Unruh says at the top of page 8, in connection with LOC2, that:
``If it were true that one could deduce solely from the
fact that a measurement had been made at L that some relations
on the right must hold, then I would agree that this requirement
would be reasonable.''
Of course, one needs not solely the assumption that the measurement
L2 is made in L: one needs also LOC1, and the assumptions that
some predictions of quantum theory are valid. But from these alone
one can deduce the validity of the premise of LOC2:
PREM-LOC2: L2-->[(R2^R2+)-->((R1/R2)-->(R1-))].
There is no assumption or requirement on what the outcome of L2 is:
that is certainly a key point in my proof.
So Unruh appears here to be conceding that my LOC2 is OK!
There are two essential points. The first is that the assumption
in PREM-LOC2 pertaining to L is merely that L2 be performed.
The second is that the statement in the square brackets refers
only to events and hypothetical events in R.
It is, of course, an absolutely central feature of logic that
the statements themselves convey their meanings: one does not
have to probe through the hundreds of assumptions that may go
into proving that some statement is true to say what it means:
each statement in the logical universe under consideration is
supposed to have a meaning before one tries to prove
whether it is true or false. Indeed, many different proofs might
be given that some statement is true, or that it is false.
As an example, consider two real numbers x and y. Let S be the
assertion that ``x is positive''. S refers only to x, not to y.
But in a context defined by ``y is positive'', and ``x=y'' one can
prove the validity of S. This proof refers to y. But
the MEANING of ``x is positive'' does not involve y.
This simple example illustrates that the proof of a statement
can refer to things beyond what the statement itself refers
to. Analogously, in our case the statement in the square bracket
refers only to events and hypothetical events in R, and it is
proved to be valid in the context defined by LOC1, and the
validity of some predictions of quantum theory, provided only
that L2 be performed.
LOC2 then asserts that this statement that refers only to
events and possibilities in R, and that is true provided
only that L2 is carried out at the specified later time, would
be true also if the later free choice in L had gone the other way.
Later on page 8 Unruh says that the assertion (a prediction of
QM)
(L2^R2^g)-->(L2^R2^c)
has two possible interpretions. But in a sound logic each
statement has only one meaning. The meaning of the above
statement is this:
``If L2 is performed and R2 is performed and the outcome of
R2 is g, then L2 is performed and R2 is performed and the
outcome of L2 is c.''
The meaning of the statement does not depend on whether it
happens to be true or false in some context, or how it was
derived.
Unruh goes on in this way conflating the meanings
of statements with how they are proved to be valid
under certain conditions. But this is not how logical
reasoning goes. His assrtions violate the basic principle
of logic that each statement must have a well defined meaning
independently of how it is proved in some context.
It should be emphasised that we are dealing here only
with possibilities for setting up experiments, and
noting their observable outcomes. Bohr often said
that no special intricacies arise at this level, and
he rejected special logics. Heisenberg explicitly
rejected logical positivism.
Unruh claims that this basic principle of logic cannot
be maintained in the quantum context even at the level
of our classically formulated descriptions of which
experiments we can set up, and what outcomes we might
observe. Bohr certainly did not appeal to any idea of
the failure of logic in his discussions with Einstein
of matters essentially similar to the one being
discussed here.
Unruh's essential claim is that in the quantum context
the meaning of the statement in square brackets in
PREM-LOC2 cannot be considered to refer only to (real
and hypothetical) events on the right. He
claims that because the proof of the validity
of PREM-LOC2 involves an intermediate step in which
the value of the outcome of L2 is fixed (by the
combination of the assumptions that R2 is performed
and gives outcome g, and that the prediction of quantum
theory mentioned above is valid), the meaning
of the statement in the square bracket is tied to that
outcome on the left.
This is certainly not the case in classical modal logic,
where referents of meanings are separable from the
referents of proofs of validity within some context.
But Unruh claims that in the quantum context the
assertion that the statement in square brackets refers
only to things on the right ``is simply another form of
realism.''
Unruh does not prove that this is true. I think it is
false.
The widely accepted idea in quantum theory, stemming from
Dirac, is that no outcome is specified in nature until the
experiment is performed, and that under the condition that
the measurement is performed Nature selects some outcome.
Within that quantum framework the statement in square
brackets says that there is a constraint on Nature's selection
process that is such that in any case for which R2 is
performed and Nature's selection for the outcome is R2+,
if, instead, R1 had been performed then Nature's selection
would have been R1-. This is the natural meaning within the quantum
universe of the statement within the square brackets, and it
refers only to events and hypothetical events in R. There is
no need to abandon of the basic principle of logic, which
is that the statements must have well defined meanings.
This property was proved within a certain context (formed by
LOC1 and some predictions of quantum theory) by an argument
that referred to things on the left. But the proved
property itself, described above, does not refer to things
on the left.
The property was prove under the condition that L2 was
performed, without imposing any condition as to which
outcome appears in L.
Since the choice to perform L2, not L1, was made later
than all of Nature's possible relavant selections
in R, any dependence of ``the constraints on those
selections'' upon which choice is made in L would
constitute some sort of influence of that choice upon
constraints pertaining to earlier possible selections.
There is no implication here that the unperformed experiment
R1 has a fixed result: there is merely the deduced result
that Nature's selection process in R must be constrained
in a particular way if the later free choice is L2, plus the
locality idea that the outcome of this later free choice
should have no effect on the constraints upon Nature's
selections pertaining to the earlier times.
I believe that this example shows how my arguments conform
to the basic logical principle that the meaning of each
statement be well defined (in a way that matches the words
of the statement) without bringing in any idea of reality
that is alien to quantum theory.