Principles of Knowledge, Belief and Conditional Belief

  • Guillaume Aucher
Part of the Logic, Argumentation & Reasoning book series (LARI, volume 3)


We review and examine in this paper the validity of the different axioms (and inference rules) of knowledge and belief and relating knowledge to belief which have been proposed in the epistemic !epistemic logic literature. In doing so, we encounter many of the problems that epistemic !epistemic logic has had to face in its relatively short (modern) history and provide relevant pointers for more details. We also contribute to this area by providing conditions under which the notion of belief can be formally defined in terms of knowledge , and vice versa. We also prove that certain convoluted axioms dealing only with the notion of knowledge can be derived from understandable interaction axioms relating knowledge and conditional belief !conditional .


Logic Logic Inference Rule Accessibility Relation Conditional Belief Epistemic Luck 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I thank Manuel Rebuschi and Franck Lihoreau for helpful comments on this paper. I also thank the anonymous English native speaker referee for detailed comments.


  1. E. Adams, The Logic of Conditionals. Volume 86 of Synthese Library (Springer, Dordrecht, 1975)Google Scholar
  2. C.E. Alchourrón, P. Gärdenfors, D. Makinson, On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50(2), 510–530 (1985)CrossRefGoogle Scholar
  3. H. Arlo-Costa, Qualitative and probabilistic models of full belief, in Logic Colloquium’98, Prague, ed. by S. Buss, P. Hajek, P. Pudlak. Lecture Notes on Logic, vol. 13 (1999)Google Scholar
  4. S. Artemov, M. Fitting, Justification logic, in The Stanford Encyclopedia of Philosophy, Fall 2011 edn., ed. by E.N. Zalta (2011)Google Scholar
  5. G. Aucher, A combined system for update logic and belief revision, in Proceedings of the PRIMA, Auckland, ed. by M. Barley, N.K. Kasabov. Volume 3371 of Lecture Notes in Computer Science (Springer, 2004), pp. 1–17Google Scholar
  6. G. Aucher, Interpreting an action from what we perceive and what we expect. J. Appl. Non-Class. Log. 17(1), 9–38 (2007)CrossRefGoogle Scholar
  7. G. Aucher, An internal version of epistemic logic. Studia Logica 1, 1–22 (2010)CrossRefGoogle Scholar
  8. A.J. Ayer, The Problem of Knowledge (Penguin books, London, 1956)Google Scholar
  9. A. Baltag, S. Smets, Conditional doxastic models: a qualitative approach to dynamic belief revision. Electron. Notes Theor. Comput. Sci. 165, 5–21 (2006)CrossRefGoogle Scholar
  10. A. Baltag, S. Smets, A qualitative theory of dynamic interactive belief revision, in Logic and the Foundations of Game and Decision Theory (LOFT 7), ed. by G. Bonanno, W. van der Hoek, M.J. Wooldridge. Texts in Logic and Games, vol. 3 (Amsterdam University Press, Amsterdam, 2008a), pp. 9–58Google Scholar
  11. A. Baltag, S. Smets, The logic of conditional doxastic actions, in Texts in Logic and Games, vol. 4 (Amsterdam University Press, 2008b), pp. 9–31Google Scholar
  12. P. Blackburn, M. de Rijke, Y. Venema, Modal Logic. Volume 53 of Cambridge Tracts in Computer Science (Cambridge University Press, Cambridge/New York, 2001)Google Scholar
  13. O. Board, Dynamic interactive epistemology. Games Econ. Behav. 49, 49–80 (2004)CrossRefGoogle Scholar
  14. D. Bonnay, P. Egré, Inexact knowledge with introspection. J. Philos. Log. 38(2), 179–227 (2008)CrossRefGoogle Scholar
  15. C. Boutilier, Conditional logics of normality: a modal approach. Artif. Intell. 68(1), 87–154 (1994)CrossRefGoogle Scholar
  16. J. Broersen, M. Dastani, J. Hulstijn, Z. Huang, L.W.N. van der Torre, The BOID architecture: conflicts between beliefs, obligations, intentions and desires, in Proceedings of the Fifth International Conference on Autonomous Agents, Montreal, 2001, pp. 9–16Google Scholar
