what is the logical form of probability attribution in qm

8/11/2019 What is the Logical Form of Probability Attribution in QM

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2/26

WHAT

IS

THE LOGICAL FORM OF

PROBABILITY

ASSIGNMENT IN QUANTUM

MECHANICS?*

JOHN F. HALPINt

Department of Philosophy

Oakland University

The

nature

of quantum

mechanical

probability

has often

seemed

mysterious.

To shed some light on this

topic,

the

present paper analyzes

the

logical

form

of

probability

assignment in quantum mechanics. To begin the

paper,

I

set out and

criticize several attempts to analyze

the

form.

I

go

on

to

propose a new forn

which utilizes a

novel, probabilistic

conditional and

argue that this

proposal is,

overall, the best rendering of the quantummechanical probability assignments.

Finally, quantum mechanics aside, the discussion here has

consequences for

counterfactual

logic,

conditional

probability,

and

epistemic probability.

Most

of the

interesting

and

difficult

interpretive

issues

in

quantum me-

chanics

(QM)

are

closely

tied to

that

theory's probabilistic

nature: QM,

rather than making

unequivocal/deterministic predictions, assigns prob-

abilities to

the

possible

results

of

observation

or

measurement. For

this

reason QM

is

said

to be an indeterministic

theory.

However, the prob-

ability assignments of QM, and in fact, probabilities in general, arephilo-

sophically vexing. Indeed,

the

question

of

the

nature of

quantum

me-

chanical

probability

assignments

constitutesa

significantpart

of the

problem

of

interpretingQM.

That

question

is to be

pursued

in

this

paper.

As

I

see

it,

this

question

of

quantum

mechanical

probability

assignment

has two

parts:

What

sort of

interpretation

should

we

give

to

probability

as

it occurs

in QM?

What is the logical form of the probability assignments of QM?

The

first

of these

questions

is

the fundamental one. It

is the

traditional

question asking,

for

example,

are

quantum

mechanical

probabilities

de-

grees

of belief? relative

frequencies? propensities?

or

perhaps something

nonclassical?

An

answer

will

explain

the

meaning

of

"probability"

as

used within

QM.

But the second

question

is

also of

importance;

as

we

*Received February

1988;

revised

January

1989.

tI

want to

thank

Arthur

Fine,

Alan

Nelson,

two

anonymous

referees for

Philosophy of

Science, and, especially, Paul Teller for comments on an earlier draftof this paper. These

were

extremely

valuable. Research

for this

paper

was

supported

by

a

Faculty

Research

Grant from Oakland

University.

Philosophy f Science,

58

(1991) pp.

36-60.

Copyright X

1991

by

the

Philosophy

of

Science Association.

36

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3/26

LOGICAL FORM OF PROBABILITY ASSIGNMENT 37

will

see,

it

is in some

ways

a

preliminary

to

the first

of

these

questions.

And, as well, this question

of

logical

form is

surprisingly difficult.

The main emphasis of this paper is the second question. The goal here

is

to see how best to think about the probability assignments of QM to

make the

best sense of these

assignments.

So here

I

attempt to analyze,

or,

in

case

of

vagueness, explicate,

the

logical

form of

quantum me-

chanical probability

attributions.

To

begin,

I

set out and

criticize several

attempts

to

analyze the

form.

I

go

on to

propose a

new form

which utilizes

a

novel, probabilistic

conditional and

argue

that this

proposal is, overall,

the best

rendering

of

the

quantum

mechanical

probability assignments.

Finally, quantum mechanics aside,

the discussion here

has consequences

for counterfactual logic, conditional probability, and epistemic proba-

bility.

1. Introduction.

I

have

claimed that

the

question

of

the

logical

form

of

quantummechanical probability assignments

is

difficult. To

begin to see

the

difficulty,

consider

a

quantum

mechanical

example.

A

particle, say

an

electron,

is

assigned

a state

description

or

"wave

function", If,

a

com-

plex valued

function

defined on

3-space. According

to

the

quantum the-

ory,

a certain

function

of

f,

I112-

W*f

=

the

complex conjugate

of

If

times If, gives

the

probability density

for

that

particle's position.

Roughly, this

means that

the

electron

is

most

likely

to be found in

regions

where

1I12

is

large.

The

straightforward

or

naive

reading

of

such a

claim

is

that,

for

all

regions

V

of

space,

the

integral

over

V of

11,12

ives

the

probability

that the

electron is

in

V.

But,

in

general,

there

is a

difficulty

with

this

straightforward endering:

on the received

view

of

QM,

that

is,

the

Copenhagen interpretation,

the

electron is

typically

not in

any region V,

at

least not

in

any

small

region

V. According to this view, it is, with a few exceptions, only when a

position

measurement

is

made that

a

particle

can be said to

occupy

a

specific region

of

space. Moreover,

the

same

is

said about

all

interesting

physical quantities:

such

quantities typically

have

no

values until

mea-

sured.'

Now,

it

would be

wrong

to

assign

nonzero

probability

to

some-

thing

which is

certainly

false.

So, assuming

the

received view of

QM,

because

a

typical

electron

certainly

is

not

in

V

before measurement,

it

cannot

have a

nonzero

probability

of

being

in

V

before

measurement.

Hence,

the

straightforward reading

of

quantum

mechanical

probability

ascription-as assigning probabilities that a particle possesses a position

(or other)

value-cannot

on

the

received view be true in

general,

for

example,

cannot

be true

before measurement.

The

received

view has

gar-

1QM assigns probabilities

not

only

for

the

physical quantity position,

but also

assigns

probabilities

for

the values of other

physical quantities,

for

example, angular

momentum.



4/26

38

JOHN F.

HALPIN

nered impressive support

in

the last twenty years through the work

of

John Bell and others.2

This

leads one to ask: If, for example, quantum

mechanical position probabilities are not probabilities (in the-straightfor-

ward, naive sense) that a particle has

a

position, what are they?

In a standardQM textbook, Eugen Merzbacher notes that

I

12

"is pro-

portional to the probability

that

upon

a measurement

of

its

position the

particle

will be found

in

[a] given

volume element"

(1970, 36).

But this

remark eaves unclear the

notion

of

probability-upon-measurement.

In an-

other standardtext Albert Messiah writes:

. .

.since

the wave

function .

.

. has a certain

spatial extension

one

cannot attribute o a quantum particle a precise position; one can only

define the

probability

of

finding

the

particle

in

a

given region

of

space

when one carries

out a

measurement

of position. (1976, 117,

Messiah's

emphasis)

Later he makes

a more

general

statement:

One

. . . abandons the fundamental

postulate

of

Classical

Physics,

according

to which all

the various

quantities belonging

to a

system

take

on well-defined values at

each

instant

of

time. One can

only

determine for each of these variables a statistical distribution of val-

ues,

which

is

the

probability

law

of the

results

of

measurement

in

the

eventuality

that

such

a

measurement

is

performed. (1976,

294,

Messiah's

emphasis)

I

take it that Messiah's eventualities may

be

hypothetical

or

counterfactual

eventualities,

and that

the probability assignments "upon"

measurement

or "in

the eventuality

that

a measurement

is

performed"

are

probabilities

given

the

(possibly counterfactual) assumption

that a measurement is

per-

formed. (The "probability aw" assigns probabilities even in the absence

of

actual measurement.)

