A hyperintensional semantics for natural language is proposed which is agnostic about the question of whether propositions are sets of worlds or worlds are sets of propositions. Montague’s theory of intensional senses is replaced by a weaker theory, written in standard classical higher-order logic, of fine-grained senses which are in a many-to-one correspondence with intensions; Montague’s theory can then be recovered from the proposed theory by identifying the type of propositions with the type of sets of worlds and adding an (...) axiom to the effect that each world is the set of propositions which are true there. Senses are compositionally assigned to linguistic expressions by a categorial grammar with only two rule schemas, based on the implicative fragment of intuitionistic linear propositional logic, and a fully explicit grammar fragment is provided that illustrates the compositional assignment of sense to a variety of constructions, including dummy-subject constructions, infinitive complements, predicative adjectives and nominals, raising to subject, ‘tough-movement’, and quantifier scope ambiguities. Notably, the grammar and the derivations that it licenses never make reference to either worlds or to the extensions of senses. (shrink)

A wide array of syntactic phenomena can be categorized as being either direction-sensitive (e.g. coordination) or direction-insensitive (quantification and medial extraction). In the realm of categorial grammar, many frameworks are engineered to handle one class of phenomena at the expense of the other. In particular, Lambek-inspired frameworks handle direction-sensitivity elegantly but struggle with cases of direction-insensitivity, whereas in linear grammars, the situation is just the opposite. One reasonably successful attempt to unify the insights of both types of grammar and allow (...) for the analysis of direction-sensitive as well as direction-insensitive phenomena is hybrid type-logical categorial grammar (HTLCG), which augments the set of syntactic categories of an ordinary linear grammar with the directional slashes of a Lambek grammar. In this paper, the complementary nature of Lambek and linear grammars, with respect to direction-sensitive and direction-insensitive phenomena, is expounded on, as is the nature of how HTLCG attempts to reconcile the two systems. The paper further discusses an alternative to HTLCG—a linear grammar which adds Lambek-like capabilities by augmenting the string-combining (phenogrammatic) mechanism instead of the definition of syntactic categories (the tectogrammar). In particular, LCGϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {LCG}_\phi $$\end{document} (linear categorial grammar with phenominators) adds a form of subtyping based on phenominators—a class of linear functions which specify various modes of (phenogrammatic) combination. Most importantly, an algorithm is provided for embedding HTLCG into LCGϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {LCG}_\phi $$\end{document}, thereby demonstrating that LCGϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {LCG}_\phi $$\end{document} can handle direction-(in)sensitivity at least as well as HTLCG. (shrink)

This paper argues for the idea that in describing language we should follow Haskell Curry in distinguishing between the structure of an expression and its appearance or manifestation . It is explained how making this distinction obviates the need for directed types in type-theoretic grammars and a simple grammatical formalism is sketched in which representations at all levels are lambda terms. The lambda term representing the abstract structure of an expression is homomorphically translated to a lambda term representing its manifestation, (...) but also to a lambda term representing its semantics. (shrink)

In this paper we will discuss the first order multiplicative intuitionistic fragment of linear logic, MILL1, and its applications to linguistics. We give an embedding translation from formulas in the Lambek Calculus to formulas in MILL1 and show this translation is sound and complete. We then exploit the extra power of the first order fragment to give an account of a number of linguistic phenomena which have no satisfactory treatment in the Lambek Calculus.

Categorial grammar is well-known for its elegant analysis of coordination enabled by the flexible notion of constituency it entertains. However, to date, no systematic study exists that examines whether this analysis has any obvious empirical advantage over alternative analyses of nonconstituent coordination available in phrase structure-based theories of syntax. This paper attempts precisely such a comparison. We compare the direct constituent coordination analysis of non-canonical coordinations in categorial grammar with an ellipsis-based analysis of the same phenomena in the recent HPSG (...) literature. We provide a set of empirical evidence, consisting of cases in which non-canonical coordinations interact with scopal operators of various sorts, which systematically falsifies the predictions of the latter, ‘linearization-based’ ellipsis approach to coordination. We propose an alternative analysis in a variant of categorial grammar called Hybrid Type-Logical Categorial Grammar. The proposed framework builds on both the Lambek-inspired variants of categorial grammar and a more recent line of work modelling word order via a lambda calculus for the prosodic component. The flexible syntax–semantics interface of this framework straightforwardly captures the interactions between non-canonical coordinations and scopal expressions, demonstrating the broader empirical payoff of the direct constituent coordination analysis of non-canonical coordinations pioneered by Steedman :523–568, 1985; Linguist Philos 13:207–263, 1990) and Dowty hitherto not explicitly recognized in the literature. (shrink)

