System with Generalized Quantifiers on Dependent Types for Anaphora

System with Generalized Quantiﬁers on Dependent Types for Anaphora

Justyna Grudzinska´ Marek Zawadowski Institute of Philosophy Institute of Mathematics University of Warsaw University of Warsaw Krakowskie Przedmiescie´ 3, 00-927 Warszawa Banacha 2, 02-097 Warszawa [email protected] [email protected]

Abstract pronouns as quantifiers; as in the systems of dynamic semantics context is used as a medium sup- We propose a system for the interpreta- plying (possibly dependent) types as their poten- tion of anaphoric relationships between tial quantificational domains. Like Dekker’s Pred- unbound pronouns and quantifiers. The icate Logic with Anaphora and more recent mul- main technical contribution of our pro- tidimensional models (Dekker, 1994); (Dekker, posal consists in combining generalized 2008), our system lends itself to the compositional quantifiers with dependent types. Empir- treatment of unbound anaphora, while keeping a ically, our system allows a uniform treat- classical, static notion of truth. The main novelty ment of all types of unbound anaphora, in- of our proposal consists in combining generalized cluding the notoriously difficult cases such quantifiers (Mostowski, 1957); (Lindstrom,¨ 1966); as quantificational subordination, cumula- (Barwise and Cooper, 1981) with dependent types tive and branching continuations, and don- (Martin-Lof,¨ 1972); (Ranta, 1994). key anaphora. The paper is organized as follows. In Section 2 we introduce informally the main features of our 1 Introduction system. Section 3 sketches the process of English- The phenomenon of unbound anaphora refers to to-formal language translation. Finally, sections 4 instances where anaphoric pronouns occur outside and 5 define the syntax and semantics of the sys- the syntactic scopes (i.e. the c-command domain) tem. of their quantifier antecedents. The main kinds of unbound anaphora are regular anaphora to quan- 2 Elements of system tifiers, quantificational subordination, and donkey 2.1 Context, types and dependent types anaphora, as exemplified by (1) to (3) respectively: The variables of our system are always typed. We (1) Most kids entered. They looked happy. write x : X to denote that the variable x is of type X and refer to this as a type specification of the (2) Every man loves a woman. They kiss them. variable x. Types, in this paper, are interpreted as (3) Every farmer who owns a donkey beats it. sets. We write the interpretation of the type X as X . k k Unbound anaphoric pronouns have been dealt Types can depend on variables of other types. with in two main semantic paradigms: dynamic Thus, if we already have a type specification x : semantic theories (Groenendijk and Stokhof, X, then we can also have type Y (x) depending 1991); (Van den Berg, 1996); (Nouwen, 2003) and on the variable x and we can declare a variable y the E-type/D-type tradition (Evans, 1977); (Heim, of type Y by stating y : Y (x). The fact that Y 1990); (Elbourne, 2005). In the dynamic seman- depends on X is modeled as a projection tic theories pronouns are taken to be (syntactically π : Y X . free, but semantically bound) variables, and con- k k → k k text serves as a medium supplying values for the So that if the variable x of type X is interpreted as variables. In the E-type/D-type tradition pronouns an element a X , Y (a) is interpreted as the are treated as quantifiers. Our system combines ∈ k k k k fiber of π over a (the preimage of a under π) aspects of both families of theories. As in the E- { } type/D-type tradition we treat unbound anaphoric Y (a) = b Y : π(b) = a . k k { ∈ k k }

