The foundations of probability and quantum mechanics

PETER MILNE THE FOUNDATIONS QUANTUM OF PROBABILITY AND MECHANICS ABSTRACT, Taking as starting point two familiar interpretations of probability, we develop these in a perhaps unfamiliar way to arrive ultimately at an improbable claim concerning the proper axiomatization of probability theory: the domain of definition of a point-valued probability distribution is an orthomodular partially ordered set. Similar claims have been made in the light of quantum mechanics but here the motivation is intrinsically probabilistic. This being so the main task is to investigate what light, if any, this sheds on quantum mechanics. In particular it is important to know under what conditions these point-valued distributions can be thought of as derived from distribution-pairs of upper and lower probabilities on boolean algebras. Generalising known results this investigation unsurprisingly proves unrewarding. In the light of this failure the next topic investigated is how these generalized probability distributions are to be interpreted. We begin with two interpretations of probability, one objective, one subjective (or, perhaps better, one physical, one epistemic). Both have been suggested as ways of understanding the probabilities that occur in quantum mechanics. Both are interpretations in the literal sense in that they have sufficient intrinsic mathematical structure that one can derive the characteristics of probability from them. Also both are "classical" in so far as the domain of definition of each interpretation is boolean. However, neither interpretation gives rise to the usual Kolmogorov axioms. Instead both yield axioms for distribution-pairs of upper and lower probabilities. The derivation of these axioms takes up w In w we briefly consider an application of upper and lower probabilities to classical physics. The heart of the paper lies in w and w In the first of these we consider the algebraic properties of the class of events or propositions, respectively, on which upper and lower probabilities coincide under the two interpretations. In both cases the general structure is that of an orthomodular, partially ordered set. It need not be a lattice. Nevertheless, since the lattice of subspace of a Hilbert space is orthomodular, the obvious next step (w is to investigate whether the point-valued probability distributions that occur in Journal of Philosophical Logic 22: !29-168, 1993. 9 1993 Kluwer Academic Publishers. Printed in the Netherlands. 130 PETER MILNE quantum mechanics can be interpreted as derived from distributionpairs of upper and lower probabilities over boolean algebras, thus permitting an uncontroversially classical interpretation of quantum mechanics, whether objectivist or subjectivist. Extending results due to Zierler and Schlessinger, we find that for most quantum systems this cannot be done. In w we give a brief but very general proof that there can be no phase-space representation of the vast majority of quantum systems employing upper and lower probabilities. Lastly (w we propose a generalized conception of probability that leads to objective and subjective interpretations of quantum mechanics. This generalized theory is non-classical in that the domain of definiton of a point-valued probability distribution is taken to be an orthomodular set. It is the natural generalization of our findings in w w TWO I N T E R P R E T A T I O N S OF P R O B A B I L I T Y We start with two interpretations of probability: relative frequencies of attributes in infinite sequences, and betting quotients subject to the constraint of weak coherence. I am not greatly concerned to defend this choice. Both approaches are sufficiently familiar. Moreover, if we are to obtain interesting conclusions concerning the formalization of probability we can only start from interpretations with a good deal of mathematical structure. With this desideratum in mind relative frequencies constitute the best objectivist candidate, betting quotients the best subjective option. We are not committed to the view that our chosen interpretations are definitions of probability. Whether one wants to say that physically meaninful probabilities are to be understood in terms of relative frequencies of attributes in infinite sequences is, in the present context, immaterial provided one grants that relative frequency serves as a reliable guide to the algebraic constraints satisfied by physically real probabilities. Similarly, whether one wants to say that degrees of belief are determined by rationally acceptable betting quotients for and against the truth of propositions is of no consequence provided one allows that rationally acceptable betting quotients reveal the algebraic structure of epistemic probabilities. On the other hand, PROBABILITY AND QUANTUM MECHANICS 131 whereas the majority of philosophical accounts of probability seek to validate something like the standard K o l m o g o r o v axioms for a pointvalued probability distribution, we do take the two interpretations sufficiently seriously not to impose a formal theory of probability on them, whether or not they are read definitionally. This said, we commence with an examination of relative frequencies of attributes in infinite sequences. We consider infinite sequences of trials. In each trial various attributes are determinately either instantiated or not. We may either suppose that the attributes themselves form a boolean algebra or consider the boolean algebra generated by the set of elementary events constituted by the instantiation of the attributes. We shall ignore the distinction and speak of the attribute space. The attribute space is a boolean algebra. Limits of relative frequency of attributes in an infinite sequence of trials need not exist. Indeed even when limits exist for the attributes a and b no limit need exist for their joint occurrence, i.e. for a/x b, or for the occurrence of one or other or both, i.e. for a v b. However, if a and b are disjoint attributes, so that at most one may be instantiated in any trial, a limit exists for a v b when limits exists for both a and b. What exists for every attribute are the lira sup and lira inf of relative frequency (see Figure 1). When, and only when, lira sup and lira inf coincide does a limit of relative frequency exist for the attribute in question. Given a space of attributes, which we assume to be _ ~ no>/n / \- sup. ~m(a)/m -)Z" x 7 m7 m >~n V/---~ Fig. t. 132 PETER MILNE closed u n d e r b o o l e a n o p e r a t i o n s , we define u p p e r a n d lower p r o babilities thus: P*(a) = lim sup # n(a)/n, P*(a) - lim inf # n(a)/n, the limits being t a k e n as n tends to infinity, a n d # n(a) being the n u m b e r o f i n s t a n t i a t i o n s o f a t t r i b u t e a in the first n trials. THEOREM 1. Upper and lower probabilities, defined by P*(a) P,(a) -- l i m s u p #n(a)/n = = limn~oo sup {m >~ n: #m(a)/m}, -- lim inf #n(a)/n = = l i m n . ~ i n f {m >~ n: #m(a)/m}, satisfy these conditions: 0 ~< P*(a) ~< P*(a) ~< 1; P , ( a c) = P*(O) = 1 O, P*(a), P,(1) P*(a r = = 1 - P*(a); 1; P*(a) + P*(b) ~ P*(a v b) ~< P*(a) + P*(b) ~< ~< P*(a v b) ~< P*(a) + P*(b) when a and b are incompatible. The domain of attributes is boolean; c is the complementation operation, v the join operation, and 0 and 1 the minimal and maximal elements, respectively, of the boolean algebra in question. Proof. P*(a) = lim inf # n(a)/n a n d P*(a) = lim sup # n(a)/n. Clearly 0 ~< P*(a) ~< P*(a) ~< 1. F o r all n ~ ~+, #n(a r = n #n(a), so sup {m ~> n: #m(aC)/m} = sup {m /> n: 1 -- #m(a)/m} = 1 -- inf {m >/ n: #m(a)/m}, thus P*(a c) = 1 - P*(a). By b o o l e a n n e s s o f the d o m a i n a c~ = a, hence P . ( a ~) = 1 - P*(a). 0 is the a t t r i b u t e no trial instantiates, 1 is i n s t a n t i a t e d by every trial, so for all n ~ ~ + , #n(O) = 0 a n d # n ( 1 ) = n, hence P*(O) = 0 a n d P*(1) = 1. W h e n a a n d b are i n c o m p a t i b l e #n(a v b) = #n(a) + P R O B A B I L I T Y AND Q U A N T U M M E C H A N I C S 133 # n(b) for all n e IN+, so sup {m >~ n: #rn(a v b)/m} = sup {m >~ n: #rn(a)/rn + + #m(b)/m} <~ sup {m /> n: #m(a)/m} + + sup {rn ~> n: #rn(b)/rn}, and sup {m i> n: #m(b)/m} = sup {m /> n: # m ( a v b)/m - - #m(a)/m} <~ sup {m >~ n: # m ( a v b)/m} - - inf {rn /> n: #m(a)/m}, hence P*(a) + P*(b) ~ P*(a v b) ~< P*(a) + P*(b). Similarly, P*(a) + P*(b) ~< P*(a v b) ~< P*(a) + P*(b). QED For the subjective/epistemic interpretation we consider bets on the truth of propositions expressible in some unspecified language. The logic of propositions is classical, i.e. boolean] A bet on the proposition a at betting quotient p and stake S pays (1 - p)S to the bettor if a is true and p S to the bookmaker if false. We consider only bets with non-negative stakes. We suppose that when a rational person finds some bet acceptable any other bet on which the pay-offs are not worse under all logically possible combinations of truth values to the propositions in question is also acceptable. This dominance constraint guarantees that bets are treated extensionally. We suppose further that mere size of stake does not affect acceptability, that the total amount staked on any family of bets is always finite, and that a bet in which no money changes hands in any circumstances - the null bet - is acceptable. A rational person does not enter into a family of bets which are such as to guarantee the loss of at least some minimum positive amount no matter what transpires. This is the constraint of weak coherence. Upper and lower probabilities are introduced thus: P*(a) = inf {p: p is an acceptable betting quotient at which to act as bookmaker for a bet on a}, P.(a) -- sup {p: p is an acceptable betting quotient at which to place a bet on a}. 