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\today \hfill LBL-29836 \\
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{\large \bf Bell's Theorem in an Indeterministic Universe}
\footnote{This work was supported by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Division of High
Energy Physics of the U.S. Department of Energy under Contract
DE-AC03-76SF00098.
DB acknowledges the financial support of the South African Foundation for Research and Development and the
University of Natal.
}
\vskip 9pt
Donald Bedford \\[9pt]
{\em Department of Physics\\
University of Natal\\
Durban, South Africa}\\
and\\[9pt]
Henry P. Stapp\\[9pt]
{\em Theoretical Physics Group\\
Physics Division\\
Lawrence Berkeley Laboratory\\
1 Cyclotron Road\\
Berkeley, California 94720}
%affiliation for faculty:
%{\em Department of Physics\\
% University of California\\
% and\\
% Theoretical Physics Group\\
% Physics Division\\
% Lawrence Berkeley Laboratory\\
% 1 Cyclotron Road\\
% Berkeley, California 94720}
\end{center}
\vskip .5in
\begin{abstract}
A variation of Bell's theorem that deals with the indeterministic case is
formulated and proved within the logical framework of Lewis' theory of
counterfactuals.
The no-faster-than-light-influence condition is expressed in terms of Lewis'
`would' counterfactual conditionals.
Objections to this procedure raised by certain philosophers of science are
examined and answered.
The theorem shows that the incompatibility between the predictions of quantum
theory and the idea of no faster-than-light influence cannot be ascribed to any
auxiliary or tacit assumption of either determinism or the related idea that
outcomes of unperformed measurements are determinate within nature.
In addition, the theorem provides an example of an application of Lewis'
theory of counterfactuals in a rigorous
scientific context.
\end{abstract}
\end{titlepage}
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{\bf Disclaimer}
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{\it Lawrence Berkeley Laboratory is an equal opportunity employer.}
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\strut\vskip 9pt
\noindent {\bf Introduction}
The theorem of Bell [1964] in its original form tells us a great deal about the
kind of world we live in.
Given the validity of certain predictions of quantum theory it shows, within a
deterministic context, that our world cannot be a world of local properties
connected by influences that propagate no faster than light: it cannot be the
kind of world that Einstein believed it to be.
A key assumption of the original Bell's theorem is that the world is governed by deterministic laws.
This assumption runs counter to orthodox quantum
thinking.
Hence the conclusion usually drawn by physicists is simply that the world is
indeterministic.
This leads to the question of whether in an indeterministic context the predictions of quantum theory can be
reconciled with the idea that there are no faster-than-light influences of any kind.
If this idea fails in both the
deterministic and indeterministic contexts then
its failure cannot be evaded
by invoking the idea of indeterminism.
The present paper gives a detailed analysis of a recent proof by Stapp [1990]
of a variation of Bell's theorem.
This version is specifically designed to work in an indeterministic universe.
The analysis of it offered here is based on Lewis' theory of counterfactuals,
and it deals, among other things, with a question raised by Redhead [1987], and by
Clifton, Butterfield, and Redhead [1990, henceforth CBR]
about the logical correctness of using `would' counterfactual conditionals to
express, within an indeterministic context, the condition that there be no
faster-than-light influence of any kind.
The critique by Redhead and CBR is based nominally on the theory of
counterfactuals constructed by Lewis [1973, 1986].
Their conclusion is that use of `would' counterfactuals in this context is not
permissible, and that some assumption akin to determinism is apparently needed
in order to prove the theorem.
Actually, the conclusion of Redhead and CBR is not derived from Lewis' theory
itself: those authors make several critical changes.
We show in section two that a straightforward application of the Lewis theory itself confirms
the validity of the counterfactual statements in question,
and thus permits, in principle, the use of this counterfactual formulation of the
no-faster-than-light-influence condition within the Lewis framework.
We then examine the arguments advanced by CBR to justify their alteration of
Lewis' theory, and point out two errors.
Section three gives a line-by-line exegesis of the proof of Stapp [1990]
from the standpoint of Lewis' theory, and confirms its validity within that
framework.
The Lewis theory itself is nominally formulated within a deterministic context.
That restriction leads to a certain amount of arbitrariness and awkwardness.
For example, particular rules must be devised to determine the `closeness of
worlds', and the theory can then entail, on the basis of perhaps miniscule
differences of closeness of worlds,
that a certain physical condition definitely `would occur' in a particular
situation.
But a miniscule difference in closeness of worlds is a tenuous foundation upon
which to base the strong conclusion that a particular physical
condition definitely `would occur'.
There is also the awkward problem that in order to be able even to speak about
accessible alternative possible worlds Lewis must allow `miracles', in which the
laws of nature, which are the proper basis of the validity of `would'
statements, are temporarily suspended.
But then, of course, everything depends upon what happens during this hiatus.
These difficulties do not arise in our indeterministic context: what happens
tomorrow is controlled in part by chance, and hence no violation of physical law
is needed to generate alternative possible worlds.
There are, nevertheless, in our context, some supervening strict laws, and our
`would' statements are expressions solely of absolute necessities entailed by
these proposed strict physical laws.
No other conditions or conventions are invoked.
Thus, within this indeterministic setting, fortified by strict supervening
physical laws, the Lewis theory yields conclusions that have a quality of
certainty that is not normally attainable from an application of that theory
within a strictly deterministic universe.
We proceed now to an analysis of the formulation of the
no-faster-than-light-influence condition in terms of Lewis `would'
counterfactual conditionals, and then to the examination, within the framework of
Lewis' theory, of the proof of the indeterministic version of Bell's theorem.
\newpage
\noindent{\bf 2. Lewis Analysis of the No-Faster-Than-Light-Influence Condition.}
\strut\vskip 9pt
The experimental situation under consideration is a generalization of the
spin-correlation experiment considered by Bohm and by Bell.
It has been discussed by Greenberger, Horne, and Zeilinger [1989], by Clifton,
Redhead, and Butterfield [1990], and by David Mermin [1990].
In this experimental arrangement there are three particles with correlated
spins, and each particle is measured in one of three spacetime regions $R_i$,
$i=1,2,$ or 3.
The three regions are mutually spacelike separated.
In each region $R_i$ a choice is made to do either measurement $X_i$ or
measurement $Y_{i}$.
It is assumed that this choice is completely free, or random, and causally
unconnected to anything that has gone before.
After this choice is made nature must select, for the chosen measurement, {\it
one} outcome, which must be either plus one or minus one.
This selection is also assumed to be random, subject however to constraints
imposed by the predictions of quantum theory.
The name given to the outcome of an $X_i$ measurement is $x_i$, and the name
given to
the outcome of a $Y_i$ measurement is $y_i$.
Thus if an $X_i$ measurement is performed then nature must select one of the two
possibilities $x_i = +1$ or $x_i =-1$.
The possible worlds can be represented with the aid of a set of branching lines,
as shown in Fig. 1.
