Date: Sat, 18 Jul 1998 05:29:53 -0700 (PDT)
From: Henry Stapp
To: finkel@thsrv.lbl.gov, mermin@msc.cornell.edu,
peres@photon.technion.ac.il, RGRIF@cmu.edu, shimony@buphy.bu.edu,
unruh@physics.ubc.ca, vaidman@post.tau.ac.il
Subject: Reply to Shimony
Dear Abner, July 18, 1998
Many thanks for sending your very tentative draft of your commentary
on the Stapp-Mermin ``debate''. You suggested that I might comment on it,
if I wished. I do believe that some comments at this early stage
will help to put your work on a course that will illuminate these
discussions better. There are, in the present early form of your paper,
some elements that I believe will obscure the issues, rather than
illuminate them.
You give in your section II an introduction to some of the logical
terminology.
[I shall use here---because this is a simple TXT document---the
symbol -> to stand for what in your text and my AJP paper was written as a
double-barred arrow, and 0--> for the counterfactual implication symbol.]
You say:
``In ref 4, p.301, Stapp says `-> stands for *implies*, the strict
conditional'. The strict conditional is usually understood to mean the
relationship between an antedent and a consequent that holds when the latter
is deducible by pure logic and mathematics from the former. Stapp's actual
usage in ref 4, however, seems to be that p->q, if and only if q is deucible
from p together with other premisses, that may be be explicit or tacit. Among
these premisses are the assumptions stated in (1), (2), and (3)
[these are my explicitlty stated assumptions of (1), validity of some simple
predictions of QT, (2), explicity stated locality conditions, (3), the
general idea that physical theories can cover a variety of special instances
that can be imagined to be created by free choices of experimenters]
The class of these assumptions, explicit or tacit, I shall refer to as A.
And if `p->q' occurs in a larger conditional statement of the form
`r->(p->q)' then another of the premises that may by used in the derivation
of q is r.''
Two points: First, I claim that there are no assumption beyond the ones
that I have explicitly stated. Thus in lines 1,2, and 3, I explicitly state
LOC1. and two predictions of QM. In line 4, which I say comes via logic from
lines 1, 2, and 3, I do not explicitly included statements 1, 2, and 3 among
the premises: line 4 is supposed to be understood to be a consequence of 1, 2,
and 3. So your clarification may be useful to a reader who does not infere
this fact from my assertion that line 4 comes from lines 1, 2, and 3. The
explicitly stated assumptions 1, 2, and 3 are, to be sure, *implicit* in
line 4: the full statement would be very long, as it would have to include
the statements of (1), (2), and (3). But those assumptions *are* explicitly
stated, and hence are not implicit in the sense of never being explicitly
stated, or of not being worked logically into the proof.
The second point pertains to the meaning of the `strict conditional', as
contrasted to the `material conditional'. I quote from a letter to me from
David Lewis, who is perhaps the pre-eminent authority on such matters:
``A *truth-functional* conditional `if A then C' is true at world w iff A
is A is false at w or C is true at w, regardless of what might happen at
any other possible worlds. A *strict* conditional `if A then C' is true
at w iff, for every world v that is possible relative to w, either A is false
at v or C is true at v. The term `material' and the hook symbol =| [lazy U]
are conventionally reserved for the truth-functional conditionals.
...the early logicians... used the word `implies' as their English reading for
the truth-functional conditional. Of course that's preposterous, and what is
defensible---up to a point---is to use the word `implies' as a reading for
the strict conditional. At least those early logicians had the excuse that
much of the time they were talking about propositions of pure mathematics,
for which presumably there's no variation of truth value from one world to
another. so the difference between the strict and truth-functional coinditions
doesn't matter. But those who came after and continued to read truth-funtional
condition as `implies' had no such excuse.''
This is all pretty relevant because I had in an earlier version of my paper
with Bedford said the my symbol =| [lazy U] meant ``material conditional''
and that by THAT reading Lewis could make no sense of my argument. But M.
Redhead pointed out to him that what I meant by that symbol was rather the
strict conditional. Lewis urged me to change my notation, which I did. He
said that with the strict conditional meaning he agreed with my formulation
of the locality condition.
I do not see in your formulation of the distinction that Lewis found so
crucial of the distinction between `strict' and `material' conditionals.
The point is that modern logicians, as opposed to `early logicians', tie
conditionals to the world in which they are issued, in order to distinguish
different possible meanings of `If A then C'.
We want to use reason in science. The actual facts of the world can be
important in applications, and we may want a statement of `implication'
to cover not just the one actual world---perhaps only by accident---but
all possible worlds in which `the laws of nature that are true in this world
continue to hold true.'
It may be true in our world that `If X is a molecule then the molecular
weight of X is less that M', where M is some very large number,
without its also being true that
`X is a molecule' implies `The molecular weight of X is less than M'
in the strict conditional sense.
