Abstract
This paper surveys recent dynamic logics of knowledge and belief for a single agent. Many of the recent developments in this area have been driven by analyzing concrete examples. These range from toy examples, such as the infamous muddy children puzzle, to philosophical quandaries, such as Fitch’s Paradox , to everyday examples of social interaction . Different logical systems are then judged, in part, on how well they conform to the analyst’s intuitions about the relevant set of examples. But this raises an important methodological issue: Implicit assumptions about what the actors know and believe about the situation being modeled often guide the analyst’s intuitions. In many cases, it is crucial to make these underlying assumptions explicit. The primary goal of this paper is to demonstrate how this “meta-information ” can be made explicit in the formal models of knowledge and belief .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This important book has recently been reissued and extended with some of Hintikka’s latest papers on epistemic !epistemic logic (Hintikka 2005).
- 2.
- 3.
- 4.
Cf. the very interesting discussion of higher-order evidence in the (formal) epistemology literature (Christensen 2010).
- 5.
See Parikh and Ramanujam (2003), Sect. 6, for a discussion of Gricean norms in this context.
- 6.
See Pacuit and Simon (2011), and references therein, for a logic to reasoning about what agents know about a protocol, or plan, that they are executing.
- 7.
The phrasing “epistemic indistinguishability”, although common in the epistemic !epistemic logic literature, is misleading since, as a relation, “indistinguishability” is not transitive. A standard example is: A cup of coffee with n grains of sugar is indistinguishable from a cup with n + 1 grains; however, transitivity would imply that a cup with 0 grains of sugar is indistinguishable from a cup with 1,000 grains of sugar. In this context, two states are “epistemicly indistinguishable” for an agent if the agent has the “same information ” in both states. This is indeed an equivalence relation.
- 8.
To be more precise, the key notion here is frame definability: A frame is a pair ⟨W, R⟩ where W is a nonempty set and R a relation on W. A modal formula is valid on a frame if it is valid in every model based on that frame. It can be shown that some modal formulas have first-order correspondents P where for any frame ⟨W, R⟩, the relation R has property P iff \(\varphi\) is valid on ⟨W, R⟩.
- 9.
It is a convention in this literature that going down according to ⪯ corresponds to being more plausible. This is just a convention which can be easily changed.
- 10.
- 11.
This is not true for multiagent languages with a common knowledge operator. Nonetheless, a recursion axiom-style analysis is still possible, though the details are beyond the scope of this paper, see van Benthem et al. (2006).
- 12.
Of course, one could drop this assumption and assume that the protocol remains fixed. I do not pursue this line of inquiry here.
References
T. Ågotnes, N. Alechina, The dynamics of syntactic knowledge. J. Log. Comput. 17(1), 83–116 (2007)
C.E. Alchourrón, P. Gärdenfors, D. Makinson, On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50, 510–530 (1985)
R. Aumann, Interactive epistemology I: knowledge. Int. J. Game Theory 28, 263–300 (1999)
A. Baltag, S. Smets, Conditional doxastic models: a qualitative approach to dynamic belief revision, in Proceedings of WOLLIC 2006, Electronic Notes in Theoretical Computer Science, Stanford University, eds. by G. Mints, R. de Queiroz, 2006, Vol. 165, pp. 5–21