\newpage
\strut\vskip 7.5in
{\bf Fig. 1.} Branching possibilities.
\newpage
In any possible world history a choice between up and down
$ (X$ and $Y)$ is made at each of
the three left-hand branch points, and, in each of the three branches thus chosen,
a selection is made between the branch with value $+1$ and the branch with value
$-1$.
No selection is made by nature for the outcome of any unperformed measurement.
Thus the possible world histories are $2^6$ in number.
The situation described above is indeterministic by construction.
The question immediately under consideration is whether, under the further condition that
there can be no faster-than-light influence of any kind, the following statement
is, according to Lewis' theory, true:
$$
X_1 \Rightarrow [(X_2 X_3 \wedge (x_1=+1)) \Rightarrow ((Y_2Y_3) \bxa (x_1 =+1))].
\eqno(2.1)$$
The symbol $\Rightarrow$ represents ``implies'' (strict conditional), $\wedge$
represents ``and'',
and $\bxa$ represents Lewis' counterfactual conditional.
In words this statement reads: If the measurement $X_1$ were to be
performed in $R_1$ then [if (the measurements $X_2$ and $X_3$ were to
be performed
in $R_2 \cup R_3$
and the outcome of the measurement $X_1$ in $R_1$ were to be $x_1
=+1$) then (if the measurements $Y_2$ and $Y_3$ were to be
performed in $R_2 \cup R_3$, instead of $X_2$ and $X_3$, then
the outcome of the
measurement $X_1$ in $R_1$ {\it would be} $x_1=+1)$].
This statement is a formulation within Lewis' theory
of the idea that the outcome selected by nature in a region cannot depend
upon which measurements will {\it later} (in some frame
of reference) be freely chosen and performed in the far-away regions.
The strict conditionals that precede the counterfactual conditional entail
that to establish the truth of statement (2.1) it is sufficient to establish the
truth of the counterfactual conditional, $((Y_2Y_3)\bxa (x_1=+1))$,
at all worlds $w$ in which $X_1, X_2$,
and $X_3$ are performed, and the outcome of the $X_1$ measurement is $x_1 =+1$.
These worlds $w$ are the worlds limited by the conditions shown in Fig. 2.
\newpage
\strut\vskip 3in
\begin{quotation}
{\bf Fig. 2.} The conditions that define the set of worlds $w$ at which the
counterfactual conditional $((Y_2Y_3)\bxa (x_1=1))$ is required to be true.
\end{quotation}
\newpage
\strut
The counterfactual conditions imposed by the counterfactual assertion are
represented in Fig. 3.
\strut\vskip 4in
\begin{quotation}
{\bf Fig. 3} The counterfactual conditions.
\end{quotation}
Statement (2.1) is true, according to Lewis' theory, if {\it every} world $w$
that satisfies the conditions shown in Fig. 2 is closer to {\it some} world $w'$
that satisfies the conditions shown in Fig. 4(a) than to any world $w''$ that
satisfies the conditions shown in Fig. 4(b).
\newpage
\mbox{ }
\strut \vskip 4.5in
\begin{quotation}
{\bf Fig. 4.} According to Lewis' theory, statement (2.1) is true if {\it every}
world $w$ satisfying the conditions shown in Fig. 2 is closer to {\it some}
world satisfying conditions (a) than to {\it any} world satisfying conditions
(b).
\end{quotation}
Lewis [1986] gives rules of priority (saliency) for determining the closeness of possible
worlds:
(1) It is of the first importance to avoid big widespread diverse violations of
law.
(2) It is of the second importance to maximize the spatio-temporal region
throughout which perfect match of particular fact prevails.
(3) It is of the third importance to avoid even small localized simple
violations of law.
(4) It is of little or no importance to secure approximate similarity of
particular fact, even in matters that concern us greatly.
In our case there are no violations of law. So the most important thing is to
maximize the spatio-temporal region throughout which perfect match of particular
fact prevails.
We may expand the region of perfect match between worlds satisfying the
conditions of Fig. 2\ and of Fig. 4 in the way shown in Fig. 5.
\newpage
\strut\vskip 5in
\begin{quotation}
{\bf Fig. 5} The spacetime region of exact match between worlds satisfying the
factual conditions shown in Fig. 2 and worlds satisfying the counterfactual
conditions shown in Figs. 4(a) and 4(b) are the regions lying to the left of the
dotted lines in (a) and (b), respectively.
\strut\vskip 9pt
The reason that an {\it exact} match can be obtained in case (a) is that the
alterations made in $R_2\cup R_3$ {\it cannot}, according to the
no-faster-than-light-influence condition, influence the events in $R_1$ any way.
Thus, even if there are chance events in $R_1$, there are possible worlds (i.e.,
worlds conforming to law) that {\it match exactly in} $R_1$ {\it in spite of the
differing choices made in} $R_2\cup R_3$.
Thus for any $w$ satisfying the conditions of Fig. 2 there will be some possible
world $w'$ satisfying the conditions of Fig. 4(a) that exactly matches $w$ in
$R_1$.
For this world $w'$ the spacetime region of exact match between $w'$ and $w$ can
be expanded as shown in Fig. 5(a).
This world $w'$ is, according the Lewis' rules, closer to $w$ than any world
satisfying the conditions of Fig. 4(b).
Consequently the statement (2.1) is true, according to Lewis' rules, under the
condition that there be no faster-than-light influence of any kind.
If this latter condition were not valid then it is not clear, without further
stipulations, whether the region of exact match could be expanded in the way shown
in Fig. 5(a) or not.
We conclude that a straightforward application of Lewis' theory leads to the
truth of statement (2.1), under the condition that there can be no
faster-than-light influences of any kind.
That is, Lewis' theory affirms, under this condition, the nondependence of the
outcome selected in $R_1$ upon the `future' free choices in $R_2$ and $R_3$.
The reason that the analysis of CBR fails to yield this intuitively mandated result is that CBR adopt
rules of saliency regarding closeness of worlds that differ from those of
Lewis.
The CBR rules explicitly assign null saliency to the condition $x_1=+1$ in Fig.
5.
By this alteration of the Lewis rules CBR exclude at the outset any possibility
of making
the outcome selected in one region independent of free choices that, in some frame of
reference, will be made at some later time.
Thus
CBR exclude
from the outset the possibility we wish to examine.
Two arguments are given by CBR for assigning null saliency to the condition
$x_1=+1$.
Shortly after introducing the Lewis truth conditions for the `would'
counterfactuals they assert that this condition $[x_1=+1]$ ``should obviously
not be salient for similarity; because if it were, then a world with response
$a_I([x_1=+1])$ will always be judged closer to $w_@$ than worlds where this
response is $-a_I([x_1 =-1])$.
{\it Ipso facto} we will have made the conditional $i$ (Eq. (2.1)) come out
true by fiat; which will undermine the ... argument which seeks to infer $i$'s
truth from the independent principle LOC (no faster-than-light influence)''.
