a new approach to probabilities in mechanics

R A C H E L W A L L A C E

A N E W A P P R O A C H TO P R O B A B I L I T I E S IN M E C H A N I C S

In this paper a general theory of probabilities is sketched, suitable for any system of mechanics. I show that the classical probabilities of classical mechanics arise as a special case, and that the general non-classical probabilities display the striking features of quantum statistics. In particular, the rules for conditionalising in quantum mechanics turn out to be in- stances of the general conditionalising rules developed here.

According to this analysis the probabilities of mechanics are generated from the propositional structure of a mechanical theory. A general account of mechanical propositions and of the logic used to express complex mechanical descriptions, is given in the first part of the paper. I argue that the logic is essentially classical, except that it has a generalised negation. It is important to stress that the same propositional logic is used by any mechanical theory. What distinguishes a classical theory of mechanics is the structure of its states. A mechanical state is analysed here in logical terms, and is crucial to the account of probabilities. The special properties of classical states give rise to classical probabilities, while in general 'peculiarities' arise. These turn out to be familiar features of quantum statistics.

So I suggest that the odd features of quantum probabilities are in fact common to any non-classical theory, and that these peculiarities should be understood as rising from the structure of the theory, not from the peculiar nature of subatomic reality. It is the structure of states used by a theory which gives rise to non-classical probabilities. This is just to suggest that quantum theories are inadequate in a well-defined sense, and that the peculiarities of quantum descriptions are a symptom of this inadequacy.

The paper is divided into four sections, entitled Propositions and the Logic LEt, Mechanical States, Probabilities, and Conclusions for Me- chanics, respectively.

Erkenntnis 16 (1981) 243-262. 0165-0106/81/0162-0243 $02.00. Copyright �9 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

244 RACHEL WALLACE

I. PROPOSITIONS AND THE LOGIC LEr

Each mechanical theory contains a set E r of elementary propositions, which give rise to a logic L E r of complex formulae. The propositions of a theory are not primitive, but are generated from primitives as follows.

Each mechanical theory T specifies a set M r of magnitudes. Informally these are understood as the particular measurable items named by a theory, for example, 'position-of-x'. Formally each magnitude m in M r is associated with a Borel set of real numbers, called the value-set of m, Vm.

Vm contains the m-values, intuitively understood as the possible outcomes of 'ideal' measurements of m. The set E r of elementary propositions of T are ordered pairs of form (m, A), where m is a magnitude in M r and A is a Borel subset of Vm. Intuitively proposition p = (m, A) is understood as 'the magnitude m has an actual value which lies in A'.

Each theory specifies laws interrelating magnitudes in M r and hence propositions in E r. These are the 'Laws of Coexistence' (see e.g. van Fraassen 1970), such as Newton's Second Law "force = mass x acceler- ation". These primitive laws vary according to the theory considered, and hence the structure of set E r will depend on the theory T.

There is however some structure which is assumed to hold in any set of elementary propositions. For example, one assumes that there is at least a binary inference relation among the elementary propositions of any theory. This relation is assumed at least a partial ordering of propositions, hence

the set E r is at least a poset, E r = (Er , ~< ). Although the particular relation ~< on E r depends on T, there are some

inference relations assumed to hold in any theory. Most important are those generated by set-inclusion among Borel subsets of each value-set. That is, these relations are given by the assumption that A1 _c A2 iff(m, A1) ~< (m, d2), where A1, A2 are both Borel subsets of Vm. Propositional meet and join with respect to this relation, are generated by set intersection and union among the Borel subsets of each Vm. In addition a relative comple-

ment operation is defined on E r by setting (m, A)• = d f (m, Vm - A),

where - is set complementation in Vm. It follows from the properties of set operations that the propositional operations are Boolean. That is, each subset of 'm-propositions' in E r, those propositions which all Concern the same magnitude, form a Boolean algebra. The whole elementary system Er = ( E r , <~ • is thus at least a union of Boolean subalgebras for any mechanical theory T.

P R O B A B I L I T I E S I N M E C H A N I C S 245

The logic of complex theoretical descriptions will be generated from set E T by means of truth-value assignments which are consistent with the relations among these propositions. Formally, a consistent truth-value as- signment, or valuation, is defined as follows:

DEFINITION. A valuation h of theory T is a function h: E T --~ {t,f, u} such that

(i) (ii) (iii)

h(tg) = t i f f h ( p • = f ; I fp ~> q, then if h (p) = t, h (q) = t and if h (q) = f , h (p) = f ; h conforms to the laws of T.

h is a 3-valued function since it is not assumed that all descriptions given by a theory must be bivalent truth-value assignments to all propositions. Ele- ments t and f of the co-domain of h represent truth-values true and false respectively, while u represents a truth-value 'gap', or 'undecided' value. By condition (i) h is consistent with the relative complement operation. Intuitively, if it is true that the actual value of m lies in A, then it is false that m has an actual value outside this set d. By the Boolean properties of • among m-propositions, (p • • = p, hence the converse of (i) also holds, and h (p) = f iff h (p • -- t. Condition (ii) gives the usual condition for consistency with an inference relation. Condition (iii) will generate various additional formal constraints on h, according to the particular theory considered. (The following evaluation symbols will be useful: one writes ~ hP

i fp is true in h,~' hP i f p is false in h, and& hP i fp is undecided in this valuation.)