A. Baltag, S. Smets, The logic of conditional doxastic actions, in New Perspectives on Games and Interaction, eds. by R. van Rooij, K. Apt. Texts in Logic and Games (Amsterdam University Press, Amsterdam, 2008a), pp. 9–31
A. Baltag, S. Smets, A qualitative theory of dynamic interactive belief revision, in Logic and the Foundation of Game and Decision Theory (LOFT7), eds. by G. Bonanno, W. van der Hoek, M. Wooldridge. Texts in Logic and Games (Amsterdam University Press, Amsterdam, 2008b), pp. 13–60
A. Baltag, S. Smets, ESSLLI 2009 course: dynamic logics for interactive belief revision (2009), Slides available online at: http://alexandru.tiddlyspot.com/#[[ESSLLI%2709%20Slides]]
A. Baltag, L. Moss, S. Solecki, The logic of common knowledge, public announcements and private suspicions, in Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 98), Evanston, ed. by I. Gilboa, 1998, pp. 43–56
P. Battigalli, M. Siniscalchi, Strong belief and forward induction reasoning. J. Econ. Theory 105, 356–391 (2002)
P. Blackburn, M. de Rijke, Y. Venema, Modal Logic (Cambridge University Press, Cambridge, 2002)
O. Board, Dynamic interactive epistemology. Games Econ. Behav. 49, 49–80 (2004)
G. Bonanno, Axiomatic characterization of AGM belief revision in a temporal logic. Artif. Intell. 171, 144–160 (2007)
G. Bonanno, Belief change in branching time: AGM-consistency and iterated revision. J. Philos. Log. 41(1), 201–236 (2012). Manuscript
G. Bonanno, P. Battigalli, Recent results on belief, knowledge and the epistemic foundations of game theory. Res. Econ. 53(2), 149–225 (1999)
C. Boutilier, Conditional Logics for Default Reasoning and Belief Revision. PhD thesis, University of Toronto, 1992
C. Boutilier, Iterated revision and minimal revision of conditional beliefs. J. Philos. Log. 25, 262–304 (1996)
A. Brandenburger, The power of paradox: some recent developments in interactive epistemology. Int. J. Game Theory 35, 465–492 (2007)
D. Christensen, Higher-order evidence. Philos. Phenomenol. Res. 81(1), 185–215 (2010)
C.B. Cross, ‘Can’ and the logic of ability. Philos. Stud. 50(1), 53–64 (1986)
A. Darwiche, J. Pearl, On the logic of iterated belief revision. Artif. Intell. 89, 1–29 (1997)
C. Dégremont, The Temporal Mind. Observations on the Logic of Belief Change in Interactive Systems. PhD thesis, Institute for Logic, Language and Computation. (DS-2010-03, 2010)
P. Egré, B. Bonnay, Inexact knowledge with introspection. J. Philos. Log. 38(2), 179–228 (2009)
D. Elgesem, The modal logic of agency. Nord. J. Philos. Log. 2, 1–46 (1997)
R. Fagin, J. Halpern, Y. Moses, M. Vardi, Reasoning about Knowledge (MIT, Cambridge, 1995)
J. Gerbrandy, Bisimulations on Planet Kripke. PhD thesis, University of Amsterdam, 1999
A. Grove, Two modellings for theory change. J. Philos. Log. 17, 157–170 (1988)
J. Halpern, Reasoning About Uncertainty (MIT, Cambridge, 2005)
J. Halpern, Y. Moses, Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990)
J. Halpern, L. Rego, Reasoning about knowledge of unawareness revisited, in Proceedings of Theoretical Aspects of Rationality and Knowledge (TARK’09), Stanford, ed. by A. Heifetz, 2009
V. Hendricks, Editor, special issue: “8 bridges between formal and mainstream epistemology”. Philos. Stud. 128(1), 1–227, 2006
V. Hendricks, Mainstream and Formal Epistemology (Cambridge University Press, Cambridge, 2006)
J. Hintikka, Knowledge and Belief: An Introduction to the Logic of the Two Notions (Cornell University Press, Ithaca, 1962)
J. Hintikka, Knowledge and Belief: An Introduction to the Logic of the Two Notions (with an Introduction by V. Hendricks and J. Symons) (King’s College Publications, London, 2005)
J. Horty, Agency and Deontic Logic (Oxford University Press, Oxford/New York, 2001)
T. Hoshi, Epistemic Dynamics and Protocol Information. PhD thesis, Stanford University, 2009