That
argument is incorrect: the derivation of the truth of statement (2.1) from
Lewis' rules of saliency makes essential use of the no-faster-than-light-influence
assumption.
Later on, at the beginning of their section (2.2), CBR return to this problem,
or, rather, to the closely related problem of the events that {\it follow} the
event $[x_1 =+1]$ or $[x_1=-1]$.
Referring to Lewis' discussion of a problem concerning the ``holocaust that would
have occurred if Nixon had pressed the button'' (the future similarity objection)
CBR elect to exclude as nonsalient everything that occurs at, or later than, the
time of the selection of $x_1=+1$ or $x_1=-1$.
But this is {\it not} Lewis' way of resolving this problem.
Lewis resolves this problem by applying precisely the rules of saliency cited
above.
The alternative CBR rule of excluding from saliency the very event whose
occurrence is supposed to be deduced as being necessary is, on the face of it, a
dangerous and unreasonable move: it is not at all surprising that it leads to
unreasonable conclusions.
Lewis' rules are far more discriminating, and lead here to the intuitively
expected result that what occurs `now' is independent of free choices that, in some frame of reference, will be made at some future
time.
Thus the Lewis theory does provide a logical framework within which one can
examine the question of the logical consistency of assuming both this
no-faster-than-light-influence condition and the validity of the predictions of
quantum theory.
As Lewis emphasizes, the rules of closeness must be constructed so as to reflect
the conditions of the situation under consideration.
In the present situation the results such as (2.1) are consequences of the
no-faster-than-light-influence condition.
However, those results do not fully exhaust this condition.
In the proof of the theorem we shall need two other consequences of this
no-faster-than-light-influence condition.
The first one is this: let
$M_1, M_2$, and $M_3$ be three alternative possible triplets of measurements,
and let $P_i$ be the assertion that $M_i$ is performed.
Let $p$ be a possible outcome of $M_1$, and let $P{(p)}$ be a proposition that
depends upon $p$, but makes no reference to $M_2$ or its outcome.
Then the proposed rule of inference asserts that
$$
\eqalignno{
[(P_1\wedge p)&\Rightarrow (P_2 \bxa (P_3\bxa P(p)))]\cr
& \Rightarrow [(P_1\wedge p)\Rightarrow (P_3\bxa P(p))].&(2.2)\cr}
$$
This rule is called `Elimination of eliminated conditions'.
The antecedent in (2.2) asserts, under condition $(P_1\wedge p)$, that if $M_2$ were to
be performed, then it must be the case that if, instead of $M_2$, rather $M_3$
were to be performed then it must be the case that $P(p)$ holds.
The condition that $M_2$ be performed is completely superceded by the
counterfactual condition that, instead of $M_2$, rather $M_3$ be performed.
No trace of $M_2$ remains because it is asserted not to be performed, and no
outcome of $M_2$ is ever mentioned. Moreover, in the present context the choice
of whether or not to do $M_2$ is considered to be a free or random choice.
It appears to us that in a general Lewis context the rule of inference
(2.2) ought to hold under these conditions in which the choice between $M_1, M_2$,
and $M_3$ is free or random, and no reference to
(the eliminated part of) $M_2$ survives in $P(p)$.
The second needed relationship is this:
$$
\eqalignno{
&[((X_2X_3)\wedge (x_2x_3=p))\Rightarrow ((Y_2Y_3)\bxa (y_2y_3=- p))]\cr
&\Rightarrow [(X_2X_3\wedge (x_2x_3=p))\Rightarrow (Y_2X_3\bxa (Y_2Y_3\bxa
(y_2y_3=-p)))]&(2.3)\cr}
$$
An analogous relationship associated with another combination of measurements is
also used.
Again, it is plausible that the no-faster-than-light-influence condition
should allow the change from $X_2X_3$ to $Y_2Y_3$ to be separated in this way
into two steps, since the two parts of the change are confined to two spacelike
separated regions.
\newpage
\noindent{\bf 3. Lewis Analysis of the Proof of the Theorem}
\strut\vskip 9pt
In this section the Stapp [1990] version of Bell's theorem is analyzed by casting it into Lewis' framework.
The context of the theorem is, however, somewhat different from those to which Lewis' theory is usually applied.
A key difference is that the theorem pertains to an indeterministic universe.
Consequently there is no need to violate any law:
tomorrow's choice can be one thing or another without violating any law.
Thus it can be demanded, and is demanded, that there be no violation of any physical law.
Although the general setting is indeterministic there are some proposed strict physical laws.
These consist precisely of the consequences of the no-faster-than-light-influence
conditions described in section two, and the four
predictions of quantum theory mentioned
in the statement of the theorem.
Each of these four predictions asserts that, under a certain condition, a corresponding {\it exact correlation} among outcomes in the
three regions will prevail.
We now state the theorem of Stapp [1990] and its proof, transcribed into the Lewis notation.
This will be followed by a line-by-line exegesis of the proof, within the Lewis framework.
Since all statements occurring within the proof proper are necessarily true in the set of worlds restricted by the proposed
physical laws we will not use Lewis' special notation $\square P$ to denote a necessarily true proposition $P$.
The notation is this:
$X_i \equiv $ a measurement of type $X$ is performed in the spacetime region
$R_i$
$Y_i \equiv$ a measurement of type $Y$ is performed in the spacetime region $R_i$
$x_i \equiv$ the variable corresponding to an $X$-type measurement performed in $R_i$
$y_i \equiv$ the variable corresponding to a $Y$-type measurement performed in $R_i$.
These two variables $x_i$ and $y_i$ have no numeric value except
under the condition that the cited measurement be performed (see
below), and only {\it one} of any two mutually incompatible measurements
can be performed.
Lower case letters $m,n, p, q$, and $r$ represent numbers,
each of which is allowed to be either plus one or minus one.
\strut \vskip 9pt
\noindent{\underline{Theorem}}
{\underline{Hypotheses:}}
1) The choices between $X$ and $Y$ made by the three experimenters can be
treated as independent free variables.
2) For each $i$, if $X_i$ then $x_i$ will acquire a single value, either plus one or minus one.
3) For each $i$, if $Y_i$ then $y_i$ will acquire a single value, either plus one or minus one.
4) The following quantum predictions hold:
$$
\eqalignno{
X_1X_2X_3 &\Rightarrow (x_1 x_2 x_3 =-1)&(3.1)\cr
X_1Y_2Y_3&\Rightarrow (x_1y_2y_3=+1)&(3.2)\cr
Y_1X_2Y_3 &\Rightarrow (y_1x_2y_3=+1)&(3.3)\cr
Y_1 Y_2X_3 &\Rightarrow (y_1y_2x_3 =+1)&(3.4)\cr}
$$
{\underline{Conclusion:}
The no-faster-than-light-influence-condition cannot be maintained.
