e l e v a n t . We use a s i m i l a r strategy to show that, for C e [A], D e [B], we obtain C&D e [A&B] . Let T = {C&D: C e [A] , D e [B]} ; Y = {f(C&D): C&D e T} . Then by a p a r a l l e l argument to the one employed above we show that ( i ) A&B f-C&D and ( i i ) C&D (-A&B . Whence r £ [A&B] , and the required result follows. (f) We write, defining a singularly operation on IT , 220 VIII.10.11 [A]' = [IA]. Again, we employ the strategy used i n (e) to show that, i f [A] = [B], then [TA] = [IB], Assume B e [A]. We set A = {-\c: C e [A]} ; x = {f('1C) : C e [A]} . Since (a) x f" f(C) for each C e [A], and we also have f[A] f-f(A.B) , i t follows that (3) X |*-f(A,B). Then by tree 10.5, x , IA (- IB . (3) [B] f(A,B) Tree 10.5 f(1A,B) f(B,lA) IA VIII.10.12 nB We now prove that ^ 0 N = <1T» $> v» A> ' » 0,1> i s a t r a n s i t i v e p a r t i a l Boolean algebra Proof. of VI.4.1. The s a t i s f a c t i o n of clauses (b), ( c ) , (e) and (f) i s guaranteed by our d e f i n i t i o n s . We need to show that 1(3 Q N s a t i s f i e d clauses (a) - (h) (a) Since QN i s sound, f [ * ] , 1 * )}• ". Thus there are at least two elements of 11, namely ["] and [ V ] . (d) By the ~|-compatibility r u l e , f[A] [ i f ( A , " ) , for any A; using the commutation and T-construction rules we obtain f[A] \- f(A,^~) , for any A. Thus, for a l l [A] e U, we have VIII.10.13 [A]$r] ; [A]$[V] . (g) Assume [A]$[B] , [B]$[C] , [C]$[A] . then [A] v [B] = [AvB] , and, since for any D e [A], E e [B] we have DvE e [AvB], i t follows that f[AvB] f- f(DvE) for any such D,E. Using the v-analysis (commutation) and selection we obtain 2 2 1 f[AvB] f- f(D) for any D e [A]; f[AvB] |- f(E) for any E e [B]. Whence, by the compatibility assumptions, ( i ) f[AvB] J- f(A,B) ; ( i i ) f [AvB],f [C] f- f(B,C) ; ( i i i ) f[AvB],f[C] \- f(C,A) . From ( i ) , ( i i ) and ( i i i ) , v-construction yields f[AvB],f[C] hf(AvB,C). Thus ([A] v [B])$[C]. S i m i l a r l y , ([A] A [B])$[C]. T r i v i a l l y , using then-rule and the compatibility of [A] and [B], we obtain f["\A],f[B] h f(A,B) , and so, v i a the 1+ ru l e , f[1A],f[B] h f OA.B) . Whence [TA]$[B] . (h) I f [A]$[B] , [B]$[C] , [C]$[A] , then the Boolean polynomials i n [A], [B], [C] forma Boolean algebra with minimum 0 and maximum 1 . Proof: We show f i r s t , for any A, V I I I . 1 0 . 1 4 [AMA] = [•*-] ; [AvIA] = ft-*-] . Consider the sets T = {A&1A,"} ; $ = {f(A&1A),f(")} Then (i) <S>,A&1A[- * ( t r i v i a l l y ) , and ( i i ) $, A [• A&1A (by tree 1 0 . 6 ) f(A&iA) Tree 1 0 . 6 f(A&*lA,-N-) A&""A Thus T c [-] and so [A&lA] - ["] . We know from trees 8.7 and 8.8 that Av A i s a theorem of QN. Hence (from(b) , above) AvIA e ["!-*•], i . e . [Av7A] = [*J-*»] Now assume that [A], [B] and [C] are pairwise compatible. Using (g), above, we can easily prove by induction that any two Boolean polynomials i n [A], [B] and [C] are compatible. The axioms of Boolean 222 algebra can be shown to hold for any three such elements of U; i t w i l l s u ffice to show that they hold for [A], [B], [C]. In each case, i f d,e are Boolean polynomials i n [A], [B], [C], to show that d = e we prove that [D] = [ E ] , where D and E are the formu-lae of Q which correspond to the polynomials d and e. Since, as we have seen, any two Boolean polynomials i n [A], [B], [C] are compatible, we know that this correspondence exists ( i . e . that [A] A [ B ] = [A&B], etc.). To prove the ide n t i t y [D] = [ E ] we show that D and E are Q-equivalent (see VIII.10.5). Now consider the following cases: (1) D = AvB E = BvA (2) D = A&B E = B&A (3) D = Av(BvC) E = (AvB)vC (4) D = A&(B&C) E = (A&B)&C (5) D = (A&B) vB E = B (6) D = (AvB)&B E = B (7) D = Av(B&C) E = (AvB)&(AvC) (8) D = A&(BvC) E = (A&B)v(A&C) (9) D = (A&U)vB E = B (10) D = (Av">A) &B E = B VIII.10.15 (Lemma 1) I f [A], [B] and [C] are pairwise compatible, Then D and E are Q-equivalent i n each of the cases (1) - (10) above. 223 Proof. Assume that [A], [B], [C] are pairwise compatible. To show that D E E i n each of the cases ( l ) - ( 8 ) , we apply theorem VIII.6.5 (q.v.). In each of these cases we can show that D° \- ^ E° and E° U „D°. Further, we can produce economical derivation trees to ' N show each of these r e s u l t s , such that, i n cases (1), (2), (5) and (6) every formula occurring i n the tree i s constructed from A° and B° using the connectives & and v, and i n cases (3), (4), (7) and (8) every for-mula i s constructed from A 0, B° and C° using these connectives. Now we know from (e), above, that i f G e [A] and He [B], then GvH e [AvB] and G&H e [A&B]. Whence, by selection and analysis, we obtain, i n cases ( l ) - ( 8 ) , f[D],f[E] hf(G) for a l l G e [A]; f[D],f[E] f- f (H) for a l l H e [B], and i n cases (3), (4), (7) and (8), ad d i t i o n a l l y , f[D],f[E] |- f(I) for a l l I e [C]. We again invoke the compatibility condition; from this assumption f[A],f[B] M ( A , B ) ; f[B],f[C] hf(B,C); f[C],f[A] V f (C,A) and these r e l a t i o n s , together with those obtained above, y i e l d , i n cases (D-(8) f[D],f[E] M(A,B) and, i n cases (3), (4), (7) and (8), additio n a l l y , f[D],f[E] f~f(B,C) f[D], f[E] H(C,A) The construction rules now guarantee that f[D],f[E] h f ( J , K ) , where, i n cases (1), (2), (5) and (6), J,K are any formulae constructed from A and B, and i n cases (3), (4), (7) and (8), J , K are any formulae con-structed from A, B and C. 224 Thus i n cases (l)-(8) there are economical derivation trees to show that D° I- „E° and E° U_TD° , and for each pair of formulae J , K occurring i n any such tree we have f[D],f[E] j- f(L,K). By theo-rem VIII. 6.5 i t follows that, i n cases (l)-(8) f [D] ,f [E] ,D V* E and f [D] , f [E] ,E pD and so, i n each of these cases the formulae D and E are Q-equivalent. Case (9). e We use the fact that [A&1A] = [«*•]. Then, by (e) , [(A&lA]vB] = P"vB], and i t i s t r i v i a l to prove that f [-^vB],f [B] ,-A.vB V-B and f [>-vB] ,f [B] ,B h-^vB Case (10) From the identity [AvnA] = [-|-*-] we obtain, s i m i l a r l y , [(AVA)&B] = [T^&B], and again the Q-equivalence of 1*-&B and B i s easily shown. Thus i n each of the ten cases D and E are Q-equivalent. Use of VIII.10.6 now concludes the proof of clause (h). Since clauses (a)-(h) of VI.4.1 are s a t i s f i e d , i s a par-t i a l Boolean algebra. We define an ordering r e l a t i o n < on II, such that VIII.L0.1 [A] < [B] i f f (i) f[A],A hB and ( i i ) f[A] hf(A,D) for each D e [B ]. Note that, i f [A] < [B], then ( i i i ) f[A] V f(D) for each D e [B]. The proof i s immediate using the commutation and selection rules. We use ( i i i ) to show that the choice of representative element i s i r r e -levant i n the d e f i n i t i o n of <. For assume C e [A], D e [B], f[A],A hB, F [A] h f (A,E) for each E e [B ]. Then f[B],B |-D, and so, 225 by ( i i i ) , f[A],B Y D. But also f[A],C h A and, by assumption, f[A],A|-B; whence f[A],C |~ D. Further, since C e [A], we have f[A]C|-A, f[A]jA:}-C, and so, given f[A] }- f(A,E), by the replacement rule we obtain f[A] h f(C,E). (See tree 10.1, replacing C by E, B by C i n the tree.) The r e l a t i o n < i s cle a r l y r e f l e x i v e . I f [A] < [B] and [B] < [A], then f[A],f[B],A |~B and f [A[ ,f [B] ,B h A. Thus A and B are Q-equivalent, and [A] = [B]. This shows that < i s antisymmetric. To show t r a n s i t i v i t y , assume that [A] < [B] and [B] < [C]. Then, by ( i i i ) , f[A] f-f(D) for a l l D e B, and since f[B],B YC, we obtain f [A] ,B |-C. But, by assump-t i o n , f[A],A l-B, and so (i) f[A],A h C. Further since, f[B] Y f (E) for a l l E e [C]: then f [ C ] , C f-E, and by the result we have just o b t a i — ned f[A],C |-E. We have already shown that f[A],B f-C, and so we know that f[A],B |-E, and by our i n i t i a l assumptions, that f[A],A |—B. We now apply the t r a n s i t i v i t y rule (see tree 10.7), to obtain, for any a r b i t r a r i l y chosen member E of C, ( i i ) f[A] h f(A,E). From (i ) and ( i i ) [A] < [C]. [A] [B] Tree 10.7 ! f(A) B E f(A,E) Thus nis p a r t i a l l y ordered by <. VIII.10.17 [A] < [B] i f f [A] A [B] = [A] i f f [A] v [B] = [B]. We prove that [A] < [B] + [A] v [B] = [B] -> [A] A [B] = [A]+ [A] * [B]. (1) Assume [A] < [B]. Then [A] and [B] are compatible from ( i i ) , and [A] v [B] = [AvB]. We show that B and AvB are Q-equivalent; then, by VIII.10.6 i t follows that [B] = [AvB]. 226 Since, by assumption, f[A],A |-B, from tree 10.8 we see that f (A,B) ,f [A] ,AvB f-B. But we know that f[A] j- f (A,B) ; thus f[A],AvB Y B. By (e), CvD e [AvB], for a l l C e [A], D e [B] whence f[AvB] Y f(CvB) for a l l C e [A], and so, by the v-analysis and selec-t i o n rules, f[AvB](-f(C) for a l l C e [A]. Thus f[AvB],AvB |~B, and so f[AvB],f[B],AvB YB. Since f [AvB ] (- f (A,B) , we have, t r i v i a l l y , f [AvB] ,f [B] ,B |-AvB , and so B and AvB are Q-equivalent. f[A] A Tree 10.8 f(A,B) AvB B B B (2) Assume [A] v [B] = [B]. Then [AvB] = [B]. To prove that [A] A [B] = [A], we show that A&B and A are O-equivalent. I t i s obvious that f [A&B] ,f [A] ,A&B Y A. By assumption, f[B],AvB|-B, and so, from tree 10.9, f (A,B) ,f [B] ,A Y A&B. But f [A&B ] j-f (A,B) ( t r i v i a l l y ) , and, by the argument used i n (1), above, we see that f[A&B] Y f(D) for each D e [B]. Thus f[A&B],f[A],A Y A&B, and so A&B and A are Q-equivalent. f(A,B) A Tree 10.9 A v B f [ B ] f(A,B) A B A&B 227 (3) Assume [A] A [B] = [A]. Then [A] = [A&B]. We now show that ( i ) f[A],A hB, ( i i ) f[A] (-f(A,D) for a l l D e [B]. By our assumption, f[A],A h A&B. Since A&B |*B, ( i ) follows immedia-tely. To show ( i i ) , r e c a l l from (e) that, for a l l D e [B], A&D e [A&B]. Then, i f D e [B], from our assumption f[A] f- f(A&D,A&D). By selection and &-analysis we obtain ( i i ) . Thus [A] < [B]. Theorem VIII.10.17, together with the t r a n s i t i v i t y of ^, shows that ^ Q N i s a t r a n s i t i v e p a r t i a l Boolean algebra. VIII.11. The System QL. The system QL i s a sequent calculus which resembles the system CL. Using the d e f i n i t i o n of sequent given by VIII.2.7, and the d e f i n i t i o n of a sequent tree given i n Chapter I I I . 6 , we define a derivation of QL as follows. VIII.11.1 A sequent tree constitutes a derivation of a sequent §, r + A i n the system QL i f f (i) $, r -*• A i s the end sequent of the tree; ( i i ) a l l i n i t i a l sequents of the tree are of the form A -> A or " -»-; ( i i i ) each inference figure i s an instance of one of the schemata displayed below. (I) Structural. D i l u t i o n $, r -> A $, r -> A f ( A , A) , $, r A , A r , A A 228 Cut (i) 90 ?, r ->• A , A f(c ,D ),f(A,r), *,r -> A , r, A - > A $, r A $ , f ( r ) , r ' , A ^ A ( i i ) f(A, r ' ) , $, r -> A where £' c r (II) Operational. &+ >, r - > A , A >, r • > B , A &- <s>,r , A - > A f (A,B) , $, r + A&B, A f ( B , r ) ,f ( A , r ) , $, r, A & B - > A $, r, B - > A f ( A , r ) ,f ( B , D , $ , T , A & B - > A v+ >, r - A , A f ( B , A ) ,f (A,B) , $, T -> AvB,A $ T By A f (A, A ) ,f (A,B) , $, T AvB , A v- $, T,A -> A $, r,B -»- A f(A,T) , f ( B , D , *, T,AvB^ A -1+ $ , r , A ->• A f(C rD A),f(A,D,f(A,A),$ , r -> "1A,A «, r - > A , r f ( c rD A), $, r, nA - A (III) Proviso Manipulation. Commutation f (A,B) , <2>, T -* A Analysis f(B,A), $, r -> A f(A,B), *, r + A f (AvB) , $, T -> A Selection f(A), $, r -> A f (A,B) , $, r -> A f (A,B) , $, r -> A f (A&B) , $, r -»- A 90, The formulae C„ , D. are defined below. r A 229 f (A,B) , $, T -> A f (nA,B) , $, r -> A Construction f(A&B,C), <S>, T ->- A Elimination f(A,B),f(B,C),f(C,A), $, r A f (AvB,C) , $, T -> A f(A,B) ,f(B,C),f(C,A) , * , r -> A f (nA,B) , $, r A f(A,B), $, r + r Replacement f (A,B) , $, T -> A *,f(D,A-»-C $,f (r) ,C -> A f(B,C), o, r A T r a n s i t i v i t y f(A,C) , *, T -»• A *,f(r),A+B <S>,f(r),B + C f (A) , $, T -> A Di l u t i o n $, T -> A ^ - c o m p a t i b i l i t y f ( A , " ) , $, T -> A f (A), o, r A f (A ) , $, r + A (weak) f ( A , B ) , $ , T , A , B ->- A (strong) f (q ±) , $ , T + A <s> r A , B - > A $ r A VIII.11.2 i s a derived rule of QL i f f $' ,r*, -> A ' we can construct a sequent tree whose i n i t i a l sequents are a l l either axioms of QL, or of the form $, T -> Aand whose end sequent i s $ ' I" ->A' , and i n which each inference figure i s an instance of one of the above schemata. 230 Operational rules for " 3 " appear as derived rules: 3 + $ T,A + B , A f(C rD A),f(A,B),f(A , r),f(A,A), $, T + A 3 B , A D- $ , r -> A , A $ , r , B -> A f ( C r D ^ , f ( A , D , f ( B , r ) , * , r , A 3 B + A We also have two derived proviso manipulation rules involving " 3 * . Analysis f ( A , B ) , $ , T -> A f (A 3 B ) , $ , r -> A Construction f(A 3 B,C), $, r -> A f ( A , B ) , f ( B , C ) , f ( C , A ) , $, T + A There are further derived rules corresponding to those of QN. Ref l e x i v i t y f (A,A) , $, T -»- A f(A), $, r -> A Deducibility f ( A , B ) , $ , T -> A $,f(r),A -> B f ( A ) , * , r -> A Negation rules f (A) , $, T -* A f O A ) , $ , T ->- A f O A ) , $, r •> A f(A), <s>, r -> A Synthesis f (A&B) , $, T -> A • f (AvB) , $, T -> A f(A sB) , $,T -> A f (A,B) , $, T + A f (A,B) , $, T -> A f (A,B) , $, T A Elimination O ) f P ) , $, T -»- A $ , r A We now define C , D . Let = {A,,...A } , T A 1 n A = {B-J...B } . We assume that, i f i 4 j , then A. 4 A. and B. 4 B.. 231 VIII.11.3 C f = A1&(A2&(...&An)...) D, = A..v(B„v(. . .vB ) .. .) A 1 2 m Since we may index the members of r and A i n n! and m! different ways, respectively, C^ , and are not uniquely defined by a given r and A . However, where £(C^,T)^) appears i n a derivation tree, the choice of indexing functions i s ir r e l e v a n t , as I w i l l l a t e r show. We discuss f i r s t what i t i s for a proviso to be absorbable. VIII.11.4 A proviso P can be absorbed by $ u T ( $ u T can absorb P; P J L S absorbable by$ uT ) i f f , for any set A of formulae of Q, p, $, r •> A VIII.11.5 VIII.11.6 $, r -> A i s a derived rule of QL. $ u T can absorb ¥ i f f $ u r can absorb each P e T P i s absorbably by Ou T i f f P i s absorbable by $ u f ( r ) . Proof. Clearly, i f P u $, or P = f(q^) , or there are formulae A,B e f (not necessarily d i s t i n c t ) and P = f(A,B), then (by the elim-ination rules i n the second and t h i r d cases) P can be absorbed ny O u r - In these cases we say that P can be immediately absorbed by $ u T- S i m i l a r l y , y can be immediately absorbed by $ u Y i f f each proviso i n ¥ can be immediately .absorbed by $ u T. Now, .with the ex-ception of the-proviso d i l u t i o n rule and-the elimination rules, a l l the proviso manipulation rules enable us to; replace a. single proviso P "either by..another proviso P^ or by three provisos P^> P2' P3* These in turn may be replaced by further application of the proviso manipula-232 l a t i o n rules. In general, we say that a set ¥ of provisos replaces P i f f , for any $, Y , A , p, o, r A v, $ , r -> A i s a derived rule of QL. By inspection of the derivation rules, we see that ( x ) P can be absorbed by * u Y i f f either P can be immediately absorbed by $ u V, or some set ¥ of provisos which replaced P can be immediately absorbed by $ u T. From ( x ) i t follows immediately that P i s absorbable by $ u f(T) i f P i s absorbable by $ u Y. We can show the converse e a s i l y , given the proviso d i l u t i o n and weak elimination rules. To confirm ( x ) , note that use of the structural and opera-t i o n a l rules does not allow us to absorb provisos, as the following argument shows. For l e t us assume that there i s a proviso f ( A , B ) , and A I T , f ( A , B ) i $. Thus f ( A , B ) cannot be immediately absorbedby $ u r However, we may, by using the d i l u t i o n r u l e , or the &- or v-rule or a combination of these, be able to derive a sequent f ( A,B) , $ ' , r ' , A A from f ( A,B) , $, Y A . In this case, i f B e l " (or i f B = A ) , f ( A , B ) can be immediately absorbed by $', I" , A . In other words, i t may be the case that f ( A,B) , $ , T ->- A $ ' , r ' , A ->• A ' 233 i s a derived rule of QL. R e c a l l , however, that, i f f(A,B) i s to be absorbed by $ u T. , then f (A,B) , $, I" $, r -> A must be a derived rule of QL, for any A . Whence, i f we are to absorb f(A,B) by the method outlined above, we need to show 0' , I" ,A - A ' $, r + A to be a derived rule of QL. But, once a formula A has appeared i n the antecedent of a sequent i n a derivation tree, i t can only disappear from the antecedent by an application of one of the cut rules, or by application of the "! + rule. Inspection of these rules shows that, i n each case, the set of provisos f(A,r') appears i n the antecedent of the lower sequent after an application of the rule. Further, B e T', and so f(A,B) e f ( A , r ' ) . Thus we have f a i l e d to absorb f(A,B). This shows that by applying the rules of QL to members of V we cannot absorb any proviso which could not have been absorbed other-wise; neither can we do so by applying proviso manipulation rules to members of $. For any such rules which we do apply w i l l need to be reversible ( l i k e the commutation r u l e , or the analysis, replacement and t r a n s i t i v i t y r u l e s ) , and any absorbtion which takes place as a result of applying the rule could equally be achieved by applying the reverse rule to the proviso which i s being absorbed. 