Some intuitive background
The hidden argument theory is built on attractive premises. Instead of explaining the behavior of demonstratives by deploying (for example) sui generis operators that take a definite description and make it directly referential, the hidden argument theorist claims that demonstratives involve the same semantic operations as other determiners. She also promises to explain a wider range of data than her competitor; her analysis is designed to cover both familiar deictic uses of demonstratives as well as non-deictic uses.
As we have just seen, however, the hidden argument theory as described so far is not empirically adequate. We can fix this problem without giving up the advantages of the two-argument architecture if we modify some of the details. In order to appreciate the way the required modifications work, we will have to look more closely at the possible syntactic and semantic structures that might be associated with the demonstratives from (4) and (5). Before doing that, however, it will be helpful to establish some context, by taking a broad look at the role demonstratives typically play in our communicative practices.
In many familiar cases, demonstratives supply answers to the question ‘which one?’ Consider a butcher-shop vignette:
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(17)
A: Number 49?
B: Yes, I’d like a rib-eye steak, please.
A: Which one?
B: That one, please. (pointing)
This dialogue illustrates an extremely common pattern of use of deictic complex demonstratives. By saying ‘rib-eye steak’, B calls attention to a particular class of individuals, and by pointing, selects one from among them. Very frequently, when we use complex demonstratives deictically, a set of candidate referents is provided by the predicate from which the demonstrative is formed, and we pick out one by way of a gesture, or by its salience, or whatever.
The idea that complex demonstratives serve to pick one object out from a set of candidates has long been prominent in the philosophical literature. Witness, for example:
A not implausible view about the reference of expressions of the form ‘that F’, is that such expressions refer to F’s which have somehow been ‘discriminated’ from all other F’s. After all, when a speaker uses an expression of the form ‘that F’ to refer to a particular F, there is an implication to the effect that the intended F is somehow ‘discriminated’ with respect to all other F’s. (Reimer 1991, p. 178)
The foregoing suggests a way of distinguishing the felicitous (4, repeated) from the infelicitous (5, repeated):
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(4)
That guy who wrote Waverley also wrote Ivanhoe.
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(5)
#That author of Waverley also wrote Ivanhoe.
At first glance, it seems like (4) should admit an interpretation on which the relative clause is used to perform the kind of restricting work that would ordinarily be done by a pointing gesture; there are many individuals that satisfy the predicate ‘guy’, and the relative clause provides a way of selecting just one from among them. On the other hand, there does not appear to be any way of saying the same thing about (5), since there is no way of restricting the predicate ‘author of Waverley’, which already picks out a single individual. Indeed, a common reaction to the contrast between examples (4) and (5) is to point out this difference.Footnote 12
Standard versions of the hidden argument theory are not well-positioned to implement an explanation along these lines, however. Neither King nor Elbourne offer a specific analysis of the structure of restrictive relative clauses, but it is clear that both authors subscribe to some version of the familiar picture on which a relative clause and the noun that it modifies combine to form a single property-type constituent.Footnote 13
As we have said, for King (2001), the English determiner ‘that’ is a quantifier expression, the behavior of which can be modeled using the following fragment of a formal language:Footnote 14
The propositional frame expressed by [[That\(\xi \ \Sigma ]\Psi \)] in c is [[THAT\(_{f(c),h(c)}\ \xi \ \Sigma ^\prime ]\Psi ^\prime ]\), where f is a function from contexts to propositional frames and h is a function that maps each context \(\langle i,w,t\rangle \) to either J, the property of being jointly instantiated, or J\(_{wt}\) the property of being jointly instantiated in w,t, where if \(f(c) = [\xi = *\ o]\) or \([o =*\ \xi ]\), for some individual o, then h(c) = J\(_{w,t}\). Otherwise, h(c) = J; and THAT\(_{f(c),h(c)}\) is the result of saturating the second and third argument places in the 4-place relation expressed by ‘that’ (i.e., THAT: \(\_\_\_\) and \(\_\_\_\) are uniquely \(\_\_\_\) in an object and it is \(\_\_\_\)) with f(c) and h(c) respectively (i.e., \(\_\_\_\) and f(c) are uniquely jointly instantiated/jointly instantiated in w,t in an object and it is \(\_\_\_\)); and with \(\Sigma ^\prime \), \(\Psi ^\prime \) as above. (King 2001, p. 165)
On this treatment, the syntactically-realized arguments \(\Sigma \) and \(\Psi \) saturate two of the four places associated with the determiner ‘that’. Adapted to fit English, and applied to the case of (18), for example, King’s approach would see the determiner’s two syntactic arguments providing the properties of being a hominid who discovered fire and the property of being a genius:
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(18)
That hominid who discovered fire was a genius.
