Dear Jerry, 24 July 1998
Your comments probably get at the key issue behind this whole contoversy.
But the exchange was rather long. I think it wiil be useful to condense it in order
to focus on the essential message.
Your key passages are:
> It seems to me that the discussion could be clarified if we were
> REALLY CLEAR about the meaning of counterfactuals, so let me first
> consider the situation to which LOC1 is meant to apply. In this situation,
> what really happens is that L2 is measured, with result c, and that R2 is
> also measured. We want to discuss both the meaning, and the truth, of
> the counterfactual "If R1 had been measured instead, result c would
> have been found for L2", which I will type as (R1) []--> (c).
>
> Counterfactual statements can be discussed by considering "hypothetical
> worlds;" here the actual world (I denote by Aworld) is:
>
> Aworld: L2 measured, result c found; R2 measured.
>
> and let me consider two possible hypothetical worlds (denoted Hworld1 and
> Hworld2):
>
> Hworld1: L2 measured, result c found; R1 measured.
> Hworld2: L2 measured, result d found; R1 measured.
>
> The counterfactual (R1) []--> (c) is true if Hworld1, but not Hworld2,
> is the hypothetical world which we should associate with Aworld.
> But no physical principle (such as quantum theory or relativity) can tell
> us that; physical principles operate within each world (ie, we demand that
> each world individually obeys quantum theory), but since hypothetical worlds
> are just fictions which we invent, there is no fact of the matter as to
> which one is "correct."
> According to David Lewis, if Hworld1 is considered to be closer to Aworld
> than is Hworld2, then the counterfactual (R1) []--> (c) is true, but if
> Hworld1 and Hworld2 are considered to be equidistant from Aworld, then
> that same counterfactual is false. This illustrates that the
> counterfactual implication []--> is only well-defined after we specify a
> distance relation among worlds. I therefore propose indicating in the
> notation what distance relation we are using. I will write [F]--> for
> the counterfactual implication with Hworld1 closer to Aworld than is Hworld2,
> and [B]--> for the counterfactual implication with Hworld1 and Hworld2
> equidistant from Aworld. Then the counterfactual "(R1) [F]--> (c)" is true,
> and the counterfactual "(R1) [B]--> (c)" is false.
> Now to consider the situation for Stapp's fifth step, let me specify an
> actual world and three hypothetical worlds (as before, I simplify the
> discussion by pretending that there are no other hypothetical worlds to
> consider):
>
> Aworld: L2 measured, result c found; R2 measured, result g found.
> Hworld1: L2 measured, result c found; R1 measured, result f found.
> Hworld2: L2 measured, result d found; R1 measured, result e found.
> Hworld3: L2 measured, result c found; R1 measured, result e found.
>
> Hworld3 by itself is not consistent with quantum theory, so I will not
> consider it further. As before, let "[F]-->" be the counterfactual
> implication where worlds which agree with Aworld on the Left are considered
> closer to Aworld than worlds that do not (and so Hworld1 is considered
> closer to Aworld than is Hworld2), and let "[B]-->" be the counterfactual
> implication in which Hworld1 and Hworld2 are considered equidistant from
> Aworld. Then "(R1) [F]--> (f)" is true, but "(R1) [B]--> (f)" is not.
> So to discuss Stapp's proof further, I will now assume that "[F]-->" is
> what is meant by counterfactuals.
> It is easy to show that if in the actual world it is L1 (not L2) which is
> measured, the analogous counterfactual is not true. Thus if I define a
> statement I call SF:
>
> SF := "{R2 and g} imply { (R1) [F]--> (f) }",
>
> we have the following:
>
> * If L2 is measured, SF is true.
> * If L1 is measured, SF is not true.
We can now ask whether these two starred statements indicate a violation of
> locality.
> Many of the previous discussions have broken this question into
> several, namely What does LOC2 say? and Is LOC2 violated? and
> Can LOC2 properly be called a locality condition? I think it simpler
> to directly ask whether locality is violated.
> I suspect that everyone would agree that, to the extent that SF refers only
> to the Right side, locality is violated. And indeed, if you look at
> the definition of SF I have written several lines above, you will see the
> letter R (twice), while the letter L does not appear. However, the symbols
> [F] do make implicit reference to the result on the Left.
> The string of symbols I am writing "(R1) [F]--> (f)" can be expanded
> to read "If R1 were measured, and if the result on the Left were the
> same as in the actual world, then the result of R1 would be f."
> And of course this meaning is not changed by rejecting my notation,
> and by using "[]-->" for the same meaning for which I use "[F]-->".
>
> To conclude: the argument of Stapp shows that the truth of a statement
> such as SF, which does not make explicit reference to anything on the Left,
> can nevertheless depend on which measurement is chosen on the Left. This is
> a violation of locality, if locality is taken to mean that the truth of SF
> must be independent of any choice made on the Left. However, it does not
> show that the truth of a statement which has NOTHING to do with the Left can
> depend on a choice made on the Left.
>
> Let me now put it another way. If I look at the definition of SR~, namely
>
> R2^g -> (R1 -> f ),
>
> I understand what that means without having to say anything about the
> Left. But if I look at the corresponding statement SR, namely
>
> R2^g -> (R1 []--> f),
>
> in order to understand the meaning I have to know what the counterfactual
> implication means. Well "R1 []--> f" means " f is true in those worlds
> closest to Aworld in which R1 is true," but in order to evaluate the
> distance of a given hypothetical world from Aworld, I have to say what is
> going on at the Left.
>
> If you do not agree with this, Henry, here is a challenge: can you give a
> complete definition of SR (including, of course, saying exactly what the
> counterfactual means) without making any reference whatsoever to any
> position other than the Right?
>
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:
"would be" would make no sense if changes other than the consequences of
making the specified change were allowed.
The tie-in to Lewis is made by asserting that the world v' that differ from v
in this minimal way are closer to v that all others.
The issue that you are raising is that although only things in R are explicitly
mentioned in the statement R1 []-->f, the exclusion of gratuitous changes
may bring in condition c.
The first line of my proof is LOC1:
L2^c^R2->{R1 []-->c]
This certainly involves c explicitly.
Two applications of QM, and logic lead to
LOC1^QM^L2->[(R2^g)->{R1 []-->f}]
SR: (R2^g)->[R1 []--> f]
is true at world w iff for every world v that is possible relative to w
either [R1 []--.f] is true at v or R2^g is false at v.
There is no explicit reference here to anything outside R.
But reference to c was used in the proof of SR under condition L2.
If LOC2 is to make sense, SR must have a meaning that makes no reference
to c: sufficient conditions for the truth of SR must involve nothing but `facts'
located in R.
SR is supposed to mean that if one passes from a world in which R2^g
holds, to one in which R1 is performed, and permits no changes that are not
consequences of this specified change, then the outcome of R1 must be f.
The general locality property that gives LOC1 as a special case is that the
the change from R2 to R1 should involve no change outside the forward light
cone of region R. This is the limitation on the hypothetical worlds v' that
enter into definition of R1 []--f. That general condition does not refer to
outcome c. The reference to c comes into the proof of SR via the condition
L2^R2^g and QM. It is not part of the definition of SR: the condition that limits
the hypothetical worlds v' that must be considered in proving R1 []-->f , under
the conditions R2^g is a generic condition that pertains to everthing that might
happen in region L: nothing at all is allowed to change there. The particular
value c is not specified in SR: it enters into the proof only when L2 is brought in,
in conjunction with the conditions R2^g and QM. SR is formulated without
mentioning c.