134 PETER MILNE We obtain the formal structure of epistemic probability by considering what conditions must be satisfied by the upper and lower probabilities o f a rational person. THEOREM conditions: 2. So defined, upper and lower probabilities satisfy these 0 ~< P*(a) ~ P*(a) ~ 1; P*(a c) = P*(0) = 1 - P*(a),P*(a ~ 0, P,(1) = = 1 - P*(a); 1; P*(a) + P*(b) ~ P*(a v b) ~< P*(a) + P*(b) P*(a v b) ~< P*(a) + P*(b) when a and b are incompatible. The algebra of propositions is boolean: c is the propositional negation operator, v the disjunction operator, and 0 and 1 are the necessarily false and the necessarily true propositions, respectively. Proof. Let S*(a) = {p: p is an acceptable betting quotient at which to act as b o o k m a k e r for a bet on a}; let S,(a) = {p: p is an acceptable betting quotient at which to place a bet on a}. By dominance and the acceptability o f the null bet 1 ~ S*(a), 0 ~ S*(a). Also, by dominance, i f p ~ S*(a) and p ~< q then q ~ S*(a); i f p ~ S*(a) and p /> q then q e S*(a). Hence P*(a) and P*(a) are well defined. Obviously, then, 0 ~< P*(a) and P*(a) ~< 1. Let p and q be arbitrary m e m b e r s of S*(a) and S*(a) respectively. The net pay-offs for two bets, with the same stake S, at these betting quotients, are (1 - p)S - (1 - q)S = (q - p)S when a is true and - p S + qS = (q - p)S when a is false. By weak coherence (q - p)S >1 0 for all S ~> 0, hence q >~ p. Thus P , ( a ) ~< P*(a). Acting as b o o k m a k e r for a bet on a with stake S at betting quotient p is extensionally equivalent to placing a bet on a c - the classical negation of a - at betting quotient 1 - p at the same stake, hence P*(a c) = 1 - P*(a). Since classically a c~ = a we have immediately that P*(a ~) = 1 - P*(a). By acceptability of the null bet 1 is an acceptable betting quotient at which to place a bet on 1, the tautologous proposition, hence PROBABILITY AND QUANTUM MECHANICS 135 P*(1) = 1. As 1 -- 0c, the negation of the necessarily false proposition, P*(0) = 1 - P . ( I ) = 0. When a and b are incompatible, bets on a and b at betting quotients p and q respectively, both with stake S, yield net pay-offs to the bettor of (1 - (p + q))S when a v b is true, and - ( p + q ) S when it is false. By extensionality then, P*(a) + P*(b) ~< P*(a v b) and P*(a v b) ~< P*(a) + P*(b). Again when a and b are incompatible, if one places a bet on a v b at betting quotient p and one acts as bookmaker for a bet on b at betting quotient q, the stake being S in each case, the net pay-offs are (1 - ( p - q))S when a is true and ( p - q )S when it is false. By extensionally this is the same as placing a bet on a at betting quotient p - q with stake S. Hence P*(a v b) - P*(b) ~< P*(a), and so P*(a v b) ~< P*(a) + P*(b). Similarly, P*(a) + P*(b) ~< P*(a v b). QED - On both interpretations of probability we obtain a distribution pair assigning upper and lower probabilities over a boolean algebra, in the objective case the attribute space, in the subjective the algebra of propositions expressible in some language. On either scheme for their introduction distribution-pairs satisfy the same axioms. Henceforth, unless otherwise indicated, we consider a distribution-pair to be any pair, (P*, P.), of functions from the domain, B, of a boolean algebra, B, into the interval [0, 1], that satisfies these conditions: 0 ~< P*(a) ~< P*(a) ~< l; P*(a ~) = P*(O) = 1 - P*(a),P*(a c) = O, P,(1) = 1 - P*(a); 1; P*(a) + P*(b) ~< P * ( a v b ) ~< P*(a) + P*(b) ~< P*(a v b ) ~< P*(a) + P*(b) when a and b are incompatible. Here ~ is the complementation operation, v the join operation, and 0 and 1 the minimal and maximal elements, respectively, of the boolean algebra B. 136 PETER MILNE w AN APPLICATION TO CLASSICAL PHYSICS Standard probability theory takes a probability space to be a triple (X, B, P) where X is some set, B is a a-algebra, and hence a boolean algebra, o f subsets of X, and P is a countably additive probability distribution over the members of B. In general it is not the case that B is fi(X), the powerset algebra containing all subsets of X. Indeed it is well known that there are cases in which it would, for given B and P, be impossible to extend P to a distribution over all subsets. 2 However, when we admit distribution-pairs of upper and lower probabilities no such impossibility impedes extension of the point-valued distribution. T h a t is, given a standard probability space (X, B, P) there is a distribution-pair (P', P,) defined over fi(X) such that P'(b) = P,(b) = P(b) for all b in B. In fact there is a canonical extension defined by: P*(a) = inf {P(b):b e B and a c_ b}; P . ( a ) = sup {P(b) :b ~ B and b _c a} for all a in fi(X). We prove a more general result. T H E O R E M 3. I f B" = (B', <~', 0~,, IB,, t ) is a boolean algebra, with domain B', partial order <~", minimal and maximal elements 0B,, 1B', respectively, and complement t, and i is an injective homomorphism from B" into the boolean algebra B = ( B, <~, 0a, 1B, c) then the distribution-pair (P', P,) defined on B' extends under i to a distribution-pair (P", P,,) of upper and lower probabilities on B. (P', P,) extends under i to (P", P " ) / f P"(i(b)) = P'(b) and P,,(i(b)) -- P,(b), for all b ~ B'. We define the canonical extension (P*, P . ) of (P', P,) to B under i thus: P*(a) = inf {P'(b) :b e B' and a ~< i(b)}; P*(a) = sup {P,(b):b ~ B' and i(b) <~ a}; We must show that (P*, P*) is a distribution-pair defined on B and that it extends (P', P,) to B under i. Proof. As i(0a,) = 0B and i(1R,) = 1B P* and P* are well-defined; also 0 ~< P*(a) and P*(a) ~< 1. And P*(0B) = 0, P*(1B) = 1. If i(b) ~ a <~ i(c) then, by injectivity of i, b ~<' c, and, as c = b v '(c /x 'bt), P,(b) ~< P,(b) + P'(c /x 'b'~) ~< P'(c). 3 Consequently, P*(a) ~< P*(a). P*(a ~ = sup {P,(b):b ~ B' and i(b) ~< a c} = 1 - inf PROBABILITY AND QUANTUM MECHANICS 137 {P'(b'~):b e B' and a ~< i(b'~)} = 1 - P*(a), since b t t = b for all b in B'. Similarly, P*(a c) = 1 - P*(a). Next we show that P*(a) + P*(b) ~< P*(a v b) ~< p . ( a ) + P*(b) ~< P*(a v b) ~< P*(a) + P * ( b ) w h e n a and b are incompatible, i.e. when a ~< b ~ We do this in four steps. (i) Let c, d e B', i(c) <<, a, i(d) <<. b. Then i(c) <~ a <~ b ~ <~ i(d) c = i(d'~), and so c 4 ' d~'. Also, i(c v ' d ) = i(c) v i ( d ) <~ a v b. So P,(c) + P,(d) ~< P'(c v ' d ) ~< P*(a v b). And so P*(a) + P*(b) ~< P*(a v b). (ii) Let c, d e B ' , i(c) <<, a v b, b <~ i(d). Then i(c A ' d'f) = i(c) A i ( d ) c ~ (a v b) A b c = a. A s c = (c A ' d ) v (c A ' d ~ ' ) , P , ( c A' d#) ~> P'(c) - P'(c A' d) /> P'(c) - P'(d). So P'(e) - P'(d) ~< P*(a). And so P*(a v b) ~< P*(a) + P*(b). (iii) Let c, d e B ' , a v b <~ i(c), i ( d ) <~ a. Then i ( d ) <~ i(e), and so d ~<' c. Consequently e = d v (c A' dt), and thus P'(c) - P,(d) >~ P'(c A 'd?). N o w i(c A' d{) = i(c) A i ( d ) ~ >~ (a v b) A a c = b (as b ~< a~ So P'(c) - P'(d) ~> P*(b). And so P , ( a ) + P*(b) ~< P*(a v b). (iv) Let c, d e B', a <~ i(c), b <~ i(d). Then a v b <~ i(c) v i ( d ) = i(c v ' d). So P*(a v b) ,G< P'(c v ' d) -G< P'(c) + P'(d A' c~') <~ P'(c) + P'(d). And so P*(a v b) ~< P*(a) + P*(b). Lastly we show that (P*, P , ) extends (P', P,) to B under i. F o r b, c e B', b ~<' c just in case i(b) ~ i(c) and P'(b) ~< P'(b) + P,(c A' b t ) ~ P'(c), hence P*(i(b)) = P'(b). Similarly, P*(i(b)) = P,(b). Q E D Letting (P', P,) be the pair (P, P) and i the identity map in Theorem 3 we obtain immediately C O R O L L A R Y 3.1 Where (J(, B, P) is a standard probability space P extends to a distribution-pair defined o n / ~ ( X ) . In fact, as Theorem 3 m a k e s clear, we n e e d only suppose B to be a boolean sub-algebra o f </a(X), c , c), and P to be finitely additive f o r this result to hold good. Given two distribution-pairs (P', P,) and (P', P-) defined over the same domain we say that (P', P,) is a refinement o f (P', P-) when for all dements a in the d o m a i n P-(a) ~< P,(a) ~< P'(a) ~< P"(a). Distribution-pairs on a c o m m o n domain are partially ordered under the refinement relation. We say that (P', P-) can be indefinitely 138 PETER MILNE refined if a refinement (P', P,) exists in which P' = P,. The canonical extension is the coarsest extension of P to ~(X). This follows from C O R O L L A R Y 3.2. The canonieal extension is the coarsest extension of (P', P,), defined on B' to B under i. Proof Where (P", P,,) on B extends (P', P,) on B' under i, if a ~< i(b) then P'(b) = P"(i(b)) ~> P',(i(b) A a ~ + P"(a) >~ P"(a). Hence P*(a) >~ P"(a). Similarly, P*(a) ~< P,,(a). QED Theorem 3 is our first result of relevance to the philosophy of physics. Classical physics uses phase space to represent the motions of systems of bodies, usually particles. If there are n bodies phase space is 6n dimensional, each point representing the three components of position and of momentum of all n particles in some possible configuration of the system. Classical thermodynamics makes use of probability distributions that assign probabilities to regions of phase space as functions of volume, where volume is determined by Lebesgue measure. But, granted the Axiom of Choice, there are regions of phase space to which no Lebesgue measure can be assigned, and consequently no probability. Not zero probability - no meaningful probability at all. From a physical point of view this restriction to Lebesgue measurable subsets of phase space may well seem arbitrary - why should some regions be treated differently from others? What we have shown is that a point-valued probability distribution on phase space can be (canonically) extended to a distribution-pair that assigns upper and lower probabilities to all regions of phase space. And since upper and lower probabilities have a natural interpretation in the two analyses of probability with which we began, recourse to a physically unmotivated restriction can be avoided. 