Re: Halting Problem for Humans

abo says...

Daryl McCullough wrote:

Okay, that's the background. Now here are the questions:

Peter is asked: Will Daryl answer "yes"?
Daryl is asked: Will Peter answer "no"?
....

So, it is clearly impossible for both Daryl and Peter to
have perfect ability to predict the future behavior of the
other.

This formulation seems to be completely about the limitations
of knowledge and reasoning ability. There is nothing paradoxical
going on. And in particular, there doesn't seem to be the
escape clause: Daryl (or Peter) knows the answer, but he can't
say it. If he knows the answer, there is nothing at all preventing
him from saying it. One player's answer has no causal effect on
the other player.

I'm not sure I agree. As you state about the one person game, there
may be a difference in what either believes and what is said.

There *could* be a difference, but part of the assumption
here is that Peter and Daryl are both attempting to answer
correctly, and that they know that the other is attempting
to correctly, etc.

In such a circumstance, there is no reason for there to
be a difference between what one believes and what one
says. It can never be to one's advantage to refrain from
saying what one believes.

So it seems to be possible, contra your claims of impossibility,
that both Daryl and Peter do have the ability to predict the
future behaviour of the other. If, for instance, both are told
that Daryl will be rewarded if he answers "yes" and Peter will
be rewarded if he answers "no", then Daryl will answer "yes"
and Peter will answer "no", regardless of the
sense of the answer.

But they *weren't* told that. I stipulated that each is
attempting to answer correctly. If you want to spell out
rewards and punishments, we can say that each is rewarded
for making a correct answer and punished for making an
incorrect answer. Making no answer produces neither
reward nor punishment.

Sure there are situations where Daryl won't know what Peter will answer
- the simplest situation would be that Daryl doesn't know the question
Peter is answering and doesn't know anything about Peter, in which case
he has no grounds for believing anything about Peter's answer;

But I stipulated that each *does* know the question given to the
other, and that they know as much as they like about the other's
motivations, intelligence, reasoning ability, etc.

so he cannot know. But there's nothing special here. So, if your
claim is simply about the limitations of knowledge, then there
are simpler examples.

An example in which someone lacks a certain piece of information
is *not* an example of a limitation of knowledge, because a lack
of information can be remedied by giving the person more knowledge.
In the scenario as I have stipulated it, there *is* no remedy.
No amount of knowledge given to Peter and Daryl will be enough.

--
Daryl McCullough
Ithaca, NY

.

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