Common knowledge (logic)

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.^[1] It can be denoted as ${\displaystyle C_{G}p}$.

The concept was first introduced in the philosophical literature by David Kellogg Lewis in his study Convention (1969). The sociologist Morris Friedell defined common knowledge in a 1969 paper.^[2] It was first given a mathematical formulation in a set-theoretical framework by Robert Aumann (1976). Computer scientists grew an interest in the subject of epistemic logic in general – and of common knowledge in particular – starting in the 1980s.^[1] There are numerous puzzles based upon the concept which have been extensively investigated by mathematicians such as John Conway.^[3]

The philosopher Stephen Schiffer, in his 1972 book Meaning, independently developed a notion he called "mutual knowledge" (${\displaystyle E_{G}p}$) which functions quite similarly to Lewis's and Friedel's 1969 "common knowledge".^[4] If a trustworthy announcement is made in public, then it becomes common knowledge; However, if it is transmitted to each agent in private, it becomes mutual knowledge but not common knowledge. Even if the fact that "every agent in the group knows p" (${\displaystyle E_{G}p}$) is transmitted to each agent in private, it is still not common knowledge: ${\displaystyle E_{G}E_{G}p\not \Rightarrow C_{G}p}$. But, if any agent ${\displaystyle a}$ publicly announces their knowledge of p, then it becomes common knowledge that they know p (viz. ${\displaystyle C_{G}K_{a}p}$). If every agent publicly announces their knowledge of p, p becomes common knowledge ${\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p}$.