4 On the two interpretations the impossibility of extending P to a point-valued distribution has different meanings. On the relative frequency side it means that increasing the number of trials will not decrease the spread of relative frequency to zero for all attributes. On the epistemic side it means that the belief state cannot be indefinitely refined. (If a belief state cannot be refined indefinitely then afortiori for some propositions fair betting quotients do not exist.) PROBABILITY AND QUANTUM MECHANICS 139 w THE ALGEBRA OF COINCIDENT UPPER AND LOWER PROBABILITIES Both the interpretations of probability of w give rise to distributionpairs of upper and lower probabilities over boolean algebras. For some, but not necessarily all, elements of the boolean algebra upper and lower probabilities coincide. (On the epistemic interpretation notice that even when P*(a) = P*(a) a fair betting quotient for a need not exist.) When the upper and lower probabilities of a coincide we say that a has a point-valued probability. Those elements of the domain that have a point-valued probability under some given distribution-pair form what I, in vague imitation of Dynkin, shall call a p-field. The immediate task is to determine the algebraic structure of p-fields. There are a number of ways to explain boolean algebras. The most common is to think of a boolean algebra as a structure closed under the unary operation of complementation and the binary operations of meet and join. Thus to each element a there corresponds its complement a c, to each pair of (not necessarily distinct) elements a and b there correspond the two elements a A b (the meet of a and b) and a v b (the join of a and b). The algebra contains at least the two elements 0 and 1, which satisfy the conditions: a /x a c = 0, a v a ~ = 1. The operations /x and v are associative and commutative, and each distributes over the other (by commutativity, on both left and right). Also (a /~ b) v b = b and (a v b) /x b = b. The two best known examples of boolean algebras arise when we consider (i) the subsets of some given set with ~ A, and v interpreted as (relative) complement, intersection and union respectively, with 0 as the empty set and 1 as the given set itself, and (ii) the propositions expressed by the sentences of some language with c, A, and v interpreted as classical negation, conjunction and disjunction respectively, 0 as the absurd proposition, true under no circumstances, and 1 as the tautologous proposition, true in all possible circumstances. (It is common but not universal practice in philosophical logic to identify the proposition a sentence expresses with the set of possible worlds in which the sentence is true; (ii) then collapses to a special case of (i).) 140 PETER MILNE On a boolean algebra characterized as above we can introduce a new relation - ~< - by stipulation: a ~< b, if, and only if, a /x b = a. The relation ~< is provably reflexive, antisymmetric and transitive, i.e. it is a partial order. Moreover, 0 and 1 are, respectively, least and greatest elements under ~<, a A b is the ~<-greatest element less than or equal to both a and b, and a v b is the ~<-least element greater than or equal to both a and b. In any partially ordered set meets and joins can be characterised thus: the meet of a and b, should it exist, is the greatest element less than or equal both to a and to b; the join of a and b, should it exist, is the least element greater than or equal both to a and to b. Given two elements of a paritally ordered set there is no guarantee that one or the other or both the meet and the join exist. By definition a lattice is a partially ordered set in whch a meet and a join, characterised in terms of the ordering relatiton, exist for every pair of elements in the set. In a partially ordered set with least and greatest elements, 0 and 1 respectively, an element a has a complement just in case there is some element b in the set such that a /x b = 0, a v b = 1. A partially ordered set, and hence a lattice, is complemented if greatest and least elements exist and every element has a complement. Complements need not be unique, but in a lattice in which the operations of meet and join, introduced via the partial ordering, are distributive complements are unique when they exist. Obviously, in the light of our initial characterization of a boolean algebra, every boolean algebra is, under the ordering <~, a complemented distributive lattice. Less obviously, every complemented distributive lattice is a boolean algebra in which meets and joins are introduced via the partial ordering. This gives us a second characterisation of boolean algebras. Namely, we start from the lattice, taking the partial ordering and complementation as primitives, and define joins and meets. For the purposes of this paper it is advantageous to concentrate on this approach. Going back to our two examples, the lattice of subsets of a given set is partially ordered by set-inclusion (the subset relation), the propositions of a language are partially ordered by logical entailment (a ~< b if, and only if, a entails b). The intersection of two sets is the largest set that PROBABILITY AND QUANTUM MECHANICS 141 is a subset of both sets. The conjunction of two propositions is the logically weakest proposition that entails both propositons. 5 Given either an attribute space or the algebra of propositions expressed in some language and a distribution-pair of upper and lower probabilities defined over them we wish to consider the p-field comprising exactly those elements whose upper and lower probabilities coincide. Recall that upper and lower probabilities satisfy the superand sub-additivity condition: P*(a) + P*(b) ~< P*(a v b) ~< P*(a) + P*(b) ~< P*(a v b) ~< P*(a) + P*(b) when a and b are incompatible. In a boolean algebra incompatibility may be expressed in various ways. Since we want to use the lattice formulation the most appropriate for present purposes is this: a and b are incompatible just in case a ~< b c.6 Since in any boolean algebra a ~< b when, and only when, b~ ~< a ~, and a ~ = a, the incompatibility relation, so defined, is symmetric, as it should be. Let L be the set of elements in the p-field obtained from a boolean algebra B = (B, ~<, ~) under some assignment of upper and lower probabilities, whatever the provenance of that assignment. 7 In view of the above-given axioms governing lapper and lower probabilities, L satisfies these closure conditions: OB~L; I ~ L ; i f a ~ L then a ~ ~ L. Also T H E O R E M 4. I f any three o f a, b, a v b, and a A b belong to L then so does the fourth. Proof. It follows easily from the super- and sub-additivity axiom (i) that if, c, d E L a n d c ~< d c t h e n c v d ~ L , and(ii) t h a t i f c , d e L , d = c v e and e ~< cc then e ~ L. Consequently, i r a , b and a /x b ~ L t h e n a c /~ b ~ L , a s b = (a /x b) v (a ~ /x b ) , a n d s o a v b ~ L a s a v b = a v (a c A b ) . I f a , b a n d a v b ~ L t h e n a c /x b E L , as 142 PETER MILNE a v b = a v (a ~ /x b), and so a A b E L as b = (a /x b) v (a ~ /x b).Ifa, a A banda v beLthena c ix b ~ L , a s a v b = a v (a ~ A b), and so b ~ L as b = (a A b) v (a c v b). Similarly, if b, a v banda v beLthenaeL. QED Since a /x b = OB when a ~< b c these conditions entail what is often a useful constraint: C O R O L L A R Y 4.1. I f a <<. b c a n d any two o f a, b and a v b belong to L , then so does the third. As we shall see below (Examples 1 and 2), we cannot weaken the antecedent in this condition. F o r example, if a and b are both in L but are not incompatible we have no information on whether either their meet or their join belongs to L, although we do know that if one does then so does the other. The three conditions above obtain on the p-field containing those elements of some boolean algebra having point-valued probabilties. The conditions are expressed using the order-relation and operation of complementation defined on the boolean algebra B. Meets and joins are characterised in terms of the order-relation on that algebra. A p-field is the domain of definition of a pointed-valued assignment of probabilities determined by a distribution-pair defined over a boolean algebra. In seeking the algebraic structure of p-fields we want to characterise the domain of a point-valued assignment without appeal to the boolean algebra underlying it. The characterization we seek is internal - in other words, it should appeal to relations that obtain only between elements of the p-field in question. In order to find such a characterization we first introduce ~<', the restriction of ~< to L, the set of elements of the p-field. As ~< is a partial order on B, the domain of the boolean algbra B, ~<' partially orders L. By the first closure condition on L, 0B and 1B are, respectively, ~<'-least and ~<'-greatest elements of L. We can define meets and joins in L in terms of ~<'. Since L is only a subset of B neither meet not join need exist, and where either does for some pair of elements it need not be the same as in the original algebra. On the other hand, by the second closure condition on L, the complements of PROBABILITY AND QUANTUM MECHANICS 143 elements o f L are also in L. Since incompatibility is defined in terms o f the partial order and complementation if a and b are incompatible in the original algebra they remain so as elements o f L, provided both belong to it. i.e. if a, b E L and a ~< b ~ then a ~<' b c, as b ~ ~ L. Also, if a and b are incompatible in B their original join also belongs to L and is their join relative to ~<'. Symbolically, using v ' to represent joins defined relative to ~<', we have that a v b ~ L and a v b = a v ' b. We k n o w that if a ~ L then a c ~ L. In B we have that a ~< b if, and only if, b ~ <~ a c , a n d t h a t a ~c = a. A l s o : a A a c = OB, a v a ~ = 1H. Consequently L's elements inherit their own complement operation from B - denote it t - and these conditions obtain: a ~<' b if, and only if, b t ~ ' at; at+ = a; a /~ ' a t = 0B, and a v ' a t = lB. The operation t is just the restriction of c to L. By the third condition t is a complement on L. As all three conditions are satisfied t is an o r t h o c o m p l e m e n t a t i o n operation. We have then that the structure L = ( L , ~<', t ) is an o r t h o c o m p l e m e n t e d partially ordered set. We can say more about the structure induced on the p-field L. T H E O R E M 5. Every p-field L = ( L, <~", t ) is orthomodular, i.e. f o r any a a n d b in L, i f a <~' b then b = a v '(b /x 'at). P r o o f I f a, b ~ L , a ~ ' b in L, then a ~< b in B = (B, ~<, c), and b = a v (b /, aC). Also b A a c ~< a ~ so, by (ii) in the p r o o f o f T h e o r e m 4, b /x a c ~ L. Hence b A ' a t exists in L and b = a v ' (b ,', 'at). QED We have established that L is an orthomodular set, at least under some acceptations o f that locution - there appears to be no standard t e r m ) N o t e well, an o r t h o m o d u l a r set: we have not established that the p-field L is an o r t h o m o d u l a r lattice. Indeed, we can show that on either o f our chosen approaches to probability o r t h o m o d u l a r sethood is the most we can claim in general. There is no necessity that L should be a lattice. 144 PETER MILNE EXAMPLE 1. Consider the infinite sequence with basic attributes encoded by the numbers 1, 2 . . . . . 6: 1,2,3,4,5,6,4,5,6,1,2,3,1,2,3,1,2,3, 4,5,6,4,5,6,...,4,5,651,2,3,!,2,3 18 elements 4,5,6,4,5,6,...,4,5,6~ "v" ..... 1,2,3V 36 elements 1,2,3 .... 