When a demonstrative sentence is used in a context in which the speaker does not intend to refer to a particular object, the h function returns the property of being jointly instantiated, and the f function a trivial property, like the property of being self-identical. Those properties saturate the remaining two argument places associated with ‘that’, and the upshot is that (18) expresses truth conditions that we might paraphrase as follows:
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(19)
The property of being a hominid who discovered fire and the property of being self-identical are uniquely jointly satisfied by an object x and x is a genius.
As noted in the previous section, this ingenious treatment makes exactly the right predictions with regard to the class of data that motivated it, i.e., non-deictic uses of demonstratives like the one from (18).Footnote 15 Crucially, however, by treating the relative clause ‘who discovered fire’ and the noun ‘hominid’ as together providing the determiner with a single property, the treatment leaves us no way to implement the idea that the relative clause serve as a restrictor on the set of hominids. So, where the subtler pattern evidenced by Wolter’s data is concerned, we are left with no way of making the required discrimination.
Elbourne’s version of the hidden argument theory runs into the same difficulty. Elbourne says that the hidden argument associated with a demonstrative expression is supplied by an index on the determiner:
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(20)
[[that\(_i\)] F]
With regard to an assignment that maps i to the property G, (20) is interpreted as though it were equivalent to:
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(21)
the x: [F(x) & G(x)]
This treatment suggests that the demonstrative from (18) would be represented as per:
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(22)
[[that\(_i\)] hominid-who-discovered-fire]
To derive a non-deictic interpretation, (22) would have to be evaluated with regard to a variable assignment that maps the demonstrative index to a property like the property of being self-identical. Such an assignment would result in something equivalent to:
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(23)
the x: [hominid-who-discovered-fire(x) & self-identical(x)]
Regardless of whether we take a King-style or an Elbourne-style approach to the hidden argument theory, as long as we think that the relative clause and the noun it modifies combine to form a single argument, we will have no way of distinguishing the felicitous (4, repeated) from the infelicitous (5, repeated):
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(4)
That guy who wrote Waverley also wrote Ivanhoe.
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(5)
#That author of Waverley also wrote Ivanhoe.
Regardless of how exactly we say the determiner works, that is, there will be no way to avoid the fact that the property contributed by ‘guy who wrote Waverley’ is the same as the property contributed by ‘author of Waverley’ (modulo an irrelevant gender presupposition introduced by ‘guy’). In order to avoid this problem, we need some way of separating a noun from the relative clause that modifies it; with regard to our example (4), that is, we need some way of treating ‘guy’ and ‘who wrote Waverley’ as two separate arguments for the determiner.
The structure of relative clause
As it turns out, a syntactic configuration that would allow us to do exactly what is needed has long been the topic of discussion among linguists. Ross (1967) described a structure for restrictive relative clauses that according to Stockwell et al. (1973) was the standard for the time. On that structure, a determiner combines with a noun to form a constituent, which in turn combines with the relative clause:
Partee (1975) argued that this configuration—which, because of the syntactic category names of the era, came to be known as the ‘NP-S’ configuration—would violate compositionality, and could thus be ruled out on semantic grounds. The problem, as she saw it, was precisely that ‘guy’ and ‘who wrote Waverley’ do not form a constituent. If ‘the’ is understood along familiar lines, and if ‘guy’ and ‘who wrote Waverley’ simply pick out the properties of being a guy and having written Waverley, respectively, this means there will be no way of deriving the expected extension:
If there is only one guy, the higher DP from (24) will have a truth-value as its extension, instead of picking out the unique author of Waverley. If there is more than one guy, the extension of (24) will be undefined. Neither of these results is acceptable.