  17. H.-N. Castañeda, Review of ‘knowledge and belief’. J. Symb. Log. 29, 132–134 (1964)CrossRefGoogle Scholar
  18. P. Cohen, H. Levesque, Intention is choice with commitment. Artif. Intell. 42, 213–261 (1990)CrossRefGoogle Scholar
  19. M. de Rijke, A note on graded modal logic. Studia Logica 64(2), 271–283 (2000)CrossRefGoogle Scholar
  20. F. Deschene, Y. Wang, To know or not to know: epistemic approach to security protocol verification. Synthese 177(Supplement 1), 51–76 (2010)Google Scholar
  21. D. Dubois, H. Prade, Possibilistic logic, preferential model and related issue, in Proceedings of the 12th International Conference on Artificial Intelligence (IJCAI), Sydney (Morgan Kaufman, 1991), pp. 419–425Google Scholar
  22. R. Fagin, J.Y. Halpern, Belief, awareness, and limited reasoning. Artif. Intell. 34(1), 39–76 (1987)CrossRefGoogle Scholar
  23. R. Fagin, J. Halpern, Y. Moses, M. Vardi, Reasoning About Knowledge (MIT, Cambridge, 1995)Google Scholar
  24. K. Fine, In so many possible worlds. Notre Dame J. Form. Log. 13(4), 516–520 (1972)CrossRefGoogle Scholar
  25. L. Floridi, The logic of being informed. Logique et Analyse 49(196), 433–460 (2006)Google Scholar
  26. N. Friedman, J.Y. Halpern, Modeling belief in dynamic systems, Part I: foundations. Artif. Intell. 95(2), 257–316 (1997)Google Scholar
  27. N. Friedman, J.Y. Halpern, Plausibility measures and default reasoning. J. ACM 48(4), 648–685 (2001)CrossRefGoogle Scholar
  28. P. Gärdenfors, Knowledge in Flux (Modeling the Dynamics of Epistemic States) (Bradford/MIT, Cambridge, 1988)Google Scholar
  29. M. Georgeff, A. Rao, Asymmetry thesis and side-effect problems in linear time and branching time intention logics, in Proceedings of the Twelfth International Joint Conference on Artificial Intelligence (IJCAI-91), Sydney, 1991, pp. 498–504Google Scholar
  30. E. Gettier, Is justified true belief knowledge? Analysis 25, 121–123 (1963)CrossRefGoogle Scholar
  31. P. Gochet, Logique épistémique et philosophie des mathématiques, chapter Un problème ouvert en épistémologie: la formalisation du savoir-faire. Philosophie des sciences. Vuibert, 2007Google Scholar
  32. P. Gochet, P. Gribomont, Epistemic logic, in Twentieth Century Modalities. Handbook of the History of Logic, vol. 7 (Elsevier, Amsterdam, 2006), pp. 99–195Google Scholar
  33. J.Y. Halpern, Should knowledge entail belief? J. Philos. Log. 25(5), 483–494 (1996)CrossRefGoogle Scholar
  34. J. Halpern, Reasoning About Uncertainty (MIT, Cambridge, 2003)Google Scholar
  35. J. Halpern, Y. Moses, Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990)CrossRefGoogle Scholar
  36. J. Halpern, Y. Moses, A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54, 311–379 (1992)Google Scholar
  37. J.Y. Halpern, R. Pucella, Modeling adversaries in a logic for security protocol analysis, in Formal Aspects of Security, London, ed. by A.E. Abdallah, P. Ryan, S. Schneider. Volume 2629 of Lecture Notes in Computer Science (Springer, 2002), pp. 115–132Google Scholar
  38. J.Y. Halpern, R. Pucella, Dealing with logical omniscience: expressiveness and pragmatics. Artif. Intell. 175(1), 220–235 (2011)CrossRefGoogle Scholar
  39. J.Y. Halpern, L.C. Rêgo, Reasoning about knowledge of unawareness revisited, in Proceedings of the TARK, Berkeley, ed. by A. Heifetz, 2009, pp. 166–173Google Scholar
  40. J.Y. Halpern, D. Samet, E. Segev, Defining knowledge in terms of belief: the modal logic perspective. Rev. Symb. Log. 2, 469–487 (2009a)CrossRefGoogle Scholar
  41. J.Y. Halpern, D. Samet, E. Segev, On definability in multimodal logic. Rev. Symb. Log. 2, 451–468 (2009b)CrossRefGoogle Scholar
  42. A. Heifetz, M. Meier, B. Schipper, Interactive unawareness. J. Econ. Theory 130(1), 78–94 (2006)CrossRefGoogle Scholar
