Re: Penrose vs the Robot

Rupert says...

>Daryl McCullough wrote:

>The reason Penrose can't star the sentence "Penrose will never star
>this sentence" is because self-reference is involved; Penrose's
>behaviour involves making a prediction about Penrose's behaviour.

Yes, both Penrose' statement and the robot's statement are
self-referential, in exactly the same way. Neither Penrose
nor the robot is able to follow the rule that he star
each and every sentence that he unassailably believes,
and for exactly the same reason: it is inconsistent to
do so.

>In the case of the robot, the sentence is merely a statement about
>numbers. So it looks like we need a different sort of explanation of
>why the robot can't star the sentence.

It doesn't look that way to me.

Let G_R = "The robot will never star this sentence."
Let G_P = "Penrose will never star this sentence."

The explanation for why Penrose cannot star G_P is *exactly*
the same for the explanation for why the robot cannot star
G_R. In both cases, the person or robot can reason: If I
star that sentence, I will have made the sentence false.
Therefore, I can't star the sentence and follow the rule
that I only star sentences that I unassailably believe.

The exact same reasoning works in both cases. Now, let's
take a look at a corresponding arithmetical statement.
Suppose we hand Penrose and the robot an arithmetical
statement G_arith and we tell Penrose

G_arith <-> Penrose will star G_arith

and we tell the robot

G_arith <-> The robot will star G_arith

Based on this information, can Penrose star G_arith,
or not? Can the robot? I don't see any difference in
principle between the situation for the robot and the
situation for Penrose.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

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