what is the logical form of probability attribution in qm
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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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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.
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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.
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LOGICAL FORM OF PROBABILITY ASSIGNMENT 39
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.
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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
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LOGICAL FORM OF PROBABILITY ASSIGNMENT 41
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
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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).
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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.
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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-
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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
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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
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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.
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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
-
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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):
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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.
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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.
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LOGICAL FORM OF PROBABILITY ASSIGNMENT 55
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
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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-
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8/11/2019 What is the Logical Form of Probability Attribution in QM
23/26
LOGICAL FORM OF PROBABILITY ASSIGNMENT 57
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