This paper describes a prototype implementation of a Labelled Deduction System for natural language interpretation, where interpretation is taken to be the process of understanding a natural language utterance. The implementation models the process of understanding wh-gap dependencies in questions and relative clauses for a fragment of English. The paper is divided in three main sections. In Section 1, we introduce the basic architecture of the system. Section 2 outlines a prototype implementation of wh-binding and indicates its potential for explanation (...) of linguistic phenomena, and in Section 3 we briefly set the model within a large theoretical perspective, comparing it to other type-logical approaches to natural language analysis. (shrink)

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for string and tree (...) languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars. (shrink)

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of linear context-free rewriting systems. This (...) suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)

The relationship between Lexical-Functional Grammar (LFG) functional structures (f-structures) for sentences and their semanticinterpretations can be formalized in linear logic in a way thatcorrectly explains the observed interactions between quantifier scopeambiguity, bound anaphora and intensionality.Our linear-logic formalization of the compositional properties ofquantifying expressions in natural language obviates the need forspecial mechanisms, such as Cooper storage, in representing thescoping possibilities of quantifying expressions. Instead, thesemantic contribution of a quantifier is recorded as a linear-logicformula whose use in a proof will establish the (...) scope of thequantifier. Different proofs can lead to different scopes. In eachcomplete proof, the properties of linear logic ensure thatquantifiers are properly scoped. (shrink)

Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose $$\text {NL}_\lambda $$, which is NL supplemented with a single structural inference rule (“abstraction”). Abstraction closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form of (...) the abstraction inference, there has been some doubt that $$\text {NL}_\lambda $$ should count at a legitimate substructural logic. This paper argues that $$\text {NL}_\lambda $$ is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames. (shrink)

Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose \, which is NL supplemented with a single structural inference rule.ion closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form of the abstraction inference, there (...) has been some doubt that \ should count at a legitimate substructural logic. This paper argues that \ is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames. (shrink)

This paper introduces λ-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λ-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λ-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable.

The paper develops Lambda Grammars, a form of categorial grammar that, unlike other categorial formalisms, is non-directional. Linguistic signs are represented as sequences of lambda terms and are combined with the help of linear combinators.

The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which finite sequences of lambda terms are the basic data structures, (...) pairs of terms syntax, semantics for example. These sequences then combine with the help of simple generalizations of the usual abstraction and application operations. This theory, which I call Lambda Grammars and which is closely related to the independently formulated theory of Abstract Categorial Grammars (de Groote, 2001; de Groote, 2002), in fact is an implementation of Curry’s ideas: the level of tectogrammar is encoded by the sequences of lambda-terms and their ways of combination, while the syntactic terms in those sequences constitute the phenogrammatical level. In de Groote’s formulation of the theory, tectogrammar is the level of abstract terms, while phenogrammar is the level of object terms. (shrink)

In this paper we discuss a new perspective on the syntax-semantics interface. Semantics, in this new set-up, is not ‘read off’ from Logical Forms as in mainstream approaches to generative grammar. Nor is it assigned to syntactic proofs using a Curry-Howard correspondence as in versions of the Lambek Calculus, or read off from f-structures using Linear Logic as in Lexical-Functional Grammar (LFG, Kaplan & Bresnan [9]). All such approaches are based on the idea that syntactic objects (trees, proofs, fstructures) are (...) somehow prior and that semantics must be parasitic on those syntactic objects. We challenge this idea and develop a grammar in which syntax and semantics are treated in a strictly parallel fashion. The grammar will have many ideas in common with the (converging) frameworks of categorial grammar and LFG, but its treatment of the syntax-semantics interface is radically different. Also, although the meaning component of the grammar is a version of Montague semantics and although there are obvious affinities between Montague’s conception of grammar and the work presented here, the grammar is not compositional, in the sense that composition of meaning need not follow surface structure. (shrink)