References in:
Language, Lambdas, and Logic
In R. Oehrle & J. Kruijff (eds.), Resource Sensitivity, Binding, and Anaphora (Studies in Linguistics and Philosophy 80). Dordrecht: Kluwer Academic Publishers. pp. 2354 (2003)
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This paper introduces λgrammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λgrammars are multidimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λterms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable. 

In this paper we discuss a new perspective on the syntaxsemantics interface. Semantics, in this new setup, is not ‘read off’ from Logical Forms as in mainstream approaches to generative grammar. Nor is it assigned to syntactic proofs using a CurryHoward correspondence as in versions of the Lambek Calculus, or read off from fstructures using Linear Logic as in LexicalFunctional Grammar (LFG, Kaplan & Bresnan [9]). All such approaches are based on the idea that syntactic objects (trees, proofs, fstructures) are (...) 



We provide firstorder axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes. 

The Montagovian hypothesis of direct modeltheoretic interpretation of syntactic surface structures is supported by an account of the semantics of binding that makes no use of variables, syntactic indices, or assignment functions & shows that the interpretation of a large portion of socalled variablebinding phenomena can dispense with the level of logical form without incurring equivalent complexity elsewhere in the system. Variablefree semantics hypothesizes local interpretation of each surface constituent; binding is formalized as a typeshifting operation on expressions that denote (...) 

A number of general points behind the story of this paper may be worth setting out separately, now that we have come to the end.There is perhaps one obvious omission to be addressed right away. Although the word “information” has occurred throughout this paper, it must have struck the reader that we have had nothing to say on what information is. In this respect, our theories may be like those in physics: which do not explain what “energy” is (a notion (...) 

We present Logical Description Grammar (LDG), a model ofgrammar and the syntaxsemantics interface based on descriptions inelementary logic. A description may simultaneously describe the syntacticstructure and the semantics of a natural language expression, i.e., thedescribing logic talks about the trees and about the truthconditionsof the language described. Logical Description Grammars offer a naturalway of dealing with underspecification in natural language syntax andsemantics. If a logical description (up to isomorphism) has exactly onetree plus truthconditions as a model, it completely specifies thatgrammatical (...) 

Preface This book is about semantics and logic. More specifically, it is about the semantics and logic of natural language; and, even more specifically than ... 

This important book provides a new unifying methodology for logic. It replaces the traditional view of logic as manipulating sets of formulas with the notion of structured families of labelled formulas with algebraic structures. This approach has far reaching consequences for the methodology of logics and their semantics, and the book studies the main features of such systems along with their applications. It will interest logicians, computer scientists, philosophers and linguists. 