At 09:53 29/07/04 -0700, Hal Finney wrote:
Tell me again where I am going wrong.
OK.
Consider each of these examples:
117. q
...
191. Bp
...
207. p - q
Now, we will say that the machines believes something if it is one of
its theorems, right? So we can say that the machine believes q, it
believes Bp, and it believes p-q, right? We could equivalently say
it believes q is true, etc., but that is redundant. If it writes x
down as a theorem, we will say it believes x, which is shorthand for
saying that it believes x is true. The is true part has no real
meaning and does not seem helpful.
We also have this shorthand Bx to mean the machine believes x. So we
(not the machine, but us, you and I!) can also write, Bq, BBp and B(p-q),
and all of these are true statements, right?
Until here you are right.
The problem arises when we start to use this same letter B in the
machine's theorems. It is easy to slide back and forth between the
machine's B and our B. But there is no a priori reason to assume that
they are the same. That is something that has to be justified.
The problem arises here, indeed. And I disagree with what you are saying.
We take, by personal choice, the total NAIVE STANCE toward the machine.
That means that by definition Bx means for us, and for the machine
that the machine believes x. Exactly like when the machine believes
(p - q), it means, like for us (-p v q), that p is false or q is true.
If the machine believes Bp, it just means that the machine believes it
will believe p. In case the machine print Bp and then never print p, it
will means (for us) that the machine has a false belief.
Focus on 207. p-q for a moment. We know that, according to the machine's
rules, this theorem means that if it ever writes down p as a theorem,
it will write down q.
Take care. p-q is also true in case p is false (by propositional logic),
it could means that the machine believes -p.
Let us look at the following example: with f denoting any contradiction
(that is f can be seen as an abbreviation of (p -p))
The machine obeys to classical propositional calculus (CPC). Thus
the machine believes all proposition f-p (with any p).
And what you said is correct, that will entails that if the machine
believes f, then it will believe p, and then, if we add the assumption
that the machine is consistent, it will never believes f, that is Bf is false
for the machine, and so we know then that [Bf - what-you want]
will always be true (because we know also the CPC.
Therefore it is true that Bp-Bq.
Yes. Because the machine obeys CPC.
This is simply
another way of saying the same thing! Bp means that p is a theorem,
by definition of the letter B, in the real world. And similarly Bq
means that q is a theorem. Given that p-q is a theorem, then if p is
a theorem, so is q. Therefore it is true that B(p-q) - (Bp - Bq).
This is not a theorem of the machine, it is a truth in the real world.
Right.
What I want to say is that 207. p-q means Bp - Bq. It means that if
the machine ever derives p, it will derive q. This is a true statement
about the operations of the machine. It is not a theorem of the machine.
OK, but what is a theorem by the machine, is p-q, independently
that *we* know this entails Bp - Bq. So your expression
p-q means Bp - Bq, is misleading. The naive stance is that the
machine believes p-q. (or, if we want to insist, that the machine
believes p-q is true, but as you said this does not add anything.
Actually p-q could be false, and the machine could have false beliefs,
in which case both p-q and (p-q) is true are false.
When we talk about what something means, I think it has to be what it
means to us, not what it means to the machine.
Why? If you talk with any platonist, you should better keep the naive
stance. If not it is like you suspect some problem with the platonist
brain, and you will no more talk about the same thing with him.
It will be much harder for you to show him that he is doing a mistake
for exemple.
When the machine writes
117.q, it doesn't mean anything to the machine.
Why? With the naive stance, it means the machine believes q.
For perhaps unknown reason to us the machine believes q. Perhaps
the machine has make a visit to the KK island; and some native
told her that if I am a knight then q. Or more simply the native said
1+1 = 2, and then latter said q.
I have said that the machine believes all classical tautologies, and that
if the machine believes X, and then X-Y, then the machine believes Y.
But I NEVERsaid that the machine believes ONLY the tautologies.
The machine can have its personal life and got some personal
non logical beliefs (non tautological belief) on its own. Like in some
of the problem I gave where machine develops beliefs on the knight/kanve
nature of the natives.
To us it means that the
machine believes q, or that the machine believes q is true.
This is right, but keep in mind that it means something for the machine.
It means q, or q is