10 Proceedings of the EACL 2014 Workshop on Type Theory and Natural Language Semantics (TTNLS), pages 10–18, Gothenburg, Sweden, April 26-30 2014. c 2014 Association for Computational Linguistics One standard natural language example of such a 1957) further generalized by Lindstrom¨ to a dependence of types is that if m is a variable of so-called polyadic quantifier (Lindstrom,¨ 1966), the type of months M, there is a type D(m) of see (Bellert and Zawadowski, 1989). To use a the days of the month m. Such type dependencies familiar example, a multi-quantifier prefix like can be nested, i.e., we can have a sequence of type is thought of as a single two-place ∀m:M |∃w:W specifications of the (individual) variables: quantifier obtained by an operation on the two single quantifiers, and it has as denotation: x : X, y : Y (x), z : Z(x, y). m:M w:W = R M W : a M : Context for us is a partially ordered sequence of k∀ |∃ k { ⊆ k k × k k { ∈ k k type specifications of the (individual) variables b W : a, b R w:W m:M . and it is interpreted as a parameter set, i.e. as a { ∈ k k h i ∈ } ∈ k∃ k} ∈ k∀ k} set of compatible n-tuples of the elements of the In this paper we generalize the three chain- sets corresponding to the types involved (compat- constructors and the corresponding semantical op- ible wrt all projections). erations to (pre-) chains defined on dependent types. 2.2 Quantifiers, chains of quantifiers 2.3 Dynamic extensions of contexts Our system defines quantifiers and predicates polymorphically. A generalized quantifier Q is In our system language expressions are all defined an association to every set Z a subset of the in context. Thus the first sentence in (2) (on the power set of Z. If we have a predicate P de- most natural interpretation where a woman defined in a context Γ, then for any interpreta- pends on every man) translates (via the process de- tion of the context Γ it is interpreted as a scribed in Section 3) into a sentence with a chain k k subset of its parameter set. Quantifier phrases, of quantifiers in a context: e.g. every man or some woman, are interpreted as follows: every = man and Γ m:M w:W Love(m, w), k m:mank {k k} ` ∀ |∃ somew:woman = X woman : X = . k k { ⊆ k k 6 ∅} and says that the set of pairs, a man and a woman The interpretation of quantifier phrases is fur- he loves, has the following property: the set of ther extended into the interpretation of chains of those men that love some woman each is the set quantifiers. Consider an example in (2): of all men. The way to understand the second sen- (2) Every man loves a woman. They kiss them. tence in (2) (i.e. the anaphoric continuation) is that every man kisses the women he loves rather than Multi-quantifier sentences such as the first sen- those loved by someone else. Thus the first sentence in (2) are known to be ambiguous with tence in (2) must deliver some internal relation be- different readings corresponding to how various tween the types corresponding to the two quanti- quantifiers are semantically related in the sen- fier phrases. tence. To account for the readings available for In our analysis, the first sentence in (2) extends such multi-quantifier sentences, we raise quanti- the context Γ by adding new variable specifica- fier phrases to the front of a sentence to form tions on newly formed types for every quantifier (generalized) quantifier prefixes - chains of quan- phrase in the chain Ch = - for the ∀m:M |∃w:W tifiers. Chains of quantifiers are built from quanti- purpose of the formation of such new types we in- fier phrases using three chain-constructors: pack- troduce a new type constructor T. That is, the first formation rule (?,..., ?), sequential composi- sentence in (2) (denoted as ϕ) extends the context tion ? ?, and parallel composition ? . The se- | ? by adding: mantical operations that correspond to the chain- constructors (known as cumulation, iteration and T T tϕ, m : ϕ, m:M ; tϕ, w : ϕ, w:W (tϕ, m ) branching) capture in a compositional manner cu- ∀ ∀ ∃ ∃ ∀ mulative, scope-dependent and branching read- The interpretations of types (that correspond to ings, respectively. quantifier phrases in Ch) from the extended con- The idea of chain-constructors and the cor- text Γϕ are defined in a two-step procedure using responding semantical operations builds on the inductive clauses through which we define Ch Mostowski’s notion of quantifier (Mostowski, but in the reverse direction.

