In this paper we present a semantics for our approach based on social commitments (SCs) and arguments for conversational agents. More precisely, we propose a logical model based on CTL* and on dynamic logic (DL). Called Commitment and Argument Network, our formal framework based on this approach uses three basic elements: SCs, actions that agents apply to these SCs and arguments that agents use to support their actions. The advantage of this logical model is to bring together all these elements and the relations existing between them within the same framework. Our semantics makes it possible to represent the dynamics of agent communication. It also allows us to establish the important link between SCs as a deontic concept and arguments. CTL* enables us to express the temporal characteristics of SCs and arguments. DL enables us to capture the actions that agents are committed to achieve.