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In this paper we develop a semantics of our approach based on commitments and arguments for conversational agents. We propose a logical model based on CTL* (Extended Computation Tree Logic) and on dynamic logic. Called Commitment and Argument Network (CAN), our formal framework based on this hybrid approach uses three basic elements: social commitments, actions that agents apply to these commitments and arguments that agents use to support their actions. The advantage of this logical model is to gather all these elements and the existing relations between them within the same framework. The semantics we develop here enables us to reflect the dynamics of agent communication. It also allows us to establish the important link between commitments as a deontic concept and arguments. On the one hand CTL* enables us to express all the temporal aspects related to the handling of commitments and arguments. On the other hand, dynamic logic enables us to capture the actions that agents are committed to achieve.