The Common Knowledge of Formula Exclusion

A multi-partition with evaluations is defined by two sets S and X, a collection P, . . . ,P of partitions of S and a function ψ : S → {0, 1} . To each partition P corresponds a person j who cannot distinguish between any two points belonging to the same member of P but can distinguish between different members of P. A cell of a multi-partition is a minimal subset C such that for all j the properties P ∈ P and P ∩ C 6= ∅ imply that P ⊆ C. Construct a sequence R0,R1, . . . of partitions of S by R0 = {ψ−1(a) | a ∈ {0, 1}} and x and y belong to the same member of Ri if and only if x and y belong to the same member of Ri−1 and for every person i the members Px and Py of P containing x and y respectively intersect the same members of Ri−1. Let R∞ be the limit of the Ri, namely x and y belong to the same member of R∞ if and only if x and y belong to the same member of Ri for every i. For any set X and number n of persons there is a canonical multi-partition with evaluations defined on a set Ω such that from any multi-partition with evaluations (using the same X and n) there is a canonical map to Ω with the property that x and y are mapped to the same point of Ω if and only if x and y share the same member of R∞. We define a cell of Ω to be surjective if every multi-partition with evaluations that maps to it does so surjectively. A cell of a multi-partition with evaluations has finite fanout if every P ∈ P in the cell has finitely many elements. All cells of Ω with finite fanout are surjective, but the converse does not hold.