Wednesday, March 26, 2008

Absolute Generality 22: Parsons on varieties of Russell's paradox

Parsons, however, doesn't think that the principal problems about quantifying over everything arise from a supposed commitment to metaphysical realism but are "logical difficulties ... [which] arise from considering how sentences or discourses containing quantifiers are interpreted. This apparently innocent talk of interpretation turns out to have considerable weight." Why?

Here's how I think the dialectic goes in the compressed but elegant Section 3 of Parsons's paper (with some changes in notation):

  1. Quantifiers are standardly interpreted as ranging over some domain, predicates are interpreted by subsets of the domain etc. A domain is understood to be a set. In standard set theory, no set contains absolutely everything. (Going for a set + classes theory just shuffles the problem upstairs.) So quantifications aren't over absolutely everything.
  2. But in fact, Parsons says, it isn't specific issues about sets or classes that generate the type of difficulty we encounter here. For consider any style of semantic interpretation for one-place predicates that assigns the open wff F the entity E(F), and which tells us that 'Fa' is true just when o I E(F), where o is the denotation of a, and I is some appropriate relation. (So if E(F) is a property, I is the instantiation relation; if E(F) is an extension, I is set membership; and so on.) Now, suppose that the language in question can itself talk about the entity E(F) and the relation I, so that now -- within the language itself -- we have 'Fa' is true iff 'a I E(F)' is true. Now consider the one-place predicate 'R' defined so that 'Rx' iff 'not-(x I x)', and suppose a is the term E(R). Then, we'll have 'Ra' is true iff 'a I E(R)' iff 'a I a' iff 'not-Ra'. Contradiction. So either there just is no such object as E(R), in which case we have a problem about giving a familiar sort of semantics for the language: or it is not available in the domain of quantification to be picked out, and the language's quantifiers don't range over everything.
  3. But ahah! Maybe the trouble in (1) comes from the idea that semantic interpretation requires us to assign an entity to be the domain. Recall, e.g., Cartwright's familiar animadversions against what he calls the All-in-One principle, the idea that a domain is another object, additional to the objects it contains. And maybe the trouble in (2) comes from the idea that semantic interpretation requires us to assign an entity to be the interpretation of a predicate. Recall, e.g. the possibility of a metaphysics-light Davidsonian style of interpretation where predicates are interpreted by translation. [Then the residue of the generalized Russell paradox, with E(F) being simply F, and R the 'true of' relation is just a familiar sort of semantic paradox. This indeed will lead us to say that the 'true of' had better not be in the language being interpreted on pain of paradox. "So," says Parsons neatly, "the interpretation does require 'ideology' not present in the language interpreted, but it does not require an expansion of ontology. So far so good for the idea that the domain of the variables includes absolutely everything."]
  4. But what, however, if one wants to generalize about Davidson-style interpretations (though, as Parsons notes, it is a moot question when we really need to). Do we get back to the sort of contradiction that we met when considering the ontologically loaded notion of interpretation deployed in (1) and (2)?
  5. If we are going self-consciously to relativize interpretative truth-theories (in a way that Davidson doesn't) preparatory to generalizing about them, then we'll have clauses for a predicate P like this '(for all o), P is true of o according to interpretation I iff Fo'. Now suppose that an interpretation can itself be an object which P can be true of. And put Ro iff not-(P is true of o according to o). Now consider an interpretation J such that (for all o) P is true of o according to interpretation J iff Ro iff not-(P is true of o according to o). Identify o with the interpretation J and we have a contradiction again. [Thus Williamson's version of the Russell paradox argument.]
  6. One response is to continue to allow that J is an object but conclude that it can't fall into the range of the quantifiers, so that the quantifiers can't be running over absolutely everything. So we again get an argument against absolutely general quantification, even though we are no longer thinking that interpretations as themselves ontologically loaded and as assigning objects as domains to quantifiers or entities as interpretations to predicates.

So far, so good! But, as I just said, that's only one response to the Williamson argument. It isn't the only possible one. Parsons mentions (at least) one other line of response at the end of his Section 3, though concludes that "the friends of absolute quantification" face difficulties in the other direction(s) too. But why?

Well, here things get a bit murkier. I'll need to think for a while more ...!

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