Results 1 
6 of
6
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 ..."
Abstract

Cited by 239 (5 self)
 Add to MetaCart
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
An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
Abstract

Cited by 139 (16 self)
 Add to MetaCart
Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Labeled deduction in the composition of form and meaning
 IN H.J. OHLBACH & U. REYLE (EDS.) LOGIC, LANGUAGE AND REASONING. ESSAYS IN HONOR OF DOV GABBAY, PART I
, 1999
"... In the late Fifties, Jim Lambek has started a line of investigation that accounts for the composition of form and meaning in natural language in deductive terms: formal grammar is presented as a logic — a system for reasoning about the basic form/meaning units of language and the ways they can be pu ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
In the late Fifties, Jim Lambek has started a line of investigation that accounts for the composition of form and meaning in natural language in deductive terms: formal grammar is presented as a logic — a system for reasoning about the basic form/meaning units of language and the ways they can be put together into wellformed structured configurations. The reception of the categorial grammar logics in linguistic circles has always been somewhat mixed: the mathematical elegance of the original system ([Lambek 58]) is counterbalanced by clear descriptive limitations, as Lambek has been the first to emphasize on a variety of occasions. As a result of the deepened understanding of the options for ‘substructural ’ styles of reasoning, the categorial architecture has been redesigned in recent work, in ways that suggest that mathematical elegance may indeed be compatible with linguistic sophistication. A careful separation of the logical and the structural components of the categorial inference engine leads to the identification of constants of grammatical reasoning. At the level of the basic rules of use and proof for these constants one finds an explanation for the uniformities in the composition of form and meaning across languages. Crosslinguistic variation in the realization of the formmeaning correspondence is captured in terms of structural inference packages, acting as plugins with respect to the base logic of the grammatical
A KripkeJoyal Semantics for Noncommutative Logic in Quantales
"... abstract. A structural semantics is developed for a firstorder logic, with infinite disjunctions and conjunctions, that is characterised algebraically by quantales. The model structures involved combine the “covering systems” approach of KripkeJoyal intuitionistic semantics from topos theory with ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
abstract. A structural semantics is developed for a firstorder logic, with infinite disjunctions and conjunctions, that is characterised algebraically by quantales. The model structures involved combine the “covering systems” approach of KripkeJoyal intuitionistic semantics from topos theory with the ordered groupoid structures used to model various connectives in substructural logics. The latter are used to interpret the noncommutative quantal conjunction & (“and then”) and its residual implication connectives. The completeness proof uses the MacNeille completion and the theory of quantic nuclei to first embed a residuated semigroup into a quantale, and then represent the quantale as an algebra of subsets of a model structure. The final part of the paper makes some observations about quantal modal logic, giving in particular a structural modelling of the logic of closure operators on quantales.
Robert Goldblatt Grishin Algebras and Cover Systems for Classical Bilinear Logic
"... Abstract. Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use a ..."
Abstract
 Add to MetaCart
Abstract. Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has firstorder quantifiers and infinitary conjunctions and disjunctions.
Extended Lambek calculi and firstorder linear logic Richard Moot
"... The Syntactic Calculus (Lambek 1958) — often simply called the Lambek calculus, L, — is a beautiful system in many ways: Lambek grammars give a satisfactory syntactic analysis for the (contextfree) core of natural language and, in addition, it provides a simple and elegant syntaxsemantics interf ..."
Abstract
 Add to MetaCart
The Syntactic Calculus (Lambek 1958) — often simply called the Lambek calculus, L, — is a beautiful system in many ways: Lambek grammars give a satisfactory syntactic analysis for the (contextfree) core of natural language and, in addition, it provides a simple and elegant syntaxsemantics interface.