Abstract not found

Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, in a light example-based manner, where this elegant logical paradigm stands when confronted with the wear and tear of reality. Starting from a brief history of the Lambek tradition since the 1980s, we discuss three main issues: (a) the fit of the lambda calculus engine to characteristic semantic structures in natural language, (b) the coexistence of the original type-theoretic and more recent modal interpretations of categorial logics, and (c) the place of categorial grammars in the complex total architecture of natural language, which involves - amongst others - mixtures of interpretation and inference.

‘Possible worlds semantics’ for modal logic is a widely used term, sometimes with ominous metaphysical connotations, but what does this style of modeling involve today? We discuss three main issues, using epistemic logic as a running example, and drawing upon both mathematical results and practices in the expertise of working researchers. Our first question is a foundational one: how does one associate a type of model with a language, and what considerations affect that choice? Our focus is on invariance and definability results, familiar from the mathematical and computational tradition, though less so in philosophy. The second question is less deep, but maybe even more challenging in practice: once we have chosen a type of models for a language, how does one select and then maintain models appropriate to concrete scenarios of application? While there is a lot of ‘art’ to this in the literature, there is very little ‘science’ of model construction for modal logics. We show how this works in dynamic epistemic logics, and identify some current challenges for a true ‘modeling theory’ as opposed to the more abstract usual ‘model theory’. Finally, we discuss the pervasive tension between ‘thin’ and ‘thick ’ worlds in modal logic, using examples from game theory, and pointing out how the contrast can be made fruitful.