11 Step 1. We define fibers of new types by inverse (3) Every farmer who owns a donkey beats it. induction. To account for the dynamic contribution of modi- Basic step. fied common nouns (in this case common nouns For the whole chain Ch = m:M w:W we put: ∀ |∃ modified by relative clauses) we include in our T := Love . system -sentences (i.e. sentences with dummy ϕ, m:M w:W ∗ k ∀ |∃ k k k quantifier phrases). The modified common noun Inductive step. gets translated into a -sentence (with a dummy- ∗ quantifier phrase f : F ): Tϕ, = a M : b W : k ∀m:M k { ∈ k k { ∈ k k Γ f : F d:DOwn(f, d) a, b Love w:W ` |∃ h i ∈ k k} ∈ k∃ k} and for a M and we extend the context by dropping the speci- ∈ k k fications of variables: (f : F, d : D) and adding T ϕ, w:W (a) = b W : a, b Love new variable specifications on newly formed types k ∃ k { ∈ k k h i ∈ k k} for every (dummy-) quantifier phrase in the chain Step 2. We build dependent types from fibers. Ch∗: T T ϕ, w:W = a ϕ, w:W (a): T T k ∃ k {{ } × k ∃ k tϕ,f : ϕ,f:F ; tϕ, d : ϕ, d:D (tϕ,f ), [ ∃ ∃ a Tϕ, ∈ k ∀m:M k} The interpretations of types (that correspond to the Thus the first sentence in (2) extends the con- quantifier phrases in the Ch∗) from the extended T text Γ by adding the type ϕ, m:M , interpreted context Γϕ are defined in our two-step procedure. T ∀ as ϕ, m:M (i.e. the set of men who love some Thus the -sentence in (3) extends the context by k ∀ k T ∗ T T women), and the dependent type ϕ, w:W (tϕ, m ), adding the type ϕ,f:F interpreted as ϕ,f:F ∃ ∀ k k interpreted for a Tϕ, as Tϕ, (a) (i.e. the set of farmers who own some donkeys), ∈ k ∀m:M k k ∃w:W k (i.e. the set of women loved by the man a). and the dependent type Tϕ, (tϕ,f ), interpreted ∃d:D Unbound anaphoric pronouns are interpreted for a Tϕ,f:F as Tϕ, (a) (i.e. the set of ∈ k k k ∃d:D k with reference to the context created by the fore- donkeys owned by the farmer a). The main clause going text: they are treated as universal quantifiers translates into: and newly formed (possibly dependent) types in- T T crementally added to the context serve as their po- Γϕ tϕ,f : ϕ,f:F tϕ, : ϕ, (tϕ,f ) ` ∀ |∀ ∃d ∃d:D tential quantificational domains. That is, unbound Beat(tϕ,f , tϕ, ), anaphoric pronouns theym and themw in the sec- ∃d ond sentence of (2) have the ability to pick up and yielding the correct truth conditions Every farmer quantify universally over the respective interpreta- who owns a donkey beats every donkey he owns. tions. The anaphoric continuation in (2) translates Importantly, since we quantify over fibers (and not into: over farmer, donkey pairs), our solution does h i not run into the so-called ‘proportion problem’. Γ T T ϕ tϕ, m : ϕ, tϕ, w : ϕ, (tϕ, m ) ` ∀ ∀ ∀m:M |∀ ∃ ∃w:W ∀ Dynamic extensions of contexts and their interpretation are also defined for cumulative and Kiss(tϕ, m , tϕ, w ), ∀ ∃ branching continuations. Consider a cumulative where: example in (4):

T T = tϕ, m : ϕ, tϕ, w : ϕ, (tϕ, m ) (4) Last year three scientists wrote (a total of) five k∀ ∀ ∀m:M |∀ ∃ ∃w:W ∀ k articles (between them). They presented them R Tϕ, : a Tϕ, : { ⊆ k ∃w:W k { ∈ k ∀m:M k at major conferences. b Tϕ, (a): a, b R { ∈ k ∃w:W k h i ∈ } ∈ Interpreted cumulatively, the first sentence in (4) T (a) T , tϕ, w : ϕ, (tϕ, m ) tϕ, m : ϕ, translates into a sentence: k∀ ∃ ∃w:W ∀ k } ∈ k∀ ∀ ∀m:M k} yielding the correct truth conditions: Every man Γ (T hrees:S, F ivea:A) W rite(s, a). kisses every woman he loves. ` Our system also handles intra-sentential The anaphoric continuation in (4) can be inter- anaphora, as exemplified in (3): preted in what Krifka calls a ‘correspondence’

