Wednesday, February 29, 2012

Sketches towards a theory of quantifiers and quantifier variance

Quantifier variance theorists think that there can be multiple kinds of quantifiers. Thus, there could be quantifiers that range over only fundamental entities, but there could also be quantifiers that range over arbitrary mereological sums. I will call all the quantifiers with a particular range a "quantifier family". A given quantifier family will include its own "for all x (∀x)" and "for some x (∃x)", but may also include "for most x", "for at least two x", and similar quantifiers. I will, however, not include any negative quantifiers like "for no x" or "for at most seven x", or partially negative ones like "for exactly one x". I will also include "singular quantifiers", which we express in English with "for x=Jones". In fact, I will be working with a language that has no names in it as names are normally thought of in logic. Instead of names, there will be a full complement of singular quantifiers, one for each name-as-ordinarily-thought-of; I am happy to identify names with singular quantifiers for the purposes of logic.

Say that quantifier-candidates are operators that take a variable and a formula and return a formula in which that variable is not open. Consider a set F of quantifier-candidates with a partial ordering ≤, where I will read "QR" as "R is at least as strong as Q", and with a symmetric "duality" relation D on F. There is also a subset N of elements of F which will be called "singular". Then F will be a quantifier family provided that

  1. There is a unique maximally strong operator ∀
  2. There is an operator ∃ dual to ∀
  3. If Q is dual to R then it can be proved that QxP iff ~Rx~P
  4. ∃ is minimally strong
  5. If R in F is at least as strong as Q in F, then from RxP one can prove QxP
  6. From P one can prove QxP for any Q
  7. From ∀xP one can prove P (note: open formulae can stand in provability relations)
  8. If Q is singular, then Q is self-dual and Qx(A&B) can be proved from QxA and QxB
We may need to add some more stuff. But this will do for now.

One can set up a model-theory as well. A domain-model for a quantifier family will include a set O of "objects", and a set S of sets of subsets of O, such that if E is a member of S, then E is an upper set, i.e., if a subset A of O is in E, then so is any superset of A as long as it is still a subset of O. A member of S will be called an "evaluator". To get a model from a domain-model, we add the usual set of relations. An interpretation I in a given model for a language with a quantifier family will then involve an ordinary interpretation of the language's predicates, plus an assignment of quantifiers to members of S subject to the constraints that (a) if QR, then I(R) is a subset of I(Q), (b) I(∀) is the evaluator {O}, (c) if Q is dual to R, then A is a member of I(Q) if and only if the complement OA is not a member of I(R), (d) if Q is singular, then I(Q) is a filter-base. We can then define truth under I using the basic idea that QxP is true if and only if the set of objects o such that o satisfies P when put in for x is a member of I(Q) (this should be all done more carefully, of course).

(The interpretation of a name is always an ultrafilter. If we wanted to, we could restrict names to being interpreted as principal ultrafilters, in which case names would correspond to objects, but I think things are more interesting like it is.)

Ideally, we'd want to make sure we have soundness and completeness at this point. I'm basically just making this up as I go along, so there may be a literature on this, and if there is, there will presumably be results about soundness and completeness in it. And maybe we need more rules of inference and maybe I screwed up above. This is just a blog post. Moreover, we might want some further restrictions about how particular quantifiers, like "for most", are interpreted (the above just constrains it to having an evaluator between the evaluators for ∀ and ∃). The point of the above is more to give an example of what a formal characterization of a quantifier family might look like than to give the correct one.

But now it is time for some metaphysics. The notion of a quantifier family is a purely formal one. Moreover, the model-theoretic notion of interpretation that I used above won't be helpful for quantifier variance discussions because it talks of "sets of objects", whereas what "metaphysically counts as an object" varies between quantifier families.

It is easy to come up with quantifier families and perverse interpretations such that under such an interpretation, we would not want to count the members of the quantifier family "quantifiers". Nor would it be a quantifier variance thesis to say that there are many such families and interpretations, since that there are such is not controversial.

I think a Thomist can give an answer: a quantifier family in the formal sense is a bona fide quantifier family provided that the family is analogous to some privileged family of quantifiers, say quantifiers over substances. In other words, the different kinds of existence are defined by analogy to existence proper. This won't satisfy typical quantifier variance folk, as I think they don't want a privileged family of quantifiers. But that's the best I can do right now.


Alexander R Pruss said...

Sounds like what I was grasping towards is the idea of generalized quantifiers.

The ones that interest me are monotone increasing (I said I was excluding negative or partly negative ones).

Alexander R Pruss said...

Let me add that I think it's central to this post that quantifier variance must come along with name variance (it's well-known that if you have two quantifiers with the usual intro and elim rules involving names, then you can prove them identical. I think name variance implies something like predicate variance. A predicate like "has a mass of 2kg" needs to be interpreted differently in the case of different names. This interpreting job is done by the name quantifiers.

Suppose we have one quantifier family over simples and the other over mereological sums. If "For x=John" is a singular quantifier of the second family, then "For x=John, x has a mass of 2kg" might get the interpretation: "For x=John, the sum of the mass-energies of the simples in x is 2kg."

And if we let our sets have ur-elements, we get set-variance, too.