Since
is a coarsening of 
i for each i 
N, K i( ( 
) ).
Hence, K 1N ( ( ) ),
and since by definition K i( ( ) ) = { | i( ) 
( ) } = ( ),
| This is a file in the archives of the Stanford Encyclopedia of Philosophy. |
()
is common knowledge for the agents of N at
.
Proof.
Since is a coarsening of i for each i N, K i( ( ) ).
Hence, K 1N ( ( ) ),
and since by definition K i( ( ) ) = { | i( ) ( ) } = ( ),
K 1N ( 
iK i(
Applying the recursive definition of mutual knowledge, for any m 
1,
K mN (
iK i ( K m - 1N (
iK i(
so, since ( )
, by definition we have K *N ( ( ) ). 
| Peter Vanderschraaf peterv@cyrus.andrew.cmu.edu |