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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
Multimodal Linguistic Inference
, 1995
"... In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resourc ..."
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Cited by 42 (6 self)
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In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...
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
A modal walk through space
 JOURNAL OF APPLIED NONCLASSICAL LOGICS
, 2002
"... We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and ..."
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Cited by 33 (5 self)
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We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.
Continuation semantics for the Lambekâ€“Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
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Logical Patterns in Space
 University of Amsterdam
, 1999
"... In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra ..."
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Cited by 18 (5 self)
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In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra iss'e style games between patterns in space. Finally, we consider spatial languages of increased logical power in the direction of geometry. Also, Intelligent Sensory Information Systems, University of Amsterdam 1 Contents 1 Reasoning about Space 3 2 Topological Structure: a Modal Approach 4 2.1 The topological view of space . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Special properties of topological spaces . . . . . . . . . . . 6 2.1.3 Structure preserving mappings . . . . . . . . . . . . . . . 7 3 Basic Modal Logic of Space 8 3.1 Topological language and semantics . . . . . . . . . . . . . . . . 8 3.2 Topologi...
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.
In Situ Binding: A Modal Analysis
, 1996
"... In this paper we compare two multimodal deconstructions of the in situ binder q(A; B; C), proposed in [9] for the scoping of generalized quantifiers. The wrapping analysis of [13] is shown to be of limited generality: it restricts the occurrence of generalized quantifier expressions to associative ..."
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Cited by 7 (1 self)
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In this paper we compare two multimodal deconstructions of the in situ binder q(A; B; C), proposed in [9] for the scoping of generalized quantifiers. The wrapping analysis of [13] is shown to be of limited generality: it restricts the occurrence of generalized quantifier expressions to associative environments  environments where sensitivity for constituent structure is lacking. We propose an alternative deconstruction where the wrapping operation is independent of resource management assumptions about the structural context. The analysis is based on the general theory of structural control proposed in [8]: the interaction principles for the wrapping operation are finetuned in terms of unary modal control devices.