Wednesday, December 19, 2007

Absolute Generality 16: Lavine on the problems, continued

(3) "The third objection to everything is technical and a bit difficult to state, and in addition it is relatively easily countered," so Lavine is brief. I will be too. Start with the thought that there can be subject areas in which for every true (∃x)Fx -- with the quantifier taken as restricted to such an area -- there is a name c such that Fc. There is then an issue whether to treat those restricted quantifiers referentially or substitutionally, yet supposedly no fact of the matter can decide the issue. So then it is indeterminate whether to treat c as having a denotation which needs to be in the domain of an unrestricted "everything". And so "everything" is indeterminate.

Lavine himself comments, "the argument ... works only if the only data that can be used to distinguish substitutional from referential quantification are the truth values of sentences about the subject matter at issue". And there is no conclusive reason to accept that Quinean doctrine. Relatedly: the argument only works if we can have no prior reason to suppose that c is operating as a name with a referent in Fc (prior to issues about quantifications involving F). And there is no good reason to accept that either -- read Evans on The Varieties of Reference. So argument (3) looks a non-starter.

(4) Which takes us to the fourth "objection to everything" that Lavine considers, which is the Skolemite argument again. Or to use his label, the Hollywood objection. Why that label?

Hollywood routinely produces the appearance of large cities, huge crowds, entire alien worlds, and so forth, in movies ... the trick is only to produce those portions of the cities, crowds, and worlds at which the camera points, and even to produce only those parts the camera can see -- not barns, but barn fa├žades. One can produce appearances indistinguishable from those of cities, crowds, and worlds using only a minisule part of those cities, crowds, and worlds. Skolem, using pretty much the Hollywood technique, showed that ... for every interpreted language with an infinite domain there is a small (countable) infinite substructure in which exactly the same sentences are true. Here, instead of just producing what the camera sees, one just keeps what the language "sees" or asserts to exist, one just takes out the original structure one witness to every true existential sentence, etc.

That's really a rather nice, memorable, analogy (one that will stick in the mind for lectures!). And the headline news is that Lavine aims to rebut the objections offered by McGee to the Skolemite argument against the determinacy of supposedly absolutely unrestricted quantification.

One of McGee's arguments, as we noted, appeals to considerations about learnability. I didn't follow the argument and it turns out that Lavine too is unsure what is supposed to be going on. He offers an interpretation and readily shows that on that interpretation McGee's argument cuts little ice. I can't do better on McGee's behalf (not that I feel much inclined to try).

McGee's other main argument, we noted, is that "[t]he recognition that the rules of logical inference need to be open-ended ... frustrates Skolemite skepticism." Lavine's riposte is long and actually its thrust isn't that easy to follow. But he seems, inter alia, to make two points that I did in my comments on McGee. First, talking about possible extensions of languages won't help since we can Skolemize on languages that are already expanded to contain terms "for any object for which a term can be added, in any suitable modal sense of 'can'" (though neither Lavine nor I am clear enough about those suitable modal senses -- there is work to be done there). And second, Lavine agrees with McGee that the rules of inference for the quantifiers fix (given an appropriate background semantic framework) the semantic values of the quantifiers. But while fixing semantic values -- fixing the function that maps the semantic values of quantified predicates to truth-values -- tells us how domains feature in fixing the truth-values of quantified sentences, that just doesn't tell us what the domain is. And Skolemite considerations aside, it doesn't tell us whether or not the widest domain available in a given context (what then counts as "absolutely everything") can vary with context as the anti-absolutist view would have it.

So where does all this leave us, twenty pages into Lavine's long paper? Pretty much where we were. Considerations of indefinite extensibility have been shelved for later treatment. And the Skolemite argument is still in play (though nothing has yet been said that really shakes me out of the view that -- as I said before -- issues about the Skolemite argument are in fact orthogonal to the interestingly distinctive issues, the special problems, about absolute generality). However, there is a lot more to come ...

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