## Tuesday, October 06, 2009

### CAI and SCQ

I've posted a new and expanded version of my paper arguing that composition as identity doesn't settle the special composition question. It's here; thoughts welcome.

Jeff said...

That's an interesting paper. Let me see if I've got the central thrust.

You argue that the Sider-Merricks argument relies on a logic with distinguished singular variables. This reliance makes claims like

if x=X then nec x=X

more loaded than they might appear. Fair enough; so let's make things more explicit by speaking a language with just plural variables, and instead introduce a new predicate, One, to do the work of marking singularity. So then Sider and Merricks' central assumption is what we might call "Number Rigidity":

if One(X) then nec One(X).

Now, ordinarily, I don't think this would be an objectionable assumption. We normally think of pluralities as essentially having their particular members, and so in particular we think it is essential to a plurality how many members it has.

You seem to think that the CAI-ist has a special reason to doubt this, but I'm not convinced. It sounded like your main argument for this was something like this: if CAI-restrictivism is true, then Number Rigidity is false; so the CAI-ist had better not assume Number Rigidity. But why shouldn't the CAI-ist believe Number Rigidity on the same grounds as the rest of us, and give up restrictivism accordingly? (That's just part of what a plurality is, we might say.) At any rate, that's precisely the position that Sider and Merricks implicitly advocate, so it seems like you're begging the question here. Am I missing something?

(There's a deeper issue here that makes things confusing. How can a CAI-ist consistently use the concept One at all? After all, a many-one identity claim would look like

X=Y and One(X) and ¬One(Y)

which violates Leibniz's law. So just to stay consistent, the CAI-ist is going to have to do some fancy footwork here, and it seems like that is going to have to have consequences for this discussion.)

Ross Cameron said...

Hi Jeff,

Yeah, I think the CAIist should reject Number Rigidity.

Why accept Number Rigidity? You say: "We normally think of pluralities as essentially having their particular members, and so in particular we think it is essential to a plurality how many members it has."

But I don't think that reasoning is good if CAI is true (or at least, I don't think we have reason to think that it is good). It only follows from the fact that the plurality of the Xs essentially has the Xs as members that the Xs essentially has the number of members that it has if each sub-plurality of the Xs essentially has the number of members it has: but to assume that begs the question.

If, for example, the Xs are three individuals A, B, C, but C could have been a plurality of individuals D, E and F, then the Xs could have been 5 individuals: A, B, D, E and F. But this *wouldn't* mean that the Xs doesn't have its members essentially: in both these circumstances, the Xs has exactly A, B and C as members, since C=D,E,F. Of course, if CAI is false, then essentiality of members entails essentiality of number - but the CAI theorist has to justify why we should still hold the latter claim if CAI is true, since I've given a reason to think it fails if CAI is true.

On the deeper issue: I don't think it's true that if we have a many-one identity Xs=A that (One(A)) is true but (One(Xs)) false. If A is an individual, they are both true, if not then they are both false. Part of the point is that I don't think you can read this off of the syntactic form.

Jeff said...

Hi Ross,

Let's keep speaking the plural-only language, and take membership as basic (write "X el Y"). Then I guess the inference from "Member Rigidity"

if X el Y then nec(X el Y)

to Number Rigidity relies on (the necessitation of) two premises. First, every One is a member (of its singleton plurality, say), and second, every member is a One. Why should the CAI-ist give up either of these claims?

As you point out, the CAI-ist holds that a member can also be Many (and so in general the "number of elements" is not well-defined—I was sloppy about that in my last comment)—but if One and Many are not contraries, then so what?

On the "deeper issue": I was glossing "Many" as "not One"—which wasn't very fair in this context; point taken. The right thing to say isn't that One-ness raises inconsistency for the CAI-ist; it's just trivialized.

Ross Cameron said...

Why should they accept that (necessarily) every member is a One? I realise I'm being annoying and just asking the question back, but it seems to me that the CAIist can coherently deny this, so I want to know why they believe it.

And really, it's the coherence of denying it that really matters to me. All I want to insist is that there's nothing just about CAI in and of itself that gets us universalism. Some other premise is necessary: maybe it's necessary that every member of a plurality is a one. Since one can, I say, coherently uphold something that deserves to be called CAI and deny this, CAI alone isn't doing the work. Even if CAIists accept it, it's still an extra premise.

Jeff said...

