# Inverse Linking, Possessive Weak Definites and Haddock Descriptions: A Unified Dependent Type Account

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## Abstract

This paper proposes a unified dependent type analysis of three puzzling phenomena: inversely linked interpretations, weak definite readings in possessives and Haddock-type readings. We argue that the three problematic readings have the same underlying surface structure, and that the surface structure postulated can be interpreted properly and compositionally using dependent types. The dependent type account proposed is the first, to the best of our knowledge, to formally connect the three phenomena. A further advantage of our proposal over previous analyses is that it offers a principled solution to the puzzle of why both inversely linked interpretations and weak definite readings (in contrast to Haddock-type readings) are blocked with certain prepositions.

## Keywords

Dependent type Inverse linking Possessive weak definite Haddock description## 1 Introduction

This paper focuses on three types of complex DPs: inverse linking constructions (May 1978, 1985; Larson 1985; Zimmermann 2002; Kobele 2010), possessive definite descriptions (Löbner 1985, 2011; Poesio 1994; Barker 2005; Peters and Westerståhl 2013) and Haddock descriptions (Haddock 1987; Champollion and Sauerland 2011; Sailer 2014; Bumford 2017). Inverse linking constructions, e.g., *a representative of every country*, allow the inversely linked interpretation in which the embedded quantifier phrase (*every country*) takes scope over the embedding one (*a representative*). Scoping out of DP islands represents an unresolved difficulty for standard LF-based scope-assignment strategies (see, e.g., Kratzer and Heim 1998; Zimmermann 2002). Possessive definite descriptions (possessive definites) and Haddock descriptions (Haddock definites) pose a puzzle for standard theories of definiteness based on uniqueness. Possessive definites, e.g., *the student (sister, son) of a linguist*, can be used in a situation in which linguists in question have more than one student (sister, son)—they carry no implication that only one individual satisfies the content of the description. Haddock definites, e.g., *the rabbit in the hat*, can be used successfully in contexts with multiple salient hats and multiple salient rabbits as long as there is a single rabbit-in-a-hat pair—they carry no standard requirements that there be exactly one hat, with exactly one rabbit in it. All three phenomena are nominal constructions involving quantification and relations that are not functional but (possibly) one-to-many (‘representing’, ‘being a sister of’, etc.). We shall argue that dependent types are well suited to modeling the interaction of non-functional relations and quantification.

Dependent type theoretical frameworks (Martin-Löf 1975; Martin-Löf and Sambin 1984; Makkai 1995) have been used to model linguistic phenomena of central importance, e.g., unbound anaphora (Ranta 1994; Fernando 2001; Cooper 2004; Grudzińska and Zawadowski 2014; Bekki 2014), lexical phenomena such as selectional restrictions and coercions (Asher 2011; Luo 2012; Chatzikyriakidis and Luo 2017b), adjectival and adverbial modification (Chatzikyriakidis and Luo 2017a) and long-distance indefinites (Grudzińska and Zawadowski 2017d). This paper argues that the three puzzling readings (inversely linked interpretations, weak definite readings in possessives and Haddock-type readings) provide another compelling example of application of dependent types to the study of natural language semantics. Specifically, we propose that the three readings under consideration have the same underlying surface structure [introduced and independently motivated in Zimmermann (2002) and Barker (2005)], and that the surface structure postulated can be interpreted properly and compositionally using dependent types. Other directly compositional (non-movement) accounts of the phenomena in question have been proposed [for inverse linking, see Hendriks (1993); Barker (2002); for possessive weak definites, see Löbner (1985); Barker and Shan (2014); for Haddock definites, see Haddock (1987); Bumford (2017)]. However, to the best of our knowledge, our proposal is the first to formally connect the three phenomena. A further advantage of our dependent type account over previous directly compositional analyses is that it offers a principled solution to the puzzle of why both inversely linked interpretations and weak definite readings (in contrast to Haddock-type readings) are blocked with certain prepositions, e.g *with*, as in *someone with every known skeleton* [example from May and Bale’s (2005)] or *the student with a brown jacket* [example from Poesio’s (1994)].

