Re: Penrose vs the Robot

abo says...

>Suppose (as Rupert said in another part of this thread) that, given
>a sentence, a particular robot can star or not star it. Given a
>robot, is the set of all starrable sentences r.e.? I don't think
>you can assume this to be the case, if the robot is not a closed
>system, i.e. it receives input and the input
>can help to determine whether a sentence is starred or not.

It depends on what you mean. Let T be the set of sentences Phi such
that there exists an input sequence S that will convince the robot
to star Phi. Then T is an r.e. set. But T will include some
counterfactual cases, that is, sentences Phi such that in some
possible world (different from this one) the robot will star Phi.
The set T of sentences Phi such that the robot will eventually star
Phi in *this* world may not be r.e. (if the robot lives forever).

>Input (for simplicity, think of this as a video camera which gets
>digitized) is non-computable; it's not r.e. Or at least the claim
>that it is r.e. is a huge assumption and not to be taken lightly.

Sure, the robot (if it lived forever) could receive a non-r.e.
sequence of inputs. But if we take the union of the robot's
star sentences over all possible input sets, the result will
be r.e.

Of course, this union theory could be contradictory, I suppose.
Maybe in one possible world, I convince the robot to star Phi,
and in a different possible world, I convince it to star not-Phi.
If that's the case, perhaps neither should be considered an
"unassailable belief", if it is possible to be convinced of
its negation.

>Two robots, with the same program but put in the same room, may reach
>different conclusions about what is starrable or not.

That's true, but would they reach *inconsistent* conclusions?
If so, then one of the two conclusions was incorrect. If not,
then we can take the union, and have a more powerful, yet r.e.
theory.

>Now I guess one could say that the robot's program should be allowed to
>star a sentence, indicating unassailable belief, only if it does *not*
>depend on input. But this gets tricky, since concept formation (the
>concept of "natural number") does I think depend on input.

Yes, I guess in two different possible worlds, the robot may
*interpret* the same strings in different (and contradictory)
ways. But lets suppose that we start off with the robot knowing
and understanding the language of Peano Arithmetic. So any
divergence based on inputs would never cause the robot to
have a nonstandard interpretation of the basic formulas
of arithmetic. Then in that case, I think we could suppose
that the union theory should be consistent.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Penrose vs the Robot
    ... >> star that sentence, I will have made the sentence false. ... "The robot will never star this sentence." ... What's Penrose going to do then? ... Daryl McCullough ...
    (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
    ... >>Daryl McCullough wrote: ... >>> the same for the explanation for why the robot cannot star ... >>> star that sentence, I will have made the sentence false. ... What's Penrose going to do then? ...
    (sci.logic)
  • 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 ...
    (sci.logic)