Various 3-valued logics could be generated from the 3-valued valuations defined above. These correspond to the various conditions which might be used to define logical connectives. However only the 3-valued Lukasiewicz logic is appropriate to express the descriptions of mechanical theories, since only this logic has primitive connectives 7 and 3 which are suitable to express operation • and relation < respectively, on the elementary system ET. These Lukasiewicz connectives are defined by the schemata

7

t f f t u u

t f u

t f u

t t t

t u t

246 RACHEL WALLACE

7 is appropriate to express the semantic properties of operation • by condition (i) of the definition of valuation. Similarly by condition (ii) the

has properties suitable to express relation ~<. In addition the valid Lukasiewicz hook is itself a partial ordering of equivalent propositions, and hence is suitable to express an inference relation. 1

The logic L E t of mechanical theory T, is the logic generated from set E r by means of the connectives defined above. Wffs of the logic will thus be strings of the form 7p, p 3 q, 7 ~ and so on (where variabl6s p, q, r refer to elementary propositions, and ~, ~, range over all wffs). Nonprimitive connectives are introduced by definition. Thus it is useful to define ~ ct = df

ct 3 7 ~ , ~ v fl = df(~t 3fl) 3fl, ct.fl = d f 7 ( 7 ~ v 7f l )and ct = f l = df (~ 3 fl)" (fl 3 ct). These are denial, disjunction, conjunction and the biconditional respectively. Note particularly that denial is a bivalent connective, which is false when the subformula is true, and true otherwise.

The logic LEr is essentially classical, in that it is represented algebraically by a distributive lattice (Rasiowa 1974). That is, implication, conjunction and disjunction are all classical Boolean connectives. However L E t is a generalisation of classical logic when it comes to negation. Algebraically negation is represented by an involution which satisfies the de Morgan identities - this corresponds to the fact that Double Negation and the de Morgan rules hold in the logic. However negation is not represented by a lattice complement, since Excluded Middle and Non Contradiction fail in logic LEr. For this reason the algebra representing LEr is not a Boolean algebra, but is a distributive lattice without an orthocomplement.

So LE T is essentially classical with respect to implication, but has a generalised negation operation. However negation too can be understood as essentially classical, if it is noted that it is algebraically a relative orthocomplement. Every proposition in LE T has a classical negation defined relative to a subsystem of the logic. (For details, see Wallace 1979) In this sense the logic LE T is locally a classical system. One can in fact regard it as having a natural relatedness relation. (Wallac e 1979) 2

Since I have argued that every mechanical theory uses a distributive logic with this generalised negation, my account contradicts the 'quantum logical' view which holds that quantum mechanics uses one which is non- distributive. This issue cannot be argued here. However it is worth point- ing out that the original argument given by Birkhoff and yon Neumann, which purports to show that distribution fails in quantum mechanics,

PROBABILITIES IN MECHANICS 247

shows instead only that either distribution or complementation must fail. (Birkhoff and von Neumann, 1936). A logic such as L E t which is distributive but only relatively orthocomplemented, will do just as well in the example they consider.

II. MECHANICAL STATES

A state in classical mechanics is determined when every magnitude of a theory is assigned a particular value. Thus a classical state corresponds to a 2-valued valuation, assigning truth-values to every elementary proposition. In general however states cannot be characterised in this way. For some theories may have no 2-valued valuations. (According to Kochen and Specker, 1967, there are quantum theories of this kind.) One must therefore find a general characterisation of state which is applicable to any theory, and which has the bivalent valuation as a special case. An appropriate definition is obtained if one notes that 2-valued valuations are maximal with respect to the set of elementary propositions which they find true. Clearly such maximal valuations can be found in any theory. Intui- tively these are the valuations which give the 'fullest possible~ descriptions

of reality. To make the definition of state precise it is convenient first to define

DEFINITION. The elementary_truth-set of h, Th E = df {p e Er: ~ hP}" The truth-set o f h, Th = df {~ e LET: ~ h ~ }.

