Negative Existentials and Empty Names

"Pegasus does not exist." The sentence is true, and saying it is a paradigm of what ontologists do — most of the section's positions consist in denying that something exists.

It is also, on the most natural semantic theory, impossible. If the meaning of a name is its bearer, and "Pegasus" has no bearer, then the sentence has no subject and expresses nothing. If it does express something, that something is about Pegasus — so there is a Pegasus for it to be about, and the sentence is false.

This is the problem of negative existentials, and it matters here because the section's entire practice presupposes a solution.

The problem stated

The difficulty is a small inconsistent set:

1. "Pegasus does not exist" is true.
2. A subject–predicate sentence is true only if its subject denotes.
3. "Pegasus" does not denote.

The solutions differ over which to deny, and each denial has been developed into a full theory.

Russell: eliminate the name

The classical solution denies (2)'s applicability by denying that "Pegasus" is a genuine subject at all. Ordinary proper names are disguised definite descriptions — "Pegasus" abbreviates something like "the winged horse captured by Bellerophon" — and descriptions are analysed away:

$$\neg \exists x(Px\wedge \forall y(Py\to y=x))$$

The sentence says that nothing uniquely satisfies the description. It has no singular term, so nothing needs to denote, and it is true. This is the manoeuvre Quine calls the razor sharp enough to cut through Plato's beard, and it is why the criterion of commitment can hold that names carry no ontological weight.

Its cost is the description theory of names, which Kripke's Naming and Necessity dismantled. Names are rigid designators and are not synonymous with any description: Gödel would still be Gödel had he not proved incompleteness, whereas "the man who proved incompleteness" would then pick out someone else. If names are not descriptions, they cannot be eliminated in Russell's way, and the solution loses its foundation.

Free logic: allow empty terms

Deny (2) directly. In a free logic, singular terms need not denote, and the sentence is regimented with an explicit existence predicate:

$$\neg E!p$$

"Pegasus" is a genuine name that happens to denote nothing, and the sentence says exactly what it appears to say. This is the tidiest formal solution and it is compatible with Kripke's semantics for names.

What it does not do is explain the semantics: how does a name with no bearer contribute to the meaning of a sentence? Free logic provides a well-behaved formal system and leaves the philosophical question open.

Meinong: let the name denote

Deny (3). "Pegasus" denotes an object that does not exist, and the sentence ascribes non-existence to it. The truth conditions are as transparent as they look, and no paraphrase is required.

This is the Meinongian's strongest case, because negative existentials are exactly what the theory was built to handle. It is also where the theory's cost is clearest: the sentence is true because there is something that lacks existence, which requires the being/existence distinction the rest of the section's method denies.

Pretense: deny the assertion is straightforward

Deny (1), or rather qualify it. On Walton's account, uttering "Pegasus does not exist" is a move that steps outside a pretence: one engages the myth far enough to invoke it, then disavows it. What is communicated is roughly that the pretence involving "Pegasus" is not answerable to anything real.

The account handles the intuitive force of the utterance well — there is something performatively odd about denying existence, and the pretense theory explains the oddity. It is criticised for making a plain assertion into a complicated speech act, and for difficulty with embedded occurrences: "if Pegasus does not exist, then the myth is false" is a conditional, not a performance.

Metalinguistic: change the subject

A different denial of (2): the sentence is not about Pegasus but about the name. "Pegasus does not exist" says that "Pegasus" does not refer — or, in the more sophisticated versions, that the associated file, dossier, or practice of using the name has no object at its origin.

The reading is natural for some cases ("there is no Santa Claus" plausibly is about a story and a word) and forced for others. It also faces the objection that the two claims differ in modal profile: the name might not have existed while Pegasus still would not have existed, so they cannot be the same claim.

Comparison

No option is cost-free, and the current situation is that the free-logical treatment is standard as a formalism while the semantic question remains open. That is a tolerable position for the section's purposes: what the method needs is a way of writing down denials of existence, and free logic supplies it.

Where negative existentials sit

This belongs with the status question, as a check on the section's own vocabulary. Every eliminativist thesis in the preceding pages — there are no universals, no numbers, no composite objects, no tables — is a negative existential, and if such claims were unstatable the disputes would be unstatable too.

The connection to the theory of meaning runs deep, and the book's treatment of reference, truth-conditions, and what a semantic theory must deliver is in the remark on semantics and the theory of meaning. The moral for ontology is the one this folder has been accumulating: the logical apparatus that looked like a neutral instrument turns out to embody substantive commitments, and choosing it is part of the ontology rather than a preliminary to it.