Quantifier Variance and Deflationism

The most influential modern descendant of Carnap's position is Eli Hirsch's quantifier variance: the thesis that there are many equally good meanings that the existential quantifier could have, that rival ontologies employ different ones, and that their disputes are therefore verbal — disagreements about which language to speak, not about what there is.

The thesis is Carnapian in conclusion but not in argument. It does not need the analytic/synthetic distinction; it needs only the claim that no candidate quantifier meaning is metaphysically privileged.

The thesis

Consider the Special Composition Question. The universalist says there is an object composed of the Eiffel Tower and your left shoe. The nihilist says there are no composite objects, only simples arranged variously. They appear to disagree flatly about what exists.

Hirsch's diagnosis: each speaks a language whose quantifier ranges over a different domain, and each can translate the other's sentences into their own without loss of information.

The translation is systematic and truth-preserving in both directions. Every fact statable in one language is statable in the other. What differs is which things the languages count as things — and that, Hirsch argues, is a matter of how the quantifier is used, not a matter of fact.

The principle of charity does the argumentative work. If a competent speaker sincerely and reflectively asserts "there are no tables," and there is an interpretation of their language on which that assertion is true, charity requires us to adopt it. So the nihilist speaks truly in nihilese, the universalist speaks truly in universalese, and neither is making a mistake. There is no further question of who is really right, because "really" would have to be asked in some third language, and there is no privileged one.

Putnam's conceptual relativity

Putnam's version, which preceded Hirsch's, uses a deliberately simple example. Take a world containing three individuals: $x_1$, $x_2$, $x_3$. How many objects are there?

A Carnapian who counts only the individuals says three. A Polish logician who accepts mereological sums counts $x_1,x_2,x_3,x_1+x_2,x_1+x_3,x_2+x_3,x_1+x_2+x_3$ — seven. Both descriptions are complete, consistent, and describe the same world. "How many objects are there?" therefore has no answer independent of a choice of conceptual scheme.

Putnam supported this with the model-theoretic argument, which draws on the Löwenheim–Skolem results and the general fact that a consistent theory has many models. If our total theory, including all its theoretical and observational constraints, can be satisfied by wildly different interpretations of its vocabulary, then nothing in the theory determines which interpretation is intended — and appealing to a further theory of reference only pushes the problem up a level. The relevant machinery is in the pages on model theory and the metalanguage hierarchy, and the semantic issues in the remark on semantics and the theory of meaning.

This is the same structure as ontological relativity: Quine's proxy functions show that permuting the domain systematically preserves all evidence, so the reference of our terms — and thus the identity of the objects quantified over — is underdetermined. It is a striking feature of the dialectic that the deflationist's best argument is one that Quine, the arch anti-Carnapian, supplied.

Replies

The joint-carving reply. Sider's response is that quantifier variance requires there to be no metaphysically privileged quantifier meaning, and this is exactly what should be denied. Just as some predicates carve nature at the joints and others do not — green rather than grue — some logical vocabulary may carve at the joints too. If "$\exists$" has a uniquely natural meaning, then even if the nihilist's language is internally consistent, it fails to describe reality in its own terms, and the universalist is right in the sense that matters. The position is developed on the ontological realism page.

The metalanguage objection. To state quantifier variance one must say: there are several equally good quantifier meanings. In which language is that said? If in a neutral metalanguage, the thesis presupposes a privileged standpoint of exactly the kind it denies. If in one of the candidate languages, it is merely one more parochial claim. This is a familiar difficulty for relativisms, and it is not obviously fatal — the variantist may say the metalanguage is itself just one more language, and its claims are true in it — but the position becomes harder to state without appearing to undercut itself.

Are the alternatives really languages? Hawthorne and others press whether "nihilese" is a genuine language a community could speak. The translations get long very quickly, and it is unclear that the nihilist's paraphrases preserve inferential role rather than merely truth value. If the translation manual breaks down anywhere, the dispute is not merely verbal.

Not every dispute is a good candidate. The composition debate is unusually amenable to the treatment because the translation scheme is clean. It is much harder to run for mathematical objects: the nominalist does not have a systematic translation of platonist mathematics — that is precisely what the nominalist programs are still trying to build, with mixed success. So even a sympathetic reader should treat quantifier variance as a diagnosis to be argued case by case rather than a general dissolution of ontology.

Verbal disputes

Underlying all of this is a question worth separating: what makes a dispute verbal? The usual test is that a dispute is verbal when the parties would agree on all the facts once each other's terms were clarified, and their apparent disagreement is entirely due to using a word differently.

Applying this to ontology is delicate. Two parties may agree on every non-ontological fact and still take themselves to disagree about what there is — and the deflationist's claim is precisely that in such a case there is nothing left to disagree about. The realist's reply is that "what there is" is itself a fact, not exhausted by the others. The dispute about whether the dispute is verbal is not itself verbal, which is at least a reassuring sign that something substantive is at issue somewhere.

Where this sits

Quantifier variance is the status question answered in the negative, and it is answered by a claim about the quantifier — which is why the innocuous-looking question of whether "exists" is univocal turns out to carry so much weight. Note that the variantist and the ontological pluralist both deny that there is one privileged quantifier, but draw opposite morals: the pluralist concludes that reality has several modes of being to be described, the variantist that there is nothing for the description to get right.

The next page takes up the other modern deflationary route, which reaches a similar conclusion without any claim about the quantifier: easy ontology.