Supplement to Common Knowledge
Proof of Proposition 2.11
Proposition 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.