Box A contains $100 and box B contains either nothing or $1 million.

In Raymond Smullyan's latest book of puzzles, "The Riddle of Scheherazade,"
he gives a new variation of Newcomb's paradox that does not contain a
predictor!  It may shed new light on the paradox and perhaps resolve it.

First, I'll give a modified version of Smullyan's description of the
Newcomb paradox  (Smullyan's description actually uses a chest with
2 drawers instead of 2 boxes and he uses different amounts of money
than is usually the case.  This is irrelevant to the paradox so I've
translated it back into the usual formulation.  Other than that, the
rest is from Smullyan).

Suppose there are two boxes.  Box A contains $100 and box B contains
either nothing or $1 million.  Suppose I give you the option of taking
whatever is in both boxes or taking just the money in box B.  Which
would you choose?  Obviously, you should take both boxes since you
will get $100 more.  Is there any further information that I could
provide that would make you change your mind and believe that you will
get more money if you take just box B than if you take both boxes?
"This sounds ridiculous, doesn't it?  I'm sure that many of you would
be willing to bet that *no* added information could make you change
your mind on that!  But wait!"  What I didn't tell you was that amount
of money placed in box B was determined by a super-predicting being
(a human, an alien, a machine, or a god) that is a perfect (or near
perfect) predictor.  If the being predicted that you would choose
both boxes, then nothing was placed in box B but if the being
predicted that you would choose just box B, then $1 million was
placed in box B.  Would this added information change your mind?  I'm
sure that many of you will say yes, on the simple grounds that you
now realize that if you take both boxes, the predictor would have
predicted this and you will get only $100 whereas if you take only
box B, you will get $1 million. But yet, doesn't this contradict the
equally obvious fact that the money is already *there*, and there
is more money in both boxes together than in box B alone?  This is the
paradox.

The following is Smullyan's new predictorless version of the paradox.

Again there is two boxes.  Box A contains $100 and box B contains
either nothing or $1 million dollars and again you are to choose
either both boxes or just box B.  Suppose we are given the following
proposition.

"Either you will choose both boxes and there will be $100 in box A and
nothing in box B, or you will choose just box B and there will be
$1 million in it (and $100 in box A)."

Note that I have said nothing about any predictor! (A Newcomb
predictor implies the proposition but the proposition is more general and
makes no reference to a predictor.)

Is this proposition consistent?  Smullyan first proves that the
proposition is inconsistent and then proves that it is consistent.

Proof that the proposition is inconsistent.
Regardless of whether box B contains nothing or $1 million dollars,
there is more money in both boxes than there is in just box B.
Therefore, if you open both boxes, you will find more money than if
you open just box B.  On the other hand, it follows from the
proposition that if you open just box B, you will find more money
($1 million) than if you open both boxes ($100).  This is clearly
a contradiction, and so the proposition is inconsistent.

Proof that the proposition is consistent.
If there is any possibility that the proposition could be true, then
the proposition must be consistent (since an inconsistent proposition
could never be true).  Now it is certainly possible that you open both
boxes and find $100 in box A and nothing in box B in which case the
proposition is validated.  Likewise, it is possible that you open
just box B and find $1 million dollars in in it (and later find out
that box A has $100 in it) which again validates the proposition.
Since the proposition can be validated, then it must be consistent.

"And so, I have proved that the proposition is both consistent and
not consistent, and that certainly is a paradox, isn't it?"

Smullyan precedes his new variation with the following curious statement.

"Some have proposed that this [Newcomb's] paradox proves that there cannot be
such a thing as a perfect predictor. I don't agree with that solution at all!
In fact, it is possible to reformulate the essential idea behind the
paradox leaving the predictor out entirely!  So here is my variant..."

Smullyan's variant is offered as a refutation of the claim that Newcomb's
paradox rules out a perfect predictor, instead I find that his variant
does the opposite!  That is, it provides the strongest and sharpest
argument yet that the Newcomb predictor cannot exist!  Suppose the
predictor exists.  This implies the given proposition which is both
consistent and inconsistent. This is a contradiction and hence, the
assumption that the predictor exists is false.  This argument has nothing
to do with the physical possibility/impossibility of the predictor nor
with free will/determinism.  It proves(?) on purely logical grounds that
the predictor cannot exist.

Does this finally resolve the paradox?  Is there some significant element
missing in Smullyan's version?  Other thoughts and comments?

-- Glenn C. Rhoads <rhoads@sonora.rutgers.edu>

If it is put this way, it reminds me of Feynman's story about Dennis
and the indestructible boxes to illustrate the principle of energy
conservation.

So, we could formulate the above as a conservation law: "The sum of
the number of open A boxes and the money in B boxes divided by $1
million minus the number of open B boxes is constant", i.e.,

        #(open A boxes) - #(open B boxes)
                + (money in B boxes)/($1000000) = const

In a world where this "law" holds, prediction is in fact possible, if
we know the initial value of the conserved quantity, which in our case
would be 0.