W. Jamroga, T. Ågotnes, Constructive knowledge: what agents can achieve under imperfect information. J. Appl. Non-class. Log. 17(4), 423–475 (2007)
P. Lamarre, Y. Shoham, Knowledge, certainty, belief and conditionalisation, in Proceedings of the International Conference on Knowledge Representation and Reasoning, Bonn, 1994, pp. 415–424
D. Lewis, Counterfactuals (Blackwell, Oxford, 1973)
F. Liu, Changing for the better: preference dynamics and agent diversity. PhD thesis, Institute for Logic, Language and Computation (ILLC), 2008
A. Nayak, M. Pagnucco, P. Peppas, Dynamic belief revision operators. Artif. Intell. 146, 193–228 (2003)
E. Pacuit, S. Simon, Reasoning with protocols under imperfect information. Rev. Symb. Log. 4, 412–444 (2011)
E. Pacuit, R. Parikh, E. Cogan, The logic of knowledge based obligation. Synthese 149(2), 311–341 (2006)
R. Parikh, R. Ramanujam, A knowledge based semantics of messages. J. Log Lang. Inf. 12, 453–467 (2003)
J. Plaza, Logics of public communications, in Proceedings, 4th International Symposium on Methodologies for Intelligent Systems, Charlotte, ed. by M.L. Emrich, M.S. Pfeifer, M. Hadzikadic, Z. Ras, 1989, pp. 201–216. (republished as Plaza (2007))
J. Plaza, Logics of public communications. Synthese 158(2), 165–179 (2007)
B. Rodenhäuser, A logic for extensional protocols, J. Appl. Non-class. Log. 21(3–4), 477–502 (2011)
H. Rott, Shifting priorities: simple representations for 27 itereated theory change operators, in Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg, eds. by H. Lagerlund, S. Lindström, R. Sliwinski, Vol. 53. Uppsala Philosophical Studies, 2006, pp. 359–384
Y. Shoham, K. Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations (Cambridge University Press, Cambridge/New York, 2009)
R. Stalnaker, The problem of logical omniscience I. Synthese 89, 425–440 (1991)
R. Stalnaker, On the evaluation of solution concepts. Theory Decis. 37(42), 49–73 (1994)
R. Stalnaker, Iterated belief revision. Erkentnis 70, 189–209 (2009)
J. van Benthem, Dynamic logic for belief revision. J. Appl. Non-class. Log. 17(2), 129–155 (2007)
J. van Benthem, The information in intuitionistic logic. Synthese 167, 329–348 (2009)
J. van Benthem, Logical Dynamics of Information and Interaction (Cambridge University Press, Cambridge, 2011)
J. van Benthem, J. van Eijck, B. Kooi, Logics of communication and change. Inf. Comput. 204(11), 1620–1662 (2006)
J. van Benthem, J. Gerbrandy, T. Hoshi, E. Pacuit, Merging frameworks for interaction. J. Philos. Log. 38(5), 491–526 (2009)
H. van Ditmarsch, Prolegomena to dynamic logic for belief revision. Synth. Knowl. Ration. Action 147, 229–275 (2005)
H. van Ditmarsch, W. van der Hoek, B.P. Kooi, Dynamic Epistemic Logic, Synthese Library (Springer, Dordrecht, 2007)
H. van Ditmarsch, S. Ghosh, R. Verbrugge, Y. Wang, Hidden protocols, in Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), Groningen, ed. by K. Apt (ACM Digital Library, 2011)
Y. Wang, Epistemic Modelling and Protocol Dynamics. PhD thesis, CWI, 2010
T. Williamson, Knowledge and Its Limits (Oxford University Press, Oxford, 2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pacuit, E. (2014). Procedural Information and the Dynamics of Belief. In: Rebuschi, M., Batt, M., Heinzmann, G., Lihoreau, F., Musiol, M., Trognon, A. (eds) Interdisciplinary Works in Logic, Epistemology, Psychology and Linguistics. Logic, Argumentation & Reasoning, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-03044-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-03044-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03043-2
Online ISBN: 978-3-319-03044-9
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)