## Monday, August 25, 2014

### Basics of Quantifier Variance

When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.

Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms.

In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:

∃x1∃x2...∃xn((x1≠x∧ ... ∧ x1≠xn+1) ∧ (x2≠x3 ∧ ...  x2≠xn+1) ∧ ... ∧ (xn≠xn+1))

(B) could be done pretty easily too, but having a translation of (A) is enough to characterize the dispute between compositionalists and anti-compositionalists. Compositionalists assert (A) whereas anti-compositionalists deny it. So it seems that the two parties are disagreeing here, and thus only one view can be correct.

However, appearances are misleading, or at least so the quantifier variantist says. Quantifier variantists assert that, in the hands of a compositionalist, (A) will mean something different than in the hands of an anti-compositionalist. Thus, when an anti-compositionalist denies (A) and asserts not-(A) she is saying something different than if the compositionalist were to deny (A) and assert not-(A). Hence, the two parties are not actually disagreeing but merely talking past one another.

It was important to translate (A) as we did into our sparse quantified language. The translation makes it clear that, assuming identity, negation, and conjunction are not ambiguous, the only possible connective left for the two parties to disagree about in their interpretation is the existential quantifier. Hence the name 'quantifier variance'. According to the quantifier variantist the existential quantifier can mean something different depending on the context of assertion.

On the face of it this may seem like a trivial thesis. Of course we can interpret the symbol '' however we want. But quantifier variance is saying more than this. First off, for ease of discussion, I will refer to the existential quantifier plus the putative compositionalist interpretation by 'c'. I will refer to the existential quantifier plus the putative anti-compositionalist interpretation by 'a'. At the expense of some precision, I will talk as though these symbols are actually different quantifiers. Really though they just refer to the normal '' symbol along with the allegedly different meanings assigned to it.

With that said, quantifier variance isn't just a restatement of the triviality that '' can have multiple interpretations. It is also saying that 'c' and 'a' behave the same logically speaking insofar as they both have the same logical rules of use associated with them. These include rules like existential generalization and existential instantiation; these are legal inference patterns for both quantifiers. They also behave syntactically as quantifiers (and thus cannot be used as names, predicates, etc.).

Most importantly, quantifier variance claims that, under each quantifier, claims about the world are objectively true or false. Under 'c' the claim that there are n+1 things is true, while under 'a' the claim that there are n+1 things is false. This is not because the way the world is is somehow indeterminate or dependent on one's perspective or conceptual scheme. It's simply because both of these meanings are equally good at describing the way the world is while at the same time remaining different ways of doing so. The two quantifiers are saying different but equally true things.

Quantifier variance says that this is what happens in metaphysical disputes between compositionalists and anti-compositionalists. Both sides are asserting true things about the world, it's just that their evaluative attitudes are not about the same proposition because of their respectively different quantifier meanings.

In my next post I will try to elaborate on and criticize the thesis of quantifier variance on the basis of some semantic considerations. In particular, I will look at the tension which comes from the quantifier variantist's simultaneously affirming:

(i) the different quantifiers behave the same logically; and

(ii) the different quantifiers have different meanings.