234 That concluded the proof of VIII.11.6. VIII.11.7 I f $ r -»- A i s a derivable sequent, then f ( A ) can be absorbed by $ u r . To prove t h i s , we can use induction on the length of a derivation tree of QL. Clearly, for any axiom, A -> A, the result holds; inspection of the rules shows that, i n each case, i f the result holds for the upper sequent(s), then i t holds for the lower sequent. The proof i s straightforward and i s omitted. Note however, that i n the case of the cut rules, and of the &-, v- and T+ rules we need to use VIII.11.6 as a lemma, and that each of the proviso manipulation rules i s dealt with i n the same way: I take the commutation rule as an example. Assume that P e f ( A ) can be absorbed by f(A,B), $, T ; then tree 11.1 shows that P can be absorbed by f(B,A),$ , r . P,f(B,A), $, r -y A Tree 11.1 f(A,B),P,f(B,A), $, r + A f(A,B),f(B,A), $, r + A f(B,A), $, r + A The theorem may be restated as follows. VIII.11.8 We have as metarules of QL, f(A,B), 4, r + A $ , T •+ A,B, E $, r -+ A f(A), $, r -»• A $, r - > A , E WhereE i s any $, r -> A ( f i n i t e set of formulae. 235 These are the rules of "elimination by deduction" We now examine again the proviso f(C^,T)^) which appears i n the cut rule and the netation rules of OL. As before, l e t = {A ,...jA^}, = {B^,...,B ml . Then for r and A we construct indexed sets of formulae {CL >r , { D r } A . such that each member of £C.} i s the form 3 r C. = A. &(A. &(...&A. )...) (1 ^  j ^ n!) J 1 2 n and each member of {D }, i s of the form r A D = B. v(B. v(...vB. )...) . (1 < r < m!) r X l X2 We stipulate additionally that, i f i 4 i, , then A. 4 A. and B. 4 B. ; cl D 1 1 , 1 1-. a b a b further, i f j 4 k, then C. 4 C , and 3 k i f r 4 s, then D 4 D . r s We now prove the following theorem. VIII.11.9 f(C,,D ) i s absorbably by f(C.,D ), f(r ) ,f(A) for a l l k s j r j,k (1 < j,k < n!) and a l l r,s (1 < r,s < m!) To prove this we observe f i r s t that the following lemma holds: for any j,k (1 < j,k < n!), f(T),C. — C i s a derivable sequent; likewise, for 3 k any r,s (1 < r,s < m!), f(A),D r ->- Dg i s a derivable sequent. Ap p l i -cations of the replacement rule now y i e l d the desired result. Proof of Lemma: Inspection of the d i l u t i o n rules shows that (a) f(T ) , r •+ T and f(A), A ->• A are derivable sequents. By induction, using the construction rules we can show that, i f C i s any multiple conjunction involving some or a l l of the formulae of ( i . e . any formula of the form 236 A. &(A. &(...A. ) . . . ) , then (b) f(T) can absorb f(A.,C) (1 < i < n). X l X2 X a 1 S i m i l a r l y , i f D i s any multiple disjunction involving some or a l l of the formulae of A , then (c) f ( A ) can absorb f(B i SD) (1 < i < m). We use (a) and (b), together with applications of each of the &+ and &- rules to show that f(r),C. C, i s a derivable sequent 3 k (1 < j,k < n!), and (a) and ( c ) , together with applications of each of the v+ and v- rules to show that f ( A ),D -> D i s a derivable sequent r s (1 < r,s < m!) . Now consider sets $, r, A as they appear i n the (schemata of the) cut rule and the twonegation rules of OL. In each of these rules a proviso of the form f(C_.,Dr) appears i n the lower sequent. We know that (a) $ u T can absorb f(r) , by the weak elimination rule; we can also show that (b) i n each of the three cases, the set of the remaining provisos and formulae i n the antecedent of the lower sequent can absorb f ( A ) , as follows. Case ( i ) (Cut Rule). The upper right sequent of the inference figure i s of the form $, r -> A,A and i s a derivable sequent. From VIII.11.7 we know that $ , T can absorb f ( A u {A}); whence $,r can absorb f ( A ) . Case ( i i ) ( -*VRule) . The proof i s sim i l a r . Case ( i i i ) (1+ Rule). 237 Since $, T , A -> A i s a derivable sequent, we know that f ( A ) i s absorbable by $, T , A . Then, by VIII.11.6, f ( A ) can be absorbed by $ , f ( A , T ) , f ( T ) , and hence, by VIII.11.6 again, f ( A ) can be absorbed by $ , f ( A , r ) , r. We can now show that: VIII.11.10 I f f(Cj,D r) appears i n the lower sequent S of an inference figure as a result of the application of the cut rule or one of the negation rules, we may derive from S another sequent S', l i k e S but containing f(C. ,D ) where S contains f(C.,D ), for a l l j,k k' s 3 r (1 < j,k < n!) and a l l r,s (1 < r,s < m!) Given VIII.11.9, and the results (a) and (b) above, tree 1.2 serves as a proof of this r e s u l t , i n the case of the " I - rule. The other cases can be proved s i m i l a r l y . $, r -> A , A f ( c.,D r), $ , r , 1A •» A f(C.,D ) ,f(C, , D o ) , f ( D , f ( A ) , * , r . l A ^ A j r K s  f(C k,D g) , f ( D , f ( A ) , $ , r , 1 A A f ( c k , D g ) , $, r . i A ^ A Theorem VIII.11.10 casts into a formal mould the statement made e a r l i e r that, when f(Cp,D A) appears i n a derivation tree, the choice of indexing functions of r and A i s irrelevant. 238 VIII.12 The Soundness of QL. The system QL i f sound, that i s , VIII.12.1 I f $, T -> A i s a derivable sequent, then $, T \z k . Proof. To prove this i t w i l l s u f f i c e to show (i) a l l the axioms of QL are Q-valid sequents; ( i i ) a l l the inference rules of QL preserve Q-validity, i.e • 5 i f the upper sequents i n an inference figure are Q-valid, then so i s the lower sequent. VIII.12.1 w i l l then follow from (i) and ( i i ) by induction on the length of a tree. i f $, r -> A i s the lower sequent of the inference figure, we c a l l an interpretation a* which interprets $ u r compatibly (within a t r a n s i t i v e p.B.a.) a relevant interpretation. In each case, also, we assume the upper sequents of each figure to be Q-valid. D i l u t i o n Rules ( i ) Di l u t i o n on the right. $ u T compatibly, whence, by the assumption, a* interprets A compatib-l y , and since a* also interprets f(A,A), a* interprets {A} u A compa-t i b l y . Whence, again using the assumption, A a * ( r ) < v a*(A) < a*(A) v v a*(A). Thus f(A,T),$ , T -> A,A i s a Q-valid sequent. (i) holds, t r i v i a l l y . We prove ( i i ) by cases. In each case, Let a* be a relevant interpretation. Then a* interprets 239 (2) D i l u t i o n on the l e f t . Let a* be a relevant interpretation. a* interprets {A} u $ u r compatibly; thus i t interprets O u r compatibly, and so, by assumption, a* interprets A compatibly. Also, a*(A) A A a * ( D < A a * ( r ) < v a*(A) . Thus $, r,A -»- A i s a Q-valid sequent. Cut Rule ( i ) . Let a* be a relevant interpretation. Since a* i n t e r -prets $ u T u f(A,T) u{f(C r,D A)} compatibly, then (a) a* interprets $ u T compatibly, and (b) a*(A) i s compatible with each member of a*(T). Thus a* interprets $ u V u {A} compatibly, and so, by assumption, a* interprets A compatibly. Also we have, using (a) and the assumption, a*(A)$(va*(A)). Again using the assumption, we see that both A a * ( D < a*(A)v va*(A) and Aa*(T) A a*(A) ^  v a*(A). Now a*(A) i s compatible with both A a*(T) and va*(A). Also, since a* interprets f ( C r , D A ) , ( A a *(T))$(va*(A)). Thus we may apply II.5.7 to obtain A a*(T) < va*(A) . I t follows that f (Cr,D ) , f (A,T) , $, T -> A i s a Q-valid sequent. Cut Rule ( i i ) . ' Let a* be a relevant interpretation. Then a* interprets f(A,T') u $ u r compatibly. Now r ' c r , and so a* i n t e r -prets $ u f(T ) u r ' u {A} compatibly. By the induction hypothesis, <&,f(r), r',A |:A: whence a* interprets A compatibly and A a * ( r ' ) A a*(A) < va*(A) 240 By the induction hypothesis again, <S>, T \z A, and so Aa*(r) < a*(A) We see that Aa*(r) = Aa*(T) Aa*(A) < Aa*(r') A a*(A) < v a*(A) I t follows by t r a n s i t i v i t y that f ( A , r ' ) , 0, r -> A i s a Q-valid sequent. In treating the operational rules, I appeal several times to results from Chapter I I . These r e s u l t s , of course, are for Boolean algebras. We may none the less apply them to p a r t i a l Boolean algebras i f we know that the requisite compatibility conditions are met, that i s , provided that wherever an expression of the form "av b" or "aAb" occurs i n the course of the proof of a given result (where a and b are Boolean polynomials), we know that a$b. &+ Rule. Let a* be a relevant interpretation. Since a* interprets $ u T u {f(A,B)} compatibly, a* interprets {A,B} compatibly; whence A&B e D a & - ~y t b e assumption, since a* interprets $ u T compatibly, a* interprets both {A} u A and {B} u A compatibly; thus a* interprets {A&B} u A compatibly. Further, we have both Aa*(T) < va*(A) v a*(A) and A a*(T) < v a*(A) v a*(B) By inspection of the proof of 11.5.8(b), we see that we do not need (Aa*(T))$ a*(A) or (A a * 0)$a*(B) i n order to apply this result to the present case. I t follows that Aa*(T) < Aa*(A) A (a*(A)A a*(B)) = v a*(A) va*(A&B) . Thus f(A,B), $, T -* A&B,A i s a Q-valid sequent. 241 &- Rule. (Ideal with one case only; the other follows, by p a r a l l e l reasoning). Let a* be a relevant interpretation. Then a* interprets $ u T u {A} compatibly, and so, by the assumption, a* interprets A compatibly. Further, since a* interprets f(A,B), a*(A&B) = a*(A)Aa*(B). We have seen already that a*(A)$(Aa*(F)) and a*(A)$a*(B). Since a* interprets f(B,T) we also have a*(B)$(Aa * (T )). Thus we can apply I I . 5.6(b) to obtain Aa*(T)Aa*(A&B) = A a * ( D A (a*(A) A (a*(A) A a*(B)) < Aa*(r) A a*(A) < v a * ( A ) . We see that f (B, T) , f (Ar) , $, T,A&B -> A i s a Q-valid sequent. v+ Rule. Let a* be a relevant interpretation. Then a* interprets $ u T compatibly, {A,B} compatibly and {B,D} compatibly for a l l D e A . By our assumption, i t follows that a* interprets {A}u A compatibly; thus a* interprets {AvB} u A compatibly. Also, Aa*(r) < a*(A)v v a * ( A ) < (a*(A) v a*(B)) v v a*(A) = a*(AvB)v v a * ( A ) Thus f (B, A ) , f (A,B) , $, T, •+ AvB , A i s a Q-valid sequent. v- Rule. Let a* be a relevant interpretation. Then a* interprets 0 u T u {A} and $ u T u {B} compatibly. Thus, from the assumption, a* interprets A compatibly, and further, both Aa*(T) Aa*(A) < v a8 ( A ) and A a*(T) Aa*(B) < v a*(A). The compatibility relations which hold enable us to employ 11.5.8(a); whence 242 Aa * ( r ) Aa*(AvB) = Aa*(T) A(a*(A) v a*(B)) < v a * ( A ) . We conclude that f(A,T),f(B,T), $, T,AvB -> A i s a Q-valid sequent. 1+ Rule. Let a* be a relevant interpretation. Then a* i n t e r -prets f(A,T) and so a*(A) i s compatible with a l l members of a*(T). Also, since a* interprets $ u T compatibly, a* interprets $ u T u {A} compatibly; thus, by the assumption, a* interprets A compatibly. Also by assumption, Aa*(T) A a*(A) < va* ( A ) . We have already established that (Aa*(T))$a*(A). We know that a* interprets A compatibly, and since a* also interprets f(AA ), i t follows that (v a * ( A))$a*(A). In addition, since a* interprets f ( C r , D A ) , we have (Aa*(T))$(va*(A)). Thus we may apply II.5.9 to obtain Aa*(T) < v a*(A) v (a(A))' = v a * ( A ) v a * p A ) . Thus f(C fD A) ,f (A,T) ,f (A.,A) , $, T + nA, A i s a Q-valid sequent. 1 - Rule. Let a* be a relevant interpretation. Then a* interprets $ u r compatibly, and so, by assumption, a* interprets {A} u A compatib-l y , and a. f o r t i o r i , a* interprets A compatibly. Also i t follows that a*(A)$(va*(A)); since a* interprets f ( c r 5 D A ) w e have (Aa*(r)) $(va*(A)) ; t h i r d l y , since a* interprets r u compatib-l y , we have (A a* ( r) ) $a* (A) . By assumption, Aa*(r) < va*(A) v a*(A). The compatibility relations required i f we are to apply II.5.9 are s a t i s -f i e d , and so, using that theorem and II.2.12. Aa * ( r ) A a*(lA) = A a * ( r ) A (a*(A))* < va* ( A ) . Thus f(C ,D ), $, TjlA A i s a Q-valid sequent. 243 Provisos Rules. We deal separately with the replacement and t r a n s i t i v i t y rules. To show the other proviso manipulation rules sou,d we observe that any relevant interpretation a* interprets the provisos of the upper sequent of each rule. This follows from the recursive d e f i n i t i o n of an i n t e r -pretation (VIII.3.1) and the d e f i n i t i o n of a p a r t i a l Boolean algebra (VI.4.1). Replacement Rule. Let a* be a relevant interpretation. Then a* interprets $ u V compatibly, whence a* interprets $ u f(E ) . Further, since a* interprets f(B,C), a* interprets C. Now, by assumption, $,f(T),C — A i s Q — v a l i d , and so a* interprets A and a*(C) * a*(A). Also, by assumption, $,f(r),A + C i s Q-valid, and so a*(A) < a'-(C). Thus a*(A) = a*(C). But a*(B)$a*(C). We see that a*(A)$a*B), and so a* interprets f(A,B) and the result follows. T r a n s i t i v i t y Rule. Let a* be a relevant interpretation. Then a* interprets $ u r compatibly, and so a* interprets 0 u f ( r ) . Further, since a* interprets A, and, by assumption, $,f(F),A •+ B i s Q-valid, a* interprets B and a*(A) < a*(B). Again, since $,f(T),B -> C i s Q-valid, a* interprets C and a*(.B) < a*(C). By t r a n s i t i v i t y of the p.B.a., a*(A) < a*(C), and so a*(A)$a*(C). Thus a* interprets f(A,C) and the result follows. We conclude that QL i s sound. 244 VIII.13 The Completeness of QL. We show that QL i s weakly complete: the proof i s s i m i l a r to that of the weak completeness of QN, and i s here given i n outline. Let 0 be a set of provisos of 0. We define a set L of for-mulae of Q as follows. VIII.13.1 A e E $ i f f , for any T, A f ( A ) , 0, T -»- A $, r -> A i s a derivable rule of QL. Note that q. e Z. for a l l i e N, and " e E^, "l * e E. Consider the r e l a t i o n - on Z l 5 such that VIII. 13.2 A = B i f f $,A -> B and $,B -»• A are both derivable sequents of QL Clearly - i s symmetric, and, since A A i s an axiom of QL, i t i s r e f l e -xive. We can use the cut rule ( i i ) to show that - i s t r a n s i t i v e . Thus - i s an equivalence r e l a t i o n on E^ and by means of i t we may effect a p a r t i t i o n of E . Let [A] be the equivalence class containing the for-mula A. We define a re l a t i o n of compatibility on thus. VIII.13.3 [A]$[B] i f f f (A,B) , $, V •+ A $, r -> A i s a derivable rule of QL, where T and A are arbitrary sets of formulae of Q. 245 This i s reflexive (by the r e f l e x i v i t y rule) and symmetric (by the commu-tation r u l e ) . We can use the replacement rule to show that the choice of representative elements of [A] and [B] i s irrelevant. For A,B e E , AvB e E $ and A&B e E $ i f f [A]$[B]. We de-fine p a r t i a l operations, v, A , on E / = : VIII.13.4 [A] v [B] = [AvB] provided [A]$[B]: otherwise these operations [A] A [B] = -[A&B] are undefined, dt T r i v i a l l y , A e L i f f ^A e I . and we write, $ $ VIII.13.5 [A]' = .[1A] dt thus defining a unary operation on T.^/-. A l l three of these operations are w e l l defined, i n the sense that the specified equivalence class i s independent of our choice of representative element of [A] and [B], We can now show that VIII. 13. 6 ^ $ = < E<j/~ ' V ' A>'' f ' t* 1"! 5* i s a t r a n s i t i v e p a r t i a l Boolean algebra. The ordering r e l a t i o n < on this algebra i s such that VIII.13.7 [A] < [B] i f f $,A +B i s a derivable sequent. Now consider the case when $ = f(A). Let a* be the i n t e r -pretation of Q within $ $ such that a*(B) = [B], for a l l B e E $ and B e D . . i f f B e L . a* $ Then f (A) -> A i s a derivable sequent i f f f(A) , 1 " A i s a derivable sequent i f f a*[l~] = a*[A]. I f f(A) -»- A i s a v a l i d sequent, than, for any interpretation a* within any t r a n s i t i v e p.B.a., i f a* i n t e r -prets A, then a*[A] i s the maximum element of that algebra. We con-clude that 246 VIII.13.8 f(A) |= A implies f(A) ->- A i s a derivable sequent. VIII.14. QL as a L o g i s t i c Calculus. Prawitz has suggested that Gentzen's L-calculus for c l a s s i c a l l o g i c may be regarded as a metacalculus for the de d u c i b i l i t y r e l a t i o n 91 i n the N-calculus CN . A s i m i l a r claim cam be made for QL with respect to ON. Since both QN and QL are weakly complete, we already know that VIII.14.1 f(A) h A i f f f(A) |= A i f f f(A) + i s a derivable sequent. The two p r i n c i p a l theorems of this section generalise this r e s u l t : they show that $, T J-1 A i f f $, T -> A i s a derivable sequent. In fact, each derivable sequent of QL expresses a de d u c i b i l i t y r e l a t i o n of QN. Notice, however, that, although we may have $, V f-i P for some proviso P, we may never derive a sequent with a proviso i n the succedent. Indeed, by the d e f i n i t i o n of a sequent of 0, there are no such sequents. Thus QL i s concerned only with the d e r i v a b i l i t y of formulae from premisses, given certain provisos. And this i s surely appropriate: the provisos serve to express conditions that must be met i f a d e d u c i b i l i t y r e l a t i o n i s to hold; i t i s only i n c i d e n t a l l y , as i t were, en route to the derivation of a formula, that we derive provisos i n QN. _ See Prawitz (1965), p.90. 247 The theorems which follow show the equivalence of the two c a l c u l i . VIII. 14.2 $, T f-A implies $, r -»• A i s a derivable sequent. Proof. The proof i s by induction on the length of a derivation tree of QN. I f A i s the end formula of a derivation tree of QN of length 1, then $ i s empty and T = {A} . In this case we know that A + A i s a derivable sequent, since A -* A i s an axiom of QL. Now, as the induction hypothesis, assume that whenever a d e r i -vation of A from $ uT exists constituted by a tree of QN of length less than or equal to n, $, T 4 i s a derivable sequent. Further, assume that, for some formula A, a derivation of A from sets $, V of provisos and formulae exists constituted by a tree of QN of length n+1. We show that, i n this case also,$ , T -»• A i s a derivable sequent. We know that, since A i s the end formula of a derivation of QN of length greater than 1, A i s the lower formula of an inference figure which i s an instance of one of the following schemata: &+, &--f+, v-, "+, . The upper formulae of this figure are a l l end for-mulae of trees of length less than or equal to n. We consider f i r s t the cases when the upper provisos of this figure are a l l i n i t i a l pro-visos of the tree. 24 8 Case &+. A = B&C and i s the lower formula of an &+ inference f i -gure. By assumption, f(B,C) e $, and by the induction hypothesis, $, T -> B and $, r -»• C are both derivable sequents. Then by the &+ rule of QL, $, r + A i s a derivable sequent. Case &-. A i s the lower formula of an &- inference figure, with upper formula A&B. Then, by the induction hypothesis, $, T -»• A&B i s a derivable sequent. Tree 14.1 shows that A -> A $, T -> A i s a f (A) ,f (B) ,A&B -> A derivable sequent. Note f (A,B) ,A&B A „ -, , -, n Tree 14.1 that I have omitted a f (A&B) ,A&B -»• A commutation step, and have A&B -> A telescoped the applications $, T -> A&B $,f(T),A&B -»- A of the proviso d i l u t i o n f (A&B) , $, T -> A $, T -> A&B rules. $, r -> A The other &- rule i s dealt with s i m i l a r l y . Case v+. A = BvC, and i s the lower formula of a v+ inference figure with upper formula B. By assumption, f(B,C) e $ (thus $ can absorb f(C)) and by the induction hypothesis $, r ->• A i s a derivable sequent. Then, by the v+ rule of QL, $, T -> A i s a derivable sequent. The other v+ rule i s dealt with s i m i l a r l y . 