In the rest of the paper we shall show that our semantic framework with dependent types provides us with uniform and adequate mechanisms to model the three puzzling phenomena. The structure of the paper is as follows. In Sect. 2, we explain the main features of our semantic framework [as presented in Grudzińska and Zawadowski (2014, 2017a)], most notably the concept of type dependency and type-theoretic notion of context. Section 3 puts forth our dependent type analysis of inverse linking constructions [key parts of this section were already presented in Grudzińska and Zawadowski (2017b), here we expand on this material with more technical details and explanations]. In Sect. 4, we present a new treatment of possessive weak definites, and in Sect. 5, we propose our dependent type solution to Haddock’s puzzle. Section 6 offers a principled answer to the puzzle of why inversely linked interpretations and weak definite readings are blocked with certain prepositions (in contrast to Haddock-type readings). Section 7 explains the remaining details of our compositional analysis, before concluding in Sect. 8.

## 2 Semantics with Dependent Types

In this section, we give an introduction to the main features of our semantic framework with dependent types (emphasizing those relevant for this paper), and explain how common nouns (sortal and relational), quantifier phrases (QPs) and predicates are to be interpreted in this framework.

### 2.1 Context and Type Dependency

*e*of all entities—strictly speaking, it has also another ‘auxiliary’ basic type

*t*of truth values, and a recursive definition of functional types). The variables of our system are always typed: \(x:X, y:Y, \ldots \); types are interpreted as sets: \(\Vert X\Vert , \Vert Y\Vert , \ldots \) In our system, types can depend on the variables of other types, e.g., if

*x*is a variable of the type

*X*, we can have type

*Y*(

*x*) depending on the variable

*x*. The fact that

*Y*is a type depending on

*X*is to be modeled as a function:

*Y*(

*x*) is interpreted as the fiber \(\Vert Y\Vert (a)\) of \(\Vert \pi \Vert \) over \(a\in \Vert X\Vert \) (the inverse image of the singleton \(\{a\}\) under \(\Vert \pi \Vert \)). If

*X*is a type and

*Y*is a type depending on

*X*, we can also form dependent sum types \(\varSigma _{x:X} Y(x)\) (whose interpretation consists of pairs \(\langle a, b\rangle \) such that \(a\in \Vert X\Vert \) and \(b\in \Vert Y\Vert (a)\)) and dependent product types \(\varPi _{x:X} Y(x)\) (whose interpretation consists of functions which assign to each \(a\in \Vert X\Vert \) an element of \(\Vert Y\Vert (a)\)). For the purposes of this paper, we only need \(\varSigma \)-types; \(\varSigma _{x:X} Y(x)\) is to be interpreted as the disjoint sum of fibers over elements in \(\Vert X\Vert \):

*x*of type

*X*precedes the declaration of a variable

*y*of type

*Y*(

*x*):

*Y*depends on the variable

*x*of type

*X*; type

*Z*depends on the variables

*x*and

*y*of types

*X*and

*Y*, respectively; and types

*X*and

*U*are constant types (i.e., they do not depend on any variables).

^{1}Having a context \(\varGamma \) as above, we can declare a new type

*T*in that context and a new variable of this type in that context, and extend the context by adding this new variable declaration. The empty context \(\emptyset \) is a valid context. If \(\varGamma \) is a valid context,

*t*is a new free variable (i.e., \(t \not \in FV(\varGamma )\)), and

*T*is a type in the context \(\varGamma \), then the extension of \(\varGamma \) by (a free variable of) the type

*T*, denoted \(\varGamma , t: T\), is also valid.

*x*:

*X*in \(\varGamma \) a set \(\Vert X\Vert \), and to every edge \(\pi _{Y,x} : Y \rightarrow X\), a function \(\Vert \pi _{Y,x}\Vert : \Vert Y\Vert \rightarrow \Vert X\Vert \), so that whenever we have a triangle of edges (as on the left), the corresponding triangle of functions commutes (i.e., \(\Vert \pi _{Z,x}\Vert = \Vert \pi _{Y,x}\Vert \circ \Vert \pi _{Z,y}\Vert \)):

The interpretation of the context \(\varGamma \), the dependence diagram, gives rise to the parameter space \(\Vert \varGamma \Vert \) which is the limit^{2} of the dependence diagram (i.e., the set of compatible *n*-tuples of the elements of the sets corresponding to the types involved).^{3} For the context \(\varGamma = x: X, y: Y(x)\), its parameter space \(\Vert \varGamma \Vert \) is \(\Vert \varSigma _{x:X} Y(x)\Vert \).

