FOM: second-order logic is a myth

[Robert Black]

Steve on 8 March:

> > OR You insist that plural quantification is just a sneaky way of talking
> > about sets.
>
>Yes, that is my preferred option, at least in those cases when plural
>quantification can't be straightforwardly paraphrased in terms of
>singular quantification.  Your Napoleon example is one such case.  (By
>the way, this example seems a little stilted.  I have never heard
>anyone except an academic philosopher talk this way.  Do you have an
>example that is less artificial?)

Boolos's papers, particularly 'To Be is to Be a Value of a Variable (or to
Be Some Values of Some Variables)' have several examples, for example, one
I've cited already, Geach's 'Some critics admire only one another'.  If we
take this to mean:

for some X (for some x Xx & for all xy (Xx & Axy -> x‚y & Xy))

then it's not first orderizable, as is shown by taking Axy to mean (x=0 v
x=y+1), which gives us a sentence true in all nonstandard models of
arithmetic but false in the standard model.

However, I'm not sure that naturalness in English is much to the point.
After all, any reasonably complicated sentence of *first-order* logic with
multiple quantifications tends to be unexpressible in natural English,
because English anaphoric pronouns will take you so far and no further.
e.g. try saying 'for some x for all y (for some z (Ayz & Azx & -Azy) ->
Axy)' in English.  'Someone admires everyone who admires someone who
admires them but not them.'???  So natural expressibility in English had
better not be the criterion of the logical.  I think Boolos has shown that
monadic second-order quantification is expressible by plural quantification
in English to roughly the extent to which first-order quantification is
expressible by 'some', 'all' etc. in English.  It's much harder to express
polyadic second-order quantification in English, but I take that to be a
fact about English rather than a fact about the bounds of logic.

>there is a distinction between sets in
>the real world (sets of cards on a table, for instance) and sets in
>the sense of ZF.  It may be that real-world sets play a role in much
>everyday reasoning, but ZF-sets include a lot of higher infinities
>that have no obvious real-world counterpart.  What is the real-world
>counterpart of aleph_omega?  The connection of aleph_omega to
>real-world sets seems very tenuous.
>
> > and by the topic-neutrality criterion set theory will be part of
> > logic.
>
>I certainly don't concede that.

No, neither you should: I overstated it.  What I should have said is that
some things which you would regard as set-theoretical will by the
topic-neutrality criterion be part of logic.  I'm not arguing, for example,
that the topic neutrality criterion would mean that claims about the height
of the set-theoretic hierarchy are settled by logic.

What is built in to second-order logic is not the hierarchy generated by
transfinite iteration of the power set operation, but rather just the
principle of separation.  Given a domain of objects over which we quantify
in first-order logic, second-order logic allows quantification over
arbitrary collections of those objects (or arbitrary 'ways of taking' the
objects, or arbitrary Fregean concepts, or however you fancy putting it).
If you give the second-order logician kappa objects, he suddenly finds
there an extra 2^kappa things to quantify over, but then he stops.  He
doesn't, qua logician, generate the cumulative hierarchy.
>
>My point is that set theory is not part of logic.  To make this
>absolutely clear, we would probably need to go into the history of set
>theory.  We would need to discuss how set theorists arrived at the
>current understanding of ZF et al as first-order theories, with
>logical axioms (i.e. the common axioms and rules of predicate
>calculus) and non-logical axioms (i.e. the specifically set-theoretic
>axioms).

If you go into the history, I think you'll find that Zermelo (1) worked
with Urelemente, so we are (at lest initially) talking about sets of cards
on the table, though we then of course construct a transfinite hierarchy
and (2) had a *second-order* axiom of separation which only later got
Skolemized into the first-order schema we're familiar with today.  (I'm
talking about separation rather than replacement not merely for historical
reasons, but because replacement also has a job to do in jacking up the
height of the universe and in our conceptualization of set theory ought to
be considered separately for this reason.)  We are agreed that operating in
first-order logic is the sensible way of *studying* sets, but I still think
that we *conceptualize* sets using second-order notions - and that for
example it's our acceptance of second-order separation that justifies our
adoption of the (weaker) first-order schema.  (And of course I'd say the
same about the induction axiom/schema in second/first-order PA.)

[I'm trying to remain neutral here on the issue of whether, if you're doing
second-order set theory, you should regard the second-order variables as
ranging only over sets, or whether, with Boolos, you allow yourself to
assert things like 'some sets are such that every non-self-member is one of
them'.  I think there are arguments on both sides of this question, but
it's an issue which only arises in the very special case of applying
second-order logic not just to natural numbers, or reals
(order-completeness), or genealogy, or critics, or any other set-sized
domain of objects but to (the universe of all) sets.]

Robert



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

tel. 0115-951 5845