Re: Halting Problem for Humans


They are completely well-defined questions. Either Peter's response
will
be "no", or it won't. Either Daryl's response will be "no" or it won't.
There is no third possibility.

Yes, that's right. I should have reflected on it more carefully.
Please, consider the following.

If you want to prove anything about limits of human knowledge, you have
to suppose your proof goes through for the the best possible human
thinkers and knowers.

So suppose Peter and Daryl are ideal thinkers and knowers; then each of
them knows they are. We do not need to suppose they are honest and
observe the rules of the game when answering; it suffices if each of
them knows that the answer of the other will be in some way determined
by his attempt to know. The situation that they are honest and they
both know they both are is a particular case of my approach.

Each of them will try to figure out what the other will think, but
since each of them knows the other will do the corresponding, each of
them will be compelled to consider his own attempt to answer while
performing that attempt.

So, ideal thinkers would find themselves trapped in circularity. And
knowing the other is also trapped won't help them. None of them will
ever work out what the other is going to say because it would imply
taking into account his own thinking in order to decide what his own
thinking should be.

Exactly this I said was phenomenologically impossible. So, what I
thought worked against your proposal actually works for it.

Now suppose one (or both) of our ideal thinkers does not observe the
rules of the game and answers even if he doesn't know the correct
answer. Well, that won't change the fact that no one knows what the
other will answer. Fortuitous success is here of no interest, I think.

By splitting into a two isolated person game the single person game you
have produced a beautiful piece where the 'escape' clause seems out of
order. It seems there is a possible situation in which ideal knowers
won't be able to know. I do think your example shows something
interesting.

I believe circularity lies in the core of the halting problem, for
humans and for machines. But I think it is quite different for humans;
for humans it is a question of 'intentional circularity': for humans
the relevant fact is that no intentional act can be its own
intentional object. E. g. when I'm thinking that it will rain I cannot
be thinking at the same time that I'm thinking it will rain. It is easy
to see that if an intentional act were its own intentional object, that
object would be of infinite complexity.

I think you have shown that an ideal thinker is not just incapable of
always having his own thought as an object but also incapable of always
having as an object the thought of other ideal thinker: no one can set
his thought at a higher level than the other's, as no one can set his
thought at a higher level than itself.

Now, there is neither logical nor scientifical proof that human
behavior is only determined by physical causality and that mental
states are causally inert. This can only be a philosophical (or
ideological) assumption.

Suppose the ideal attempt of our ideal thinkers could be represented as
a physical process completely determined by physical causes. Then, if
each of them is provided with a computer that has previously collected
sufficient information about the other's brain (and a relevant part of
its environment) and can calculate physical causality, each of them
should be able to know what the other would say.

Nevertheless, this seems impossible: what would prevent them from
answering what they know?

Regards





Daryl McCullough wrote:
LauLuna says...

There is a crucial difference between the case you propose and the
halting problem. The halting problem is a well defined question while
your questions to Peter and Daryl are not.

Not really. Imagine that Peter and Daryl are both robots,
and their behavior is completely specified by a computer
programs, and that both Peter and Daryl have access to
each other's programs. In that case, it is exactly analogous
to the halting problem.

In the case of real humans, it only becomes fuzzier because
we lack perfect knowledge about the mechanisms by which each
of us makes decisions.

Each of them is circularly defined. Trying to complete the text of the
question to Peter (which I suppose you think contextually completed)

The question was already complete. Peter was asked whether
Daryl's next utterance will be "yes". Daryl was asked whether
Peter's next utterance will be "no". That's a complete question.

we would obtain:

'Will Daryl answer 'yes' to the question whether you will answer 'no'
to the question whether Daryl will answer 'yes' to the question
whether...?'

Yes, you can expand any question into an equivalent longer question.
The point is, there is no ambiguity about what each is asked to do.

'Will Daryl answer 'yes' to the question whether you will answer 'no'
to this question?'

Now the self-reference only makes the circularity still more evident.

What difference does it make whether it is self-referential or not?
There is no ambiguity in what each is being asked to do.

If they were completely described by computer programs, then
it is clear what is being asked:

Daryl is being asked: Will Peter's response to string S be "no"?
Peter is being asked: Will Daryl's response to string S be "yes"?

where string S is just the description of the game. There is no
uncertainty or ambiguity here, other than the fact that (in the
case of humans) we each only have partial knowledge of the behavior
of others.

The halting problem contains no circularity in its definition because
it poses a purely syntactical or mechanical question with no semantic
or intentional dimension.

There is no semantic dimension here, either. Human beings are physical
systems that work through natural laws. If we had perfect understanding
of those laws (and of the initial conditions), then we would know what
Peter's response will be (or else, his response could be nondeterministic,
in which case we could make probabilistic predictions).

In contrast, and using phenomenological terms, we can say that no
intentional act of thinking can be its own intentional object; call
'PNS' that proposition; so according to PNS, it is not the case that
any intentional act is a possible object for any thinking subject.

I'm not sure what you are talking about. The question of what
Peter's response to the game will be is a purely physical question.
There is a causal chain from the act of Peter being told the rules
to Peter producing certain sounds from his mouth. We only have partial
knowledge of that causal chain, but I don't see how in principle there
is any difference between asking a question about the response of
a human being and asking a question about the behavior of a computer
program.

This is why your questions to Peter and Daryl cannot be well defined
questions for none of them:

They are completely well-defined questions. Either Peter's response will
be "no", or it won't. Either Daryl's response will be "no" or it won't.
There is no third possibility. (Well, actually, the third possibility
is that they will not answer, or will answer using an illegal reply
such as "I don't know"---but in that case, it is false that the answer
is "yes" and it is also false that the answer is "no").

trying to answer them correctly would force them to consider their
own attempt to answer and this would be phenomenologically impossible.

No, that's not correct. For Daryl to answer the question, he only
needs to know how *Peter's* mind works. It's not self-referential
in that respect. Daryl only needs to know what Peter *believes*.
Those beliefs may not be correct.

Maybe we've met here an essential difference between thinking behaviors
and machines: only the latter are always possible objects for the
former.

I think that's incorrect.

--
Daryl McCullough
Ithaca, NY

.

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