Persistence and minimality in epistemic logic

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 the operator □. Relative to an arbitrary modal system, we give three characterizations of minimal information and provide conditions under which these characterizations are equivalent. We then study information orders based on bisimulations and Ehrenfeucht–Fraïssé games. Moving to the area of epistemic logics, we show that for one of these orders almost all systems trivialize the notion of minimal information. Another order which we present is much more promising as it permits to minimize with respect to positive knowledge. In S5, the resulting notion of minimal knowledge coincides with well‐established proposals. For S4 we compare the two orders.

[1]  Krister Segerberg,et al.  An introduction to modal logic : the Lemmon notes , 1977 .

[2]  Robert Stalnaker,et al.  A Note on Non-Monotonic Modal Logic , 1993, Artif. Intell..

[3]  Miroslaw Truszczynski,et al.  Minimal Knowledge Problem: A New Approach , 1994, Artif. Intell..

[4]  Joseph Y. Halpern.  A Theory of Knowledge and Ignorance for Many Agents , 1997, J. Log. Comput..

[5]  J.F.A.K. van Benthem,et al.  Modal logic and classical logic , 1983 .

[6]  Wolfgang Lenzen,et al.  Recent work in epistemic logic , 1978 .

[7]  D. Scott,et al.  Intensional Logic, preliminary draft of initial chapters by EJ Lemmon, July 1966, Nowadays available as An Introduction to Modal Logic (American Philosophical Quarterly Monograph No. 11) edited by K. Segerberg , 1977 .

[8]  P. T. Geach,et al.  KNOWLEDGE AND BELIEF: An Introduction to the Logic of the Two Notions , 1963 .

[9]  Sally Popkorn.  First Steps in Modal Logic , 1995 .

[10]  Wiebe van der Hoek,et al.  Honesty in partial logic , 1995, Stud Logica.

[11]  Brian F. Chellas.  Modal Logic: Normal systems of modal logic , 1980 .

[12]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[13]  Robert C. Moore.  Semantical Considerations on Nonmonotonic Logic , 1985, IJCAI.

[14]  Moshe Y. Vardi.  A Model-Theoretic Analysis of Monotonic Knowledge , 1985, IJCAI.

[15]  Robin Milner,et al.  Algebraic laws for nondeterminism and concurrency , 1985, JACM.

[16]  Siobhan Chapman.  Logic and Conversation , 2005 .

[17]  Johan van Benthem,et al.  Back and Forth Between Modal Logic and Classical Logic , 1995, Log. J. IGPL.

[18]  M. Fitting.  Intuitionistic logic, model theory and forcing , 1969 .

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  Jan Jaspars.  A Generalization of Stability and its Application to Circumscription of Positive Introspective Knowledge , 1990, CSL.

[21]  Max J. Cresswell,et al.  A companion to modal logic , 1984 .

[22]  Hector J. Levesque,et al.  All I Know: A Study in Autoepistemic Logic , 1990, Artif. Intell..

[23]  Ronald Fagin,et al.  An internal semantics for modal logic , 1985, STOC '85.

[24]  Kit Fine,et al.  Normal forms in modal logic , 1975, Notre Dame J. Formal Log..

[25]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[26]  Jan Jaspars,et al.  Fundamentals of partial modal logic , 1996 .

[27]  Sally Popkorn.  First steps in Modal Logic: Löb logic , 1994 .

[28]  Jerzy Loś,et al.  Quelques Remarques, Théorèmes Et Problèmes Sur Les Classes Définissables D'algèbres , 1955 .

[29]  Kurt Konolige,et al.  On the Relation Between Default and Autoepistemic Logic , 1987, Artif. Intell..

[30]  Joseph Y. Halpern,et al.  Towards a Theory of Knowledge and Ignorance: Preliminary Report , 1989, NMR.