# Sergei Artemov: Generic Common Knowledge

Computational Logic Seminar
August 28, Time 2:00 – 4:00 PM, Room 3309.
Title: Generic Common Knowledge

Abstract: In four points of origin, each coming from a different body of motivations:
David Lewis in “Convention,” 1969;
John McCarthy in “On the model theory of knowledge,” 1970-78;
Robert Aumann in “Agreeing to Disagree,” 1976;
S.A. in “Justified Common Knowledge,” 2006.
the common knowledge operator C(F) was intended to capture the iterative knowledge “A knows that B knows that A knows … F” which, in the classical environment, is informally represented by the “infinite conjunction” I(F)
F & E(F) & E^2(F) & E^3(F) & … & E^n(F) & …
Here E^n(X) is the n-th iteration of the statement “every agent knows X.”

Despite differences in modes of presentation these approaches all agree that
C(F) => I(F), (*)
thus postulating that “common knowledge” yields iterative knowledge. However, there is no consensus concerning a stronger assumption that common knowledge should be equivalent to iterative knowledge:
C(F) I(F), (**)
Moreover, Lewis, McCarthy, and Artemov define C(F) as (*) whereas Aumann opted for (**), which subsequently became a dominant model for common knowledge, in particular, in Game Theory.

We suggest calling C(F) satisfying (*) Generic Common Knowledge, GCK. Note that whereas the standard common knowledge for a given Kripke model M is unique and determined by the reachability in M, there might be many logically non-equivalent GCK operators. In particular, the Universal Knowledge of F, U(F), meaning that F holds at all nodes, is an instance of GCK.

We argue that using GCK instead of common knowledge can be beneficiary in some meaningful situations. Public announcements of non-epistemic facts, the standard vehicle of obtaining common knowledge, actually yield GCK (in particular, Universal Knowledge) rather than common knowledge. In games of perfect information in belief revision setting, the standard common knowledge of rationality’ assumption does not guarantee that there is no irrationality in the system’ and hence should be replaced by some instance of GCK.

Generic Common Knowledge is easier to axiomatize and analyze, whereas the standard applications of common knowledge to epistemic scenarios work equally well with GCK.