12 fashion (Krifka, 1996). For example, Dr. K wrote 4 System - syntax one article, co-authored two more with Dr. N, who 4.1 Alphabet co-authored two more with Dr. S, and the scientists that cooperated in writing one or more articles The alphabet consists of: also cooperated in presenting these (and no other) type variables: X,Y,Z,...; articles at major conferences. In our system, the type constants: M, men, women, . . .; first sentence in (4) extends the context by adding type constructors: , , T; individual variables: x, y, z, . . .; the type corresponding to (T hrees:S, F ivea:A): P Q predicates: P,P ,P ,...; t : T , 0 1 ϕ,(T hrees,F ivea) ϕ,(T hrees:S ; F ivea:A) quantifier symbols: , , five, Q ,Q ,...; ∃ ∀ 1 2 interpreted as a set of tuples three chain constructors: ? ?, ? , (?,..., ?). | ? T = k ϕ,(T hrees:S ,F ivea:A)k 4.2 Context = c, d : c S & d A & c wrote d {h i ∈ k k ∈ k k } A context is a list of type specifications of (indi- The anaphoric continuation then quantifies univer- vidual) variables. If we have a context sally over this type (i.e. a set of pairs), yielding the desired truth-conditions The respective scientists Γ = x1 : X1, . . . , xn : Xn( xi i Jn ) h i ∈ cooperated in presenting at major conferences the respective articles that they cooperated in writing. then the judgement Γ: cxt 3 English-to-formal language translation ` We assume a two-step translation process. expresses this fact. Having a context Γ as above, Representation. The syntax of the representa- we can declare a type Xn+1 in that context tion language - for the English fragment consid- ered in this paper - is as follows. Γ Xn+1( xi i Jn+1 ): type ` h i ∈ S P rdn(QP , . . . , QP ); → 1 n MCN P rdn(QP ,..., CN ,...,QP ); where Jn+1 1, . . . , n such that if i Jn+1, → 1 n ⊆ { } ∈ MCN CN; then Ji Jn+1, J1 = . The type Xn+1 depends → ⊆ ∅ on variables xi i Jn+1 . Now, we can declare a QP Det MCN; h i ∈ → X ( x ) Det every, most, three, . . .; new variable of the type n+1 i i Jn+1 in the → h i ∈ CN man, woman, . . .; context Γ → P rdn enter, love, . . .; → Γ xn+1 : Xn+1( xi i Jn+1 ) Common nouns (CNs) are interpreted as types, ` h i ∈ and common nouns modified by relative clauses and extend the context Γ by adding this variable (MCNs) - as -sentences determining some (pos- ∗ specification, i.e. we have sibly dependent) types.

Disambiguation. Sentences of English, con- Γ, xn+1 : Xn+1( xi i Jn+1 ): cxt trary to sentences of our formal language, are of- ` h i ∈ ten ambiguous. Hence one sentence representa- Γ0 is a subcontext of Γ if Γ0 is a context and a sub- tion can be associated with more than one sentence list of Γ. Let ∆ be a list of variable speciﬁcations in our formal language. The next step thus in- from a context Γ, ∆0 the least subcontext of Γ con- volves disambiguation. We take quantiﬁer phrases taining ∆. We say that ∆ is convex iff ∆0 ∆ is − of a given representation, e.g.: again a context.

P (Q1X1,Q2X2,Q3X3) The variables the types depend on are always and organize them into all possible chains of quan- explicitly written down in specifications. We can tifiers in suitable contexts with some restrictions think of a context as (a linearization of) a partially imposed on particular quantifiers concerning the ordered set of declarations such that the declara- places in prefixes at which they can occur (a de- tion of a variable x (of type X) precedes the dec- tailed elaboration of the disambiguation process is laration of the variable y (of type Y ) iff the type Y left for another place): depends on the variable x. Q1x1:X1 Q2x2:X2 The formation rules for both Σ- and Π-types are | P (x1, x2, x3). Q3x3:X3 as usual.

13 4.3 Language 2. Sequential composition of pre-chains Quantifier-free formulas. Here, we need only Γ Ch ~ : p-ch, Γ Ch ~ : p-ch predicates applied to variables. So we write ` 1 ~y1:Y1(~x1) ` 2 ~y2:Y2(~x2) Γ Ch Ch : p-ch 1 ~y1:Y~1(~x1) 2 ~y2:Y~2(~x2) Γ P (x , . . . , x ): qf-f ` | ` 1 n provided ~y (~y ~x ) = ; the specifications of to express that P is an n-ary predicate and the 2 ∩ 1 ∪ 1 ∅ the variables (~x ~x ) (~y ~y ) form a con- specifications of the variables x , . . . , x form a 1 ∪ 2 − 1 ∪ 2 1 n text, a subcontext of Γ. In so obtained pre-chain: subcontext of Γ. the binding variables are ~y ~y and the indexing Quantifier phrases. If we have a context Γ, y : 1 ∪ 2 variables are ~x ~x . Y (~x), ∆ and quantifier symbol Q, then we can 1 ∪ 2 3. Parallel composition of pre-chains form a quantifier phrase Qy:Y (~x) in that context. We write Γ Ch ~ : p-ch, Γ Ch ~ : p-ch ` 1 ~y1:Y1(~x1) ` 2 ~y2:Y2(~x2) Ch ~ Γ, y : Y (~x), ∆ Qy:Y (~x) : QP Γ 1 ~y1:Y1(~x1) : p-ch ` Ch ~ ` 2 ~y2:Y2(~x2) to express this fact. In a quantifier prase Qy:Y (~x): provided ~y (~y ~x ) = = ~y (~y ~x ). the variable y is the binding variable and the vari- 2 ∩ 1 ∪ 1 ∅ 1 ∩ 2 ∪ 2 ables ~x are indexing variables. As above, in so obtained pre-chain: the binding variables are ~y ~y and the indexing variables Packs of quantifiers. Quantifiers phrases can 1 ∪ 2 are ~x ~x . be grouped together to form a pack of quantifiers. 1 ∪ 2 The pack of quantifiers formation rule is as fol- A pre-chain of quantifiers Ch~y:Y~ (~x) is a chain lows. iff ~x ~y. The following ⊆ Γ Qi yi:Yi(~xi) : QP i = 1, . . . k ` Γ Ch ~ : chain Γ (Q ,...,Q ): pack ` ~y:Y (~x) ` 1 y1:Y1(~x1) k yk:Yk(~xk) expresses the fact that Ch is a chain of quan- where, with ~y = y , . . . , y and ~x = k ~x , we ~y:Y~ (~x) 1 k i=1 i tifiers in the context Γ. have that y = y for i = j and ~y ~x = . In so i 6 j 6 ∩ S ∅ Formulas, sentences and -sentences. The for- constructed pack: the binding variables are ~y and ∗ mulas have binding variables, indexing variables the indexing variables are ~x. We can denote such and argument variables. We write ϕ (~z) for a pack P c to indicate the variables involved. ~y:Y (~x) ~y:Y~ (~x) a formula with binding variables ~y, indexing vari- One-element pack will be denoted and treated as ables ~x and argument variables ~z. We have the a quantifier phrase. This is why we denote such a following formation rule for formulas pack as Qy:Y (~x) rather than (Qy:Y (~x)). Pre-chains and chains of quantifiers. Chains Γ A(~z): qf-f, Γ Ch~y:Y~ (~x) : p-ch and pre-chains of quantifiers have binding vari- ` ` Γ Ch~y:Y~ (~x) A(~z): formula ables and indexing variables. By Ch~y:Y~ (~x) we de- ` note a pre-chain with binding variables ~y and in- provided ~y is final in ~z, i.e. ~y ~z and variable ⊆ dexing variables ~x so that the type of the variable specifications of ~z ~y form a subcontext of Γ. In − yi is Yi(~xi) with i ~xi = ~x. Chains of quantifiers so constructed formula: the binding variables are are pre-chains in which all indexing variables are S ~y, the indexing variables are ~x, and the argument bound. Pre-chains of quantifiers arrange quantifier variables are ~z. phrases into N-free pre-orders, subject to some A formula ϕ (~z) is a sentence iff ~z ~y ~y:Y (~x) ⊆ binding conditions. Mutually comparable QPs in a and ~x ~y. So a sentence is a formula without free ⊆ pre-chain sit in one pack. Thus the pre-chains are variables, neither individual nor indexing. The fol- built from packs via two chain-constructors of se- lowing quential ? ? and parallel composition ? . The chain | ? formation rules are as follows. Γ ϕ~y:Y (~x)(~z): sentence ` 1. Packs of quantifiers. Packs of quantifiers are pre-chains of quantifiers with the same bind- expresses the fact that ϕ~y:Y (~x)(~z) is a sentence ing variable and the same indexing variables, i.e. formed in the context Γ. We shall also consider some special formulas Γ P c : ~y:Y~ (~x) pack that we call -sentences. A formula ϕ (~z) is a ` ∗ ~y:Y (~x) Γ P c ~ : p-ch -sentence if ~x ~y ~z but the set ~z ~y is possibly ` ~y:Y (~x) ∗ ⊆ ∪ −

14 ˇ not empty and moreover the type of each variable Γ such that one variable speciﬁcation tΦ0,ϕ : TΦ0,ϕ in ~z ~y is constant, i.e., it does not depend on vari- was already added for each pack Φ P acks − 0 ∈ Ch∗ ables of other types. In such case we consider the such that Φ0

Ch Γ0 TΦ,ϕ( tΦ0,ϕ Φ0 P acksCh ,Φ0

Γ0, tΦ,ϕ : TΦ,ϕ( tΦ ,ϕ Φ P acks ,Φ < Φ): cxt Γ ϕ (~z): -sentence h 0 i 0∈ Ch∗ 0 Ch∗ ` ~y:Y (~x) ∗ The context obtained from Γˇ by adding the new to express the fact that ϕ~y:Y (~x)(~z) is a -sentence ∗ variables corresponding to all the packs P acksCh formed in the context Γ. ∗ as above will be denoted by Having formed a -sentence ϕ we can form a ∗ new context Γϕ defined in the next section. Γϕ = Γˇ T(Ch ~ A(~z)). Notation. For semantics we need some notation ∪ ~y:Y (~x) for the variables in the -sentence. Suppose we At the end we add another context formation ∗ have a -sentence rule ∗ Γ Ch ~ A(~z): -sentence, Γ Ch~y:Y (~x) P (~z): -sentence ` ~y:Y (~x) ∗ ` ∗ Γϕ : cxt We define: (i) The environment of pre-chain Ch: Then we can build another formula starting in the Env(Ch) = Env(Ch~y:Y~ (~x)) - is the context defining variables ~x ~y; (ii) The binding variables context Γϕ. This process can be iterated. Thus − in this system sentence ϕ in a context Γ is con- of pre-chain Ch: Bv(Ch) = Bv(Ch~y:Y~ (~x)) - is the convex set of declarations in Γ of the binding structed via specifying sequence of formulas, with the last formula being the sentence ϕ. However, variables in ~y; (iii) env(Ch) = env(Ch~y:Y~ (~x)) - the set of variables in the environment of Ch, i.e. for the lack of space we are going to describe here only one step of this process. That is, sentence ϕ ~x ~y; (iv) bv(Ch) = bv(Ch ~ ) - the set of − ~y:Y (~x) in a context Γ can be constructed via specifying biding variables ~y; (v) The environment of a pre- -sentence ψ extending the context as follows chain Ch0 in a -sentence ϕ = Ch~y:Y (~x) P (~z), de- ∗ ∗ noted Envϕ(Ch0), is the set of binding variables Γ ψ : -sentence in all the packs in Ch∗ that are <ϕ-smaller than all ` ∗ packs in Ch0. Note Env(Ch0) Envϕ(Ch0). If Γ ϕ : sentence ⊆ ψ ` Ch0 = Ch1 Ch2 is a sub-pre-chain of the chain | For short, we can write Ch , then Env (Ch ) = Env (Ch ) ~y:Y (~x) ϕ 2 ϕ 1 ∪ Bv(Ch1) and Envϕ(Ch1) = Envϕ(Ch0). Γ Γ ϕ : sentence ` ψ ` 4.4 Dynamic extensions 5 System - semantics Suppose we have constructed a -sentence in a ∗ context 5.1 Interpretation of dependent types The context Γ Γ Ch A(~z): -sentence ` ~y:Y~ (~x) ∗ x : X(...), . . . , z : Z(. . . , x, y, . . .): cxt ` We write ϕ for Ch~y:Y~ (~x) A(~z). We form a context Γϕ dropping the specifica- gives rise to a dependence graph. A dependence tions of variables ~z and adding one type and one graph DGΓ = (TΓ,EΓ) for the context Γ has variable specification for each pack in P acks . types of Γ as vertices and an edge π : Y X Ch∗ Y,x → Let Γˇ denote the context Γ with the specifica- for every variable specification x : X(...) in Γ tions of the variables ~z deleted. Suppose Φ and every type Y (. . . , x, . . .) occurring in Γ that ∈ P acksCh∗ and Γ0 is an extension of the context depends on x.

15 The dependence diagram for the context Γ is an Envϕ(Ch0) is a sub-context of Γ disjoint from the association : DG Set to every type X convex set Bv(Ch ) and Env (Ch ), Bv(Ch ) is k − k Γ → 0 ϕ 0 0 in T a set X and every edge π : Y X a sub-context of Γ. For ~a Env (Ch ) we de- Γ k k Y,x → ∈ k ϕ 0 k in E a function π : Y X , so that fine Bv(Ch ) (~a) to be the largest set such that Γ k Y,xk k k → k k k 0 k whenever we have a triangle of edges in EΓ, πY,x as before π : Z Y , π : Z X we have ~a Bv(Ch0) (~a) Envϕ(Ch0), Bv(Ch0) Z,y → Z,x → { } × k k ⊆ k k π = π π . k Z,xk k Y,xk ◦ k Z,yk Interpretation of quantifier phrases. If we have The interpretation of the context Γ, the param- a quantifier phrase eter space Γ , is the limit of the dependence dia- k k gram : DGΓ Set. More specifically, Γ Q : QP k − k → ` y:Y (~x) Γ = x : X(...), . . . , z : Z(. . . , x, y, . . .) = and ~a Env (Q ) , then it is interpreted k k k k ∈ k ϕ y:Y (~x) k as Q ( Y (~a)) ( Y (~a ~x)). ~a : dom(~a) = var(Γ), ~a(z) Z (~a env(Z)), k k k k ⊆ P k k d { ∈ k k d Interpretation of packs. If we have a pack of πZ,x (~a(z)) = ~a(x), for z : Z in Γ, x envZ quantifiers in the sentence ϕ k k ∈ } where var(Γ) denotes variables specified in Γ and P c = (Q1 ,...Qn ) env(Z) denotes indexing variables of the type Z. y1:Y1(~x1) yn:Yn(~xn) The interpretation of the Σ- and Π-types are as and ~a Env (P c) , then its interpretation with ∈ k ϕ k usual. the parameter ~a is

5.2 Interpretation of language P c (~a) = (Q ,...,Q ) (~a) = k k k 1y1:Y1(~x1) nyn:Yn(~xn) k Interpretation of predicates and quantifier sym- n bols. Both predicates and quantifiers are inter- A Yi (~a ~xi): πi(A) Qi ( Yi (~a ~xi), preted polymorphically. { ⊆ i=1 k k d ∈ k k k k d If we have a predicate P defined in a context Γ: Y for i = 1, . . . , n } x1 : X1, . . . , xn : Xn( xi i Jn]) P (~x): qf-f π i h i ∈ ` where i is the -th projection from the product. Interpretation of chain constructors. then, for any interpretation of the context Γ , it k k 1. Parallel composition. For a pre-chain of is interpreted as a subset of its parameter set, i.e. quantifiers in the sentence ϕ P Γ . k k ⊆ k k Quantifier symbol Q is interpreted as quantifier Ch 1~y1:Y~1(~x1) Q i.e. an association to every1 set Z a subset Ch0 = k k Ch2 ~ Q (Z) (Z). ~y2:Y2(~x2) k k ⊆ P Interpretation of pre-chains and chains of quan- and ~a Envϕ(Ch0) we define tifiers. We interpret QP’s, packs, pre-chains, and ∈ k k chains in the environment of a sentence Envϕ. Ch1 ~ ~y1:Y1(~x1) (~a) = A B : This is the only case that is needed. We could kCh2~y :Y~ (~x ) k { × interpret the aforementioned syntactic objects in 2 2 2 their natural environment Env (i.e. independently A Ch (~a ~x ) and 1~y1:Y~1(~x1) 1 of any given sentence) but it would unnecessarily ∈ k k d B Ch2 ~ (~a ~x2) complicate some definitions. Thus having a ( -) ~y2:Y2(~x2) ∗ ∈ k k d } sentence ϕ = Ch~y:Y (~x) P (~z) (defined in a con- 2. Sequential composition. For a pre-chain of text Γ) and a sub-pre-chain (QP, pack) Ch0, for quantifiers in the sentence ϕ ~a Env (Ch ) we define the meaning of ∈ k ϕ 0 k Ch0 = Ch Ch 1~y1:Y~1(~x1) 2~y2:Y~2(~x2) Ch0 (~a) | k k and ~a Envϕ(Ch0) we define Notation. Let ϕ = Ch P (~y) be a - ∈ k k ~y:Y~ ∗ sentence built in a context Γ, Ch0 a pre-chain used Ch1 ~ Ch2 ~ (~a) = in the construction of the ( )-chain Ch. Then k ~y1:Y1(~x1)| ~y2:Y2(~x2)k ∗ 1 This association can be partial. R Bv(Ch0) (~a): ~b Bv(Ch ) (~a): { ⊆ k k { ∈ k 1 k

