FOM: second-order logic is a myth

[Stephen G Simpson]

Jacques Dubucs 20 Feb 1999 04:42:15 writes
 > the probleme is that in our case, namely higher-order geometry, B
 > can be logical consequence of E without being conclusion of a
 > "logical reasoning" starting from E 

I assume here that by higher-order geometry you mean something like
Hilbert's well-known axioms for geometry, which are second-order and
categorical.  If so, then I would dispute your statement, in the
following way.

Among f.o.m. researchers, it's widely recognized that `second-order
logic' (involving quantification over arbitrary subsets of or
predicates on the domain of individuals) is not really a well-defined
logic, because it involves many hidden assumptions.  In particular, it
depends on the underlying set theory.  Do we want to assume the axiom
of choice, or not?  What about V=L?  What about large cardinals?  Etc
etc.  It is well known that these decisions concerning the underlying
set theory are unavoidable, in the sense that they can easily affect
the set of second-order validities.  For example, there are sentences
in the language of second-order arithmetic whose truth-values depend
on the underlying set theory, in the sense that they are independent
of ZFC, or even ZFC + GCH.

This means that, if we make all of the hidden assumptions of
`second-order logic' explicit, then we are in the realm of set theory,
with all its attendent difficulties.  In modern f.o.m. research, the
most successful way to deal with such difficulties has been to
formalize set theory in the usual Zermelo-Fraenkel manner, as a
(first-order!) theory in the predicate calculus, where the
independence phenomena can be fruitfully studied.  This is the
framework used by, for example, Shelah and his collaborators in their
many papers on `second-order logic'.  In such a framework, the
categoricity of Hilbert's second-order axioms for geometry is simply a
theorem of ZFC.  The same holds for the categoricity of the well-known
second-order axioms for the real number system, etc etc.

In short, what I am saying is that one must embed `second-order logic'
(as well as `third-order logic', `fourth-order logic', etc.) into
first-order logic, in order to make all the hidden assumptions
explicit.  Putting the point even more concisely and provocatively:
`second-order logic' is a myth!

This is a vindication of what I in 18 Feb 1999 20:53:06 called `the
logicicist thesis'.

Once we have embedded `second-order logic' into first-order logic, we
then see that, contrary to your remark, B is a logical consequence of
E if and only if it is the conclusion of a chain of logical reasoning
from E.  This is a well-known property of (first-order) predicate
calculus: G"odel's completeness theorem.

 > (of course, the objection does not apply at all to first-order or
 > "weak-second-order" geometry Steve Simpson evoked in a recent
 > posting).

Actually, the so-called `weak second-order logic' is very different
from first-order logic.  In particular, it is not axiomatizable unless
we use something like an omega-rule, beyond first-order logic.  The
set of validities of `weak second-order logic' is Pi^1_1 rather than
recursively enumerable.  Thus, although `weak second-order logic' is
more absolute than full `second order logic', it is still subject to
some of the same set-theoretic difficulties.

 > Note of course that the "full" second order geometry with his
 > assumptions of continuity is of course taken by the
 > neo-transcendantalists as the "true" geometry

It seems to me that the neo-transcendentalists have not taken adequate
account of the well-known analysis of `second-order logic', which I
presented above.

-- Steve