# Is there an approach to quantification theory that construes quantifiers as subset creators?

In formal semantics, the theory of generalized quantifiers analyzes quantifiers (e.g. all, some, no, most, few, etc.) in terms of sets to sets. The meaning of the quantifier some, for instance, is understood in terms of two sets, say, set F and set G; some, then, denotes the intersection of F and G (F ⋂ G), as expressed in the copular sentence Some Fs are G. This meaning is sometimes visualized using Venn diagrams. One circle, labeled as set F, is drawn in such a manner that it overlaps with another circle, labeled set G. The precise area where the two circles overlap is the intersection of F and G and is then understood as representing the meaning of the quantifier some.

For me, this sort of approach to the meaning of quantifiers is counterintuitive. The difficulty in my view is that to understand the meaning of a given quantifier, one has to acknowledge the meaning of the main predicate of the sentence in which the quantifier appears. Thus, to represent the meaning of the noun phrase All men in the sentence All men are mortal, one has to make reference to the meaning of the predicate are mortal. This is counterintuitive, since the meaning of the noun phrase All men is for me just as complete as, say, the meaning of a constant such John or the president.

My question concerns the potential of an alternative approach to the meaning of quantifiers. Instead of sets to sets, one views quantifiers as subset creators, that is, in terms of subsets of sets. One would still access set theory to interpret the meaning of quantifiers, but instead of acknowledging two distinct and independent sets that do or do not overlap to a greater or lesser extent, one acknowledges that the one set is a subset of the other. Thus if we consider the meaning of Some men in the sentence Some men are happy, we understand it in terms of the following Venn diagram:

The quantifier some serves in this case to establish a subset of the members that make up the set of men. The quantifier two, e.g. two men, serves to establish a subset with a cardinality of two of the set of men; the quantifier most, e.g. most men, serves to establish a subset of the set of men such that that subset includes more than half of the men, etc.

Has this sort of approach to the meaning of quantifiers already been developed by anyone? If not, why not? Is there a reason why such an approach would not work? Any and all commentary and guidance in this area will be much appreciated.

• Quite possibly missing something but have to ask whether these options are really so different - to say that two sets overlap is to say that there is a subset of the first that is also a subset of the second, isn't it? If so, some men does designate a subset of men, and it's just that you don't know which until you hear the rest of the sentence. How are you going to draw a subset of set A which is also a subset of B except by showing A and B as overlapping? All men is cheating because there is only one "subset" it can possibly be, so you don't have to wait for the rest to know which it is. Commented Jun 30, 2021 at 23:07
• @rchivers "some men does designate a subset of men, and it's just that you don't know which until you hear the rest of the sentence." No. In the traditional theory of generalized quantifiers, "some men" does not denote a subset. It denotes a set of sets. But "to say that two sets overlap is to say that there is a subset of the first that is also a subset of the second, isn't it?" - yes, in the particular case of "some" whose meaning is overlap, those sets that are among the specified set of sets will also automatically also be subsets of the man set, for the reason you give. Commented Jun 30, 2021 at 23:10

I should perhaps first clarify how the traditional theory works:

Quantifiers with a slot for a subject restriction, like some, every, do not denote sets of sets. They denote binary relations between sets: [[some]] = the set of all pairs of sets F, G which overlap.

It is the quantifiers combined with a subject noun phrase that denote sets of sets: [[some men]] = the set of all sets G which intersect with the set of men. The quantified subject is formed by plugging in the subject noun phrase in for the first argument F in the quantifier relation.

The quantified sentence is then formed by plugging in the verb phrase for the second argument G: [[some men are mortal]] = true iff the set of mortals is among the set of sets which intersect with the set of men, i.e., true iff men and mortals intersect, and false otherwise.

Unrestricted quantifiers that can not accommodate another noun phrase and straightforwardly combine with a verb phrase, like something, someone, are of the same type as a restricted quantifier + subject restriction, they denote sets of sets.

In the computation of [[some men]], there is no explicit reference to the set of mortals. [[some men]] is simply the set of all sets which intersect with the men. It does not have to be established at this point which of these sets coincides with what is comprised by being mortal, or any predicate for that matter. The sets are anonymous. [[some men]] consists of any conceivable collection of individuals so long as it has at least one element with the men in common; these arbitrary sets exist independently of linguistic expressions that describe them, many of them may not even fit a single concise description at all. It is only when combining the quantified subject with the verb phrase in question that [[mortal]] has to be evaluated, in order to establish whether what it extends to coincides with one of those sets that intersect with the set of men, i.e., whether what man comprises and what mortal comprises have at least one element in common. But before putting together the subject and the verb phrase, [[mortal]] is nowhere needed in the definition of [[some men]].

