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.[1]: 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.[2] 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".[3]

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, G={a,b}), if they both have muddy face (viz. Ma∧Mb), both of them know that there is at least one muddy face. Written formally, let p=[∃x∈G(Mx)], and then we have Kap∧Kbp. However, neither of them know that the other child knows ((¬KaKbp)∧(¬KbKap)), which makes p=[∃x∈G(Mx)] mutual knowledge. Now suppose if Alice tells Bob that she knows p (so that Kap becomes common knowledge, i.e. CGKap), and then Bob tell Alice that he knows p as well (so that Kbp becomes common knowledge, i.e. CGKbp), this will turn p into common knowledge (CGEGp⇒CGp), which is equivalent to the effect of a public announcement "there is at least one muddy face".

## See also

* Elephant in the room
* The Emperor's New Clothes
* Distributed knowledge

## External links

* Steven Pinker. "The Stuff of Thought: Language as a window into human nature". RSA. https://www.youtube.com/watch?v=5S1d3cNge24&t=46m02s.

## References

1. ↑ Osborne, Martin J., and Ariel Rubinstein. A Course in Game Theory. Cambridge, MA: MIT, 1994. Print.
2. ↑ Peter Vanderschraaf, Giacomo Sillari (2007). Common Knowledge. Stanford Encyclopedia of Philosophy. Accessed 18 November 2011.
3. ↑ Stephen Schiffer, Meaning, 2nd edition, Oxford University Press, 1988. The first edition was published by OUP in 1972. Also, David Lewis, Convention, Cambridge, MA: Harvard University Press, 1969. For a discussion of both Lewis's and Schiffer's notions, see Russell Dale, The Theory of Meaning (1996).

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