Specifically the asserted truth of statements such as
$$
\eqalignno{
((X_1X_2X_3) \wedge (x_1=p))&\Rightarrow ((X_1Y_2Y_3) \bxa (x_1=p))&(3.5)\cr
((Y_1X_2X_3)\wedge (x_2x_3 =- p))&\Rightarrow ((X_1X_2X_3) \bxa (x_2x_3=- p))&(3.6)\cr
((X_1Y_2Y_3) \wedge (y_2y_3=p))&\Rightarrow ((Y_1Y_2Y_3) \bxa (y_2y_3 =p)),&(3.7)\cr}
$$
and of (2.2) and (2.3), leads to a logical contradiction.
{\underline{Proof}}
Conditions (3.1), (3.5), and (3.2) entail, for any $p$,
the sequence of syllogisms
$$
\eqalignno{
((X_1X_2X_3) &\wedge (x_2x_3=- p))&\cr
&\Rightarrow ((X_1X_2X_3)\wedge (x_1x_2x_3=-1) \wedge (x_2x_3=-p))&(3.8)\cr
&\Rightarrow ((X_1X_2X_3)\wedge (x_1=p))&(3.9)\cr
&\Rightarrow ((X_1Y_2Y_3)\bxa ((x_1y_2y_3 = 1)\wedge (x_1=p)))&(3.10)\cr
&\Rightarrow ((X_1Y_2Y_3)\bxa (y_2y_3 =p)).&(3.11)\cr}
$$
Thus (3.1), (3.5), and (3.2) entail that
$$
\eqalignno{
((X_1X_2X_3)&\wedge (x_2x_3=-p))\cr
&\Rightarrow ((X_1Y_2Y_3)\bxa (y_2y_3 =p)).\hbox{\hskip 1in}&(3.12)\cr}
$$
By using (3.6), (3.12), and (3.7) one obtains the sequence of syllogisms
$$\eqalignno{
((Y_1X_2X_3) &\wedge (x_2x_3=-p))\cr
&\Rightarrow ((X_1X_2X_3)\bxa (x_2x_3 =-p))&(3.13)\cr
&\Rightarrow ((X_1X_2X_3)\bxa ((X_1X_2X_3) \wedge (x_2x_3=-p)))&(3.14)\cr
&\Rightarrow ((X_1X_2X_3)\bxa ((X_1Y_2Y_3)\bxa (y_2y_3=p)))&(3.15)\cr
&\Rightarrow ((X_1X_2X_3)\bxa ((X_1Y_2Y_3) \bxa
((X_1Y_2Y_3)\wedge (y_2y_3=p))))&(3.16)\cr
&\Rightarrow ((X_1X_2X_3)\bxa ((X_1Y_2Y_3) \bxa ((Y_1Y_2Y_3) \bxa
(y_2y_3=p)))).&(3.17)\cr}
$$
This sequence of syllogisms gives
$$
\eqalignno{
((Y_1X_2X_3)&\wedge (x_2x_3=- p))\cr
&\Rightarrow ((Y_1Y_2Y_3)\bxa (y_2y_3 =p)).\hbox{\hskip 2in}&(3.18)\cr}
$$
Relations (3.18) and (3.12) combine to give (with $\vee =$ ``or'')
$$
(X_1 \vee Y_1)\Rightarrow[((X_2X_3)\wedge (x_2x_3=-p))\Rightarrow
((Y_2Y_3)\bxa (y_2y_3 = p))].\eqno(3.19)
$$
But the conditions of the experiment entail that $(X_1 \vee Y_1)$ is necessarily true.
Hence (3.19) gives
$$\eqalignno{
((X_2X_3)&\wedge (x_2x_3 =-p))\cr
&\Rightarrow ((Y_2Y_3)\bxa (y_2y_3 = p)).\hbox{\hskip 2.25in}&(3.20)\cr}
$$
Starting from (3.3) and (3.4), instead of (3.1) and (3.2), one obtains in the same way, for any $q$,
$$
\eqalignno{
((Y_2X_3)&\wedge (y_2x_3=q))\cr
&\Rightarrow ((X_2Y_3)\bxa (x_2y_3=q)).\hbox{\hskip 2.25in}&(3.21)\cr}
$$
Condition (3.20) entails that for any $m, $ $n$, and $r$,
$$
\eqalignno{
[(X_2&\wedge (x_2=m)) \wedge (X_3\wedge (x_3 = n))]&\cr
&\Rightarrow [(Y_2Y_3)\bxa (y_2y_3 =- mn)]&(3.22)\cr
&\Rightarrow [ (Y_2\wedge (y_2=r)\wedge X_3)\bxa ((Y_2Y_3)\bxa (y_3=- mnr))].&(3.23)\cr}
$$
This entails that
$$
\eqalignno{
[(X_2&\wedge (x_2 =m))\wedge (X_3\wedge (x_3 =n))]\cr
&\Rightarrow [((Y_2\wedge (y_2=r))\wedge (X_3\wedge (x_3=n)))\bxa
((Y_2Y_3) \bxa (y_3 =- mnr))].\cr
&&(3.24)\cr}
$$
On the other hand, (3.21) entails that
$$
\eqalignno{
[(Y_2 &\wedge (y_2=r))\wedge (X_3\wedge (x_3=n))]\cr
&\Rightarrow [((Y_2Y_3)\wedge (y_3=- mnr))\bxa ((X_2Y_3)\bxa
(x_2=- m))].\hbox{\hskip .75in}&(3.25)\cr}
$$
Insertion of this relationship into (3.24) entails that
$$
\eqalignno{
[(X_2 &\wedge (x_2 = m))\wedge (X_3 \wedge (x_3 = n))]\cr
&\Rightarrow [((Y_2\wedge (y_2=r))\wedge X_3)\bxa
((Y_2Y_3) \bxa [(X_2Y_3) \bxa (x_2=-m)))].\hbox{\hskip 5pt}&(3.26)\cr}
$$
The numbers $n$ and $r$ appearing here are arbitrary, and appear only once in this relationship.
Thus by virtue of our premises, which demand that if $X_3$ then $(x_3 =+ 1) $ or
$(x_3 =-1)$, and if $Y_2$ then $(y_2=+1)$ or $(y_2=-1)$,
the conditions in (3.26) involving $n$ and $r$ are necessarily fulfilled for some pair $(n,r)$.
Thus (3.26) reduces to
$$
\eqalignno{
[X_2&\wedge (x_2=m)\wedge X_3]&\cr
&\Rightarrow [(Y_2X_3)\bxa ((Y_2Y_3)\bxa ((X_2Y_3)\bxa (x_2=-m)))]
\hbox{\hskip .75in}&(3.27)\cr}
$$
It is argued that this leads to a contradiction.
\strut \vskip 9pt
\noindent{\underline{Exegesis}}
Appendix A gives the derivation of two rules of inference that are valid in the Lewis theory, and are used extensively in this
exegesis:
$$
(C\Rightarrow D)\Rightarrow [(B\bxa C)\Rightarrow (B\bxa D)]\eqno(3.28)
$$
and
$$
(B\Rightarrow D)\Rightarrow [(B\bxa C)\Rightarrow (B\bxa (C\wedge D))].\eqno(3.29)
$$
The justification within Lewis' framework of the various lines of the proof then goes as follows:
(3.8): Use (3.1).