Th E contains the elementary propositions true in h, while Th contains all wffs true in this valuation. Similarly one can define an elementary falsity- set Fh E to be the set of elementary propositions false in h, while Fh contains all wffs false in this valuation. An ordering on Hr , the set of all valuations of theory T, is induced by set-inclusion as follows:

DEFINITION. h 2 contains hi, h~ ~_ h 2 iff Thl E ~_ Th2 E.

One valuation contains another if it assigns at least the same elementary truth-values. It is with respect to containment that a state must be maximal.

248 RACHEL WALLACE

DEFINITION. h is a state of T i f f h ~ H r and there is no h' ~ Hr : h _ h'.

No other valuation of a theory can make the same elementary truth-value assignments as a state, and also make more.

Bivalent valuations will always be states, although states will not in general be bivalent. In the special case where a theory has all states bivalent, the theory is called classical. A theory without this property is non-

classical. Classical theories have many important properties, as will be seen below.

I now consider a relation among the valuations of any theory, which will specify the order in which these may be used to describe unchanged realities. This will be called the successor relation. The successors of valuation h in HT, are to be those valuations which might be used immediately after h to describe the same unchanged physical system.

To determine the successor relation, two assumptions are introduced. First is that of Successive Non-Contradiction: h 2 may not immediately follow h 1 to describe the same reality, if Thl ~ c~ Fh2 E # 0. A successive valuation does not contradiet immediately preceding elementary truth- values. No proposition is true in the first, but false in the second, valuation. The second assumption is that of Conservation o f Information: If h 2 immediately follows h 1 to describe the same reality, and if Th '~ = Thl E u

Th2 e for some valuation h', then h 2 = h'. That is, if preceding elementary truth-values can be consistently added to those of the succeeding valuation, then these elementary truth-values are preserved. According to this assumption, information is not simply 'lost' in succeeding valuations.

Combining these two Principles, one defines the following relation on

HT:

D E F I N I T I O n . h 2 is a successor of h~, h2S h~ iff Thl E ~ Fh2 ~ = 0, and if there is an h' ~ Hr: Th 'E = Th~ E, then h z = h'.

A successor of h does not contradict elementary truth-values assigned by h, and it preserves these assignments where possible. One assumes that only successors of h may be used after h, to describe the same unchanged system.

Classical theories turn out to have a special successor relation. For since all states of such theories are bivalent, there must always be an h' such that Th 'e = Thl E u Th2 r when Thl e" ~ Fh2 e = 0. Thus one establishes


RESULT 1 : h2S h i iff h x ~ h 2 in any classical T.

In classical theories succession coincides with containment. Thus in a classical theory, a sequence of successive valuations will be a

sequence in which elementary truth-sets increase. I f h 1 . . . . . h, is a series whereh i+ lSh i , 1 ~< i < n, then T h l e ~_ The e ~ . . . ~ Th , e b y resu l t 1

above, Furthermore such a classical series must always ultimately end with

a state, since by result 1 these alone have no non-trivial successors. Thus successive descriptions of a single reality can always be understood in classical theories, as successive determinations of some unique classical

state. It follows that the states of classical theories can be supposed in one- one correspondence with the realities described. In classical theories the

single term 'state ' may be used both for the maximal valuations of the

theory, and for the physical states-of-affairs which these describe. In non-classical theories however, result 1 above does not hold, and our

understanding of states and of succession must be carefully revised. Since

states are not always bivalent, succession does not coincide with containment, and so one cannot assume that elementary truth-values are simply added in successive valuations. In non-classical theories a series h~ . . . . , h,

of successive valuations is no longer a series in which elementary truth-sets increase until a final bivalent state is established. It may be the case in such a series that Th i ~_ Th i + ~ for some i < n, but this need not be so. An elementary proposition may be true in one valuation, but undecided in the next member of the series. Furthermore this proposition may actually be false in some later valuation. In no theory can a successive valuation con-

t radic t a n immediately preceding elementary truth-value, but only in the special case of a classical theory does it follow that all successive valuations actually agree on preceding elementary assignments. In every theory con-

sistency must be local, in the sense that immediate successors do not assign conflicting elementary truth-values. But only in classical theories can one assume that consistency is also global , in the sense that no members in a series of successors will contradict elementary assignments. 3

One important consequence of the nature of succession in non-classical theories, is the reduced significance of the non-classical state. Since succession does not coincide with containment, states can have non-trivial successors. Thus distinct states of a non-classical theory can be used to describe a single unaltered state-of-affairs. The unique classical correspon-

250 RACHEL WALLACE

dence between states and realities, is lost in non-classical theories. It therefore becomes important to distinguish states, which are just maximal descriptions given by a theory, from the real states-of-affairs which these describe. The single term 'state' cannot be used for both in non-classical theories, without great ambiguity. Many quantum mechanical 'paradoxes' arise from the confusion of theoretical states, with the realities they describe.