I suppose that's technically true, but it's such an unassuming assumption that it seems a bit miserly to deny it to Sider and Merricks. CAI doesn't imply much at all strictly on its own; the interesting claim was that CAI together with an independently attractive plural logic implies lots of standard mereology. (Anyway, that was the way I saw the dialectical terrain.) And it seems reasonable (to me, anyway) to grant the member-One link as part of that logical package.

I like the way giving up some syntactic privileges brings to light the particular assumptions at work here, but I'm not seeing a reason to doubt the assumptions.

Ross Cameron said...

I guess I just want to see the assumptions motivated by the metaphysics, and while I can see how they might be motivated if CAI is false, I think there's positive reason to doubt them if CAI is true. I mean, if I think into the headspace of CAI, to the best of my ability, I want to reject the additional assumptions that take you from CAI to universalism, because they seem to me unjustified.

Still, I'm happy enough if we can make it clearer just *what* you need to say to get from CAI to universalism. Whether you actually believe it or not - that I care less about!

Ross Cameron said...

But let me ask you something Jeff. So the principle we're agreeing does the work is: necessarily, every member is a one. That is, I take it, no one member could have been many members and not also one member. So it sounds very close to the principle in the paper I label the essentiality of individuality.

Now, in the paper I don't actually reject this. I say: why should I accept it, given that I'm being asked to deny that mere pluralities of things couldn't have been a one? That is: if I'm being asked to accept that a mere plurality of things identical to no one could have been, why shouldn't I also accept that an individual could have been a mere plurality of things, in which case a singleton plurality could have been a many plurality without a change in members. And I am being asked to accept the former, since I'm a restrictivist and I'm being asked to accept Poss Comp.

So it's not so much that I'm saying the principle in question is false, but that I can't see why I would accept both it and another essential part of the argument at the same time. Do you think this helps my case?

(Apologies if I'm a bit unclear just now. I'm kind of sick.)

Jeff said...

So that line of reasoning was the one that originally sounded question-begging to me. Let me see if I can flesh this out.

You say that you (the CAI restrictivist) are being asked to accept that a "mere many" could have been a one. I don't think that's quite right. What you're being asked to accept is (say) that these scattered atoms could have been a one. Whether these scattered atoms are a mere many is what is at issue in the debate, and Sider and Merricks are arguing that you should not think they are.

You really shouldn't hold that mere pluralities could be individuals, nor that individuals could be mere pluralities. It seems to me that you're mistaking an assumption used for reductio (that these things do not compose) for an assumption on which the argument relies. (Better not to frame the argument as a reductio, to better avoid this confusion. A simple version:

Premise 1: poss Compose(X)
Premise 2: nec (Compose(X) iff One(X))
Premise 3: nec (One(X) then nec One(X))
poss nec One(X) follows by several applications of K.
One(X) follows by B.
Compose(X) follows from Premise 2 and T.)

It's true that the restrictivist CAI-ist has a reason to reject Premise 3, but that doesn't mean the CAI-ist tout court should—since the CAI-ist shouldn't be a restrictivist.

Is that making sense? (I'm kind of sick, too.)

Ross Cameron said...

It makes sense, but I don't buy it. I just can't see that there's any independent reason to accept premise (3). Of course the CAI-ist universalists accept it and the CAI-restrictivists reject it, so now the question is: what's the reasonable independent starting point? (It would be easy for each side to simply say to the other 'you're begging the question' - but that's not very interesting.) As far as I'm concerned, the reasonable starting point is agnosticism: I can see a metaphysics on which the premise is true, one on which it is false - and I for one at least don't have any pre-theoretic inclinations between the two. In which case, I think, we need to have some reason one way or the other. That one metaphysics settles SCQ doesn't seem to me like a convincing reason, at least no if one has some reason to think that SCQ *shouldn't* be settled in favour of universalism. So to me it looks like the debate comes down to what it does if CAI is false: trading intuitions and theoretical virtues to decide whether to accept the universalist CAIist metaphysics or the restrictivist CAIist metaphysics.

Jeff said...

Ok, so do you at least agree that you haven't offered a positive reason for the CAI-ist to doubt Premise 3? That was all I was going for, so far.

At least for dialectical purposes, I was thinking the burden was on you to give some reason for the CAI-ist to think Premise 3 is false. The main reason for this is just that Premise 3 is a standard assumption of any standard modalized plural logic (implicit in the use of sorted variables), and it's something your opponents Sider and Merricks take for granted. It's nice to make the assumption more explicit, but that doesn't in itself make it look any dimmer—even assuming CAI. A thing is essentially a thing. Why should a thesis about the nature of composition call that into question? So it seems like you haven't seriously undermined Sider and Merricks' argument (though you have clarified it) unless you have a substantive argument against Premise 3.

