From stapp@thsrv.lbl.gov Wed Jul 29 22:18:36 1998
Date: Wed, 29 Jul 1998 22:07:15 -0700 (PDT)
From: Henry Stapp
To: JLFinkelstein@lbl.gov
Cc: mermin@msc.cornell.edu, RGRIF@cmu.edu, shimony@buphy.bu.edu,
unruh@physics.ubc.ca, vaidman@post.tau.ac.il
Subject: Re: Condensed reply to Jerry
On Sun, 26 Jul 1998, Jerry Finkelstein wrote:
>
>
> Dear Henry,
>
> Good. We seem to be saying the same things (although perhaps using slightly
> different words to do it).
>
> and your answer was
>
> > I take the general meaning of A[]-->B to be this:
> >
> > A[]-->B
> >
> > is true at world v iff B is true in every world v' that is identical
> > to v apart from
> > consequences of changing from v to the world v' in which A is true.
> >
> > I assume that the change from R2 to R1 is achieved in a specified
> > way that involves only things in the immediate neighborhood of the
> > preparation of the experiment in R.
> >
> > This definition accords with the intuitive meaning of "would be":
> > when one specifies "If R1 were performed, instead of R2, then something"
> > --"would be so", one clearly means that one changes ONLY R1, and the
> > consequences of that change, but not, gratuitously, a lot of other things
> > as well...
>
> So you are saying that part of the definition of the counterfactual is that
> one only considers worlds in which there are no "gratuitous" changes from
> Aworld, which implies in particular that there are no "gratuitous" changes on
> the Left. In other words, if you know all the properties of world v *at R*,
> and you also know all the properties of world v' *at R*, you still do not
> know whether "A[]-->B" requires that B be true in v' (because you don't know
> whether there are "gratuitous changes" at L).
> This is the sense in which I mean that the definition of the
> counterfactual makes reference to the Left, and is why I say that the
> statement "A []--> B" does refer to the Left, even if both A and B are
> defined completely on the Right.
>
>...
> So the definition of SR does ... involve a
> "generic condition that pertains to...region L."
>
> Jerry
>
The issue that you raise about a hidden reference to region L, arising
from the meaning of "apart from consequences of changing..."
is a very important one that needs a precise answer. Although I hate to
turn people off with technical definitions that might seem to have the
potentiality of burying the essential points in a maze of formalism,
still they really are needed to allow one to deal clearly with subtle
points, such as the one you raise. One simply must be able to express
things precisely.
I follow the path blazed by logicians interested in scientific discourse.
I say that "a conceivable world w' is accessible from a conceivable world
w " iff the laws of nature are the same in w and w'. Here the laws of
nature are the predictions of QT.
"A=>B" is true in w iff for each w' accessible from w either
A is false in w' or B is true in w'.
Following the same format:
"A[]-->B" is true in w iff for each w' accessible from w either
w' differs from w by facts that are not necessary consequences
of the change in w required to make A true in w', or
B is true in w'
In the present context I suppose that there is a specified local
change corresponding to the change R2->R1, and a specified local
change corresponding to the change L2->L1.
An equivalent definition for "A[]-->B" is:
"A[]-->B" is true in w iff, for each w' that is accessible from w,
if w' differs from w only by facts that are necessary
consequences of the change in w required to make A true in w',
then B is true in w'.
LOC1 is the assertion that no factual change that is a necessary
consequence of (R2-->R1) lies outside the forward lightcone from R.
LOC2 is the assertion that the property of being, within a certain
context, a fact in R that is a necessary consequence of (R2-->R1) cannot
depend on which experiment is chosen LATER in region L.
My claim is that QT^LOC1^L2=>SR,
where
SR=:={(R2^g)=>[R1[]-->f]},
and that
LOC2=>{[QT^LOC1^L2=>SR]=>[L1[]-->SR]}.
Modal logic has been reduced here to ordinary logic. There is no
mystery: one deduces all possible conclusions by ordinary
logic.
Definitions:
R2 is true iff experiment R2 is performed. (etc.)
f if true iff outcome f appears to the observers of the
experiment R1, of which f is a possible outcome. (etc.)
f is meaningful only in a world in which the experiment of which it is a
possible outcome is performed. (etc,)
No two incompatible measurements can be performed in any allowed
world.
The statement (QT^LOC1^L2)=>{(R2^g)=>[R1[]-->f]} is true in w expands into
For each w' that is accessible to w
if QT^LOC1^L2^R2^g is true in w'
then for each w'' that is accessible to w'
if w'' differs from w' only by facts that are necessary consequences of
(R2->R1)
then f is true in w''.
This is proved by using in w' the prediction of QT L2^R2^g=>c,
invoking LOC1 to conclude that c is true in w'', and
invoking in w'' the prediction of QT L2^R1^c=>f
to conclude that f is true in w''.
Then the truth of f in w'' is, in this context, a fact in R that
is a necessary consequence of (R2->R1).
{(QT^LOC1^L2)=>[(R2^g)=>(R1[]-->f)]}=>{L1[]-->[(R2^g)=>(R1[]-->f)]}
is true at v expands into
For, each v' that is accessible to v,
if (QT^LOC1^L2^R2^g)=>SR is true at v'
then
for each v'' that is accessible from v'
if v'' differs from v' only by facts that are
necessary consequences of (L2->L1)
then
for each v''' that is accessible from v''
if R2^g is true in v'''
then
for each v'''' that is accessible to v'''
if v'''' differs from v''' only by facts that are
necessary consequences of (R2->R1)
then f is true in v''''.
My claim is that if LOC2 is true then this condition is satisfied.
One must the prove, under the stated conditions, and LOC2, the assertion
for each v'''' that is accessible to v'''
if v'''' differs from v''' only by facts that are
necessary consequences of (R2->R1)
then f is true in v''''.
Recall that
LOC2 is the assertion: the property of being, within a certain
context, a fact in R that is a necessary consequence of (R2-->R1) cannot
depend on which experiment is chosen LATER in region L.
Recall also, from above, that
Then the truth of f in w'' is, in this context, a fact in R that
is a necessary consequence of (R2->R1).
The *present* context is specified by the chain of conditions that have
led to this final assertion that needs to be validated, in order for the
long statement to be true. But this context is the same as the one
in which the truth of f IS a fact in R that is a necessary consequence
of (R2->R1), apart from the free choice made LATER in L. Thus invoking
LOC2 completes the proof.
Note that LOC2 is not just an analog of LOC1: it is a higher order
assertion, which claims that a fact in R that is a *necessary
consequence* of the switch (R2->R1) under LATER condition L1,
cannot be altered by changing that later free choice, without
*some* sort of influence acting backward in time.
Getting back to your claim that SR does have some sort of implicit
dependence on region L, I would say that LOC1^SR has an explicit
dependence on L, because LOC1 asserts that the change (R2->R2)
has no necessary consequences in L.
But condition LOC1 holds only under the condition L2: it is not used
under condition L1. In the proof, the final claim is that SR holds under
condition L1, not that LOC1^SR holds. The transition to L1 makes use of
the very different sort of locality assumption LOC2, whose roots and
application I have explicitly exhibited. The only invariance that is
required under the LATER change from L2 to L1 is the invariance of a FACT
in R that in one of the two alternative cases, namely L2, is a fact in R
that is a necessary consequence of the change from R2 to R1.
Henry