For more technical details and explanations about our notion of context, we refer the reader to our system presented in Grudzińska and Zawadowski (2017a).

### 2.2 (Relational) Common Nouns as (Dependent) Types

*man*,

*country*) are modeled as predicates/one-place relations (expressions of type \(\langle e,t \rangle \)). In our framework they are treated as types, e.g.,

*man*is modeled as the type

*M*(

*an*) and extends the context by adding a new variable

*m*of the type

*M*, with

*M*being interpreted as the set \(\Vert M\Vert \) of men. In the Montagovian setting relational common nouns (e.g.,

*representative*,

*sister*) are interpreted as two-place relations (expressions of type \(\langle e, \langle e,t \rangle \rangle \)). Our framework allows us to treat relational common nouns as dependent types, e.g.,

*representative*(as in

*a representative of a country*) is modeled as the dependent type:

*r*of type

*R*(

*c*):

*C*as the set \(\Vert C\Vert \) of countries, then we can interpret

*R*as the set of the representatives of the countries in \(\Vert C\Vert \), i.e., as the set of pairs:

*a*can be recovered as the fibers of this projection (the inverse images of \(\{a\}\) under \(\Vert \pi _{R,c}\Vert \)):

*a*only. To form the set of all representatives, we need to use \(\varSigma \) type constructor which takes the sum of fibers of representatives over countries in \(\Vert C\Vert \):

### 2.3 QPs and Predicates in Context

*y*is the

*binding variable*and the variables \(\mathbf {x} = x_1, \ldots , x_k\) are

*indexing variables*. We have a polymorphic interpretation of quantifiers. A generalized quantifier associates to every set

*Z*a subset of the power set of

*Z*:

*E*Montague (1973). For example,

*some country*denotes the set of subsets of

*E*:

*some country*is interpreted over the type

*Country*(given in the context), i.e.,

*some country*denotes the set of all non-empty subsets of the set of countries:

*some representative of France*:

*P*defined in a context \(\varGamma \):

*x*in

*Y*(

*x*) is outside the scope of the second binding occurrence of

*x*next to \(Q_1\). A global restriction on variables is that each occurrence of an indexing variable be preceded by a binding occurrence of that variable—free undeclared variables are illegal (for more on this restriction, see Grudzińska and Zawadowski 2017a). As a consequence, this principle considerably restricts the number of possible readings for sentences involving multiple quantifiers and will have a major role to play in our account of the three puzzling phenomena to be discussed in the sections to follow: inversely linked interpretations, weak definite readings in possessives and Haddock-type readings.

## 3 Inverse Linking Constructions

- (1)
A representative of every country

*inverse reading*in which

*every country*outscopes

*a representative*. Under standard LF-based analyses, the inverse reading of ILCs is attributed to the application of quantifier raising (QR). QR replaces the QP

*every country*with the coindexed trace (\(t_1\)), and adjoins it at DP

*mother*,

*head*,

*age*) and relational common nouns (e.g.,

*representative*,

*sister*,

*son*). Relational nouns involve relations that are not functional but (possibly) one-to-many. Under our analysis, relational nouns are treated as dependent types. For example, as explained in detail above,

*representative*(as in

*a representative of a country*) is modeled as the dependent type:

*r*of type

*R*(

*c*):

*c*:

*C*,

*r*:

*R*(

*c*), we get the inverse ordering of quantifiers:

*c*(in

*R*(

*c*)) is outside the scope of the binding occurrence of that variable (see our global restriction on variables introduced above). By making the type of representatives dependent on (the variables of) the type of countries, our analysis forces the inversely linked reading without positing any extra scope mechanisms.

*of every country*is right-adjoined to the QP consisting of the head nominal (

*a representative*). The head nominal

*representative*is modeled as the dependent type:

*r*of type

*R*(

*c*):

*country*is modeled as the type

*C*. The preposition

*of*is a purely syntactic marker signalling that

*country*is a type on which

*representative*depends

^{4}The interpretation of the structure:

*a*only. We use \(\varSigma \) type constructor to form the type of all representatives \(\varSigma _{c:C} R(c)\). The complex DP

*a representative of every country*is interpreted as the complex quantifier living on the set of all representatives \(\Vert \varSigma _{c:C} R(c)\Vert \):

*surface reading*in which

*a representative*outscopes

*every country*.

^{5}To capture this reading, we propose that

*representative*be treated standardly as a predicate, interpreted as the subset of \(\Vert P(erson)\Vert \times \Vert C(ountry)\Vert \) (and not as a dependent type). Quantifying universally the

*country*-argument, we obtain a type whose interpretation is the set of individuals who represent all the countries. By quantifying existentially over this set, we get the surface reading. More specifically, we propose the following analysis of the surface reading for ILC:

The QPP stands in the sister position to the head nominal *representative*. Our proposal is that the relational noun *representative* is now interpreted as the subset of \(\Vert P(erson)\Vert \times \Vert C(ountry)\Vert \). The complex NP (noun modified by the postnominal QPP) *representative of every country* is then interpreted as the type/set of individuals who represent all the countries: \(\{p: \{c: \langle p, c\rangle \in \Vert Represent\Vert \} \in \Vert \forall \Vert (\Vert C\Vert )\}\), and the DET *a* quantifies existentially over this set, yielding the surface ordering of quantifiers.

*a man from every city*. Our solution is that a sortal noun like

*man*can undergo a ‘sortal-to-relational’ shift, resulting in the (IR)-structure:

Partee and Borschev (2012) emphasize the permeability of the boundary between sortal an relational nouns and the fact that nouns can often be coerced to cross this border. In this case, the relational use of the sortal noun *man* can be coerced by the presence of the locative preposition *from*. Such prepositions specify the local position or origin of an entity and since entities do not occur at more than one place simultaneously, the dependency *c* : *C*, *m* : *M*(*c*) is likely to be a preferred interpretation for *man from* (for any city, there is a set (fiber) of the men from that city). Our analysis of the above structure is then exactly like the one described for the inverse reading of *a representative of every country*.

- (a)
One person [RC who was famous] [QPP from every city] was invited.

- (b)
\(\sharp \) One person [QPP from every city] [RC who was famous] was invited.

*x*is such that one famous person from

*x*was invited, while sentence (b) is semantically odd—it only allows a surface reading saying that one person who came from every city and who was famous was invited. Inverse readings are possible when QPPs follow RCs [as in (a)], while non-final QPPs give rise to surface readings only [as in (b)]. This asymmetry is unexpected on the LF-based analysis since all postnominal modifiers (QPPs, RCs) have the same syntactic status, i.e., they stand in sister position to the head nominals. Based on this argument, Zimmermann proposes a different structure for the inverse reading, where the QPP (

*of every country*) is not a regular postnominal modifier [as in (SR)] but is right-adjoined to the QP

*a representative*[as in (IR)]. On the semantic side, Zimmermann proposes to derive inverse readings via a non-compositional reinterpretation procedure (for more on this, see Zimmermann 2002) In our analysis, we adopt Zimmermann’s position that ILCs are structurally ambiguous at surface structure: the two readings derive from the two surface structures distinguished by the syntactic position of the QPP. The advantage of our dependent type account over Zimmermann’s analysis is that it interprets the surface structure for the inverse reading in a fully compositional way. We shall now see that the same two structures were independently introduced and motivated for possessive definites.

## 4 Possessive Definites

*of*-PP, as in (2):

- (2)
The student (sister, son) of a linguist

^{6}Uniqueness theories assume that definite determiner requires uniqueness (relativized to the discourse situation), i.e., the object referred to by a definite description must be the only object (in the discourse situation) that satisfies the content of the description. The uniqueness requirement is standardly taken to be a presupposition and

*the*is defined as a partial function

*P*—defined only if there is exactly one

*x*such that

*P*(

*x*) (\(\exists !x P(x)\))—to the unique

*x*such that

*P*(

*x*) (\(\iota x P(x)\) is a referring expression standing for the unique individual who satisfies

*P*). We propose to use the following type-theoretic variation of this definition (one that lifts

*the*to the quantifier-type interpretation and does not introduce partiality into the system)

On the strong reading (SR), *student* combines with the *of*-phrase resulting in a property *P* corresponding to the set of individuals who are students of a linguist; *the* combines then with this set to yield one individual from this set (Barker 2005). Similarly, under our dependent type theoretical analysis, the complex NP (noun modified by the *of*-PP) *student of a linguist* is interpreted as the type/set of individuals who are students of a linguist: \(\{s: \{l: \langle s, l\rangle \in \Vert Student_{of}\Vert \} \in \Vert \exists \Vert (\Vert L\Vert )\}\), and the DET *the* quantifies over this set, yielding the singleton consisting of a single student (if there is a unique student of a linguist), and the empty set, otherwise.

*the*first combines with the relational noun

*student*, and then with the

*of*-PP. On the semantic side, the idea is that the interpretation of

*the*is shifted (via the Geach rule) and combined with the interpretation of

*student*via function composition, and only then applied to the interpretation of the

*of*-PP. This order of combination gives the same result as in the previous analysis (corresponding to the (SR)-structure). Barker claims that the difference between the two analyses consists in the presuppositions triggered. In the (SR)-case, the requirement is that the complex NP as a whole (

*student of a linguist*) describes a unique object in the discourse situation. In the (WR)-case, the requirement is that the referent of the relational

*student*is determined uniquely relative to the possessor argument (

*a linguist*); a successful use of a PD ‘is one that provides enough information for the listener to pick out the intended kind of object’, i.e., the student of a linguist that the speaker has in mind. While no formalization of this intuitive analysis of the (WR)-case was proposed in Barker (2005), our dependent type analysis provides a way of expressing this intuition formally. Under our account, the head nominal

*student*is modeled as the dependent type:

*s*of type

*S*(

*l*):

*the student of a linguist*is interpreted as the complex quantifier living on \(\Vert \varSigma _{l:L} S(l)\Vert \) (an element of \(\Vert \varSigma _{l:L} S(l)\Vert \) is a pair \(\langle a, b\rangle \) such that \(a\in \Vert L\Vert \) and \(b\in \Vert S\Vert (a)\)):

*a*in \(\Vert L\Vert \) such that the fiber over

*a*, \(\Vert S\Vert (a)\), contains only one student (that the speaker has in mind). Otherwise, it denotes the empty set.

Two more remarks are in order. First, in this section we only considered PDs with an indefinite possessor (object of the genitive *of*) such as *the student of a linguist*. In the literature there is a disagreement over whether weak definite readings are also available for PDs with a definite possessor such as *the student of the linguist*. Contra to Poesio (1994), Barker claims that PDs with definite possessors allow the weak interpretation (Barker 2005). We shall come back to this problem after discussing our account of Haddock definites. Second, some authors observed that ILCs and PDs are systematically subject to the so-called ‘narrowing’, i.e., the universal quantifier of the ILC *a representative of every country* is restricted to countries that delegate (some) representatives, and the existential quantifier of the PD *the student of a linguist* is restricted to linguists who have students (Barker 1995; Champollion and Sauerland 2011; Peters and Westerståhl 2013; Sailer 2014). The narrowing mechanism is sometimes viewed as a pragmatic process of accommodation (Lewis 1979; Champollion and Sauerland 2011) and sometimes built into the truth conditions of the possessive definite (Peters and Westerståhl 2013) or the inverse linking construction (Sailer 2014). In this paper, we do not integrate narrowing into our account. In particular, we shall show that our analysis allows us to derive Haddock-type readings without the recourse to narrowing in either semantic or pragmatic sense [unlike in the proposals in Champollion and Sauerland (2011); Sailer (2014)].

## 5 Haddock Definites

- (3)
The rabbit in the hat

*the rabbit in the hat*presupposes that the number of hats that contain exactly one rabbit is exactly one. In Bumford’s dynamic account (Bumford 2017),

*the rabbit in the hat*has a stronger presupposition that the number of hats that contain at least one rabbit is exactly one. Under the former analysis

*the rabbit in the hat*can be used felicitously in the context in which one hat contains one rabbit and another hat contains two rabbits, while under the latter analysis the use of the description is infelicitous in this context (since it has three rabbit-hat pairs). It will be shown that both variants can be accounted for using our dependent type analysis.

*the hat*is interpreted as the strong definite:

*The hat*comes with the standard requirement that there be a unique hat. The head nominal

*rabbit*combines with the

*in*-phrase resulting in the type/set of rabbits in the hat; the DET

*the*requires that this set be a singleton—thus we get Bumford’s absolute reading saying that there is exactly one hat, with exactly one rabbit in it.