72 elements Limits of relative frequency exist for all and only the following attributes (sets): O {1, 4}, {1, 5}, {1, 6}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6), {1, 3, 5, 6}, {2, 3, 4, 5), {2, 3, 4, 6}, {2, 3, 5, 6}, (1, 2, 3, 4, 5, 6}. When ordered by set-inclusion the resulting p-field is not a lattice. The sets {1, 4) and {1, 5} have no join - both are subsets of both {1, 2, 4, 5} and {1, 3, 4, 5} but no element of the p-field includes both and is included in both these four-element sets. (In turn, { 1, 2, 4, 5) and {1, 3, 4, 5} have no meet.) QED EXAMPLE 2. Consider the following assignments of upper and lower probabilities to the subsets of { 1, 2, 3, 4, 5, 6}. If a contains exactly one element P*(a) = 1/12, P*(a) = 1/4; if two elements P*(a) = P*(a) = 1/3; if three elements P*(a) = 5/12, P*(a) = 7/12; if four elements P*(a) = P*(a) = 2/3; if five elements P*(a) = 3/4, P*(a) = 11/12, P*(O) = 0; P.({1, 2, . . . , 6}) = 1. The assignments, we may suppose, are derived from the betting quotients deemed acceptable by some agent. We argue that any family of bets entered into by the agent employing said betting quotients is weakly coherent. Note first that were the agent to act as bookmaker for a bet on a at betting quotient p ~< P*(a), then, for a fixed stake S, the pay-offs to her would be no better than on bets at betting quotients at which she agrees to act as bookmaker. Similarly for the placing of bets at betting quotients greater than or equal to P*(a). Hence were the distribution-pair (P*, P*) to admit an incoherent family of bets so too would the pair (P, P) where P = (P* + P*)/2. In particular, P, PROBABILITY AND QUANTUM MECHANICS 145 thought of as furnishing fair betting quotients, would admit an incoherent family of reversible bets (bets concerning which the agent is indifferent as to the side she takes). But P is a standard pointvalued probability distribution (in fact, the uniform distribution over the subsets of {1, 2 . . . . ,6}). By de Finetti's version of the Dutch Book Argument P admits no such incoherent family.9 Therefore, as claimed, bets made in accordance with the distribution-pair (P*, P*) are weakly coherent. This established, we note that P* and P. coincide on all, and only, those subsets with an even number of members. Hence the resulting p-field L contains only those subsets. It is not a lattice: {1, 2, 3, 4} and { 1, 2, 3, 5} both belong to L but they have no meet in L under the partial ordering induced on L by set-inclusion: {1, 2} and {2, 3} are each subsets of both these sets but there is no set with an even number of members that contains both these and is contained in both {1, 2, 3, 4} and {I, 2, 3, 5}, as there would have to be if a meet existed. These sets do have a join, namely {1, 2, 3, 4, 5, 6}. Correspondingly, {1, 2} and {2, 3} have a meet in L, namely O, but no join. QED Even when the p-field is a lattice we can make no stronger general claim than that it is orthomodular. The modular law - a v (b /x (a v c)) = (a v b) A (a v c) -- and afortiori the distributive law - a v (b /x c) = (a v b) /x (a v c) - may fail to obtain. In consequence complements need not be unique despite the uniqueness of complements in the parent boolean algebra. EXAMPLE 3. Consider the infinite sequence with attributes again encoded by the numbers 1, 2 , . . . , 6: 1,2, 1 , 3 , 4 , 5 , 4 , 6 , 1,2, 1,3, 1,2, 1,3, 4, 5, 4, 6, 4, 5, 4, 6, 1, 2, 1 , 3 , . . . , 1,2, 1,33 24 elements 4,5,4,6,...,4,5,4,6, k -y- 1,2, 1,3 . . . . . J 24 elements 72 elements 4,5,4,6,...,4,5,4,6,1,2,1,3,..., Y 7'2 elements 1,2, 1,3, -.g 1,2, 1 , 3 ~ . . . "v 216 elements 146 PETER MILNE Limits of relative frequency exist for all and only the following attributes (sets): O, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {1, 5, 6}, {2, 3, 4}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {2, 3, 5, 6}, {1, 2, 3, 4, 5, 6}. Ordered by set inclusion the resulting p-field is a nonmodular (hence non-distributive) orthomodular lattice. Non-modularity follows by noticing that {2, 5} v'({1, 5, 6} /x '({2, 5} v ' {1, 2, 4, 5})) = {2, 5} v'({1, 5, 6} A' {1, 2, 4, 5}) = {2, 5} • {1, 2, 4, 5} = ({2, 5} v ' {1, 5, 6}) /x'({2, 5) v ' {1, 2, 4, 5}). QED EXAMPLE 4. Consider the p-field that results from the following distribution of upper and lower probabilities to the subsets of {1, 2, 3, 4, 5, 6}. Single-element sets receive the lower probability 1/12, upper probability 1/4. P*({1, 6}) = P*({1, 6}) = P*({2, 5}) = P*({2, 5}) = P*({3, 4}) = P*({3, 4}) = 1/3; the other two-element sets receive lower probability 1/4, upper probability 5/12. P*({1, 2, 3}) = P.({1, 2, 3}) = P*({4, 5, 6}) = P.({4, 5, 6}) = 1/2; the other threeelement sets receive lower probability 5/12, upper probability 7112. The upper and lower probabilities of four- and five-element sets are fixed in accordance with the complement rule - P*(a) + P*(a c) = 1. By the argument of Example 2 bets placed in conformity with the pair (P*, P.) are weakly coherent. The resulting p-field is a nonmodular (hence non-distributive) orthomodular lattice. Non-modularity follows by noticing that {1, 6} v'({1, 2, 3} ^'({1, 6} v ' {1, 2, 5, 6})) = {1, 6} v ' ( { l , 2, 3} /x' {1, 2, 5, 6}) = {1, 6} va {1, 2, 5, 6} = ({1, 6} v ' {1, 2, 3}) ^ ' ({1, 6} v ' {1, 2, 5, 6}). Complements are not unique: {1, 2, 3} is a complement of {1, 6}, as is {2, 3, 4, 5}. The resulting lattice - it is a lattice - is the smallest orthomodular nonmodular lattice. QED The p-field generated by a distribution-pair may, of course, be a boolean algebra. One family of such cases arises via the canonical extension (see w of a possibly only finitely additive probability distribution over a boolean sub-algebra of a power-set algebra. THEOREM 6. Where B = (B, ~_, c) is a boolean sub-algebra of the power-set algebra over X, and P is a finitely additive probability distribution over B the p-field generated by the canonical extension of P to j~(X) is a boolean algebra. PROBABILITY AND QUANTUM MECHANICS 147 Proof. Let L be the p-field in question and suppose that a and b belong to L, the domain of L. Let c, d, e and f be any four not necessarily distinct members of B, the domain of B, such that c c a, d _c b, a ~ e and b _~ f. As c n d ~_ a n b and c n d E B, P(ccTd) ~< P*(acTb). Similarly, P ( c u d ) ~< P*(aub). Also, P*(anb) ~< P ( e ~ f ) and P*(aub) ~< P ( e u f ) . Now P ( c u d ) + P ( c n d ) = P(c) + P(d) and P ( e u f ) + P ( e u f ) = P(e) + P ( f ) , so consequently P(c) + P(d) ~< P*(a~b) + P*(a~b) ~ P*(aub) + P*(anb) ~< P(e) + P ( f ) . Hence P*(a) + P*(b) ~< P , ( a u b ) + P,(acTb) ~< P*(aub) + P*(acTb) ~< P*(a) + P*(b). But a and b belong to L so P,(a) = P*(a), P*(b) --P*(b), from which it follows that a u b and a n b both belong to L. We also have that a ~ ~ L just in case a e L, and that O ~ L and X e L. Hence the complementation, meet and join operations on L are the genuine boolean operations inherited from/~(X) ordered by set inclusion and L is a boolean subalgebra of (~(X), _c, ~). QED Neither of our chosen accounts of probability encourages us to believe that the domain of the point-valued probability assignment derived from a distribution-pair should be closed under countable joins, even of mutually incompatible elements of the domain. Nor is there any reason to suppose that when the join of countably many mutually incompatible elements does belong t ~ the p-field the assignment is countably additive. E X A M P L E 5. Consider the sequence constituted by the natural numbers under their normal ordering. The singleton sets {n}n~ form a jointly exhaustive and mutually exclusive family of attributes. Defining upper and lower probabilities in terms of relative frequencies we find that P*({n}) = P*({n}) = 0, and P , ( N ) = P*(N) -- 1. So for all n E N, {n} belongs to the p-field L generated by (P*, P*), as does N. Letting P be the restriction of P* to L, we see immediately that countable additivity fails for P: 1 = P(N) -r 2n~~ P({n}) = 0. Now let a be the set {n e N: 22m ~< n < 22~'+~ for some m E N}. P*(a) = 1/3 and P*(a) = 2/3, hence a q~L. Suppose that a join exists in L for the family {n}n~a. Let b be this join, so b is the _c_least element of L that includes a. a is a proper subset of b so let rn ~ b-a. Then P * ( b - { m } ) = P*(b) = P*(b) = P*(b-{rn}), so b-{m} ~ L and 148 PETER MILNE a _c b-{m}, contradicting the minimality of b. Therefore L is not closed under countable joins of mutually incompatible members. QED EXAMPLE 6. We consider a set of mutually contradictory and jointly exhaustive propositions indexed by the natural numbers. We suppose that our token agent agrees to bets so that, in the notation of Theorem 2, where a is a finite subset of N S*(a) = {0}, S*(a) = [0, 1]; where a is cofinite, i.e. has a finite complement, S*(a) = [0, 1], S*(a) = {l}; where a is neither finite nor cofinite S.(a) = {0}, S*(a) = { l }. As we now show, these sets of acceptable betting quotients admit no weakly incoherent family of bets. Since the agent loses nothing on bets on sets that are neither finite nor cofinite we may exclude them from consideration, so that we consider only bets on finite and cofinite sets. Suppose, for a "worst case scenario", that the agent acts as bookmaker for best on all and only finite sets, places bets on all and only cofinite sets, and in the first case the betting quotient is uniformly 0, in the second uniformly 1, so that under no circumstances does the agent gain. By extensionality this arrangement is equivalent to the agent acting as bookmaker for bets on all and only the singleton sets at betting quotient 0. Let Sn be the stake for the bet on {n}. Then if n indexes the one true proposititon the agent pays out Sn and receives nothing, returning a net loss of S,. Thus no matter which is the true proposition she makes a loss. However we stipulated that the total staked on any family of bets is finite. As the S,'s are all non-negative and En~ Sn < o% S, ~ 0 as n ---, oo. Hence, as weak coherence demands, there is no minimum positive amount that the agent must lose. The acceptable betting quotients admit no weakly incoherent family of bets. They give rise to this distribution pair of upper and lower probabilities of/;(N): if a is finite, P*(a) = P*(a) = 0; if a is cofinite P*(a) = P*(a) = 1; if a is neither finite nor cofinite then P*(a) = 0 and P*(a) = 1. The resulting p-field, L, contains exactly the finite and cofinite subsets of N. Let P be the restriction of P* to L. Then P({n}) = 0 for all n e N, but P(N) = 1, so P is not countably additive. Now let a be the set of even numbers. The countably infinite family {n},~a of mutually contradictory