Common knowledge (logic)

Common knowledge is a special kind of <a href="/facts/Knowledge/joozoaup">knowledge</a> for a group of <a href="/facts/Agent-based_model/C3C1vR8s">agents</a>. 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 <a href="/facts/Ad_infinitum/1LemHjUl">ad infinitum</a>. It can be denoted as 
 
 
 
 
 C
 
 G
 
 
 p
 
 
 {\displaystyle C_{G}p}
 
.
The concept was first introduced in the philosophical literature by <a href="/facts/David_Kellogg_Lewis/8aG4Ex6V">David Kellogg Lewis</a> in his study Convention (1969). The sociologist Morris Friedell defined common knowledge in a 1969 paper. It was first given a mathematical formulation in a <a href="/facts/Set_theory/1ptfsvUP">set-theoretical</a> framework by <a href="/facts/Robert_Aumann/xWXaDKhB">Robert Aumann</a> (1976). <a href="/facts/Computer_science/lN3pEyPz">Computer scientists</a> grew an interest in the subject of <a href="/facts/Epistemic_logic/kRIBmdp5">epistemic logic</a> in general – and of common knowledge in particular – starting in the 1980s.<a href="https://en.wikipedia.org/wiki/Common_knowledge_(logic)#endnote_CSTexts">[1]</a> There are numerous <a href="/facts/Logic_puzzle/Q1LLT6TR">puzzles</a> based upon the concept which have been extensively investigated by mathematicians such as <a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a>.
The philosopher <a href="/facts/Stephen_Schiffer/2no3hkza">Stephen Schiffer</a>, in his 1972 book Meaning, independently developed a notion he called "<a href="/facts/Mutual_knowledge/6BuNsI3e">mutual knowledge</a>" (
 
 
 
 
 E
 
 G
 
 
 p
 
 
 {\displaystyle E_{G}p}
 
) which functions quite similarly to Lewis's and Friedel's 1969 "common knowledge". If a trustworthy announcement is made in <a href="/facts/Dynamic_epistemic_logic/Dc2hcT2H">public</a>, 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" (
 
 
 
 
 E
 
 G
 
 
 p
 
 
 {\displaystyle E_{G}p}
 
) is transmitted to each agent in private, it is still not common knowledge: 
 
 
 
 
 E
 
 G
 
 
 
 E
 
 G
 
 
 p
 ⇏
 
 C
 
 G
 
 
 p
 
 
 {\displaystyle E_{G}E_{G}p\not \Rightarrow C_{G}p}
 
. But, if any agent 
 
 
 
 a
 
 
 {\displaystyle a}
 
 publicly announces their knowledge of p, then it becomes common knowledge that they know p (viz. 
 
 
 
 
 C
 
 G
 
 
 
 K
 
 a
 
 
 p
 
 
 {\displaystyle C_{G}K_{a}p}
 
). If every agent publicly announces their knowledge of p, p becomes common knowledge 
 
 
 
 
 C
 
 G
 
 
 
 E
 
 G
 
 
 p
 ⇒
 
 C
 
 G
 
 
 p
 
 
 {\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p}
 
.

Common knowledge (logic) open-in-new

Common knowledge (logic)