So,

at

least

on

the

standard

view, quantum

mechanical

probabilities

are

2Bell's (1965) argument

shows that

the

hypothesis

that

all

physical quantities

have

values

leads to statistical predictions different

from

those

of QM. (Bell assumes a statistical

lo-

cality condition,

that measurement

results are

statistically independent

of what

measure-

ments are made

at distant locations, plus

he in

effect makes an idealization

about

mea-

surements,

that measurement

devices are

perfectly

efficient and

accurate,

in order to

derive

the conflict with QM.) Now,

the

quantum

mechanical statistics are

confirmed

in

the lab-

oratory, hence

the

hypothesis

that all

physical quantities

have values

is

often

taken to be

proven false. However, ArthurFine and others have shown thatreasonable possessed value

models exist (models

which respect

the

observed quantum

mechanical

statistics) as

long

as detector inefficiencies

can be

assumed. But even

if

Fine's models are

correct,

we

cannot

read

the

quantum

mechanical probability

attributions

as

being

about

possessed

values. On

Fine's models,

the

possessed

values are not

distributed in

accordance with the

quantum

mechanical

statistics, (Bell's

theorem assures

this).

Only

the observed

values are

so

dis-

tributed. See Fine (1982)

for some

possessed

value models.



5/26


in some way conditional on a measurement. But how so? There have

been a number of

attempts to say exactly what

this means.

In what fol-

lows, I try to show that the extant proposals are all problematic.

2.

A

Survey of Proposals.

As

we

have

seen, the probability assign-

ments

of

QM

are conditional

upon

the occurrence of a

hypothetical

or

counterfactual measurement. Now,

the

resources

for

clearly formulating

conditional assertions

are at hand.

First, mathematicians have developed

the notion

of

a

conditional

probability, Pr(A/B),

read "the

probability of

A

given that B", and defined as P(A&B)/P(B) where P, a probability

measure,3

gives the

unconditional probabilities. Also, logicians have de-

veloped the semantics for counterfactual conditionals. Take an example:

If the Earth's mass were larger, its gravitational attraction would be

greater.

This is

to be analyzed as follows: Under certain (counterfactual) circum-

stances

in

which the antecedent

holds,

the

consequent

is

also true.

Such

counterfactual

ituations

are

usually spelled

out

in

terms

of

possible worlds.

Possible

world

semantics

are

still

controversial,

but as a

provisional

def-

inition

we

may take the following. (An

A-world

is

defined to

be a possible

world at which A is true.)

(0) A

>

B

is true

at a world w

if

and

only

if

B

is

true at the

A-worlds

most

similar4 to w.

On

this

analysis,

the above counterfactual about

the Earth is

presumably

true because the

worlds most similar to

ours which contain

a more mas-

sive Earth will

obey

the law

of

gravitation

and hence

will

involve

greater

gravitational

attraction.

Three

facts

about conditionals

will

be of

interest

here.

First,

on the

theory of conditionals just sketched, A > (B&C) is equivalent to (A >

B)&(A

>

C):

A

>

(B&C)

is

true

at

world w iff

(B&C)

is

true at the

A-

worlds

most

similar to

w iff

both

B

and C

are true at

the

A-worlds most

similar to w

iff both

A

>

B

and

A

>

C

are true at w.

Secondly,

Modus

3A probability measure,

P,

is defined so

that 0 ' P(A)

'

1, P(A

V

-A)

=

1, and P(A

V

B)

=

P(A)

+

P(B)

if

P(A&B)

=

0.

Furthermore,

the domain

of

definition of

P

is

to

be

a Boolean algebra. Moreover,

P

is sometimes taken to

be defined on a

sigma field of

propositions, a set closed under countable disjunction and conjunction. If so, countable

additivity is assumed. However, these additional

properties

of a

probability measure are

not pertinent to the present project.

'The notion of similarity

here

must

be construed in the appropriate way. As I see

it,

"most similar" comes to "similar enough in relevant respects". For one nice attempt to

spell

out this notion

see Lewis

(1979).

Those

skeptical

of

the

similarity theory

of coun-

terfactualsshould substitute

the

word

"relevant"

for

"similar"

in the above

definition. That

change of wording helps to emphasize

that much more needs to be said in order to clarify

the

operative

notion of

this

definition.



6/26

40 JOHN F. HALPIN

Ponens

is

a valid inference

for

the conditionals described:

if A

and

A

>

B

are true

at

w, then

B is true

at the

A-worlds

most similar to w; but by

assumption w is an A-world, so it must be an A-world most similar to

itself (nothing else could

be

any

more similar to

itself ), hence

B is

true

at w. So B follows from A and A

>

B.

Finally,

still

assuming the theory

of

counterfactuals ust sketched, con-

ditional excluded middle

(CEM),

(A

>

B) V(A

>

-B),

can fail

to

be

true.

For

example,

it is

plausibly

false for the

well-known

Bizet-Verdi case:

Either

if

Bizet

and Verdi were

compatriots,

they

would

have been

French; or

if Bizet

and

Verdi were

compatriots, they

would have

been

Italian.

Intuitively,

this claim

is

false because

neither

disjunct

is

true;

had

Bizet

and Verdi been

compatriots, they might

have

been

either

French or

Ital-

ian. The analysis just given seems to

bear out this

intuition.

This is so

because there presumably

are antecedent

satisfying

worlds

most

similar

to the actual world at which both men are French and some just as similar

at which both

are

Italian.

So, assuming

the

usual

truth

functional defi-

nition of

disjunction, CEM

fails

to be

valid

for

the

semantics

of

>

just

sketched.

Now,

as

noted,

our

definition

of the

counterfactual

(0)

is

controversial.

Still,

the

consequences just

derived are

generally accepted.

It is

these

noncontroversial consequences which are

most

important

n

what follows.

However, there

is

one

exception

to

my

claim

about

the

noncontroversial

nature

of

these consequences: Stalnaker's

theory

of

the

conditional does

validate CEM. I want to digress briefly to describe this exception and the

reason to stand by (0).

On

the

original similarity theory

of

counterfactuals,

Stalnaker's

(1968),

A

>

B

is

(nontrivially)

true

iff

B

is

true

at

the

A-world, s(A),

most

similar

to the actual.

(Here

s is called a selection

function; intuitively s(A)

is

the

world

or

situation that

would hold

if

A

were

true.)