249 Case v-. A i s the lower figure of a v- inference figure. Then there are formulae B, C and a set A of formulae, such that A c r and (i) by assumption f(B , A ) c $ and f(C , A ) c $, and ( i i ) by induction hypothesis $, T ->• BvC, $, A,B -»- A and $, A,C A are a l l derivable sequents. $, A,B -> A $, A,C -> A Tree 14.2 $, A,BvC -»• A $. T BvC $,f ( D , A,BvC A f (BvC,A) , $, r - > A f(B,C),f(B , A ),f(C , A ) , $, r - A f(B,C), $ , T ->• A f (BvC) , $ , r-> A $ , r -> BvC ** $ , r - > A Tree 14.2 now shows that 0, r A i s a derivable sequent. Note that at * we use the generalised form of the v-construction rule of QL, and at ** the metarule of elimination by deduction. Case A = and i s the lower formula of a "+ inference figure. Then there i s a formula B such that, by the induction hypothesis, both $, T -»- B and <£>, r + IB are derivable sequents. Tree 14.3 now shows that $, T * i s a derivable sequent. 250 B B Tree 14.3 f (B) ,B,1B -> f(B),f("),B,1B + f(B),B,1B », r -> B $, r -> TB f (B,~1B) ,B&"»B f(B,^B), $, r -y B & 1 B f ( B & 1 B ) ,B&~iB f(B), $, T -> B & 1 B $ , T -> B B&^B $, r B S - B $,f(r),B&iB -> " $, r -+ ~ Case "- (i ) A = *lB and i s the lower formula of a "- inference figure. Then, by our assumption, f(B,T) c $ and, by the induction hypothesis, 0, T,B + " i s a derivable sequent. Tree 14.4 shows that $, r ->- ^ B is a derivable sequent. s r ,B " $,f(r u { B } ) , Tree 14.4 $ ' F ' B f ( c r ) , f ( B , r ) , $ , r -+ I B $, r ->• ^ B : Note that the construction and weak elimination rules guarantee that f(Cp) i s absorbable by T. Case ~- ( i i ) A i s the lower formula of a "-inference figure of the second kind. By assumption, f(A,T) c $, and, by the induction hypothesis, $, r, "*A -> * i s a derivable sequent. Tree 14.5 now shows that $, T -> A is a derivable sequent. 251 -> * A -> A s r , i A + * $ , f ( r ) , ~ + f(A,1A) -> 1A,A $, r, nA - f ( A , 1 A ) , " H A - A f(c ) , f ( A , ), *, r -*• - ^ A T « A - » • A Tree 14.5 $' F * ^ A »,f(r),TlA*A <s>, r - > A Note that (at*) I have omitted the steps by which ~ » T A absorbs f ( A , 1 A ) . We now deal with the case i n which the upper provisos of the inference figure are not i n i t i a l provisos of the tree. That i s , within the framework of our overall inductive proof we now show (by induction) that any such upper proviso can be absorbed by $ u V. Note that we are retaining the o r i g i n a l induction hypothesis, and the assumption that a derivation i n QN of A from $ u V exists constituted by a derivation teee of length n+1. From this assumption, i f P -is any proviso occurring as an upper proviso i n the inference figure of which A i s the lower for-mula, then $, T f-P, and the tree constituting the derivation of P from $ u T i s of length less than or equal to n. Thus we are assuming the induction hypothesis to hold for any limb of such a tree which branches from a formula B. The result we want i s proved by showing the general result that, provided the o r i g i n a l induction hypothesis holds, i f a derivation of a proviso P from $ u V e x i s t s , constitured by a tree of QN a length no greater than n, then $ u r can absorb P. 252 We show this by induction on the length of the derivation of P. I f the length of this derivation i s 1, then P e $ or p = f ( q ^ , for some propositional variable q^. In either case, P can be absorbed by <£> u I". Now assume as a secondary induction hypothesis that, whenever we have a derivation of a proviso P from $ uT constituted by a tree of length no greater than k (k < n - l ) , $ u T can absorb P. Further, assume that, for some proviso P, $, r \- P, and a derivation of P from $ u T exists constituted by a tree of length k+1. Then P i s the lower proviso of an inference figure which i s an instance of one of the thirteen proviso rules. Since there are rules of QL exactly corresponding to the f o l -lowing rules of QN, a l l these cases can be dealt with s i m i l a r l y : ^ - c o m p a t i b i l i t y , selection, commutation, analysis and construction. As an example I take the case of an &-construction rule. P = f(A&B,C) and i s the lower proviso of an &-construction inference figure of QN. Then we know that $, V' |» f (A,B) , $, Y f-f(B,C) and $, T (*• f (C,A) , and, by the secondary induction hypothesis, $ u T can absorb f(A,B), f(A,C) and f(C,A). From the &-construction rule of QL, $ u r can absorb P. We now treat the remaining four cases. 253 (Introduction) P = f(A) and i s the lower proviso of a proviso introduction figure. Then $, r \- A, and, hy the o r i g i n a l induction hypothesis,0, Y p A i s a derivable sequent. I t follows immediately from VIII.11.7 that $ u V can absorb f(A). (Assumption) P = f(A,B) and i s the lower proviso of an assumption figure. Then we know that since no undischarged i n i t i a l formulae stand above A, either A e T, or ¥ [• A and $ u Y can absorb V. S i m i l a r l y , either B e V, or x (" B and $ u T can absorb x (where x I s a set of provisos). I f A e V and B e T, then, by virtue of the weak elimination rule of QL, $ u V can absorb f(A,B). Assume that A I Y, i . e . , that f | - A and $ u Y can absorb ¥. Then by the o r i g i n a l induc-tion hypothesis, ¥ A i s a derivable sequent. Tree 14.6 now shows that, i n this case, f(A,B) can be absorbed by $ u Y provided that $u Y can absorb f(B). f n + n f (A,B) , 0 , T ^ A " ¥ -> A f ( A , B ) , T, *, r -> A f, $,f(T),A ¥, * , f ( r ) , "7" ->- A f ( B , T ) , V, o, r -> A f (B) , ¥, $, r -> A f ( B ) , $, r -* A Tree 14.6 Now either B e Y, i n which case f ( B ) can be absorbed by $ u Y, or x | ~ B and $ u T can absorb x- In the l a t t e r case also, f ( B ) can be absorbed by $ u Y, as tree 14.7 shows. 254 f ( B ) , 4 , r •+ A f D -»-. f(B, A) , 4 , r + A . + x - » • B f ( B , ~ ) , x , 4 , r - A x. * , f ( r ) , B - » • T A x, 4 , f ( r ) , n ~ + B f r , i A ) , x , « , r -> A f C ) , x , * , r -+ A X , 4 , r ->• A Tree 14.7 4 T -> A Thus i n each case 4 u T can absord f(A,B). (Replacement) P = f(A,B) and there i s a proviso f(B,C) such that 4 , T f-f(B,C), and also a set ¥ of provisos such that Y,A f-C and Y,C L. A. Note that, from the secondary induction hypothesis (a) 4 u T can absorf f(B,C). From the o r i g i n a l induction hypothesis, (b) y,A •> C and (c) YjC -»- A are derivable sequents. Since some members ¥ of may be derived from r by the assumption rule (see the r e s t r i c t i o n s on the replacement rule of QN), we need to invoke the secondary i n -duction hypothesis to make good the claim that (d) $ u r can absordT . Given (a), (b), ( c ) , and (d), tree 14.8 shows that 4 u A can absorb f ( A , B ) f(A,B), 4 , T -> A ¥,A + C V,C - A f(A,B), T, 4 , T -> A T, 4,F(T),A -> C Y, $,F(r),C - A f(B,C), Y, 4 , T + A Tree 14.8 4 , T A 4 , T -> A 255 (Transitivity) P = f(A,C); we have §, T |— f(A) , and there exist a formula B and a set ¥ of provisos such that ¥,A |»-B and Y,B f- C . As i n the previous case, from our various induction hypotheses we know that $ u T can absorb both f(A) and ¥, and that ¥,A ->• B and T,B -> C are derivable sequents. We can now construct a sequent tree, s i m i l a r to tree 14.8 but using the t r a n s i t i v i t y rule of QL where that used the replacement rule , to show that $ u T can absorb f(A,C). Thus i n each case, i f a derivation i n QN of P from 0 u Texists of length k+1 (k < n-1), then P i s absorbable by $ u V. Whence, given the o r i g i n a l induction hypothesis, i f $, T |- P by virtue of a deriva-tion tree of length no greater than n, then $ u T can absorb P. Returning to the o r i g i n a l induction step, we see that, i f $, T |~ A by virtue of a derivation tree of length n+1, then 0, r-* A i s a derivable sequent. Since a l l derivations i n QN are of f i n i t e length, i f follows that 0, T \~A implies $, T -> A i s a derivable sequent. To prove the converse of VIII.14.2 we need to consider the general case of a sequent^, r A where A can contain any ( f i n i t e ) number of sentences. Let A be the set of sentences {B,,...,B} . I m (We assume that, i f i ^ j , then B 4- B.). We construct the sentence DIS 72 (A) as follows . Consider each possible disjunction D. such that J "92 Compare the d e f i n i t i o n of CON(T) i n Section VIII.8 256 D. = B. b(B. v(...vB. )...) (1 < j < m!) 2 X l 12 Xm and, i f i 4 i , , then B. 4 B. a b i x , a b By s t i p u l a t i n g some alphabetical ordering of the symbols of the language Q we can arrange that the numerical subscripts j on the formulae D_. correspond to a systematic ordering of the set {D_.: 1 ^ j s m!}. We now write: VIII.14.3 DIS(A) = £ D.,&(D_&(.. .&D .)...) dt l z m! and also DIS(0) =,£ " where 0 i s the empty set. dt We can now show that VIII. 14.4 i f $, T -> A i s a derivable sequent, then $, T |- DIS(A) . I give the outline of the proof, which i s by induction on the length of a sequent tree. T r i v i a l l y , the result holds for each axiom of QL. To prove the induction step we assume that the result holds for the upper sequent(sO of a permitted inference figure and prove that a derivation tree of QL can be constructed to show that i t holds for the lower sequent also. The proof i s again t r i v i a l i n a l l cases but those of d i l u t i o n on the r i g h t , c u t ( i ) , &+, v+, + and -. A s p e c i f i c example w i l l show 73 the strategy of the proof i n these s i x remaining cases Let = C,D and the inference figure i n question be an &+ figure of QL. Then we assume that $, r L DLS. {A,C,D} and $, r pDIS{B,C,D}. Via the analysis, selection and commutation rules of QN, this yields 93 I employ here the handwinkenbeiweismethode, to give i t i t s German t i t l e . 257 f(A,B), $, T \-V f o r a 1 1 P e f(S u {A,B } ) Whence, by the construction rules we get (*) f(A,B), $, T |-f(E,F) Where E and F are any formulae we can construct from A,B,C and D, using the sentential connectives of Q. By repeated applications of the &- rule of QN we obtain from our assumptions. 0, T f-Av(CvD) $, T f-Bv(CvD) From (*) and VIII.6.5 i t follows that f(A,B), $, Th (A&B)v(CvD) Similarly f (A,B) , $, Y |- (A&B)v(dvC) f(A,B), 0, r |-Cv(Dv(A&B)) f(A,B) $ T |- Cv((A&B)vD) f(A,B), $, T (-Dv((A&B)vC) f(A,B), $, T Dv(Cv(A&B)) Using (*) again, together with the &+ rule of QN, we obtain f (A,B) , $, T f-DIS {A&B,C,D} as required. The following i s an immediate corollary of VIII.14.4. VIII. 14.5 I f $, T -y A i s a derivable sequent, then 0, V \- A . Notice that VIII.14.6 $, T -v A i s a derivable sequent i f f $, T -> DIS (A ) i s a derivable sequent. Proof. The l e f t to right conditional i f given by VIII.14.4 and 258 VIII.14.2. To prove the converse, we f i r s t use the d i l u t i o n rules, and the v- and &- rules of QL, together with the construction rules where necessary, to show that f(A),DIS(A) A i s a derivable sequent. Then the analysis, selectionand commutation rules allow us to derive the sequent f(DIS(A)), DIS(A) -> A. Whence (by selection and weak elimination) we see that DIS (A) -»- A i s a derivable sequent. Now as-sume that $, r -> DIS (A) i s a derivable sequent; i t follows by cut rule ( i i ) that $, T -> A i s a derivable sequent. I t i s simple to show that, for any interpretation a* of Q within any t r a n s i t i v e p a r t i a l Boolean algebra, VIII.14.7 DIS(A) e D . i f f A c D . a* — a* and, i f A c D then a*(DIS(A)) = va*(A) From VIII.14.2-7, i t follows that VIII.14.8 QL i s strongly complete i f f QN i s strongly complete. I t i s not known whether either of the cut rules of QL can be eliminated. The method of proof which Gentzen employed successfully for a c l a s s i c a l sequent calculus f a i l s for cut rule ( i ) , and, i n view of the fact that i n Q the semantic consequence operator i s not a closure operator (see Section 3), i t seems unlikely that this r u l e , at least, i s eliminable from QL. 259 Chapter IX. The Logical Systems of Kochen and Specker. IX.1. The System LS. The quantum log i c developed i n the previous chapter i s based on the work of Kochen and Specker, and the systems I have presented may be compared with those they produce; these are described i n two papers, "Logical Structures A r i s i n g i n Quantum Mechanics" and "The Calculus of 94 P a r t i a l Propositional Functions". I w i l l refer to these systems as LS and PP1, and discuss each i n turn. LS uses, as primitive connectives, a single binary connective "v" and the singulary connective "~". Other connectives, "&", "=>" 95 and are defined i n an orthodox way. There i s a denumerable set of propositinal variables. Each member of this set i s a formula of the language, as i s any formula constructed from a f i n i t e number of them using the connectives. Such formulae I w i l l c a l l the c l a s s i c a l formulae df the language. In the proof theory, the symbol "IT" i s added to the language, and any expression of the form ""il (A^,... jA^) 1 , where A^,...,A are c l a s s i c a l formulae, i s also a formula of the language. I w i l l c a l l formulae of the l a t t e r type quasiformulae when I need to distinguish them from c l a s s i c a l formulae. Only c l a s s i c a l formulae and quasiformulae are formulae of the language. Note that only c l a s s i c a l formulae can be 94. These are Kochen and Specker (1965a) and (1965b), respectively. 95. Here and elsewhere I adjust Kochen and Specker's notation to align i t with my own. 260 linked with the sentential connectives. We write £ for the set of c l a s s i c a l formulae, Z* for the set of a l l formulae of the language. Like the formulae of Q , the ( c l a s s i c a l formulae of this language may brought, v i a an interpretive function, into correspondence with the elements of a p.B.a. B . However, rather than associating with each sequence a* of elements of B a function mapping a set of formulae into B, Kochen and Specker associate a function [A] with each formula A of their language, which takes as arguments sequences of members of B , and maps these sequences onto elements of,B. The two procedures are formally related ina very simple way, as follows. For c l a r i t y , I w i l l here distinguish between a sequence a* of members of 5 , and the associated interpration of Q, denoting the l a t t e r by [a*]. Let A be a c l a s s i c a l formula, a* a sequence of members of B . ' Then IX.1.1. a* e Dom[A] i f f A e Dom[a*] (= D ) i f a* e Dom[A] , then [A]a* = [a*]A There are no functions associated with the quasiformulae, and accordingly they play no semantic role i n the lo g i c . Since, for c l a s s i c a l formulae, the correspondence betwen.the two procedures i s exact, we may discuss the semantics of LS i n terms of i n t e r -pretations within p a r t i a l Boolean algebras. For Kochen and Specker, a c l a s s i c a l formula i s v a l i d i n a p.B.a. B_ i f f a*(A) = 1 for every i n t e r -pretation a* within that p.B.a. which interprets A. A i s q-valid 261 i f f A i s v a l i d i n a l l p.B.a.'s. We may compare d e f i n i t i o n VIII.3.8, under which A i s v a l i d i f f i t i s v a l i d i n a l l t r a n s i t i v e p.B.a.'s. In accordance with the discussion at the end of Chapter VIII.3, the more gen-e r a l d e f i n i t i o n of q - v a l i d i t y given by Kochen and Specker goes togther with a more r e s t r i c t e d d e f i n i t i o n of semantic entailment (or "consequence", i n their terms). I f r c E* , then A i s a consequence of r'( T | A ) i f f a*(A) = 1 for every interpretation a* within any p.B.a. such that a*(B) = 1 for each c l a s s i c a l formula B e.T. One difference which may be noted i s that Kochen and Specker do not require that a p.B.a. have more than one element: the i r d e f i n i t i o n allows the degenerate case when 0 = 1 . This i n turn means that, for them, {q^, ~q^} r*A holds degenerately, as i t were, rather than vacuously. The quasiformulae appear i n the system of proof, LS. Let T be a set of c l a s s i c a l formulae. XI.1.2 A sequence G%. ... ,C of formulae i s T-admissible i f f , for I n a l l i , 1 < i < n , (1) there i s a formula A e r which contains B as a subformula, and C^ = 1I(B,B) or (b) there i s a formula A e which contains B^ v B^ as a subformula, and C^  = H(B^,B2) or (c) there ex i s t indices i , , ... , i where 1 < i , < i 1 m k (k = l,...,m), and C^ follows from C^ ,...,C. 1 m by one of the following rules. R l 1T(A1,... ,An) (where 1 < i , j < n) U(A1,Aj) 262 R2. If (A- ,A1) ,IT (A_ .A ) ,... ,f (A. ,A ) ,... ,fl(A ,A ) 1 1 1 Z ' x 3 n n 11^,... ,An) (The premise consists of the n 2 formulae 1f(A^,A..) such that 1 < i , j < n) R#. H(A 1,A 2) , A2-<-^ A3 iKA^A^ R4. H(~A1,A2) iKA-^) R5. ir(A 1 5A 2,A 3) H(A 1vA 2,A 3) SI. 11 (A- ,...,A ) • 1 n 3(A 1,...,A n) where 3 ( q ^ , . . . , 0 ^ ) i s a C-valid formula. S2. A± A± => A 2 T2 IX.1.3. A i s q-provable i f f there i s a sequence of formulae C^,...,Cn which i s {A}-admissible and contains A. As Kochen and Specker point