III. PROBABILITIES

It is now possible to define two different kinds of conditional probability. In non-classical theories these are quite distinct, but in classical theories they both collapse onto the familiar classical conditional.

The first probability will be called a predictive conditional, since it is to give the 'likelihood' that a wff will be found true in the next valuation describing a system, given some initial description. That is, probh (0t) is to tell us how likely it is that 0t will be true when we next look at, or next describe, the system described by h. Formally this will be defined as a measure of the successor states of h, which find wff ~t true.

The second probability will be called a ratio conditional, since it is to depend on the proportion of states which share certain truth-assignments. That is, prob(0~/fl) tells us how likely ~ is to be also true in a state in which is true. This probability makes a statement about shared truth-assignments, and is not essentially a prediction, as in the first case above.

The predictive conditionals will be taken as fundamental here, and the ratios can later be defined in terms of them. To formally define these probabilities one must first construct the sets over which they are to be measures. For any wff ct, valuation h, one defines:

DEFINITION. The successor-set of h, S t h = df {h' e H r : h' is a state & h ' S h } ;

The a-successor-set of h, Sth ~ = df {h' e S t h : ~ ~}; The successor-fieM of h, F h = df {Sth ~ : ~ ~ L E t } .

S t h is the set of all states which are successors of the valuation h, while Sth ~

is that subset of successor-states, which also find wff ct true. The field Fh

contains all the 0~-successor-sets, for each ct in the logic L E t .


One shows that F h is indeed a field, as follows. Clearly F h is a set of subsets of St h by the definitions above. Furthermore there are connectives in L E r which correspond to set operations among these subsets. Thus disjunction and conjunction correspond to union and intersection, since Sthot v fl = Sth~ u Sth # and also Sth "'~ = Sth ~ • Sth ~. Although negation does not correspond to set complement, there is a connective in the logic which does. This is denial ~ (defined in section I), which is such that Sth ~" = -Sth ~. SO F h is closed with respect to set operations. Lastly one can show that F h contains set S t h by noting that p 3 p for example is a tautology of L E r, and hence SthP ~P = Sth .

Thus Fh = (Fh, u, n, - ) is a field of subsets of Sth, and hence one may construct the Kolmogorov probability space (Sth, Fh, #h) where Ph is a Kolmogorov probability measure on field F h. The conditional probability function probh, mentioned at the outset of this section, will be defined in terms of this measure, as follows

DEFINITION. The conditional probability o f ct given h, prob h (ct) = df #h (Sth ~) for any ~ in L E t , h in Hr . This probability is a measure of the states which are sucessors of h and which find �9 true.

An 'absolute' probability function may be introduced as a special case. This is conditional on the valuation ho which makes_only trivial truth-valtte assignments, h o is defined by setting Th o = d f {~ ~ L E t : ~h ct for all h

Hr}.

DEFINITION. The absolute probability o f ~, prob (~t) = df prob h (~) o

Intuitively this is just the probability of ~ given no initial information about the reality described.

The fundamental predictive conditional probability, prob h (~), given earlier, may be re-expressed using the following natural correspondences between wffs of the logic L E r, and valuations in H r.

DEFINITION. h, is the characteristic valuation for ct, iff

Th~ = {fl ~ LEr: ~ h ~ 3 fl for all h in H r }

252 RACHEL WALLACE

~h is the characteristic wf f for h, iff 7h is the conjunction of all elementary propositions in Th (i.e. all members of Th ~)

h~ finds true ~ and all valid consequences of 0~, but makes no further truth assignments. ~h is the conjunction of all the elementary propositions which are true in h. Using these terms one may re-express conditional probabilities in terms only of wffs or valuations, as follows

DEFINITION. prob~ (fl) = d f probh (fl) problc (h) = df prob~, (~,h)

The ratio probability can now be introduced by the definition

DEFINITION. probh (~t/fl) = df probh (~-fl) / probh (fl); and in particular prob (~/fl) = df prob (~. fl) / prob (fl).

This is a ratio of absolute probabilities. The expression is familiar as the standard 'conditional' probability of ordinary statistics, and indeed one can show that in classical theories this ratio does coincide with the predictive conditional defined above.

RESULT 2: prob h (0t) = prob (0t/yh) in any classical theory; and prob, (fl) = prob (fl/~) in any classical theory.

