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.

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)

Doing Worlds with Words throws light on the problem of meaning as the meeting point of linguistics, logic and philosophy, and critically assesses the possibilities and limitations of elucidating the nature of meaning by means of formal logic, model theory and model-theoretical semantics. The main thrust of the book is to show that it is misguided to understand model theory metaphysically and so to try to base formal semantics on something like formal metaphysics; rather, the book states that model theory (...) and similar tools of the analysis of language should be understood as capturing the semantically relevant, especially inferential, structure of language. From this vantage point, the reader gains a new light on many of the traditional concepts and problems of logic and philosophy of language, such as meaning, reference, truth and the nature of formal logic. (shrink)

Portable Grammar Format (PGF) is a core language for type-theoretical grammars. It is the target language to which grammars written in the high-level formalism Grammatical Framework (GF) are compiled. Low-level and simple, PGF is easy to reason about, so that its language-theoretic properties can be established. It is also easy to write interpreters that perform parsing and generation with PGF grammars, and compilers converting PGF to other formats. This paper gives a concise description of PGF, covering syntax, semantics, and parser (...) generation. It also discusses the technique of embedded grammars, where language processing tasks defined by PGF grammars are integrated in larger systems. (shrink)

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)

We expose some basic elements of a style of programming supported by functional languages like Haskell by relating them to a coherent set of notions and techniques from Curry’s work in combinatory logic and formal systems, and their algebraic and categorical interpretations. Our account takes the form of a commentary to a simple fragment of Haskell code attempting to isolate the conceptual sources of the linguistic abstractions involved.

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)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

This paper reminds, puts in order, sketches and also initiates some researches from the field of logic and philosophy of language. It lays emphasis on the logical-linguistic and ontological developmental lines originated with Polish researchers. The author discusses two opposite orientations of the former line in the process of formalization of language, called here nominalistic and Platonistic. The paper mentions the author's result (1989; 1991) concerning theoretical equivalence of two axiomatic approaches to language syntax which take into consideration these two (...) different philosophical orientations, respectively. It also introduces certain formal-logical foundations of language semantics, discusses and develops the problem of the syntactic and semantic categorial agreement, i.e. the agreement of the syntactic category of each language expression with the ontological category assigned to the reference of this expression. The author, in particular, gives a solution to this problem for quantifiers. (shrink)

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)

In this paper we consider the relations existing between four deductive systems that have been called categorial grammars and have relevant connections with linguistic investigations: the syntactic calculus, bilinear logic, compact bilinear logic and Curry''s semantic calculus.