out, the rule SI i s a shema of schemata: they suggest that i t could be'replaced by a f i n i t e number of ordinary schemata, for instance tose corresponding to the axioms of P r i n c i p i a Mathematica. The notion of d e r i v a b i l i t y i s derived rather d i f f e r e n t l y . I f T u{A} c "E* , then IX.1.4.. - TY A i f f there i s a sequence of formulae C^,...,Cn> such that A = C and, for a l l i , 1 < i < n , n 263 (•a) C± € r or (b) CL = ^(q^'l^) » f° r some propositional variable, ; or (c) there ex i s t indices, i 1 5 . . . , i , where 1 < 1, < i , 1 m k (k = l,...,m) and C. follows from C. ,...,C. 1 m by one of the given rules. In this proof theory the quasiformulae act much l i k e the provisos of Q. The quasiformula 1F(A^,...,A ) takes the place of the set f(T) of provisos of Q (where T = {A^,...,An? ): as R l and R2 show, H(A^,...,A ) i s deductively equivalent to the n 2 quasiformulae IKA^A ) ( 1 < i , j < n) taken together. The rules R l - R5 of LS allow us' to perform manipulations on quasiformulae which correspond to those performed on the provisos of S by the proviso rules of QL and QN: R l and R2 license moves corresponding to those permitted by the commutation and selection rules, while R3, R4 and R5 correspond d i r e c t l y to the replacement, ~-analysis and v-construction rules. Of course, there i s no rule of LS corresponding to the t r a n s i t i v i t y rules of QN and QL. Kochen and Specker show that LS i s sound and weakly complete: i . e . that A i s q-provable i f f A i s q-valid. They also claim that the system i s strongly complete, i . e . that, for a c l a s s i c a l formula A , r \- A i f f A i s a consequence of T. However, this claim may be challenged. Let T = {q^,~q)} , A = q^ . Then degenerately, as we have noted, r [= A . But consider a derivation of q^ from {q^,~q^} . The formula i s not C-valid, nor can i t be derived from a set of C-valid formulae involving only q^. Thus q 9 can only appear i n the proof as a result of the application of 264 S2 (modus ponens). We must assume then, that the derivation includes a c l a s s i c a l formula with as subformula, and that the major premise of the f i n a l application of S2 either involves both q^ and , or i s i t s e l f derived from some formula which involves them both. However, to introduce any formula of this kind into a derivation of LS, we require H (<l-j_»* ' B u t ^^1*^2^ i s neither available as an assumption, nor may we derive i t , since i t s derivation would involve recourse to R3, which i n turn would requi-e us to have e a r l i e r derived a formula i n both q^ and q 2 , and our a b i l i t y to do this l a s t i s the point at issue. We conclude that r \~f-A , and hence that LS i s not strongly complete. IX.2. The Calculus PP1. In their second paper, Kochen and Specker use a language with denumerably many propositional variables, a propositional constant "±" and one binary connective "=>" The propositional constant, which i s to be read as the absurd sentence, has two functions. F i r s t , we may use i t to express negation: we define ~A == A o i , and then define connectives "v" and "&" i n an dr orthodox way. I f we do t h i s , the set of formulae of this l o g i c i s the same as that of Q. Secondly, "l" may also be used to express the content, of a proviso of Q. Semantically, the proviso f(A) i s precisely equivalent to the formula i A of Q, and f(A,B) to X => (A = B) , as I now show. C l a s s i c a l l y , J L = A i s true for any formula A , but i n quantum log i c l =>: A i s only interpreted provided A i s interpreted; 265 i n f a c t , a* i s an interpretation within a p.B.a. which interprets A i f f a* interprets x 3 A i f f a*(x 3 A) =1 i f f a*(f(A)) = 1 i f f a* interprets f(A) . S i m i l a r l y , a* i s an interpretation within a p.B.a. which interprets A and B compatibly i f f a* interprets l = (A = ) i f f a*(± => (A 3 B)) = 1 i f f a* interprets f(A,B) i f f a*(f(A,B)) = 1 . We may also note, that i n QN f (A) \- ± 3 A , 1 3 A|- f (A) and f (A,B) h ± 3 (A => B) , ±3 (A3B) h f (A,B) . As with LS, the semantics of PP1 can be discussed without d i s t o r -tio n i n terms of interpretations within p a r t i a l Boolean algebras. Kochen and Specker define the notions of q - v a l i d i t y and semantic e n t a i l -ment as i n their f i r s t paper, and also show that A i s q-valid i f f ± 3 A| A (compare VIII.3.15). In the proof theory of PP1 there are seven rules of inference, as follows. IX.2.1 R l Af- i 3 A R2 1 3 (A 3 B) L. j. 3 B R3 (A 3 x):3 x h A R4 x 3 (A 3 B) f- A 3 (B 3 A) R5 1 3 (B 3 C) , 1 3 (A 3 B) , x 3 (A 3 C) |-(A 3 (B 3 C) ) 3 ((A 3 .B) 3 ( A 3 C) ) R6 A , A 3 Bf- B R7 1 3 (A 3 B) , B 3 C , C 3 B |- x, 3 (A 3 C) Then a rule A ,... ,A f- C i s a derivable rule of PP1 i f f . . there 1 m n :  exists a sequence C-,...,C of formulae of PP1 such that each C I n x ( i < n) i s either one of the formulae A,,...,A or follows from 266 formulae C. ,...,c. ( i . < i ; j = l,...,k ) by one of the rules R1-R7 X l \ 2 A formula A i s provable, i n PP1 i f f X => A h A i s a derivable rule of PP1 Inspection of these rulse shows that R l acts l i k e the assumption rule of QL, and R2, R4 and R7 l i k e the selection, commutation and replace-ment rules of QN. R3, R4 and R5 may be compared with the axiom system 96 for propositional l o g i c suggested by Church: A l A => (B => A) A2 (A 3 (B 3 C)) 3 ((A 3 B) 3 (A => C)) A3 ((A a J.) = x) a A R6 i s of course modus ponens. Kochen and Specker show that A i s provable i f f a i s q-valid. Further, the calculus PP1* obtained by adjoining to R1-R7 the axiom schema l 3 q^ i s claimed to be strongly complete, that i s , i t i s claimed that r^=A implies that i f - A i s a derivable rule of PP1*. To show that the additional axioms are needed, Kochen and Specker argue as follows. Clearly 1 f° q^ holds i n a l l p a r t i a l Boolean algebras, as 97 there i s no a* such that [ l ] a * = 1 . However, i f - " ^ i s ..not a derivable rule (of PP1). For, i f the variable q^ does. not.occur i n the premise of the rules R1-R7, neither does i t occur i n the conclusion. 11— A i s therefore only derivable.for formulae A not containing q^. 96 See Church (1956) p.72. (He states them as axioms rather than as schemata.) 97. Or, as we would say, there i s no interpretation a* such that a*(i) = 1 . Note that Kochen and Specker here overlook the degenerate case, but the argument goes through none the less. 267 But now consider the following. We can say that and q^ are connected i n a formula i f f A = q^ q^ or A = q^ => q^ or q^ => q^ occurs as a subformula of A or q^ = q^ occurs as a subformula of A . Inspection of the rules shows that q^ and q^ can be connected i n the conclusion of a rule only i f they are connected i n one of the premises, or (R7) one of them i s interderivable with a t h i r d formula which i s con-nected to the other i n a premise. Now q^ i s not interderivable with any formula not containing q^ , by soundness), so that we see that q^ and q 2 can be connected i n the conclusion of a sule i f they are connected i n one of the premises or i f one of them i s connected to a formula containing the other i n one of the premisses. Since q^ and q^ are connected neither i n any of the axioms, nor i n the formula l, q^ q^ i s not derivable from l . But on Kochen and Specker*s semantics, l| q^ => q^ . ' Thus the system PP1 i s not strongly complete. IX.3 Conclusions. The sa l i e n t points of difference between the Kochen and Specker systems and those developed i n Chapter VIII may now be summarised. As we have observed, the semantic notions that Kochen and Specker use, . p a r t i a l although the consider interpretations within all/Boolean algebras rather than l i m i t i n g attention to t r a n s i t i v e p.B.a.'s, confine attention to those interpretations which assign the unit element to the formula being interpreted. Their d e f i n i t i o n of semantic entailment i s on that account less general than that adopted i n Chapter V I I I . 3 , and i s open to the c r i t -icism made i n that section of the proposed d e f i n i t i o n V I I I . 3 . 9 ( a ) . Also, they define no notion comparable to that of an admissible valuation of 268 Q,(see Chapter VIII.4); i n fact they use the phrase "A i s true" (under a certain interpretation a* ) synonympusly with "a*(A) = 1". In terms of proof theory, the elegance of their systems PP1 and PP1* contrasts with the comparatively cumbersome machinery of QN and QL. In p a r t i c u l a r one can admire the economies derived from the dual function of the absurd sentence i n these c a l c u l i . However, this formal elegance i s achieved at the cost of some i n t u i t i v e immediacy. Among the aims of adopting a natural deduction system are those of making each move of a deduction i n t u i t i v e l y plausible and of rendering the dedctive force of each connective e x p l i c i t . In the case of quantum l o g i c the f i r s t of these aims may not be f u l l y r e a lised, but i n a system l i k e QN or QL the role of meaningfulness conditions i s v i s i b l e at each step of teh deductive process. Note f i n a l l y that the counterexamples offered to the claims of strong completeness of LS and PP1* do not speak against the strong com-pleteness of QN and QL, since, given the d e f i n i t i o n of semantic e n t a i l -ment we employ, q n~q n (=?*q9 and . J l | t A l 1 => q 0 269 Chapter X. The Nature of Quantum Logic. X.1 Jauch 1s Views. Greechie and Gudder conclude a paper e n t i t l e d "Quantum Logics" by l i s t i n g four tasks which they consider "of prime importance". The l a s t of these i s the question I now turn to: i t i s "to explain 98 the meaning of the word logic i n the t i t l e of this paper." Logic, as generally understood, seeks, i n t e r a l i a , to systematise our notion of what i t i s for an argument to be v a l i d and to make e x p l i c i t what i s involved when we say that one proposition follows from another: at r i s k of c i r c u l a r i t y we can say that i t deals with the l o g i c a l relations that hold between (sets,- of) sentences. Does quantum l o g i c address i t s e l f to the same issues, or, as Friedman and Glymour suggest, i s the term "quantum l o g i c " a "rather misleading 99 t i t l e " for a "purely algebraic investigation"? At least one theoretician, J.M. Jauch, has denied that there i s anything but a formal resemblance between ordinary ( c l a s s i c a l ) l o g i c ' and quantum l o g i c . I w i l l propose an assessment of quantum l o g i c at odds with h i s : I w i l l suggest that quantum l o g i c , l i k e c l a s s i c a l l o g i c , deals with sentences and the relations between them. However, before doing so I w i l l examine his views i n some d e t a i l , not only because his book, Foundations of Quantum Mechanics, has been a seminal text for those taking a logico-algebraic approach to the subject"*"^, but because 98. 99. 100. Greechie and Gudder (1973), p.170 Friedman and Glymour (1972), p.18. See Jauch (1968). In the rest of this chapter I w i l l just give 270 an extended critique of these views offers a useful route to some.of th interpretive problems of quantum theory. These problems, i n turn, are just those we need to address i n a discussion of our central concern: see what we mean when we talk of quantum l o g i c . He writes (p.77) , The propositional calculus of a physical system has a certain s i m i l a r i t y to the corresponding calculus of ordinary l o g i c . In the case of quantum mechanics, one often refers to this analogy and speaks of quantum log i c i n contradistinction to ordinary l o g i c . This has unfor-tunately caused such confusion that we s h a l l add a few words of explanation here to avoid any misunderstanding. The calculus introduced here has an e n t i r e l y different meaning from the analogous calculus used i n formal l o g i c . Our calculus i s the formalisation of a set of empirical relations which are obtained by making measurements on a physical system. I t expresses an objectively given property of the physical world. I t i s thus the formalisation of empirical facts, induct-i v e l y arrived at and subject to the uncertainty of any such fact. The calculus of formal l o g i c , on the other hand, i s obtained by making an analysis of the meaning of propositions. I t i s true under a l l circumstances and even tautologically so. Thus, ordinary l o g i c i s .a page number when quotiig from this book. I adjust Jauch's notation to bring i t into l i n e with my own. 271 used even i n quantum mechanics of systems with a pro-po s i t i o n a l calculus vastly different from that of f o r -mal l o g i c . The two need have nothing i n common. I t turns out, however, that, i f viewed as abstract structures, they have a great deal i n common without being i d e n t i c a l . Now i t i s doubtful whether the "confusion" Jauch refers to would be lessened i f the term "quantum l o g i c " were replaced everywhere i n the l i t e r a t u r e by "propositional calculus of quantum mechanical systems". Clearly, however, Jauch wants to claim that he i s using the phrase "propositional calculus" i n a way which i s s i g n i f i c a n t l y different from the way i n which i t i s used i n , say, an elementary l o g i c text. As we s h a l l see, this claim i s open to dispute. I t i s certainly true that Jauch defines the term "proposition" i n an idiosyncratic way which few logicians would recognise. For Jauch, a proposition i s a p a r t i c u l a r kind of experiment, s p e c i f i c a l l y an experiment which permits "only one of two alternatives as an answer", or, as he c a l l s i t , a "yes-no experiment"."'"^"'" This description i s modified l a t e r to the following (p.74): "We define a proposition as a class of physical yes-no experiments, a l l of which measure the same proposition ( s i c ) . " We can rewrite the d e f i n i t i o n so that i t i s no longer c i r c u l a r , but we may s t i l l ask why Jauch chooses to define "proposition" as he does. 101. C f . p. 73: "We s h a l l also refer to yes-no experiments simply as propositions of a physical system." Other writers on quantum l o g i c (e.g. Greechie and Gudder (1973)) use "question" where Jauch uses "proposition". 272 For when we examine the behaviour of these oddly defined e n t i t i e s , we fi n d that they are barely distinguishable from their l o g i c a l counterparts. They may be linked by connectives, and Jauch himself enjoins us to i n t e r -pret "a n b" as "a and b", "a u b" as "a or b" and "a"' as "not a" (p. 77). His propositions, l i k e those of an interpreted l o g i c a l language, can be true or f a l s e , and "we say a implies b, and write a £ b , i f , whenever a i s true, i t follows that b i s true too."(p.74) The conditions for the truth of "a n b" and "a u b" also look f a m i l i a r : "a n b i s true i f and only i f both a and b are true;" (p.75) "a u b i s true i f a or b, or both, are true." (p.76)The l a t t e r condition i s weaker than the corresponding c l a s s i c a l condition, which contains " i f f " i n place of " i f " ; this d i f f e r -ence i s s i g n i f i c a n t , as we s h a l l see. Another departure from the bivalent semantics of c l a s s i c a l l o g i c occurs when Jauch distinguishes " f a l s e " from "not true" (p.77), but such oddities are scarcely enough to warrant the claim that what i s being developed i s not a logi c but something d i f f e r e n t . Nevertheless t h i s i s the assertion he makes; to j u s t i f y i t Jauch must maintain that, despite his use of terms l i k e "implies", "and", "or", "not"-, "true" and " f a l s e " , his "propositional calculus, does not deal with the l o g i c a l relations between sentences. That i s , he must deny either (a) that there are sentences which correspond to his "propositions", or (b) that the relations with which his calculus deals are l o g i c a l ones. 102. Gardner, looking no further than the truth conditions for "a nb" and the conditions under which a c b holds, suggests that Jauch's calculus "effects no revision i n ~ l o g i c at a l l . " Indeed, for him " t h i s notion of implication obviously pre-supposes standard truth-functional l o g i c . " (Gardner (1971), p.513) These claims are manifestly false ^ unless we are to translate "presupposes"as "derives from"j but the fact that they can be made shows that Jauch's use of the vocabulary of log i c does not help to convince commentators that his work belongs i n another f i e l d . 273 Only i f he can argue successfully against (a) or (b) (or both) can he regard his use of the vocabulary of l o g i c merely as the exploitation of an analogy. His d e f i n i t i o n of a proposition as a class of experiments i s i m p l i c i t l y a denial of (a); contra (b) he suggests (b') that we should regard the relations between propositions as " s t r u c t u r a l properties" of a system, (p.74) However, I hope to show(i) that any argument mounted against (a) i s unsuccessful, and ( i i ) that (b) and (b') are mutually compatible. X.2. The Case Against Jauch. A number of authors w r i t i n g on quantum l o g i c have used the term "proposition" to cover a notion which seems non-linguistic. For instance, Birkhoff and von Neumann define an "experimental proposition" concerning a physical system as a "subset of the observation spaces 103 associated with that system." Bub moves (sometimes within the same sentence) from talk of "propositional structures" to talk of'event structures", as though "proposition" were synonymous with "event". "^^ His use of "proposition" t i e s i n with Jauch Ts as follows. An event i s a; set of possible outcomes of experiments"*"^5; assume, for s i m p l i c i t y that i t i s the set {a^,a^,a^} from an experiment E. This set, obviously, i s a subset of the set of a l l possible outcomes of E. Then 103. 