# Modelling the quantifier theory in a cognitive theory of language structure

Any cognitive theory needs to be able to model the meanings of words such as all, each, every, some, most, many. If the representations in (11) are reasonable ways of modelling the, then it should be possible to model the in a way analogous to a restricted set in any cognitive theory that has a way of modelling a proportional quantifier like most. Hudson (2007) gives representations for cardinal sets (two dogs) and for every. Every is a bit complicated as it is singular in the morphosyntax and semantically plural, so I shall not present those analyses here; instead, I shall work up an analysis of proportional quantification by looking at the semantics of conjoined NPs. Hudson (2007: 34-35) presents analyses of the three sentences/interpretations in (13).

(13) a. John and Mary bought a house together. (collective)

b. John and Mary each bought a house. (distributive)

c. John or Mary bought a house. (arbitrary member of the set)

The diagrams that follow are taken from Hudson (2007: 34-35); the first diagram shows an analysis of the collective interpretation of John and Mary bought a house. I take each of the diagrams in turn, and explain how they capture the intended meaning.

In the diagram Figure 4, the agent of the 'buying' event (the arrow labelled "Er") has as its argument a concept which is a set.[1] The set is defined explicitly: its nature as a set is shown by the inheritance link from the category 'set' to the '1', which indicates that there is just one set that is the Er of the 'buying', showing that there is only one act of buying. The set is also defined ostensively, in that it is the instance of the type 'set' which has 'John' and 'Mary' as its members and which is the Er of this particular instance of the type 'buying'. In the latter way, the diagram shows that the buying event is a token of 'buying', not the type itself. In summary, the diagram says that the house was bought by the set of John and Mary.

A distributive interpretation is a bit more difficult to represent. In order to capture distributive conjunction, we need to show that it is both members of the set that buy the house and that there are two buying events, therefore.

The diagram Figure 5 says that the Er of 'buying' is the typical member of the set; the set is, like the previous set, defined as a set with two members, John and Mary. Since both John and Mary are instances of the typical member (John Isa typical member and Mary Isa typical member) they both inherit the relation to the Er of the event. The diagram does not permit me to show the way in which there are two events; in order to do this, it is necessary to show more of the verb's

Figure 4. Collective interpretation of John and Mary bought a house

Figure 5. Distributive interpretation of John and Mary bought a house

Figure 6. Distributed interpretation of John and Mary bought a house with plural events

semantics as well. In Figure 6, I add this next part of the complexity. Figure 6 draws on the analysis of Two researchers wrote three articles in Hudson (2007: 230).

In Figure 6, the referent of bought is shown as the typical member (i.e. any instance) of a set which Isa (is an instance of) a buying event where John Isa the typical member and Mary Isa the typical member. In this case, the cardinality of the set is defined by the cardinality of the set of house buyers so, just as there are two buyers, there are two buying events. The analysis relies on WG's intramental notion of reference, which allows verbs to refer, so that the conceptual structure permits the same tools to quantify over events and things.

The previous two diagrams force a universal quantification, because both John and Mary are involved in the house buying, either jointly or separately. What if we have a set conjoined by or rather than and? Hudson (2007: 35) says, "The effect of changing and to or is much the same as that of changing universal to existential quantification, because we change from 'every member' to 'some member'." As we shall see, this difference is important because the forces both a universal and an existential quantification, whereas a(n) only involves existential quantification.

In Figure 7, an arbitrary member of the set is chosen as the agent of 'buying'; I have shown that arbitrary member with a dotted line to show that it is not the same as the other two 'member' relations: 'member 1' and 'member2' define the set the set is the set of John and Mary. The third member relation, 'm3' in the diagram is just an arbitrary member that is bound to one or other of the actual members, but crucially neither of the actual members Isa m3. The dotted line is a notational convenience to show this arbitrary property.

The way the existential quantification works is through inheritance: "any node X always means 'every X', [...] if X isa Y, then it is merely 'some Y', so its properties are not inherited by other instances of Y" (Hudson 2007: 33). By linking the Er

Figure 7. John or Mary bought a house

of 'buying' to an instance of a relation which is a member of the set, the diagram shows that the buyer is one of John and Mary as long as the "1" which is the Er of the 'buying' event, and "m3", is bound to John or Mary.

So how do we extend this to the? One way is to exploit the treatment of sets developed for collective and distributive conjunction above, and for or, and to build a semantics in set terms. This would give a WG account of Neale's restricted quantifier treatment of definite articles, which could be extended to other proportional quantifiers. Although our intramental semantics does not allow us to exploit Neale's truth statements, we can, however, also take an insight from them, and set a cardinality statement on our sets for singular and plural the: in this way, like Neale I should be able to use the same basic representation for singular and plural the, with the difference between them being limited to a statement of set size.