(3.9): Use algebra.
(3.10): Use (3.5), then (3.2) and (3.29).
(3.11): Use algebra and (3.28).
(3.12): Restatement of (3.11).
(3.13): Use (3.6).
(3.14): Use (3.29).
(3.15): Use (3.12) and (3.28).
(3.16): Use (3.28) for the first $\bxa$, and (3.29) for the second $\bxa$.
(3.17): Use (3.28) for the first two $\bxa$, and (3.7).
(3.18): Use (2.2) twice.
(3.19): Use Cor. of Lemma A5 and $((X_1 \Rightarrow P)\wedge
(Y_1\Rightarrow P))\Rightarrow ((X_1\vee Y_1)\Rightarrow P)$.
(3.20): Use Modus Ponens: $[(A\Rightarrow B)\wedge A] \Rightarrow B$.
(3.21): Repeat steps (3.8) to (3.20).
(3.22): Use $(A\Rightarrow B)\Rightarrow [(A\wedge C)\Rightarrow B]$.
(3.23): Use Lemma B1 and algebra and (3.28) twice.
(3.24): Use the Corollary to Lemma A3.
(3.25): Repeat with new variables the steps (3.20) to (3.23).
(3.26): With the identifications
\hskip .5in $A = [(Y_2\wedge (y_2=r))\wedge (X_3\wedge (x_3=n)]$
\hskip .5in $B=[Y_2Y_3]$
\hskip .5in $C=[(y_3=-mnr)]$
\hskip .5in$D=[(X_2Y_3)\bxa (x_2=-m)]$
\hskip .5in the results (3.24) and (3.25) assert that,
\hskip .5in under the condition
$(X_2\wedge (x_2=m))\wedge (X_3\wedge (x_3=n))$,
\hskip .5in $A\bxa (B\bxa C)$
\hskip .5in and
\hskip .5in $A \Rightarrow [(B\wedge C) \bxa D)]
$
\hskip .5in The rule (3.29). then gives
\hskip .5in$A\bxa[(B\bxa C)\wedge ((B\wedge C)\bxa D)]$.
\hskip .5in This statement is asserted by Lemmas A4 and A1 to entail
\hskip .5in $A\bxa (B\bxa D)$.
Then Lemma A4 yields (3.26).
(3.27): The removal of the condition $(y_2=r)$ is discussed in appendix B.
The
\hskip .5in removal of the condition $(x_3 =n)$ is obtained by
using $[((A\wedge B)\Rightarrow C) \wedge$\\
\strut\hskip .70in $((A\wedge \overline{B})\Rightarrow C)]\Rightarrow (A\Rightarrow C)$.
Completion of proof: Use (2.2) twice to obtain
$$
((X_2X_3)\wedge (x_2=m))\Rightarrow ((X_2Y_3)\bxa (x_2=- m)).\eqno(3.30)
$$
This contradicts the no-faster-than-light-influence condition
$$
(X_2X_3 \wedge (x_2 =m))\Rightarrow ((X_2Y_3) \bxa (x_2 = m)).\eqno(3.31)
$$
\newpage
\noindent{\bf 4. Discussion}
We consider here several issues raised by referees.
The first question concerns ``determinism''.
The occurrence in (2.1) of the word ``would'' raises the question of
whether determinism is being tacitly assumed.
The answer is that (2.1) is saying only that the result of a certain local
measurement is plus one under a condition that includes the statement that
the result of this very same local measurement is plus one: the two values
of plus one are the result of one and the same local measurement.
{\it There is no condition placed upon how nature came to deliver the result
plus one, or about how that result was related to what came before.}
The the altered conditions pertain to what {\it will come later}, in
some frame of reference.
The statement (2.1) says that the (random) {\it later } choices of which
measurements will be made in $R_2$ and $R_3$ can have no influence upon the
selection of a value for $x_1$ made by nature at the earlier time.
Moreover, the notion of ``no influence'' makes no appeal to determinism: the
notion that the changes $X_2X_3 \rightarrow Y_2Y_3$ made in $R_2 \cup R_3$
``do not influence'' whatever happens in $R_1$ is expressed as the condition
that what happens in $R_1$ must at least be {\it allowed} to be
unaffected by the change in $R_2 \cup R_3$:
there are {\it possible worlds} where there is no change in $R_1$. (See the
argument below Fig. 5.)
This way of formulating the no-faster-than-light-influence condition is an
appropriate condition in an indeterministic context, and is the no-influence
condition that has been used by the second-named author for many years.
Thus the present proof, though technically very different, rests on the same
basic assumptions as those earlier works.
Arthur Fine [1989] has discussed these ``would'' and ``could'' issues, and
has concluded that the ``would'' formulations of the locality condition
tacitly assumes something akin to determinism, and that the ``could''
version is too weak to do the job.
Fine's argument that the ``would'' formulation entails some use of the
notion of determinism is based on a consideration of what conditions are
necessary in order to pass from the assertion of ``no influence'' to the
assertion that ``things would have been just the same''.
He claims, first, that this passage requires the supplementary principle that
``where nothing relevant to an outcome changes the outcome itself could not
change'', and, second, that this principle is based on the idea of determinism.
This argument does not seem to cover to the following example: ``If a
radioactive
decay was just observered in my laboratory, and in one minute
a certain atom in the sun will
disintegrate, then even if that atom in the sun will not disintegrate in one
minute the radioactive decay {\it would} still have just been observed
in my laboratory.
The assumption that this statement is true does not {\it require}
determinism: it {\it could} be imposed even within the context of an
indeterministic theory of the universe.
We can consistently imagine an indeterministic theory of the universe
in which the principle that the past is not altered by completely random
future events is strictly enforced.
Our no-faster-than light-assumption is essentially the assumption that
completely random future decisions do not alter past events.
For, according to the theory of relativity, the complete lack of any causal
connection between a cause and a possible effect that is located in a
spacelike direction from the cause is the same as the complete lack of
causal connection in the case that the cause is situated (still spacelike from
but) slightly {\it later } than the possible effect.
The idea of the complete lack of causal dependence {\it in both directions}
is brought out, intuitively, by considering --- for each of the two possible
directions of the possible causal connection --- a frame in which the event
identified as the cause is situated slightly {\it later} than the event
identified as the possible effect.
{\it This } is the intuitive idea of ``no influence'' that we wish to
formalize, and that Lewis's modal analysis so satisfactorily captures.
Fine considers also the ``could'' version of the ``no influence'' condition
(which asserts merely that the nearby outcome ``could be'' one and the same
outcome, independently of which random decision is eventually made
faraway). He concludes that this assumption too weak to do the job.
For, as he correctly observes, the ``could'' version can entail at most only
that the predictions of quantum theory ``could'' be violated.
