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Categorial Type Logics
 Handbook of Logic and Language
, 1997
"... Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource mana ..."
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Cited by 254 (5 self)
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Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntaxsemantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation
Structural Control
 SPECIFYING SYNTACTIC STRUCTURES, PATRICK BLACKBURN, MAARTEN DE RIJKE (EDS.)
, 1988
"... In this paper we study Lambek systems as grammar logics: logics for reasoning about structured linguistic resources. The structural parameters of precedence, dominance and dependency generate a cube of resourcesensitive categorial type logics. From the pure logic of residuation NL, one obtains L, N ..."
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Cited by 42 (8 self)
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In this paper we study Lambek systems as grammar logics: logics for reasoning about structured linguistic resources. The structural parameters of precedence, dominance and dependency generate a cube of resourcesensitive categorial type logics. From the pure logic of residuation NL, one obtains L, NLP and LP in terms of Associativity, Commutativity, and their combination. Each of these systems has a dependency variant, where the product is split up into a leftheaded and a rightheaded version. We develop a theory of systematic communication between these systems. The communication is twoway: we show how one can fully recover the structural discrimination of a weaker logic from within a system with a more liberal resource management regime, and how one can reintroduce the structural flexibility of a stronger logic within a system with a more articulate notion of structuresensitivity. In executing this programme we follow the standard logical agenda: the categorial formula language is enriched with extra control operators, socalled structural modalities, and on the basis of these control operators, we prove embedding theorems for the two directions of substructural communication. But our results differ from the Linear Logic style of embedding with S4like modalities in that we realize the communication in both directions in terms of a
Formal grammar and information theory: Together again?
 PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY
, 2000
"... In the last 40 years, research on models of spoken and written language has been split between two seemingly irreconcilable traditions: formal linguistics in the Chomsky tradition, and information theory in the Shannon tradition. Zellig Harris had advocated a close alliance between grammatical and i ..."
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Cited by 30 (0 self)
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In the last 40 years, research on models of spoken and written language has been split between two seemingly irreconcilable traditions: formal linguistics in the Chomsky tradition, and information theory in the Shannon tradition. Zellig Harris had advocated a close alliance between grammatical and informationtheoretic principles in the analysis of natural language, and early formallanguage theory provided another strong link between information theory and linguistics. Nevertheless, in most research on language and computation, grammatical and informationtheoretic approaches had moved far apart. Today, after many years on the defensive, the informationtheoretic approach has gained new strength and achieved practical successes in speech recognition, information retrieval, and, increasingly, in language analysis and machine translation. The exponential increase in the speed and storage capacity of computers is the proximate cause of these engineering successes, allowing the automatic estimation of the parameters of probabilistic models of language by counting occurrences of linguistic events in very large bodies of text and speech. However, I will argue that informationtheoretic and computational ideas are also playing an increasing role in the scientific understanding of language, and will help bring together formallinguistic and informationtheoretic perspectives.
Analyzing the Core of Categorial Grammar
, 2001
"... Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modal ..."
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Cited by 11 (4 self)
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Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic operators. We will show how we can obtain Lambek residuated triple n, = and of binary operators, and how the operators 3 and 2 introduced by Moortgat in [16] are indeed their unary counterpart.
Constants of Grammatical Reasoning
 Constraints and Resources in Natural Language Syntax and Semantics
, 1999
"... This is a screen version, enhanced with some dynamic features, of the paper that has appeared under the same title in Bouma, Hinrichs, Kruij# & Oehrle (eds.) Constraints and Resources in Natural Language Syntax and Semantics. CSLI, Stanford, 1999. You can use the # and # keys to move through ..."
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Cited by 11 (2 self)
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This is a screen version, enhanced with some dynamic features, of the paper that has appeared under the same title in Bouma, Hinrichs, Kruij# & Oehrle (eds.) Constraints and Resources in Natural Language Syntax and Semantics. CSLI, Stanford, 1999. You can use the # and # keys to move through the document. The # sign at the bottom of the screen brings you back from a hyperlink. Contents # Contents 1 Cognition = computation, grammar = logic . . . . . . . . . . . . . . . . . . 4 1.1 Grammatical resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Composition: the form dimension . . . . . . . . . . . . . 12 1.1.2 Composition: the meaning dimension. . . . . . . . . . 16 1.1.3 Lexical versus derivational meaning. . . . . . . . . . . . 19 1.2 Grammatical reasoning: logic, structure and control . . . . . 20 2 Patterns for structural variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1 English relativization: right branch extraction . . . . . . . . . . 31 2.2 Dutch relativization: left branch extraction . . . . . . . . . . . . . 38 2.3 Dependency: blocking extraction from subjects . . . . . . . . . 43 3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Contents # 1. Cognition = computation, grammar = logic Within current linguistic frameworks a rich variety of principles has been put forward to account for the properties of local and unbounded dependencies. Valency requirements of lexical items are checked by subcategorization principles in HPSG, principles of coherence and completeness in LFG, the theta criterion in GB. These are supplemented by, and interacting with, principles governing nonlocal dependencies: movement and empty category principles, slash featu...
Galois Connections in Categorial Type Logic
, 2001
"... The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and ..."
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Cited by 11 (4 self)
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The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and prooftheoretic groundwork. In x3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quanti er expressions, and to describe a typology of polarity items.
Lambek Calculus with Nonlogical Axioms
 Language and Grammar, Studies in Mathematical Linguistics and Natural Language
, 2002
"... We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associa ..."
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Cited by 8 (6 self)
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We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages; here we give a new proof of this fact. We also obtain similar results for systems with permutation and n ary operations.
Anaphora and Quantification in Categorial Grammar
"... The paper proposes a type logical reformulation of Jacobson 's ([9]) treatment of anaphoric dependencies in Categorial Grammar. To this end, the associative Lambek Calculus is extended with a new connective. In the first part, its proof theory is introduced and the logical properties of the res ..."
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Cited by 6 (2 self)
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The paper proposes a type logical reformulation of Jacobson 's ([9]) treatment of anaphoric dependencies in Categorial Grammar. To this end, the associative Lambek Calculus is extended with a new connective. In the first part, its proof theory is introduced and the logical properties of the resulting calculus are discussed. In the second part, this system is applied to several linguistic phenomena concerning the interaction of pronominal anaphora, VP ellipsis and quantifier scope. Finally, a possible extension to crosssentential anaphora is considered. 1
Symmetric categorial grammar
 Journal of Philosophical Logic
, 2009
"... is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left a ..."
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is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the CurryHoward derivational semantics, and structurepreservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning. 1 1