You suggest that my meaning of `Y 0--> Z' is that this statement is true
if and only if Y is false and Y->Z in the strict conditional sense.
This really misses the mark: I doubt very much that you can make such a
meaning part of a consistent logical scheme (See below)
Each conditional, counterfactual or not, is deemed to be issued from a world.
The assertion `Y 0--> Z' issued from a world v asserts that the
condition for the truth of Z is satisfied in any world v' that differs
from v only by the consequences of doing Y instead of doing the contrary
thing that was done in v.
Your formulation leaves out the essential aspect of the counterfactual
assertion, namely that it is issued from one world v, but is about what
``would be'' true in all worlds v' that differ from v in certain
specified ways. In particular,
the phrase ``instead of'' carries the specific connotation that nothing
in the counterfactual world v' is changed relative to v except for direct
consequences of the specific change demanded by the specified
counterfactual condition Y.
The compound statement A->(Y 0--> Z) is issued in some world w and it
asserts that in each world v that is possible relative to w (i.e.,
in all possible worlds v that obey the same laws of nature as w)
if A is true in v then (Y 0-->Z) is true in v.
I hope you will see in this formulation both the meaning of `implies'. in
the strict conditional sense of *entailment* or *compulsion*, and the
`compulsive' or `entailment' aspect of the verb ``would'' in
the statement Z would be true: the meaning of `Y 0-->Z' is that Z is
*forced* to be true in v' by the fact that nothing in v' is changed r
elative to v except for the direct consequences of the change demanded
by the specified counterfactual condition Y.
Of course, the change demanded by the counterfactual condition Y can be
implemented in an infinitude of ways in the real world, and hence logicians
had to bring in that awkward and not completely defined idea of `simlarity'
or `closeness' of worlds.
But in our case that awkwardness is not needed! This is because no matter
how this free choice is implemented in the later region, the locality idea
that we want to assert says that the effects of that change should not
affect anything outside the forward light cone from that later region.
Hence nothing in the other experimental region can be changed. So the
general, but loose, notion `similarity of' can, in our context, be r
eplaced by the notion of causal `effect of'. (See below)
You begin your section III (Critique of Stapp) with the phase:
``Stapp considers an ensemble of two-component systems...''
That is not correct: I have always insisted that I am considering
one single actual system, which in this case is a world w in which
L2^R2^g is true. The predictions of QT I take to be a theoretical idea:
indeed, QT is itself a theoretical idea.
You discuss first the first line of my proof, which is my statement if my
locality condition LOC1:
(1) (L2^c^R2)->[R1 0--> (L2^c^R1)]
You make three comments, (a), (b), and (c):
Your first claim, (a) and (b), is that (1) is simply true: it is not a
locality condition because it is a logical truth.
This shows that your suggested meaning for `Y 0--> Z' is not mine:
it is certainly not a logical truth that changing R2 to R1 leaves the
outcome c undisturbed. That is a physical condition/constraint.
In (c) you say that it is disturbing that the total set of premisses in (1)
are *inconsistent*. That does indeed seem to be a huge problem with
you suggestion about how `Y 0--> Z' is to be interpreted. The crucial
point is that the phrase ``instead of'' means that one is talking about
some world, or worlds, OTHER THAN the actual world. So one must spell out
in detail what connection is supposed to hold between these different
possible worls in order both to capture the intended meanings of the
words, and to arrive at a logically consistent logic.
In this connection it is my claim that Lewis's modal logic is a good
model of the logic to use here. It might be good to look at my article
with Bedford [Synthese 102: 139-164, 1995] for a discussion of that
formalism and its connection to locality.
I do not expect physicists to accept without examination my claim that
Lewis' logic is just right in a quantum context. But it is just wrong-headed
to ignore what generations of experts have learned, and to start making
assumptions that lead to contradictions, rather than looking at
how professional logicians avoid such contradictions.
I do claim that one can arrive at the correct meanings by examining very
carefully what the words ought to mean in a quantum context, and, indeed. I
myself arrived at the `correct' formalism in this way, before I learned
about the works of the philosophers. So this is not a matter of just
accepting on faith what philosophers tell us, but rather a matter of
thinking very carefully about the situation from the perspective of a
quantum physicist, and demanding that the adopted meanings must lead to a
consistent logic.
Now you suggest that in order to proceed without getting into logical
contradictions there are two ways to proceed: (a) keep your definition
but change my argument, (which is your preference) or (b) change your
definition of `Y 0--> Z' (e.g., to adopt mine)
It seems to me that in order to evaluate my argument you must do the latter:
otherwise you will be criticizing your own proof, not mine.
Your proposed meaning of ` Y 0--> Z ' is ` Y is false and Y->Z '.
