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.

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Peter Vanderschraaf
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