104. 105. Birkhoff and von Neumann (1936), p.2. Bub (1974); see, for instance, pp.93, 106, 107 and 119. I b i d , p.32. 274 with this event we can associate a yes-no experiment i n which the experimental arrangement i s just that required for E, and which i s said to give the result "Yes" whenever the result of E i s a^ or a^ or a^. This yes-no experiment i s a proposition i n Jauch's sense.of the word. Clearly we can generalise this analysis to show that to every event there corresponds a yes-no experiment, and conversely. The propositional structures Bub favours are p a r t i a l Boolean algebras, whereas Jauch deals with, a l a t t i c e ; however,this difference i s irrelevant to our main concern, which i s to show that i n each case the propositional structure involved i s a l o g i c a l structure i n the t r a d i t i o n a l sense. Unlike Jauch, Bub does not deny t h i s , but neither dogs he e x p l i c i t l y endorse i t . My comments w i l l be s p e c i f i c a l l y directed at Jauch's l a t t i c e of propositions; they can be suitably modified to apply also toa p a r t i a l Boolean algebra. The elements of Jauch's l a t t i c e are equivalence classes of yes-no experiments. Corresponding to each such experiment E we may form the sentence S(E), which reads, "Experiment E conducted on the system yields a positive r e s u l t . " Thus to each equivalence class of experiments (or proposition i n Jauch's sense) there w i l l correspond an equivalence class of sentences (or proposition i n the l o g i c a l sense). The mere possibility of this simple one-to-one mapping of experiments onto sentences i s s u f f i c i e n t to show that, even by defining a proposition as he does, Jauch i s not able to deny (a) (see p. ). Now, given the operations of u, n and ' on the set of experi-ments, we may define connectives "v", " A " and "~" on the set of correspond-275 ing sentences. We f i r s t define an equivalence r e l a t i o n on the l a t t e r set. X.2.1. A E B i f f A and B are sentences corresponding to members of the same equivalence class of experiments. Now we write, for experiments E, F, X.2.2 S(E) y S(F) = S(E u F) S(E) A S(F) = S(E n F) ~(S(E)) = d f S(E') I f we now define an ordering r e l a t i o n on the set of sentences to corres-pond to the r e l a t i o n c on the set of experiments, we get a l a t t i c e of equivalence classes of sentences isomorphic to Jauch's l a t t i c e of experimental propositions. I t remains to show (b): that the l a t t i c e theoretic relations we obtain are recongnisably l o g i c a l - r e l a t i o n s . What we have done so far i s to take a limited class of atomic sentences, a l l of the form "Experiment E conducted on the system yie l d s a positive r e s u l t " , together with three connectives "v", " A " and "~" which are used to form complex sentences. We have also provided rules which t e l l us that the complex senteces we form are equivalent to certain atomic sentence's I f we are to obtain a logic what we now need to do i s to give rules by which we may deduce certain sentences from others ( i . e provide a proof theory) and to give a systematic account of how the truth of certain sentences depends on the truth of others ( i . e . provide a semantics). The l a t t e r i s supplied by Jauch himself. For uncontroversially. we may require that S(E) i s true i f f experiment E conducted on the system 276 would y i e l d a positive r e s u l t , and i f we predicate truth of experiments, as Jauch does, this becomes, X.2.3. S(E) i s true i f f E i s true Now the l a t t i c e r e l a t i o n , £ , on the l a t t i c e of experiments, i s i n -timately connected with the truth of these experiments; Jauch writes, as we have noted: X.2.4. E c F i f , whenever E i s true, i t follows that F i s true too. We see that, i f we regard this l a t t i c e as a l a t t i c e of sentences, then the r e l a t i o n which orders the l a t t i c e i s the f a m i l i a r one of entailment; i n fact X.2.4 i s j u s t clause (b) of Friedman and Glymour's S3 for quantum l o g i c . A l t e r n a t i v e l y we can take a proof theoretic route. Following, e.g., Hacking, or Friedman and Glymour, we can devise a proof theory, that i s we can (syntactically) specify a set of axioms and rules of inference, . i n which the l a t t i c e r e l a t i o n appears as the r e l a t i o n of d e d u c i b i l i t y . I f we do so, then the considerations above show that on the intended interpretation this proof theory w i l l be sound. Exactly how the semantics, on the one hand, or the proof theory, on the other, i s to be developed i s not germane to the present discussion. What I take this argument to have shown i s t h i s : i f we associate with each experiment E the sentence S(E), then to the relations 277 with which Jauch's calculus deals there correspond relations between these sentences which are, prima f a c i e , l o g i c a l . Now there may be further c r i t e r i a which have to be s a t i s f i e d before we bestow this t i t l e on them: I discuss this i n section 6; i n the meantime, however, we have shown (a); i f * * (provisionally) we accept (b) , then the d i s t i n c t i o n Jauch makes between a > (propositional calculus and a lo g i c collapses. X.3. Operations on the La t t i c e of Propositions. The objection could s t i l l be made that the move by .which we introduced the connectives "v", " A " , and "~" i n the la s t section, together with the rules (X.2.2) which govern thei r l o g i c a l behaviour, i s wholly a r t i f i c i a l . The connective " A " , for instance, i s introduced to correspond to the operation n on the l a t t i c e of experiments. But the experiment E n F turns out to be related i n a rather complicated way to the two experiments E and F; thus to presuppose that the sentence S(E Q F) w i l l be related straightforwardly to the two sentences S(E) and S(F) seems an unwarranted and overly strong assumption. Yet such a simple connection i s implied when we write S(E) A S(F) = S(E n F) •To meeT .this' 6BtjeTc.'t'I6n we" nee'd' f i r s t ' ro see how the experiments E', E n F and E o F are related to the experiments E and F. For any experiment E, there i s an experiment E' (the complement of E). The appar-atus required for E' i s the same as that required for E; i n fact the experiment E' i s the same as E, but for the fact that we regard E T as giving a positive answer exactly when E gives a negative answer and vice 278 versa. Thus the sentence ~(S(E)) i s simply the (strong) negation of S ( E ) , and the objection cannot be maintained as f a r as the connective "~" i s concerned. The binary connectives are more problematic. Jauch wishes the (experimental) proposition E n F to be true ( i . e . the experiment E n F to y i e l d a p o s i t i v e r e s u l t ) exactly when both E i s true and F i s true. As we have seen, t h i s y i e l d s the f a m i l i a r c l a s s i c a l truth conditions f o r S(E) A S ( F ) . However, i f E and F are two experiments which we may perform on the same system, we cannot always conduct them sumultaneously; and, whereas i n c l a s s i c a l physics we can devise experiments to measure any given p a i r of observables i n such a way that the r e s u l t s we get w i l l not depend on the order i n which the experiments are performed, i n quantum physics t h i s cannot be done i f the observables i n question do not commute. To deal with t h i s problem Jauch proceeds as follows. He f i r s t introduces the notion of a f i l t e r , that i s , " a yes-no experiment which serves to s e l e c t the value of a measurable quantity" (p.73). Then "we construct the f i l t e r for the proposition E n F by using an i n f i n i t e sequence of a l t e r n a t i n g f i l t e r s f o r the propositions E and F respectively. ... The proposition E n F i s then true i f the system passes t h i s f i l t e r , and i t i s not true otherwise." (p.75) An example w i l l i l l u s t r a t e what i s involved. In the Stern-Gerlach experiment a beam of s i l v e r atoms was passed through an inhomogeneous magnetic f i e l d 106 between two curved magnetic pole pieces. The beam was s p l i t i n two 106. i - . f . Merzbacher (1961), pp.277-9. Note that i n the present-ation of Jauch's views which follows I assume with Jauch that the p r o j e c t i o n postulate holds ( c . f . von Neumann (1932), Ch. I I I . 3 ) . Recent work by Bub and van Fraassen casts doubt on t h i s postulate (see, for instance, Bub (1977). 279 according to the values of the component of spin along the axis of the apparatus. With the axis of the apparatus horizontal, the two emergent beams corresponded to the two possible values of the x-component of spin, S = +V2 and S = -11'2. We can imagine a Stern-Gerlach apparatus i n which one of the emergent beams (the beam for which S^ = - ^ 2 ) say) i s blocked off. Then any emerging electron w i l l have a positive x-component of spin, and the apparatus i s thus a f i l t e r which selects the value S x = Let this be the experiment E. (See Figure 22.) A sim i l a r apparatus, rotated through 90° would select the value of the y-component of spin, S^ = +1/2- C a l l t h i s the f i l t e r F. Then the proposition E n F i s the "experiment" which uses an i n f i n i t e sequence of pairs of Stern-Gerlach apparati, each pair set up as i n Figure 23. We Eiay notice i n passing that this procedure ensures that, i f E and F are experiments performed to select s p e c i f i c values of two 6b6ervables, A and B, with no eigenvectors i n common, as i n the example above, then E n F can never be true. For assume that E selects a value a for observable A, while F selects a value b for observable B. Then the systems which pass the f i l t e r E w i l l be i n that eigenstate of A corresponding to eigenvalue a (for s i m p l i c i t y we assume no degen-eracy) ; they w i l l then not be i n an eigenstate of observable B, and so (from V.7.3) X.3.1. p(val(B) = b) 4 1 That i s , the' probability of such a system passing the f i l t e r F w i l l be less than one. However, any which do pass through f i l t e r F w i l l then be in the eigenstate of B corresponding to eigenvalue b, and so'the pro b a b i l i t y Figure 21. Experiment E (schematically). 281 Figure 22. 282 of their passing the second f i l t e r for E i s given by X.3.2. p(val(A) = a) < 1 For i l l u s t r a t i o n , consider the example above. An atom passing through the f i r s t Stern-Gerlach apparatus i s i n the state x^ .. We know from V.8.11 that, for such a system, X.3.3. P (val(S ) = +V2) = V 2 < 1 S i m i l a r l y , for any system which passes the second f i l t e r , X.3.4. P (val(S ) = +-/2) =  l l l < 1 y_l_ x Now imagine a sequence of f i l t e r s set up i n accordance with Jauch's pro-cedure to determine whether E n F i s true. whenever a system passes through any one f i l t e r , the probability of i t s passing through the next i s less than one; since the sequence of f i l t e r s i s i n f i n i t e , the probab-i l i t y of the system passing the whole composite f i l t e r i s zero, and so E n F cannot be true. Jauch uses a sim i l a r procedure to arrive at the (experimental) proposition E u F. He regards u as a defined operation; that i s : X.3.5. E u F = (E' n F')' dt Thus the idealised experimental arrangement which I have just outlined for the experiment E n F, consisting of an i n f i n i t e number of pairs of Stern-Gerlach apparati, could also be used for the experiment G u H where G tests whether the system has negative x-component of spin and H whether i t has negative y-component. Then a sim i l a r apparatus i s used for G as for E, and since an atom enters the left-hand (x.) beam ( i . e ; 283 E i s true) i f and only i f i t does not enter the right hand (x_) beam ( i . e . G i s not true), and vice versa, we have X.3.6. G' = E and, likewise, H* = F Thus X.3.7. G* n H' = E n F I t follows that the complement of E n F , i . e . the experiment which uses the same apparatus as E n F but which i s regarded as y i e l d i n g a positive result whenever E n F yields a negative r e s u l t , and vice versa, i s the experiment G u H. We can now see why Jauch uses a weaker version of the truth conditions for G u H than the c l a s s i c a l version. I t may wel l be the case that, for a certain system, neither G nor H would y i e l d a positive r e s u l t ; that i s , for a given atom we may find that measurements of and both y i e l d the value +11'2. However, as we noted, (E n F)' = (G' n H')' = G u H , and so G u H i s always true. Thus i t i s not necessary, but merely s u f f i c i e n t , that either G or H be true i n order that G u H be true. Note i n c i d e n t a l l y that i t follows that Jauch cannot consist-ently write, as he does, "E u F has the l o g i c a l significance of E or F"; i f he wishes to r e s t r i c t the term " l o g i c " to mean c l a s s i c a l l o g i c . We may regard this inconsistency as evidence that Jauch should have heeded his own injunction to preserve an absolute d i s t i n c t i o n between his pro-pos i t i o n a l calculus and l o g i c . A l t e r n a t i v e l y , however, we can accept Jauch's weakened account of the l o g i c a l s i n i f i c a n c e of E u F, note that 284 i n an important respect i t d i f f e r s from the c l a s s i c a l account, and conclude that we are dealing with a bona f i d e , but non-classical, logic. Let us now return to the objection raised at the beginning of this section, that the introduction of connectives i n Section 2, and thus the construction of a l a t t i c e of sentences isomorphic to the l a t t i c e of experiments was a wholly a r t i f i c i a l move. Two responses may be made to th i s . The f i r s t i s to point out that, however arbitrary the introduction of thesesconnectives may appear, yet their behaviour i n many ways t a l l i e s with that of the "&", "v" and 'V of c l a s s i c a l l o g i c . In other words, we may argue on the basis of the results we obtain when we use them. The other response i s more r a d i c a l . I t i s to argue that the a r t i f i c i a l i t y enters the analysis, not when we introduce these connectives to obtain a l a t t i c e of sentences, but when Jauch i n s i s t s that the fundamental structure with which he i s dealing i s a l a t t i c e of experiments. , On this account, i t i s not the reduction of a complex r e l a t i o n between experiments to a simple one between sentences which i s a r b i t r a r y , but the devising of idealised and complicated combin-ations of experiments to r e f l e c t very simple relationships within the theory of quantum mechanics. X.4. An Inadequate Positivism. Jauch's emphasis on the experiment as a physical notion i s just one manifestation of the e x p l i c i t l y p o s i t i v i s t metaphysic which he adopts. (See i n pa r t i c u l a r his Chapter 5.1.) Throughout his discussion of the propositional calculus there i s a neglect of the role of theoretical con-. 285 cepts i n physics, and a correspondingly distorted analysis of the res-pective functions of mathematics and experience i n the generation of a theory. According to Jauch, "General laws are arrived at by induction from observed facts. ... These general laws are formulated as axioms i n a mathematical language." (p.70) I t i s the set of axioms thus obtained which constitutes the theory. The notion of a theory as a set of formal versions of physical laws, each of which may be "expressed i n the following form: i f a system S i s subject to conditions A, B, ... , then the effects X, Y, ... can be observed," (p.71) i s patently absurd, neglecting as i t does the part played by theoretical concepts i n showing the systematic i n t e r -connections between disparate kinds of experimental r e s u l t s , not to mention the p o s s i b i l i t y of providing a theoretical explanation for these physical laws. Jauch himself describes i t as "somewhat schematic" (p.71), but his comment that i t "would have to be refined to be accurate" may most charitably be described as euphemistic. The same neglect of theory occurs when he desribes his pro-p o s i t i o n a l calculus as the "formalisation of a set of empirical relations which are obtained by making measurements on a physical system. ... I t i s thus the formalisation of e m p i r i c a i i f a c t s . " (p.77) For consider what i s i m p l i c i t i n the assertion that experiments form a l a t t i c e . Inter a l i a there i s the assumption that a p a r t i a l ordering of yes-no experiments ex i s t s : thus the r e l a t i o n of implication, £ , i s assumed to be t r a n s i t i v e . Stronger s t i l l i s the assumption that the experiment E n F described i n the previous section constitutes a greatest lower bound for E and F . with respect to this ordering. 