Journal of Logic, Language and Information

, Volume 24, Issue 1, pp 65–93 | Cite as

Expressive Power of “Now” and “Then” Operators

  • Igor YanovichEmail author


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.


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



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.


  1. Areces, C., Blackburn, P., & Marx, M. (2001). Hybrid logics: Characterization, interpolation and complexity. The Journal of Symbolic Logic, 66(3), 977–1010.CrossRefGoogle Scholar
  2. Areces, C., Figueira, D., Figueira, S., & Mera, S. (2011). The expressive power of memory logics. Review of Symbolic Logic, 4(2), 290–318.CrossRefGoogle Scholar
  3. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic, volume 53 of Cambridge tracts in theoretical computer science. Cambridge: Cambridge University Press.Google Scholar
  4. Blackburn, P., & Seligman, J. (1995). Hybrid languages. Journal of Logic, Language and Information, 4, 251–272.CrossRefGoogle Scholar
  5. Blackburn, P., & Seligman, J. (1998). What are hybrid languages? In M. Kracht, M. de Rijke, H. Wansing, & M. Zakharyaschev (Eds.), Advances in modal logic (Vol. 1, pp. 41–62). Stanford: CSLI Publications.Google Scholar
  6. Blackburn, P. and van Benthem, J. (2007). Modal logic: A semantic perspective. In Blackburn et al., (Eds.) chapter 1. Amsterdam: Elsevier.Google Scholar
  7. Blackburn, P., van Benthem, J. F., and Wolter, F., editors (2007). Handbook of modal logic, volume 3 of Studies in logic and practical reasoning. Elsevier.Google Scholar
  8. Cresswell, M. (1990). Entities and indices. Dordrecht: Kluwer.CrossRefGoogle Scholar
  9. Cresswell, M. (1991). In defense of the barcan formula. Logique et Analyse, 135–136, 271–282.Google Scholar
  10. Fara, D. G. (2008). Relative-sameness counterpart theory. The Review of Symbolic Logic, 1(2), 167–189.CrossRefGoogle Scholar
  11. Fitting, M. and Mendelsohn, R. L. (1998). First-order modal logic, volume 277 of synthese library. Kluwer, Dordrecht.Google Scholar
  12. Gabbay, D. M. (1981). An irreflexivity lemma with applications to axiomatizations of conditions on linear frames. In U. Mönnich (Ed.), Aspects of philosophical logic (pp. 67–89). Dordrecht: Reidel.CrossRefGoogle Scholar
  13. Goranko, V., & Passy, S. (1992). Using the universal modality: Gains and questions. Journal of Logic and Computation, 2(1), 5–30.CrossRefGoogle Scholar
  14. Grädel, E., & Otto, M. (1999). On logics with two variables. Theoretical Computer Science, 224, 73–113.CrossRefGoogle Scholar
  15. Kamp, H. (1971). Formal properties of “now”. Theoria, 37, 227–273.CrossRefGoogle Scholar
  16. Lewis, D. K. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, 65(5), 113–126.CrossRefGoogle Scholar
  17. Meyer, U. (2009). ‘Now’ and ‘then’ in tense logic. Journal of Philosophical Logic, 38(2), 229–247.CrossRefGoogle Scholar
  18. Percus, O. (2000). Constraints on some other variables in syntax. Natural Language Semantics, 8, 173–229.CrossRefGoogle Scholar
  19. Recanati, F. (2007). Perspectival thought: A plea for (moderate) relativism. Oxford: Oxford University Press.CrossRefGoogle Scholar
  20. Saarinen, E. (1978). Backward-looking operators in tense logic and in natural language. In J. Hintikka, I. Niiniluoto, & E. Saarinen (Eds.), Essays on mathematical and philosophical logic (pp. 341–367). Dordrecht: Reidel.Google Scholar
  21. Schlenker, P. (2003). A plea for monsters. Linguistics and Philosophy, 26, 29–120.CrossRefGoogle Scholar
  22. ten Cate, B. D. (2005). Model theory for extended modal languages. PhD thesis, ILLC, University of Amsterdam.Google Scholar
  23. van Benthem, J. F. (1977). Tense logic and standard logic. Logique et Analyse, 20, 41–83.Google Scholar
  24. Verkuyl, H. (2008). Binary tense, volume 187 of CSLI lecture notes. Stanford: CSLI Publications.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute of LinguisticsUniversität TübingenTübingenGermany

Personalised recommendations