Existence-Likeness

I would like to take a stab at defining “existence-like” as characterized in Eklund (2008).

An expression is existence-like if it is translated by “exists” in English.

For starters, this handles the cases of disputes about mereology. If L has some formal mereology, with formal criteria for when things have a fusion, and s is the sentence in L stating that there is no fusion that is a table of a certain number of particles "arranged table-wise", then I think we would translate s in English with something such as "There is no table composed of x, y, z particles." The quantifier in L is translated in English with a "there is" equivalent to "exists".

The definition also handles a number of cases related to deviant logics. Consider a schema with branching quantification such as:

For all w there is an x,
(1) such that F(w, x, y, z).*
for all y there is a z,

This might seem to present a problem, since different theorists will translate (1) different ways. Proponents of branching quantification – and especially proponents of branching quantification as a resource for schematizing the logical form of actual English sentences – will translate sentences of this form more-or-less homophonically (perhaps adding some punctuation or inflection marks). Proponents of classical FOL, such as Quine, will probably translate it with a sentence of the form:

(2) For all x there is an f, and for all y there is a g, such that F(x, f(x), y, g(y).

So a potentially existence-like expression, such as the branched existential quantifier in (1), will be translated multiple ways in English, depending on the syntactic and ontological commitments of the translator. But this does not seem actually to be a problem for our definition of “existence-like”, since both the classical logician and the proponent of branching quantification translate the “there is” of (1) with a “there is”. Where they differ is on the range of values of the existentially quantified variables.

Another problem is the theorist who thinks that “there is” and “exists” are not synonymous or equivalent, or that existential quantifications are not existence claims. This view can manifest itself in a number of ways. The most obvious case would be one in which a theorist uses a language in which the “existential” quantifiers are interpreted substitutionally and are intended to help regiment some fragment of English discourse containing “there is”, but no fragment containing “exists”. Given her preferred regimentation, and the existence of the name “Pegasus” in English, she might consider the following true:

(3) There is at least one winged horse.

The classical logician, who prefers to regiment “there is” discourse with objectual quantification alone, has a few options here. She can translate the substitutionally regimented counterpart of (3), and other sentences of similar logical form, just as (3), but with “there is” interpreted objectually. She can treat the translatum as either true or false depending on her preferred theory and interpretation of fictions and empty names in English. She can also employ semantic ascent, translating the substitutionally regimented counterpart of (3) as:

(4) “There is at least one winged horse” is true.

Again, she can evaluate (4) based on her preferred theory of truth. Perhaps she thinks “true” is implicitly relativized to something more precise than English; perhaps she thinks (3) isn’t truth-apt.

Now, in the case of (3), if the translation is interpreted objectually, then I think it is clear that the existential quantifier in the substitutionally regimented counterpart of (3) satisfies our definition of “existence-like”. Even if (3) is viewed as true, but elliptical (as required by certain theories of fiction or empty names), I can’t see how even the fully explicit interpretation could lack the expression “there is”. And if there is a “there is” in the fully explicit version of (3), I don’t see how we could deny that that “there is” is the translation of the existential quantifier in the substitutionally regimented counterpart of (3), as required by our definition of “existence-like”.

If we take the route of semantic ascent, things get a little weirder. After all “there is” does occur in (4), but it is mentioned, not used. Still, my intuition is that “there is” is the translation in (4) of the substitutionalist’s existential quantifier. One might want to say that “‘there is’”, and not “there is” is the translation, but “‘there is’” does not occur in (4), strictly speaking. (There is no close-quotation mark after the “there is” in (4).) Alternatively, assume that, if a word in a sentence is not treated by a translation as elliptical, or as an auxiliary term in a construction-yielding particle phrase, then part of the translation of the sentence is a translation of the word. Then, since the substitutionalist’s existential quantifier is not being treated as elliptical or auxiliary in the translation (4), there must be a translation in (4) of that quantifier. But that translation obviously has to be “there is”.

(Note that we encounter a similar sort of situation when the classical logician translates second-order quantification. If she doesn’t want to translate second-order quantification objectually into set theory, then she will most likely treat it metalinguistically. For instance, she might translate “There is a P and an x such that P(x)” as “There is a “P” such that ‘There is an x such that P(x)’ is true”. The second-order quantification over P becomes metalinguistic quantification over “P”.)

Recall that our substitutionalist distinguishes between “there is” and “exists” as between substitutional and objectual quantification. What happens when she wants to translate from another substitutional language into English? She will translate existential quantification with “there is”, but not with “exists”. We have seen that, in these cases, the classical logician, who treats “there is” and “exists” as equivalent, can use either of these translations. In this case, then, not only is it unclear what translatum sentence to use (which is not necessarily a problem for our definition of “existence-like”), it is also unclear whether the translatum should contain “exists” as the translation of a given non-English expression. Since the questions about whether and when to use substitutional or objectual quantification are fundamental ontological questions, and we are defining “existence-like” in order to help answer these questions, it would be silly to require that we settle the questions about whether and when to use substitutional or objectual quantification in order to apply our definition correctly. I think we need to modify the definition. The only modification I can think of that works (and is also the simplest) is this:

An expression is existence-like if it is translated either by “exists” or “there is” in English.

This solves our problem, since both the substitutionalist and the classical logician satisfy this definition. This also solves a similar problem, which we haven’t explored, for Meinongian non-English languages.

This solution might create a problem of its own, however. The substitutionalist and the Meinongian have sought to create an interesting distinction between “exists” and “there is”. To some extent, our modified definition erases that distinction. We just observed that we don’t want our definition to trivially settle ontological questions. Has our modified definition done just that?

I think not. The point of a definition of existence-likeness is to enable us to survey what meanings it is theoretically possible to assign to “exists” and “there is” when doing ontology. From a Carnapian point of view, we could say that the point was to find out what terms in what languages have semantic rules that we can use to explicate “exists” and “there is” in English. Our definition indicates that we can use (separately) the Meinongian and the substitutional rules for “there is”, as well as other rules. But surely if these rules determine existence-like uses for expressions, and the rules do not prohibit the use of other rules for distinct expressions (such as the Meinongian or objectual “exists”), then our definition does not prohibit the use of both these sorts of expressions to state an ontology. Simply because we could use the Meinongian rules for “there is” to explicate “exists” does not mean that we could not use distinct rules to explicate “exists” a different way in the same language. So our definition is not problematic for that sort of reason.

* - I'm having trouble writing branching quantifiers on Blogger. The two quantifier strings on the different lines are supposed to be on two different branches.

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?
___________

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).