Brandenburger and Keisler formalize the above argument in order to prove a very strong impossibility result about the existence of so-called assumption-complete structures. We need some notation to state this result. It will be most convenient to work in qualitative type spaces for two players (Definition 2.7). A qualitative type space for two players (cf. Definition 2.7. The set of states is not important in what follows, so we leave it out) is a structure \(\langle \{T_A, T_B\}, \{\lambda_A, \lambda_B\}\rangle\) where