Variables vs. Parameters in the Interpretation of Natural Language

This paper compares two systems of functional type logic that have been applied to the analysis of meaning composition in natural language: Montague's Intensional Logic IL and its extensional substratum Ty2 of two-sorted type theory. The two systems differ in their treatment of reference and quantification over indices (like possible worlds or times): whereas the denotations of IL-formulae (inter alia) depend on indices as parameters, their Ty2 -counterparts contain explicit free and bound variables for them. Building on earlier results, it is argued that, appearances to the contrary, the two systems are largely equivalent; that any differences in expressivity are irrelevant to said applications; and that the equivalence also extends to variations of the systems that make use of multiple indices (as in mixed systems of modal and temporal interpretation) or additional dimensions (as in standard accounts of context dependence).