for The Grice Club
A point of clarification on how I (as a universal non-expert)
take Hilbert and Carnap on semantics of formal systems.
The intention is not that semantics is DISPENSED WITH in
favour of exclusively syntactic methods.
The intention is that the semantics is PROVIDED FOR by
syntactic means.
The two are not the same generally but I believe they are
similar in that specific respect.
For Hilbert its best to think of a first order axiomatic
theory. Let us assume that somehow we have obtained
an understanding of the semantics of first order logic, so we
know what it is for a formula to be true in some
"interpretation" or structure.
We give a meaning to a first order language by specifying
which are the intended interpretations.
This we do by supplying a set of first order formulae such
that the set of intended interpretations is the set of all
interpretations which satisfy all the formulae.
We call these formulae the axioms.
Hilbert's formalism (in this aspect, not in relation to his
foundational program) consists in the view that the meaning
of a first order language should be determined EXCLUSIVELY by
such means, and should not be supplemented by any informal
descriptions of the intended interpretations.
In judging the tenability of this position you should bear
in mind that Hilbert adopted this position before the
incompleteness results, and he believed that all
mathematical problems were soluble.
For that to be the case, all mathematical theories would
have to have complete axiomatisations, and if that were the
case the axioms would precisely determine the intended
interpretations. However, Godel showed that this could not
be.
Carnap's case is more complex, because Carnap was not wedded
to any particular logic, and so he could not just talk about
axiomatic semantics. His conception of formal system is in
terms of rules which determine the logic from scratch.
Carnap also does not have the excuse of doing this work
before Godel's incompleteness results, so he should have
realised that the formalisation of semantics could not be
fully accomplished.
However, when he saw Godel's work, he does not seem to have
seen that aspect of it. What he saw was the magic of
arithmetisation which got him out of the difficulty which
Tarski had put him in with the idea that a metalanguage had
to be different to the object language. He saw that he COULD
use some languages (e.g. arithmetic) as their own
metalanguage. This definitely worked for syntax, and
following Hilbert, one could use syntactic rules to fix the
semantics.
There are some mistakes in here, but there was never any
intention (to the best of my knowledge) either on the part
of Hilbert or on the part of Carnap, to dispense with
semantics. The intension was to dispense with INFORMALITY.
(almost as misguided).
The most fundamental problem here is the one illuminated by
the incompleteness results, that formality never quite gets
things right on the button, there is always a little bit
more to be had.
When Carnap leaves syntax behind and talks as if he were not
reducing semantics to syntax, he still doesn't realise that
some informality is indispensible if you want to get the
semantics right on the button.
That's my take, as an enthusiastic non-expert, its probably
not all right, I'd be glad to hear what I have wrong.
How it fits in with Grice I leave for JLS.
Roger Jones
Posts
Excellent post, Roger Bishop Jones!
ReplyDeleteI'll read and re-read it.
For the time being, my gut reaction: congrats!
--- I'm saddened Hilbert died before Goedel's discovery and as you say it's good to think that Carnap had no excuse on that front.
It's excellent to have your correction or shall I say refinement as to "disposing" with: not semantics per se, but "informality". And I'm glad (in a way) that you see a purely formal semantics would be a chimaera.
I don't know, but soon will find out, how Grice fits in all this.
He confesses he was never wedded to anything, but that first-order predicate logic with identity (implicating, but not with modal operators) was okay with him, and thus is his System G-HP.
He tended to view the "semantics" bits -- and he was writing in 1969 -- as along Benson Mates's lines in "Elementary Logic" (Oxford). Anything too more formal or less elementary he would not have considered too non-elementarily, or something.
In Mates, the semantics is provided pretty much "informally" (although it seemed a heck of a lot of formality when I had to digest that thing as a student). So it's models M, under interpretations I, in set-theoretical terms. The meta-language is meta-logic (This book on "Meta-Logic" was my logic teacher's night-stand book).
Grice provides some "informal" yet pretty much "formal" (for me) accounts of (Ex) and (x) in these terms. He is concerned with vacuous readings (as it were) of these quantifiers, so he is particular about how the "semantics" for those quantifiers should be given.
While some of his tenets in "Logic and Convesation" (page 1-2 of this lecture) suggest an axiomatic approach, he would have been aware of Goedel's results so he couldn't have been meaning a purely formal axiomatic treatment for calculi that Goedel proved to be undecidable. Etc. These comments are not supposed to exceed some limits, so I'll stop here.
Cheers,
JL