Philosophy 730                                                                                                                                                                                 W. Lycan
Fall, 2007

An alternative idea about numbers etc.

        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.
        Suppose we're not Platonists about numbers, and we want to account for quantification over numbers without incurring Quinean commitment.
        Well:  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.
        Thunderous 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.)
        I reply, 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 stop me.
        But, as Lycan himself has prominently asked, what is the noncommissive one supposed to mean?  It's not the backwards E as Quine understands it.
        Right, 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.
        Notice 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) ....
        N.b., only one of those solutions--(4)--is Meinong's.  So unless I embrace (4), I have not hopped in bed with Meinong.
        But:  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?
        No.  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 modal ontologist!

        So, the 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 you?
        (What's 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 elsewhere. 
        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.