Supplement to Common Knowledge
Proof of Proposition 2.12
Proposition 2.12.Consider a population P and a proposition A such that (i) A is a reflexive common indicator in P that x and (ii) A is a reflexive common indicator in P that each member of P reasons faultlessy. Suppose A holds and each agent in P reasons faultlessly. Then there is common (actual) belief in P that x.
Proof. (Cubitt and Sugden 2003)
1. | R_{i} A | (from RCI1 and the assumption that A holds) | |
2. | A ind_{i} R_{j} A | (from RCI2) | |
3. | i reasons faultlessly | (assumption) | |
4. | A ind_{i} (j reasons faultlessly) | (from RCI3) | |
5. | A ind_{i} x | (from RCI3) | |
6. | R_{i} x | (from 1 and 5, using CS1) | |
7. | B_{i} x | (from 3, 6, and the definition of "faultless reasoning") | |
8. | R_{i} (A ind_{i} x) | (from 5, using RCI4) | |
9. | A ind_{i} (R_{j} x) | (from 2 and 8, using CS5) | |
10. | A ind_{i} (R_{i} x ∧ (j reasons faultlessly)) | (from 4 and 9, using CS3) | |
11. | (R_{j} x ∧ (j reasons faultlessly)) entails B_{j} x | (from definition of "faultless reasoning") | |
12. | A ind_{i} B_{j} x | (from 1 and 11, using CS1) | |
13. | R_{i} B_{j} x | (from 1 and 12, using CS1) | |
14. | B_{i} B_{j} x | (from 3, 13, and the definition of "faultless reasoning") | |
15. | R_{i} (A ind_{j} B_{k} x) | (from 12, using RCI4) | |
16. | A ind_{i} (R_{j} B_{k} x) | (from 2 and 15, using C5) |
And so on, for all i, j, k, etc. in P. Lines 7, 14, 7n (n > 2) establish the theorem.