That's all I've really been going for so far. I went a little beyond that, by offering the argument from essentiality of membership. (1) Any thing is a member of its singleton plurality; (2) any plurality has its members essentially; and (3) necessarily any member is a thing; therefore, any thing is necessarily a thing.

You challenged the third premise of the argument. I'm not sure what the best way is to support it, since it's pretty close to the bottom. Plural logic comes with notions of plurality, member, and thing in its basic repertoire. There are some claims about their relationships that are constitutive of plural logic; we should think they are true insofar as we believe in plural logic at all. One of these claims, I would have thought, is that if X is a member of Y, then X is a thing and Y is a plurality. (That's such a basic supposition that it's normally encoded all the way into the syntax.) And that's necessary, since it's a logical truth.

If you think that CAI gives us a reason to doubt this member-thing link, then it looks to me like you think CAI gives us a reason to give up plural logic altogether. That may be right, but it's a strong claim, and one that requires stronger support than just the thought that agnosticism is a "reasonable independent starting point". Is this making sense?

Now, as it happens, I do think CAI makes serious trouble for plural logic, not by pushing us to give it up, exactly, but by weakening it to the point of triviality—since effectively all pluralities are things. But that's rather different from your conclusion.

Ross Cameron said...

I don't agree that I've given no positive reason to doubt premise 3. I think I've shown a metaphysic on which it fails and which satisfies what I think are all the intuitions driving CAI. That, I think, is reason to doubt it. Secondly, I've tried to argue that intuitions one might have had (independently of thinking about CAI) against universalism, still carry over if CAI is true - so insofar as you have those intuitions at all, you should still reject the claim that takes you from CAI to universalism. (That won't move one who just doesn't have them, of course - but the dialectic is that I'm trying to show how the CAIist restrictivist needn't me motivated to abandon restrictivism.)

I just don't find it at all motivating that premise 3 is "a standard assumption of any standard modalized plural logic". That may be so: but to my mind, that's just because we're used to thinking that CAI is false. On that assumption, there's no conceptual room for thinking that some individuals could have been mere manys and some mere manys individuals. But once CAI is accepted, I think there is such conceptual room, and I think this should lead us to rethink the logic. The fact that it's the standard logic carries no weight at all, to me: the logic has to be justified by the metaphysics - or at the very least, not undermined by it - and I think it is undermined by it. (Cf. those who argue for the Barcan formulae on the grounds that they're entailed by the simplest quantified modal logics. Who cares? If the best modal metaphysics undermines BF, revise your logic - revising your metaphysics because you like the logic is exactly the wrong way round. Now I don't deny that there are some principles of logic that are deal breakers - perhaps law of non-contradiction - but I don't think the principle in question is one. (Modalised principles, I would think, are never going to be deal breakers - they pack too much metaphysics in to have that status.) Also Cf. the 'logic of mereology'. The standard logic of mereology simply settles things in favour of universalism - it has that as an axiom. I think that doesn't give us even a pro tanto reason to accept universalism. The metaphysics should drive the logic of mereology, not the other way around. Again, this is a case where too much metaphysics is built into the logic to let the logic drive the metaphysics. I think the same goes for your standard modalised plural logic. Mostly, but not solely, due to the modalising part.

Part of the problem that I raise in the paper is that I think that if the CAIist doesn't grant me the resources to raise the problem I raise then I have no way of distinguishing them from the fancy talking nihilist. So there are two views I understand: one where the metaphysics is nihilist and I have a logic that allows me to say that for any things they have a sum, namely themselves. And one where the metaphysics really has both simples and complexes, and the complexes really are the simples, and it's an open question what collections of simples are identical to some complex.

So I can understand Jeff* who is, at bottom, a mereological nihilist but who has a logic and a way of talking that allows him to sound like a universalist; and I can understand Jeff** who thinks that CAI is offering a different metaphysics than the nihilist, and who sees that this metaphysics can be universalist or restrictivist, and who accepts the extra principle that takes him to universalism. (I understand it, but don't understand his reasons for accepting it - and I think he needs one, and I don't think it can be 'because my logic says so'.) But I don't understand Jeff*** who thinks he has a non-nihilist metaphysics and who doesn't think it's even an option to have a restrictivist metaphysics.

I think you're Jeff** (well, you would be if you accepted CAI!), so at least I understand you. I just think it takes more to justify the extra step than you do.

This has been really helpful btw - thanks!

