Quantifiers and Scope in Pregroup Grammar Claudia Casadio Faculty of Psychology - Dept. of Philosophy University “G. D’Annunzio” Chieti
[email protected] - www.unich.it/∼casadio.html
Abstract In this paper we propose a geometrical representation of quantified noun phrases and their scope properties in the framework of Pregroup Grammar ([6][7][8]) and by means of the planar proof nets of noncommutative linear logic ([1][3][4]).
1
Preliminaries
From the point of view of logical semantics [12] quantifiers are functions from properties into propositions, and their scope properties are unambiguously determined by the logical form. On the other hand, natural language quantifiers remain in situ, taking scope over arbitrary large contexts. The following sentence with quantified subject and object is a typical example: (1) Every astronomer loves some star. This is the simplest case of scope ambiguity, with two readings distinguished by the scope of the quantifiers every, some. Under the first reading, every astronomer may love a different star. In this reading, the quantifier phrase every astronomer is said to take wide scope and some star to have narrow scope. Under the second reading, there must be some star such that every astronomer loves her (Venus, or Marilin Monroe). The second reading, where the object quantifier takes wide scope over the subject quantifier, entails the first reading. The two readings can be represented both in non-commutative linear logic or bilinear logic (bill [1],[5]), and compact bilinear logic (cbill [4],[7]). The second system has been applied to linguistics by J. Lambek giving rise to the new form of syntactic calculus called Pregroup Grammar ([6],[7],[8]). The 1
crucial difference between the two systems is that the former is based on two operations, sum + and product • (corresponding to the connectives par ℘ and times ⊗ of linear logic), while the latter defines just one operation, the compact product (see [4][8] for details). In this paper we will apply the pregroup grammar to the analysis of scope structures, defining suitable logical types for predicates and quantifiers within compact bilinear logic, but we will also show that equivalent results hold for non-commutative linear logic.
2
The calculus of Pregroups
A pregroup {G, . , 1, ` , r , →} is a partially ordered monoid in which each element a has a left adjoint a` , and a right adjoint ar such that a` a → 1 → a a` a a r → 1 → ar a where the dot “.”, that is usually omitted, stands for multiplication with unit 1, and the arrow denotes the partial order ([6][7]). In linguistic applications the symbol 1 stands for the empty string and multiplication is interpreted as concatenation. As shown in [8] adjoints are unique and it is proved that 1` = 1 = 1r , (a · b)` = b ` · a ` , (a · b)r = b r · a r , b` → a` a→b a→b br → ar ` ` r r `` `` b →a , b →a , a →b , arr → brr . The following also hold ar` = a = a`r , a`` a` → 1 → a` a`` ,
ar arr → 1 → arr ar . The first equation introduces the cancellation of double opposite adjoints; then we have the rules for contracting and expanding identical double left and double right adjoints. We work with the pregroup freely generated by a partially ordered set of basic types. From each basic type a we form simple types by taking single or repeated adjoints: . . . a`` , a` , a, ar , arr . . . . Compound types or just types are strings of simple types. One (or more) types are assigned to each word in the dictionary and the only computations required are contractions (C) and expansions (E) : 2
(C) (E)
a` a → 1 , a ar → 1 , 1 → a a` , 1 → ar a .
For the purpose of sentence verification expansions are not needed, but only contractions, combined with some rewriting induced by the partial order (as proved in [2][8]). Developing a pregroup grammar pg for a language consists in two main steps: (i) assign one or more (basic or compound) types to each word in the dictionary; (ii) check the grammaticality and sentencehood of a string of words by a calculation on the corresponding types, where the only rules involved are contractions, ordering postulates of the form α → β (α, β basic types) and metarules, language specific conditions on types introduced in the lexicon.