members of L has no join in L since there is no least cofinite set containing all the even numbers. QED PROBABILITY AND QUANTUM MECHANICS 149 On the basis of upper and lower probabilities defined over a boolean algebra we arrive at a point-valued probability assignment over an orthomodular set. Joins and meets may or may not exist for pairs of elements from that set. Suppose a ix' b does exist for some pair of elements a and b. What is its significance? To take a concrete case, suppose upper and lower probabilities are distributed over the propositions of some language. The logic of the language is classical so that the propositions form a boolean algebra when partially ordered by logical entailment. Exact, point-valued degrees of belief are assigned to some of the propositions. As we now know this subset forms an orthomodular set under the partial ordering and the inherited orthocomplementation operation. The orthocomplement of a proposition is just its classical negation. Question: for propositions a and b what is the significance of a /x' b, supposing that it exists? To answer we have only to recall how joins and, specifically in this case, meets are defined, a /x' b is logically the weakest proposition on which upper and lower probabilities coincide that entails both a and b (and so is entailed by every proposition on which upper and lower probabilities coincide that entails both a and b). Perhaps one could think of /x" as a new logical connective, but notice it is tied to the distribution-pair of upper and lower probabilities; if the distribution-pair changes it may change too. a A' b may, but need not, be identical with a /x b, in which case it is just the ordinary, classical conjunction of a and b. But beware: this is a special case. Usually a /,' b's significance is exhausted by the definition of meets in terms of the partial order, inherited from the original boolean algebra, and dependent on the distribution-pair of upper and lower probabilities. There is nothing more to it than that. l~ We end this section with a probabilistic derivation: T H E O R E M 7. Where (P*, P,) is a distribution-pair on the boo&an algebra B = (B, <~, c), we have that t f P , ( a ) = 1 then, f o r any b in L, P*(a /x b) = P * ( b ) a n d P * ( a A b) = P*(b). P r o o f 0 ~ P*(a c ,,, b) ~< P*(a ~ /x b) ~< P*(a ~ /x b) + P*(a c /x b ~) ~< P*(a ~) = I - P*(a) ~ O. Hence, as P*(a /x b) + P , ( a c /x b) P,(b) ~ P*(a /x b) + P*(a ~ A b), P*(a /x b) = P*(b). Similarly, as P*(a /x b) + P*(a ~ /\ b) ~< P*(b) ~< P*(a /x b) + P*(a ~ /x b), P*(a A b) = P*(b). QED 150 PETER MILNE As an immediate consequence we find that the J a u c h - P i r o n condition holds for all point-valued probability assignments derived from distribution-pairs of upper and lower probabilities over boolean algebras. That is, COROLLARY on the p-field L (P*, P*) on the P(a) = P(b) = 7.1. Where P is the point-valued probability assignment = ( L, <~', "~) determined by the distribution-pair boolean algebra B = (B, <,, ~ for any a, b ~ L, if 1 then a ix' b exists and P(a A' b) = 1. w UP P E R A N D LOWER P R O B A B I L I T I E S A N D Q U A N T U M MECHANICS On the basis of probabilistic and classical considerations we have arrived at point-valued probability assignments over orthomodular sets. Let us then introduce an at this point purely formal notion of a probability space. The pair (L, P ) is a probability space when L = (L, ~<', t ) is an orthomodular set and P: L --, [0, 1] is a probability assignment satisfying these constraints: 0 ~< P(a) ~ 1; P(IL) = 1; if a ~<'bt then P(a v ' b ) = P(a) + P(b). In the literature P is often called a state when L is an orthomodular lattice. It is known that there are orthomodular lattices that admit no states. This is in contrast with standard probability theory: every boolean algebra admits a (full set of two-valued) probability distribution(s). Our enquiry thus far prompts this question. Can any probability space be regarded as derived from a distribution-pair of upper and lower probabilities over a boolean algebra in the way exhibited in w An affirmative answer would be of the greatest importance for the interpretation of quantum mechanics. The lattice of closed subspaces of a Hilbert space is orthomodular (modular if the space is finite dimensional), and the probabilities that enter into quantum mechanics are easily construed as determining a probability assignment over PROBABILITY AND QUANTUM MECHANICS 15l the lattice of closed subspaces. In other words quantum mechanics deals with probability spaces as the notion has been defined here. An affirmative answer would allow a straightforward classical/realist interpretation of quantum mechanics, for the Hilbert space describing a quantum mechanical system would be embeddable in a boolean algebra and the point-valued probability assignment on the lattice of closed subspaces derivable from some distribution-pair of upper and lower probabilities over that algebra. Upper and lower probabilities are of course open to both physical and epistemic interpretation, as in w The short answer is decisively negative, as Example 7 shows. The probability assignment there contravenes the J a u c h - P i r o n condition, hence, by Corollary 7.1, cannot be derived in the manner of w from a distribution-pair over a boolean algebra. E X A M P L E 7. Let L = ~L, ~<', t ) be the lattice M02, known as the Chinese lantern, and illustrated in Figure 2. Let P be the assignment P(a) = P(b) = P(1L) = l, P(at) = P(b'~) = P(0L) = 0. As a A' b = 0L this distribution contravenes the J a u c h Piron condition. QED This negative answer is in itself not of great moment as far as the interpretation of quantum mechanics is concerned, for there is no reason to suppose that the majority of probability assignments that arise there contravene the J a u c h - P i r o n condition. To be exact, it is a consequence of Gleason's Theorem that for quantum systems whose Hilbert space representation requires a space of dimension 3 or 1 a @ ~ 0 Fig2., a+ 152 PETER MILNE greater the probability assignments that arise satisfy the J a u c h - P i r o n condition. This need not be so when the space is of dimension 2. H However, it is indeed the case that for most quantum mechanical systems there is no boolean algebra into which the lattice of closed subspaces can be embedded in the manner in which the derivation in w would suggest most appropriate. This requires explanation. In moving from a boolean algebra to an orthomodular set we respected the partial order and complementation operation on the parent algebra. Sometimes we introduced new meets and joins but when a and b are incompatible in the boolean algebra their join is in the orthomodular set and is their join there. I.e. when a, b ~ L, a ~< b c then a v b ~ L and a v ' b = a v b. Moving in the opposite direction, that is from an orthomodular set to a boolean algebra, the most natural question to raise is whether there exists a boolean algebra into which the partial order on the set can be embedded so that the set's orthocomplements are respected, i.e. their images are complements in the boolean algebra, and the embedding is additive. That is, given the orthomodular set L = (L, ~<', ] ) we want to know whether there exists a boolean algebra B = (B, ~<, c) and a function i : L ~ B such that: (i) a ~<' b if, and only if, i(a) <~ i(b); (ii) i(df) = i(a)C; (iii) i(a v ' b) = i(a) v i(b) when a ~<' bt. We call the function i an additive boolean embedding. To explain this terminology, the function i: L ~ B is an embedding if a 4 ' b when, and only when, i(a) <~ i(b); it is sub-boolean if i(a'~) <~ i(a)C; it is boolean if i(a4f) = i(a)C; it is additive if i(a v ' b) = i(a) v i(b) when a <~' bt. As immediate consequences of these definitions we have that if i: L ~ B is order-preserving and sub-boolean then i(0L) = 0B; if it is also additive then it is boolean just in case i(le) = lB. The negative result is that there are modular and orthomodular lattices for which no additive boolean embedding into a boolean algebra exists. In particular this is true of the lattices of closed subspaces of finite-dimensional Hilbert spaces, over the complex numbers, of dimension greater than 2. These lattices together with those PROBABILITY AND QUANTUM MECHANICS 153 generated by the closed subspaces of infinite-dimensional separable complex Hilbert spaces, are employed in the description of quantum mechanical systems. In the interests of brevity let us call them all, without prejudice to quantum systems whose Hilbert-space representation is of dimension 2, quantum lattices. We shall call a quantum lattice finite-dimensional when it is the lattice associated with a finitedimensional HilberlL space. In short, then, no additive boolean embedding of a finite-dimensional quantum lattice into a boolean algebra exists (Corollary 8.1.). In consequence we have that countably additive probability assignments over quantum lattices are not derived in the way of w from distribution-pairs of upper and lower probabilities over boolean algebras (Corollary 8.2.). The restriction to countably additive distributions is really no restriction at all. All the probability distributions that occur in quantum mechanics are countably additive. 12 T H E O R E M 8. Let L = <L, ~<', t> be an orthomodular set, B = <B, <~, c > a boolean algebra, and i: L ~ B an additive, boolean order preserving function. Then a dispersion-free, i.e. a two-valued, probability assignment exists on L. Proof As B is a boolean algebra there exists a homomorphism of boolean algebras h : B ~ {0, 1} from B into the two-membered boolean algebra, whose domain is {0, 1}] 3 The function hoi: L -~ {0, 1} is a two-valued probability assignment on L. QED C O R O L L A R Y 8.1. No additive, boolean order preserving function from a finite-dimensional quantum lattice L into a boolean algebra B exists. Proof Gleason's Theorem entails that no finite-dimensional quantum lattice admits a two-valued probability distribution] 4 QED C O R O L L A R Y 8.2. A countably additive probability assignment to the lattice of closed subspaces of a separable real or complex Hilbert space, being fully determined by the values it assigns to the one-dimensional closed subspaces, must assign a non-zero probability to at least one onedimensional closed subspace and thereby induces a probability distribution on a finite-dimensional closed subspace of dimension greater than 2. 