286 In p a r t i c u l a r we can question the assumption which i s at work when the experiment E n E i s devised, that to each yes-no experiment there corresponds a f i l t e r for the value of the measurable physical quantity involved. These assumptions appear i n the context of an empirical approach which seeks to provide a general treatment of physics applicable to any theory. The analysis i s reminiscent of A r i s t o t l e ' s theory of tragedy, produced after the fact of A t t i c tragedy and admirable as a c r i t i c a l exam-ination of that genre, by purporting to have universal v a l i d i t y . Certainly, when applied to c l a s s i c a l physics of to quantum mechanics Jauch's assumptions appear innocuous enough, but this i s because the approach has been taken with these p a r t i c u l a r theories i n mind. In other words, the propositional calculus i s not a mathematical structure which i s the "formalisation of empirical facts", which i n turn "are obtained by making measurements on a physical system", but a structure yielded by the theory from which experimental predictions and the r e l a t i o n -ships between them can be deduced. For even i f we grant that the set of yes-no propositions can be appropriately syructurdd as a l a t t i c e , another, more t e l l i n g , c r i t -icism of Jauch's approach i s t h i s : i n the absence of theoretical considerations, what determines the conditions to be l a i d upon the l a t t i c e ? S p e c i f i c a l l y , without a p r i o r commitment to the , theory ofquantum mechanics, the choice of a l a t t i c e structure for quantum l o g i c which i s isomorphic to the l a t t i c e s of subspaces of H i l b e r t space seems 107 en t i r e l y arbitrary. Yet certain conditions, l i k e completeness and 107 A s i m i l a r point i s made by Gardner (1971) (p.516). 287 modularity, are imposed on the l a t t i c e of propositions precisely because they y i e l d such a structure. P a r t i c u l a r l y i n the case of completeness, this leads Jauch away from a s t r i c t l y empiricist formulation of quantum mechanics, for we find appearing i n the l a t t i c e non-realisable i d e a l elements of the kind /^E^ ( i e I) » where the indexing set I need not even be denumerable Thus the p o s i t i v i s t approach which Jauch takes-is far from free of theoretical assumptions, and to that extent f a i l s i n i t s aim. Further, i n an adopting i t , rather than an overtly theoretical approach, 107. Later writings by Jauch and Piron suggest an experimen-t a l arrangement for /^\A. which at f i r s t sight seems more p r a c t i c a l than an extension of the arrangement for A n B described here. (See Jauch and Piron (1969), Piron (1972), Piron (1977); the quotation below i s from the f i r s t of these, pp.429-30) I f a. ( i e l , some index set) i s any family of yes-no experiments then one can define another such experiment denoted by Ila. by the following procedure: one chooses at random one of the a. ( i e I) and measures i t . The result i s the value of Ila Let a be any yes-no experiment. We denote.by &_ = {a} the class of a l l such experiments which are equivalent to a. ... We denote by iS.a. the equivalence class {Ila^} . But since our choice of which experiment to perform i s arbitrary, the results of a succession of such experiments w i l l be inconsistent, for reasons which have nothing to do with the p r o b a b i l i s t i c nature of quantum mechanical predictions. We could presumably overcome this problem by taking an i n -f i n i t e sequence of a r b i t r a r i l y chosen experniments, but that would make the procedure as idealised as before. 288 we exchange a number of mathematically simple notions for their tortuously defined experimental counterparts: compare, for instance,' the baroque constructions which y i e l d the meet and j o i n of two experiments with the elementary definitions of these operations on the set of subspace of Hilbe r t space. Since theory i s inescapable, we may at least ask that i t be simple. X.5. Theory as Description. By i t s e l f , s i m p l i c i t y i s only part of what i s at issue.' By choosing to work with a p a r t i a l Boolean algebra of experiments rather than a weakly modular l a t t i c e , Jauch could sidestep the problem of interpreting E n F , when E and F cannot be simultaneously performed, and many of the complexities of his account could be avoided. More important i s the s t a r t i n g point of the discussion, which for Jauch i s a set of experiments rather than the theory of quantum mechanics. And i n arguing against Jauch's approach I have not shown that an analysis of quantum mechanics from a theoretical rather than an empirical standpoint w i l l resolve the problem posed at the beginning of Section 3, that i s , the question of the a r t i f i c i a l i t y of associating sentential connectives of a language with the operations on an algebraic structure. Both analyses employ such a structure; we have, on the one hand, Jauch's l a t t i c e of propositions - less misleadingly, his l a t t i c e of experiments -and, on the other, a p a r t i a l Boolean algebra of subspaces of Hil b e r t space, but we may s t i l l ask: can this structure be seen as an algebra of sentences more naturally on the theoretical approach than on the experimental one? What, come to that, i s involved i n a "t h e o r e t i c a l approach"? 289 This approach i s i m p l i c i t i n e a r l i e r chapters and i t s formal expression has been summarised i n Chapter V. What I w i l l do here i s to review that material less formally, and see whther there i s an interpre-tation of quantum mechanics to which this approach commits us. Under-lying this approach i s the assumption that the theory i s , i n some sense, descriptive of physical real i t y . ' Now this assumption i s , of course, made by anyone who takes a r e a l i s t position vis--a v i s theoretical terms; as we s h a l l see, however, we may make i t without being forced to adopt such a position: the word "descriptive" i s s u f f i c i e n t l y e l a s t i c to allow for substantial disagreement about the status of the theoretical terms which quantum mechanics employs. We take as a st a r t i n g point the idea of the state of a system. When i n a given theory we specify the state of a certain system, we are providing a description of that system couched i n the language of that theory; i n quantum mechanics the.state of a system i s represented 109 mathematically by a vector i n Hilbe r t space. Note that to say that we describe a microsystem by specifying i t s state i n this way. need not commit us to regarding these state vectors as elements of r e a l i t y : we can, i f we wish, maintain that to provide such a description i s merely to supply a maximal set of predictions about the experimental results which w i l l be observed when the system interacts with various pieces of apparatus. Many of these predictions w i l l be s t a t i s t i c a l i n nature; i n fact the categorical predictions which are made, e.g., "Experiment E w i l l 109 I deal here with pure states; more generally, any state i s represented by a s t a t i s t i c a l operator on the space. 290 y i e l d result X", "Experiment,-E w i l l not y i e l d result X", are l i m i t i n g cases of these p r o b a b i l i s t i c predictions. According to the theory, these two kinds of categorical predictions can be made when the vector representing the state of the system f a l l s within a certain subspace or within the or-thogonal complement of that subspace, respectively. Thus each subspace corresponds to a set of possible states of the system, and when the vector representing the state f a l l s within that subspace, a part i c u l a r class of experiments w i l l y i e l d a positive result with certainty. Whether or not a given experiment f a l l s within that class i s predicted by the theory. The set of subspaces has the structure of a (non-Boolean)--algebra, and so we may, i f . we wish, i d e n t i l y each element i n this algebra with a class of experiments. Observe how, by taking this approach, we have finished up with the structure which Jauch takes as a st a r t i n g point.' Also, for com-parison, we can see how Jauch introduces the idea of a state of a system i n a way consonant with the empiricist and instrumentalist elements i n his presentation. For Jauch, the state of a system " i s a measurable qu quantity" (p.93). However, i t i s a measurable quantity i n a somewhat Pickwickian sense: to assess i t we need to know the r e s u l t s , not only of a selection of yes-no experiments on the system, but of a l l possible such experiments, since i t i s a "probability function p(E) defined on a l l the propositions" (p.94). I m p l i c i t i n Jauch's approach i s the view, shared by F i n k e l s t e i n ^ t h a t one should not "elevate the state vector to 110. Finkelstein (1972), pp.59-60 291 the position of a substitute element of r e a l i t y . " None the le s s , even on this account, the state vector s t i l l offers a summary of the predict-able experimental behaviour of a system; as such i t s t i l l describes the system i n the terms which the theory permits. Thus as long as we include microsystems i n our ontology (as Jauch does"'""'""'") we should regard statements of the form, "For the system S, x e L", where x Is the state vector of the system, L a subspace of the appropriate Hil b e r t space, as descriptive of such systems; i n 112 fact we may regard this as the appropriate form of these descriptions.' Sentences of this kind I have already called " Q-propositions" (see Chapter VI.1). I f we construct a language i n which to talk about micro-systems, these sentences w i l l be the atomic sentences: to each such sentence w i l l correspond a pa r t i c u l a r subspace, and vice versa. Consider two such sentences, each descriptive of a pa r t i c u l a r physical system: "For the system S, x e Lp" and "For the system S, x e L Q " - C a l l them P and Q respectively. Then i f we l i n k them by a connective ".,,* ", we obtain P * Q , which we may c a l l R . We presumably want to regard R also as a sentence descriptive of the system. But this i s to demand that R be of the form, "For the system S, x e L ", where R i s some' K subspace. Now i f we want to maintain the one-to-one correspondence we have set up between sentences and subspaces, then any connectives we i n t r o -duce into the language must mirror some operation on the set of subspaces: the usefulconnectives w i l l then be those corresponding to simple operations 111 On p.71 he writes, "The main problem i n the selection of empir-i c a l material i s the separation of relevant from irrelevant conditions!,"! This i s often made possible by i s o l a t i n g a s u f f i c i e n t l y simple part from the rest of the physical world and studying the properties of the isolated part alone. We s h a l l c a l l such an 292 l i k e meet and j o i n . We therefore introduce connectives "A" and "V", and give the i r interpretation by f i a t , as follows: X.5.1. P A Q d f For the system S, x e ( L p A-L ) P v Q d f For the system S, x e ( L p v L^) At this point we can, i f we want, seek the "operational umeanings" of each of these connectives interms of idealised experiments 113 l i k e those Jauch describes. And, as Putnam says, Provided that one does not s l i p over from the view that 'operational d e f i n i t i o n s ' are a useful h e u r i s t i c device for getting a better grasp of a theory, to the view that they r e a l l y t e l l us what theoretical terms mean (and that theoretical statements are then mere shorthand for statements about measuring operations), no harm results. Looking at Jauch's work from this viewpoint, we find that the "operational meanings" which his experiments associate with "A" and "v" t a l l y very closely with those of their c l a s s i c a l counterparts, "and" and "or". This i s not surprising: the operations A and v on the set of subspaces of a Hilb e r t space are analogous to the set-theoretic operations of intersection and union, and we expect the connectives corresponding to 113. Putnam (1969), p.236. Putnam i s discussing here the work of Fin k e l s t e i n , whose approach has much i n common with that of Jauch: se Finkelstein (1969), (1972). isolat e d part a physical system. The simplest physical systems are those which consist of just one elementary p a r t i c l e , i f we disregard i t s interactions with other p a r t i c l e s . " 112 For the view of the connectives i n quantum l o g i c which follows, see also Fine (1972), pp.14-20. 293 them to behave accordingly. i n fact we may say that these connectives are the versions of "and" and "or" appropriate to the quantum domain. This analysis holds good whether we regard the algebraic struc-ture of the set of subspaces of Hi l b e r t spaces of H as an orthomodular l a t t i c e or as a p a r t i a l Boolean algebra. Note that, i f A and v are p a r t i a l operationsoon Sp(H), then P A Q and P v Q are only meaningful when Lp A and Lp v are defined, i . e . , when Lp$L^. Whichever structure we choo^se, the construction of an algebra of sentences isomorphic to the algebra of subspaces of Hi l b e r t space follows at once from the way we have set up a language to talk about microsystems. The interpretation of the connectives by d e f i n i t i o n , which seemed a r b i t -rary when the fundamental structure was thought of as a l a t t i c e , now follows naturally from two considerations. We require that ( i ) i n l i n e with quantum theory, each sentence descriptive of a system i s equivalent to a Q-proposition; ( i i ) i f we take two sentences, each descriptive of a physical system, and l i n k them with a connective, then the resu l t i n g com-plex sentence i s also descriptive of that system. We can now see how the l o g i c a l relations between Q-propositions can also be regarded as the "st r u c t u r a l properties" of a system, i n Jauch's phrase. For to say that a microsystem has a p a r t i c u l a r struct-u r a l property i s just to say that the appropriate H i l b e r t space for the system i s of a par t i c u l a r kind; for instance, a s p i n - 1 ' ^ p a r t i c l e i s described by reference to a Hilbert space of 2 dimensions, whereas a spin-1 p a r t i c l e requires a Hi l b e r t space of 3 dimensions. . r. And we have defined a correspondence between Q-propositions and subspaces of H i l b e r t space i n a way which ensures that the l o g i c a l relations among the 294 sentences descriptive of a given microsystem correspond exactly to the algebraic relations among the subspaces of the appropriate Hilbert space. Thus (b) and (b'),on p. , are e n t i r e l y compatible. X.6. Quantum "Logic" as Logic In the previous section I outlined how one might construct a language i n which to talk about microsystems; i n the next I show how such a language, TH, i s related to the formal language discussed i n Chapter VIII. From that chapter we have an account of the "quantum l o g i c a l " relationships which hold between the formulae of Q„ and we can investigate the corresponding relationships between the sentences of However, i n the meantime I return to a pair of questions raised e a r l i e r : are these relations l o g i c a l r e l a t i o n s , and i s i t s u f f i c i e n t that our "quantum l o g i c " give an account of them i n order to qualify as a logic? I would answer "Yes" to both questions, and say that the same would be true for a system based on l a t t i c e s rather than on p.B.a.'s. 114 ' However, there are dissenting views. For instance, Jauch and Piron suggest that a lattice-based quantum " l o g i c " i s not a l o g i c because there can be no connective i n the language which captures the metalinguistic r e l a t i o n of d e r i v a b i l i t y between sentences, and to which corresponds an operation on the orthomodualr l a t t i c e under which the l a t t i c e i s closed. "^""^  This they contrast with the c l a s s i c a l case, i n which we have A(-B i f f 114. Jauch and Piron (1970). 115. For a discussion of l a t t i c e conditionals, see Hardegree (1975). 295 A 3 B is valid, and a*(A o B) = (a*(A))' v a*(B) for any interpreta-tion a* within a Boolean algebra. Their view rests on a mistaken belief that a l l deduction must proceed by modus ponens. Certainly, where a connective exists which•captures this relation, then modus ponens appears as a rule of inference, but i t does not follow that a calculus lacking this 116 rule " i s not a logic since i t does not admit any deduction." While there are presentations of classical propositional logic, for instance, in which modus ponens i s the only rule of inference,"'""'"^  there are others in which i t i s one of many; in fact, in the system CN i t appears as an afterthought, in the form of a derived rule. Thsu even for the classical case i t i s absurd to say that "without such a rule ... inferences are 118 impossible." Note that the force of their contention, feeble at best, i s further reduced when i t i s applied to s quantum logic based on p.B.A.'s. In QN, for instance, modus ponens, like most sules of CN, goes through given certain provisos, and we have A, A => B h B More interesting to the questions at hand are considerations like 119 those urged by Hacking. He sees logic as primarily a "science of deduction", and l i s t s five features which he would expect to find in a deducibility relation. With one exception they are shared by classical logic and the quantum logic developed in Chapter VIII. In QN and QL (as in CN and CL) we have, (i) Reflexivity: A |_A ( i i ) Permutation: i f T ,A,B, |-C then r,B,Ar"C 116. Jauch and Piron (1970), p.174. 117. See, for instance, Church (1956) or Kleene (1967). 118. Jauch and Piron (1970), p.173. 119. Hacking (1977), pp. 10-12. As he points ou, these are not the defining characteristics of a deducibility relation. 