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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 286 (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
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 50 (7 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...
Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
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Grammar and Logical Types
 In 7th Amsterdam Colloquium
, 1990
"... This paper represents categorial grammar as an implicational type theory in the spirit of Girard's linear logic, and illustrates linguistic applications of a range of typeconstructors over and above implication. The type theoretic perspective is concerned with a correspondence between the logic ..."
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Cited by 23 (8 self)
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This paper represents categorial grammar as an implicational type theory in the spirit of Girard's linear logic, and illustrates linguistic applications of a range of typeconstructors over and above implication. The type theoretic perspective is concerned with a correspondence between the logic of types, and computational operations over the objects inhabiting types. In linguistic applications this correspondence is between rules of grammar which are theorems of type inference, and compositional operations in the various algebras in which linguistic objects, i.e. signs, are assumed to have dimensions: syntax, semantics, etc. Ruletorule description is familiar from Montague Grammar, but the idea here is to classify signs with structured types satisfying universal type laws determined by the semantics of the type connectives, in contrast to classification by categories satisfying stipulated rules. On this scheme an object language is to be specified by a type assignment to its finite vocabulary: a formal grammar is just a lexicon, plus perhaps some improper type axioms, and a grammar formalism is just a metalanguage of types with its uniform logic and interpretation is each linguistic dimension. The aim is to develop a language of types which has sufficient transparency, sensitivity, and generality to implement interesting descriptions of natural language. The paper will illustrate sentence grammar, and also use of the semantic term algebra as a functional programming language for presentation of lexical semantics. 1
Compositionality as an empirical problem
 In Chris Barker and Pauline Jacobson (eds.) Direct Compositionality
, 2007
"... Gottlob Frege (1892) is credited with the socalled “principle of compositionality”, also called “Frege’s Principle”, which one often hears expressed this way: Frege’s Principle (socalled) “The meaning of a sentence is a function of the meanings of the words in it and the way they are combined synt ..."
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Cited by 21 (0 self)
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Gottlob Frege (1892) is credited with the socalled “principle of compositionality”, also called “Frege’s Principle”, which one often hears expressed this way: Frege’s Principle (socalled) “The meaning of a sentence is a function of the meanings of the words in it and the way they are combined syntactically.” (Exactly how Frege himself understood “Frege’s Principle ” is not our concern here; 1 rather, it is the understanding that this slogan has acquired in contemporary linguistics that we want to pursue, and this has little further to do with Frege.) But why should linguists care what compositionality is or whether natural languages “are compositional ” or not? 2.1.1 An “Empirical Issue”? Often we hear that “compositionality is an empirical issue ” (meaning the question whether natural language is compositional or not)—usually asserted as a preface to expressing skepticism about a “yes ” answer. In the most general sense of Frege’s Principle, however, the fact that natural languages are compositional is beyond any serious doubt. Consider that:
LambdaGrammars and the SyntaxSemantics Interface
, 2001
"... types in this paper are built up from ground types s, np and n with the help of implication, and thus have forms such as np s, n((np s)s), etc. A restriction on signs is that a sign of abstract type A should have a term of type A in its ith dimension. The values of the function : for ground t ..."
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Cited by 19 (1 self)
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types in this paper are built up from ground types s, np and n with the help of implication, and thus have forms such as np s, n((np s)s), etc. A restriction on signs is that a sign of abstract type A should have a term of type A in its ith dimension. The values of the function : for ground types can be chosen on a per grammar basis and in this paper are as in Table 2. For complex types, the rule is that (AB) = A B . This means, for example, that np(np s) = np(np s) = (t)((t)t) and that np(np s) = e(e(st)). As a consequence, (2c) should be of type np(np s). Similarly, (2a) and (2b) can be taken to be of type np, (3a) and (3b) are of types np s and s respectively, etc. In general, if M has abstract type AB and N abstract type A, then the pointwise application M(N) is de ned and has type B.
On the semantic readings of proofnets
 Proceedings of formal Grammar
, 1996
"... A la mémoire de ..."
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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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Cited by 14 (2 self)
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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
Quantification And Scoping: A Deductive Account
"... In this paper, we argue that the grammatical scopings of quantifiers should be treated by deductive methods. In support of this position, we offer a logical treatment of almost all previously proposed substantive constraints on quantifier scoping, including those imposed by coordinate structure, co ..."
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Cited by 12 (0 self)
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In this paper, we argue that the grammatical scopings of quantifiers should be treated by deductive methods. In support of this position, we offer a logical treatment of almost all previously proposed substantive constraints on quantifier scoping, including those imposed by coordinate structure, control verbs, unbounded dependency constructions, anaphoric dependency and nested dependent quantifiers. These are correctly captured by a handful of linguistically motivated and logically natural inference schemes for quantification, coordination and unbounded dependency, combined with the previously motivated function introduction and elimination schemes of categorial logic. In addition, we argue that phrasestructure and transformational accounts of similar phenomena at best provide an approximation of the logical approach.