Penrose's reply to Chalmers

In "Minds, Machines, and Mathematics", Chalmers criticizes an argument
presented by Roger Penrose in "Shadows of the Mind".

http://psyche.cs.monash.edu.au/v2/psyche-2-09-chalmers.html

He paraphrases the argument as follows:


'(1) Assume my reasoning powers are captured by some formal system F
(to put this more briefly, "I am F"). Consider the class of statements
I can know to be true, given this assumption.


(2) Given that I know that I am F, I know that F is sound (as I know
that I am sound). Indeed, I know that the larger system F' is sound,
where F' is F supplemented by the further assumption "I am F".
(Supplementing a sound system with a true statement yields a sound
system.)


(3) So I know that G(F') is true, where this is the Gödel sentence of
the system F'.


(4) But F' could not see that G(F') is true (by Gödel's theorem).


(5) By assumption, however, I am now effectively equivalent to F'.
After all, I am F supplemented by the knowledge that I am F.


(6) This is a contradiction, so the initial assumption must be false,
and F must not have captured my powers of reasoning after all.


(7) The conclusion generalizes: my reasoning powers cannot be captured
by any formal system.'


Chalmers contends that the flaw in this argument lies in a
misidentification of the source of the contradiction. He says the
source of the trouble is not the hypothesis that I know I am F, but the
assumption that I know that I am sound.


He argues for this conclusion from some general considerations about
any system that is capable of reasoning about its own beliefs. Let
"|-S" mean that the system unassailably believes S, where S is a
sentence in a language which includes at least the first-order language
of arithmetic, and a predicate symbol B, where B("A") means that the
system believes A. (The quotes around A indicate that we are talking
about A's Gödel number rather than A itself; by a slight abuse of
notation for convenience Chalmers omits them, and so shall I from now
on).


He then shows how the following four inference rules/axioms, together
with those of Peano artihmetic, yield a contradiction via Gödel-type
reasoning:


(1) If |- A, then |- B(A).
(2) |- B(A_1) & B(A_1 -> A2) -> B(A2)
(3) |- B(A) -> B(B(A))
(4) |- not B(false)


This argument, Chalmers claims, would apply to any belief system
capable of discoursing about its own beliefs, whether computational or
not. So he says the source of the contradiction is not that the belief
system is computational, as Penrose claims, but (4), i.e. the
assumption that the belief system unassailably believes in its own
soundness.


Penrose replies in Sec. 3.8 of


http://psyche.cs.monash.edu.au/v2/psyche-2-23-penrose.html


that the contradiction can be averted by restricting the output of B to
P-sentences (pi-1 sentences), while still allowing there to be
"internal musings" about more complex sentences.

He writes:

'Is there anything wrong in B "believing in the soundness of B"?
Nothing whatever, if we interpret this in the right way. The important
thing is that B is allowed only to make assertions about P-sentences.
It can use whatever procedures it likes in its internal musings, but
all its outputs must be assertions as to the validity of particular
P-sentences. If we apply the diagonal procedure that Chalmers and
McCullough refer to, then we get something which is not a P-sentence,
and is accordingly not allowed to be part of this belief system's
output.'

It's hard to be clear about exactly what Penrose is proposing here. He
might, for example, be proposing that we restrict axioms (1) and (2) to
the cases where the arguments of the predicate B are all P-sentences,
and abandon axiom (3). But in that case it's not entirely clear how his
own Gödelian argument against the extension of B being recursively
enumerable goes through. Is there a way to get Penrose's argument to go
through while blocking Chalmers' paradox?

.

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