An alternative idea about
For a few years now I've had a
little idea about quantification over numbers and such, that's a close
relative of but an alternative to Thomas' substitutional
approach. In particular, it counts as a solution to his
"Puzzle." I'd be surprised if someone hadn't put it forward
already, but I haven't seen it in print.
we're not Platonists about numbers, and we want to account for
quantification over numbers without incurring Quinean commitment.
There are things that don't exist.
That's a near-Moorean fact. So there are (at least) two
quantifiers or anyway existential expressions in English. Now, we
can say: There are numbers, prime ones and large ones and
irrational ones and imaginary ones, but they don't exist.
Quinean remonstrance. Isn't this just the dreaded Wyman
position? Quine says, Lycan has still damn well quantified over
numbers and he's ontologically stuck with them, even if he
hypocritically says they "don't exist." Worse, has Lycan hopped
in bed with Meinong? We all know about Meinong. (Lycan
himself, and no less a figure than DKL, have suggested that Meinong's
view is in the end unintelligible.)
first (before we get to Meinong), read my lips: There are two quantifiers in English,
one ontologically/existentially commissive and the other not. Get
used to it. There are two. There just are. I'm using
the noncommissive one on numbers. I stipulate that, and you can't
Lycan himself has prominently asked, what is the noncommissive one
supposed to mean? It's
not the backwards E as Quine understands it.
it's not. (To coin a phrase, I'll give Quine the backwards E and try
not to use it again; I still have the noncommissive quantifier. I
can use a backwards N to notate it.) What does it mean,
then? I don't know.
That's a terrible philosophical problem. But, n.b., it's not my
problem qua philosopher of mathematics. It's a problem in modal
ontology, not about arithmetic.
that any number of solutions to that problem have been offered by
metaphysicians of modality: (1) The N quantifier is
substitutional (Marcus, Hofweber). (2) N can be paraphrased away
in terms of counterfactuals or something (Kripke). (3) N ranges
over Ersatzes, like sets of sentences or structured sets of properties
(Carnap, Hintikka, Adams, Plantinga, Stalnaker, Lycan). (4) N is
primitive, as is the backwards E; tough darts (Meinong). (5) What
I'm calling noncommissive isn't, modally speaking; rather, the narrower
existential expression restricts it to worldmates of ours
(Lewis). (6) N is used fictionally (Rosen). (7) N occurs as
part of a pretense, or with an illocutionary force different from that
of standard assertion, or the like (Walton, Currie). (8) ....
only one of those solutions--(4)--is Meinong's. So unless I
embrace (4), I have not hopped in bed with Meinong.
A key thing to grasp here is that every one of those solutions is terrible, even if none of the
others is quite as bad as Meinong's. So isn't Lycan committed to
the disjunction of a bunch of terrible theories? A disjunction of
falsehoods is a falsehood, so isn't Lycan refuted right there?
I repeat: It's a near-Moorean fact that there are things that
don't exist. That fact doesn't entail a disjunction of
falsehoods. If each of the solutions I listed is indeed false,
then whatever is the true account wasn't on the list, and no one knows
what it is. Too bad. It's lucky that (right now) I'm
only a philosopher of mathematics and not a
idea is that deflationary philosophers of mathematics have tended to
couch their views in more specific terms, e.g. by saying that numerical
quantification is substitutional or by being fictionalists, and thereby
incurred objections. But, I say, the objections are really to the
more specific interpretations of N, not to interpreting numerical
quantifiers as N in the first place. I can't think of any good
objection to interpreting numerical quantifiers as N. Can
awful is that by training and instinct, I'm an old rock-ribbed Quinean,
and the view I'm suggesting sounds suspiciously like Hintikka's, the
one he puts in terms of "ideology" as opposed to ontology, which I have
derided in print as the merest sleazy cop-out. I'd try to
distinguish them, but Hintikka's remarks are too sketchy for me to get
a proper grip on.)
I have discussed some of
this with Thomas (thanks, Thomas), and it's worth dwelling briefly on a
few of our differences.
First, as Thomas has
pointed out in correspondence, I haven't actually shown that
there are two quantifiers--not,
at any rate, real quantifiers in logical form.
Right, I haven't, but that is
part of my dialectical design. I am deliberately using
"quantifier" in a general and superficial enough sense that I hope I'm
not saying anything that should be controversial. Someone who
urged a pretense view, e.g., would not be disagreeing with me but
would be offering a particular theory/solution to the Meinongian
puzzle, a theory about one of the surface quantifiers.
Second, even if there are two
quantifiers, why should we think that the one that ranges over numbers
is the noncommissive one? That's an empirical question. To
answer it, you'd have to look specifically at number words--which might
turn out to be referential in function.
Good point, and I agree it will
be important to make the empirical investigation of number words (on
which, see Thomas' wonderful "Number
Determiners, Numbers, and Arithmetic"). But remember, for now
we are only supposing that we
don't want to incur real ontological commitment to numbers. I
have no argument against Platonism that hasn't been made
Also (important:), even if contra
Thomas number words did turn out to be referential in function, my
argument would apply after his had ceased to. For the names of
fictional characters etc. are
referential in function, but it remains true that the fictional people
do not exist. So my approach is free of that limitation.
Third, Thomas gives a substantive,
positive theory of quantification over numbers. Isn't that better
than my merely dialectical wimpery?
Other things being equal, yes.
But remember my corresponding dialectical advantage: One who
gives a substantive positive theory thereby
incurs objections, that are most
likely objections only
to the more specific interpretation of N, not to interpreting
numerical quantifiers as N in the first place. It remains for my
opponent to come up with a
good objection to, per se,
interpreting numerical quantifiers as N.