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.

Return to Common Knowledge

Copyright © 2022 by
Peter Vanderschraaf
Giacomo Sillari <>

Open access to the SEP is made possible by a world-wide funding initiative.
The Encyclopedia Now Needs Your Support
Please Read How You Can Help Keep the Encyclopedia Free