The set of relevant deals is the set of deals that are publicly indistinguishable from the actual deal of cards. These are the deals that players still have to consider while reasoning about card possession. In the initial state of the game, the cards have been dealt but the players have not acquired knowledge about other players' cards. The point of the model is the actual deal. Accessibility is indistinguishability of deals.
Propositional atoms describe that a player holds a card.
It expresses that exactly one of the relevant card deals must be actually the case. We also characterize the knowledge game state where the cards have been dealt but where the players have not yet turned their cards. Game actions in knowledge games consist of card requests and responses to those requests, plus some other moves, such as announcing, or guessing, that you have won. The remaining players see that a card is being shown, but cannot see which card. The question is a set of possible answers.