Fine sees nothing troubling about that: ``After all any experiment could
fail to satisfy any set of predictions.
We hardly need any principle of strong locality to learn that''.
However, in situations such as the one considered here, in which all of the
relevant predictions are predictions that particular correlations occur with
certainty (i.e., with probability unity) the assertion that these
predictions ``could'' fail is, from a logical or theoretical perspective,
precisely the assertion that it is false that these predictions must
necessarily be true:
it is false that these predictions must hold in all cases.
But this is just the assertion that the quantum laws fail.
Thus Fine's argument does not succeed in showing that the ``could'' version
of the no-faster-than-light-influence assumption is compatible with the
assumption that the quantum laws hold.
We conclude that nothing in Fine's arguments contradicts our results.
\newpage
\noindent{\bf 5. Conclusions}
1. David Lewis's theory of counterfactuals provides a rigorous logical
framework within which one can formulate, in an
indeterministic context, the physical concept of no faster-than-light
influence of any kind.
2. The theory can be applied to the experimental situation described by
Greenberger, Horne, and Zeilinger.
This situation involves three spacetime regions that are mutually spacelike
separated.
In each region a choice is to be made to perform one or the other of two
mutually incompatible
measurements.
These choices are treated as independent free variables.
It is assumed that for either of the alternative possible local measurements
$M$ performable in a region, {\it if} $M$ is chosen then nature will
select some single outcome of $M$.
This selection is considered to be random, subject to two supervening strict
laws.
The first law is connected to the fact that for four of the eight possible
combinations of
measurements quantum theory predicts
an exact correlation among the three outcomes.
The first postulated supervening physical law asserts that, for each of these
four alternative possible combinations of
measurements, {\it if} that combination is chosen and performed {\it then} the
corresponding quantum prediction must hold.
The second postulated supervening physical law is that there can
be no faster-than-light influence of any kind.
This condition is formulated as the demand that nature's selection in each one
of the three regions must be independent of which
choices will (later, in some frame of reference) be made by the experimenters
in the other two regions.
3. It is shown that, within the Lewis framework
supplemented by two reasonable rules of inference, these assumptions lead
to a contradiction.
\newpage
\noindent{\bf References}
Bell, J.S. [1964]: ``On the Einstein Podolsk Rosen Paradox'',
{\it Physics}, {\bf 1}
195-200.
Clifton, R.K., Butterfield, J.N., and Redhead, M.G. [1990]: `Nonlocal Influences and Possible Worlds', {\it British Journal for the
Philosophy of Science}, {\bf 41} 5-58.
Fine, A. [1989]: ``Do correlations need to be explained'' in {\it
Philosophical Consequences of Quantum Theory}, eds. J.T. Cushing and E. McMullin.
Notre Dame: Notre Dame University Press. p. 187-188.
Lewis, D. [1973]: {\it Counterfactuals}. Oxford: Blackwell Press.
Lewis, D. [1986]: {\it Philosophical Papers Vol.II.} Oxford:
Oxford University Press.
Redhead, M.G. [1987]:
{\it Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics.}
Oxford: Clanendon Press.
Stapp, H.P. [1990]: ``Quantum Measurement and the Mind-Brain Connection'',
in {\it Symposium on the Foundations
of Modern
Physics 1990}, eds. P. Lahti and P. Mittelstaedt.
Singapore: World Scientific.
\newpage
\noindent{\bf Appendix A}
The following six lemmas are general consequences of Lewis'
definition of the symbol $\bxa$.
We denote by \{$P$\} the set of worlds in which the proposition $P$ is true. The set $\phi$ is the empty set. The converse of $P$
is denoted by $\overline{P}$.
\underline{Lemma A1}
$$
(C\Rightarrow D)\Rightarrow [(B\bxa C)\Rightarrow (B\bxa D)]\eqno(A.1)
$$
\underline{Proof}
\underline{Case 1: $B=\phi$}
In this case $(B\bxa D)$ is true by definition.
But then the consequent of (A.1) is true by definition.
But then (A.1) is true by definition.
{\underline{Case 2: $B\neq \phi$}}
In this case the antecedent in (A.1) asserts that every element of \{$C$\} is
an element of \{$D$\}.
The antecedent of the consequent asserts that the point $w$ from which the statement
$(B\bxa C)$ is issued is closer to some point of \{$B\wedge C$\} than to any point of
\{$B\wedge \overline{C}$\}.
These two suppositions are indicated in Figs. 6(a) and 6(b), respectively.
\strut\vskip 3in
\begin{quotation}
{\bf Figure 6.} Figure (a) indicates that every $C$-world is a $D$-world.
Figure (b) indicates that every $(B\wedge \overline{C})$-world
is further from $w$ than some $(B\wedge C)$-world.
\end{quotation}
The consequent of the consequent of (A.1) asserts that the relation 7(b) continues to hold if $C$ is replaced by $D$.
This is obvious because \{$B\wedge D$\} includes every point in \{$B\wedge C$\} whereas every point in \{$B\wedge \overline{D}$\} is
included in \{$B\wedge \overline{C}$\}.
\noindent\underline{Lemma A2}
$$
\eqalignno{
(B&\Rightarrow D)\Rightarrow\cr
&[(B\bxa C) \Rightarrow (B\bxa (C\wedge D))]&(A.2)\cr}
$$
\underline{Proof}
\noindent{\underline{Case 1. $B=\phi$}}
\noindent Here the reasoning is the same as in case 1 of Lemma Al.
\noindent {\underline{Case 2. $B\neq \phi$}}
The statement (A.2) asserts that if every $B$-world is a $D$-world then
if the relationship shown in Fig. 7(a) holds, then the
relationship shown in Fig. 7(b) must hold.
\strut\vskip 3in
\begin{quotation}{\bf Figure 7.} Diagram (a) asserts that some $(B\wedge C)$-world is closer
to $w$ than any $(B\wedge \overline{C})$-world.
Diagram (b) asserts that some $(B\wedge C\wedge D)$-world is
closer to this world $w$ than any $(B\wedge \overline{[C\wedge D]})$-world.
\end{quotation}
If \{$D$\} includes \{$B$\} then \{$B\wedge C$\} is
the same as \{$B\wedge C\wedge D$\}.
The set \{$B\wedge \overline{[C\wedge D]}$\}
is the union of the sets \{$B\wedge \overline{C}$\} and \{$B\wedge \overline{D}$\}.
The latter set is empty, if \{$D$\} includes \{$B$\}.
Thus the sets in Fig. 7(a) are identical to the sets Fig. 7(b), under the
condition $B\Rightarrow D$, and the lemma is proved.
\noindent {\underline{Corollary}}
$$
(B\Rightarrow D)\Rightarrow (B\bxa D)
$$
\underline{Proof} \ \ Let $C$ in the lemma be $B$.
Then $(B\bxa C)$ is necessarily true, and $(B\bxa (C\wedge D))$
is equivalent to $(B\bxa D)$.