  43. D. Hemp, The KK (knowing that one knows) principle, in The Internet Encyclopedia of Philosophy, ed. by J. Fieser, B.H. Dowden (J. Fieser, Martin, 2006)., Oct 2006
  44. V. Hendricks, Mainstream and Formal Epistemology (Cambridge University Press, Cambridge, 2005)CrossRefGoogle Scholar
  45. V. Hendricks, J. Symons (eds.), 8 Bridges Between Formal and Mainstream Epistemology, vol. 128 (Springer, Dordrecht, 2006)Google Scholar
  46. A. Herzig, Modal probability, belief, and actions. Fundamenta Informaticae 57(2–4), 323–344 (2003)Google Scholar
  47. J. Hintikka, Knowledge and Belief, an Introduction to the Logic of the Two Notions (Cornell University Press, Ithaca/London, 1962)Google Scholar
  48. S. Kraus, D. Lehmann, Knowledge, belief and time, in Automata, Languages and Programming, ed. by L. Kott. Volume 226 of Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 1986), pp. 186–195Google Scholar
  49. S. Kraus, D.J. Lehmann, M. Magidor, Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44(1–2), 167–207 (1990)CrossRefGoogle Scholar
  50. H.E. Kyburg, Probability and the Logic of Rational Belief (Wesleyan University Press, Middletown, 1961)Google Scholar
  51. P. Lamarre, Y. Shoham, Knowledge, certainty, belief, and conditionalisation (abbreviated version), in Proceedings of the KR, Bonn, 1994, pp. 415–424Google Scholar
  52. N. Laverny, J. Lang, From knowledge-based programs to graded belief-based programs, Part II: off-line reasoning, in Proceedings of the IJCAI, Edinburgh, 2005, pp. 497–502Google Scholar
  53. K. Lehrer, T. Paxson, Knowledge: undefeated justified true belief. J. Philos. 66, 225–237 (1969)CrossRefGoogle Scholar
  54. W. Lenzen, Recent Work in Epistemic Logic. Acta Philosophica Fennica, vol. 30 (North Holland, Amsterdam, 1978)Google Scholar
  55. W. Lenzen, Epistemologische Betractungen zu [S4;S5]. Erkenntnis 14, 33–56 (1979)CrossRefGoogle Scholar
  56. W. Lenzen, Knowledge, belief, and subjective probability: outlines of a unified system of epistemic/doxastic logic, in Knowledge Contributors, ed. by V.F. Hendricks, K.F. Jørgensen, S.A. Pedersen, J. Symons, D. Dalen, T.A.F. Kuipers, T. Seidenfeld, P. Suppes, J. Woleński. Volume 322 of Synthese Library (Springer, The Netherlands, 2004), pp. 17–31Google Scholar
  57. H. Levesque, A logic of implicit and explicit knowledge, in Proceedings of the AAAI-84, Austin, 1984, pp. 198–202Google Scholar
  58. I. Levi, The logic of full belief, in The Covenant of Reason: Rationality and the Commitments of Thought (Cambridge University Press, Cambridge/New York, 1997), pp. 40–69Google Scholar
  59. D. Lewis, Convention, a Philosophical Study (Harvard University Press, Cambridge, 1969)Google Scholar
  60. E. Lorini, C. Castelfranchi, The cognitive structure of surprise: looking for basic principles. Topoi Int. Rev. Philos. 26(1), 133–149 (2007)Google Scholar
  61. W. Lycan, On the Gettier problem, in Epistemology Futures, ed. by S.C. Hetherington (Oxford University Press, Oxford/New York, 2006), pp. 148–168Google Scholar
  62. D. Makinson, P. Gärdenfors, Relations between the logic of theory change and nonmonotonic logic, in The Logic of Theory Change, Konstanz, ed. by A. Fuhrmann, M. Morreau. Volume 465 of Lecture Notes in Computer Science (Springer, 1989), pp. 185–205Google Scholar
  63. J.-J.Ch. Meyer, W. van der Hoek, Epistemic Logic for AI and Computer Science (Cambridge University Press, Cambridge, 1995)CrossRefGoogle Scholar
  64. J.-J.Ch. Meyer, F. de Boer, R. van Eijk, K. Hindriks, W. van der Hoek, On programming KARO agents. Log. J. IGPL 9(2), 245–256 (2001)CrossRefGoogle Scholar
  65. R.C. Moore, Possible-world semantics for autoepistemic logic, in Proceedings of the Non-monotonic Reasoning Workshop, New Paltz, 1984, pp. 344–354Google Scholar