16 ~c Bv(Ch ) (~a,~b): ~b,~c R then we deﬁne sets { ∈ k 2 k h i ∈ } ∈ ~ Ch2 ~ (~a, b) Ch1 ~ (~a) T (~a) Ch (~a) k ~y2:Y2(~x2)k } ∈ k ~y1:Y1(~x1)k } k ϕ,Chi k ∈ k ik Validity. A sentence for i = 1, 2 so that

~x : X~ Ch ~ P (~y) T (~a) = T (~a) T (~a) ` ~y:Y k ϕ,Ch0 k k ϕ,Ch1 k × k ϕ,Ch2 k is true under the above interpretation iff if such sets exist, and these sets ( Tϕ,Ch (~a)) are k i k undeﬁned otherwise. P ( Y~ ) Ch Sequential decomposition. If we have k k k k ∈ k ~y:Y~ k

Ch0 = Ch1 ~ Ch2 ~ 5.3 Interpretation of dynamic extensions ~y1:Y1(~x1)| ~y2:Y2(~x2) Suppose we obtain a context Γϕ from Γ by the fol- then we put lowing rule Tϕ,Ch (~a) = ~b Bv(Ch1) (~a): Γ Ch A(~z): -sentence, k 1 k { ∈ k k ` ~y:Y~ (~x) ∗ ~ ~ T Γ : cxt ~c Bv(Ch2) (~a, b): b,~c ϕ,Ch (~a) ϕ { ∈ k k h i ∈ k 0 k } Ch2 (~a,~b) where ϕ is Ch~y:Y~ (~x) A(~z). Then ∈ k k } For ~b Bv(Ch1) we put ˇ T ∈ k k Γϕ = Γ (Ch~y:Y~ (~x) A(~z)). ∪ T (~a,~b) = ~c Bv(Ch ) (~a,~b): k ϕ,Ch2 k { ∈ k 2 k Γ From dependence diagram : DGΓ Set k − k → ~b,~c T (~a) we shall define another dependence diagram h i ∈ k ϕ,Ch0 k } Step 2. (Building dependent types from fibers.) Γϕ = : DGΓϕ Set If Φ is a pack in Ch , ~a Env (Φ) then we k − k k − k → ∗ ∈ k ϕ k put Thus, for Φ P ackCh∗ we need to define Γ ∈ T ϕ and for Φ < Φ we need to define Tϕ,Φ = ~a Tφ,Φ (~a): ~a Envϕ(Φ) , k Φ,ϕk 0 Ch∗ k k {{ }×k k ∈ k k T T [ T πTΦ,ϕ,t : Φ,ϕ Φ ,ϕ Φ0

T (~a) Bv(Ch0) (~a) 6 Conclusion k ϕ,Ch0 k ⊆ k k It was our intention in this paper to show that This is done using the inductive clauses through adopting a typed approach to generalized quan- which we have defined Ch∗ but in the reverse di- tification allows a uniform treatment of a wide ar- rection. ray of anaphoric data involving natural language The basic case is when Ch0 = Ch∗. We put quantification. T ( ) = P k ϕ,Chk ∅ k k Acknowledgments The inductive step. Now assume that the set The work of Justyna Grudzinska´ was funded by T (~a) is defined for ~a Env (Ch ) . the National Science Center on the basis of de- k ϕ,Ch0 k ∈ k ϕ 0 k Parallel decomposition. If we have cision DEC-2012/07/B/HS1/00301. The authors would like to thank the anonymous reviewers for Ch 1~y1:Y~1(~x1) valuable comments on an earlier version of this pa- Ch0 = Ch per. 2~y2:Y~2(~x2)

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