You then say that ``If Stapp does not accept it, then it is hoped that
he will provide another.''
I am surprised that you should think that I have not made my meaning clear.
But, at the risk of being overly pedantic, let me do so here.
Each conditional, ordinary (->) or counterfactual ( 0-->), is supposed to
be issued in (or from) some possible world, and a conditional that is true
when issued in a world w is said to be true at w.
(A->B) is true at w iff B is true for every possible world v in which:
(i) A is true, and
(ii) the laws of nature are the same as in w.
So we see here the idea of `implies' as one of `entailment' by the laws of
nature, as contrasted to a mere accidental combination of facts.
This is both the way that scientist want to use language in their
discussions with their colleagues, and the way that ordinary people want
to use language when they use `If A then C' in the sense of `implies'---
i.e., in the sense that means
``In each possible world where A is true, C also is true.''
But normally, in science, we want the `possible worlds' to be
restricted to the class of worlds in which the laws of nature are the same
as in the world we are in: here `possible world' is always meant to be
limited in this way. `possible' means `physically possible'.
(Y 0--> Z) is true at v iff Z is true at every possible world v'
that differs from v only by the consequences of changing to `Y is true'
the contrary assertion that is true in v
Lewis would say that (Y 0--> Z) is true at v iff there is a possible
world v' in which Y and Z are both true such that, for every world v''
in which Y is true and Z is false, v is more similar to v' than to v'':
|v-v'| < |v-v''|.
Lewis's concept of `more similar to' is not pinned down precisely.
But it is pinned down enough to do the job here:
Lewis asserts that v is more similar to v' than to v'' if the space-time
region in which v' is identical to v properly contains---i.e., extends
further (into the future) than---the space-time region in which v and v''
are identical. Neither of these two regions of identicalness can cover the
region associated with Y, because Y is true in both v' and v'', but
false in v.
But if Z is true in v, and if no matter how the change to `Y is true' is
implemented in the Y region nothing will change in the Z region (the
region associated with Z), then there is a v' such that the region of
identicalness of v and v' will include this Z region, and also the region
where v and v'' are identical. But v cannot be identical to v'' in the Z
region because Z is true in v, by assumption, but false in v'' by
definition. Thus Lewis's conditions for `more similar to' are met:
(Y 0--> Z) is true in a world v in which Z is true, under this condition
that the change to `Y is true' does not change anything in the (earlier)
region where the facts pertaining to the truth of Z are located.
Actually, the situation is MORE satifactory in the quantum context than in
the classical one for two reasons:
1) In the quantum context we may demand that two world be identical prior
to some time t yet be different later, due to a different free choice made
later that t: classical determinism would forbid this.
2) The `more similar to' rules are somewhat arbitrary, and were merely
extracted from our normal linguistic conventions, in an imprecise way:
that ``would'' does not really encompass the full meaning of compulsion
or physical/logical necessity. But in our quantum case one really does get
physical/logical necessity, given the physical ``demand'' that the
``consequences'' of changing from `Y is false' to `Y is true' are confined
to the forward light-cone of the region where this change is made.
The combined assertion
A->(Y 0--> Z)
issued from world w asserts that for all possible worlds v such that
(i) A is true,
and
(ii) The laws of nature are the same as in w,
Z is true in each possible world v' that is identical to v apart
from the effects of implementing the change from v to v' demanded by the
condition that Y be true in v'.
One sees here both the `implies' character of the meaning of the symbol ->,
in the sense that the implication say something is true at ``all
possible worlds'' that satisfy some conditions, and the `implies' character
of the meaning also of the symbol 0-->, in that it says that Z would/must
be true in all possible worlds that meet the specified conditions.
Here the entailment or complusion cannot be expected to hold unless
A contains some assumption (like a locality condition) that curtails v'
relative to v. Lewis uses `more similar to'. But for quantum
science I use `consequences of'. This makes the asserted connection
a consequence of the putative physical principles, rather than of an
arbitrary rule.
You object to my premise LOC2. It asserts that if it is true (under
the general assumptions of LOC1 and QM) that
L2-> SR,
where the defining condition for SR to be true pertains only to
possible experiments that could be performed is the earlier region R, and
possible outcome of these experiments, then IF the choice between L2 and L1
can have no effects whatsoever on any of these observable things in R then
if L1 is performed, instead of L2, SR must continue to be true also under
that counterfactual condition L1.
When evaluating counterfactual statements one must enlarge the idea of the
`facts' that underlie the truth-values of these statements to include the
things in the hypothetical world that generalize the `actual facts'
in the actual world. If we interpret `facts' in the general way [as
distinguished from `actual facts'] then LOC2
(L2->SR)->[L1 0--> SR]
is a consequence of the `demand' that all the facts in R be undisturbed by
changing L2 to L1.
