[FOM] The Lucas Penrose Thesis

[Hartley Slater]

Panu Raatikainen previously wrote:

>Slater and others adhere 2 [A human can, but a machine cannot nail
>down the intended intepretation].
>Maybe they're right, but much more needs to be said in its
>defense. And if we can tell a more specified story on how exactly a human
>can do that, it may turn out that there is nothing there that a more
>sophisticated machine could not imitate.

but then has to admit:

>one way to view the issue is to note that we can fix the
>intended interpretation from the perspective of model theory, which
>assumes some amount of set theory. But so can a machine which is not
>restricted to the language of arithmetic but can use the language of set
>theory.
>Of course, FO set theory has its own non-standard models...

I still do not think he appreciates the dimension of the problem,
anymore than Bob Hadley does, even though the latter realises:

>the generation of a sentence is not equivalent to
>proving its truth,

The point is that it is *not* the sentence which has the property of
truth; at the very least it is the sentence when interpreted a given
way.  But the problem with that is that standard twentieth century
logic has had no way to write the sentence when interpreted a given
way as a referential phrase referring to an object which might be the
subject of a truth predication. What is true, in the Goedel case, for
instance, is not '(x)Fx' but that all natural numbers are F.  Humans
certainly can, like machines, utter sentences like '(x)Fx' -
sentences which are then put in quotes - but they also do something a
machine cannot, namely use sentences like '(x)Fx' to state things
about models (in this case the standard model) - the sentences are
then not in quotes. In terms of the operation of a machine it would
have to not only utter a sentence, but *mean by it* one thing rather
than another.  But it lacks any capacity to mean anything.

Forgetting this is exactly why, for instance, people get into such a
tangle with The Liar Paradox. They think that the Liar sentence
(let's call it 't'), on the self-referential interpretation is
paradoxically both true and false.  But what one is involved with, if
one chooses the self-referential interpretation is not a syntactic
identity in direct speech,
t = 't is not true',
but a statement about content in indirect speech, namely
t says that t is not true.
This is clearly true only on a certain semantic interpretation, and
so it is not just an extensional remark about the syntactic object 't
is not true'.  The distinction makes it plain that what is true or
false is not the bare sentence 't is not true', but the statement
made when it is used a certain way, namely to say that t is not true.
The inability of the logical tradition to represent such a
propositional referring phrase as 'that t is not true' has made it
seem that what is true or false on the given interpretation is still
the (mentioned) sentence, but only the sentence in use, preceded by
'that', refers to the item that has the truth value.

As a result sentences, in themselves, are neither true nor false, so
that t is not true is definitely true, in this case.  In fact the
full truth is that t is neither true nor false.  It follows that
there is no longer any problem with Strengthened Liars, either.
Certainly the question naturally arises about what to say in
'strengthened' cases, where, for instance, one has
s = 's is neither true nor false'.
But first of all there is no need to take the sentence thus defined
to speak about it itself being neither true nor false.  For
sentences, by themselves, have no voice, and the 's' in quotes could
be interpreted differently.  If anyone chooses to interpret this
sentence self-referentially that is therefore an additional
(intensional) matter beyond the extensional, direct-speech identity.
And what is true in that self-referential case is simply that s is
neither true nor false, which is not paradoxical in any way, since to
say that is not to say that what is true is s, i.e. 's is neither
true nor false'. It is to say that what is true is what s says on the
self-referential interpretation.


-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater