Supplement to Common Knowledge
Proof of Proposition 2.11Proposition 2.11.
If A holds, and if A is a common reflexive indicator in the population P that x, then there is common reason to believe in P that x.
Proof. (Cubitt and Sugden 2003)
|1.||Ri A||(from RCI and the assumption that A holds)|
|2.||A indi Rj A||(RCI2)|
|3.||A indi x||(RCI3)|
|4.||Ri x||(from 1 and 3, using CS1)|
|5.||Ri (A indj x)||(from 3, using RCI4)|
|6.||A indi Rj x||(from 2 and 5, using CS5)|
|7.||Ri Rj x||(from 1 and 6, using CS1)|
|8.||Ri (A indj Rk x)||(from 6, using RCI4)|
|9.||A indi Rj (Rk x)||(from 2 and 8, using CS5)|
|10.||Ri (Rj (Rk x))||(from 1 and 9, using A1)|
|11.||Ri (A indj Rk (Rl x))||(from 9, using RCI4)|
And so on, for all i, j, k, l etc. in P. Lines 4, 7, 10, 3n+1 (n > 3) establish the theorem.