296 ( i i i ) Contraction: i f Y, A,A|-B then r,Af-B (iv) (a) Thinning ( i n the antecedent): i f . T (-A then r,B)-A (v) T r a n s i t i v i t y : i f A(-B and B L C , then A|-C Only (iv)(b) requires modification; this i s the rule for thinning i n the succedent (and thus i s relevant to QL but not to QN). I t also f a i l s for the i n t u i t i o n i s t sequent calculus, and so we may question whether i t should feature as a general characteristic of a de d u c i b i l i t y r e l a t i o n . Within QL i t goes through with the addition of a set of provisos i n the antecedent (see VIII.11.1). Note that ( i i ) and ( i i ) are guaranteed by specifying that f- i s a r e l a t i o n between sets of formulae and formulae. Of the l o g i c a l constants, hacking writes, ... anything defined by a rule of inference l i k e Gentzen,s i s a l o g i c a l constant. ... Of course the exact boundary delends e n t i r e l y on what s h a l l count 120 as " l i k e " Gentzen,s rules. Now the connectives of Q were hardly defined by Gentzen rules. The motivation for constructing quantum l o g i c i s primarily semantical, and i t i s the correspondence between the connectives of the language and the algebraic operations on B(H) which i s our primary concern when we do so. Nevertheless, i t i s not f a n c i f u l to see the deductive role of "&" i n ^  defined by the introduction and elimination rules of the system QN. The language (£, and the l o g i c a l system QN were designed to include 120. Hacking (1977), p.8. 297 the notion of compatibility: thus i t i s not surprising that a proviso bearing on compatibility finds i t s way into the &+ rul e . And the rules as a group show clea r l y the deductive force, not only of the connectives of Gl, but also of the compatibility conditions which we express v i a the provisos. Thus f a r , any constraints which Hacking lays on the concept of a l o g i c are met by quantum l o g i c . But Hacking also suggests that a Cut-Elimination Theorem should be provable for the l o g i s t i c calculus associated with a l o g i c ; such a theorem i s apparently provable i n OM(H), 121 but, as we have noted, i t may not be provable for QL. How serious a consideration i s this? The philosophical importance of the Cut Elimination Theorem i s that i t ensures that the t r a n s i t i v i t y r e l a t i o n i s not just a r b i t r a r i l y b u i l t into the system of proof. The d e f i n i t i o n of a derivation i n QN, for instance, l i k e that for CN, ensured that the r e l a t i o n j- for these systems i s t r a n s i t i v e , and clearly the cut rules themselves do the same for QL and CL. The e l i m i n a b i l i t y of "Cut" from CL, and the equivalence of that calculus to CN, together guarantee that there i s no connective or propositional constant of IP which acts as a "run-about inference t i c k e t for CN, that i s , whose introduction and elimination rules enable.us to derive an arbitrary formul B from any arbitrary formula A."*"22 Since CN i s sound, we know that no such constant can ex i s t ; the significance of the Cut-Elimination Theorem i s that i t establishes this result without 120. 121. Hacking (1977) , p.8. See P r i o r (1960). 298 recourse to the semantics of P . The lack of a Cut-Elimination Theorem for QL merely means that we must go outside proof theory to establish a sim i l a r result for quantum lo g i c . Thus i t does not invalidate the.'claim that we are dealing with a bona fide l o g i c , but rather t e l l s us that quan-tum l o g i c i s not just a science of deduction, with which Hacking would (probably) agree. X. 7. The Language Formalisation of the ideas presented i n the previous sections w i l l allow us to answer the question with which the chapter began. By analogy with the procedure of Chapter IV.2, we consider, for a given microsystem, the language W, whose sentences are the Q-propositions descriptive of that microsystem. Let H be the appropriate Hilbert space for the system, Sp(H) the set of subspaces of H , and B(H) the p a r t i a l Boolean algebra of subspaces of H. Then M = <SW),V1A> , where S m i s the set of sentences of V^ the set of admissible. valuations of Hi. S^ , i s defined recursively: X.7.1(a) (b) I f L e Sp(H), then " [ L ] " i s a sentence of I M ; i f A and B are sentences of flA^ then r(A v B)' and r(A A B)1 are sentences of (c) i f A i s a sentence of then so i s ^ A*1 ; (d) nothing i s a sentence of M, except by virtue of (a), (b) or (c) ab ove. The admissible valuations of lYl map into {0,1} 299 X.7.2. v i s an admissible valuation for M/ (v e V ) i f f —x — —x there i s a vector x € H , and v i s the function v :S ->Z_ , -^ x —x 2 such that, for any subspaces L , M of H and sentences A, B of W/, (a) v ([L]) = 1 i f f x e L —x and, of v^A) = ^ ( [ L ] ) and v^B) = ^ ( [ M ] ) , (b) v (A v B) = v ([L] v [M]) = v ([L v M ] ) -x x —x v (A A B) = A T ( [ L ] A [ M ] ) = v ([L A M ] ) (c) v (~A) =-v (~TiL]) = v ( [ L ' ] ) X X A , Clause (a) t e l l s us that each atomic sentence [L] of M, i s to be read as the Q-proposition x £ L , i . e . that Tit i s an interpreted language, and clauses (b) and (c) guarantee that each complex sentence of ffy i s semantically equivalent to some Q-proposition. Since the states of the system are represented by vectors of H , the admissible valuations of TH/ are state-induced. We now construct an algebra of Q-propositions isomorphic to the p a r t i a l Boolean algebra of subspaces of Hi l b e r t space. Then every u l t r a f i l t e r U on B(H) i s the p r i n c i p a l f i l t e r generated by some one-dimensionalsubspace L of H . I f xTT:Sp(H)—Z„ i s the character-X u z. i s t i c function on Sp(H) associated with U , and x £ , then c l e a r l y , x for arbitrary M £ Sp(H), X.7.3 ^ ( [ M ] ) = X Y ( M ) Since i s a t r a n s i t i v e . p a r t i a l Boolean algebra we can provide interpretations of Qj within tH^ and use these interpretations to furnish 300 admissible valuations of Q>. Now the elements of ( l i k e those of B^ i n Chapter IV.3) have a fixed interpretation: i t follows that an i n t e r -pretation a* of Cb within B^ i s an interpretation of the formal language (XJ into mathematical English, s p e c i f i c a l l y into the language appropriate to quantum theory. Also, i f a* i s an interpretation of within B^ and v^ i s an admissible valuation of Wl, then, from X.7.3, X.7.4 a*.v i s an admissible valuation of CL —x To summarise: we have an interpreted language , whose sen-tences are the Q-propositions descriptive of a given microsystem. The algebra of these Q—propositions i s an algebra isomorphic to the t r a n s i t i v e p.B.a. of subspaces of a Hil b e r t space appropriate to that system. We also have a formal language and an associated system of l o g i c , designed precisely so that GL may receive interpretations within such algebras. This l o g i c sets i t s e l f t r a d i t i o n a l goals: i t deals with sentences and the relations between them. I t i s the lo g i c which bears the same relationship to the propositions of quantum mechanics as does c l a s s i c a l l o g i c to those of Newtonian mechanics. 301 Chapter XI. The Scope and Limitations of Quantum Logic. XI. 1 P r o b a b i l i t i e s and Truth Values. I have presented a number of systems of quantum l o g i c , placing special emphasis on those systems which involve the p a r t i a l Boolean algebra of subspaces of a Hi l b e r t space; I have also drawn attention to a class of sentences occurring within quantum theory to which these systems of log i c may be applied. But these .logics have d i s t i n c t l i m i t -ations. In at least one clear sense they do not achieve for quantum mech-anics what c l a s s i c a l l o g i c does for Newtonian mechanics. Further, they neither commit us to a s p e c i f i c interpretation of quantum theory, nor do they resolve the paradoxes associated with that theory. Their scope i s also l i m i t e d , and any claim that acceptance of quantum mechanics requires us to revise the laws of l o g i c should be greeted with caution. These limitations of quantum l o g i c are the subject of my f i n a l chapter. We have seen that the language & receives an interpretation within the p a r t i a l Boolean algebra of Q-propositions, that i s , within the p a r t i a l algebra of theoretical statements about a given microsystem. To each such Q-proposition there corresponds an equivalence class of. Q-statements about the system (see Ch.VI.1): corresponding to the Q-proposition "x e L" are just those statements of the form "Val(M ) e S" which are certain i f and only i f the state of the system l i e s within L. For each such statement X describing the outcome of an experiment on the system we have, X i s certain i f f L = p(X) ; (VI.1.2) . 302 and we can, again by analogy with the procedure outlined i n Ch.IV.4, construct a language W g for a system, whose atomic statements are Q-statemetns and whose connectives correspond to the operations on this p a r t i a l algebra B A s Now l e t X be a Q-statement about the system belonging to the equivalence class [X] , and h be the isomorphism from to By^ , so that XI. 1.1 h[X] = [ p(X)] Then we may extend the domain of any valuation, v^ of % to include the set of Q-statements, so that, for the arbitrary Q-statement X, XI.1.2 v (X) = v (h[X]) = v [ p(X)] X X X These functions then serve as admissible valuations for s Thus far the p a r a l l e l with the Newtonian case holds. However, as we have already noted (p. ), the categorical predictions of quantum theory are merely l i m i t i n g cases of p r o b a b i l i s t i c predictions. I f X i s a Q-statement about a system which i s i n state x , then, from VI.1.2. and XI.1.2, XI.1.3 v (X) = 1 i f f X i s certain. — x Thus, for instance, for an electron i n the state x +, XI.1.4 v (val (S ) = +1/2) = 0 — x + y since i t i s not certain that an experiment on this p a r t i c l e to measure 303 i t s y-component of spin w i l l y i e l d the resul t = + 1/2» In fac t , from quantum theory we have, XI.1.5 p (val(S ) = + 1/ 2)= -I2 (V.8.11-13) x+ y Immediately we see that the information supplied by quantum l o g i c i s markedly less than that offered by quantum theory. This i s i n contrast with the c l a s s i c a l case. The bivalent semantics of c l a s s i c a l l o g i c are appropriate for Newtonian mechanics, according to which every N-statement pertaining to a system i n a given state i s either true or fa l s e . Quantum theory, on the other hand, i s inherently p r o b a b i l i s t i c ; the bivalent semantics adopted for the languages tW and lK>g do not allow us to distinguish Q-statemtns with ne g l i g i b l e probability from those whose probability approaches (but does not equal) certainty. We can f a i r l y e a s i l y adjust the semantics so that we not only assign different truth values to statements X and Y when p(X) = 1 and p(Y) < 1 (or vice versa), but also evaluate d i f f e r e n t l y statements X , Y and Z such that p(X) = 1 > p(Y) > 0 = p(Z) We do this by moving to a three-valued l o g i c obtained as follows. I describe the move i n terms of the language WL of Q-propositions; the corresponding move for the language 1^ i s obvious. Let be a p a r t i a l Boolean algebra of Q-propositions for a system, and U be the X u l t r a f i l t e r on B^ generated by the atom of B^ corresponding to the 304 vector x e H. We define an ultraideal I corresponding to U by x x writing XI.1.6 I = , {[M]1: [M] e U } x df x In a Boolean algebra, the union of an u l t r a f i l t e r and the corresponding ultraideal i s the set of a l l members of the algebra, and their inter-section is empty. In the case of a p.B.a. the two are mutually ex-clusive, but exhaustive only when i t is also a Boolean algebra. We now define a language ftl^ whose sentences are just those of , but whose admissible valuations are trivalent. These valuations, which I c a l l T-admissible valuations, are defined as follows. Let U x be an u l t r a f i l t e r on B^ , I the corresponding ultraideal. Consider the function {0,1,T} such that, for any sentence [M] of WL^ , XI.1.7 (a) w ([M]) = 1 i f f [M] e U —x x w ([M]) = 0 i f f [M] e I .X. 2V w ([M]) = T i f f [M] i U and [M] I I X X X (b) and (c) as in X.7.2 (b) and (c). XI. 1.8. w is a T-admissible valuation of tK,„ i f f there —x 3 is an u l t r a f i l t e r U on BM and w i s a x ~ —-x function defined in accordance with XI.1.7. For any subspace M e Sp(H) and sentence [M] e S , XI.1.9 v ([M]) =1 i f f w ([M]) = 1 i f f x e ft X X Such a move, from bivalent to trivalent valuations, would enable us to distinguish within the semantics between the two possible cases that may occur i f x I M , that i s , the case when x e M' and the case when 305 x i s skew to M. ( i f P^ i s the projection operator onto M , then P^x = 0 i n the f i r s t case, and P„x > 0 i n the second.) Note that on this semantics M the connective "~" of i s truth functional, and should be read as 122 diametrical negation. We have the truth table: A ~A 1 0 T M 0 1 Clearly a sim i l a r emendation could be made to the d e f i n i t i o n of - 123 an admissible valuation for K>. • The conditions under which a formula of &\j would be assigned a truth value "1" would remain unchanged, as would those under which i t received no evaluation. Thus the modification would e n t a i l no adjustment to VIII.4.5, which relates the notion of entailment to that of an admissible valuation. However, this enrichment of the semantics for 1 ^ and for Cb would not get us very far. A l l those Q-statements to which, for a system i n a given state, quantum theory assigns a probability lying i n the open i n t e r v a l (0,1) would be assigned the value "T" by the 122. Note i n . t h i s regard Dummett (1959), p.106: Finer d i s t i n c t i o n s between dif f e r e n t designated (truth) values or different undesignated ones [than a bivalent semantics supplies], however naturally they come to us, are o n l y . j u s t i f i e d i f they are needed i n order to give a truth-functional account of the formation of complex statements by means of operators. 123. Seg"Ch.VIII.3. In the following discussion I.refer to the semantics outlined i n that chapter as bivalent, and the "modified i n neither case do I regard "N" (= "No evaluation") as a truth value. 306 appropriate T-admissible valuation. We would s t i l l have lost a considerable amount of the information provided by the.theory. Jauch and Piron make a similar comment while discussing the three-valued quantum 124 logic proposed by Reichenbach. For Reichenbach, elementary propositions about quantum mechanics should admit three truth values: true, false and undetermined. In view of the fact, however, that the state of a system attributes to each yes-no experiment a probability function p(a) with 0 ^ p(a) S 1 , i t seems more natural, once one has passed beyond the ordinary double-valued logic, to consider 'quantum logic' as an infinite-valued logic. But for the quantum logics I have been examining i t is not clear how one would set about constructing an infinite-valued semantics. In fact, the structure of these logics leads us to ignore the probabil-i s t i c features of quantum thoery, and to focus on a particular class of Q-statements with probabilities one or zero. Within Newtonian mechanics, the bivalence of the logic of N-statements was linked to an interpretation of the theory, under which we thought of each system as having just those properties vouched for by the N-statements which were true of that.system. The fact that the obvious semantics for the quantum logics reviewed here are bivalent or trivalent may be seen as a failure of those logics to come to grips with a fundamentally probabilistic theory',- or as evidence of the conservatism of an approach which seeks to retain' the notion of a "property" of a physical system in the face of a theory within which that concept is as anachronistic as a Beefeater within MI5. 124. Reichenbach (1944) and Ch.VTII.5; also Jauch and Piron (1970) 307 XI.2. States and Properties. In quantum theory we cannot move straightforwardly from t a l k of the,state of a system to talk of i t s properties, as we can i n the. Newtonian case. Indeed i t i s not clear i n what sense we can talk of "properties" of a microsystem at a l l . In the c l a s s i c a l case, since knowledge of the state of a system allows us to predict with certainty the outcome of any experiment on i t , we can ascribe to the system certain properties; these properties are values of observable physical magnitudes l i k e k i n e t i c energy, which are, i n p r i n c i p l e , measurable by such experi-ments. Values of a l l observable physical magnitudes can be simultaneously assigned to a system, and the state of the system i t s e l f i s specified i n terms of such properties, the positions and momenta of the p a r t i c l e s which comprise i t . In quantum mechanics this i s no longer true. The state vector (or s t a t i s t i c a l operator which describes the state) i s not among the observable physical magnitudes for the system (pace, Jauch) , and two si m i l a r systems prepared i n the same way.and i n the same quantum mechanical state, may w e l l y i e l d d i f f e r e n t results when the same exper-iment i s performed on each. As we saw i n Chalter X, a beam of p a r t i c l e s a l l i n state x + i s s p l i t i n two by an experiment designed to discriminate between between p a r t i c l e s with positive and negative y-components of spin. I t could even be argued that talk of "discriminating" between these p a r t i c l e s begs several questions, since i t carries the implication that, p r i o r to the experiment, some of the p a r t i c l e s had a positive y-component of spin while others had a negative y-component. On some interpretations of quantum theory we should accept such an implication (e.g. on the 308 s t a t i s t i c a l interpretation and the Rayski interpretation), but on the orthodox view we should r e f r a i n from doing so. On this view we can either say that the notion of a property of a physical system i s inappropriate at the quantum l e v e l , or that the properties of microsystems hang together i n a peculiar way, so that 125 certain kinds of properties are mutually exclusive. We may note i n this regard that i n quantum theory angular momentum, for instance, i s defined just i n terms of commutation r e l a t i o n sV^ XI.2.1 A vector operator J i s an angular momentum i f f i t s Cartesian components are observables s a t i s f y i n g commutation relatio n s : [Jx»Jy] = i J z c y c l i c a l l y where [J ,J 1 = J J - J J x y x y y x Thus i f we regard values of components of angular momentum as properties we must accept the fact that only one of the three can be specified at any one time. The p a r t i c l e i n spin state x + has no property corresponding to i t s y-component of spin: these two properties may be said to be incompatible. An interpretation of quantum theory which makes use of this attenuated concept of a property I w i l l c a l l a "neo-classical" i n t e r -pretation. 