Re: Penrose vs the Robot

Rupert says...

>Daryl McCullough wrote:
>> Rupert says...
>>
>> >Penrose's suggestion was that the robot could prefix a star to those
>> >sentences it believed unassailably. Are you suggesting there might be
>> >things the robot unassailably believes but doesn't prefix a star to?
>>
>> Yes, I'm suggesting that. If I instructed Penrose to prefix a star
>> to the sentences he believes unassailably, and then asked him about
>> the following sentence:
>>
>> Penrose will never prefix a star to this sentence.
>>
>> he would be unable to comply with the rules. The same thing is true
>> of the robot when he is confronted with the arithmetical sentence
>> G <-> the robot will never prefix a star to G.
>
>This has some plausibility when applied to the sentence "Penrose will
>never prefix a star to this sentence." But the robot's sentence will be
>equivalent to the unsolvability of a certain Diophantine equation.
>Surely the robot can decide whether or not it unassailably believes
>that, and always prefix a star if it does.

Why can't Penrose decide whether he believes the statement

Penrose will never prefix a star to this sentence.

and then prefix a star if he does? The answer to both questions
is the same: There is *no* way for Penrose to comply with the
rule that he prefix a star to each sentence that he unassailably
believes, and there is no way for the robot to comply with the
rule that he prefix a star to each Diophantine equation that
he unassailably believes.

The question is whether *Penrose* can star that statement.

The original question was whether Penrose' brain can be
functionally equivalent to some computer program. To the
extent that Penrose is doubtful that this is possible, the
functionally equivalent computer program would *also* have
to be doubtful. How can that doubt be maintained?

Well, to make things symmetrical between Penrose and computer
program, lets suppose that there are *two* identical robots,
a red one and a blue one. The red one is controlled by Penrose
from within a sealed booth, and the blue one is controlled by a
computer that is programmed to be a simulation of Penrose
inside his booth. Both Penrose and his simulacrum believe
themselves to be the real Penrose.

Now, the question is: Is there some arithmetical sentence
that the red robot can star, but the blue one cannot? I don't
see how Penrose could construct such a sentence. Let's suppose
that both Penrose and the computer simulation are told the
computer program used by the simulation. So they each
can formulate the r.e. theory T = the set of all arithmetical
sentences Phi such that the robot controlled by the simulation
will eventually prefix by a star. Let They can each construct
an arithmetical statement G such that

G <-> the robot controlled by the simulation will never
prefix a star to sentence G

But can Penrose claim that G is unassailably true? I don't
see how. If T is a consistent extention of PA, then G is
true, but how would Penrose figure out that G was a consistent
extention of PA? Penrose claims to know that his unassailable
beliefs are sound, but how could he ever know whether T was
exactly equal to his unassailable beliefs?

I don't see how Penrose proposes to be able to star statements
that the computer simulation is incapable of starring.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Penrose vs the Robot
    ... I could also prefix a star to those statements of mine I think I ... > Unassailabality of beliefs should imply some sort of soundness ... How about for the hypothetical robot? ...
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  • Re: Penrose vs the Robot
    ... >>Penrose's suggestion was that the robot could prefix a star to those ... Are you suggesting there might be ...
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  • Re: Penrose vs the Robot
    ... >The reason Penrose can't star the sentence "Penrose will never star ... >In the case of the robot, the sentence is merely a statement about ...
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  • Re: Penrose vs the Robot
    ... >Penrose's suggestion was that the robot could prefix a star to those ... Yes, I'm suggesting that. ...
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  • Re: Penrose vs the Robot
    ... >> The robot will never star this sentence. ... it comes to Penrose believing his own soundness. ... Penrose will never star this sentence. ... Daryl McCullough ...
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