FOM: second-order logic is a myth

[dubucs]

20 Feb 1999 Dubucs wrote:

> > 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
>
Simpson replied:

>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.

	I suppose that the last word of your penultimate line is rather
"dependant" ?
>
>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.

1. What is "widely recognized among the f.o.m. searchers" is a point of
sociology, from which nothing follows except sociology. But keeping this
viewpoint for an instant, I observe that this wide majority is not
unanimity: look for example at Shapiro's remarkable book "Foundations
Without Foundationalism. A Case for Second-Order Logic" (Oxford: Clarendon
Press, 1991). Moreover, I can't consider mathematical pieces as Henkin's
famous 1950 paper "Completeness in the theory of types" (JSL, XV-1950,
81-91) as contributions to mythology. Is your opinion really accepted by
all the members of FOM or a staggering majority of them ?
>
>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.

2. Prof. Simpson, the echoe of the completness theorem has reached me. But
once again the only question that matters in our dispute is the following.
Let E Hilbert's second-order system, and B a geometrical truth (i.e. a
sentence which is true in the unique model of E). After your manoeuver of
"embedding", is there, yes or no, a set of formal (recursive) rules such
that, starting from E, one arrives to B by means of a finite number of
applications of the rules ? If it's the case, could you present this set of
rules to FOM's readers ? If it's not the case, where is your vindication ?

3. It's very amusing to be taken here as the average continental
philosopher. Could you imagine that all French philosophers are not
impenitent post-modernists, always the knife down on the reason ? From this
side of the Ocean, we are far of considering all American philosophers as a
regiment of Feyerabendians ...

My first posting (Feb 11) was just designed to denounce abusive claims
concerning irreducibility of geometry to logic. But I realize that the
whole debate is somehow polluted by an indetermination affecting the very
notion of logic.

As regards your "logicist" thesis, I find the designation very misleading.
It's evokes Frege's project of defining arithmetical notions on the basis
of logical ones, and of deriving arithmetical truths on the basis of
logical truths. The realizability of this program is highly controversial
as arithmetic is concerned, and I can't anyway imagine an extension of it
to geometry.

The "logicization" of geometry is that: building a system of axioms which
describe precisely, rigourously, completely and without "intuitive" or
unanalyzed residue the structure of space. That's Hilbert immortal
achievement. Now, "logicization" as you conceive it would require also that
the geometrical truths (the sentences that are true in this well-defined
structure) are mechanically derivable from the axioms which describe the
structure. This second requirement is beyond reach. It arises from a too
narrow concept of logic, according which the only genuine logic is the
first-order one.

To conclude on a philosophical note: irrationalism arises often from a too
narrow conception of the rationality ...

Jacques Dubucs
IHPST CNRS Paris I

Jacques Dubucs
IHPST CNRS Paris I
13, rue du Four
75006 Paris
Tel (33) 01 43 54 60 36
    (33) 01 43 54 94 60
Fax (33) 01 44 07 16 49