Then Modus Ponens yields the corollary.
\noindent {\underline{Lemma A3}}
$$
[(B\bxa C)\wedge (B\bxa D)] \Rightarrow [(B\wedge C)\bxa D]\eqno(A.3)
$$
\underline{Proof} The antecedent says that the world $w$ in which the statement
is issued is closer to some $BC$-world $w'_1$ than to any $B\overline{C}$-world,
and that $w$ is closer to some $BD$-world $w'_2$ than to any $B\overline{D}$-world.
The consequent asserts that some $(B\wedge C\wedge D)$-world $w'_3$ is closer to
$w$ than any $(B\wedge C\wedge\overline{D})$-world.
Take $w'_3$ to be either $w'_1$ or $w'_2$, whichever is closer to $w$, or either one if they are equally close.
If $w'_3$ is $w'_2$, and is also a $BC$ world then the consequent is true.
If $w'_3$ is $w'_1$ then, since it is no further than $w'_2$ from $w$,
the consequent is again true.
It is not possible for $w'_3$ to be both $w'_2$ and a $B\overline{C}$- world
because $w'_1$ is closer to $w$ than any
$B\overline{C}$-world.
\noindent{\underline{Corollary}}
$$
\eqalignno{
[(A&\Rightarrow (B\bxa C))\wedge (A\Rightarrow (B\bxa D))]\cr
&\Rightarrow [A\Rightarrow ((B\wedge C)\bxa D)]&(A.3')\cr}
$$
\underline{Proof}\ \ For any $w\epsilon \{A\}$, in the consequent,
let this same $w$ be used in both parts of the antecedent. Then the
lemma ensures that the truth of the antecedent in (A.3$'$) will entail the truth of the consequent.
In the application of this corollary to (3.24) one makes the identifications
$$
\eqalignno{
A&=[(X_2\wedge (x_2=m))\wedge (X_3\wedge (x_3=n))]\cr
B&=[Y_2\wedge (y_2=r)\wedge X_3]\cr
C&=[(x_3=n)]\cr
D&= [Y_2Y_3\bxa (y_3=-mnr)]\cr}
$$
Then $[A\Rightarrow (B\bxa C)]$ is entailed by the no-faster-than-light-influence condition, and $[A\Rightarrow (B\bxa D)]$ is entailed by
(3.23).
\noindent{\underline{Lemma A4}}
$$
[(B\bxa C)\wedge ((B\wedge C)\bxa D)]\Rightarrow (B\bxa D)\eqno(A.4)
$$
\underline{Proof}
The antecedent in (A.4) states that: (1) the world $w$ from which the statement
(A.4) is issued is closer to some $(B\wedge C)$-world $w'_1$ than to any
$(B\wedge \overline{C})$-world, and (2),
this point $w$ is closer to some $(B\wedge C\wedge D)$-world $w'_2$ than
to any $(B\wedge C\wedge \overline{D})$-world.
It must be shown that this $w$ is closer to some $(B\wedge D)$-world
$w'_3$ than to any $(B\wedge \overline{D})$-world.
There are two cases. If $w'_1$ lies in $B\wedge D$ then $w'_3$ can
be taken to be $w'_1$ or $w'_2$, whichever is closer to $w$, or either
one if they are equally close.
Then this $w'_3$ must be closer to $w$ than any point of
\{$B\wedge \overline{C}$\}, and closer to $w$ than any point of \{$B\wedge
C\wedge \overline{D}$\}, and hence closer to $w$ than
any point of \{$B\wedge \overline{D}$\}, as required.
If $w'_1$ lies outside of $D$ then it must be further from $w$ than $w'_2$.
Hence one can again take $w'_3$ to be the closer of $w'_2$ and $w'_1$ to $w$, namely $w'_2$, and apply the earlier argument.
\noindent{\underline{Corollary}}
$$
\eqalignno{
[(A&\Rightarrow (B\bxa C)\wedge (A\Rightarrow((B\wedge C)\bxa D)]\cr
&\Rightarrow [A\Rightarrow (B\bxa D)]&(A.4')\cr}
$$
\underline{Proof}
Same as proof of the corollary to Lemma A3.
\noindent{\underline{Lemma A5}
$$
\eqalignno{
&[((A\wedge B)\Rightarrow ((A\wedge C)\bxa D))\wedge ((A\wedge B)\Rightarrow (C\bxa A))]\cr
&\Rightarrow [A\Rightarrow (B\Rightarrow (C\bxa D))]&(A.5)\cr}
$$
\underline{Proof}
The antecedent says that each world $w$ of \{$(A\wedge B)$\}
is closer to some world $w'_1$ in \{$A\wedge C\wedge D$\} than to any
world in $\{A\wedge C\wedge \overline{D}\}$;
and is closer to some world $w'_2$ in \{$A\wedge C$\}
than to any world in \{$\overline{A} \wedge C$\}.
The consequent says that each world $w$ of \{$A\wedge B$\} is closer to some
world $w'_3$ in \{$C\wedge D$\} than to any world in
\{$C\wedge \overline{D}$\}.
If $|w'_1 - w|\leq |w'_2-w|$ take $w'_3=w'_1$.
Then $w$ will be closer to $w'_3\epsilon \{C\wedge D\}$
than to any world in either $\{ A\wedge C\wedge \overline{D}\}$ or
$\{\overline{A}\wedge C\}$.
Hence $w$ will be closer to $w'_3$
than to any world in $\{ C\wedge \overline{D}\}$, as
required.
If $|w'_2 -w|< |w'_1-w|$ then $w'_2\epsilon \{A\wedge C\}$ must lie in
$\{A\wedge C\wedge D\}$, and hence in \{$C\wedge D$\}.
So take $w'_3=w'_2$.
The world $w$ is then closer to $w'_3$ than to any world in either
\{$\overline{A}\wedge C$\} or
\{$A\wedge C\wedge \overline{D}$\}, and hence than to any world in
\{$C\wedge \overline{D}$\}, as required.
\noindent\underline{Lemma A6}
$$
\eqalignno{
&[(A\Rightarrow ((B\wedge b)\bxa C))\wedge (A\Rightarrow ((B\wedge \overline{b})\bxa C))]\cr
&\Rightarrow (A\Rightarrow (B\bxa C))&(A.6)\cr}
$$
\underline{Proof}
The antecedent says that any world $w$ in $A$ is closer to some world $w'_1$ in \{$B\wedge b\wedge C$\} than to any world in
\{$B\wedge b \wedge \overline{C}$\}; and to some world $w'_2$ in $\{B\wedge \overline{b} \wedge C\}$ than to any world in
$\{B\wedge \overline{b} \wedge \overline{C}\}$.
The consequent says that any world $w$ in $A$ is closer to some world $w'_3$ in \{$B\wedge C$\} than to any world in
\{$B\wedge\overline{C}$\}.
