Old and new riddles on concept sharing

We ask whether social interaction demands sharing social concepts. We illustrate our point by depicting possible situations that emerge when two individuals play chess. We formalize our hypothesis in First Order Logic and we show that the very idea of sharing social concepts poses an interesting challenge both from the standpoint of knowledge representation and of philosophical conceptual analysis. By endorsing a minimal notion of interaction, we conclude that sharing social concepts is not necessary for social interaction. Then, we relate our view to Wittgenstein's and Kripke's "Rule-following Considerations".