Ross Cameron said...

One other minor point. I don't actually think it's fair to say that Ted and Trenton take the logic in question for granted. At least not explicitly; and I can't even see anything that implicitly commits them to it. Certainly not in Trenton's paper, which doesn't go into CAI in much detail, but I don't think in Ted's either. And even if it is there, I still think it's fair for me to present my case as a refutation of their argument if they don't give us any positive reason to accept it - and I don't think they do.

Jason said...

Jeff, I take it you're putting a lot of weight on the "standard formulations" of plural logic. But it's not clear to me (at least) how much weight these can bear. In a non-CAI setting, we can make do with a single existential quantifier (taken to range over pluralities, where a plurality can be a "singleton" plurality, as it were, although that's not encoded in the syntax anywhere) and the plural-to-plural "among". Then we define something like "Y is a (single) thing" as "AX(X among Y --> X = Y)".

If you take this logic as primitive, once you adopt CAI you find you can't define "Y is a single thing" like this anymore (for the sorts of reasons that, I assume, you're pointing to in your 10:35 comment). But we can imagine a CAI-ist who started out not believing in CAI, preferring his plural logic of this sort (it is cleaner in a lot of ways, not building in a singular/plural distinction into the syntax, for instance). Then he comes to believe CAI. So he knows he needs some more ideology; suppose he adopts something like Ross's predicate "One(Y)" for this purpose.

Two questions: (1) Is there any reason to think that a CAI-ist who came at things from this direction isn't using plural logic anymore? And (2): if not, is there any reason to think that this fellow ought to adopt your premise 3 for his new primitive "One"?

If the answers are "no" and "no", then it looks to me like Ross can say that the apparent status of premise 3 as somehow being tied to the logic of plural quantification is a historical-accidental illusion. If we start with first-order logic and then slap some plural resources on top of that, 3 looks (as you say) built into the syntax; but if we come at it from another direction, it doesn't look so un-give-upable.

Jeff said...

Ross,

I'm getting a better handle on some methodological differences between us. For instance, I do think that the fact that universalism is logically nice (and hence "standard") is a pro tanto reason to accept it; and I'm happy to let logic drive metaphysics quite a ways, when that logic earns its salt by being sufficiently useful. But never mind that.

So I agree with this much: the place for the CAI restrictivist to push back against the Sider-Merricks argument is at the "essentiality of individuality", which can be coherently doubted. I guess I was expecting something stronger (the possibility of coherent doubt is a pretty low bar, and "refutation" suggests something stronger to me), but this is certainly a worthwhile point, and I accept it.

I have one more question, though:

On that assumption, there's no conceptual room for thinking that some individuals could have been mere manys and some mere manys individuals. But once CAI is accepted, I think there is such conceptual room, and I think this should lead us to rethink the logic.

I don't think I understand what role CAI is playing in this. Set CAI aside—(and so "mere many" is the same as just "many")—couldn't I just as easily coherently imagine that this individual could have been many things? (Perhaps this table could have been scattered atoms, and not a single thing at all.) And if that's unattractive, then I don't understand how adding a thesis about composition makes it any more attractive.

Jeff said...

Jason,

That helps a lot. So normally we have two ways of talking plurally. We can take "Member" as primitive, where Members are always Things; or we can take "Among" as primitive, and define "Thing" as you say. Ordinarily it doesn't matter which way we go.

But when we adopt CAI, we have choices in how we go on using the word "Thing".

We could talk about "Member-Things". This leads us, along Sider-Merricks lines, to the conclusion that every plurality is a Thing. Now it looks like we're universalists.

We could talk about "Among-Things". Then it quickly follows that the Things are just the mereological atoms. Now it looks like we're nihilists.

Or we could introduce "Thing" as a new primitive. Now it looks like we could be organicists or whatever—it depends on what special metaphysics we add to prop up this use of "Thing".

It now seems to me that the right thing to say is that the CAI-ist doesn't get a determinate notion of a Thing—at least, not one that comes from logic, the way it classically does. They have two notions, which are equally good. On one concept of a Thing, the CAI-ist is a nihilist; on the other, the CAI-ist is a universalist. You might say the Special Composition Question isn't so much answered as deflated—the CAI-ist doesn't really believe in the metaphysical distinctions you need to make the question have a determinate answer.

Unless the CAI-ist takes the third option, introducing something new to hang the question on—which is what you (Ross) assume they need to do. But on this way of looking at it, that avenue seems pretty unmotivated to me. What work is this new, special primitive "Thing" being introduced to do, which can't be done by one of the other two notions? Primitive Singularity raises questions, but doesn't answer any—at least none that I can see.

