Categorial grammar, modalities and algebraic semantics

Categorial grammar, modalities and algebraic semantics Koen Versmissen Onderzoeksinstituut voor Taal en Spraak Universiteit Utrecht Trails 10 3512 JK Utrecht Netherlands koen. versmissen@let, ruu. nl Abstract This paper contributes to the theory of substructural logics .that are of interest to categorial grammarians. Combining se- mantic ideas of Hepple [1990] and Mor- rill [1990], proof-theoretic ideas of Venema [1993b; 1993a] and the theory of equational specifications, a class of resource-preserving logics is defined, for which decidability and completeness theorems are established. 1 Introduction The last decade has seen a keen revival of investi- gations into the suitability of using categorial gram- mars as theories of natural language syntax and se- mantics. Initially, this research was for the larger part confined to the classical categorial calculi of Aj- dukiewicz [1935] and Bar-Hillel [1953], and, in partic- ular, the Lambek cMculus L [Lambek, 1958], [Moort- gat, 1988] and some of its close relatives. Although it turned out to be easily applicable to fairly large sets of linguistic data, one couldn't real- istically expect the Lambek calculus to be able to ac- count for all aspects of grammar. The reason for this is the diversity of the constructions found in natural language. The Lambek calculus is good at reflect- ing surface phrase structure, but runs into problems when other linguistic phenomena are to be described. Consequently, recent work in categorial grammar has shown a trend towards diversification of the ways in which the linguistic algebra is structured, with an accompanying ramification of proof theory. One of the main innovations of the past few years has been the introduction of unary type connectives, usually termed modalities, that are used to reflect certain special features linguistic entities may pos- sess. This strand of research originates with Morrill [1990], who adds to L a unary connective O with the following proof rules: F,B,F'FA [mL] OF~-A [oR] F, OB, F ~ b A OF b DA OF here denotes a sequence of types all of which have O as their main connective. The S4-1ike modal- ity o is introduced with the aim of providing an ap- propriate means of dealing with certain intensional phenomena. Consequently, O inherits Kripke's pos- sible world semantics for modal logic. The proof sys- tem which arises from adding Morrill's left and right rules for [] to the Lambek calculus L will be called Lb. Hepple [1990] presents a detailed investigation into the possibilities of using the calculus L• to account for purely syntactic phenomena, notably the well- known Island Constraints of Ross [1967]. Starting from the usual interpretation of the Lambek calculus in semigroups L, where types are taken to denote subsets of L, he proposes to let D refer to a fixed subsemigroup Lo of L, which leads to the following definition of its semantics: [oAf = [A]n Lo As we have shown elsewhere [Versmissen, 1992] 1 the calculus LD is sound with respect to this seman- tics, but not complete. This can be remedied by 1This paper discusses semigroup semmatics for …

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