  66. R.C. Moore, Logic and Representation. CSLI Lecture Notes (Center for the Study of Language and Information, Stanford, 1995)Google Scholar
  67. Y. Moses, Y. Shoham, Belief as defeasible knowledge. Artif. Intell. 64, 299–321 (1993)CrossRefGoogle Scholar
  68. E. Pacuit, Procedural information and the dynamics of belief, in Dialogue, Rationality, Formalism. LEUS (Springer, 2014)Google Scholar
  69. D. Prichard, Epistemic luck. J. Philos. Res. 29, 193–222 (2004)Google Scholar
  70. A. Rao, M. Georgeff, Modeling rational agents within a BDI-architecture, in Proceedings of Knowledge Representation and Reasoning (KR & R-91), Cambridge, ed. by R. Fikes, E. Sandewall (Morgan Kaufmann, 1991), pp. 473–484Google Scholar
  71. H. Rott, Stability, strength and sensitivity: converting belief into knowledge. Erkenntnis 61, 469–493 (2004). doi:10.1007/s10670-004-9287-1CrossRefGoogle Scholar
  72. C. Sakama, M. Caminada, A. Herzig, A logical account of lying, in Logics in Artificial Intelligence, ed. by T. Janhunen, I. Niemelä. Volume 6341 of Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2010), pp. 286–299Google Scholar
  73. Y. Shoham, K. Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations (Cambridge University Press, Cambridge/New York, 2009)Google Scholar
  74. W. Spohn, Ordinal conditional functions: a dynamic theory of epistemic states, in Causation in Decision, Belief Change, and Statistics, vol. 2 (Reidel, Dordrecht, 1988a), pp. 105–134Google Scholar
  75. W. Spohn, A general non-probabilistic theory of inductive reasoning, in Proceedings of the UAI, Minneapolis, ed. by R.D. Shachter, T.S. Levitt, L.N. Kanal, J.F. Lemmer (North-Holland, 1988b), pp. 149–158Google Scholar
  76. R. Stalnaker, On logics of knowledge and belief. Philos. Stud. 128, 169–199 (2006)CrossRefGoogle Scholar
  77. J. van Benthem, Dynamic logic for belief revision. J. Appl. Non-Class. Log. 17(2), 129–155 (2007)CrossRefGoogle Scholar
  78. J. van Benthem, Logical Dynamics of Information and Interaction (Cambridge University Press, Cambridge/New York, 2011)CrossRefGoogle Scholar
  79. W. van der Hoek, Sytems for knowledge and belief. J. Log. Comput. 3(2), 173–195 (1993)CrossRefGoogle Scholar
  80. W. van der Hoek, J.-J.Ch. Meyer, Graded modalities in epistemic logic, in Proceedings of the LFCS, Tver, ed. by A. Nerode, M.A. Taitslin. Volume 620 of Lecture Notes in Computer Science (Springer, 1992), pp. 503–514Google Scholar
  81. H. van Ditmarsch, Prolegomena to dynamic logic for belief revision. Synthese 147, 229–275 (2005)CrossRefGoogle Scholar
  82. H. van Ditmarsch, W. van der Hoek, B. Kooi, Dynamic Epistemic Logic. Volume 337 of Synthese Library (Springer, Dordrecht, 2007)Google Scholar
  83. H.P. van Ditmarsch, J. van Eijck, F. Sietsma, Y. Wang, On the logic of lying, in Games, Actions and Social Software, ed. by J. van Eijck, R. Verbrugge. Texts in Logic and Games, LNAI-FoLLI (Springer, 2011)Google Scholar
  84. B. van Linder, W. van der Hoek, J.-J.Ch. Meyer, Formalising abilities and opportunities of agents. Fundamenta Informaticae 34(1–2), 53–101 (1998)Google Scholar
  85. F. von Kutschera, Einführung in die Intensionale Semantik (W. de Gruyter, Berlin, 1976)Google Scholar
  86. F. Voorbraak, As Far as I know. Epistemic Logic and Uncertainty. PhD thesis, Utrecht University, 1993Google Scholar
  87. G. Wheeler, Is there a logic of information? J. Exp. Theor. Artif. Intell. (special issue on Inforgs and the Infosphere: Themes from Luciano Floridi’s Philosophy of Artificial Intelligence) (2012)Google Scholar
  88. T. Williamson, Knowledge and Its Limits (Oxford University Press, Oxford/New York, 2000)Google Scholar
  89. M. Wooldridge, Reasoning About Rational Agents (MIT, Cambridge, 2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Rennes 1RennesFrance

Personalised recommendations