# Expressive Power of “Now” and “Then” Operators

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## Abstract

Natural language provides motivation for studying modal backwards-looking operators such as “now”, “then” and “actually” that evaluate their argument formula at some previously considered point instead of the current one. This paper investigates the expressive power over models of both propositional and first-order basic modal language enriched with such operators. Having defined an appropriate notion of bisimulation for first-order modal logic, I show that backwards-looking operators increase its expressive power quite mildly, contrary to beliefs widespread among philosophers of language and formal semanticists. That in turn presents a strong argument for the use of operator-based systems in the semantics of natural language, instead of systems with explicit quantification over worlds and times that have become a de-facto standard for such applications. The popularity of such explicit-quantification systems is shown to be based on the misinterpretation of a claim by Cresswell (Entities and indices, Kluwer, Dordrecht, 1990), which led many philosophers and linguists to assume (wrongly) that introducing “now” and “then” is expressively equivalent to explicitly quantifying over worlds and times.

## Keywords

“Now” operator Backwards-looking operators Bisimulation First-order modal logic Hybrid logic## Notes

### Acknowledgments

I am grateful to Patrick Blackburn, Benjamine George, Salvador Mascarenhas, Philippe Schlenker and Wolfgang Sternefeld for discussions of the material, and to the anonymous reviewers, whose comments have helped to significantly improve the paper both in form and in substance. All remaining errors are, of course, my own. The final stages of the work on the paper have been partially supported by the Alexander von Humboldt foundation, which is hereby gratefully acknowledged.

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