Bach and Cooper (1978), however, showed that this objection could be avoided, by describing a simple way of making the NP-S structure produce the expected compositional outcome. Their solution was to effectively raise the type of the determiner, by inserting a variable over properties into its semantic representation. When a relative clause occurs in the NP-S configuration, they say, instead of:
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(26)
\(\llbracket \mathbf {the} \rrbracket \) = \(\lambda f. \iota x: f(x)=1\)
the determiner is interpreted as though it introduced a resource variable, R:
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(27)
\(\llbracket \mathbf {the} \rrbracket \) = \(\lambda f.\iota x: f(x)=1\) and \(R(x)=1\)
A construction-specific composition principle allows the property picked out by the relative clause to provide the value of the R variable, so that the semantic value for (24) turns out to be:
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\(\iota x: x\) is a man and x wrote Waverley
Bach and Cooper argued that the NP-S structure was required to explain the composition of relative clauses in Hittite.Footnote 16 Where English is concerned, however, they saw their work as a proof of concept, since the data involving definite descriptions that they considered are data that could just as easily be handled using the structure that is standard today, on which a noun and a relative clause combine to form a constituent.
Even if English data were the only data we had access to, it would be reasonable to take the contrast between (4) and (5) to provide precisely the kind of argument Bach and Cooper would have needed to vindicate their proposal—their structure for relative clause fills a theoretical role the standard structure cannot.Footnote 17 After all, regardless of how exactly we explain the difference between the demonstratives that license non-deictic interpretations and the ones that do not, it is hard to see how the puzzle could possibly be solved if there were no way of separating the NP from the relative clause. If ‘guy’ and ‘who wrote Waverley’ form a constituent, that constituent will pick out a property which, for all the determiner cares, is the same as the property picked out by ‘author of Waverley’.
Importantly, however, the contrast between the English examples (4) and (5) is not the only reason for thinking that relative clauses sometimes occur in the familiar low configuration, and sometimes high, in the NP-S configuration. In fact, Lin (2003) and del Gobbo (2003) argue that the NP-S analysis is required to make sense of a parallel contrast in the interpretations associated with certain demonstrative constructions in Mandarin, a contrast that Huang (1982) credits Chao (1968) with first remarking on.Footnote 18 Consider:
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(29)
neiben wo zuotian mai de shu
that I yesterday buy DE book
‘That book, which I bought yesterday’
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wo zuotian mai de neiben shu
I yesterday buy DE that book
‘The book that I bought yesterday’
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na-yi-ge [chouyan de] ren
that-one-CL smoke DE person
‘That person that smokes’
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[chouyan de] na-yi-ge ren
smoke DE that-one-CL person
‘The person that smokes’
Huang (1982) maintained that the relative clause from (29) is used non-restrictively, to add parenthetical information about the extension of a a deictic demonstrative, while the relative clause from (30) plays a restrictive role, helping to determine the extension of the demonstrative phrase. Lin (2003) offers a variety of syntactic and semantic arguments against the idea that the relative clause in (29) is non-restrictive, but he describes a broad agreement among authors about the fact that the first syntactic configuration systematically produces deictic interpretations, while the interpretations associated with the second are non-deictic.
This is especially interesting for our purposes because the word order in Mandarin provides a non-speculative way of determining how high the attachment site of a relative clause is. Unlike in English, that is, the claimed syntactic difference between the NP-S relative clause and the standard structure is manifest on the surface. Lin and del Gobbo analyze Huang’s demonstrative-initial constructions using the following structure:
In the relative clause-initial constructions, on the other hand, they claim that the demonstrative combines first with an unmodified NP and later with the relative clause:
How can the semantic derivation for (34) proceed compositionally? And why should a structure like (33) produce a deictic interpretation, while structure (34) is interpreted non-deictically? Lin and del Gobbo answer both questions at once by employing the same machinery Bach and Cooper used to avoid Partee’s challenge about compositionality. Lin writes:
Context-dependency can be nicely captured by introducing to the usual translations of determiners an extra property variable whose value is filled by a variable assignment. However, if there is overt linguistic material denoting a property around the property variable, the variable can be filled by that property... (Lin 2003, p. 230)
In other words, Lin and del Gobbo treat the demonstrative determiner in roughly the same way as King and Elbourne, claiming that it performs a semantic operation not on one property, but two.Footnote 19 If (33) provides the right structure for (31), then when the noun and the relative clause form a constituent, they jointly occupy only one of the two argument places introduced by the demonstrative determiner, making the second available for contextual saturation. On the other hand, when the structure of the demonstrative expression allows ‘overt linguistic material’ (here in the form of a high relative clause) to supply a distinct property-type argument, there is no work left for the context to do, and the expected interpretation turns out like a definite description.Footnote 20
The upshot for us is that Mandarin demonstrative constructions involving relative clauses provide a model for understanding their English counterparts. As we will see, if we combine the idea that English relative clauses can occur in either of the two structures described here with the idea that the demonstrative determiner introduces presuppositions involving both uniqueness and anti-uniqueness, we can explain the puzzling data we began with.Footnote 21
A job for presupposition
We turned our attention to questions about the structure of relative clause because we wanted a way to explain the contrast between the English sentences (4) and (5). We thought we might be able to make progress on that contrast by saying that the predicative material from the former sentence is structured in a way that allows one of the determiner’s arguments to perform a kind of restriction operation on the other, while the material from the latter sentence is not. We needed something like the NP-S structure to make that explanation compositionally possible. Now that we see that there are independent reasons for thinking that at least some relative clauses indeed occur in that configuration, we are in a position to pull the rest of the pieces together.
