Talk:Generalized quantifier

Distinct linguistic bias of this page

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)

An explanation that I've heard re "sentences can't be of type t": If they were, then all sentences with the same truth value would mean the same thing. -- UKoch (talk) 22:41, 10 March 2011 (UTC)

I'm late here but I wanted to say, I think this page does need more linguistics than set theory, either that, or the page Quantifier (linguistics) needs to stop redirecting here and should be unblanked.

This page is sorely lacking from a linguistics and languages standpoint. Furthermore it's grossly lacking in any non-English examples or comparisons of how other languages and language families approach quantities. 2A02:C7C:C47D:8700:9DD3:7D69:CC2D:31DE (talk) 18:09, 29 April 2023 (UTC)

Suggestions for the "Type theory" section

I was reading recently in the "Type_theory" section of the version or "revision" of this article which is currently the most recent one ... namely, the 22:10, 4 April 2022‎ version or "revision" of this article.

Some of the use of language there struck me as being unrigorous in a way that might be acceptable in some contexts, but is probably not appropriate in an article whose "Talk:" page says (in part)

This article is within the scope of WikiProject Mathematics

For one example, there is a bullet item that says:

• Expressions of type t denote a truth value, usually rendered as the set $\{0,1\}$, where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of type t are sentences or propositions.

In my opinion, what it should say (and what would be true) would be [more like]

• An expression of type t has a truth value, which is an element of the set $\{0,1\}$, where 0 stands for (or "means") "false" and 1 means "true". Some examples of expressions that are considered to "have" a truth value of that kind (or, "of that type" ... that is, of the type t) are: sentences and propositions.

There might (also) be some other places in that section of this article with [similar or 'analogous' instances of having] some "room for improvement".

This (the above "bullet item" example) is just one example.

I was almost about to "be bold" and just go edit the article to insert this change (the above mentioned change). However, ... I thought it might be better to elicit some comments first.

*** Any advice or other comments would be appreciated. ***

Mike Schwartz (talk) 18:44, 8 July 2022 (UTC)