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Dynamic Epistemic Logic as a Substructural Logic

  • Guillaume Aucher
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 5)

Abstract

Dynamic Epistemic Logic (DEL) is an influential logical framework for reasoning about the dynamics of beliefs and knowledge. It has been related to older and more established logical frameworks. Despite these connections, DEL remains, arguably, a rather isolated logic in the vast realm of non-classical logics and modal logics. This is problematic if logic is to be viewed ultimately as a unified and unifying field and if we want to avoid that DEL goes on “riding off madly in all directions” (a metaphor used by van Benthem about logic in general). In this article, we show that DEL can be redefined naturally and meaningfully as a two-sorted substructural logic. In fact, it is even one of the most primitive substructural logics since it does not preserve any of the structural rules. Moreover, the ternary semantics of DEL and its dynamic interpretation provides a conceptual foundation for the Routley & Meyer’s semantics of substructural logics.

Keywords

Dynamic epistemic logic Substructural logics Update Ternary relation  Dynamic inference 

Notes

Acknowledgments

I thank Olivier Roy and Ole Hjortland for organizing and inviting me to an inspiring workshop on substructural epistemic logic in Munich in February 2013. Also, I thank Johan van Benthem and Igor Sedlar for comments on an earlier version of this article. Finally, I thank Sean Sedwards for checking the English of this article.