> Proof that the proposition is inconsistent.
>
> Regardless of whether box B contains nothing or $1 million dollars,
> there is more money in both boxes than there is in just box B.
> [...]

This is a well-hidden assumption which, of course, may be wrong.  In
light of the above, we could say that the quantity is initially in the
closedness of box B and transforms into money quantity when we free
some quantity by opening B.  So this assumption, while intuitive, may
not reflect "reality" in a world where such a strange quantity is
conserved.

Of course, if you want to read something more entertaining about
conservation, go and read Feynman's story about Dennis' boxes.

-- Rolf Strebel <strebel@dioxide.inf.ethz.ch>

I decided a little while ago that the main problem in Newcomb's paradox is
that the existence of _any_ predictor contradicts the idea that there is a
choice to be made, so the assumptions are inconsistent. The appeals that
such a predictor might be possible is completely irrelevant to the actual
root of the paradox. Anyways....

> Note that I have said nothing about any predictor! (A Newcomb
> predictor implies the proposition but the proposition is more general and
> makes no reference to a predictor.)

But there is still a predictor hiding in there. The proposition is,
itself, a predictor of an outcome. That might be the source of the
confusion.

> Proof that the proposition is inconsistent.
> [...]

The inconsistency comes from the confusion of a comparison across separate
instances of the choice and the outcome in any one particular case. The
above proposition predicts the outcome of any given choice. It says that,
in the set of possible outcomes, only a specific subset will occur. The
statement in the "proof", however, says that given a prior arrangement,
one choice will result in more money that the other -- it says that given
any outcome that includes "chose B", given a prior arrangement, there also
exists an outcome that results in more money and that it will be a "chose
both" outcome. The fact that this is true does not have any bearing on
whether or not such an outcome might actually is "valid".

Put another way, suppose we have a set of objects with two labels. The
first is "chose B"/"chose both". The second is the amount of money in box
B. Thus, the four elements in our set are

    B, 0
*** B, 1000000
*** both, 0
    both, 1000000

The proposition states that only the starred elements will occur in
reality. This does not invalidate them as theoretical outcomes, however.
The proof is merely the statement that given the outcomes with a certain
amount in box B, the "chose both" element will result in more money.
Whether or not this outcome will occur in reality is completely
irrelevant.

-- Aaron Bergman <abergman@premier1.net>

Did you quote exactly from Smullyan?? Because there is a fairly obvious
error in the reasoning that doesn't seem like an error he would ever make.
Let me explain.

[ Editor: Dreier quotes the set-up and the "proof that the proposition
is inconsistent" given above. ]

This one contains the fairly obvious mistake.

There are two propositions demonstrated on the assumption.

   P1 If you open both boxes, you will find more money than if you open just B.

   P2 If you open just box B, you will find more money than if you open both.

But these are not inconsistent!!! P1 and P2 are perfectly consistent.

The only confusing aspect is in the interpretation of the conditionals.
But suppose we take them to be *material* conditionals (i.e., the regular
old truth functional conditionals we meet in propositional logic all the
time). Then P1 is true if its antecedent is false; P2 is also true if its
antecedent is false. So P1 is true if you open just box B! And P2 is true
if you open both boxes! So obviously they are consistent.

Is there some other interpretation of the conditionals according to which
(i) they both really do follow from the premisses, and (ii) they are
inconsistent?

(Bold Hypothesis: no.)

-- Jamie Dreier <pl436000@brownvm.brown.edu>

[ Editor: Rhoads replied (his reply was not collected) that P1 is not
part of the proposition.  Dreier answered: ]

> P1 is *not* part of the proposition!

Part of what proposition?

I'd better explain more explicitly.

You gave a 'proof', from Smullyan, that the proposition is inconsistent.
(By 'the proposition' here we mean the bit I have set off from the margin
with the three occurrences of the symbol '#'. Let me call that proposition
'#' from now on.)

### "Either you will choose both boxes and there will be $100 in box A and
### nothing in box B, or you will choose just box B and there will be
### $1 million in it (and $100 in box A)."

The 'proof' proceeds by deriving two propositions from #. The two
propositions derived I have labeled 'P1' and 'P2'.

The 'proof' claims that P1 and P2 are inconsistent with each other. But
this is wrong. They are consistent with each other.

> The proposition has two clauses (connected with an or)
> 1) you will choose both boxes and there will be $100 in box A and nothing
>    in box B
> 2) you will choose just box B and there will be $1 million in it

Right, those are the two disjuncts of #.

> P2 is essentially the second clause, but P1 is not the first clause.

Fine. However, P1 is the proposition that you yourself stated in the
'proof' as one of two propositions that are alleged to be mutually
inconsistent. Here, I will extract your exact words:

"Therefore, if you open both boxes, you will find more money than if you
open just box B."