154 PETER MILNE Proof. The proof follows easily from these observations. Given an orthomodular set L = <L, ~ ' , t> and a ~ L/{0L} let L. = <La, 4 , # > , w h e r e L a = { b ~ L : b <~'a}, ~ = ~<'c~L~ x La, and b # = a A' b t for all b e La. Then L. is an orthomodular set and meets and joins in La are identical with those in L. If L is a boolean algebra so is L a. If P : L -~ [0, 1] is a probability distribution on L and P(a) 0 then Q:L~ --* [0, 1] is a probability distribution on La, where Q(b) = P(b)/P(a) for all b e L~. And if i: L --* B extends P to the distribution-pair (P', P,) on the boolean algebra B = <B, ~<, c >, then i ~ L a x Bi(a~ extends Q to the pair (Q', Q,) on Bi(a), where Q'(b) = P'(b)/P(a) and Q'(b) = P,(b)/P(a) for all b e Bi(~). QED A result due to Zierler and Schlessinger shows that if we weaken the constraints on what is to count as an acceptable mapping of an orthomodular set into a boolean algebra an embedding always exists. For present purposes Zierler and Schlessinger's result may be stated thus: given any orthomodular set L = <L, ~<', t> there exists a boolean algebra B = <B, ~<, ~ and a boolean embedding i : L ~ B of L in B (Zierler and Schlessinger, Theorem 2.1). In other words they evade the consequences of Gleason's Theorem by dropping the additivity condition on the embedding. 15 Given their result our next step is to ask whether a probability assignment on a quantum lattice L extends to a distribution-pair over some boolean algebra B into which L is mapped by a boolean embedding. A probability assignment P : L ~ [0, 1] extends under the function i : L --* B to a distributionpair (P', P,), defined on B, just in case P'(i(b)) = P,(i(b)) = P(b). Again our ambition is thwarted. Extending another theorem of Zierler and Schlessinger's we arrive at this limitative assertion: given a quantum lattice L and a countably additive probability assignment P on L there is no extension of P by a distribution-pair under a boolean embedding of L into a boolean algebra. This follows from Theorem 9, Theorem 8, and Corollary 8.2.16 T H E O R E M 9. The probability assignment P : L --* [0, 1] over the orthomodular set L = <L, <<.', t> extends to a distribution-pair (P', P,) on B, the domain o f the boolean algebra B = <B, <~, c>, under a subboolean order-preserving function i : L ~ B only if there is an additive, PROBABILITY AND QUANTUM MECHANICS 155 boolean order-preserving map h o f L into a boolean algebra B' = (B', ~ , # 5 . Proof. Suppose that the probability distribution P : L ~ [0, 1] extends to a distribution-pair (P', P,) on B under a sub-boolean order-preserving function i : L -~ B. Let J = {a ~ B: P'(a) = 0}. We show first that J is a proper ideal in B. If a ~< b and b ~ J then 0 ~< P'(a) ~< P'(a) + P,(b A a c) <~ P'(b) = 0, hence a ~ J. If a, b E J t h e n 0 <~ P'(a v b) <~ P'(a) + P'(b A a c) <~ P'(a) + P'(b) = 0, hence a v b ~ J. Thus J is an ideal. J is a proper ideal since P'(1B) = 1 # 0. (The fact that J is an ideal yields an alternative p r o o f o f the Jauch - Piron condition (Corollary 7.1).) As J is a proper ideal there is a boolean algebra B / J = ( B / J , 4 , # ) and a h o m o m o r p h i s m o f boolean algebras j : B ~ B/J. The elements o f B / J are the equivalence classes induced on B by the relation of probabilistic equivalence under (P', P,), elements a and b of B being probabilistically equivalent under (P', P,) just in case P'((a A b ~ v (b A aC)) = 0. j maps each element of B to the equivalence class to which it belongs; j ( a ) = 0Bu just in case a ~ j.17 Obviously joi is order-preserving. Also it is sub-boolean, for j o i ( a t ) j(i(a) c) = (/oi(a))#. We now show that joi: L -~ B / J is additive. Let a, b ~ L, a ~<' b~'. Then i(a v ' b ) = i(a) v i(b) v (i(a v ' b ) A i(a) r A i(b)C). As (P', P,) extends P under i, P'(i(a v 'b)) = P(a v 'b) = P(a) + P(b) = P,(i(a)) + P,(i(b)). But, since i(a) <~ i(b'~) <~ i(b) c, P,(i(a)) + P,(i(b)) ~< P,(i(a) v i(b)) <. P,(i(a) v i(b)) + P'(i(a v ' b ) A i(a) c A i(b) ~ <~ P'(i(a v'b)), whence P'(i(a v ' b ) A i(a) ~ A i(b) ~) = 0. Thus i(a v ' b) A i(a) ~ A i(b) ~ ~ J, and s o j ( i ( a v " b) A i(a) ~ A i(b) ~ = 0BU. Consequently, joi(a v ' b) = jo(i(a) v i(b) v (i(a v ' b) A i(a) ~ A i(b)C)) = joi(a) v joi(b) v jo(i(a v ' b ) A i(a) ~ A i(b) c) = joi(a) v joi(b). Lastly we prove t h a t j o i is boolean by showing thatjoi(1L) = 1B/j. 1 = P(1L) = P*(i(IL)) ~< P*(i(1L)) + P*(i(1Ly) ~< P*(M = l, hence P*(i(1L) ~) = 0. Thus i(1L) is probabilistically equivalent to 1B under (P*, P*). Sojoi(1L) = j(1B) = 1B/~. QED H o w helpful is Zierler and Schlesinger's embedding result, the failure of probabilities to extend from q u a n t u m lattices notwithstanding? Let S be some physical system whose quantum-mechanical description by 156 PETER MILNE the Hilbert space H gives rise to a quantum lattice L. Let us say that under an embedding i of L into a boolean algebra B = (B, ~<, c> an element b of B/{0B} is quantum-mechanically realisable if i(a) <. b for some a e L/{0L}. The motivation behind the condition of quantummechanical realizability is quite straightforward. Thinking of the elements of B as physical events in some broad sense, the condition is intended to capture the idea that event b is quantum-mechanically realisable if some state of S unfailingly brings about b. We find that given a boolean embedding i of L into a boolean algebra B there are elements of B that are not quantum-mechanically realisable. To be more specific, for some pair of mutually disjoint subspaces a and b, despite the fact that no state of the system S belongs to both a and b the join of their images under the embedding is not empty, i.e. i(a) A i(b) ~ 0 B (Theorem 10). The join subsumes the image of no state of S, ~s and so, as already asserted, B must contain elements that are not quantum-mechanically realisable. Some, if not all, m such elements are boolean combinations of quantummechanically realisable events, hence some boolean combinations of quantum-mechanically realisable events are not themselves quantummechanically realisable. The physical interpretation of the boolean embedding of a quantum lattice into a boolean algebra therefore engenders some heavy physical and/or metaphysical commitments. Perhaps this is why Beltrametti and Cassinelli go so far as to say, without explanation, 'It is [...] doubtful whether the existence of such an embedding represents a physically meaningful way out for hiddenvariable theories'. 2~ T H E O R E M 10. I f a boolean embedding i of the orthomodular lattice L = ( L , <<.', t> in the boolean algebra B = (B, <<.,~ satisfies this condition: l f a A' b = 0L then i(a) A i(b) = OB, then L ks a boolean algebra. Proof. Let a, b e L and suppose that a A' b = 0e, a v ' b = 1L. Then i(a) A i(b) = 0B, and so i(a) <~ i(b) ~ = i(bt). Hence a ~<' bt, i.e. b <~' at. Consequently, a t = b v ' (at A' bt) = b v ' (a v ' b)t = b v ' 1Lt = b v ' 0L = b. Thus every a in L is uniquely PROBABILITY AND QUANTUM MECHANICS 157 complemented. And so, by a theorem on orthocomplemented lattices, L is a boolean algebra. 2~ QED w THE PHASE SPACE REPRESENTATION OF QUANTUM SYSTEMS There are known difficulties facing attempts to provide phase space representations of quantum systems. One such is the joint distribution problem, which we may state thus. Quantum theory yields standard probability distributions on the Borel subsets of ~ for values of the non-commuting observables position and momentum. However, it is known that there is no probability distribution, consonant with quantum mechanics, to the Borel subsets of N2 yielding the original distributions as marginal distributions. Following Wigner, Moyal et al. a joint distribution function can be defined formally but it takes negative values. Negative values are of course anathema to the interpretations of probability with which we started this paper. 22 In this section we argue quite generally that except for exceptional cases no phase space representation, employing upper and lower probabilities, of a quantum mechanical system is possible. Let L = (L, ~<', t ) be the lattice of closed subspaces of the Hilbert space H describing some quantum mechanical system S. For a phase space representation of S we wish to find a field of subsets B = (B, G, *), i.e. a boolean subalgebra of some power-set algebra, and a map i: L ~ B such that: (i) a ~<' b if, and only if, i(a) ~_ i(b); (iv) for all a e B/{O } there exists b e L/{0L} such that i(b) ~_ a; (v) P : L --, [0, 1] extends under i to a distribution-pair (P', P,) on B. A function i : L ---, B that satisifes condition (iv) is called a dense map of L in B. The constraints (i), (iv) and (v) are quite natural. (v) requires no explanation. (i) captures the requirement that the structure of the phase space should reflect that of the Hilbert space 158 PETER MILNE representation of S with some degree of faithfulness. (iv) expresses the idea that the phase space should include no quantum-mechanically unrealisable events. In fact when B is a complete boolean algebra (i) and (iv) entail a perhaps more obvious and certainly superficially more restrictive requirement (Lemma 11.1): for all a E B, a = u {i(b): b ~ L and i(b) c_ a}. In other words, if B is complete every element of B is identical to the union of the images under i of the quantum-mechanical events that realise it. L E M M A 11.1 Let i : L ~ B be a function f r o m the partially ordered set L = ( L , <~') with least element Oe into the complete boolean algebra B = (B, <<,,~ and suppose that: (i) a <~' b if, and only if, i(a) <~ i(b); (iv) for all a ~ B/{0. } there exists b e L/{0L} such that i(b) 4 a. Then for all a ~ B, a = v { i ( b ) : b E L a n d i ( b ) <~ a}. Proof. We show first that i(0L) = 0B. For any b ~ L, if i(b) <<. i(0L) then, by (i), b <~' 0L, so b = 0L. By (iv), then, we have that i(0L) = 0B. It follows from (i) that, for any b ~ L, if i(b) <<. OB then b = 0 L. So 0B = i(0L) = V { i ( 0 L ) } = V {i(b): b ~ L and i(b) 4 0 B } . N o w l e t a ~ B t { 0 B } and l e t c = v { i ( b ) : b ~ L a n d i ( b ) <~ a } . B y (iv) the set is not empty, c exists as B is complete. Obviously c ~< a so a = c v (a ^ c C ) . L e t b ~ L , i ( b ) <~ a ^ c c . A s i ( b ) <~ a , i ( b ) <~ c by the definition of c. But i(b) <<. c~. Hence i(b) <<. OB, and so b = 0L. By (iv), a ^ c~ = 0B, and consequently a = c. QED Conditions (i) and (iv) also have these desirable consequences (Lemma 11.2): that