If I am understanding you correctly, you are proposing that quantified determiner phrases like some men do not denote sets of sets, but sets of individuals, with the requirement that this set always be a subset of the subject noun phrase. Then a quantifier must be not a relation between sets (something where you put in two sets and get out yes or no), but a function from sets to sets (something where you put in a set and get out another set): some would take a set and return some subset of it.

What would this function look like? Which subset should be assigned to the set of men such that it fits the nature of "some"? And how should it combine with the verb phrase predicate? Which relation does the set of mortals need to have with which particular subset of the men?

If a subject with the quantifier incorporated denotes a set and a verb phrase denotes another set, what is the additional rule that relates them? With the traditional theory of generalized quantifiers, a quantifier combines with two predicates and gives you 'yes' or 'no', depending on whether or not the two sets stand in the specified relation, so that the sentence can be composed to have a truth value. If your quantifier no longer does the job of comparing two sets, and instead gives you some subset of the subject noun phrase and leaves you with two sets (the quantified subject and the verb phrase), what other hidden syntactic element makes sure that these two sets combine to something which can be true or false, and how?

And how would you go about reasonably defining a quantifier like no (as in No man is mortal) while demanding that no man be a subset of the set of men, when the meaning of no is precisely that whatever is predicated in the verb phrase does not share any elements with the set of men?

I see no easy way to define the standard quantifiers and the rules for their composition in this way.

I am also not convinced that such an assignment from sets to subsets would be more intuitive than the conventional theory of generalized quantifiers.

I agree that the conception of a set of sets, as the denotation for a half-baked sentence like some men with the subject noun phrase already plugged in but the verb phrase out of the picture, is hard to make sense of intuitively. Simply put, some men describes the properties which apply to some men; assuming that a property corresponds to a set of individuals, a quantified determiner phrase should thus be a set of such sets of individuals.

But it is probably easier not to overthink this intermediate composition step and instead plug in the two sets simultaneously: some denotes a relation between two sets, namely the relation of having a non-empty intersection. You plug in the men and the mortals, and get out 'yes' if they overlap, and 'no' otherwise. all denotes a relation between two sets, namely the relation of the one set being a subset of the other. You plug in the men and the mortals, and get out 'yes' if everything that falls under the description of a man is also inside the mortal circle, and 'no' otherwise.

A quantifier with two slots compares two sets. A quantified determiner phrase with the quantifier combined with the subject collects which properties apply to the subject in the specified way. A quantified sentence with the quantifier combined with the subject and the verb phrase has a truth value, depending on whether the subject and the verb phrase stand in the specified relation. I think this makes perfect sense intuitively.

• Thank you for the insightful answer! I have read it carefully. Let me explain my motivation for posting the question. I am endeavoring to combine dependency syntax (i.e. dependency grammar, DG) with formal semantics. Most work in formal semantics relies heavily on strict binarity of branching in the syntax to address the semantics-syntax interface. Dependency syntax does not acknowledge this strict binarity of branching. Thus, formal semantics and DG tend to ignore each other. I am currently probing to see how the chasm can be bridged. Commented Jul 1, 2021 at 1:42
• One important step to bridging the chasm would be to allow for a more flexible understanding of predication. In the sentence Fido chased some cat, one wants to be in a position to view Fido chased as denoting a set, one that then intersects with the set of cats. Standard phrase structure syntax relies on unseen movement in logical form (LF) to move some cat above Fido chased to accomplish this goal. Dependency syntax is not comfortable with the unseen manipulations. Commented Jul 1, 2021 at 2:01
• In any case, I have a follow-up questions and comments. But to try to go over them here would be too difficult. Perhaps you are open to an email exchange. My email address is [email protected]. I would, for instance, like to get your views on cases like the one just mentioned, that is, cases where the quantifier is in the object noun phrase instead of in the subject noun phrase. Commented Jul 1, 2021 at 2:07
• @TimOsborne Others might benefit if you were to post further questions here rather than asking them in a private email exchange... Commented Jul 1, 2021 at 2:25
• Rchivers, you could be included in any email exchange. Just email me. I fear that the discussion would quickly become dense and technical and hence of little interest to most in this forum. Commented Jul 1, 2021 at 4:43

Yes, it's been developed in the early 1980s, read Jerry Hobbs' paper "An Improper Treatment of Quantification in Ordinary English". To combine dependency syntax with formal semantics you could use linear logic, it's been used in unification grammars so it'll work equally well for DG.