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. | R_{i} A | (from RCI and the assumption that A holds) | |
2. | A ind_{i} R_{j} A | (RCI2) | |
3. | A ind_{i} x | (RCI3) | |
4. | R_{i} x | (from 1 and 3, using CS1) | |
5. | R_{i} (A ind_{j} x) | (from 3, using RCI4) | |
6. | A ind_{i} R_{j} x | (from 2 and 5, using CS5) | |
7. | R_{i} R_{j} x | (from 1 and 6, using CS1) | |
8. | R_{i} (A ind_{j} R_{k} x) | (from 6, using RCI4) | |
9. | A ind_{i} R_{j} (R_{k} x) | (from 2 and 8, using CS5) | |
10. | R_{i} (R_{j} (R_{k} x)) | (from 1 and 9, using A1) | |
11. | R_{i} (A ind_{j} R_{k} (R_{l} 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.