In classical theories, conditional probabilities always coincide with corresponding ratios. This result is easily established in the finite case, and arises from the special property of classical theories that succession coincides with containment. (Wallace 1979)

It follows from this important result, that all conditional probabilities of classical theories, can be expressed as ratios of absolute probabilities. Thus a single probability space will suffice in classical theories, to generate all conditional probabilities. This is the space (St, F, #) = (Stho, Fh0 ,/tho ) where ho is the trivial valuation used earlier to define the absolute probability function. It follows from result 2 above that this 'absolute' space alone is required to express all conditional probabilities of a classical theory. Since set St ( = St h ) will always be the set of all states in a classical theory, one can see that ~ 'absolute' probability space is just the familiar 'phase space' of classical mechanics.

P R O B A B I L I T I E S IN M E C H A N I C S 253

In non-classical theories Result 2 does not hold. Predictive conditionals do not coincide with ratios of absolute probabilities. Thus there is no single 'phase space' of states which will generate all the conditional probabilities of a non-classical theory. Different predictive conditionals are measures over different spaces of states. It is only in the special case of a classical theory that the single absolute space will generate all probabilities of the theory.

The conditional probabilities of non-classical theories exhibit peculiarities which are very similar to the most puzzling peculiarities of quantum statistics. In non-classical theories the probability of a sequence must be computed using a rule quite unlike the expression which is used in classical statistics. Later it will be shown that this rule can be regarded as a generalised version of the rule used in quantum mechanics. And secondly the probabilities of non-classical theories exhibit measurement-dependence, even though no extra assumptions are introduced about the real nature of systems described by a theory, or the physical influence of measurement procedures.

In the following discussion attention will be limited to theories which have only a finite number of states. This allows the interesting features of probabilities to be easily discussed, while generalisations to non-finite cases are usually clear. The general expression for conditional probabilities in a theory with a finite number of states, is given by

prob h (~) = ~ , w h ( h ' )

h' E St~

where w h is a weight-function which depends on h. That is, the probabilities are in the finite case, just weighted sums of the successor-states of h in which wff ~ is true.

The expression used to compute sequential probabilities in any theory with a finite number of states, will now be derived. First consider the general expression for the probability of a 2-member sequence. This is given by

DEFINITION. prob h (ct, ]~) = df prob h (~). probh<~) (fl) where probh<~) (fl) is the conditional probability of/~, given that ct was found true of the system initially described by h. In the finite case this is

254 R A C H E L W A L L A C E

DEFINITION. probh(~) (~) = d f

XE w,`. E w.,N h' ~ S ~ h" ~ St~

E w h (h') h' ~ Sty,

That is, to compute this probability one 'counts' all the successor-states finding fl true, of successor-states of h which find ~ true. Combining these definitions:

RESULT3:probh(~,fl}= ~ E wh(h')Wh,(h") h" ~ S ~ h" ~ Sty.

One may generalise this expression to get

(G) probh (~x . . . . , ~,)

=E hl~ E St~ h2. ~ S t ~ ' h.~ ~ S~-~k

The conditional probability of a sequence is computed by 'counting' all the successor-states of all the successor-states which find the appropriate wffs true.

In the case of classical theories however, no such complex expression is needed. For in this case sequential and joint probabilities coincide. Con- sider for example the simple case of the absolute probability assigned to the 2-element sequence (~, fl}. prob (~,/3} gives the probability that ~ is first found true, then fl, given no initial condition. The appropriate expression for this probability is clearly:

DEFINITION. prob (~, fl} =d f prob (~). prob, ~)

But by result 2 above, prob~ ~) = prob (fl/~) in a classical theory, so


RESULT 4: prob (~, fl) = prob(~ .fl) in a classical theory

This result can be generalised to establish that the probability of a sequence always coincides with the probability of the corresponding conjunction in a classical theory. One consequence of this result is that the order of elements in a sequence is never relevant to its probability in a classical theory.

In general however, where succession does not coincide with containment and so result 2 does not hold, result 4 above does not hold either. In these cases the probability of a sequence will be computed using a general conditionalising rule such as (G) above. In particular it follows that the order of elements in a sequence is crucial to its probability, if a theory is non-classical.

Another feature of the general probabilities which is evident in quantum statistics is their measurement-dependence. It is a well-known 'peculiarity' of quantum mechanics, that the probability assigned to a proposition or state, varies with any measurement which is supposed to take place. This is often used to support the so-called 'disturbance' interpretations of quantum theory. Unfortunately the role of measurement in mechanical theories is an important issue that cannot be discussed in detail here. However one can point out that the measurement-dependence evident in quantum mechanics will also arise in any non-classical mechanical theory, even though

no assumptions are made about the nature of measuring processes, or about the nature of realities described by a theory. Thus a 'disturbance' interpretation is unnecessary, and the measurement-dependence simply arises from the lack of bivalent states.

