Saturday, April 19, 2008

Is Existence a First-Order Property?

I think I’ve found a not-so-bad argument for construing existence as a first-order property. Here goes:

Begin with the principle that first-order properties are those things in virtue of which a sentence of the form “x is not identical to y” is true, for some substitution of individual-symbols for “x” and “y”. So existence is a first-order property if some such statement is true solely in virtue of the truth of a corresponding sentence of the form “x exists” or “x does not exist” – or, if you like, corresponding sentences of the form “there is a z, such that x = z” and “there is no z, such that x = z.” Let c1 =def the bottom tennis ball in a certain canister that can contain as many as three tennis balls; let c2 =def the middle tennis ball in that same canister; let c3 =def the ball immediately above c1 in the canister. Now suppose that there are actually only two balls in the canister. Then c3 is above c1, but c2 is not, since c2 does not exist. It seems that ~(c2 = c3).

I would suggest that this is the case just because c2 does not exist. Supporting this contention is the apparent fact that if c2 did exist (i.e. if there were some ball, such that that ball is the middle ball in the canister; if there were some ball, such that that ball is c2), then it would be the case that c2 = c3. But if ~(c2 = c3) just because c2 does not exist, then by our initial principle about first-order properties, existence is a first-order property.

Does this work? Am I missing something? I don’t really understand the necessity of identity. Is some abuse of that principle at work here? Or is my beginning principle about first-order properties false? Or is it actually not the case that ~(c2 = c3) – say, because “~(c2 = c3)” lacks a truth-value?
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Update:

In a comment at the Maverick's, Ockham, writes: "You cannot say 'c2 =def the middle tennis ball in that same canister' and then say 'but c2 does not exist'" Here's my response, which is too long and too irrelevant to the discussion of the Maverick's post to leave on his blog.

I suppose that, in classic FOL, you can't introduce a term through a definite description that doesn't refer. FOL aside, though, do you mean to say that this is incoherent in natural English? "The middle ball in that same canister doesn't exist" does sound strange to me. Admittedly, it sounds more natural to say "There is no middle ball in the canister," which we might formalize "~(Ex)(Middle ball in the canister(x))." It doesn’t sound unnatural to introduce a proper name through that definite description, however. Nor does it sound unnatural to me to say “c2 doesn’t exist” (even though “the middle ball in the canister” is the definition of “c2”), just the same way it doesn’t sound unnatural to say that “Pegasus doesn’t exist.” The big difference between “c2” and “Pegasus” seems to be that “c2” was obviously introduced by a definite description, whereas “Pegasus” was not obviously so introduced. I’m not sure that this difference is enough to make “c2 doesn’t exist” incoherent and “Pegasus doesn’t exist” coherent. Still, we might imagine situations in which a term like "c2" is introduced – say, by a group of people who believe, incorrectly, that there are three balls in the canister – otherwise than by the definite description “the middle ball in the canister.” (Briefly: maybe the group could use a function from canisters to the middle ball in the canister. Maybe they could just start using “c2” to (try to) refer to what they believe to be the middle ball by stating facts that would be true of such a ball, were it to exist.) In any event, I’m not sure that “c2 doesn’t exist” is incoherent, and if it is, we can use other stories about how something like “c2” comes to have a certain meaning, and then tell a story similar to the one I tell about ~(c2 = c3).

4 comments:

Richard said...

I'd go for the last option, on the grounds that 'c2' does not refer. So it can't be identical or non-identical to anything. It's just nonsense.

iolasov said...

Thanks for the comment, Richard!

I'm tempted to go for the last option as well - I'm friendly to truth-value gaps, and kinda don't want to think that existence is a first-order property. But if that's the way out, it can't be simply because "c2" doesn't refer. All sorts of statements asserting identities and non-identities between non-existent relata seem to be true. Pegasus is Bellarophon's winged equine companion; Pegasus is not He-Man. There might be some relevant difference between the referential failure of "Bellarophon's winged equine companion" and that of the definition of "c2", but it's hard to say what that could be.

Richard said...

"Pegasus is Bellarophon's winged equine companion" arguably isn't literally true. Instead, we may think it's just "true in the fiction", or some such. (Granted, charitable listeners will reinterpret what's said in order to make it informative and true; but the point is that this involves turning to something other than the literal meaning.)

Unless we're Meinongian, we won't think there's some thing, Pegasus, which has the property of being Bellarophon's winged companion. There is no such thing. (And similarly for talk invoking 'c2'.) If we think such talk is true at all, we will opt for some more indirect account of its truthmakers.

iolasov said...

Well, I'm no Meinongian yet, but I guess I'm assuming here that some primitive sentences containing empty names or descriptions are literally true or true without qualification in natural English. I take sentences like "Pegasus is Bellarophon's winged equine companion" and "~(c2 = c3)" to be just the sort of sentences that satisfy this assumption, if any sentences satisfy it at all. Maybe they (or at least the latter) are just "true in the fiction", or are best understood non-cognitively, in which case using them to support claims about the logic of ordinary, literal discourse is a little fishy. Maybe my case should just be that *if* we understand these sentences to be cognitive and literal(ly true), then the referential failure of "c2" doesn't present any obvious or prima facie problem for the argument in the post.