125. Obviously, even i n c l a s s i c a l physics, certain properties are mutually exclusive: since k i n e t i c energy, E, momentum, p, and mass, m, are related by the equation E = p2/2m , a system cannot have the properties p=2 , m=l , E = 1 (in consistent u n i t s ) ; i n quantum physics, however, i t i s not possible to specify precisely the values of two non-commuting observables, unless they happen.to share an eigenvector whcih represents the system's state. 126. Lectures'by J.M.McMillan, University of B r i t i s h Columbia, Feb. i 3 t h , 1974. See also Schwinger (1965). 309 The concept i n question makes (marginally) more sense i n the example given, where, by a modest indulgence i n metaphor, we can say that the various components of anangular momentum can be thought of as different aspects of teh same physical r e a l i t y , than i t does when the incompatible pair of observables are, say, position and momentum; we may then agree with Bohm when he writes that "the concepts appropriate at the quantum l e v e l are those of incompletely defined p o t e n t i a l i t i e s , " 127 rather than properties, or with Heisenberg's remark i n simi l a r vein, that a sp e c i f i c a t i o n of the state of a system contains "statements about p o s s i b i l i t i e s , or better tendencies (potentia i n A r i s t o t e l i a n philosophy). However, despite Bohm's advocacy of the notion of p o t e n t i a l i t y , he does not forswear the use of the word "property": i n fact some of his comments are very close to the neo-classical view. He writes, We s h a l l see that a given system i s capable, i n p r i n c i p l e , of demonstrating an i n f i n i t e variety of properties that cannot a l l e x i s t i n simultaneously well-defined forms. ... These properties w i l l actually become de f i n i t e only when the object i n question interacts with an appropriate system, such as a suitable measuring apparatus, that brings about the r e a l i s a t i o n of this p a r t i c u l a r property i n a def i n i t e form. In this passage the "properties" of the neo-classical view reappear as 127. See Bohm (1951), p.626. I t i s not clear that these are his present views. 128. Heisenberg (1958), p.53. 310 "realised properties". I f we can make sense of the neo-classical concept of property (or of Bohm's notion of a."realised property), then quantum l o g i c enables us to employ i t without f a l l i n g f o u l of quantum theory. I f we regard N-statemehts as property ascriptions rather than as statements of experi-mental r e s u l t s , then, by r e s t r i c t i n g ourselves to the language which contains these statements, we are prohibited from simultaneously 129 ascribing two incompatible properties to a system. This follows from the way the connectives of ftt correspond to p a r t i a l operations on the p.B.a. 23^ (and, v i a an isomorphism, to the p a r t i a l operations on 23(H)). Because, i n the p.B.a. B(H) for the spin */2 p a r t i c l e , the subspaces spanned by the vectors x +. and y + are not compatible, the sentence "Val(S x) = + V 2 A v a l ( S y ) = +1/2" i s not a sentence of Uf . s^ Within,the neo-classical interpretation, the adoption of a deviant lo g i c i s demanded by the use of a deviant notion of property. Property ascriptions also occur within two other interpretations of quantum mechanics, the s t a t i s t i c a l interpretation and the Rayski i n t e r -pretation. On the s t a t i s t i c a l i nterpretation, i n d i v i d u a l systems are said to possess certain properties i n the same way. that the systems of Newtonian mechanics have them; however, the term "state" i s properly applied not to single systems, but to ensembles of such systems which have undergone sim i l a r preparations. In support of this view, 129. More precisely, from uttering the conjunction of two such ascriptions. 311 Ballentine writes We see that a quantum state is a mathematical representation of the result of a certain preparation procedure. Physical systems which have been subjectd to the same state preparation w i l l be similar in some of their properties, but not in a l l of them [similar in x-component of spin in the example given on p. , but not in y-component]. Indeed the physical implic-ation of the uncertainty principle ... i s that no state preparation procedure is possible which would yield an ensemble of systems identical in a l l of their physical properties. Thus i t is most natural to assert that a quantum state represents an ensemble of similarly prepared systems, but does not provide a complete description of an individual system. On different grouds Rayski also suggests that we should consider 131 a microsystem as having a l l of i t s properties at any given instant. He suggests that we must ...assume that the eigenvalues of a l l observables characterising the system are i n t r i n s i c (immanent) for the system but that the concept of state i s not i n t r i n s i c and is no more than inform-ation about the system. In this case the role of the measure-ment that yields a maximum of information about the system 130. Ballentine (1970), p.361. In the original, the phrase in parentheses reads "similar in momentum but not position in the f i r s t example." 131. Rayski (1973), p.95. I w i l l not rehearse his arguments, which are based on the invariance of the laws of quantum mechanics with respect to time reversal. 312 (allowing us to determine a certain state vector) should be understood differently: the measurement of A gives us, as information, an eigenvalue which preexisted. On neither of these interpretations i s a specification of a state a f u l l description of a physical system: on the one hand i t is a description of the aggregated features of an ensemble, and on the other a summary of available information. Both of these views are at odds with a realist interpretation of the state vector. Yet nevertheless on these interpretations too, quantum logic has a role to play. For neither Ballentine nor Rayski deny that state descriptions are a fundamental ingredient of quantum theory, and the language fK developed in Chapter X is simply a language of state descriptions. Compare, for instance, Rayski's conclusions with those of an advocate of quantum logic like 132 Putnam, who writes: (T)he notion 'state' must be used with more-than-customary caution i f quantum logic is accepted. A system has no complete description in quantum mechanics; such a thing is a logical impossibility. ... A system has a position-state and i t has a momentum-state (which is not to say ' i t has position r^ and i t has momentum r ' for any r^, r^ , but to say '(It has position r., v ... v i t has position r ).(It has momentum -r. v ... v i t J- K 1 has momentum r )', as already explained; and a system has many other 'states' besides (one for each 'non-degenerate' magnitude). These are 'states' in the sense of logically strongest consist- ent statements, but not in the sense of 'the statement which implies every true physical proposition about S'. 132. Putnam (1969), pp.227. 313 Notice that we recommend quantum l o g i c to Rayski on account of his view of a state, while Putnam arrives at his view of a state as a resu l t of moving to quantum l o g i c , or so he claims. However, one wonders why he i n s i s t s on talking about " l o g i c a l l y strongest consistent statements", rather than "strongest statements consistent with quantum theory" except to beg the question of which has conceptual p r i o r i t y . 1 For the more quantum l o g i c displays i t s usefulness i n the context of widely d i f f e r i n g interpretations of quantum emchanics, the less we can talk of those views which we must have " i f quantum l o g i c i s accepted".' Moreover, quantum log i c i s s i l e n t on.the feature of quantum 133 mechanics which most requires interpretation, the measurement problem In c l a s s i c a l phys-cs a set of d i f f e r e n t i a l equations allows us to calculate the evolution of the state of a physical system through time (see IV.1.5-6). In quantum physics the evolution of a state of a system i s given by Schrodinger's equation: i f x i s the state of a system, then XI. 2.2 ih-9x Tt = H x In this equation -h: = Planck's constant, and H i s the Hamiltonian (energy operator) for the system. H i s a unitary operator, that i s , H has an inverse and leaves the norms of vectors unchanged: we have, for a l l x, XI.2.3 |Hx| = |x| i33. This discussion of the measurement problem i s based on Bub (1977b). 314 Now consider a single spin- 1/? p a r t i c l e i n state x, ° * -  Jr entering a Stern-Gerlach apparatus which i s oriented to provide a measurement of the 134 y-component of spin. Assume further that a counting device reveals that the p a r t i c l e leaves the apparatus i n the upper (y +) beam. Then we would assume that on i t s passage throught the apparatus the p a r t i c l e ' s state evolved from state x to state y , and that the probability of this t r a n s i t i o n was equal to |(y,>x,)| 2 = V ? . But there i s no unitary transformation which produces such a t r a n s i t i o n ; the system seems to have "jumped" into i t s new state. Thus i n addition to the unitary transform-ations dealt with i n quantum theory we also have to postulate "the existence of purely stochastic physical i n t eractions between physical 135 systems" - j u s t i n those case when one of those systems happens to be a measuring apparatus. According to (one version of) the Copenhagen interpretation, "the observation i t s e l f changes the prob a b i l i t y function discontinuously; i t selects of a l l possible events the actual one that has taken place. 1 1 Now, despite Putnam's claim that, one we give up the d i s t r i b u t i v e law of c l a s s i c a l l o g i c , "every single anomaly (of quantum theory) vanishes", 137 the adoption of quantum l o g i c , per se, does not solve t h i s problem. Indeed, as the problem has been presented here, i t i s not clear what 138 quantum l o g i c has to say about i t . P n o •< i „ „ „ . . ^ ^. n ° J Possibly an interpretation which 134. See Figure X.2 135. Bub (1977b), p.4. 136. Heisenberg (1958), p.54. 137. Putnam (1969), p.226. 138. Putnam's discussion of the measurement problem as exemplified in 315 considers the partial algebraic structure of the set of subspaces of Hilbert space (I have in mind and interpretation l i k e , say, Bub's theory of 139 generalised conditional probabilities ) could provide a solution to i t , but such an interpretation would have to go beyond quantum logic. As we have seen, between interpretations of quantum mechanics quantum logic is neutral. XI.3 The Scope of Quantum Logic. We have seen that there i s no such thing as a "quantum logical interpretation" of quantum mechanics, i f we take that to mean an inter-pretation which entails and i s entailed by the adoption of quantum logic. Rather the significance of quantum logic depends on the interpretation of quantum mechanics we adopt. Whichever interpretation we choose, we find that,in the context of quantum,theory, certain kinds of sentences are related in ways which differ from those in which otherwise comparable sentences are related within Newtonian physics. The relations are those explored by quantum logic. On different interpretations this class of sentences (the class of Q-propositions) has more or less physical impor-tance, and the significance to be attached to quantum logic varies accordingly. However, a few general remarks about the scope and applic-ation of this logic can be made. I introduce these by using Haack's use-f u l taxonomy of deviant logics, and then look very briefly at the problems 139 Bub (1977a). the two-slit experiment i s crit i c i s e d by Bub, because, inter a l j a , i t ignores the i n i t i a l state of the particle and also the distance between s l i t s and screen. See Bub (1977a), fn.4, p.340. 316 raised by acceptance of a non-classical l o g i c . Unlike, say, modal l o g i c , quantum l o g i c i s proposed as a r i v a l to c l a s s i c a l l o g i c rather than as a supplement to i t . I t would be wrong to use c l a s s i c a l l o g i c for the language of Q-propositions. This change i n l o g i c i s made on pragmatic grounds: by changing our l o g i c we can construct a language of this kind i n a comparatively simple and economical way, whereas the c l a s s i c a l l y defined connectives "&" and "v" f a i l to do the job we want. However, this change should be thought of as a l o c a l rather than a global reform. The domain of discourse to which quantum l o g i c i s appropriate i s s t r i c t l y l i m i t e d , i n contradiction to Haack's claim that "the adoption of a r i v a l l o g i c , i f j u s t i f i e d at a l l , 141 should normally be global." In support of her view, she writes, the following consideration might be urged. the principles of l o g i c are characterised by an extreme: generality - what makes them l o g i c a l principles as opposed to e.g. high l e v e l physical p r i n c i p l e s , i s precisely t h e i r n e u t r a l i t y as regards subject matter. ... (Ryle proposes 'topic-neutrality' as a c r i t e r i o n for picking out the l o g i c a l constants.) I f this i s r i g h t , there i s certainly something odd about supposing that one set of l o g i c a l p r i n c i p l e s might apply to one subject matter and a r i v a l set to another. For l o g i c a l p r i n c i p l e s would be precisely those which apply to any subject matter. 140. See Haack (1974), pp.1-3, and also her discussion of quantum logics i n Ch.8, pp.148-167. 141. Haack (1974), p.46. The extended quotation which follows i s from p.45. 317 But Haack here declines to give an independent s p e c i f i c a t i o n of what i s to count as a l o g i c a l p r i n c i p l e . She argues that r i v a l logics should be global i n application, by r u l i n g out of court principles that are not topic-neutral. But to be topic neutral i s presumably to be global i n application. This i s less an argument, therefore, than a st i p u l a t i v e definition-of a l o g i c a l p r i n c i p l e . And, as we saw i n Chapter X.6, there are alternative views of l o g i c , and other character-j' isations of l o g i c a l principles to hand besides topic n e u t r a l i t y . Perhaps the d i f f i c u l t y l i e s with the term " r i v a l " . Rivals are usually i n competition with each other, but, on the view presented here, there can be no competition between c l a s s i c a l l o g i c and quantum l o g i c . : Quantum lo g i c i s merely the lo g i c appropriate to a limited class of sentences appearing within quantum theory or i t s interpretation. As van Fraassen puts i t , No law i s contradicted by the proposal of a non-standard quantum l o g i c . Rather, what is shown ip tthat- we can construct languages for which the fam i l i a r laws do not hold; hence, what i s show i s that standard l o g i c has a limited domain of application. But to say this i s not to deny the d i f f i c u l t i e s associated with this view. In this apparently e g a l i t a r i a n concourse of log i c s , c l a s s i c a l l o g i c s t i l l has a privi l e g e d position. Faced with a given mathematical structure we may promote orthomodular l o g i c , or the l o g i c 142. van Fraassen, (1974), p.603. 318 of occasionally commutative monoids, or whatever seems appropriate, but to decide what the properties of the structure are we use standard mathematics and c l a s s i c a l mathematical l o g i c . I t seems that the upstart logics can never be Logic, although an adequate demarcation of Logic from logics s t i l l needs to be supplied. The alternative i s to suggest that quantum l o g i c i s global, given that we consider i t to be a l o g i c (see Ch.X.6). I t has been suggested (by Edwin Levy) t h a t ' c l a s s i c a l l o g i c i s a special case of quantum l o g i c , just as a Boolean algebra i s a special case of a p a r t i a l Boolean algebra. But this leaves unanswered the question of why, i n abstract reasoning, including mathematics, we should regard a l l statements as compatible (compatible, that i s , i n the technical sense: obviously they are not a l l mutually consistent). Further, i f throughout mathematics c l a s s i c a l l o g i c i s i d e n t i c a l with quantum l o g i c , then we have witnessed a scarcely revolutionary take-over, and the statement that quantum l o g i c i s global i n application seems a s l i g h t l y perverse thing to say. Yet, i f there are s i g n i f i c a n t differences between the two i n this domain, then the global party finds i t s e l f i n an even odder position. For quantum theory i s developed i n mathematical terms, and the mathematics involved, although modern i n the obvious sense, uses perfectly t r a d i t i o n a l canons of proof. 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