Existence and the Quantifier

The spine page on existence set out the Fregean package: existence is not a first-order property but what the quantifier expresses. This page takes the logic seriously, because the orthodoxy has consequences and costs that only appear when the formalism is examined — and because the choice among quantificational systems is itself an ontological choice.

Objectual and substitutional quantification

Two readings of $\exists xFx$, with quite different ontological implications.

Objectual. The variables range over a domain of objects, and $\exists xFx$ is true iff some object in the domain satisfies $F$. This is the standard model-theoretic semantics, and it is what makes Quine's criterion work: to be committed to a truth of this form is to be committed to something in the domain.

Substitutional. $\exists xFx$ is true iff some substitution instance $F(t)$ is true, for some term $t$ of the language. No domain of objects is mentioned; truth is defined over sentences.

The ontological difference is exactly what the substitutionalist wants. If "there is a prime between 8 and 12" is true because "11 is prime" is true, no commitment to an object attaches, and the criterion of commitment has nothing to bite on. This is a way of accepting mathematical talk without accepting mathematical objects — a rival to the nominalist programs which does not require paraphrasing anything.

Two standard difficulties:

• Not enough names. Substitutional quantification only reaches what the language can name. There are uncountably many real numbers and countably many terms in any ordinary language, so most reals are unnameable and truths about them are lost.
• The truth predicate. Substitutional truth conditions are given in terms of the truth of sentences, so the semantics presupposes a truth predicate for the object language, and stating it requires the metalanguage hierarchy. Critics argue the account is either circular or covertly objectual at the metalinguistic level.

Whether the substitutional reading really avoids commitment is therefore contested. What is not contested is that the choice of reading determines what the criterion delivers — which means the criterion is not the neutral instrument it appears to be.

Free logic

Standard first-order logic makes an assumption that is awkward for ontology: every singular term denotes something in the domain. Consequences:

$$⊢\exists x(x=a)\forall xFx⊢Fa$$

The first is a theorem: it is provable that $a$ exists, for any name $a$. So "Pegasus does not exist" cannot even be stated with "Pegasus" as a name — which is unfortunate, since much of ontology consists of denying that things exist.

Free logic relaxes this. It is free of existence assumptions for its singular terms, admitting terms that denote nothing while remaining classical elsewhere. An explicit existence predicate is added:

$$E!a{=}_{\mathrm{df}}\exists x(x=a)$$

and universal instantiation is restricted: from $\forall xFx$ one may infer $Fa$ only given $E!a$. Now "Pegasus does not exist" is $\neg E!p$ — a straightforward, well-formed claim.

The varieties differ on atomic sentences containing empty terms:

• Positive free logic: some can be true. "Pegasus is a winged horse" comes out true, which suits the Meinongian.
• Negative free logic: all are false. "Pegasus is a winged horse" is false, since there is nothing for the predicate to be true of.
• Neutral free logic: all are truth-valueless.

Note what has happened. The Fregean package held that existence is not a predicate; free logic reintroduces $E!$ as a predicate of individuals. The tension is only apparent — $E!$ is defined from the quantifier and adds no new primitive — but it shows that the slogan "existence is not a predicate" was never quite the whole story. What is denied is that existence is a substantive property that could be part of a thing's nature, not that a predicate expressing it can be defined.

Unrestricted quantification

"What is there?" is meant to be asked without restriction. Is that possible?

The doubt comes from indefinite extensibility. Given any determinate totality of sets, one can specify a further set not in it — the Russell set of its non-self-membered members. So the concept set has no fixed extension, and quantifying over "all sets" is at best quantifying over some sufficiently large domain that could always be extended.

If this generalises to all objects, the consequences for the section's project are severe:

• The question "what is there?" would have no absolutely general answer, only answers relative to a domain.
• The criterion of commitment assumes a determinate domain for the variables to range over.
• Claims like "everything is physical" or "there are no abstract objects" become unstatable in the intended generality.

The defender of absolute generality replies that the argument overgeneralises: it shows that no set contains all sets, which is a fact about sets rather than about quantification. Quantifying over everything does not require a set of everything — the all-in-one principle, that to quantify over some objects there must be a single object collecting them, is precisely what plural quantification denies. If we can talk about some things without a set of them, we can talk about all things without a set of them.

That reply is the most important payoff of the plural resources, and it shows why the choice of logical apparatus is not a technical preliminary to ontology but part of it.

Where this sits

This page belongs to the status question, in the specific form: what does the formal machinery of the section's method actually assume? Three assumptions have now surfaced. That quantification is objectual, without which the criterion delivers no commitments. That singular terms denote, without which standard logic cannot state a denial of existence. And that quantification can be absolutely general, without which the central question is not askable as intended.

None is beyond dispute, and each has a well-developed alternative. That does not undermine the method, but it does show that the neutrality claimed for logic — that it is a common instrument all parties can use — is more limited than advertised. The remaining pages of this folder examine what the alternatives buy: Meinongianism, fictional objects, and the treatment of negative existentials.