Take $w'_3$ to be either $w'_1$ or $w'_2$, whichever is closer to $w$, or either one if they are equally close.
This satisfies the required conditions.
\newpage
\noindent {\bf Appendix B}
\noindent{\underline{Lemma B1}}\ \ Under the conditions of the theorem the
following relationship holds: For any numbers
$m, n, $ and $r$
$$
\eqalignno{
[((X_2 &\wedge (x_2=m))\wedge (X_3\wedge (x_3=n)))\cr
&\Rightarrow ((Y_2Y_3)\bxa (y_2y_3=-mn))]\cr
&\Rightarrow [((X_2\wedge (x_2=m))\wedge (X_3 \wedge (x_3=n)))\cr
&\Rightarrow ((Y_2\wedge (y_2=r)\wedge X_3)\bxa ((Y_2Y_3)\bxa ((y_2=r)\wedge
(y_2y_3=- mn))))].\cr
&&(B.1)\cr}
$$
\vskip 9pt
\noindent{\underline{Proof}}
The antecedent in (B.1) in conjunction with (2.3) entails
$$
\eqalignno{
&((X_2\wedge (x_2=m))\wedge (X_3\wedge (x_3=n)))\cr
&\Rightarrow (Y_2X_3 \bxa (Y_2Y_3\bxa (y_2y_3=-mn))).&(B.2)\cr}
$$
This statement asserts that for any world $w$ in
$(X_2\wedge (x_2=m)) \wedge (X_3\wedge (x_3=n)$ (see Fig. 8), there is a world
$w'_1=w'_1(w)$ in \{$Y_2\wedge X_3$\} where
$((Y_2Y_3)\bxa (y_2y_3 =- mn))\equiv P$ is true that is closer to $w$ than any world in \{$Y_2\wedge X_3$\} where $P$ is false.
The set of worlds $\{Y_2\wedge X_3\wedge P\}$ consists of those worlds $w'$ in \{$
Y_2\wedge X_3$\} such that $w'$ is closer to some world $w''_1=w''_1(w')$ in
$\{Y_2\wedge Y_3 \wedge (y_2y_3=-mn)\}$ than to any world in \{
$(Y_2\wedge Y_3 \wedge (y_2y_3=+mn))$\}.
Let the $Y_2X_3$ worlds and the $Y_2Y_3$ worlds
be separated into the $(y_2=r)$ worlds and the $(y_2=- r)$ worlds.
To prove the Lemma it must be shown that if $\{Y_2\wedge X_3 \wedge (y_2=r)\}$
is nonempty, then for any world $w$ in
$
\{(X_2\wedge (x_2= m))\wedge (X_3\wedge (x_3 =n))\}$ there is a world $w'_2 = w'_2(w)$ in $\{Y_2 \wedge X_3\wedge (y_2=r)\wedge
P_r\}$
such that $w$ is closer to $w'_2$ than to any world in
$\{Y_2\wedge X_3\wedge (y_2=r)\wedge \overline{P}_r\}$.
Here $P_r$ is the set of $Y_2\wedge X_3$ worlds $w'$
such that $w'$ is closer to some point $w''_2 = w''_2 (w')$ in $\{ Y_2\wedge
Y_3 \wedge (y_2=r)\wedge (y_2y_3=-mn)\}$
than to any $(Y_2\wedge Y_3)$-point outside this set.
If the point $w'_1(w)$ lies in the set where $y_2=r$ then let $w'_2=w'_1$.
This point in \{$Y_2\wedge X_3\wedge (y_2=r)\wedge P$\} will be closer
to $w''_1(w'_1)$
than to any ($y_2y_3=+mn)$-point.
If $w''_1(w'_1)$ were a $(y_2=-r)$-point then there will be a point $w''_2$
in $\{ Y_2\wedge Y_3 \wedge (y_2=r)\wedge (y_2y_3=-mn)
\}$
that is closer to
$w'_1\epsilon
\{ Y_2\wedge X_3 \wedge (y_2=r)\}$ than to any point outside $\{Y_2\wedge Y_3 \wedge
(y_2=r)\wedge (y_2y_3 =-mn)\}$,
as the Figure 8 shows, since by virtue
of the no-faster-than-light-influence condition $w'_1$
is closer to {\it some} $\{Y_2\wedge Y_3\wedge (y_2=r)\}$
point than to any $(Y_2\wedge Y_3\wedge (y_2=-r))$ point.
Thus in this case where $w'_1(w)$ satisfies $(y_2=r)$ and
$w''_1(w'_1)$ satisfies $(y_2=-r)$ the Lemma is proved.
If $w'_1$ satisfies $(y_2=r)$ and $w''_1(w'_1)$ satisfies $(y_2=r)$
then the choice $w'_2 = w'_1(w)$ satisfies the required conditions.
The conditions of the theorem entail that, under any conditions,
if $Y_2X_3$ is performed then {\it some} outcome must
appear.
Thus we have the condition that for {\it some} value $r$
$$
\eqalignno{
&((X_2\wedge (x_2 = m)) \wedge (X_3\wedge (x_3=n))\cr
&\Rightarrow (Y_2X_3 \dba (y_2 = r)),&(B.3)\cr}
$$
where $\dba$ is Lewis' `might' counterfactual conditional.
It asserts that if $Y_2X_3$ then it is false that $y_2 =-r$ would appear.
This entails that one of two conditions must hold: (1),
there is a $(Y_2\wedge X_3 \wedge (y_2=r))$-world that is closer to
$w$ than any
$(Y_2\wedge X_3\wedge (y_2 =-r))$-world; or (2), for any
$(Y_2 \wedge X_3 \wedge (y_2=r))$ world there is a
$(Y_2\wedge X_3\wedge (y_2=-r))$-world that is closer or equally close
to $w$, and vice versa.
But in this case the world $w'_1$ can be taken to be a $(y_2=r)$-world,
and the previous argument proves the lemma, for this value
of $r$ for which (B.3) holds.
For a value $r$ that does not satisfy (B.3), and hence cannot occur,
the Lemma is trivially true because $(Y_2\wedge (y_2 =r)\wedge
X_3)$ cannot be satisfied.
The trivial case does not actually play any role:
later in the proof the value of $r$
is chosen to be one such that $(Y_2\wedge (y_2=r)\wedge X_3)$ can occur.
If both values can occur (might occur) then Lemma A6 is used.
If only one value of $y_2$ can occur under the conditions imposed then $r$
is taken to be this value.
\vskip 6in
\begin{quotation}
{\bf Figure 8.} $w$ is closer to $w'_1 \epsilon \{ Y_2\wedge X_3\wedge P\}$
than to any point in $\{ Y_2\wedge X_3 \wedge
\overline{P}\}$, $w'$ is closer to $w''_1(w')\epsilon \{ Y_2\wedge Y_3\wedge
(y_2y_3=-mn)\}$ than to any $Y_2Y_3$ point with $y_2y_3=+mn$.
\end{quotation}
\end{document}