  We provide an epistemic foundation for cooperative games by proof theory via
studying the knowledge for players unanimously accepting only core payoffs. We
first transform each cooperative game into a decision problem where a player
can accept or reject any payoff vector offered to her based on her knowledge
about available cooperation. Then we use a modified KD-system in epistemic
logic, which can be regarded as a counterpart of the model for non-cooperative
games in Bonanno (2008), (2015), to describe a player's knowledge,
decision-making criterion, and reasoning process; especially, a formula called
C-acceptability is defined to capture the criterion for accepting a core payoff
vector. Within this syntactical framework, we characterize the core of a
cooperative game in terms of players' knowledge. Based on that result, we
discuss an epistemic inconsistency behind Debreu-Scarf Theorem, that is, the
increase of the number of replicas has invariant requirement on each
participant's knowledge from the aspect of competitive market, while requires
unbounded epistemic ability players from the aspect of cooperative game.