Notice

that CEM is

true

on this

theory

because

either B

or -B is true at

the world

s(A),

hence

either

A

>

B or

A

>

-B

is

true

(at

the

actual

world). However,

van

Fraassen's "The End

of

the Stalnaker

Conditional",

an

unpublished post-

script

to van Fraassen

(1982),

showed

that

Stalnaker's

(1968) theory

leads

to

Bell-like conflicts with

experiment. (The problem

arises from the

treat-

ment of counterfactuals like

"if

a measurement

of

quantity q

were

made,

the

result would

be r". On Stalnaker's

(1968) theory

a

unique

measure-

ment-of-q-world

is

determined

by

the selection

function,

hence so is

a



7/26


measurement result

r. This

definiteness

of

values, here a "counterfactual

definiteness",

is

what

is

used by

Bell

to

derive

conflicts

with

QM. See

footnote 2.)

Now, van Fraassen takes the problem just described to be a reason to

favor a revised theory, Stalnaker (1981).

The

latter account admits that

realistic contexts determine no unique selection function because there

is

in general no most similar A-world. Instead, this revision presupposes the

existence of a set

{sj}

of selection functions all of which need to be con-

sidered

in a

counterfactual analysis.

Stalnaker does so

by utilizing su-

pervaluations over this set:

a

sentence

of

counterfactual logic

is

true iff

it is evaluated as true relative to each selection function. So, for example,

A > B is true iff for all selection functions

sj

in {sj},A > B is true relative

to

si, that is,

iff for

all

j,

B

is

true at

sj(B).

Now,

CEM is true on

this

account

because, as

we

have

seen

above,

it

is

true

with

respect

to

an

arbitrary

election function

s.

Unfortunately,

much the

same

thing

can be

said about

the result derived

by

van

Fraassen;

the

conflict with

experi-

mental

results

he derived

for

the

original

Stalnaker

theory

can

be gen-

eralized to

apply

to the

revised

account. See

Halpin (1986)

for

an ar-

gument showing that

both versions

of

the

theory

lead to

conflict with both

QM

and

experimental

results.

I take it, then, that Stalnaker's theory in either version is not appro-

priate

for

the quantum

mechanical

context.

However,

the

other well-known

similarity

heory

of

counterfactuals,

David Lewis's

(1973),

has

also

seemed

suspect to many observers. On

Lewis's account

(put roughly)

A

>

B

is

nontrivially true

iff

A&B

is true at some world

more

similar to the

actual

world

than

is

any

world

making

A&

-B

true.

Now,

Lewis assumes

that

worlds

can

be more and

more

similar to the

actual world without the

existence

of

any unique

most

simnilar

world.

For

instance,

take Lewis's

example (1973, 20). Suppose

that

in the

actual

world a line

is

exactly

1"

long. Then a world in which the line is

1

/4"1

long will be less similar to

the actual

world

than

is some

world

in

which it is

1

1/5

long.

And some

world

in

which

it is

1

/6"1

long

is more similar still to the actual world

than

any

of

the

worlds

in which the

length

is

1

1/51.

And so on.

(As

Lewis

notes, these claims about similarity

will hold

only

in

some

contexts;

I

assume such a

context

in

what

follows.) Then,

on Lewis's

theory

we

can

truly say that,

for

any

n

>

0,

if

the

line were

longer

than

1"

in

length,

then

it

would be

shorter than

l/[n".

Unintuitively,

this does

not leave

any

length that

the line

might

be

if it were

longer

than 1". This

sort

of ob-

jection has been made many times; for example see Stalnaker (1981).

Our

brief discussion

so

far

suggests

that

the worlds relevant

for

coun-

terfactual

analysis, given

an antecedent

A,

should

include

more

worlds

than

just s(A). But,

to

continue the

example above,

a world in which

the

line has

length

1

/6"1, though

not as similar

to the actual

world as other



8/26

42 JOHN

F. HALPIN

worlds

in

which the length

of

the line

is

greater than

1",

is

nonetheless

a world

that might

occur.

In

some

contexts,

at

least,

it is

similar

enough

to the actual world to be worthy of consideration when the antecedent is

"the

line

is

greater

than

1"

long". So,

I

suggest

a

compromise

between

these theories which

in

effect takes the selection function

to have a set

of worlds

as values, that is, the set

of

worlds similar

enough to the actual;

this is

(0). This proposal differs

from

Lewis's

in

that we are

asked to

consider a set of

most

similar worlds

(for a given

antecedent)

even

though

some

of

these

are

not

as

similar as others to the

actual. And (0) differs

from

Stalnaker's (1981) because

on

Stalnaker's

theory

(even

as

revised),

there

is a

kind

of

counterfactual

definiteness;

that

is, relative

to

each se-

lection function "if a q-measurement is made, then r would result" is true

for

some

r

(because

each selection function

picks

out a

unique q-

measurement

world).

This counterfactual definiteness

(relativized

to a se-

lection function)

is

sufficient to

derive

Bell's conflict

with

experiment.

(Because

counterfactual definiteness

holds with

respect

to an

arbitrary

selection function

s,

one can also derive as true

relative

to s

a certain

claim C about quantum experiments. Because s

is

arbitrary, Stalnaker's

theory

takes

C to be true. But C

is

refuted

in

the

laboratory.) See Halpin

(1986).

On

the

other

hand, accordingto (0),

"if

a

q-measurement

s

made,

r would result" will typically not be true for any r because in typical

quantum

mechanical

cases,

r

does not result

in

all

of

the most similar

q-

measurement

worlds. There is

no

sense

in

which

counterfactual

definite-

ness is

true according to (0) because

on

this definition there

is

no

inter-

mediate

stage

of

truth

value

assignment

at which

sentences

are

evaluated

with

respect

to a

single q-measurement

world.

A

number

of

authors

have

suggested

versions

of

(0).

Most

important

for our

purposes

is

Wessels

(1981)

who

gives

an

analysis

for

measure-

ment counterfactuals.

Her

idea, basically,

is

that

for

antecedent "a mea-

surementof quantity q on system s occurs", we take the possible worlds

most similar to the actual to be

just

like

the

actual with

respect

to the

system's

quantum mechanical

state,

and

its

possessed

values.

Further-

more,

the influences

acting upon

the

system

in

such

most similar

possible

worlds must

be

just

like those

acting

on the

actual

system

except

for

the

influences of a measurement device

set

to

measure q.

Unfortunately this

is just a schematic

description.

It

does

not

give

a full

account

of

the

possible

worlds

and

what counts as

similarity.

We

would like

to

know

what

values

are

possessed,

and what influences

a

measurement

device

has

on a

system

in

each member

of

the set

of

most similar worlds.

These

questions

involve controversial unknowns.

A

fuller account will

require

a

complete

description

of

physically possible worlds, and

that

awaits a

solution to

the

quantum measurement

problem. Still,

Wessels's account

shows us

how to

begin

to

flesh

out

(0).