i(H) = i(IL) = X, where B is a sub-algebra of <~(x), _, ~ that i(a ^ ' b) = i(a) n i(b), PROBABILITY AND QUANTUM MECHANICS 159 from which it follows that a /x' b = 0 L if, and only if, i(a) c~ i(b) = O; and that i(a'O ~_ i(a) ~. These conditions are desirable since the meet of subspaces of a vector space coincides with set-theoretic intersection whilst orthogonal subspaces are contained in but are usually not identical with set-theoretic (relative) complements. The first condition tells us that phase space does not outrun the image of the totality of states of the system S. L E M M A 11.2. Let i: L ~ B be a function from the orthocomplemented set L = ( L, <~', t ) into the boolean algebra B = ( B, <~, ~), and suppose that: (i) a <~' b if, and only if, i(a) <~ i(b); (iv) for all a e Bt{0B } there exists b e L/{0L} such that i(b) <~ a. Then i(1L) = 1B, i(a A ' b) = i(a) /x i(b) when a A ' b exists, and i(agf) <~ i(a) c. Proof. By the first part of the p r o o f of L e m m a 11.1, which does not appeal to completeness of the boolean algebra, we have that for any b e L, if i(b) <<, OB then b = 0L. N o w let b ~ L, i(b) <~ i(1L) c. As b ~<' 1L, i(b) <~ i(1L), and so i(b) <~ Oh, thus b = 0L. Hence, by (iv), i(1L) c = 0B and so i(IL) = I b. Suppose a A ' b exists. Then a A ' b ~<' a, a A ' b ~<' b, so i(a A ' b) <~ i(a) /x i(b), by (i). Let c ~ L, i(c) <~ i(a) A i(b) A i(a A ' b) ~. Then i(c) <~ i(a) and i(c) <~ i(b), so, by (i), c ~<' a and c ~<" b. Hence c ~<' a A ' b, and so, by (i) i(c) <~ i(a A" b). But i(c) <~ i(a A" b) ~. Thus i(c) <~ 0h, and so c = 0 L. By (iv), i(a) A i(b) /x i(a A ' b) c = 0h, and consequently i(a A" b) = i(a) A i(b). Since i(b) = 0B when, only when, b = 0L, we have immediately that a /x' b = 0L if, and only if, i(a) /x i(b) = 0B, hence i(at) <~ i(a) ~ 9 QED 160 PETER MILNE A theorem that, for lack of evidence to the contrary, I shall attribute to Kenneth Kunen, tells us that for any orthomodular lattice L = (L, ~<', t ) there exists a field of subsets B forming a complete boolean algebra B = (B, ___,~) and a function i: L ~ B satisfying (i) and (iv). 23 B is unique up to isomorphism; it is called the boolean completion of L. Unfortunately, however pleasing this result may be we are stymied as far as (v) is concerned. In light of Lemma 1 1.2 and the theorems of w we have immediately that for any given quantum lattice L and a countably additive probability assignment P on L there is no extension of P by a distribution-pair under a dense embedding of L into a boolean algebra. 24 Recalling the definition of a quantum lattice this justifies the assertion that there is no phase-space representation, employing upper and lower probabilities, of the vast majority of quantum systems. w THE INTERPRETATION OF QUANTUM MECHANICAL PROBABILITIES These results concerning quantum mechanics have been depressingly negative. It would, I admit, be surprising, if careful scrutiny of essentially classical conceptions of probability were to resolve the foundational problems of quantum mechanics. One may, nevertheless, wonder why the results are negative. The explanation, as the proof of Theorem 9 shows, is that under very weak constraints on how a quantum lattice L = (L, ~ ' , t ) is mapped into a boolean algebra B = (B, 4 , c) the additivity of the probability assignment P over the lattice would force the mapping i of the lattice into a boolean algebra to be almost additive and almost boolean relative to any distributionpair extending P, were such to exist. That is, for incompatible elements c and d of L, i(c v ' d) and i(c) v i(d) would be probablistically equivalent relative to the distribution-pair (P', P,), as would i(c'~) and i(c) ~, elements a and b of B being probabilistically equivalent just in case P'((a A bc) v (a c A b)) = 0. Factoring by probabilistic equivalence would give rise to a strictly additive, strictly boolean map from the quantum lattice into the boolean algebra formed by the equivalence classes. But, by Gleason's Theorem, there is no such additive, boolean map. Consequently there can be no PROBABILITY AND QUANTUM MECHANICS 161 distribution-pair extending P. Put bluntly, the additivity of P on the quantum lattice forces i to be almost additive, and the fact that P(1L) -- 1 contributes to forcing i to be almost boolean, hence precluding the possibility of a distribution-pair over a boolean algebra extending P. As stated, only weak conditions on acceptable candidates i lead to this conclusion. These conditions are: i f a ~ ' b then i(a) <~ i(b), and i(at) <~ i(a) ~ I confess ignorance as to what happens if we weaken these constraints; I also have no idea what the physical significance of an embedding into a boolean algebra contravening one or other or both of these constraints would be. We are left with an open question: how are the probabilities that occur in quantum mechanics to be understood? One route to answering this question is to ask how our as yet purely formal probability spaces of w are to interpreted. Our examination of two classical conceptions of probability had lead to point-valued probability assignments over orthomodular sets. Probability spaces were obtained by dropping two constraints, namely that the point-valued probabilities satisfy the J a u c h - P i r o n condition, and that the orthomodular set can be embedded in a boolean algebra by an additive boolean embedding. That is, our probability spaces are the natural outcome of the classical conceptions once one seeks a purely internal characterization of the algebra of coincident upper and lower probabilities, a characterization that makes no reference to the origin of the events/propositions in some boolean family, nor to the provenance of the pointvalued probabilities. (Dropping the second constraint is the more important, for once there is no reason to suppose that arbitrary meets exists, let alone that they behave like boolean meets, there is no good reason to suppose that the J a u c h - P i r o n condition obtains.) It is, then, fair to say that we have arrived at probability spaces on the basis of probabilistic and classical considerations. This is in contrast to Suppes, who arrives at the same structures - he calls them quantum-mechanical pobability spaces, the orthomoduIar sets being 162 PETER MILNE his quantum-mechanical algbras - by considering the joint-distribution problem of quantum mechanicsY In the context of quantum mechanics there are, I suggest, two obvious possibilities regarding the interpretation of probability spaces. One might be called metaphysical/realist, the other empiricist/ instrumentalist. The first adopts the standpoint of quantum logic. It accepts that at the fundamental level the logical structure of the domain is only orthomodular; it gives up any adherence to underlying boolean algebras. Forrest would assert, and did in person, that adoption of "quantum logic" in the attribute space of a relative frequency interpretation is a much less radical departure from orthodoxy than adoption of quantum logic as the logic governing propositions. 26 Forrest's line depends on putting great weight on, for example, the distinction between 'attribute a is not instantiated on trial t', with the negation understood clasically, and the 'the attribute orthocomplementary to a is instantiated on trial t'. The possibility of drawing the distinction lies open when propositions are regarded as true or false independently of our ability to determine which, but this goes against the grain of much work in quantum logic which seeks to define propositions operationally. Also, in view of the results of w there is no way to extend the probabilities defined on the orthomodular attribute space to events of the type 'attribute a is not instantiated', with the negation understood classically. As the latter would seem to be the genuine events of interest, and the restriction of the attribute space to typically quantum mechanical events correspondingly arbitrary, the position appears at best an uneasy compromise, at worst unstable. Moreover, whatever the merits of Forrest's contention, the epistemic interpretation of probability would allow no such half-way house - there it is the genuine propositions whose logic is non-classical. (Of course embracing quantum logic by no means resolves all the interpretive problems in quantum mechanics. The measurement problem, to cite the obvious example, remains a problem.) This approach to quantum probabilities extends generally to the interpretation of probability spaces. In quantum mechanics all probabilities are point-valued. Given our starting point in w it would seem natural to introduce upper and lower probabilities, defined over PROBABILITY AND QUANTUM MECHANICS 163 orthomodular sets and interpreted either in relative frequency terms or in betting terms. Orthomodular sets are then found to be stable in the sense that the p-field derived from a distribution-pair defined on an orthomodular set in the obvious way is again an orthomodular set. (This fact can be recovered from the proofs of Theorems 4 and 5.) This re-inforces the claim that the natural domain of definition for a point-valued probability distribution is an orthomodular set. The second approach to the interpretation of probabilities in quantum mechanics recognizes that what we attempted to do in w is to find a boolean redescription of the whole distribution over the quantum lattice. Instead we should settle for the standard probability distributions - note the plural - that result when we consider the restrictions of P to the maximal boolean sub-algebras comprising the closed subspaces spanned by the eigenvectors of families of commuting observables (Kochen and Soecker's partial algebras). Taking quantum mechanics to be a theory about the results of measurement, this makes perfect sense. That there is no way to combine these distributions into one "super-distribution" over a boolean algebra is simply a reflection of the fact that there are non-commuting observables, i.e. observables that are not simultaneously measurable. This is what differentiates quantum mechanics from classical physics. On this approach P itself is a mathematical artefact, not a genuine "in the world" probability distribution. However it determines a set of standard probability distributions parametrized by families of commuting observables and these are interpreted standardly. This approach would seem to fit best with the Copenhagen Interpretation of quantum mechanics. We can say a littl[e more about this Copenhagen Interpretation. The assignment P depends on the state of the quantum system, so considering different states of the system gives rise to a family of assignments, over the same domain, parameterized by states. Incorporating this with the ideas of the last paragraph we obtain a family of standard probability distributions doubly indexed by state and family of commuting observables. These can be reconstrued as conditional probability distributions concerning the possible results of measurement, given that the system is in state s and a measurement of 164 PETER MILNE observable O is made. This assertion is to be taken quite literally. The conditional probability distributions are to be understood as derived from a standard probability space. Quantum mechanics yields theorems concerning the relations between these conditional probabilities. 