Justified common knowledge

In this paper we introduce the justified knowledge operator J with the intended meaning of J ϕ as 'there is a justification for ϕ.' Though justified knowledge appears here in a case study of common knowledge systems, a similar approach is applicable in more general situations. First we consider evidence-based common knowledge systems obtained by augmenting a multi-agent logic of knowledge with a system of evidence assertions t:ϕ, reflecting the notion 't is an evidence for ϕ,' such that evidence is respected by all agents. Justified common knowledge is obtained by collapsing all evidence terms into one modality J. We show that in standard situations, when the base epistemic systems are T, S4, and S5, the resulting justified common knowledge systems are normal modal logics, which places them within the scope of well-developed machinery applicable to modal logic: Kripke-style epistemic models, normalized proofs, automated proof search, etc. In the aforementioned situations, the intended semantics of justified knowledge is supported by a realization theorem stating that for any valid fact about justified knowledge, one could recover its constructive meaning by realizing all occurrences of justified knowledge modalities Jϕ by appropriate evidence terms t:ϕ.

[2]  Sergei N. Artëmov,et al.  Logic of knowledge with justifications from the provability perspective. , 2004 .

[3]  Edmund L. Gettier.  Is Justified True Belief Knowledge? , 1963, Arguing About Knowledge.

[4]  Melvin Fitting,et al.  The logic of proofs, semantically , 2005, Ann. Pure Appl. Log..

[5]  Lev Gordeev,et al.  Basic proof theory , 1998 .

[6]  Yevgeny Kazakov,et al.  On logic of knowledge with justifications , 1999 .

[7]  Sergei N. Artëmov,et al.  On epistemic logic with justification , 2005, TARK.

[8]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[9]  E. Pacuit.  A Note on Some Explicit Modal Logics , 2005 .

[10]  Sergei N. Artëmov.  Explicit provability and constructive semantics , 2001, Bull. Symb. Log..

[11]  Vladimir Krupski,et al.  Referential logic of proofs , 2006, Theor. Comput. Sci..

[12]  Sergei N. Artemovy.  Operational Modal Logic , 1995 .

[13]  Elena Nogina.  Logic of Proofs with the Strong Provability Operator , 1994 .

[14]  Alvin I. Goldman,et al.  A Causal Theory of Knowing , 1967 .

[15]  Sergei N. Artëmov,et al.  Integration of decision procedures into high-order interactive provers , 2006 .

[16]  N. Malcolm.  On Knowledge and Belief , 1954 .

[17]  Daniel Lehmann,et al.  Knowledge, common knowledge and related puzzles (Extended Summary) , 1984, PODC '84.

[18]  Alexey Mkrtychev,et al.  Models for the Logic of Proofs , 1997, LFCS.

[19]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[20]  Polly S Nichols,et al.  Agreeing to disagree. , 2005, General dentistry.

[21]  Vincent F. Hendricks,et al.  Active Agents , 2003, J. Log. Lang. Inf..

[22]  Sergei N. Artëmov.  Logic of Proofs , 1994, Ann. Pure Appl. Log..

[23]  Tatiana Yavorskaya.  Provability Logic with Operations on Proofs , 1997, LFCS.

[24]  Joseph Y. Halpern,et al.  Knowledge and common knowledge in a distributed environment , 1984, JACM.

[25]  David Lewis.  Convention: A Philosophical Study , 1986 .

[26]  Keith Lehrer,et al.  Knowledge: Undefeated Justified True Belief , 1969 .

[27]  Roman Kuznets,et al.  Making knowledge explicit: How hard it is , 2006, Theor. Comput. Sci..

[28]  Territoire Urbain,et al.  Convention , 1955, Hidden Nature.

[29]  Gerhard Jäger,et al.  About cut elimination for logics of common knowledge , 2005, Ann. Pure Appl. Log..

[30]  Vladimir Krupski,et al.  The single-conclusion proof logic and inference rules specification , 2001, Ann. Pure Appl. Log..

[31]  Sergei N. Artëmov,et al.  Introducing Justification into Epistemic Logic , 2005, J. Log. Comput..

[32]  Tatiana Yavorskaya,et al.  Logic of proofs and provability , 2001, Ann. Pure Appl. Log..

[33]  Laurence Bonjour,et al.  The coherence theory of empirical knowledge , 1976 .

[34]  J. McCarthy,et al.  On the model theory of knowledge , 1978 .