, Volume 191, Issue 5, pp 745–777 | Cite as

Dynamics of lying

  • Hans van DitmarschEmail author


We propose a dynamic logic of lying, wherein a ‘lie that \(\varphi \)’ (where \(\varphi \) is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula \(\varphi \). The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish two speaker perspectives: (Obs) an outside observer who is lying to an agent that is modelled in the system, and (Ag) an agent who is lying to another agent, and where both are modelled in the system. We distinguish three addressee perspectives: (Cred) the credulous agent who believes everything that it is told (even at the price of inconsistency), (Skep) the skeptical agent who only believes what it is told if that is consistent with its current beliefs, and (Rev) the belief revising agent who believes everything that it is told by consistently revising its current, possibly conflicting, beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or in the extension of that logic to belief revision.


Lying Multi-agent systems Modal logic Dynamics 



Hans van Ditmarsch is also affiliated to IMSc, Chennai, as associated researcher. I thank the anonymous reviewers of the journal Synthese for their comments, and for their persistence. I gratefully acknowledge comments from Alexandru Baltag, Jan van Eijck, Patrick Girard, Barteld Kooi, Fenrong Liu, Emiliano Lorini, Yoram Moses, Eric Pacuit, Rohit Parikh, Ramanujam, Hans Rott, Sonja Smets, Rineke Verbrugge, and Yanjing Wang. As my work on lying has a long history (from 2008 onward), I am concerned I may have forgotten to credit yet others, for which my apologies. As the editor of the special issue in which this contribution appears, Rineke had many valuable comments in addition to those by the reviewers, and was as always extremely encouraging. Emiliano spared me the embarrassment of including unsound axioms (and an incorrect action model) in the axiomatization of agent announcement logic, for which infinite thanks.