To make this point, let us suppose that measurement is some unspecified kind of procedure which establishes the truth of some non-trivial elementary proposition. That is we suppose that a measurement of magnitude m will, by definition, establish that some m-proposition (m, A) say, is true, where A # Vm. Let ~Sm be the disjunction of all such non-trivial m-propositions, so that 6m must be true when any measurement of m has taken place. Then the probability of any wff ~, given that a measurement of m is performed, will be the ratio probability prob h (~/6m), where the initial state is h. If m is measured 6m is true, and so this ratio gives the appropriate probability of ~ given that a measurement of m is performed on the system initially described by h.

256 RACHEL WALLACE

In classical theories probabilities will not vary with the measurement given. For classical states, being bivalent, assign truth-values to all elementary propositions. It follows that 6rn is true in every classical state, for any magnitude m, and hence in every case the ratio probh (a/rm) coincides with the simple conditional probh (a). However in non-classical theories, where states are not always bivalent, the wff 6m is not generally true in every state. It follows that probabilities will vary according to the magnitude measured. Ifm ~ n, then in general probh (~/6m) 4: prob h (a/fin) ~ prob h (~). Generally probabilities are measurement-dependent in this way.

IV. CONCLUSIONS FOR MECHANICS

In the earlier sections of this paper, a general probability theory was sketched, and shown to arise from the logic of a mechanical theory. It has been briefly shown that the general statistics does give way to regular classical statistics if the mechanical theory is a classical one. It has also been suggested that there are some similarities between the general probabilities derived here for non-classical theories, and the probabilities used in quantum theory. It has not however been shown that the quantum probabilities are non-classical conditionals of the kind discussed here, nor will this be attempted in this paper. I suggest only that quantum probabilities may be a special case of the general non-classical conditionals developed here. Ex- actly how the quantum probabilities are to be defined in terms of measures over successor-sets, will be merely suggested later.

However Jeffrey Bub, in an unpublished paper, has compared the general 'Wallace Rule' (G) above, to the conditionalising rules used for com- puting sequential probabilities in quantum mechanics. In particular he has compared (G) to the rule derived from von Neumann's Projection Pos- tulate for the probability of a sequence, and to Ltiders' amendment of this rule. Bub has shown that one can regard the quantum mechanical rules as successively special cases of the rule (G). His comparison does indicate the kind of definition one might expect for quantum probabilities.

The following notation will be used in the discussion of quantum theories. Magnitudes of a quantum theory T, are represented by particular operators of a Hilbert space ~r associated with the theory. Variables A, B . . . . are used for these operators, and so these correspond to variables m, n . . . . of the earlier discussion. According to quantum theory the spectra of


operators A, B (in the discrete case their eigenvalues {ea, }, {ebj } respectively) represent possible outcomes of measuring the magnitudes, and hence these spectra correspond to the value sets Vm, Vn. So particular eigenvalues correspond to particular m-values, and hence to particular m- propositions of the quantum theory. Each eigenvalue e (and hence each elementary proposition) is associated with a subspace o,~(e of the Hilbert space ~'~T, and thus with a projection operator Pe - Pe is that projection operator which has ~ffe as its range. States of quantum theories are represented by vectors in the Hilbert space )fiT, and variables if, 0 . . . . will be used to refer to them.

According to standard quantum mechanics the probability of the proposition associated with eigenvalue el, given state if, is

(Q) probe (el) = Tr(Weel)

where W is the statistical operator associated with ~k. The probability of a 2-member sequence is given by the expression

(Q2) probe (ex, ez) = Tr(WPel). Tr(W'Pe2)

According to von Neumann's Projection Postulate, the operator W' above is

k e 1

~ Tr (W Peli) Peli W ~ ~ i = 1

ke 1

S Tr (W Peli) i - - 1

where kel is the dimensionality of Jgel, and {e~i} is a basis for this subspace. Substituting in expression (Q2) above and cancelling, one concludes that

kel

probr (el, e2) = ) i Tr(WPeai)" Tr(PeaiPe2) i = l

This is generalised to give the yon Neumann Rule for conditionalising:

(V) prob~ (e I . . . . , e , )

258 RACHEL WALLACE

kel ke2 ke.

= ~ ~ "" ~ T r ( W P e x , ) ' T r ( P e ~ , P e 2 s ) ' . . . ' T r ( P e . - ~ k P e . , ) i = 1 j = l I = 1

According to Liiders' amendment of this rule, the operator W' in expression (Q2) above, is not that given by von Neumann, but is instead

W' = Pel W Pe 1

Tr (Pel W Pel)

and hence according to Liiders, the correct probability for a 2-member sequence is

probe (el, e2) = Tr (Pet W Pe 1 Pez)

Generalising one concludes that

probe (el, ..., e . ) = Tr (Pe._ 1 .. . Pel W Pe I . . . Pe ._ 1 Pe.)

