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Curry Typing, Polymorphism, and Fine-Grained Intensionality

Chapter in book
Authors Shalom Lappin
Published in Shalom Lappin and Chris Fox (eds.), The Handbook of Contemporary Semantic Theory, Second Edition, Wiley-Blackwell, Oxford and Malden MA
Pages 408-428
ISBN 978-0470670736
Publisher Wiley-Blackwell
Place of publication Oxford, Malden MA
Publication year 2015
Published at
Pages 408-428
Language en
Keywords computational semantics, type theory, natural language semantics
Subject categories Computer and Information Science, Languages and Literature


Two of the central elements of Montague semantics (Montague, 1974b) are (i) a higher-order intensional logic IL that incorporates Church’s simple theory of types (STT, (Church, 1940)), and (ii) a model theory that uses a Kripke frame semantics. The latter gives a modalized treatment of intensions based on Carnap’s view of an intension as a function from possible worlds to denotations (Carnap, 1947). These formal devices have continued to play an influential role in semantic theory even in many of the revisions of Montague semantics and the alternative semantic theories that have emerged in the past thirty years. Montague’s framework remains a seminal achievement in formal semantic theory. However, several of its foundational assumptions encounter serious problems when this framework is extended beyond the small fragment of English that Montague formalized. In this chapter I will examine several of these problems, and I will consider alternatives to Montague’s type theory and his characterization of intensions in order to deal with these problems. In Section 2, I give a brief summary of the architecture of IL, and take up some of the difficulties that it raises. Section 3 describes Property Theory with Curry Typing (PTCT, (Fox & Lappin, 2005, 2010)), a first-order semantic representation system that uses Curry typing with weak polymorphism. In Section 4, I discuss how PTCT provides a formal solution to the problem of fine-grained intensionality through its typed treatment of identity vs. equi- valence. I then extend this solution to a computational account of intensional di↵erence which accounts for intensions without using possible worlds. I ar- gue that worlds are not effectively representable, and so it is necessary to develop a cognitively viable intensional semantics on foundations that do not rely on them. Section 5 presents some programmatic ideas on how one could move beyond classical categorial semantic theories to a probabilistic system that accommodates the pervasive gradience of semantic properties. Such an approach also provides a framework for addressing the nature of semantic learning. Finally, Section 6 states conclusions and suggests some directions for future work on the issues considered in this chapter.

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