Looking backwards in type logic

Abstract

ABSTRACT Backwards-looking operators Saarinen, E. [1979. “Backwards-Looking Operators in Tense Logic and in Natural Language.” In Essays on Mathematical and Philosophical Logic, edited by J. Hintikka, I. Niiniluoto, and E. Saarinen, 341–367. Dordrecht: Reidel] that have the material in their scope depend on higher intensional operators, are known to increase the expressivity of some intensional languages and have thus played a central role in debates about approaches to intensionality in terms of implicit parameters (as in modal and tense logic) vs. variables explicitly quantifying over them. The current contribution takes a look at these operators from a type-logical perspective. It is shown that extending Gallin's ([Gallin, D. [1975. Intensional and Higher-Order Modal Logic. Amsterdam: North-Holland Pub. Company]) translation from intensional type logic (IL, Montague, R. [1970. “Universal Grammar.” Theoria 36: 373–398]) to two-sorted type theory so as to include a version of Yanovich's (Yanovich, I. [2015. “Expressive Power of Now and Then Operators.” Journal of Logic, Language and Information 24: 65–93]) backwards-looking operators, does not increase the expressive power of formulae with exclusively intensional parameters. The result, which makes use of a theorem by Zimmermann (Zimmermann, T. E. [1989. “Intensional Logic and Two-Sorted Type Theory.” The Journal of Symbolic Logic 54 (1): 65–77]), is illustrated by pertinent examples from the literature. The paper closes by indicating alternative strategies of incorporating backwards-looking operators.