Abstract
We give a general approach to characterizing minimal information in a modal context. Our modal treatment can be used for many applications, but is especially relevant under epistemic interpretations of an operator. Relative to a modal system S, we give three characterizations of minimality of a formula ϕ and give conditions under which these characterizations are equivalent. We then argue that rather than using bisimulations, it is more appropriate to base information orders on Ehrenfeucht-Fraïssé games to come up with a satisfactory analysis of minimality. Moving to the realm of epistemic logics, we show that for one of these information orders almost all systems trivialize, i.e., either all or no formulas are honest. The other order is much more promising as it permits to minimize wrt positive knowledge. The resulting notion of minimality coincides with well-established accounts of minimal knowledge in S5. For S4 we compare the two orders.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
H. Andréka, J. van Benthem & I. Németi, ‘Back and Forth Between Modal Logic and Classical Logic’, Journal of the IGPL, Vol.3, No. 5, pp. 685–720, 1995.
J. van Benthem, Language in Action. Categories, Lambdas and Dynamic Logic, North-Holland, Amsterdam, 1991.
B.F. Chellas, Modal Logic. An Introduction, Cambridge University Press, 1980.
H.-D. Ebbinghaus & J. Flum, Finite model theory, Springer-Verlag, Berlin, 1995.
J.Y. Halpern, ‘Theory of Knowledge and Ignorance for Many Agents’, in Journal of Logic and Computation, 7 No. 1, pp. 79–108, 1997.
J.Y. Halpern & Y. Moses, ‘Towards a theory of knowledge and ignorance’, in Kr. Apt (ed.) Logics and Models of Concurrent Systems, Springer-Verlag, Berlin, 1985.
J. Hintikka, Knowledge and Belief: An introduction to the Logic of the Two Notions, Cornell University Press, Ithaca N.Y, 1962.
W. van der Hoek, J.O.M. Jaspars, & E.G.C. Thijsse, ‘Honesty in Partial Logic’. Studia Logica, 56(3), 323–360, 1996.
J.O.M. Jaspars, ‘A generalization of stability and its application to circumscription of positive introspective knowledge’, Proceedings of the Ninth Workshop on Computer Science Logic (CSL’90), Berlin: Springer-Verlag 1991.
J. Jaspars & E. Thijsse, ‘Fundamentals of Partial Modal Logic’, in P. Doherty (ed.) Partiality, Modality, Nonmonotonicity (pp.111–141), Stanford: CSLI Publications, Studies in Logic, Language and Information, 1996
H.J. Levesque, ‘All I know: a study in auto-epistemic logic’, in Artificial Intelligence, 42(3), pp. 263–309, 1990.
J. Łoś, ‘Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres’, in Th. Skolem et al. (eds.), Mathematical Interpretation of Formal Systems, North-Holland, Amsterdam, 1955
R.C. Moore, ‘Semantical considerations on non-monotonic logic’, Artificial Intelligence 25, pp. 75–94, 1985
M. Vardi, ‘A model-theoretic analysis of monotonic knowledge’, IJCAI85, pp. 509–512, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
van der Hoek, W., Jaspars, J., Thijsse, E. (1998). Persistence and Minimality in Epistemic Logic. In: Dix, J., del Cerro, L.F., Furbach, U. (eds) Logics in Artificial Intelligence. JELIA 1998. Lecture Notes in Computer Science(), vol 1489. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49545-2_5
Download citation
DOI: https://doi.org/10.1007/3-540-49545-2_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65141-3
Online ISBN: 978-3-540-49545-1
eBook Packages: Springer Book Archive