References

  1. 1.
    Aucher G (2004) A combined system for update logic and belief revision. In: Barley M, Kasabov NK (eds) PRIMA, vol 3371., Lecture Notes in Computer ScienceSpringer, Berlin, pp 1–17Google Scholar
  2. 2.
    Aucher G (2008) Perspectives on belief and change. Ph.D. thesis, University of Otago - University of Toulouse.Google Scholar
  3. 3.
    Aucher G (2011) Del-sequents for progression. J Appl Non-class Logics 21(3–4):289–321CrossRefGoogle Scholar
  4. 4.
    Aucher G (2012) Del-sequents for regression and epistemic planning. J Appl Non-class Logics 22(4):337–367CrossRefGoogle Scholar
  5. 5.
    Aucher G (2013) Update logic. Research report RR-8341, INRIA. http://hal.inria.fr/hal-00849856
  6. 6.
    Aucher G, Herzig A (2011) Exploring the power of converse events. In: Girard P, Roy O, Marion M (eds) Dynamic formal epistemology, vol 351. Springer, The Netherlands, pp 51–74CrossRefGoogle Scholar
  7. 7.
    Aucher G, Maubert B, Schwarzentruber F (2012) Generalized DEL-sequents. In: del Cerro LF, Herzig A, Mengin J (eds) JELIA, vol 7519., Lecture Notes in Computer ScienceSpringer, New York, pp 54–66Google Scholar
  8. 8.
    Baltag A, Coecke B, Sadrzadeh M (2005) Algebra and sequent calculus for epistemic actions. Electron Notes Theoret Comput Sci 126:27–52CrossRefGoogle Scholar
  9. 9.
    Baltag A, Coecke B, Sadrzadeh M (2007) Epistemic actions as resources. J Logic Comput 17(3):555–585CrossRefGoogle Scholar
  10. 10.
    Baltag A, Moss L (2004) Logic for epistemic programs. Synthese 139(2):165–224CrossRefGoogle Scholar
  11. 11.
    Baltag A, Moss LS, Solecki S (1998) The logic of public announcements and common knowledge and private suspicions. In: Gilboa I (ed) TARK. Morgan Kaufmann, San Francisco, pp 43–56Google Scholar
  12. 12.
    Baltag A, Moss L, Solecki S (1999) The logic of public announcements, common knowledge and private suspicions. Indiana University, Technical report Google Scholar
  13. 13.
    Baltag A, Smets S (2006) Conditional doxastic models: a qualitative approach to dynamic belief revision. Electron Notes Theoret Comput Sci 165:5–21CrossRefGoogle Scholar
  14. 14.
    Baltag A, Smets S (2008) The logic of conditional doxastic actions. Texts in logic and games, vol 4. Amsterdam University Press, Amsterdam, pp 9–31Google Scholar
  15. 15.
    Baltag A, Smets S (2008) A qualitative theory of dynamic interactive belief revision. Texts in logic and games, vol 3. Amsterdam University Press, Amsterdam, pp 9–58Google Scholar
  16. 16.
    Barwise J (1993) Constraints, channels, and the flow of information. In: Cooper R, Barwise J, Mukai K (eds) Situation theory and its applications, vol 3. Center for the Study of Language and Information, US, pp 3–27Google Scholar
  17. 17.
    Barwise J, Perry J (1983) Situations and attitudes. MIT Press, CambridgeGoogle Scholar
  18. 18.
    Beall J, Brady R, Dunn JM, Hazen A, Mares E, Meyer RK, Priest G, Restall G, Ripley D, Slaney J et al (2012) On the ternary relation and conditionality. J philos logic 41(3):595–612CrossRefGoogle Scholar
  19. 19.
    Beall JC, Restall G (2006) Logical pluralism. Oxford University Press, OxfordGoogle Scholar
  20. 20.
    van Benthem, J (1977) Modal correspondence theory. Ph.D. thesis, University of Amsterdam.Google Scholar
  21. 21.
    van Benthem J (1991) General dynamics. Theoret Linguis 17(1–3):159–202Google Scholar
  22. 22.
    van Benthem J (1991) Language in action: categories, lambdas and dynamic logic, vol 130. North Holland, AmsterdamGoogle Scholar
  23. 23.
    van Benthem J (1996) Exploring logical dynamics. CSLI publications, StanfordGoogle Scholar
  24. 24.
    van Benthem J (2001) Correspondence theory. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 3. Reidel, Dordrecht, pp 325–408CrossRefGoogle Scholar
  25. 25.
    van Benthem J (2003) Structural properties of dynamic reasoning. In: Peregrin J (ed) Meaning: the dynamic turn. Elsevier, Amsterdam, pp 15–31Google Scholar
  26. 26.
    van Benthem J (2007) Dynamic logic for belief revision. J Appl Non-class Logics 17(2):129–155CrossRefGoogle Scholar
  27. 27.
    van Benthem J (2007) Inference in action. Publications de l’Institut Mathématique-Nouvelle Série 82(96):3–16Google Scholar
  28. 28.
    van Benthem J (2008) Logical dynamics meets logical pluralism? Australas J Logic 6:182–209Google Scholar
  29. 29.
    van Benthem J (2010) Modal logic for open minds. CSLI publications, StanfordGoogle Scholar
  30. 30.
    van Benthem J (2011) Logical dynamics of information and interaction. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  31. 31.
    van Benthem J (2011) Mccarthy variations in a modal key. Artif intell 175(1):428–439CrossRefGoogle Scholar
  32. 32.
    van Benthem J, Gerbrandy J, Hoshi T, Pacuit E (2009) Merging frameworks for interaction. J Philos Logic 38(5): 491–526. doi: 10.1007/s10992-008-9099-x. http://dx.doi.org/10.1007/s10992-008-9099-x
  33. 33.
    van Benthem J, Gerbrandy J, Pacuit E (2007) Merging frameworks for interaction: DEL and ETL. In: Samet D (ed) Theoretical aspect of rationality and knowledge (TARK XI). ILLC, Brussels, pp 72–82Google Scholar