My statement in terms of observable phenomena is to be interpreted in this way:
by observable phenomena I meant things that can in principle be obverved, like
how experiments are set up, and which outcome appear to observers. But both
the actual and hypothetical versions are to be allowed in discussing possible
worlds.
Thus when I say ``everything mentioned in SR is an observable phenomenon in
region R'' I meant these `observable' things themselves, not the logical
relationships between these `facts'.
You are correct in pointing out that the logical relationship
(R1 0--> R1+) is itself NOT such a phenomena; it is a logical relationship
between such facts. But if all the facts in R are unaffected by the
switch from L2 to L1 then the truth of the statement SR,
[R2^g->(R1 0--> f)], regarded as a statement of entailment among the
facts pertaining to R, should remain undisturbed by the switch.
To address this question more deeply one must, I believe, look at the
intended use of the theorem. Like Bell's theorem, this theorem is to used
as a constraint on theories that can imbed the various possible quantum
worlds that evolve along one branch OR ANOTHER according to which of the
free choices available to an experimenter is actually picked by the
experimenter.
There is no hidden-variable predetermination assumption that the actual
outcomes are predetermined.
There is no hidden-variable ASSUMPTION that some unperformed measurement has
some definite fixed outcome.
There is no ASSUMPTION that some unperformed measurement would
have SOME definite result if it were to be performed
Each assertion about an outcome of some actually unperformed experiment is
deduced from explicitly stated assumptions of QM, Locality, and Free Choice.
There is no EPR-type assumption of `elements of reality':
EPR gave a criteria for the existence of an `element of reality',
but the required condition---of being able to predict with certainty---
could be met in only *one or the other* of the alternative possible
experimental conditions. To reach their final conclusion of the
incompleteness of QM they had to say that
``This makes the reality of P and Q depend upon the process carried out
on the first system, which does not disturb the second system in any way.
No reasonable definition of reality could be expected to permit this.''
But this begs the question, by assuming that there is must be an
objective reality `behind the phenomona'.
The only actual realities that I accept are the actual experiments and
the actual outcome appearing to observers of that experiment. The only
hypothetical `facts' are hypothetical experiments and their possible
outcomes: no reality property is assigned to them; they are identified as
strictly hypothetical/theoretical constructs.
And no assertion of made about the outcome of an actually unperformed
measurement unless it is deduced from the explicity stated assumptions,
and an actual outcome.
Are these claims of `no extra reality assumption' violated by
the assertion of LOC2?
You claim that because the ``proof'' of SR involves outcome c of L2, that
c is somehow buried implicitly in SR, even though there is no explicit
reference in SR to anything outside of region R.
Now I am sure that you would not ordinarily say such a thing: the principal
principle of logic is that the meaning of a statement (i.e., the condition
that it be true) should exist exist apart from the possible proofs that it
is true or is false, and that this meaning should be specified by the words
in the statement: Fermat's last theorem had a well defined meaning before
it was proved. Rational thinking itself would become chaos if what
statements meant were confounded with their proofs.
Certainly, logicians have constructed modal logic in conformity with this
usual principle. So it is not inherent in the logic/rational treatment
if possible worlds that these basic principles of rational thought need
to be abandoned.
So your claim seems to be that quantum theory forces me to abanbon this
principle of rationality, when I introduce LOC2, whereas I claim that I am
in fact just strictly adhering to the normal principles.
Of course, you might elect to embark upon this dangerous course of
confounding meanings and proofs. But my proof adheres to the normal
principles of rational thought, and you have not, as far as I can see,
actually said why these principles must be abandoned at this point.
The test is this. Suppose one tries to construct a theory that does not
bring in any elements of reality or hidden-variables, but invokes LOC1,
and hence arrives at a theory in which L2->SR. This means that in the t
heoretical structure there will be a constraint: if
on the actual branch where L2^R2 are performed the outcome g appears to the
observers of the actually performed experiment R2, then in a hypothetical
branch in which the experimenter in R makes the other free choice, and p
erforms R1, instead of R2, but in which nothing else is allowed to change
unless it is a consequence of the change of R2 to R1, then one must find
that on this hypothetical branch the outcome f will certainly appear to
the observers of R1. So this necessary constraint exists, in this
theoretical model, between the actual outcome g of the actual experiment
R2 and the outcome of the hypothetical experiment R1. Suppose region L
is located much later than region R, and we impose the physical condition
that a free-choice change made in L can have no effect of any kind backward
in time. Can a consequence of changing L2 to L1 be that the outcome in R
on the hypothetical branch in which R1 is performed is changed from f to e?
Or stated conversely: if changing L2 to L1 entails that an outcome in R
that formerly was required to be f can now be e, is the model completely
compatible with the theoretical idea that every effect of changing L2 to L1
is confined to the future of L?