Ross Cameron said...

Well, I don't think all I've done is to suggest the possibility of coherent doubt: as I've said, I make the argument that intuitions against ontological exotica like gerrymandered sums carry across even if CAI is true. You might disagree, but that at least is the ambition. Don't call what I've given a refuation if you don't want to - I don't mind much about that.

On the question you asked. Suppose there can be cases of many-one identity. Then there can be cases of trans-world many-one identity (since trans-world identity is identity). So a mere many, the Xs, in w can be trans-world identical to an individual, A, in v, which means that in w the Xs are a mere many that could have been an individual, and in v A is an individual which could have been a mere many (assuming a symmetric accessibility relation for ease of exposition). Suppose identity is simply a one-one relation. Then trans-world identity is a one-one relation (trans-world identity is identity). So an individual A in w is only identical to an individual in any other world v. So A is essentially an individual. So the essentiality of individuality is true if identity is one-one, but coherently fasle if it can be many-one.

This would fail, of course, if trans-world identity isn't identity but is rather something less strict, like counterparthood. ("I might be twins" - although that wasn't what Lewis meant.) But then, if you go with counterparthood, the necessity of identity arguably fails, and so the Merricks-Sider argument will fail that way instead.

Jason said...

I'm not quite sure what you mean by "Among-Things" -- I take it you mean something like the things ranged over (plurally) by the existential quantifier (of the plural-to-plural language). I'm not sure how you get from here to nihilism, though. After all, couldn't I have gunky models of CAI in this language? (Granted, when I *started out* I talked about there being singleton assignments to the variables. Think of this as a ladder we kick away when we move from "ordinary" thinking to CAI. The point is that you'd want to think about the possible variable assignments the way you think of the "standard" models of 2nd-order logic: as including every subset of the domain, however we think about *that*.)

Also: there's a kind of chicken-and-egg worry about your strategy. You're happy to rely on something like our intuitive grip on "thing" to get a feel for the plural quantification, I take it. But once you get a grip on the logic from the non-CAI-perspective, you want to import the logic you already got, in part from that intuitive grip, into the CAI setting. But if you do that there's a serious worry that in the CAI setting, the logic formed to deal with your intuitive notion of "thing" will now be expressively inadequate. And if you think that -- which is a natural thought -- you'll want to go for the third option.

Of course, you don't have to. But consider again gunky models of "Among-things". If the language just has the standard resources (sans "One" or the like), there will be natural questions to ask about the gunky models that you just don't have the expressive resources to ask. (For instance, you might wonder if the model is isomorphic to a gunky, zero-less boolean algebra that embeds a unique non-gunky boolean algebra "at the top", as it were --- as it would if e.g. universalism in Ross's sense were true.) You're certainly free to insist that the questions don't make sense; but I take it Ross wants to argue that, if you want to respect the motivations that drove you to CAI in the first place, you'll be unsatisfied with thinking these questions misguided. So you'll need to introduce new resources --- like "One" and the like --- to have a language strong enough to express the *intuitive* content you wanted to express.

Jason said...

P.S. -- I should have made it clearer that I'm skeptical about the notion of "thing" "coming from logic", at least if by "logic" you mean something like "the formal predicate calculus". It seems to me that we have a notion of thing that helps us *design* a logic -- it's the contours of our (pre-formal) ideas about "things" that make us plump for a logic where, e.g., Fx v ~Fx is a theorem. (If we thought some things are intensional and incomplete, for instance, we might be led to deny this principle -- maybe in conjunction with a denial of existential generalization. However it goes, though, the point is that we fiddle with our logic based on our metaphysical presuppositions even at the level of thinking about things.)

Jeff said...

Ross,

Ok, I think I understand your argument better now, but I don't buy it. Certainly this is not a valid argument: red things are never identical to blue things—there can be no cases of red-blue identity. Then since trans-world identity is identity, there can be no cases of red-blue trans-world identity; no red thing could be blue. Why is the many-one case different? The question of whether "trans-world identity is identity" looks to me like a red herring.

Ross Cameron said...

But red things are identical to blue things, both across times and across worlds. If the logic of identity ruled out identity holding between a red thing and a blue thing then I think we should hold that red things couldn't be blue. But it doesn't. But on the usual way of thinking (if we're not CAIists) it does rule out identity holding between one thing and many things, so if we accept that we should hold that individuals couldn't have been non-individuals.

