FOM: Pratt on truth

[Vaughan Pratt]

From: John Mayberry <J.P.Mayberry at bristol.ac.uk>
>Of course my argument does not depend on such a premise. If you prove 
>that A follows from the axioms B,C,...,D, then you prove, not that A is 
>true, but that if B,C,...,D are true then A is true. If we are talking 
>of formal axiomatics here, then you have proved that in any 
>interpretation under which B,C,...,D are all true A is also true.

(Just to remind people of the larger context, Butz asked "How can anyone
claim that there is absolute truth out there?" and Mayberry sprang to
the defense of absolute truth, later joined by Bill Tait and Joe Shipman,
much to my surprise.  I am siding with Butz in questioning absolute truth,
whose existence and meaningfulness strikes me as *much* harder to defend
than to attack.)

If we are talking of formal axiomatics here then what has been formally
proved is A, not the hypothetical statement "B&C&...&D -> A", which is
only in your external interpretation of the situation.

But if you are going to insist on incorporating the axioms as premises
in order to make a sentence "absolutely true", where do you draw the line
between axioms and inference rules?  This tends to be a fuzzy line; often
an axiom is understood as simply a rule with no premises.  In that case
why would you be satisfied that "B&C&...&D -> A" is absolutely true when
you have omitted the inference rules?  For absolute truth of the kind you
envisage, shouldn't you also include "and assuming the inference rules
are sound" among your premises?  One problem with using your notion of
what "you have proved" is that until you can say precisely what hedging
to add, you can't even say what it is that you have proved.  Formal
proof systems do not have this problem, they are perfectly clear about
what has been proved.

Sometimes we view the rules as absolute and the axioms as relative.
But this seems to be an intuitive distinction not reflected in any
formal property of the proof system.  There is no sharp demarcation
between axioms and rules supporting such a distinction.

Stepping back from these technicalities, like Butz I find this whole
notion of absolute truth entirely unsatisfactory, and I am very surprised
to find this many supporters of it on this list.  First, assuming that
absolute truth is even meaningful, it would be very inconvenient if
every fact we believe to be absolutely true had to be hedged about with
all the circumstances responsible for that belief before we felt we had
uttered an absolutely true statement.  Second, I don't see how it *can*
be meaningful.  As Tarski showed, one cannot even *define* truth within
the system to which it is to be applied.  How can you accept any notion
of absolute truth when you can't even say in principle what truth is
for the universe of mathematics?

>Butz's view is absolute nonsense (and in this case I use the 
>adjective without any hesitation whatsoever). It is absolute, 
>double-dyed nonsense in the mouth of a mathematician.  ...
>Told of a similar species of absurdity, Dr. Johnson replied "Why Sir, if
>he really believes that there is no difference between vice and virtue,
>when he leaves our houses let us count our spoons".

I am disturbed by the illogicality of Johnson's reasoning, which Mayberry
seems to be endorsing.  I would not dream of inferring from Tarski's
result questioning the meaningfulness of truth that he or any other
logician was more likely to be dishonest than one pledged to absolute
truth, however defined.

Vaughan Pratt