*rabbit*undergoes a ‘sortal-to-relational’ shift and is modeled accordingly as the dependent type:

*r*of type

*R*(

*h*):

*the rabbit in the hat*is interpreted as the complex quantifier living on \(\Vert \varSigma _{h:H} R(h)\Vert \). Simply combining this analysis with our semantic clause for

*the*will not deliver the desired reading (in either of its variants):

*the rabbit in the hat*cannot be used felicitously in the context with multiple salient hats.

*the*is ambiguous between a presuppositional analysis (

*the*) and a quantificational analysis (

*the’*):

*the rabbit in the hat*:

*the rabbit in the hat*can be used successfully in the context with multiple salient hats. The way to obtain Bumford’s reading would be to combine the definite determiner ‘unselectively’ with the type \(\varSigma _{h:H} R(h)\) (consisting of pairs \(\langle h, r\rangle \) such that

*h*is of type

*H*and

*r*is of type

*R*(

*h*)), interpreted as the sum of fibers over elements in \(\Vert H\Vert \)

*THE*quantifies then over pairs (i.e., binds ‘unselectively’ all the variables in its scope):

*a*is a hat,

*b*is a rabbit, and

*b*is in

*a*—

*the rabbit in the hat*can be used felicitously in the context with multiple salient hats and multiple salient rabbits as long as there is exactly one pair of a rabbit and a hat such that the rabbit is in the hat.

^{7}Our analysis allows us to derive the relative reading (in each of its variants) but it relies on additional amendments: presuppositional/quantificational ambiguity in

*the*or the mechanism of unselective binding. We shall emphasize that these additional mechanisms are well motivated independently of Haddock definites.

Extending our analysis to PDs with definite possessors (e.g., *the student of the linguist*) would give an explanation of why *the student of the linguist* can be used successfully in contexts with multiple salient students and multiple salient linguists (as long as there is exactly one student-linguist pair). However, the controversial weak uses of PDs with definite possessors mentioned in the previous section cannot be explained using our analysis. Barker (2005) gives examples such as *the side of the road* which can be used felicitously in contexts in which the road in question has two equally silent sides (i.e., in contexts with multiple side-road pairs). For example, the sentence ‘It is safer to mount and dismount towards the side of the road, rather than in the middle of traffic’ is so constructed that the bicycle rider can be riding on either side of the road (left or right)—here, the speaker is perfectly aware that the road has two sides. We think that the problematic example mentioned may fall into a small group of idiosyncratic uses. In his paper, Barker discusses a well-known group of exceptions to uniqueness, e.g., ‘Could you please open the window?’ (uttered in a room with three equally silent windows), or ‘Please eat the apple (uttered when offering a bowl of apples). Barker’s claim is that such uses are idiosyncratic and have no implication for the treatment of systematic and productive uses. One clue that a use is idiosyncratic is that it does not tolerate modification, e.g., ‘Could you please open the tall window?’ is infelicitous in the context with two tall windows. Similarly, the examples of weak uses of PDs with definite possessors (adduced by Barker) may constitute a group of idiosyncratic exceptions to Haddock-type uniqueness, e.g., *the side of the road* (26,700,000 Google search results) also does not tolerate modification, as evidenced by 5 Google search results for *the side of the long road* or 4 results for *the side of the beautiful road*.

## 6 Preposition Puzzle

*a representative of every country*) and PDs (e.g.,

*the student of a linguist*) are DPs which contain a QP which is selected by the preposition. This includes genitive

*of*(connector to a relational noun) but also locative prepositions such as

*from*or

*in*. One puzzling difficulty for the existing accounts of ILCs is that some prepositions like

*with*block inverse readings. A similar puzzle arises in connection with PDs—here, an even stronger claim has been made that ‘only the true genitive

*of*systematically gives rise to weak interpretations’ (Barker 2005, p. 100). To illustrate the point, May and Bale (2005) give the example in (4):

- (4)
Someone with every known skeleton key opened this door,

- (5)
The student with a brown jacket.

*with*. As discussed in Bumford’s (2017), HDs such as

*the table with the apple*can be used successfully in contexts with multiple salient tables and multiple salient apples as long as there is exactly one pair of a table and an apple such that the table comes with the apple.