  1. Ågotnes, T., & van Ditmarsch, H. (2011). What will they say?—Public announcement games. Synthese, 179(S.1), 57–85.CrossRefGoogle Scholar
  2. Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50, 510–530.CrossRefGoogle Scholar
  3. Aucher, G. (2005). A combined system for update logic and belief revision. In Proceedings of the 7th PRIMA, LNAI 3371 (pp. 1–17). Heidelberg: Springer.Google Scholar
  4. Aucher, G. (2008). Consistency preservation and crazy formulas in BMS. In Proceedings of the 11th JELIA, LNCS 5293 (pp. 21–33). Heidelberg: Springer.Google Scholar
  5. Baltag, A. (2002). A logic for suspicious players: Epistemic actions and belief updates in games. Bulletin of Economic Research, 54(1), 1–45.CrossRefGoogle Scholar
  6. Baltag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In Proceedings of the 7th TARK (pp. 43–56).Google Scholar
  7. Baltag, A., & Smets, S. (2008a). The logic of conditional doxastic actions. In New perspectives on games and interaction, texts in logic and games (Vol. 4, pp. 9–31). Amsterdam: Amsterdam University Press.Google Scholar
  8. Baltag, A., & Smets, S. (2008b). A qualitative theory of dynamic interactive belief revision. In Proceedings of the 7th conference on LOFT, texts in logic and games (Vol. 3, pp. 13–60). Amsterdam: Amsterdam University Press.Google Scholar
  9. Bok, S. (1978). Lying: Moral Choice in Public and Private Life. New York: Random House.Google Scholar
  10. Conitzer, V., Lang, J., & Xia, L. (2009). How hard is it to control sequential elections via the agenda? In Proceedings of the 21st IJCAI (pp. 103–108). San Francisco: Morgan Kaufmann.Google Scholar
  11. d’Agostino, G., & Lenzi, G. (2008). A note on bisimulation quantifiers and fixed points over transitive frames. Journal of Logic and Computation, 18(4), 601–614.CrossRefGoogle Scholar
  12. de Bruin, L., & Newen, A. (2013). The developmental paradox of false belief understanding: A dual-system solution. Synthese. doi: 10.1007/S11229-012-0127-6.
  13. Dégremont, C., Kurzen, L., & Szymanik, J. (2013). On the tractability of comparing informational structures. Synthese. doi: 10.1007/S11229-012-0215-7.
  14. Frankfurt, H. G. (2005). On bullshit. Princeton: Princeton University Press.Google Scholar
  15. Gerbrandy, J. D., & Groeneveld, W. (1997). Reasoning about information change. Journal of Logic, Language, and Information, 6, 147–169.CrossRefGoogle Scholar
  16. Gneezy, U. (2005). Deception: The role of consequences. American Economic Review, 95(1), 384–394.CrossRefGoogle Scholar
  17. Grimm, J. L. K., & Grimm, W. K. (1812) Kinder- und Hausmärchen (Vol. 1). Berlin: Reimer.Google Scholar
  18. Grimm, J. L. K., & Grimm, W. K. (1814) Kinder- und Hausmärchen (Vol. 2). Berlin: Reimer.Google Scholar
  19. Hales, J. (2011). Refinement quantifiers for logics of belief and knowledge. Honours Thesis, University of Western Australia.Google Scholar
  20. Halpern, J. Y., & Moses, Y. (1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54, 319–379.CrossRefGoogle Scholar
  21. Hintikka, J. (1962). Knowledge and belief. Ithaca, NY: Cornell University Press.Google Scholar
  22. Hollebrandse, B., van Hout, A. & Hendriks, P. (2013). First and second-order false-belief reasoning: Does language support reasoning about the beliefs of others? Synthese. doi: 10.1007/S11229-012-0169-9.
  23. Icard, T. F., Pacuit, E., & Shoham, Y. (2010). Joint revision of beliefs and intention. In Proceedings of 12th KR. Menlo Park: AAAI Press.Google Scholar
  24. Kartik, N., Ottaviani, M., & Squintani, F. (2006). Credulity, lies, and costly talk. Journal of Economic Theory, 134, 93–116.CrossRefGoogle Scholar
  25. Kooi, B. (2007). Expressivity and completeness for public update logics via reduction axioms. Journal of Applied Non-Classical Logics, 17(2), 231–254.CrossRefGoogle Scholar
  26. Kooi, B., & Renne, B. (2011). Arrow update logic. Review of Symbolic Logic, 4(4), 536–559.CrossRefGoogle Scholar
  27. Littlewood, J. E. (1953). A Mathematician’s Miscellany. London: Methuen and Company.Google Scholar
  28. Liu, F., & Wang, Y. (2012). Reasoning about agent types and the hardest logic puzzle ever. Minds and Machines. doi: 10.1007/s11023-012-9287-x.
  29. Lutz, C. (2006). Complexity and succinctness of public announcement logic. In Proceedings of the 5th AAMAS (pp. 137–144).Google Scholar
  30. Mahon, J. E. (2006). Two definitions of lying. Journal of Applied Philosophy, 22(2), 21–230.Google Scholar
  31. Mahon, J. E. (2008). The definition of lying and deception. In The Stanford Encyclopedia of Philosophy.
  32. Meyer, J.-J. Ch., & van der Hoek, W. (1995) Epistemic logic for AI and computer science. Cambridge tracts in theoretical computer science (Vol. 41). Cambridge: Cambridge University Press.Google Scholar
  33. Plaza, J. A. (1989). Logics of public communications. In Proceedings of the 4th ISMIS (pp. 201–216). Oak Ridge: Oak Ridge National Laboratory.Google Scholar
  34. Rodenhäuser, B. (2010). Intentions in interaction. In: Informal Proceedings of 7th LOFT.Google Scholar
  35. Rott, H. (2003). Der Wert der Wahrheit. In M. Mayer (Ed.), Kulturen der Lüge (pp. 7–34). Böhlau-Verlag, Köln und Weimar.Google Scholar
  36. Roy, O. (2009). A dynamic-epistemic hybrid logic for intentions and information changes in strategic games. Synthese, 171(2), 291–320.CrossRefGoogle Scholar
  37. Sakama, C. (2011). Dishonest reasoning by abduction. In Proceedings of 22nd IJCAI (pp. 1063–1064). IJCAI/AAAI.Google Scholar
  38. Sakama, C. (2011). Formal definitions of lying. In: Proceedings of the 14th TRUST.Google Scholar
  39. Sakama, C., Caminada, M., & Herzig, A. (2010). A logical account of lying. In Proceedings of the JELIA 2010, LNAI 6341 (pp. 286–299).Google Scholar
  40. Siegler, F. A. (1966). Lying. American Philosophical Quarterly, 3, 128–136.Google Scholar
  41. Steiner, D. (2006). A system for consistency preserving belief change. In Proceedings of the ESSLLI Workshop on Rationality and Knowledge (pp. 133–144).Google Scholar
  42. Trivers, R. (2011). The Folly of Fools: The logic of deceit and self-deception in human life. New York: Basic Books.Google Scholar
  43. van Benthem, J. (2007). Dynamic logic of belief revision. Journal of Applied Non-Classical Logics, 17(2), 129–155.CrossRefGoogle Scholar
  44. van der velden, J. H. (2011). Iedereen liegt maar ik niet. Bruna.Google Scholar
  45. van Ditmarsch, H. (2005). Prolegomena to dynamic logic for belief revision. Synthese (Knowledge, Rationality & Action), 147, 229–275.Google Scholar
  46. van Ditmarsch, H. (2008). Comments on ‘The logic of conditional doxastic actions’. In New perspectives on games and interaction texts in logic and games (Vol. 4, pp. 33–44). Amsterdam: Amsterdam University Press.Google Scholar
  47. van Ditmarsch, H., de Lima, T., & Lorini, E. (2011). Intention change via local assignments. In Languages, Methodologies, and Development Tools for Multi-Agent Systems (Proceedings of 3rd LADS), LNAI 6822 (pp. 136–151). Heidelberg: Springer.Google Scholar
  48. van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic Epistemic Logic, volume 337 of Synthese Library. Heidelberg: Springer.Google Scholar
  49. van Ditmarsch, H., van Eijck, J., Sietsma, F., & Wang, Y. (2012). On the logic of lying. In Games, Actions and Social Software, LNCS 7010 (pp. 41–72). New York: Springer.Google Scholar
  50. van Emde Boas, P., Groenendijk, J., & Stokhof, M. (1984). The Conway paradox: Its solution in an epistemic framework. In Truth, Interpretation and Information: Selected Papers from the Third Amsterdam Colloquium (pp. 159–182).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.LORIA – CNRSUniversité de LorraineNancyFrance

Personalised recommendations