3
Quantifiers and scope
We extend the pregroup grammar pg introducing new basic types in the style of Montague semantics: e (entities), t (truth values). These logical types are invoked for treating specific semantical tasks, such as scope analysis, and are associated with the syntactic types involved in grammatical analysis (multidimensional lexical entries as defined in [9][10]). In particular, sentential predicates, syntactically corresponding to the category VP, are assigned the type: (er t) in cbill [ (er + t) in bill ], expressing the fact that a proposition is obtained by right adjoining to the predicate an entity occurring to the left (the subject). Venus e
is shining (er t) → t
We show how calculations over types procede from left to right in terms of appropriate links drawn under basic and simple types, where underlying links connecting two types (e.g. a br , or b` a, when a → b) are introduced to indicate the contractions allowed in the free pregroup (cf. the planar proof nets of non-commutative linear logic [3][10][11]). Subject quantifier phrases, as functions from properties to propositions, will then receive the type: (t t` e) in cbill [ (t + (t` e)) in bill ]. Every star (t t` e)
is shining (er t) → t
We analyze two arguments predicates, or transitive verbs, as taking type: (er t e` ) in cbill [ (er + t + e` ) in bill ]. 3
Galileo e
loves (er t e` )
Venus e → t
The type for object quantifier phrases is then introduced: (e tr t) in cbill [((e tr ) + t) in bill ]. Embedding the two quantifiers in the same sentence we obtain a direct representation of scope ambiguities by means of two different ways of linking the type t with its dual counterparts (the left and right adjoint). Wide scope reading of Q1 Every astronomer (t t` e)
loves (er t e` )
some star (e tr t) → t
Wide scope reading of Q2 Every astronomer (t t` e)
loves (er t e` )
some star (e tr t) → t
Challenging examples of multiple readings are provided by complex sentences with two or more nested quantifier phrases: (2) a. Every astronomer thinks that Galileo likes some star. b. Every mathematician thinks that every astronomer likes some star. c. Every astronomer thinks he likes some star. d. Galileo believes a star appeared. e. Some astronomer believes a star appeared. We may think of the wide vs narrow scope readings of the embedded quantifiers as the different ways in which information flows within the given contexts and it is interesting to obtain an explicit representation of this dimension. This can be done within the pregroup calculus (compact bilinear logic) by introducing appropriate over -links (corresponding to possible expansions). Since these links are not needed for sentence calculation, but can have applications in expressing semantic or pragmatic information, we will define them as meta-links. By means of such meta-links we can see e.g. that, in the wide scope reading of the subject quantifier, there is a path of type t starting in the subordinate predicate and ending into the initial quantifier 4
phrase. On the other and, in the wide scope reading of the object quantifier we find a different path, starting from the same source but ending into the postverbal quantifier phrase. Wide scope reading of Q2 Every astronomer (t t` e)
loves (er t e` )
some star (e tr t)
→ t
The meta-links are an interesting device for representing different scope configurations and also anaphora connections. Consider for example the case (2c) in which we have a pronoun in place of the name Galileo in (2a). The pronoun can take the universal quantifier phrase every astronomer as its antecedent and we can express this information by means of a meta-link of type e in this case.
References [1] Abrusci, V. M. (1996), ‘Lambek calculus, Cyclic Multiplicative-Additive Linear Logic, Noncommutative Multiplicative-Additive Linear Logic: language and sequent calculus’ in Proofs and Linguistic Categories: Proceedings 1996 Roma Workshop, Bologna, CLUEB, 21-48. [2] Buszkowski, W., Lambek grammars based on pregroups. In P. de Groote, G. Morrill and C. Retor´e (eds.), Logical Aspects of Computational Linguistics, 95-109, SpringerVerlag, Berlin, 2001. [3] Casadio, C. (2001), ‘Non-commutative linear logic in linguistics’, Grammars, 4/3, 1-19. [4] Casadio, C. and J. Lambek. (2002). ‘A tale of four grammars’, Studia Logica, vol. 71, 2. Special Issue edited by W. Buszkowski. [5] Lambek, J. (1995), ‘Bilinear logic in algebra and linguistics’, in Girard et al. (eds.), Advances in Linear Logic, Cambridge, Cambridge University Press, 43-60. [6] Lambek, J. (1999), ‘Type grammars revisited’, in A. Lecomte, F. Lamarche and G. Perrier (eds.), Logical Aspects of Computational Linguistics, Springer LNAI 1582, 1-27. [7] Lambek, J. (2001), ‘Type grammars as pregroups’, Grammars 4, 21-39. [8] Lambek, J., A computational algebraic approach to English grammar, Syntax, 2004. [9] Moortgat, M. (1997), ‘Categorial Type Logics’, in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, 93-177. [10] Morrill, G. (1994), Type Logical Grammar, Kluwer, Dordrecht. [11] Roorda, D. (1991), Resource Logics: Proof Theoretical Investigations, Ph.D. Dissertation, University of Amsterdam. [12] Van Benthem, J. (1991), Language in Action. Categories, Lambdas, and Dynamic Logic, North Holland, Amsterdam.
5