By straightforward manipulation, Bub shows that where ~ is pure, this may be re-expressed as the following statement of Liiders' Rule for conditionalising:

(L) probe (e 1, ..., e . ) = Tr ( P $ Pit)" Tr (P;t Ph) ..-" Tr (P,,~,_ 1

where for example, ~1 is the normalised projection of $ onto subspace o~"el, ~2 is the normalised projection of ~i onto ~e2 , and so on.

To compare the two quantum mechanical rules (V) and (L) above, with rule (G) Bub makes the following identifications. Let h = ~, a pure state. Then w h (hli) = Tr (P $ Peli), wnl (h2j) = Tr (Peli Pe2j) and so on. These are the weight-functions given by quantum theories. In addition one re- gards the basis set {elf} of J/e~ for example, as a selection of states from the set St~ 1. According to quantum mechanics the selection of particular basis {eti } depends on a measurement, M e t say. Thus set {el~} may be represented by the term [h' ~ St~l]nel. This is a subset of St~ ~ which depends


on the particular measurement M e t Using this notation and the identifications above, von Neumann's rule may be re-written for any theory as

(V') prob h (e 1 . . . . . e . )

e2 [hll e St~]M~, [hzj E Sth~t]M~ ~ [h~ e Sfh" ,]M~

Wh (h.k) n - l l

This is expressed more clearly by abbreviating

(V') p robh(e 1 . . . . . e , ) = y ... y ,

[ ]~e, [ 1~.

Wh (h l i ) . . . w h (hr, k) n - l l

To compare Lfiders' Rule, set ~ = h, ~ a pure state, and let gi = hi* 1 -%< i ~< n. Then Tr (Pr Pe~) = w h (hi*), and Tr (P~I P~2) = % , (h2*) and

�9 . , ' . ' , 1

so on. Substituting m (L) above, Bub obtains the following general expression for Ltiders' Rule:

(L') prob h (el , ..., e , ) = w h (hi*) ... Wh_, . (hn*)

= ~ " . Z wh(ha*)..,Wh , ( h , *)

[]M.I []Me.

using the same abbreviation as in (V') above. This expression makes clear that Ltiders' expression for sequential probabilities is indeed a special case of von Neumann's. According to Liiders, a single state in the successor set SfhL, is selected at each stage of the computation. The unique state hi*, say, depends on h and satisfies the requirement that Wh (hi*) =

w h (hli) any measurement Melj of e 1. Similarly h2* for

[hli E stehl]Melj

is that unique state for which Wh, . (h2*) = ~ ' Whl . (h2j) for

[h2/~ Steh~,]Me~,

260 RACHEL WALLACE

all measurements Me2k of e 2. One may regard Ltiders' Rule as that special case of von Neumann's, appropriate when unique states of this kind can be selected.

But von Neumann's Rule is itself a special case of the general rule (G) above. In the general case, all members of the successors sets St~,i_ 1 are considered at each stage of the computation, while according to von Neumann's expression only a selection of members of this set are con-

sidered - each selection [ ]Me corresponds to a basis set of the subspace �9 i k

3fie i, depending on the particular measurement Me~ of e i. Bub points out that the general rule (G) will arise as that special case of von Neumann's rule (V) in which only non-specific measurements are given. That is, if each measurement of each el is of the sort "measure el either by measurement Meil or by measurement Mei2, or by . . ." for every measurement Meu, ofei,

e, and so then every basis is considered and so in this case [ ]Me - - Sthi_ 1, rules (V) and (G) coincide.

It seems likely from this discussion that the conditional probabilities of quantum mechanics are the special case of general non-classical conditionals in which a special subset of successor-states are used to compute a probability, rather than the whole successor set. The special subsets are 'bases', containing only mutually non-successive members of the successor-sets S t h . Quantum mechanics is perhaps a non-classical theory in which basis stlbsets can be taken to represent the successor-sets, and so probabilities are generated by these subsets. According to Bub's analysis of Ltiders amendment however, quantum probabilities may sometimes even be the yet more special case in which each basis representing a successor-set is itself represented by a single state which has the special property given

earlier. The exact nature of quantum probabilities will not be analysed here.

However if it were shown, as suggested above, that these are a case of non- classical conditional probabilities of the kind developed here, then ones views of quantum theory would need to be revised. The peculiar features of quantum probabilities would then be seen to arise from the non-classical nature of quantum theory, and not for example from the peculiar nature of subatomic reality, or from the peculiarly disturbing nature of subatomic measurement.

