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. | \(\mathbf{R}_i A\) | (from RCI and the assumption that \(A\) holds) |
| 2. | \(A \indi \mathbf{R}_j A\) | (RCI2) |
| 3. | \(A \indi x\) | (RCI3) |
| 4. | \(\mathbf{R}_i x\) | (from 1 and 3, using CS1) |
| 5. | \(\mathbf{R}_i (A \indj x)\) | (from 3, using RCI4) |
| 6. | \(A \indi \mathbf{R}_j x\) | (from 2 and 5, using CS5) |
| 7. | \(\mathbf{R}_i \mathbf{R}_j x\) | (from 1 and 6, using CS1) |
| 8. | \(\mathbf{R}_i(A \indj \mathbf{R}_k x)\) | (from 6, using RCI4) |
| 9. | \(A \indi \mathbf{R}_j (\mathbf{R}_k x)\) | (from 2 and 8, using CS5) |
| 10. | \(\mathbf{R}_i (\mathbf{R}_j (\mathbf{R}_k x))\) | (from 1 and 9, using A1) |
| 11. | \(\mathbf{R}_i (A \indj \mathbf{R}_k (\mathbf{R}_l x))\) | (from 9, using RCI4) |
And so on, for all \(i, j, k, l\) etc. in \(P\). Lines \(4, 7, 10, 3n+1 (n \gt 3)\) establish the theorem.