Friday, August 29, 2014

Quantifier Variance and the Semantics of Quantifiers

In my previous post I explained the basic idea behind quantifier variance. Now I want to criticize it. In particular, I said I want to point out some problems with the quantifier variantist's simultaneously affirming the following two statements:

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

(ii) the different quantifiers have different meanings.

Let's do a little basic semantics. Let's define the truth function τ[ψ]U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:

τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}
  • (τ-)τ[∀xφ]M,s = T ⇔ for all variable assignments s′, if for all variables v, s(v) ≠ s′(v) ⇒ v = x, then τ[φ]M,s′ = 
  • (τ-)τ[∃xφ]M,s = T ⇔ for some variable assignment s′, for all variables v, s(v) ≠ s′(v) ⇒ v = x, and τ[φ]M,s = 

The definition (τ-) could of course be derived from the definition (τ-and the following definition of existential quantification (or vice versa): 

  • (-ABV): τ[∃xφ]M,s = ⇔ τ[¬∀x¬φ]M,s = T

It is easier though for illustrative purposes to use the earlier definitions.

Of course, the definitions could be worded slightly differently, but the slight variations in wording don't make a difference to the argument I want to make. Besides, at the very least, one should admit that these definitions are at least extensionally correct; in other words, they get the truth right all of the time, and thus one could adopt these definitions and get all of the same logical and semantic results as the other variant definitions.

The reason these definitions of truth for quantified phrases are important is that they are the definitions which allow us to prove soundness and completeness. This is strong evidence in their favor. The point about soundness is particularly important. We want to be able to say that our inference rules for our quantifiers are valid, otherwise we would have to abandon these inference rules! With this in mind, and in conjunction with the quantifier variantist's affirmation of (i), he should want to endorse the semantics presented here as holding true for any quantifier, be it c or a.

But this opens a big question. Where is the quantifier variance taking place? The answer seems to be (in all its quantificational irony): Nowhere! If the semantics for c and a have literally the exact same wording then how could the quantified claims (such as (A)) of a compositionalist hold true in a given model while at the same time those of an anti-compositionalist do not? 

Notice that in the definiendum clauses in (τ-) and (τ-) there are meta-language quantifiers. They range over variable assignments and variables. If there is quantifier variance then it's going to have to hold with respect to these meta-language quantifiers. So, once again: Where does the quantifier variantist think the quantifier variance is taking place? 

Presumably there is no variance when quantifying over variables, or at least I would hope not! These quantifiers seem entirely univocal. But maybe there exists variance when quantifying over variable assignments? 

A host of thorny issues arise when we ask these questions. In my next post I'll investigate the issue more. In doing so I hope to make it even more clear what quantifier variance is saying, and also lay down some criticisms based on my own view of the nature of quantification and ontological debates.

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