See, that sentence occurs in what you typed as a proof that the
proposition # is inconsistent.

> Hmmn, this argument seems to go through even with the above change.  But
> I'm still not convinced that you've shown this to be non-paradoxical.

Well, I am not claiming to show that this is non-paradoxical. I am
claiming to show only one thing: that we have seen *no proof* that the
proposition # is inconsistent. The proffered proof is flawed.

> Likewise, Smullyan also proves that the proposition is consistent.

The proposition *is* consistent. There is no proof that it is inconsistent.

> That it is his paradox, the proposition is both consistent and
> inconsistent.  If you can show that one of two proofs has a flaw
> in it, then we can put his variation out to pasture.

I *have* shown this! The 'proof' that it is inconsistent is incorrect!
That is what I showed.

>  But I'm not
> seeing any errors in his reasoning.  Exactly which step in the proof
> does not follow and why?

The step....
Well, he derives two propositions, P1 and P2, from the proposition #. The
derivations are fine. But they do not show that the proposition # is
inconsistent, because the two propositions derived are not inconsistent
with each other.

Ok?

-- Jamie Dreier <pl436000@brownvm.brown.edu>

Yes, I now see exactly what you are saying.

I agree that proposition # is not internally inconsistent with itself
(as Smullyan's misleadingly says).  What his inconsistency proof really
shows is that proposition # and proposition P1 are inconsistent with
each other.  Proposition # says that the maximum utility argument holds
while proposition P1 says that the dominance principle holds.  This is
indeed the essence of Newcomb's paradox.

That's all I have to add to your fine post.  Thanks for clearing up
the problem in Smullyan's description.

-- Glenn C. Rhoads <rhoads@paul.rutgers.edu>

In article <35DCD810.15A06B4E@mindspring.com>,
  RM Mentock <mentock@mindspring.com> wrote:
> gcrhoads@my-dejanews.com wrote:
> > This is at odds with the quote above which is why I was
> > struck by it.
>
> My point exactly.  In order for your interpretation to hold,
> he seems to contradict himself.  I'd rather try to interpret
> his words otherwise.

Smullyan's description of his own version is misleading at best.
He states a proposition.  Then proves that the proposition is both
consistent and inconsistent.  Initially I read this as saying the
proposition is *internally* consistent and *internally* inconsistent.
That reading is what I based my previous comment on (which you have
thankfully deleted).  As pointed out in another post, this is not
what Smullyan proves.  He doesn't prove that the proposition is
internally inconsistent (as he seems to say), rather he proves that
the proposition is inconsistent with the dominance principle.

While my initial take was incorrect, I still have to disagree with
Smullyan's claim that he has removed the predictor from the equation.
The essence of Newcomb's original version is the following.
A) A superpredictor makes correct predictions which implies that the maximum
utility argument holds.
B) The dominance argument holds.
The maximum utility argument and the dominance argument conflict.

Smullyan's proposition contains the prediction.  The proposition is of the
form "either X or Y will happen." X is "you will choose both boxes and
this was correctly predicted" (as indicated by there being nothing in box B).
Y is "you will choose box B and this was correctly predicted" (as indicated
by the $1 million in box B).  These correct predictions imply that the
maximum utility argument holds which conflicts with the dominance argument.

I don't view Smullyan's version as removing the predictor, but rather as
subtly hiding the fact that there is a predictor lurking about.  I was
initially hoping that Smullyan's version might shed some light on the problem
but now I'm thinking that it does nothing; that is, trying to resolve it
is just as infuriating as before for all the same reasons.

> My version is that he is just pointing out that with the
> predictor or no, the *wording* of the problem is inherently
> paradoxical.  Thus, of course, sticking in the predictor is
> still going to leave you with the paradox.  So, like I said,
> he is avoiding implying anything about the existence or non-
> existence of a predictor.
>
> Otherwise, we have a simple logic puzzle that can prove whether
> we have free-will or not.  C'mon.

I still think the predictor cannot exist (though not for the
reason I initially stated when I was misled by Smullyan's
description).

The impossibility of a predictor in no way settles the free will/
determinism debate.  Predictability does not necessarily hold
even in a deterministic world; they are not the same concepts.
In a deterministic world, it might be the case that the future
cannot be predicted in any manner faster than allowing the
Universe to unfold itself and see what happens.  Another way
that predictability could break down in a deterministic world
comes from the "butterfly effect."  We can't know that state of
the world with infinite precision so there will always be some
small roundoff error.  These small errors could blow up very quickly
making huge differences in observed phenomenon so that prediction
beyond a very short time frame becomes impossible.  This is really
by the way.

I'm not claiming that all predictions are impossible, only that a
certain type of prediction is impossible.  Specifically, those
predictions where the act of making a prediction interacts causally
with the event being predicted.  I wrote another long post about this
so I won't repeat that here.

-- Glenn Rhoads <rhoads@paul.rutgers.edu>