  34. 34.
    van Benthem J, Kooi B (2004) Reduction axioms for epistemic actions. In: Schmidt R, Pratt-Hartmann I, Reynolds M, Wansing H (eds) AiML-2004: advances in modal logic, number UMCS-04-9-1 in technical report series. University of Manchester, Manchester, pp 197–211Google Scholar
  35. 35.
    Burgess JP (1981) Quick completeness proofs for some logics of conditionals. Notre Dame J Formal Logic 22(1):76–84CrossRefGoogle Scholar
  36. 36.
    van Ditmarsch H (2005), Prolegomena to dynamic logic for belief revision. Synthese 147:229–275.Google Scholar
  37. 37.
    van Ditmarsch H, van der Hoek W, Kooi B (2007) Synthese library. In: Dynamic epistemic logic, vol 337. Springer, New York.Google Scholar
  38. 38.
    van Ditmarsch HP, Herzig A, Lima TD (2009) From situation calculus to dynamic epistemic logic. J Logic Comput 21(2):179–204CrossRefGoogle Scholar
  39. 39.
    Dunn JM, Restall G (2002) Relevance logic. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 6. Kluwer, Dordrecht, pp 1–128CrossRefGoogle Scholar
  40. 40.
    van Eijck J (2004) Reducing dynamic epistemic logic to PDL by program transformation. Technical report SEN-E0423, CWI.Google Scholar
  41. 41.
    Fagin R, Halpern J, Moses Y, Vardi M (1995) Reasoning about knowledge. MIT Press, CambridgeGoogle Scholar
  42. 42.
    Gabbay DM, Hogger CJ, Robinson JA, Siekmann J, Nute D (1998 eds) Nonmonotonic reasoning and uncertain reasoning. In: Handbook of logic in artificial intelligence and logic programming, vol 3. Clarendon Press, Oxford.Google Scholar
  43. 43.
    Gärdenfors P (1988) Knowledge in flux (modeling the dynamics of epistemic states). Bradford/MIT Press, CambridgeGoogle Scholar
  44. 44.
    Gärdenfors P (1991) Belief revision and nonmonotonic logic: two sides of the same coin? Logics in AI. Springer, New york, pp 52–54CrossRefGoogle Scholar
  45. 45.
    Hintikka J (1962) Knowledge and belief, an introduction to the logic of the two notions. Cornell University Press, Ithaca, LondonGoogle Scholar
  46. 46.
    Liu F (2008) Changing for the better: preference dynamics and agent diversity. Ph.D. thesis, ILLC, University of Amsterdam.Google Scholar
  47. 47.
    Makinson D (2005) Bridges from classical to nonmonotonic logic. King’s College, LondonGoogle Scholar
  48. 48.
    Makinson D, Gärdenfors P (1989) Relations between the logic of theory change and nonmonotonic logic. In: Fuhrmann A, Morreau M (eds) The logic of theory change. Lecture notes in computer science, vol 465. Springer, Berlin, pp 185–205.Google Scholar
  49. 49.
    Mares ED (1996) Relevant logic and the theory of information. Synthese 109(3):345–360CrossRefGoogle Scholar
  50. 50.
    Mares ED, Meyer RK (2001) Relevant Logics. In: Goble L (ed) The Blackwell guide to philosophical logic. Wiley-Blackwell, OxfordGoogle Scholar
  51. 51.
    Muskens R, van Benthem J, Visser A (2011) Dynamics. In: van Benthem JFAK, ter Meulen A (eds) Handbook of logic and language. Elsevier, Amsterdam, pp 607–670CrossRefGoogle Scholar
  52. 52.
    Nute D, Cross CB (2001) Conditional logic. Handbook of philosophical logic, vol 4. Kluwer Academic Pub, Dordrecht, pp 1–98CrossRefGoogle Scholar
  53. 53.
    Parikh R, Ramanujam R (2003) A knowledge based semantics of messages. J Logic Lang Inform 12(4):453–467CrossRefGoogle Scholar
  54. 54.
    Perry J, Israel D (1990) What is information? In: Hanson PP (ed) Information, language, and cognition 1. Columbia Press, VancouverGoogle Scholar
  55. 55.
    Ramsey F (1929) Philosophical papers. In: General propositions and causality. Cambridge University Press, Cambridge.Google Scholar
  56. 56.
    Restall G (1996) Information flow and relevant logics. Logic, language and computation: the 1994 Moraga proceedings. CSLI, Stanford, pp 463–477Google Scholar
  57. 57.
    Restall G (2000) An introduction to substructural logics. Routledge, LondonGoogle Scholar
  58. 58.
    Restall G (2006) Relevant and substructural logics. Handbook of the history of logic, vol 7. Elsevier, London, pp 289–398Google Scholar
  59. 59.
    Routley R, Meyer RK (1972) The semantics of entailment-ii. J Philos Logic 1(1):53–73CrossRefGoogle Scholar
  60. 60.
    Routley R, Meyer RK (1972) The semantics of entailment-iii. J philos logic 1(2):192–208CrossRefGoogle Scholar
  61. 61.
    Routley R, Meyer R (1973) The semantics of entailment. Stud Logic Found Math 68:199–243CrossRefGoogle Scholar
  62. 62.
    Routley R, Plumwood V, Meyer RK (1982) Relevant logics and their rivals. Ridgeview Publishing Company, AtascaderoGoogle Scholar
  63. 63.
    Urquhart AI (1971) Completeness of weak implication. Theoria 37(3):274–282CrossRefGoogle Scholar
  64. 64.
    Urquhart A (1972) A general theory of implication. J Symbolic Logic 37(443):270Google Scholar
  65. 65.
    Urquhart A (1972) Semantics for relevant logics. J Symbolic Logic, pp 159–169.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Rennes 1 – INRIARennes, CedexFrance

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