Re: Penrose vs the Robot


Daryl McCullough wrote:
> Rupert says...
>
> >Daryl McCullough wrote:
> >> Penrose makes a big point of saying that F is only assumed
> >> to be sound for arithmetical sentences (more specifically,
> >> for pi-1 sentences).
>
> >Does he? Can you give me a quote supporting that?
>
> You were the one who brought up the distinction between
> P-sentences and other sentences.
>
> >> There is no reason to believe F is sound in this broader sense.
> >> Penrose doesn't claim it---he only claims to be sound in the more
> >> limited sense (anything he believes unassailably about pi-1
> >> arithmetical sentences is true).
>
> >Well, that's not the way I interpret Penrose.
>
> Then why does he make the distinction between P-sentences
> and other sentences?
>

Because he thinks you have to restrict the range of sentences the
belief predicate applies to. Because the belief predicate for the
sentences in a particular language is not itself definable in that
language.

> >I interpret him as claiming F is sound in this broader sense.
>
> Well, if F believes itself to be sound in this broader sense,
> then it is inconsistent.
>

Yes.

> >> Well, what you said was that your system was scrapping
> >> the axiom
> >>
> >> |- B(A) -> B(B(A))
> >>
> >> That seems very implausible to me.
> >
> >The motivation is, the belief predicate for sentences in a particular
> >language might not be expressible in that language.
>
> But that's not a very plausible interpretation for Penrose'
> unassailable beliefs, because Penrose *has* a way to express
> "A is an unassailable belief"; namely, by using the phrase
> "unassailable belief".
>
> If you consider the "Godel sentence for Penrose"
>
> G <-> Penrose does not unassailably believe G
>
> then Penrose cannot consistently maintain his soundness
> with respect to sentences such as G.
>

This sentence leads to paradox. That's a reason for thinking that for
the purposes of precise mathematical argument Penrose shouldn't adopt
features of our ordinary language like having an unassailable belief
predicate which can take as arguments sentences containing itself.

> --
> Daryl McCullough
> Ithaca, NY

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