Re: Penrose vs the Robot


> Rupert says...
> Are you suggesting there might be
> >things the robot unassailably believes but doesn't prefix a star to?

Daryl McCullough wrote:
> 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.

Is there a standard name for this?
Something like "the blindspot paradox"?
This argument can be universally generalized, you know.
"Penrose" or "the Robot" might as well be an arbitrary name
not otherwise occurring in the axioms governing this discourse,
or a variable name. In that case, universal generalization (as an
inference
rule) is applicable to/after/following this proof, and thus we prove
that for ALL
reasoners r, r cannot prefix a star to the sentence
name(r) concatenated with "cannot prefix a star to this sentence."
Since we have just proved this, WE ALL believe it unassailably.
But as a proven universal generalization, it has, as provable
consequences,
each of its instances, including an instance for each of us.
This is not paradoxical YET because while the universal generalization
cannot be TRUE without all its instances being true, it (arguably) CAN
be BELIEVED without all of its instances being believed, if the
statement
itself is not ABOUT belief. But what if it is? What happens when you
replace "prefix a star to" by
"believe" x "unnassailably because" x "has been proven to him"?

None of the individual instances is paradoxical because the rest of the
community
laughs in derision at poor r's inability to believe/assert/whatever
something so
obviously true: "Sounds like a PERSONAL problem to me ;~>". The fact
that
the something is about r HIMSELF and that he therefore ought to have
BETTER
rather than weaker knowledge (than everybody else) of it only ADDS to
r's plight
and to everybody else's smug sense of superiority. But of course they
do not
REALLY feel superior; each of them knows he has his own blindspot as
well.
Thus we all settle down to an (entirely MISguided) ACCEPTANCE of the
general situation, in a "laugh and let laugh" mode. This is NO
laughing matter.
Each individual instance is problematic for and only for the individual
it is about,
but the universal generalization HAS THE EXACT SAME proof (simply
followed
by an application of universal generalization) and IT is UNIVERSALLY
problematic.
In fact, worse, if believing a UG entails believing every specific
instance of
it that's actually presented to you, it's paradoxical.

.

Relevant Pages

  • Re: Penrose vs the Robot
    ... >>>Penrose's suggestion was that the robot could prefix a star to those ... >> Penrose will never prefix a star to this sentence. ...
    (sci.logic)
  • 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? ...
    (sci.logic)
  • Re: Penrose vs the Robot
    ... >>Penrose's suggestion was that the robot could prefix a star to those ... Are you suggesting there might be ...
    (sci.logic)
  • Re: Penrose vs the Robot
    ... >Penrose's suggestion was that the robot could prefix a star to those ... Yes, I'm suggesting that. ...
    (sci.logic)