9/26

LOGICAL FORM OF PROBABILITY

ASSIGNMENT

43

Finally, we are

preparedto address

the main business

at hand:

quantum

mechanical

probabilityattribution.

A

number

of

proposals for

the

logical

form of quantum mechanical probability attributions have been given,

utilizing conditional

probabilities

and

counterfactuals.

To

help

describe

the

proposals,

I

will

use several abbreviations.

Let

R

stand

for

the

state-

ment that "value

r

results

upon

measurement".5

Let

M

stand for

"quantity

q is

measured"

Now, for all AP,q, r there is

a p such

that the following

holds:

If

f

is the state of a

system,

then

QM assigns a

probability p to

R

given that

M.

It

is

the

logical

form of such

assignments we wish

to

uncover.

Suppressing

mention

of

f

and

q,

the

Merzbacher and

Messiah

version of

this attribution

s

roughly:

(*)

R

has

probability p

given (hypothetically) that M.

I

have

seen three

construals of (*) advanced:

(a) Pr(R/M)

=

p,

so

that the attribution is a

conditional

probability given that a

measure-

ment

has

occurred,

(b)

P(M

>

R)

=

p,

so

that the attribution s an

assignment

of a

probability

to

a

conditional,

and

(c)

M

>

P(R)

=

p,

so

that the

probability

attribution

is

made

only

on

the

counterfactual as-

sumption

that a

measurement has occurred.

In

the

following sections,

I

will

try

to show

that each

of

(a),

(b), and

(c)

have

problems

as

readings

of

(*).

I

will

go

on

to

suggest

an

alternative

explication. But we should now ask, what features are desirable in an

explication

of

(*)? First,

such

an

explication

should

allow the

quantum

mechanical

probability assumptions

the best chance to

be true

over

the

widest

range

of

cases.

This is

just

a

principle

of

charity.

Secondly, be-

cause

this

explication

of

(*)

is

meant as a

preliminary

to

interpretation,

we

do

not want it to

prejudge

interpretive

issues.

That

is,

we

should,

for

the sake

of

generality, prefer

not

to

saddle

QM

with

controversial as-

sumptions.

Ideally, then,

an

explication

of

(*)

would

not

beg fundamental

questions relating

to

the

quantum

mechanical

interpretation

problem.

Fur-

thermore, such an explication should, in so far as possible, not presup-

pose

any particularmetaphysics

of

possible

worlds. As

we will

be

dealing

with semantical issues

usually

discussed

in

the

possible

worlds

frame-

work,

this constraint

will

become

important.

In

any

case,

the sum

total

5For

generality,

r can be taken to be either a

single

real

number or a

range

of

these.



10/26

44 JOHN F. HALPIN

of this second

suggestion

is

that

for

purposes

of

generality, the best ex-

plication

of

(*) will,

to

stay away

from

controversy, be neutral with re-

gard to interpretive issues. Finally, of course, a good explication will

uphold the physicist's

intuitions that

(*)

involves

probabilities given hy-

pothetical measurements.

3. Against Proposal (a).

The first

proposal, (a),

that

quantum mechan-

ical probability attributions like (*) are conditional probabilities of the

form

Pr(R/M)

=

p, is, at first blush, perhaps the

most

obvious rendering.

The probabilities

of

QM are probabilities given the performance of a mea-

surement. Conditional probabilities

of

the

form

P(R/M), at least as usu-

ally understood, arejust this. Since Kolmogoroff, such conditional prob-

abilities

have

standardly

been defined

in

terms of

standardunconditional

probabilities as described above: Pr(R/M)

=

P(R&M)/P(M). However,

so

long

as

this

definition

of

probability

is

maintained,

a

significant prob-

lem

for

proposal (a)

exists. The

argument

for this

claim comes from van

Fraassen and Hooker (1976) as

follows.

Proposal (a) presupposes

that a conditional

probability

is

defined.

For

this to be

so,

the

probability

that

m is

measured, P(M),

must also be

defined and

nonzero

(for arbitrarym).

Not

only

is

this condition

implau-

sible on the face of it, but for arbitrarym, it must fail. (There are sets

{mj}

of measurement types

such that

(i)

{mj}

s of

uncountablecardinality,

(ii)

no

two elements

of

{mj}

can be measured

at once (they are incom-

patible) andyet (iii) QM

makes

probability assignments conditioned upon

each

of

the

mj.

(For example

take the set

{mj}

to be defined so

that

mj

is

the measurement

of

spin

of

a

particle

in

direction

j,

where

j ranges

over

the set

of

all

directions between

0

and

90

degrees exclusive.)

Let

Mj

formalize

"mj

s

measured".

It is a fact of

probability theory

that

because

of

the size

of

the set

{mj},

not all the

incompatible Mj

can receive nonzero

probability. It follows that the conditional probabilities Pr(R/M) cannot

be defined

for

all

quantum

measurements

m; indeed, they

cannot be

de-

fined

for

more than

a countable subset

of

the

uncountable

set

{mj}.

So,

proposal (a)

cannot

in

general

allow us to make

sense

of

the

quantum

mechanical

probability

attributions

(*)

which are

given

for

all

mj.

Before rejecting (a), however,

we should consider

responses

which

may

be made

to the van Fraassen-Hooker

argumentjust given. First,

someone

might object

to the above

argument by suggesting

that not all

of the

un-

countable number

of

measurements

in

{mj}

are such

that

they

each

might

be performed in practice. (In practice we have no chance of being able

to

make

so

many

discriminations

for

the same

reason that

we are

not,

for

an

arbitrary

real

number

s,

able to measure

whether

or

not

an

object

is

exactly

s

units

in

length.) So,

the

objection continues, probability

as-

signments

conditioned

upon any

but the

experiments

which

might

be

per-



11/26

LOGICAL

FORM OF PROBABILITY ASSIGNMENT

45

formed

in

practice are unnecessary. Furthermore

there is no reason to

believe that uncountable

sets

of

disjoint measurements

exist which are in

practice performable. This objection assumes that the quantum mechan-

ical attributions of probability

given measurements

not in practice per-

formable may be neglected

for

the project

at

hand.

However, I would

argue

that one

should at

least try

to

make sense

of

all probability attri-

butions of QM, even

if

some

are

not

practical.

Not to do so would be

out

of

line with

our first condition

on

a good explication of quantum

mechanical

probability

assignments:

that

we

make sense of

the widest

range of cases.

The second response to the van Fraassen-Hooker

argument against (a)

is to countenance nonclassical probability measures with infinitesimal

weights. According

to this

response,

the failure

of

(a)

is

a result of

the

standard

unconditional

probabilities

assumed

in

the

definition

of

condi-

tional probability.

The van Fraassen-Hooker

argument

against (a) shows

that the

probabilities

Pr(R/Mj)

=

P(R&Mj)/P(Mj)

cannot

be defined

in

general.