However, there is a small wrinkle: like conjoined NPs, plural NPs determined by the can have either collective or distributive interpretations. If I say the researchers wrote a paper, you can interpret it either as (a) they wrote the paper together, or (b) they each wrote a paper, so in the end there were two papers. We need, therefore, to propose a semantics which captures this difference. This will be an advance on Neale (1990), which does not show the difference between collective and distributive plural definite NPs.

The first representation, in Figure 8, gives a set representation of the; I analyse the sentence The cat played. My diagram says that the Er of 'playing' is a set, which has one cat as its member.

For the diagram to capture the insights of Russell's (1905) analysis, it has to capture both the universal and existential commitments which are expressed in (3), the argument being that the sentence the King of France is bald asserts the existence of the King of France and that there is only one King of France. The universal

Figure 8. the is a quantifier, singular np

Figure 9. the is a quantifier, plural np

Figure 10. Distributive interpretation of a definite np

quantification is achieved by stating that it is the set which is the Er of playing: if it is the set, then all members of the set are quantified over. The set is defined extensively as the set which has one member, which is the cat. Existential quantification is achieved by the Isa relationship between the concept 'cat' and the node which is the member of the set. The diagram therefore gives us a network analysis which captures the same insights as Neale's restricted quantifier approach.

In the next diagram, I tackle the collective interpretation of the cats played an interpretation that says that the cats played together. This interpretation asserts that there is only one playing event.

Like the previous diagram, this analysis shows the Er of 'playing' as the set of cats. The diagram is different from the diagram for a singular definite NP in that the set size is shown to be greater than one. However, it captures the universal and the existential quantification in the same way as the previous diagram. The difference between a singular definite NP and a plural definite NP with collective interpretation is established by the cardinality of the set. The next diagram presents a distributive analysis of plural definite NPs.

Figure 10 works in the same way as the diagram for the distributed interpretation of John and Mary bought a house (each) in Figure 6: the set has a typical member which links to the Er; the actual members of the set all Isa this member. In this diagram, I have not shown the actual members the fact that this is a set with more than one member is shown in the 'size' relation. The event is also given as a set to show that what is intended is a distributive interpretation where there is a separate playing event for each cat.

So far, I have shown how it is possible to have a set-based account in Word Grammar a cognitive linguistic theory of the definite article in its three main interpretations: within a singular NP, within a plural NP interpreted collectively, and within a plural NP interpreted distributively. What about the indefinite article a(n), and its plural counterpart some?

It is obvious that a(n) needs a treatment as a quantifier. It interacts with other quantifiers in the scope phenomenon involved in the two interpretations of the sentence in (14a) given in (14b) and (14c) from Neale (1990: 119).

(14) a. Every man danced with a woman who was wearing Ray-Bans.

b. [every x: man x] ([a y: woman y&y was wearing Ray-Bans] (x danced with y))

c. [a y: woman y&y was wearing Ray-Bans] ([every x: man x] (x danced with y))

In the interpretation in (14b), there are several different women wearing Ray-Bans, and the men are dancing with different women; in the interpretation where a has wide scope in (14c), there is just one woman wearing Ray-Bans, and every man dances with her. A(N)must be a quantifier: it behaves just like one with respect to scope alternations.

Some is straightforward: it occurs with plural nouns (in several dialects of English, some, like the, can occur with both plural and singular nouns, but I shall only take the plural case here). I am not going to propose diagrams here, because they would become unfeasibly complex, and because my main focus is on showing how we could capture definiteness set-theoretically in a cognitive theory, but we can at least think about how we could analyse some.

It seems that the main point is that some involves a relation between sets. When some has a full vowel, we can analyse some cats as analogous to some of the cats, in which case we need to identify the referent of some as a subset of the set of cats.[2] Because the particular subset is not known (some cats is indefinite) it is an arbitrary subset of members of the set of cats that is identified by some cats. The treatment of some belongs in a larger scale treatment of quantifiers in WG, which is the subject of further research. However, the subset relation will figure in the treatment of all proportional quantifiers apart from those which, like the, involve universal quantification.

• [1] The semantic relations Er and Ee are glosses over the many semantic relations that verb meanings might have. The Er is the relation that relates to the referent of the Subject in the active voice; the Ee links to the referent of the Object, when the verb is transitive.
• [2] The picture is a little more complex in that when some has a schwa rather than a full vowel, it does not imply a superset, and it can also go together with mass nouns, giving rise to a part-of relation. Worse, it can also be used non-proprtionally when set off against another cardinal value: there were some not many misprints. I am grateful to an anonymous reviewer for pointing this out.