There are certainly ways of thinking about modality and identity across worlds according to which this isn't a good way of thinking, but on the standard way of thinking I think it just is. And the only disanalogy between the red-blue case and the many-one case is that no-one has a logic of identity whereby the very nature of the relation says that identity never goes from a red thing to a blue thing.

Jeff said...

Jason,

Sorry, I guess I was a bit too telegraphic. By "Among-Thing", I meant the notion of "Thing" defined in terms of "among", the way you suggested: X is an Among-Thing iff every Y among X is identical to X.

It follows that every Among-Thing is a mereological atom: if Y is a proper part of X, then Y is among things that compose X (X and Y, for instance); so by CAI Y is among X; since Y≠X, X is not an Among-Thing. So CAI implies "Among-Nihilism": every Among-Thing is a mereological atom. With me so far?

As for gunk: yes, it looks like there is a sense in which Among-Nihilism is compatible with gunk. That's because Among-Things aren't all we believe in: we also have all these pluralities. The CAI-ist's world of gunk doesn't have any Among-Things at all—but that doesn't mean it has nothing at all. There are pluralities among pluralities that never bottom out in Among-Things. Also, there is a sense in which these aren't "mere" pluralities: since they are all Member-Things. We can perfectly intelligibly ask the kind of questions you pose—they just aren't questions about Among-Things.

I'm not feeling the pull to go on to ask—ok, but what are the real things? It's like being told about relativistic mass and rest mass, and then asking—which one is really mass? Why think there should be any determinate fact of the matter? The CAI-ist should say that our pre-theoretic concept "Thing" is indeterminate between a two different options that reality gives us. Likewise, questions we posed in terms of that concept, like the Special Composition Question, are indeterminate.

You might have reasons to posit some third option as well, besides Among-Things and Member-Things, but I'm not seeing what those reasons would be.

Jason said...

Hi Jeff,

So I think I kind of ran two things together, but let me focus on the one I think is more pressing. So in a gunky model, there are no among things, yes? There are just "pluralities". But notice that I *got my grip* on the values of X being *pluralities* (back in the olden, pre-CAI-days) by thinking about them as "ranging over several things". Now I have these same variables, and I want to keep the same pre-CAI thought that they range over pluralities.

One thought I was having was something like: I'm not sure how I can do this if I don't have the resources to talk about (single) things --- and, in the CAI setting we're looking at, I don't have the resources to do this. But, you (may) reply, I *do*, since (given CAI-universalism) every plurality is also one of the single things that pluralities are pluralities of. At this point, somebody's begging a question, and I'm not sure who it is. So let's set this aside for now.

I think the more important point should have been: you seem to want to insist that we read the metaphysics off the logic -- that any "ontological" question that can't be formed using just the logical resources shouldn't be allowed as a question at all (or, at least, there's something dodgy about asking such a question and we need to work really hard to motivate adding the extra resources we need to ask it.)

But I think this can't be right, because picking a logic doesn't settle all sorts of questions about its interpretation. For instance, picking the "logic of plurals" doesn't settle whether we're dealing with plural quantification over things or (genuinely) higher-order quantification. (Even in the modal context -- it might be genuinely higher-order modal quantification over extensional higher-order ranges of values. And we know they're different, not because they have different logics (they don't, modulo some stuff about the empty set that could easily be finessed) but because their interpretations allow combinations of their resources not otherwise available.)

This means that, to ask about the interpretation of the language in question (the one with no syntactically simple first-order singular quantifier), we have to ask whether or not its variables should be taken as ranging over things (in a plural way) or not. But asking *this* question requires resources not available in the language. And if this question is OK to ask in the non-CAI setting, then the question "is every value of X also a thing?" should be a perfectly good question to ask in the CAI setting. And if it's a good question to ask, we can introduce new resources to ask it.

Jason said...
This comment has been removed by the author.
Jason said...

Quick follow-up to my last comment, in the spirit of full disclosure: I'm more sympathetic to something in the neighborhood of the "read the ontology off the logic" methodology than my last comments probably make it look. But things are pretty subtle here, and (without going into a lot of detail) I think that in Ross's paper the issue hinges on whether the CAI-language without something like "One" is sufficient to state the intuitive motivations for CAI. But also: it's a pretty substantive question as to whether this methodology is right, and it's not clear (given his dialectical aims) we can appeal to it to undermine Ross's claim of non-entailment. The comment above is partly in deference to the details I'm not going into and partly to emphasize the controversialness of the methodological claim.

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