This is

accomplished by showing

that

not

all the

P(Mj)'s

can

have

nonzero values.

But

they

can have nonzero values

if

the notion

of

a

probability

measure

P is

extended

to

allow infinitesimal

values

in

its

range.

In that

case both numerator and

denominator

of

the

definition

of

Pr(R/Mj) may have infinitesimal values, yet the fraction itself would be

a

standard,

finite real

value.

This

currently popular

counterproposal

de-

pends

on

the

field

of

nonstandard

analysis.

For a

review

of

this

possi-

bility,

see

appendix

four to

Skyrms (1980).

So, according

to

this

second

response, (a)

can

be

revitalized

if

we

as-

sign

infinitesimal

probabilities

to the

Mj's.

But

on

the

face

of

it,

this

would seem

implausible.

Are there

really probabilities

that

a measure-

ment

will occur? One

might

think that there are

not

because

said occur-

rence depends typically

upon

the

experimenter's

free choice.

Still, one

may want to think of the experimenteras a complex quantummechanical

system,

so

subject

to

probabilities.

But

the

quantum

mechanical

proba-

bilities

are conditional

probabilities

and what we need

for

the

definition

are

unconditional

probabilities

P(Mj).

Finally,

even

if

some

other source

of the infinitesimal

unconditional

probabilitiesexists,

it would seem

that

these

probabilities

of

measurements

would

sometimes

be

zero;

actual sit-

uations

would

sometimes

absolutely

rule

out

the

performance

of

a

given

measurement

mj;

hence

P(Mj)

would be

zero.

(To

take

an

extreme ex-

ample,

consider the case

of

a

universe

which

contains

only

a

few

simple

atomic

systems.

Because there is no measurement

apparatus

in such a

universe,

the

probability

of

an

m-measurement

occurring

would be

zero

for

some

if

not

all m.

So, Kolmogoroff

conditional

probabilities

are

not

defined

here, yet QM

would

still

seem

to

apply. QM assigns

a

state

to

the atomic

systems

and

so

assigns probabilities

for

measurement

results



12/26

46 JOHN F.

HALPIN

that could result if, per impossible, a measurement device had existed.)

Hence, it would

seem

that

this second response

to

the van Fraassen-Hooker

argument, the

infinitesimal probability proposal,

will

still not guarantee

that all quantum mechanical probabilitiesare defined. Moreover, if, con-

trariwise, there

is

some

way

to

make the proposal work, this would re-

quire taking a stand on several controversial interpretive issues (e.g., the

assigning

of unconditional

probabilities

o human

systems)

which

we should

avoid.

For

the

sake of

generality

and

avoiding controversy, a better ex-

plication

of

the

form of

(*)

is

in

order.

The final response to the van Fraassen-Hooker argument rejects the

Kolmogoroff

definition

of

conditional

probability. Realizing that the tra-

ditionalKolmogoroffconditionalprobabilitiesare not always defined when

we

want

them to

be,

both Karl

Popper

and Alfred

Renyii have suggested

theories

for

which conditional

probabilities

are

fundamental: by fiat they

exist,

rather

than

by

definition.

(In

a

nutshell,

these

views

stipulate that

conditional

probabilities

exist

given arbitrary

condition

A,

with

A

from

a

set

of

conditions

large enough

to

include

all

the

incompatible Mj. Fur-

thermore,

for

fixed

A,

the values

of

Pr(X/A)

for

variable

X

obey

the laws

of

the probability

calculus discussed

in

footnote

3.)

As

long

as this

option

is open

for

QM, the

van

Fraassen-Hookerargument

is

obviated because

that argumentrelies on the Kolmogoroff definition of conditional prob-

abilities

in

order to show that certain

conditional

probabilities

are

not

defined.

Indeed,

van Fraassen and Hooker

suggest

that we

understand he

probabilities

of

QM

in

Popper's way. They

show this to

be consistent

with the

quantum

mechanical

probabilities.

Van Fraassen and Hooker's

suggestion

that

quantum mechanical con-

ditional

probabilities

are

fundamental

(rather

than

defined

in

terms

of

un-

conditional

probabilities)

seems

right.

But one should still

want to

give

a more substantial

positive

account

of

what these

probabilities

are. Van

Fraassen and Hooker suggest that the conditional probabilities are prob-

abilities

of

conditionals:

Pr(R/M)

=

P(M

>

R). Though

I

have no ob-

jection

to

taking

the

probabilities

of

QM

on

the model

of

either

Renyii

or

Popper,

I

think

van Fraassen and

Hooker's explication

of

these

as

probabilities

of

counterfactuals

fails

for

two reasons.

First, taking quan-

tum mechanical

probabilities

to

be

probabilities

of

counterfactuals has

unfortunate

consequences

which

are

developed

in

the

next section.

Also,

van Fraassen and

Hooker

presuppose

Stalnaker's

theory

of

the counter-

factual

>.

As described

in

the last

section,

this

conditional would

seem

inappropriate or the context of QM; see Halpin (1986) for an argument

that Stalnaker's

theory

leads all too

easily

to

Bell-like

conflicts

with

ex-

periment. So,

if

we are

to take the

probabilities

of

QM

as

fundamental,

non-Kolmogoroff

conditional

probabilities,

hen

we need

to

say

more about

just

what

they

are.

In

section

6

below,

I

explicate

the

form of

quantum



13/26

LOGICAL

FORM OF PROBABILITY ASSIGNMENT 47

mechanical

probability assignments

in terms of

a probabilistic condi-

tional. The probabilities associated with that conditional might be taken

as fundamental conditional probabilities. If so, the proposal of section 6

can be taken to

underwrite (a). So,

I

do not want to claim that (a) is

wrong. But

I

will,

in

section

6, try

to

give

an

explication of (*) which

is

more perspicuous.

4.

Against Proposal

(b).

Let

PQM(R) symbolize

the

probability

value

assigned

to

QM

to

R

given

M.

Now, proposal (b)

states

that

a

quantum

mechanical assignment

of

probability, PQM(R)

=

p,

has

logical form Pr(M

> R)

=

p. What we will show is that from (b) one can derive a weakened,

but still undesirable,

version

of conditional excluded

middle. Assuming

(b), we have that Pr(M

>

R)

=

PQM(R). But,

as

well,

because

from

QM

we have

PQM(-R)

=

1

-

PQM(R)

and

PQM(R&-R)

=

0,

we also

have

P(M

>

R)

+

P(M

>

-R)

= 1

and P(M

>

(R&-R))

=

0,

and

so, because

A

>

(B&C) is

equivalent

to

(A

>

B)&(A

>

C), P((M

>

R)&(M

>

-R))

=

0. A version of conditional excluded

middle follows:

(CEM')

P((M

>

R)

V

(M

>

-R))

=

1.

