Talk:Generalized quantifier

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I created this page with a distinct linguistic bias. It may be better to add a more mathematical perspective as well, but I think the field of generalized quantifiers is among the few where mathematics has been influenced by linguistics. Thta` said, I'm open to suggestions. I know that the section on properties of GQs is incomplete. It doesn't discuss important properties like continuity, the strong/weak distinction, etc. Also, I wasn't entirely happy with adding long sections on type theory and the lambda calculus, but I didn't see how else to do justice to the topic. Hence: Comments and improvements are welcome! Neither 19:39, 16 November 2006 (UTC)

First, on a cursory glance I found the article good though incomplete. Second, I found the remark "After work by the philosopher Gottlob Frege, we know that sentences can't really be of type t" strange. Is that a philosophical claim that sentences are not bearers of truth, and that, for example, propositions (whatever those may be) are instead? It's stated as if that's a fact when it is highly controversial.

You mention only monadic quantifiers which are easily generalized as seen by the straightforward algebraic truth conditions to which they give rise. But what about polyadic quantifiers? Suppose we wish to say that ∃x₁x_n(A & B), where A and B share more than one free x_i and have a different number of free variables. We cannot say it is true if the intersection of A and B is nonempty, because it will always be nonempty since A and B have a different number of free variables. The satisfaction relation from which truth in a model is defined must either deal with cases in which n-tuples by convention satisfy m-ary predicates for n>m by e.g. "ignoring" elements of the tuple after the mth, or do something else that I cannot think of. E.g., we can say it is true in a model M if the intersection of {<a₁,...,a_n>: M |= A[a₁,...,a_n]} and B^M (the interpretation of B in M) is nonempty, where A has m free variables and B has n free variables, for m<n. How is this usually done? Nortexoid 02:22, 13 January 2007 (UTC)

Well, Frege's point is hard to pin down, I guess, but one way of doing it, due to Carnap, is to say that you need to treat sentences as intensional (i.e. relativized to worlds, or interpretation functions), rather than purely extensional. I don't think a thorough discussion of that belongs here, though, so feel free to delete that passage. I agree that it would seem opaque to people who don't know about it.

About polyadic quantifiers, there's some discussion of it within the linguistic literature, which I find very interesting, but in this first version of the article I chose to abstain from discussing it. Again, feel free to add stuff: my version was the first stab at an article about GQs, and there was a long standing request for one. But it surely isn't the final version. :) Neither 02:49, 14 January 2007 (UTC)