*of*and

*from*, as in

*a representative of (from) every country*, the ‘dependent component’ (

*representative*) precedes in surface linear order the component on which it is dependent (

*country*). Thus

*a representative of (from) every country*introduces the dependency:

*with*, as in

*a man with every key*, the potentially ‘dependent component’ (

*key*) follows the component on which it is dependent (

*man*). Thus the only possible dependency to be introduced is:

*m*(in

*K*(

*m*)) is outside the scope of the binding occurrence of that variable. Thus, under the analysis proposed, the inverse interpretation is unavailable to the QP in the object position of

*with*. Similarly PDs with the preposition

*of*, e.g.,

*the student of a linguist*, introduce the dependency:

*with*, as in

*the student with a brown jacket*, the potentially ‘dependent component’ (

*brown jacket*) follows the component on which it is dependent (

*student*). Thus the only possible dependency to be introduced is:

*the*, \(\exists _{j:J(s)}the_{s:S}\), is again not available because the indexing variable

*s*(in

*J*(

*s*)) is outside the scope of the binding occurrence of that variable, and so the weak interpretation is unavailable to the QP in the object position of

*with*.

^{8}Second, by contrast to both inverse and weak definite readings, Haddock-type readings are insensitive to the prepositions used. This follows from the fact that the analyses proposed for such readings are not contingent on the dependencies in question. A HD like

*the rabbit in the hat*or

*the table with the apple*denotes a single pair (either a single rabbit-hat pair or a single table-apple pair), and the semantic analyses proposed can yield the pairs in question regardless of the dependencies involved.

## 7 Details of the Compositional Analysis

- (6)
A representative of every country is bald.

\(\forall _{c:C} \exists _{r:R(c)}\;\ Bald (r)\)

- (7)
A representative of every country missed a meeting.

\(\forall _{c:C} \exists _{r:R(c)} \;\ missed \;\ \exists _{m:M}\)

*a representative of every country*. The illustration below serves to provide an intuitive explanation of the formula:

Note that a person counts as a representative only in virtue of standing in a particular relationship with some country. Now each representative *r* has an underlying person *U*(*r*), in fact two representatives can have the same underlying person (if this person represents two countries). *U* stands for a function that forgets this part of the structure that relates people to countries and yields just people. Predicate *Bald* is defined over the type/set of people—by taking the inverse image of the set of bald people under the forgetting function *U*, \(U^{-1}(\Vert Bald\Vert )\), we get the set of bald representatives. For sentence (6) to be true, the set of bald representatives must be in the denotation of the complex quantifier expression *a representative of every country*.

*Q*on type

*X*is of type \({{\mathcal {C}}}(X) = (X \rightarrow \mathbf{t}) \rightarrow \mathbf{t}\)—thus its meaning can be expressed only in continuized terms. For combining continuized predicates with QPs, we use two CPS composition rules (\({{\mathcal {P}}}(X)=X\rightarrow \mathbf{t}\) and \({{\mathcal {C}}}(X)={{\mathcal {P}}}{{\mathcal {P}}}(X)\)):

*Miss*is defined on \(P(erson) \times M(eeting)\) and interpreted as the subset of \(\Vert P(erson)\Vert \times \Vert M(eeting)\Vert \). It induces an obvious predicate

*Miss’*defined on \(R(epresentatives) \times M(eeting)\) and interpreted as the subset of \(\Vert \varSigma R\Vert \times \Vert M\Vert \), \((U \times 1)^{-1}(\Vert Miss\Vert )\). We now lift our predicate

*Miss’*to an expression of type \({{\mathcal {C}}}{{\mathcal {P}}}(\varSigma R \times M)\), and combine the continuized predicate (of type \({{\mathcal {C}}}{{\mathcal {P}}}(\varSigma R \times M)\)) with the object QP

*a meeting*(of type \({{\mathcal {C}}}(M)\)) using either (left or right) of the two \({\mathbf{CPS}}\) transforms:

*G*of type \({{\mathcal {C}}}{{\mathcal {P}}}(\varSigma R)\)) is then merged with the subject complex DP

*a representative of every country*(of type \({{\mathcal {C}}}(\varSigma R)\)), using \({\mathbf{CPS}}\) transforms again:

*G*by \(\lambda c:{{\mathcal {C}}}(\varSigma R). A(\lambda m.c(\lambda r. p'(r,m)))\)

*c*, there are two politicians who spy on someone from

*c*—separate QP cannot take scope in between the two nested QPs:

*every city*>

*two politicians*>

*someone*. Similarly,

*a meeting*cannot take scope in between

*every country*and

*a representative*—the two interleaved interpretations are not possible for sentence (7). Note that under our analysis, the inseparability of the two nested QPs falls out immediately.