I

take it that

(CEM')

is

an unfortunate

consequence

of

proposal (b).

On

any interpretation

of

the

quantum

mechanical

probabilities, objective

or

epistemic, (CEM')

will

not

in

general

hold.

Typically

in

a quantum

mechanical

world it will be a clear

physical

fact that neither

disjunct

is

true.

For

instance, suppose

an

m-measurement

is

not

performed, but that

if it were to be performed then value

r

might

result.

But also

suppose

that

some

r' #A r

might

result. So

if M

were

true,

R

might

be

true,

but

also

might

be false. Hence it

would

be

false to

say

either that

if

M

were

true, then R would be true, or that if M were true, then -R would be

true. So

the

disjuncts

are

both false.

Hence,

we will not

assign probability

one to

the disjunction.

Because

proposal (b)

leads one

to

do

so,

I

take

this

proposal

to

be

discredited.

As we noticed

in

section

2,

conditional

excluded

middle

has at least

one

defender,

Robert Stalnaker.

So

a

proponent

of

this

theory

would

be

comfortable

with

the

consequence (CEM') just

derived and

would

not

take

the

argument

to be a reductio

of

(b). However,

as was

also men-

tioned

in

section

2,

Stalnaker's

theory

of

the conditional leads all too

easily to conflict with experimental results (Halpin, 1986). Moreover,

even

if

the

case

against

Stalnaker's

theory

and

against

CEM

is

not

con-

vincing,

it should be clear that these

positions

are

deservedly

controver-

sial. It

would

be

unfortunate, then,

to saddle

QM

with

proposal (b)

and

the resultant

controversy.



14/26

48 JOHN

F. HALPIN

5. Against Proposal (c).

Philosophers

have

frequently

taken

proposal

(c) to be the best option

for

explicating

the form

of

quantum

mechanical

probability ascriptions. (For example, see Skyrms, 1982.) But there may

be a

problem

with

(c).

Consider the case

in

which

a

measurement of

m

is now performed, that

is,

M

is

now

true,

and for which

the result

of

measurement, r,

is

immediately decided,

for

example, is immediately

registered

in

the memory

of

a measurement device.

According

to

(c),

in

this case

P(R)

=

p. (This

follows

simply by

Modus

Ponens.)

Do

we want

to

say

that the

probability

of result

r is

p? Perhaps

not.

One

might argue

that after such a measurement

is

performed

and

the

result, r,

is

in,

the

truth value

of R is no

longer chancy.

Rather

it is

decidedly

true. Hence

(c) is wrong here because its consequent assigns a chance


15/26

LOGICAL FORM OF

PROBABILITY ASSIGNMENT

49

taken to be

flips

of

a

coin

by

John

Halpin

on 14

May 1987-suppose

they are HHTTHTHTT-then this

ratio for

heads

is 4/9

and

the

proba-

bility of H in

this reference class

is

4/9.)

When the

reference class is

countably infinite (and assumed

to be ordered

in

a

sequence), the prob-

ability

cannot

be defined

as a

ratio,

but instead

is

defined to

be

the limit

of these ratios as

we

think of the reference class as

growing sequentially.

We

can

make

this

precise:

Let

A,

be the number

of

A's in

the first n

trials. The relative

frequency,

Rn,

of A's

in

the first

n

trials

is

defined as

An/n. Then for the case where the number of

trials

is

finite and equal to

m, one

identifies the probability

of

A

with Rm.

Where the number of trials

is

countably infinite, the

probability

is

taken to

be

the

limit

of

Rn

as n

approachesinfinity.

In

this simplest form of the

frequency interpretation,

probabilities are

defined

in

terms

of

actually

occurring

trials.

But

surely

these cannot

be

what

QM

tells us about. It is

possible

that

quantum

mechanical

proba-

bilities differ

from

the associated ratios

(or

limits of

ratios), just

as a coin

with probability

1/2

of

coming up

heads, may

in

a class of

trials come

up

heads

only

4

times

in

9.

Indeed,

for

the case

of

a

finite number

of

trials-

and we know of

no

sequence

of

quantum

mechanical trials which is

not

finite-quantum mechanical

probabilities, which are sometimes irra-

tional, cannot in general be given as relative frequencies; an irrational

number

by

definition

cannot

be

expressed

as a

ratio.

Similarly, even

if

the reference class

is

infinite,

and the

probabilities QM

assigns

are

the

right

ones,

that the limit

of

relative

frequency

will exist and

be

equal

to

the

probability assigned by QM

is not

guaranteed.

The laws

of

large

num-

bers

only

tell us

that

a

large

difference between

the

probability

and

the

limit of

relative frequencies

is

unlikely;

it

is

not

impossible.

A hard

core

empiricist may

want, despite

these

objections,

to hold

that

probability

is

just

relative

frequency

in

actual and

(typically)

finite

ref-

erence classes. Such a proponentof the frequency view will hold that to

go beyond

the

extant

frequency

is to make an

unwarranted

idealization

(e.g.,

that

there

is

a "virtual"ensemble

to

serve as infinite

reference class

or

that there

is such a

thing

as

propensity).

This

may

be

so; however,

our

job

here

is

to

interpretQM,

and

that

theory pretty

clearly

does ideal-

ize;

for

example,

it

assigns probabilities

even

to

unmeasured

quantities

such as those discussed earlier

in

section 3.

So,

the

probabilities

of

QM

are

usually thought

of not as

relative

fre-

quencies

defined

over extant reference

classes,

but rather as the limits of

relative frequency in a "virtual"ensemble; that is, the limits that would

exist

if

hypothetical experimental

trials

(the

reference

class)

were an

in-

finite sequence.

(Such

a view

is sometimes

taken to be a sort

of

propen-

sity

view

rather

than a

frequency

view.

In

any case,

on

this

view the

quantum

mechanical

assignments

of

probability

read

as

a variant

to

(c):



16/26

50

JOHN F. HALPIN

(An infinite sequence

of

measurement

is

performed)

>

(the relative

frequency of A in the

sequence

is

the quantum mechanical probability

P).

This reading

is no

better

off

than the frequency view for actual infinite

reference classes: just as actual relative

frequencies

can

differ from prob-

abilities,

so can counterfactual

sequences.

For

example,

if

we

were to

flip a fair (probability1/2)

coin an infinite number of

times,

it

might land

heads

on

exactly the even

trials

(one way

to

get

relative

frequency

I/2)

but also might

land heads

on

every try (this

too is a

possibility).

Both of

the infinite

sequences just

described are

very unlikely

to

occur

(if the

trials are independent)

but

are, nonetheless, possible. Generally, there is

no

guarantee that the limit

of

relative frequencies

will

or would be equal

to the

probability. Again,

the laws

of

large

numbers

only

tell us

that a

difference

is

unlikely,

not

impossible.

So, though

we

may expect relative

frequency

in

the long run to be close to

the

probability, we should not

identify the two.

