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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 299 (6 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
A Nondeterministic View on Nonclassical Negations
 Workshop on Negation in Constructive Logic
, 2004
"... We investigate two large families of logics, diering from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant for \the false") by adding various standard Gentzentype rules for ..."
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Cited by 9 (6 self)
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We investigate two large families of logics, diering from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant for \the false") by adding various standard Gentzentype rules for negation. The logics in the other family are similarly obtained from LJ, the positive fragment of intuitionistic logic (again, with or without ). For all the systems, we provide simple semantics which is based on nondeterministic fourvalued or threevalued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of nondeterminism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ (with or without ) are shown to be conservative over the underlying logic, and it is determined which of them are not. 1
Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
Anaphora and Ellipsis in TypeLogical Grammar
 Proceedings of the Eleventh Amsterdam Colloquium, ILLC, University of Amsterdam
, 1997
"... The aim of the present paper is to outline a unified account of anaphoricity and ellipsis phenomena within the framework of Type Logical Categorial Grammar. 1 There is at least one conceptual and one empirical reason to pursue such a goal. Firstly, both phenomena are characterized by the fact that t ..."
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The aim of the present paper is to outline a unified account of anaphoricity and ellipsis phenomena within the framework of Type Logical Categorial Grammar. 1 There is at least one conceptual and one empirical reason to pursue such a goal. Firstly, both phenomena are characterized by the fact that they reuse semantic
Lambek calculus proofs and tree automata
 Logical Aspects of Computational Linguistics Third International Conference, LACL '98, Selected Papers, volume 2014 of Lecture Notes in Artificial Intelligence
, 2001
"... Abstract. We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the La ..."
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Cited by 5 (1 self)
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Abstract. We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the Lambek calculus, which typically use only a subset of the set of all proofs. While Lambek grammars can assign regular tree languages as structural descriptions, there exist Lambek grammars that assign nonregular structural descriptions, both when considering normal and nonnormal proof trees. Combining the results of Pentus (1993) and Thatcher (1967), we can conclude that Lambek grammars, although generating only contextfree languages, can extend the strong generative capacity of contextfree grammars. Furthermore, we show that structural descriptions that disregard the use of introduction rules cannot be used for a compositional semantics following the CurryHoward isomorphism. 1
Basic DependencyBased Logical Grammar
, 1998
"... A logical grammar is presented that employs (kinds of) dependency relations as its basic categories, rather than constituents. The aim with this dependencybased logical grammar is to provide a calculus for doing analysis based on the description of natural language as provided by Sgall et al ([46, ..."
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A logical grammar is presented that employs (kinds of) dependency relations as its basic categories, rather than constituents. The aim with this dependencybased logical grammar is to provide a calculus for doing analysis based on the description of natural language as provided by Sgall et al ([46, 45]) and Petkevic ([42]). Table of Contents Motivation 4 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Functional Generative Description 7 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Tectogrammatical Representations . . . . . . . . . . . . . . . 9 2.1 Dependency Relations . . . . . . . . . . . . . . . . . . 10 2.2 Lexical Information . . . . . . . . . . . . . . . . . . . 10 2.3 Coordination and Apposition . . . . . . . . . . . . . . 12 2.4 Contextual Boundness/Nonboundness, and Deep Word Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Gramm...
Categorial Grammars with Negative Information
 A notion in focus, de Gruyter
, 1995
"... this paper we discuss some possibilities of introducing negative information in the formalism of categorial grammar. Traditionally, formal grammars, including categorial grammars, admit positive information only. For instance, one postulates the rule S)NP,VP but not \GammaS)NP,\GammaVP; here \GammaA ..."
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this paper we discuss some possibilities of introducing negative information in the formalism of categorial grammar. Traditionally, formal grammars, including categorial grammars, admit positive information only. For instance, one postulates the rule S)NP,VP but not \GammaS)NP,\GammaVP; here \GammaA stands for the negation (complement) of category A.
Proof theory and formal grammars: applications of normalization
 In Benedikt Löwe, Wolfgang Malzkom, and Thoralf Räsch, editors, Foundations of the formal sciences II
, 2003
"... One of the main areas of interaction between logic and linguistics in the last 20 years has been the proof theoretical approach to formal grammars. This approach dates back to Lambek’s work in the 1950s. Lambek proposed to ..."
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One of the main areas of interaction between logic and linguistics in the last 20 years has been the proof theoretical approach to formal grammars. This approach dates back to Lambek’s work in the 1950s. Lambek proposed to
Meeting a modality? Restricted permutation for the Lambek calculus. ' OTS Working Paper, Onderzoekinstituut voor Taal en Spraak, Universiteit
, 1993
"... This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there are constraints on the application of some structural rules. In particular, we address the question how to add an operator to the Lambek Calculus in order to giv ..."
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This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there are constraints on the application of some structural rules. In particular, we address the question how to add an operator to the Lambek Calculus in order to give it a restricted access to the rule of Permutation, an extension which is partly motivated by linguistic applications. In line with tradition, we use the operator (∇) as a label telling us how the marked formula may be used, qua structural rules. New in our approach is that we do not see ∇ as a modality. Rather, we treat a formula ∇A as the meet of A with a special type Q. In this way we can make the specific structural behaviour of marked formulas more explicit. We define a minimal proof calculus for the system and prove some nice properties of it, like cutelimination, decidability an an embedding result. The main motivation for our approach however is that we can supply the proof system with an intuitive semantics.