This view of quantum mechanics corresponds to some extent to the views of Einstein (Einstein, Podolsky and Rosen 1935) since the pe-

P R O B A B I L I T I E S IN M E C H A N I C S 261

culiarities of quantum statistics would essentially arise from an 'incom- pleteness' in the descriptions given by quantum mechanics. The states of quantum theories are generally complete only in the weak sense that they give maximal consistent assignments of truth-values to elementary propositions. Quantum states are not complete in the stronger sense of giving exhaustive truth-value assignments to all elementary propositions. It is the lack of 2-valued valuations which supports the claim that quantum states are incomplete.

A better theory than quantum mechanics would be one that described subatomic reality with bivalent states, and hence which used a classical probability theory. However Kochen and Specker have argued (1967) that standard quantum mechanics cannot simply be augmented by 'hidden variables', to give it bivalent states. If this is correct, then quantum mechanics must be altered if it is to be complete. That is, a new theory, with different magnitudes, laws, or value-sets, would be needed to supercede quantum mechanics.

Hafod Lwyfog

A C K N O W L E D G E M E N T S

Research for this paper was partly funded by a Winifred Cullis grant from the International Federation of University Women (1976), and an Ontario Graduate Scholarship, and also by funds from the University of Western Ontario.

N O T E S

1 Note that the well-known Post and Kleene 3-valued logics are ruled out by the requirement that connectives express the structure of E r. For the Post negation is the 'cyclical' kind which for example, makes a true wff undecided, and hence by condition (i) is unsuited to express operation • The valid Kleene hook is not a partial ordering. It is not reflexive, since this hook between two undecided wffs is undecided. (Both these logics are discussed e.g. in Re- scher 1969). 2 Formally, the scope of a wff ~, S~, is defined by

DEFINITION. S~ =df{ f l~LEr : h ( ~ . 7 ~ ) 3 r a n d hfl 3 ( ~ v 7 ~ ) f o r a l l h i n H r

This term is used to introduce the following natural relatedness, or relevance relation

DEFINITION. fl is related to ~, fi R ~, iff fi ~ S

3 Clearly one could also assume the following Principle of Distinct Realities: If ha, h 2 E H r are such that Th~ c~ Fh~ 2 ~ ~), then h a and h 2 describe different realities. By appealing to this principle one could rule out a sequence h a, ..., h, in which two members contradict one another, i.e. in which Thr~ c~ Fhg ~ ~ for some i, j ~< n, as a sequence describing a single

262 R A C H E L W A L L A C E

reality. That is, all theories would be assumed globally consistent. However there seems no reason to suppose that this principle must hold for every theory. One might call theories with this property, semi-c.assical, since classical theories agree with the principle. In the present discussion it will not be assumed that all theories are semi-classical.

B I B L I O G R A P H Y

Birkhoff, G. and von Neumann, J.: 1936, 'The Logic of Quantum Mechanics' Annals of Math. 37, Reprinted in Hooker (ed.) (1975).

Bub, J. : 1974, The Interpretation of Quantum Mechanics, The University of Western Ontario Series in Philosophy of Science, vol. 3, D. Reidel, Dordrecht.

Bub, J. : 1976, 'The Statistics of Non-Boolean Event Structures', in Harper and Hooker (eds.) below.

Einstein, A. Podolsky, B. and Rosen, N.: 1935, 'Can Quantum Mechanical Description of Reality be Considered Complete?' Physical Review 47, 777-780.

Harper, W. and Hooker, C. (eds): 1976, Foundations of Probability Theory, Statistical In- ference, and Statistical Theories of Science, vol. III, U.W.O. Series in Philosophy of Science, vol. 6, D. Reidel, Dordrecht.

Hooker, C. A. (ed.): 1973, Contemporary Research in the Foundations and Philosophy of Quantum Theory, U.W.O. Series in the Philosophy of Science, vol. 2, D. Reidel, Dordrecht.

Hooker, C. A. (ed.): 1975, The Logico-Algebraic Approach to Quantum Mechanics, vol. I: Historical Evolution, U.W.O. Series in Philosophy of Science, vol. 5, D. Reidel, Dordrecht.

Kochen, S. and Specker, E. P. : 1967, 'The Problem of Hidden Variables in Quantum Me- chanics', Journal of Mathematics and Mechanics 17. Reprinted in Hooker (ed.) (1975).

Rasiowa, H.: 1974, An Algebraic Approach to Non-Classical Logics, North-Holland, Amster- dam.

Rescher, N.: 1969, Many- Valued Logic, McGraw-Hill. van Fraassen, B.: 1970, 'On the Extension of Beth's Semantics of Physical Theories', Philo-

sophy of Science (September 1970), pp. 325-339. van Fraassen, B.: 1973, 'Semantic Analysis of Quantum Logic' in Hooker (ed.) above. Wallace, R. : 1979, A Logic for Mechanics, Ph. D. Thesis, University of Western Ontario,

June.

a new approach to probabilities in mechanics

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