The

reasoning

of

the above

paragraph

is

well-known and

usually

ac-

cepted by interpreters

of

QM. So,

for

the

rest

of

this section

I

assume

that

quantum

mechanical

probability

is

not

to

be

given

a

frequency inter-

pretation. Now, probably

the most

popular

view

among philosophers

is

that

quantum

mechanical

probability

is

a kind of

propensity

I

will

call

"chance".

On this

view, probabilities

are

probabilities

for

the single case,

and are tendencies

or

dispositions

that admit

of

degree. Typically,

we

estimate

the chance

given

relative

frequencies

as evidence and

vice versa.

But chances are new sorts

of

theoretical entities that

are

not to be

defined

in

terms

of

frequency.7

We understandthese theoretical

entities

in

terms

of

what

QM says

about

probabilities

and

in

terms

of

the

probability

cal-

culus and

its

theorems, together

with the evidential relation to

relative

frequencies. In this way propensities take their place as primitives within

a theoretical network.

I

have

argued

that an

epistemic interpretation

of

probability ascription

is

not

appropriate

within

proposal

(c)

because such an

interpretation

would

suggest

that

QM

is

about belief states.

Furthermore,

I

argued

that this

leaves one

with

some

version

of

a

propensity interpretation:

as

a

relative

frequency account

will not do

here,

one should take

proposal (c)

to

be

about

chance.

Chance

applies

to the

future, however,

and

there

is no

chance

or

propensity

for

the

past

or

present

to be

different

from

the

settled

way it was or is. This is an old view that goes back at least to William

of

Ockham.

I think it

is

also a

very plausible

view

of

chance. David Lewis

puts

the

point

this

way:

7This is not

to

say

that chance

is

an

irreducible

property

of

systems;

it

may supervene

on

physical properties.



17/26

LOGICAL FORM

OF

PROBABILITY ASSIGNMENT

51

This

temporal asymmetry

of

chance

falls

into place as part of

our

conception

of

the

past

as 'fixed'

and

the future as

'open'-whatever

that may mean. The asymmetry of fixity and of chance may be pic-

tured by

a

tree. The single

trunk

is

the

one

possible past

that

has any

present chance of being actual. The

many branches are the many

possible

futures that have some present chance

of

being actual.

(1981,

277)

We can think

of

Lewis's tree structures

and

the

fixed-open

distinction

in

terms

of nomic

possibility given the present state

of

the

world: the pos-

sibilities that have a chance

at

a moment

are

those

that are

nomically

possible at that moment. Facts "about"now-say, that a coin lands heads-

are fixed and

so

not

chancy. (It

is

worth

noting

that the sort of

chance

just

describedneed not be

deeply metaphysical.

For

instance,

Brian

Skyrns

takes

chance

to

be just

a

special

case of

subjective probability,

viz.

the

subjective

probability

conditioned

upon

a

partition

of

the

possible situa-

tions. On

his

view,

if

a set

of

propositions,

{Pj},

form

the

appropriate

partition

of the set

of

all

possible

situations,

and if

PJ

is the

true

member

of

these, then chance

A

is

equal

to

the

subjective probability

of

A

given

Pi.

As

long

as

the

partition

members

fully

reflect the

state

of

the world

up until a time, intuitively the present time, then chance will apply non-

trivially only

to

statements about

the future.

In

this

way Skyrms

makes

sense

of chance as described above

but from

a

subjectivist's viewpoint.

(See Skyrms

1984, 107-109;

and

Skyrms

1988.)

Given our

understanding

of

probability

in

(c)

as

chance,

we

can

better

understand the

argument against (c) given

at the outset

of this

section.

Again, we are

to

consider the case

for

which

QM assigns

a

probability

to

result

r

(0

B"

because, by hypothesis,

the person asserting (3)

holds that

A

>

B is false and so not

probable.

Furthermore, (3)

cannot

be construed

as A

>

(B

is

probable)

that

if

Williams had faced

today's

big league pitching,

then it

would have been

probable

that he was

a

300+

hitter.

This

latter

construal

should seem

odd,

unlike

the

original,

because

had

Williams faced

the

pitching

in

question,

his

batting average

would

not have been

uncertain

or

chancy;

it would have

been

more than

probable

whetheror not he was a 300+ hitter, that is, his average would have been

known,

a settled

fact.)

Finally, we can

return to the

question:

How

is

(*),

the statement

that

R has

probabilityp given

counterfactually

that

M,

to be

interpreted?

I

will

try

to show that

if we think of

(*)

as a

quantitative

version of

might

or

would-most-likely conditionals,

then the problems

for

(a)-(c) disap-

pear. My suggestion

is

that

(*)

be understood

as

M

>P

R were

>P,

read

"would-with-probability-p",

means

roughly

"would

in a

set

of

measure

p". That is, where

P is a probability measure on s(A),

A

>P

B is true just

in case if b is the set of elements of s(A) at which B is true, then P(b)

-p. So,

for

example,

where

QM assigns probability

that

a

particle

be

found

in

region V,

if measured for

position,

the

proposal implies

the

fol-

lowing:

a measure

p

of

s(M),

the set

of most similar

position measurement

worlds,

have resultant

value

in

region

V.

And,

in

general,

the

hypothetical

probabilitiesof QM are

to

be analyzed

in

terms

of

probability

measures

on the

set of

m-measurement

worlds that

might

be.

Let me

try

to

clarify

the

suggestion

of the

last

paragraph.

One wants

to

give

truth

conditions

for

A

>P

B.

I will do

this

in

terms of

probability

measures

PS(A),

on each of the sets

s(A)

for arbitrary entences A. That is,

I assume

the

additional

structure

of

this set

of

probability

measures,

one

for

each sentence.

Now,

let

[B]

=

{w

E

s(A):

B is

true

at w}.

Then,

the

truth

condition

goes

as follows:

A

>P

B is

true

just

in

case

[B]

is

mea-

surable

and

PS(A)[B]

=

p.



21/26


The suggestion, then, for explicating the quantum mechanical proba-

bility attributions (*)

is

the following:

(d)

M >P

R.

Given

that the

probability measures, upon

which

(d)

is

based,

come

from

QM,

it

would appear

that

(d)

involves

objective probabilities, though we

will discuss an alternative

in

the next section. Furthermore, given the

discussion of the last section, it would seem that

a

frequency interpre-

tation

is not

appropriate

here.

So, finally,

it would

seem

reasonable to

take the probabilities

in

(d)

as

chances; after all, they are theoretical en-

tities that

have

meaning

for the

single

case.

I

take

it, then,

that

the prob-

abilities of (d) are chances, though this is not a necessary concomitant of

the

proposal. (If

it were

necessary,

I

would be

begging

an

important

in-

terpretive

issue that

I

promised

not

to do in an

explication

of

(*).

For-

tunately, (d) places

no

restrictions

on how its

probability

measures

should

be interpreted.)