## 8 Conclusion

In this paper, we provided a unified dependent type account of the three problematic phenomena: inversely linked interpretations, weak definite readings in possessives and Haddock-type readings. We proposed that the three readings under consideration share the same underlying surface structure and we showed that the surface structure proposed can be interpreted compositionally and properly, using our semantic framework with dependent types. Furthermore, by introducing a restriction prohibiting free undeclared variables we were able to explain the puzzle of why both inversely linked interpretations and weak definite readings are blocked with certain prepositions (in contrast to Haddock-type readings). Our dependent type account is the first, as far as we are aware, to formally connect the three phenomena and to offer a principled solution to the preposition puzzle.

## Footnotes

- 1.
We adopt the convention that the variables the types depend on are always explicitly written down in type specifications.

- 2.
By this we mean the (categorical) limit of the described (dependence) diagram in the category

*Set*of sets and functions. The notion of a limit used here is the usual category-theoretic notion. In particular, the notion of a parameter space makes sense in any category with finite limits. However, the definition we give in the text is a standard representation of this limit and does not require any knowledge of Category Theory. - 3.
For \(a \in \Vert X\Vert , b \in \Vert Y\Vert , c \in \Vert Z\Vert \), we say that a triple \(\langle a, b, c \rangle \) is

*compatible*iff \( \Vert \pi _{Y,x}\Vert (b) = a, \ \Vert \pi _{Z,y}\Vert (c) = b, \ \Vert \pi _{Z,x}\Vert (c) = a.\) - 4.
Our treatment of prepositions is inspired by Barker’s (2011). Under Barker’s analysis, the relational head nominal

*representative*denotes the*representing*-relation; the preposition*of*is analyzed as ‘semantically inert (an identity function), a purely syntactic marker that the object of the preposition is an argument of the relational head nominal’. Under our dependent type analysis, the relational head nominal*representative*is modeled as the dependent type and the preposition*of*is ‘semantically inert’ (as in Barker’s proposal), it just serves as a marker signalling that*country*is a type on which*representative*depends. - 5.
- 6.
Weak uses pose a problem also for familiarity approaches to definiteness, for a discussion see e.g., Barker (2005).

- 7.Yet another way to derive the ‘polyadic reading’ would be to take the recourse to the narrowing mechanism:where \(\Vert H_{n}\Vert \) denotes the set of hats that contain (some) rabbits.$$\begin{aligned}&\Vert the_{h:H_{n}}|the_{r:R(h)}\Vert \\&\quad = \{X \subseteq \Vert \varSigma _{h:H_n} H_n(r)\Vert : \{a \in \Vert H_{n}\Vert :\{b \in \Vert R\Vert (a): b \in X \} \in \Vert the\Vert (\Vert R\Vert (a))\}\in \Vert the\Vert (\Vert H_{n}\Vert )\}, \end{aligned}$$
- 8.
Note, however, that

*with*comes with a number of meanings, including: ‘having or possessing (something)’ and ‘accompanied by; accompanying’. If the relation expressed is one of possession, as in our example*a man with every key*, then the thing possessed depends on the possessor (as described above). If, however, the relation is that of accompanying, as in*a problem with every account*, then the accompanying entity (problem) depends on the entity to be accompanied (account). Thus the dependency introduced is*a*:*A*,*p*:*P*(*a*), forcing the inverse ordering of the QPs, and the inverse reading is available for constructions with the preposition*with*taken in the sense ‘accompanying’ (as evidenced by the examples such as ‘I had a problem with every game I played, from crashing to stupid errors’).

## Notes

### Acknowledgements

For incisive and encouraging comments on some parts of this work, the authors would like to thank the audience of *The Stockholm Logic Seminar* and three conferences: *The 18th Szklarska Poreba Workshop (Szklarska Poreba 2017)*, *The Workshop on Logic and Algorithms in Computational Linguistics 2017 (Stockholm 2017)* and *Amsterdam Colloquium 2017*.

### Funding

Funding was provided by Polish National Science Center (Grant No. DEC-2016/23/B/HS1/00734).

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