Mutual knowledge (logic)
Mutual knowledge (logic)
- Mutual knowledge** is a fundamental concept about information in [[game-theory]], (epistemic) logic, and epistemology. An event is mutual knowledge if all agents know that the event occurred. 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. [[common-knowledge-(logic)|Common knowledge]] is a related but stronger notion; any event that is common knowledge is also mutual knowledge.
The philosopher Stephen Schiffer, in his book Meaning, developed a notion he called "mutual knowledge" which functions quite similarly to David K. Lewis's "common knowledge".
Communications (verbal or non-verbal) can turn mutual knowledge into common knowledge. For example, in the Muddy Children Puzzle with two children (Alice and Bob, <math>G=\{a,b\}</math>), if they both have muddy face (viz. <math>M_{a}\land M_{b}</math>), both of them know that there is at least one muddy face. Written formally, let <math>p=[\exists x\!\in\! G(M_{x})]</math>, and then we have <math>K_{a}p\land K_{b}p</math>. However, neither of them know that the other child knows (<math>(\neg K_{a}K_{b}p)\land(\neg K_{b}K_{a}p)</math>), which makes <math>p=[\exists x\!\in\! G(M_{x})]</math> mutual knowledge. Now suppose if Alice tells Bob that she knows <math>p</math> (so that <math>K_{a}p</math> becomes common knowledge, i.e. <math>C_{G} K_{a}p</math>), and then Bob tells Alice that he knows <math>p</math> as well (so that <math>K_{b}p</math> becomes common knowledge, i.e. <math>C_{G} K_{b}p</math>), this will turn <math>p</math> into common knowledge (<math>C_{G}E_{G} p \Rightarrow C_{G} p</math>), which is equivalent to the effect of a public announcement "there is at least one muddy face".