Mutual knowledge (logic)

Mutual knowledge is a fundamental concept about information in <a href="/facts/Game_theory/zkEIj1ya">game theory</a>, (epistemic) <a href="/facts/Logic/8JnpJLtt">logic</a>, and <a href="/facts/Epistemology/NsDwpcYH">epistemology</a>. An <a href="/facts/Event_(probability_theory)/aJ0l4oQb">event</a> is mutual knowledge if all agents know that the event occurred.: 73  However, mutual knowledge by itself implies nothing about what agents know about other agents' knowledge: i.e. it is possible that an event is mutual knowledge but that each agent is unaware that the other agents know it has occurred. <a href="/facts/Common_knowledge_(logic)/DwJ82TXM">Common knowledge</a> is a related but stronger notion; any event that is common knowledge is also mutual knowledge.
The philosopher <a href="/facts/Stephen_Schiffer/2no3hkza">Stephen Schiffer</a>, in his book Meaning, developed a notion he called "mutual knowledge" which functions quite similarly to <a href="/facts/David_Lewis_(philosopher)/8aG4Ex6V">David K. Lewis</a>'s "common knowledge".
<a href="/facts/Communication/EGEkchsT">Communications</a> (verbal or <a href="/facts/Nonverbal_communication/Fq7EdXMW">non-verbal</a>) can turn mutual knowledge into common knowledge. For example, in the <a href="/facts/Induction_puzzles/DkxwhdV5">Muddy Children Puzzle</a> with two children (Alice and Bob, 
 
 
 
 G
 =
 {
 a
 ,
 b
 }
 
 
 {\displaystyle G=\{a,b\}}
 
), if they both have muddy face (viz. 
 
 
 
 
 M
 
 a
 
 
 ∧
 
 M
 
 b
 
 
 
 
 {\displaystyle M_{a}\land M_{b}}
 
), both of them know that there is at least one muddy face. Written formally, let 
 
 
 
 p
 =
 [
 ∃
 x
 
 ∈
 
 G
 (
 
 M
 
 x
 
 
 )
 ]
 
 
 {\displaystyle p=[\exists x\!\in \!G(M_{x})]}
 
, and then we have 
 
 
 
 
 K
 
 a
 
 
 p
 ∧
 
 K
 
 b
 
 
 p
 
 
 {\displaystyle K_{a}p\land K_{b}p}
 
. However, neither of them know that the other child knows (
 
 
 
 (
 ¬
 
 K
 
 a
 
 
 
 K
 
 b
 
 
 p
 )
 ∧
 (
 ¬
 
 K
 
 b
 
 
 
 K
 
 a
 
 
 p
 )
 
 
 {\displaystyle (\neg K_{a}K_{b}p)\land (\neg K_{b}K_{a}p)}
 
), which makes 
 
 
 
 p
 =
 [
 ∃
 x
 
 ∈
 
 G
 (
 
 M
 
 x
 
 
 )
 ]
 
 
 {\displaystyle p=[\exists x\!\in \!G(M_{x})]}
 
 mutual knowledge. Now suppose if Alice tells Bob that she knows 
 
 
 
 p
 
 
 {\displaystyle p}
 
 (so that 
 
 
 
 
 K
 
 a
 
 
 p
 
 
 {\displaystyle K_{a}p}
 
 becomes common knowledge, i.e. 
 
 
 
 
 C
 
 G
 
 
 
 K
 
 a
 
 
 p
 
 
 {\displaystyle C_{G}K_{a}p}
 
), and then Bob tells Alice that he knows 
 
 
 
 p
 
 
 {\displaystyle p}
 
 as well (so that 
 
 
 
 
 K
 
 b
 
 
 p
 
 
 {\displaystyle K_{b}p}
 
 becomes common knowledge, i.e. 
 
 
 
 
 C
 
 G
 
 
 
 K
 
 b
 
 
 p
 
 
 {\displaystyle C_{G}K_{b}p}
 
), this will turn 
 
 
 
 p
 
 
 {\displaystyle p}
 
 into common knowledge (
 
 
 
 
 C
 
 G
 
 
 
 E
 
 G
 
 
 p
 ⇒
 
 C
 
 G
 
 
 p
 
 
 {\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p}
 
), which is equivalent to the effect of a <a href="/facts/Dynamic_epistemic_logic/Dc2hcT2H">public announcement</a> "there is at least one muddy face".

Mutual knowledge (logic) open-in-new

Mutual knowledge (logic)