27 Unlike the previous case there may be no obvious way to extend the instrumentalist approach to the interpretation of probability spaces generally. While there are always boolean sub-algebras contained in an orthomodular set, and every element belongs to some such sub-algebra, it is not at all clear that maximal boolean subalgebras will have a straightforward physical significance, as they do in quantum mechanics. They will, however, have a special significance under an epistemic interpretation, being the maximal subalgebras within which the logical operators can be understood classically. This last remark prompts the observation that whilst the classification of interpretations of probability employing the oppositions physical/epistemic and realist/instrumentalist allow four distinct possibilities there may be reason to confine attention to only two. Methodological considerations push one towards the binary classificational realist-physical and instrumentalist-epistemic. The methodological maxim that encourages a uniform approach, other things being equal, to the interpretation of theoretical concepts implies that in the case of a probability space the non-boolean domain and the probability assignment over it are to be interpreted uniformly i.e., either both are treated instrumentally, or both are understood realistically. Therefore, if one is to deny that the use of a non-boolean domain is of anything more than instrumental value, it is natural to say as much about the probabilities defined over it, they are to be understood not as physical features of the world but rather as indicating only rational degrees of belief concerning certain physical features thereof. And if, on the other hand, the employment of a non-boolean domain is taken to indicate the non-classical logical structure of physical reality, the probabilities defined over the domain ought to be understood as physical features of reality. The methodological consideration in play here is not intended to provide conclusive grounds for adopting the binary classification. It PROBABILITY AND QUANTUM MECHANICS 165 presents a prima facie case only, appeal to it could be over-ridden, for example if preservation of classical (boolean) conceptions was of greater importance. It does show that, given the starting point of this paper, there is a case to be made for restricting attention to a relative frequency interpretation of a probability assignment over an orthomodular set of attributes, and to a rational betting odds interpretation of the derived standard distributions over the maximal boolean sub-algebras of the class of propositions in the language under consideration. But these remarks at best constitute only sketches of the possibilities for interpretation. The hard work of fleshing them out is deferred for another occasion. 28 NOTES E We consider propositions, rather than sentences of the language, as distinct sentences may express the same proposition. Given some language we can identify the algebra of propositions with the Lindenbaum algebra of the language, obtained by factoring the sentences by logical equivalence. 2 At least this is the case if the Axiom of Choice obtains. As Solovay showed, when the Axiom of Choice is dropped then, consistently with the other axioms of, say, Z e r m e l o - F r a e n k e l set theory, one may assume that every set of real numbers is Lebesgue-measurable. 3 The join v ' is defined, relative to the partial order ~<', by the stipulation that a v ' b is the ~<'-Ieast element in B" greater than or equal to both a and b. The meet A ' is defined dually. 4 The canonical extension is by no means the only distribution-pair extending P. Finer distribution-pairs exist that attribute point-values to some non-Lebesgue-measurable sets, i.e. sets with no determinate volume. The crucial question is why assignment o f probability should be restricted to regions of phase space to which one particular idealized notion of volume applies. On the basis of their very different conceptions of probability, but motivated partly by a common concern for the applicability of probability theory, de Finetti and yon Mises have both argued for the centrality of sets measurable in the sense of Peano and Jordon. For comparisons with the Kolmogorov approach see, e.g., de Finetti (1974), pp. 10-11, pp. 125-26, and especially w167 von Mises, Appendix Two, Ch. 2, w and Appendix Three; also Fine, w w 5 The mathematical details in the preceding three paragraphs come mostly from Bell and Machover, Ch. 4, w167 6 The more usual definition says that a and b are incompatible (or mutually exclusive, or disjoint) just in case a A b = 0. Distributivity guarantees their equivalence in a boolan algebra. In lattice theory it is more common to say that a and b are orthogonal when a ~< bc. We retain the term incompatible because of its connexion with probability theory. 166 PETER MILNE 7 In the sequel we shall generally take a boolean algebra B to be determined by the triple (B, ~<, c), where B is the domain of the algebra, ~< the partial order on B and ~ the complementation operation. Where necessary we denote the ~<-least and ~<-greatest elements by 0 B and In, respectively. 8 'Orthomodular poset' is another possibility. For terminology compare e.g. Beltrametti and Cassinelli, Kalmbach, and Zierler and Schlessinger. 9 See de Finetti (1980), Ch. 1. l0 In this regard compare Forrest's qualms concerning quantum logic. Forrest, Ch. 2, w 11 See Beltrametti and Cassinelli, Theorem 11.2.1, Example Q, pp. 118-9, and w 12 Beltrametti and Cassinelli, Ch. 1, w t3 See Bell and Machover, Ch. 4, w ~4 See Gleason's Theorem 4.1 and Zierler and Schlessinger's Example 3.2. Zierler and Schlessinger assert that Gleason's Theorem yields a characterization of all states (finitely additive probability assignments) over the closed subspaces of a separable real or complex Hilbert space of dimension greater than 2. In this they err. Gleason characterizes (all and) only countably additive measures. Over a finite dimensional space there is no difference since no infinite set of mutually orthogonal closed subspaces exists. On an infinite dimensional space there can be no eountably additive two-valued probability distribution. Whether a finitely additive two-valued probability distribution to the closed subspaces of an infinite-dimensional separable real or complex Hilbert space exists is not settled by Zierler and Schlessinger's argument and I do not know what the answer is. It follows from Gleason's Theorem that any such distribution must assign zero probability to all finite-dimensional subspaces. (Alternatively, this follows directly from an argument of J. S. Bell's. See Bell w or Jammer, Ch. 7, w t5 They retain it only for elements of the centre of L. a belongs to the centre of L just in case a commutes with every element of L, i.e. a = (a ^ ' b) v ' (a ^ ' b~) for all binL. 16 The proofs of Theorems 8 and 9 owe much to Zierler and Schlessinger's proofs of their Lemmata 4 . 1 - 4 . 3 and Theorem 4.1. 17 See Bell and Machover, loc. cit. ts Here we identify a ray in H (a class of unit vectors in H identical up to multiplication by a complex scalar) with the one-dimensional subspace it generates. Throughout we accept the conventional identification of rays in H with states of S. 19 In the case of Zierler and Schlessinger's minimal boolean extension of L it is all (as follows from the proof of their Lemma 2.2). 20 Beltrametti and Cassinelli, p. 175. 2~ For a proof see, e.g. Kalmbach, Ch. 1, w Proposition 7. Alternatively, where L is orthomodular, if a /x" b = 0 L only if a <~' bt then L is boolean. See, e.g., Beltrametti and Cassinelli, Theorem 14.7.1. 22 For more on negative probabilities see Forrest, Ch. 4, w and Feynman. 23 The proof of this assertion follows readily, mutatis mutandis, from Kunen's Lemma 3.3 in the light o f his Ch. 3, Ex. 15, noticing that orthomodular lattices are, in Kunen's terminology, separative. 24 By Theorems 8 and 9 and Lemma 11.2 a probability assignment over an orthomodular lattice extends to a distribution-pair under a dense embedding into a boolean algebra only if the lattice admits a two-valued probability assignment. Example 7 shows that we cannot strengthen this corollary to read 'if, and only if', for since, under a PROBABILITY AND QUANTUM MECHANICS 167 dense embedding i, i(a) ix i(b) = 0B, P,(i(a) v i(b)) >~ P,(i(a)) + P,(i(b)) = P(a) + P(b) = 2, which is absurd. (Setting P(a) = P(b) = 3/4 again leads to impossibity of extension without contravention of the J a u c h - P i r o n condition.). 25 See Suppes, {}2. Strictly, his quantum mechanical algebras are orthomodular sets whose members are sets partially ordered by inclusion. 26 See note 28, and cf. Forrest, Ch. 2, {}VII. 27 See van Fraassen and Hooker. 2~ An abbreviated version of an ancestor of the present paper was read at the Australasian Association of Philosophy Conference, University of Sydney, July 1990. My thanks to Peter Forrest, Adrian Heathcote, and Richard Sylvan for comments made on that occasion. My thanks also to an anonymous referee for forcing clarification both of the structure of the article as a whole and of points of detail. REFERENCES Bell, J. L. and M. Machover: A Course in Mathematical Logic., North-Holland, 1977. Bell, J. S.: 'On the Problem of Hidden Variables in Quantum Mechanics', Rev. Mod. Phys. 38 (1966). Reprinted in Speakable and Unspeakable in Quantum Mechanics, CUP, 1987. Beltrametti, E. and G. 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