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<Paper uid="E93-1044">
  <Title>Categorial grammar, modalities and algebraic semantics</Title>
  <Section position="3" start_page="0" end_page="377" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> The last decade has seen a keen revival of investigations into the suitability of using categorial grammars as theories of natural language syntax and semantics. Initially, this research was for the larger part confined to the classical categorial calculi of Ajdukiewicz \[1935\] and Bar-Hillel \[1953\], and, in particular, the Lambek cMculus L \[Lambek, 1958\], \[Moortgat, 1988\] and some of its close relatives.</Paragraph>
    <Paragraph position="1"> Although it turned out to be easily applicable to fairly large sets of linguistic data, one couldn't realistically expect the Lambek calculus to be able to account 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 reflecting surface phrase structure, but runs into problems when other linguistic phenomena are to be described.</Paragraph>
    <Paragraph position="2"> 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.</Paragraph>
    <Paragraph position="3"> 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 possess. 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 modality o is introduced with the aim of providing an appropriate means of dealing with certain intensional phenomena. Consequently, O inherits Kripke's possible world semantics for modal logic. The proof system which arises from adding Morrill's left and right rules for \[\] to the Lambek calculus L will be called Lb.</Paragraph>
    <Paragraph position="4"> 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 semantics, but not complete. This can be remedied by 1This paper discusses semigroup semmatics for L and LO in detail, and is well-suited as an easy-going introduction to the ideas presented here. It is available by anonymous ftp from ftp.let.ruu.nl in directory /pub/ots/papexs/versmissen, files adding.dvi.Z and adding, ps. Z.</Paragraph>
    <Paragraph position="5">  replacing the rule \[OR\] with the following stronger version: FlbOB1 ... Fo~-OB, F1,...,FnbA rl,..., F, ~- raA \[oR'\] Hepple \[1990\] also investigates the benefits of using the so-called structural modalities originally proposed in \[Morrill et al., 1990\], for the description of certain discontinuity 'and dislocality phenomena.</Paragraph>
    <Paragraph position="6"> The idea here is that such modalities allow a limited access to certain structural rules. Thus, we could for example have a permutation modality rap with the following proof rule (in addition to \[rapL\] and \[OpR'\] as before): r\[oeA, B\] ~ C r\[8, opA\] ~- C The symbol ~ here indicates that the inference is valid in both directions. The interpretation of OR would then be taken care of by a subsemigroup Lop of L having the property that x * y = y * x whenever z*Lnpory*Lop.</Paragraph>
    <Paragraph position="7"> Alternatively, one could require all types in such an inference to be boxed: F\[rapA, DpB\] I- C I r\[opB, OpA\] ~- C In this case, Lop would have to be such that z. y = y- x whenever z, y * Lop.</Paragraph>
    <Paragraph position="8"> Closely related to the use of structural modalities is the trend of considering different kinds of product connectives, sometimes combined into a single system. For example, Moortgat &amp; Morrill \[1992\] present an account of dependency structure in terms of headed prosodic trees, using a calculus that possesses two product operators instead of just one. On the basis of this, Moortgat \[1992\] sketches a landscape of substructural logics parametrized by properties such as commutativity, associativity and dependency. He then goes on to show how structural modalities can be used to locally enhance or constrain the possibilities of type combination. Morrill \[1992\] has a non-associative prosodic calculus, and uses a structural modality to reintroduce associativity at certain points.</Paragraph>
    <Paragraph position="9"> The picture that emerges is the following. Instead of the single product operator of L, one considers a range of different product operators, reflecting different modes of linguistic structuring. This results in a landscape of substructural logics, which are ultimately to be combined into a single system. Specific linguistic phenomena are given an account in terms of type constructors that are specially tailored for their description. On certain occasions it is necessary for entities to 'escape' the rules of the type constructor that governs their behaviour. This is achieved by means of structural modalities, which license controlled travel through the substructural landscape. Venema \[1993a\] proves a completeness theorem, with respect to the mentioned algebraic interpretation, for the Lambek calculus extended with a permutation modality. He modifies the proof system by introducing a type constant Q which refers explicitily to the subalgebra Lo. This proof system is adapted to cover a whole range of substructural logics in \[Venema, 1993b\]. However, the semantics given in that paper, which is adopted from Dogen \[1988; 1989\], differs in several respects from the one discussed above. Most importantly, models are required to possess a partial order with a well-behaved interaction with the product operation. In the remainder of this paper we will give a fairly general definition of the notion of a resource-preserving logic. The proof theory of these logics is based on that of Venema, while their semantics, with respect to which a completeness theorem will be established, is similar to that of Hepple and Morrill.</Paragraph>
  </Section>
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