Re: Penrose vs the Robot

Rupert wrote: 
Aatu Koskensilta wrote:

Sure. I could also prefix a star to those statements of mine I think I
believe "unassailably". I've no reason to suppose the set of the
statements I'll ever prefix with a star will be consistent.

Unassailabality of beliefs should imply some sort of soundness
condition, e.g. consistenty, 1-consistency, truth or some such in order
for Penrose's argument to go trough, which putting stars before
sentences in no way guarantees. Presumably there is in some sense an
infinite subset of sentences I would star (what ever this means exactly)
all of which are true - perhaps this is the set of sentences I
"unassailably know". I've no idea whether *this* set is (would be?) r.e.
or not.

I have grave doubts about the "set of all arithmetical truths I can
unassailably know" being definite enough a thing to say anything about
its recursion theoretic properties.


How about for the hypothetical robot? Penrose tries to argue it's a
definite r.e. set. Do you think his arguments succeed?

I'll be happy to grant - for the sake of argument, at least - that the set of sentences prefixed by a star by the robot is r.e. But as in case of humans, e.g. myself, I see no reason to suppose a sentence being prefixed by a star to be necessarily true, or the set of them 1-consistent or whatever soundness condition Penrose needs. This, obviously, means that the set need not be - in case of me or the robot - the set of arithmetical statements "unassailably believed" or "unassailably known". We'd have to have some sort of a precise definition of "unassailable belief" before we can say anything about the set of sentences "unassailably believed", e.g. whether there is any reason to suppose I can list just those sentences I "unassailably believe".

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


What reason do we have to think that the set (or the deductive closure
of the set) of sentences the robot prefixes with stars is consistent?


Penrose assumes (for the sake of reductio ad absurdum) that it's a
reasonable model of the human mind, and he thinks this entails at least
pi-1-soundness.

The set of sentences a robot "unssailably believes" might be - in case it extensionally models my "unssailable believing capability" -, by definition, sound or 1-consistent or whatever. But why should the robot have the capability of starring exactly the sentences in this set? What reason do we have to suppose humans have the capability to star exactly those sentences they unassailably believe? As I said in the previous post (quoted above), "I unassailably believe x" seems to have some of the force of some kind of a partial truth predicate, and I for one wouldn't claim to have the ability of listing the sentences to which it applies. Knowledge is classically defined as "true belief" - and luckily we needn't get into the innumerable ramifications found in books on episteomology here to make my point! - which is similar to "unassailable belief" in the following respect: if I know P, then P is true, and similarly if I "unassailably believe P", then P is true (or at least, if I "unassailably believe P", I won't "unassailably belive" any Q, s.t. P&Q is contradictory, or whatever the soundness condition we need is). I am in no position to list things I know - I am probably in error about a great deal of things - , although obviously I can list a number of things I *think* I know. What I know is true, but from my thinking I know something the something does not follow. The same applies to what I "unassailably believe".

Furthermore, some sentences, such as "1+1=2" I certainly "unassailably" believe, but I'd be hard pressed to say whether I "unassailably" believe "Cons(ZFC+a weakly compact cardinal exists)". I certainly *believe* the latter arithmetical proposition, but am unsure enough as to what "unassailably" means exactly to say whether or not I should star it. However, this uncertainty of mine does nothing to alter the presumably objective fact of my either unassailably believing (or not) the proposition.

I babble like this just to make it clear that I really think I'm in no position to star exactly those sentences I "unassailably believe" - and not merely unconvinced by Penrose's arguments - and we shouldn't expect a robot to be able do so either.

Of course, I needn't have written the above, since I think there is no such a set as "the set of arithmetical sentences A unassailably believes" for either A=robot or A=me - making debate about its r.e.-ness rather pointless from my point of view.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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? ...
    (sci.logic)
  • Re: Penrose vs the Robot
    ... *behaviors* of the robot. ... We don't have a foolproof means of determining what "unassailable beliefs" are implied by that behavior. ... statements I'll ever prefix with a star will be consistent. ... Unassailabality of beliefs should imply some sort of soundness ...
    (sci.logic)
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