The Generation of Formulas Held in Common Knowledge

Robert S. Simon

This ariticle concerns the interactive modal propositional calculus, using the multi-agent epistemic logic S5. With regard to the space of maximally consistent sets of formulas, the relations between three aspects of common knowledge are investigated: 1) whether common knowledge defined semantically is determined by the set of formulas held in common knowledge, 2) the partial order by inclusion of the sets of formulas that can be held in common knowledge, and 3) the cardinality of a generating set of formulas for those held in common knowledge. Additionally, assuming at least two agents, it is shown that the number of connected components of holding only the tautologies in common knowledge has the cardinality of the continuum.

June, 1997
Published in: 
International Journal of Game Theory 30 (2001), 1-18.