Neale (1990) presents a set-theoretic treatment of the definite article, located in the Generalized Quantifier Theory which accounts for proportional quantifiers such as many and most.3 It has been widely noted that unrestricted quantification analysed with the standard sentence connectives cannot help with words such as-many or most, which therefore have to be analysed in an alternative system. Neale extends that system to the definite article. Neale's analysis (1990: 40-43) follows a treatment in Wiggins (1980) and Barwise and Cooper (1981). Neale's argument is given here (1990: 40).

[A]s soon as we encounter a genuine binary structure, we get stuck. For instance, suppose we wish to represent (7):

(7) Most men are immortal.

What we require is a formula of the form of (8)

(8) (mostx) (man x immortal x)

where '' is a binary, truth-functional connective. Clearly '' cannot be '&' for then (7) would mean that most things are men-and-immortal. Nor can '' be '3' for then it would mean that most things are if-men-then-immortal. But since nearly everything is not a man, nearly everything is if-men-then-immortal; therefore the sentence will come out true whether or not most men are immortal (here I borrow heavily from Wiggins' succinct discussion). In fact, there is no sentential connective that captures what we require of ''; indeed, it is not possible to define 'most Fs' in first order logic at all, even if attention is restricted to finite domains. The problem is that in (8) the "quantifier" most is ranging over the entire domain of quantification rather than just those things that are men. Intuitively, we want something like the following result:

'most Fs are Gs' is true if and only if |F n G| > |F-G|

(F = the set of things that are F. FnG = the set of things that are both F and G. F-G = the set of things that are F and not-G. |FnG|is the cardinality of FnG. This means that we should treat 'most' and other natural language determiners ('some', 'every', 'all', 'no' and so on) as exactly what they appear to be: devices that combine with two simple or complex formulae (or predicates, depending on how one views matters) to form a formula.

To put it another way, most should be treated as a restricted quantifier rather than as a fully general quantifier. For most men are mortal what we need is for the whole set of men to be sorted or given at the outset. This way most will pick out a proportion from the set of men, rather than from the set of everything or objects in general. Rather than creating a complex proposition using connectives, the range of the quantifier most is restricted by the noun men.^{[1]}

(9) Most men are mortal

[Most x: menx] (mortal x)

Neale (1990: 42-3) shows that all quantifier determiners can be presented in the same way. I give some of his truth clauses immediately below. After presenting his truth clauses, Neale goes on to show how restricted quantification can capture quantifier scope phenomena as well as unrestricted quantification, and how the Theory of Descriptions can be presented in the language of restricted quantification.

Truth clauses for every, no, some, an, and most. (From Neale 1990: 42-3)

Truth clause can be translated into natural language prose as, "every x which is an F is also a G if the cardinality of F minus G is zero." That is, if you like, the material on the left of the predicate "is true" is the syntax of the quantification, and the material on the right is the semantics, in a theory where the semantics consists of the conditions under which a statement is true. To put it another way, if you assume a restricted quantification, "The semantics of definite descriptions can then be given in terms of generalized quantifier theory" (Ludlow 2007), which is to treat them as proportions of a set. Generalized quantifier theory claims that natural language quantifiers express relations between sets. This can be shown diagrammatically, as in Figure 3, below.

Neale goes on to argue that the restricted quantifier treatment allows a straightforward handling of quantifier scope interactions, which obviously any theory needs to address, and then presents a version of Russell's Theory of Descriptions, pointing out that the Theory of Descriptions does not require Frege's unrestricted quantification, and that to present it in a restricted formalisation is simply to "choose a language other than that of Principia Mathematica in which to state and apply the theory." I will come back to the representation of the in 6; before I get there, I want to address the issue of the semantics of a restricted quantification in a conceptualist theory.

