## 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 Ri A (from RCI1 and the assumption that A holds) 2 A indi Rj A (from RCI2) 3 i reasons faultlessly (assumption) 4 A indi (j reasons faultlessly) (from RCI3) 5 A indi x (from RCI3) 6 Ri x (from 1 and 5, using CS1) 7 Bi x (from 3, 6, and the definition of "faultless reasoning") 8 Ri (A indi x) (from 5, using RCI4) 9 A indi (Rj x) (from 2 and 8, using CS5) 10 A indi (Ri x ∧ (j reasons faultlessly)) (from 4 and 9, using CS3) 11 (Rj x ∧ (j reasons faultlessly)) entails Bj x (from definition of "faultless reasoning") 12 A indi Bj x (from 1 and 11, using CS1) 13 Ri Bj x (from 1 and 12, using CS1) 14 Bi Bj x (from 3, 13, and the definition of "faultless reasoning") 15 Ri (A indj Bk x) (from 12, using RCI4) 16 A indi (Rj Bk 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.