[FOM] The Lucas-Penrose Fallacy

[Bob Hadley]

Good grief!

It appears that many who have been commenting on this debate
still Mistakenly believe that it is logically impossible for 
some deterministic Turing machine (whose abilities include derivational
competence within PA or Q ) to determine the truth of a godel-sentence
for a formal deductive system that is equivalent to that machine
(or whose deductive closure includes the sentences that the machine
generates).

However, both Chihara and myself pointed out more than 15 years ago
that in this debate, is necessary to distinguish between a formal model of 
the machine *simpliciter* (i.e. the machine without its input being 
supplied) and a formal model of the machine with its input supplied.
When a Turing machine is supplied with an input set, a Larger formal
system is required to model the machine than when we are considering
just the machine simpliciter.

Both Lucas and Penrose, in their reduction arguments, assume they they 
are supplied with an input set which is a description of some formal
model that is purported to be equivalent to themselves.  Supposing each
of them to be a machine, any reasoning they engage in about this
"input set" would need to be modelled in a larger formal system than
the system they have received as input.  Ergo, any success they 
may obtain in "meta-proving" some godel-sentence for the system supplied
as input in no way establishes the conclusion of their putative
reductio arguments.  

There is a lot more to be said in relation to this issue, including
some apparently parodixical results.   If you would like to
see my 1987 paper or my Current paper on this, just let me know.

Bob Hadley
hadley at sfu.ca