Figure 3. A set interpretation of most men are mortal

In cognitive theories, we don't treat either verbs' meanings or nouns' meanings as sets. Furthermore, in a cognitive framework, the conditions under which a sentence is true do not make up its meaning. The assumption is that the relationship between a proposition and the world is mediated by perceptual, embodied experience, and that the meanings of sentences are conceptual representations. Therefore, we can borrow from Neale the notion of working with a restricted set, but we have no need of any statement of the conditions under which a proposition is true. Therefore, in a cognitive theory of quantification, we need to discuss quantifiers such as most in terms of how it scopes over a set. This is part of a mental model. The plural noun denotes a set, and most establishes a scoping relation over that set, therefore expressing a proportion: a subset of the set denoted by the plural noun. Following Neale, we can treat the as a proportional quantifier, as well as the quantifiers given above in to (*5).

Recall that in 2, I pointed out that Milsark shows that so-called definiteness effects in existential there sentences are actually effects that single out proportional quantifiers, so it makes sense to treat the as another proportional quantifier. The claim in (2b), repeated here as (10), is that a definite description denotes a set, which is understood both existentially and exhaustively.

The formula in (10) asserts the existence of the content of the set, it quantifies over all members of the set, and therefore it argues that the existence claim of Russell's the King of France is bald is actually an assertion, not a presupposition. Essentially, the point here is to do with whether definite expressions have a truth value or not. For Russell, the King of France is bald is false; for Strawson, it is neither true not false. If the existence of the King of France is presupposed, then the existence claim is not part of the semantics (although see Burton-Roberts 1989 for an interesting alternative position on presupposition). In 7.3, the argument that the existence claim is part of the semantics is part of the analysis of specificational sentences that I sketch.

Neale (1990: 45, 46) gives (11a) for singular definites with the and (11b) for plurals.

(11) a. '[the x: Fx] (Gx)' is true iff F G = 0 and F = 1 b. '[the x: Fx] (Gx)' is true iff F G = 0 and F > 1

Neale points out that '[the x: Fx] (Gx)' is definitionally equivalent to (3x)((Vy) (Fy = y=x) and Gx). He also points out, following Chomsky (1975), that the relationship between singular and plural definite descriptions comes into view in the Generalized Quantifier Treatment given in (11), because the difference is simply one of cardinality.

I haven't yet explained how treating a definite expression as containing a proportional quantifier captures the familiarity effect. We can understand this by thinking about incomplete descriptions. Kearns (2000: 97) gives the example in (12a), which has the meaning given in (12b).

(12) a. All men must report before taking leave.

b. All enlisted men now serving on this base must report before taking leave!

The example in (12a) is incomplete out of context, whereas (12b) can be used out of context because the description is complete. But (12a) is the normal expression because "the speaker or writer assumes, or presupposes, that the audience can identify the background set, either from general shared knowledge, or because the information has been given earlier in the discourse" (Kearns 2000: 80). As Kearns says (2000: 96), "To understand a proposition with a strong quantifier, the hearer must be able to identify the background set."^{[2]} To put it another way, familiarity falls out of the quantificational story because an NP with a strong or proportional quantifier expresses a quantification over a background set, identifying the relevant subset in the case of a singular definite NP, the relevant subset is a singleton set.^{[3]}

My claim in the next section is that we can develop a conceptualist or cognitive version of the restricted set treatment of proportional quantifiers (including the) and as a result compare the familiarity and the quantifier treatment of definite articles from within the terms of a single framework. The set theoretic treatment of the is predictive: if it is right, it should be possible to find linguistic contexts where a definite description has to be understood as denoting a set rather than as having a referent. I set out to show such a situation in 7. In the next section, I show how a version of this theory can be modelled in a cognitive network.

[1] Neale (1990: 41-2) argues whether the appropriate treatment is restricted quantification or binary quantification. He observes that the two systems have the same expressive power, and asserts that as restricted quantification most closely matches natural language syntax, it is the system he will prefer.

[2] Kearns' term "strong quantifier" simply means "proportional quantifier".

[3] The discussions of (11) and (12) are foregrounded to some extent by Hawkins (1978: 157167), where the notion of uniqueness is replaced by one of inclusivity in order to capture the semantics of definite plurals. Van Langendonck (1979) challenges the notion of inclusivity on the grounds that plurals can be used without all members of the set being invoked, as in the clouds were covering the moon (see also Chesterman 1991: 22-24 for discussion). I think that Van Langendonck's problem can be resolved by thinking about definiteness as existential and universal quantification over a restricted set with reference to a background set.

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