Full-Name: Joshua D. Guttman
Posted-Date: Tue, 30 Jul 91 08:32:05 -0400
Date: Tue, 30 Jul 91 08:32:05 -0400
From: guttman@org.mitre.linus
Message-Id: <9107301232.AA00678@malabar.mitre.org>
To: info-hol@edu.uidaho.cs.ted
Cc: ian@com.oracor.cambridge (Ian Sutherland)
In-Reply-To: ian@cambridge.oracorp.com's message of Sat, 27 Jul 1991 12:44:01 GM
Subject: Re: More on the choice function
Postal-Address: MITRE, Mail Stop A156 \\ Burlington Rd. \\ Bedford, MA 01730


Ian writes:

   I must say that I don't see why this bothers Josh so much :-)

I suppose that on the scale of human emotions, the intensity of this one ranks
fairly low :-)

But it seems to me undesirable in itself to have an ingredient in determining a
model that's quite independent of the signature of a language.  I can imagine
that there may be counter-balancing reasons that make it the right thing to do,
all in all.  But just on the face of it, @ seems unappealing.

Why?  Well, for starters:

One of the appealing things (in my view) about second or higher order logic,
with the full semantics, is that various traditional axiomatizations are
categorial.  For instance, (what became) the Peano axioms are categorial, as
Dedekind showed.  Similarly, there's a categorial axiomatization of the reals
(also linked to Dedekind).  When an axiomatization is categorial, you can
certainly infer that, for every sentence A, either A or not A is a semantic
consequence of the axioms.

Or can you?  Well, you certainly don't know whether (@x . x=1 or x=2) = 1.  So
either the familiar axiomatizations can't be considered categorial or
categoricity doesn't entail this sort of completeness.

And I think many other traditional notions of logic and model theory will get
mucked up in a similar way, irritatingly slightly wrong.  Maybe these problems
are more verbal than mathematical.  But in my view a consideration in accepting
a logic is that traditional meta-logical ideas should work out right.

Of course, I understand that this isn't the *only* consideration.

        Josh