On the foundation of intersubjectivity, Mikhail Bakhtin proposes the dialogism theory. It exists within and between the subjects. As a part of communication language in society, presupposition also respectively holds the relation of dialogism. The nature of presupposition is a cognitive communication dialogism interaction based on intersubjectivity. We argue that semantic presupposition and pragmatic presupposition belong to different communication subjects. Semantic presupposition is in the domain of the speaker. Pragmatic presupposition is in the domain of the communication participants. Semantic presupposition is the result from the intersubjectivity interaction with knowledge categorization (the world of reality, the world of imagination and ad hoc world) within the human cognition fields. By comparison, pragmatic presupposition is the result from the intersubjective interaction between the speaker and the hearer in communication.