Finally, I should say how my suggestion,

M >P

R, for the probability

attributions

of

QM,

fares

against

the sort

of

argument brought against (b)

and

(c). First,

the

argument against (b)

started with the

quantum me-

chanical

probability assignments,

and concluded

that

a

certain disjunc-

tion, (M >

R) V (M

> -R), has probability one, an unfortunateconse-

quence

in

general.

Let me

run

through

that sort

of

argument with (d)

assumed

as the form

of

quantum

mechanical

probability

attribution.

want

to show

that

no

undesirable

consequence

is

forthcoming.

As

before,

we

have hat

PQM(R)

=

p,

PQM(-R)

=

1

-

p,

and

PQM(R&-R)

= 0.

So,

the

suggestion (d) implies

M >P

R,

M >('-PI

-R,

and M

>0

(R&-R). By

the truth efinition

have

ust given,

this

amounts

o:

PS(M)([R])

=

P, PS(M)([-R])

=

1

-

p,

and

PS(M)([R&-R])

=

PS(M)([R]

n

[-R])

=

0.

It

follows

from these

facts

that

PS(M)([R]

U

[-R])

=

PS(M)([R

V -R])

=

1

+

(1

-

p)

=

1. So, by the truth definition for >P just given,

M

>1

(R V -R).

This

result

is not

a

problem;

indeed it

is

required by the se-

mantics I

have

given, following

without the

assumption

of

(d).

Further-

more, PQM(RV -R)

=

1.

So, the argument

that led

(b)

to

a

problematic

conclusion

leads (d) only

to

a

triviality.

Also

notice

that

the

problem

described for

(c)

cannot

arise

with

(d):

Modus Ponens

is not allowed for the

would-with-probability-p

conditional

>P,

so one

cannot,

from

M

>P

R

and

M,

derive claims

about

post-measurement

chances.

(Modus

Ponens

does

not hold

because

the

probabilitiesdescribed by

M

>P

R

relate to a measure over a set of pos-

sible

worlds; they

are not

probabilities

that

something

is

true

in

the

actual

world.) Finally,

notice

that

because

(d)

involves a

counterfactual

condi-

tional

which

is

to

be

analyzed

in

accordance with

(0),

one

might worry

that

(d) prejudges interpretive

issues or

saddles

QM

with

unfortunate



22/26

56

JOHN

F.

HALPIN

metaphysical

baggage

and controversy

over

possible

worlds

semantics.

If

(d)

did

so, it would,

like (b)

and

(c),

fail one

of

section two's tests

for

a good explication of quantummechanical probability assignment. For-

tunately, the

counterfactual analysis

of

(0)

is

generic,

leaving details

of

a theory of

counterfactuals

open.

Finally,

I should repeat

that proposal (d)

is

not

inextricably opposed

to (a),

so

long

as the

conditional

probabilities

of

(a)

can

be taken

as fun-

damental. Indeed one might

identify

conditional probabilitywith the

sort

of conditional chance

described

in

(d).

However,

before this

identification

can

reasonably

be

discussed,

one

would

need to

go

more deeply into the

analysis

of counterfactuals.

For

example,

conditional

probabilities

are

usually supposed to obey the product rule: Pr(A&B/C)

=

Pr(A/C)

Pr(B/A&C).

To

see

if the conditional

chances also

obey

the

product

rule-

that

is,

to evaluate

the claim

that

if

C

>P

(A&B),

C

>q

A,

and

(A&C)

>r

B,

then

p

=

q *

r-we would

need

to

sketch

the

details

of

how

possible

world

sets

s(C), s(A&C),

and their

measures are related.

This,

however,

goes beyond

the scope

of this paper;

we

have set aside

such

interpretive

issues as the details

of

the

analysis

of

counterfactuals. So,

I

leave a

study

of the

relationship

of

(d)

to

conditional

probability

for

another

occasion.

7. Epistemic Probability. In section 5, we rejected the possibility that

quantum

mechanical assignments

of

probability

are

personal

probabilities

because QM

is

clearly not about personal

belief states.

But

we set

aside

the alternative "Epistemic

Probabilities"

proposal

that the

quantum

me-

chanical

assignments

are

really prescriptions

or instructions

or

these

states.

On this

view, QM

indicates

appropriate

degrees

of

plausibility,

that

is,

quantum

mechanical

assignments

of

probability

give

the

degrees

of

belief

one

should have toward

statements

about measurement

results. One

way

to

read these

prescriptions

would be

a variant to

(c):

(c')

If

an

m-measurement

s

performed,

then

(one

should)

assign

de-

gree

of belief

p

to

R.

This new

proposal, (c'),

is a conditional

prescription.

(Note

that

(c')

like

most prescriptions

can

be

defeated

by

other

considerations;

for

example,

one should

not

assign personal probability

p ($ 1)

to

R

in

the case

in

which one looks

and

sees

that

R

is

true.

So, (c')

is

perhapsobjectionably

vague.

Also note

that

one

might

want to

rephrase (c') by dropping

the

words "one

should";

this would

make it

clear that

(c')

is

an

instruction,

that is, a command, ratherthan a normative claim. I prefer to leave this

vague.)

Now,

there

is

little

doubt that

we

do,

from

QM, get

information

per-

tinent to our

personal probabilities.

The

proponent

of

chance will

typi-

cally

hold that

chances

mandate

personal

probabilities

(in ways

to be

de-



23/26


scribed later

in this

section). But

the

proponent

of

(c') suggests rather

that chance doesn't

play

a

role

in

the

quantum

mechanical

probability

attributions *), but that these involve only instructions (Leeds 1984). As

we have seen in section 5, (d) is perhaps most plausible if we understand

it in terms of chance. Leeds's argument, then,

has

the potential to cut

against (d). Indeed, let us suppose for the rest

of

this section that the

probabilities

of

(d)

are chances.

I will

attempt

first to

suggest

a

pre-

sumption in favor of chance

and of

(d) over (c')

in

part by showing that

Leeds's arguments against

chance

in

QM,

and so

against (d), are not

forceful. There is surely something

of

interest

to

(c'),

a

connection be-

tween

QM

and

personal probability,

even

if

it

is

not

an

appropriate

anal-

ysis or explication of (*). Secondly, I will discuss the relation between

personal probability

and the counterfactual

probabilities suggested

in

the

last section.

Leeds gives an argument

for

the "incompatibility"

of

QM and realism

via "a skeptical attack

on

the notion

of chance

in

QM" (1984, 568). He

argues that when we

use

QM

to make statistical

predictions

or

explana-

tions, we

need

only

take into

account

the

wave function from

which the

appropriate probabilities

can be

deduced;

there

is

no

need to

mention

"chance".

I

would

argue

that

though

this is

true,

it

by

no means

excludes

chances from QM; rather

what is the logical form of probability attribution in qm

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