FOM: second-order logic is a myth

[Robert Black]

I'm with Jacques Dubucs on this one.  Of course Steve's entitled to his
neoformalist view, and I dare say it's widely shared, but it's not the only
tenable view, and it's not clear that in the long run it's really tenable
at all.

There is a 'naive' view that talk about quantification over *every
possible, however arbitrary* subset of an infinite set makes perfectly
determinate objective sense.  Intuitively, this has a lot going for it.
(Note, for example, that on Steve's contrary view 'arbitrary real number'
has no clear sense, which would surprise most working mathematicians.)  The
naive view seems initially so obvious that I personally find it hard to
explain to my students of the philosophy of mathematics just what is
supposed to be wrong with it.

Nor need the view stay naive when confronted with the results of
20th-century logic and foundations.  It then becomes the view that
second-order logic is perfectly determinate (though not recursively
axiomatizable), that CH has a determinate truth-value (though one we don't
know, and perhaps even couldn't ever know), and so on.

>From this viewpoint, let us discuss what Steve says:

>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.


Well, no.  Second-order validity doesn't on the naive view depend on
set-*theory* (what axioms we assume about sets) - it depends on the *facts*
about sets.  AC is true.  The truth-value of V=L is unknown, though
falsehood is a good guess.  Large cardinals are irrelevant, because
although large cardinal axioms in first-order ZF allow us to prove new
theorems about what goes on at lower levels, they don't affect what is true
at those lower levels.  Etc etc.  There are sentences of second-order logic
which are valid if CH is true, invalid if CH is false, so we don't know
whether they're valid or not. And so on.  As Thomas Hobbes (not a
philosopher I would normally want to cite for his views on mathematics)
said three centuries ago "Geometry hath in it somewhat like wine, which,
when new, is windy; but afterwards though less pleasant, yet more
wholesome. Whatsoever therefore is true, young geometricians think
demonstrable; but elder not".

[Note incidentally that although ZF is normally studied today (for good
reason!) as a first-order theory, it was not originally so conceived by
Zermelo.  And just as a matter of history, I think the person most
responsible for trying to get everything into first-order logic was Skolem
rather than Hilbert - I forget the details, but you'll find them e.g. in
Gregory Moore - 'The emergence of first-order logic' in Aspray & Kitcher
(ed) - History and Philosophy of Modern Mathematics.]

The sort of position I've just sketched has been taken seriously by a lot
of people who can hardly be accused of infection by postmodernism (Goedel,
Kreisel, Boolos, Shapiro).  Now of course there are arguments against this
ultrarealist position, basically working out the idea that, pace Hobbes,
mathematics can't contain undemonstrable truths, but (1) I don't think any
of them are knockdown and (2) if they do work, they are quite likely to
produce overshoot and lead to various versions of constructivism.  Just for
starters, does Steve think first-order arithmetical truth is a well-defined
notion?  The only ways I know of getting a grip on it are via
intuitively-understood second-order logic or something which will be just
as suspect from Steve's point of view for more or less the same reasons.

Incidentally, Martin Davis seems to think he agrees with Steve, but I
wonder whether he really does.  He says of the Henkin completeness proofs
that 'This was completeness relative a to a particular set of structures,
because of course, relative to the intended interpretation, it is
incomplete.'  But if Steve's right, is there any such thing as 'the
intended interpretation' of second-order logic?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845