As we have seen, King and Elbourne invite us to think of demonstratives as definite descriptions that sometimes take an identificational property as the value of a hidden argument, and sometimes a trivial property. This flexibility allows them to offer a unified analysis of both deictic and non-deictic data, but as we saw, it causes their theories to overgenerate. If we add a certain presuppositional restriction to the basic architecture of the hidden argument analysis, we can retain its breadth of application while avoiding the overgeneration problem.
Like King and Elbourne, I propose that we treat demonstratives that appear to have the form:
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(35)
that F
as though they really involved the determiner’s taking two arguments, F and G, and performing a description-type operation on them, so that (35) is interpreted:
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(36)
the x: [F(x) & G(x)]
As on the hidden argument theory, I take the property that occupies the first argument place in the schema, F, to be supplied by the predicative material from which the complex demonstrative is formed.
Instead of following the hidden argument theorist in saying that the property that occupies the second argument place, G, is always covert, I claim that it is covert in deictic cases, but overt in non-deictic cases. The fact that certain syntactic and semantic environments make an explicit second argument available, while others do not, will play an important role explaining the curious pattern in the data concerning non-deictic demonstratives.
The key difference between my proposal and the hidden argument theory and between my proposal and the proposals described by Lin and del Gobbo concerns the relationship between the two arguments taken by the determiner. Instead of letting ‘that’ return the singleton intersection of any two properties, I propose limiting its application along the following lines:
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that F = \( {\left\{ \begin{array}{ll} \ [\mathrm{the}~x {:}\, [F(x)\ \& \ G(x)]]~\mathrm{iff}~(F\cap G)\subset F \\ \ \mathrm{otherwise~undefined} \end{array}\right. }\)
(37) is meant to capture the intuition that when someone utters a (singular) complex demonstrative, the predicate she uses introduces a set of candidates from which a single individual is to be picked. Our formulation works by adding a new presupposition to the presupposition of uniqueness that is standardly supposed to be a part of the semantics of ‘the’. This presupposition requires that the demonstrative’s second argument restrict its first argument in the following sense:
Definition
A property G is a restrictor on another property F just in case the intersection of {x : F(x)} and {x : G(x)} is a proper subset of {x : F(x)}.
The imprecise formulations from (36) and (37) are meant to underscore the fact that I would prefer to remain agnostic about details that I take to be irrelevant where the primary point of this paper is concerned. So, for example, while it would be natural to think that when demonstratives are used deicitically, an identificational property occupies the syntactic position that can be occupied by a relative clause in a non-deicitc construction, I see no reason to stake a claim with regard to the question (in my discussion to follow, I will assume that structure, but nothing important turns on it).
By the same token, since my aim here is not to settle the question of Frege versus Russell, I characterize the demonstrative determiner in terms of ‘the’, without saying how exactly ‘the’ should be understood. In the discussion to follow, I assume a Fregean approach to definite descriptions so that I can talk simply about ‘the referent’ of a certain demonstrative instead of about a function from properties to truth values. For concreteness’ sake, then, I will assume that modulo a host of irrelevant details concerning the distal/proximal distinction, animacy/inanimacy requirements, and similar:
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\(\llbracket \mathbf {that} \rrbracket \) = \(\lambda f \lambda g\): the intersection of {\(x: f(x)=1\)} and {\(x: g(x)=1\)} is a proper subset of {\(x: f(x)=1\)}. \(\iota x: f(x)=g(x)=1\)
As far as I can tell, however, none of the relevant features of my proposal depend on this assumption. The restriction presupposition that is at the heart of my proposal could easily be built into either of Elbourne’s or King’s version of the hidden argument theory, as well as theories like the one described by Roberts (2002).