Tuesday, May 10, 2016

Quantifier and Predicate Variance

Suppose we decide to speak with a quantifier family (a quantifier family includes ∃ and ∀, but may also include things like "many" and "most" and "at least three", and maybe even two-place quantifiers, all ranging over the same domain) that makes arbitrary pluralities of things have a fusion, i.e., a mereologically universalist quantifier family. According to the Quantifier Variance thesis, this decision to extend quantifiers is a linguistic decision that needs merely pragmatic justification, a decision that introduces a quantifier perhaps different from the one most ordinarily used.

This sounds like a decision solely about quantifiers. Not so. For now we need to say something about the meaning of predicates applied to variables bound by these quantifiers. For instance, we need to be able to meaningfully say using the new "there is" whether there is something whose mass is 455 tons, i.e., whether ∃x(Mass(x, 455T)). (Using van Inwagen quantifiers which range over simples and organisms, unless there are alien organisms much larger than blue whales, there isn't anything of that mass.) We can give a semantics for the universalist quantifiers in terms of plural quantification, but we need to account for how much a plurality masses. Intuitively, the mass of a plurality is the sum total of the masses of the simples in the plurality (some technical problems: what if there is gunk? do we count the mass-equivalent of the energy of the bonds between the simples?), and so ∃x(Mass(x, 455T)) provided that there are ys which plurally mass 455T. I suppose extending mass in this way once one has extended the quantifiers is pretty obvious.

But not all predicates extend in an obvious way. For instance, consider the predicate that says that something is spatially extended. Does that predicate apply to the fusion of the number seven with the Empire State Building? Here we have a decision to make, roughly about what it is to be plurally extended: Do we say that for the ys to be plurally extended, each one of them must be extended, or is it enough that one of them is extended? The former decision will fit better with the intuition that extended objects are material. The latter with the intuition that occupying space is sufficient for extension. And either way, we will have some technicalities (can't a plurality of unextended points make up something extended?). Or take the causal relation. Did the people who built the Empire State Building cause the fusion of the number seven with the Empire State Building? It's hard to say. And aesthetic properties will be particularly hairy (is the fusion of Beethoven's Ninth with Michelangelo's David beautiful? is it a work of art?).

Thinking about such examples makes it clear that the linguistic decision to speak with a new quantifier family needs to come along with a correlate decision about the semantics of predicates extended to work with this new quantifier family (a decision that could, sometimes, be simply to leave a predicate underdefined or vague). This means that it is a bit misleading to talk of "Quantifier Variance". The relevant thesis is "Quantifier and Predicate Variance". (And we may also need to have "name variance", unless we consider names to be a kind of quantifier.)

Note that none of this is an issue if one creates a new quantifier merely by domain restriction. It's domain expansion that generates the problems.


Heath White said...

Shouldn't the view be that the predicate "is spatially extended" continues to have its present semantics, but gets applied to new objects? Now, at present that predicate may be undefined as applied to a composite of concrete and abstract objects; and it would be useful to have it defined if we begin recognizing such objects. But one COULD continue to say that whether the predicate applies is undefined, and one SHOULD say this barring (useful) revision of linguistic practice.

Alexander R Pruss said...

That's a possible linguistic decision, too. But it's also a decision. It seems to be a part of the ordinary meaning of "is spatially extended" that everything either is or is not spatially extended. To drop that is to shift the meaning.