The article presents two ideas for the semantics of a dispute, the model of which is based on the metaphor of “processing”. Abstract models of dispute and argumentation are formal systems that reproduce the interaction of agents. The task of logical semantics is to interpret the formalized language in which the dispute is described. The dispute as a process presupposes a finishing, which is associated with assignment of some meaning to the argumentation. The article describes three aspects of this meaning: epistemic, interactional and influentional. The combinations of the assessments desired by the agents make it possible to give a number of characteristics to the result of dispute. The end of the dispute occurs when the agents reach the desired or acceptable estimates. At the same time new arguments does not affect the change in assessments. The concept of a fixed point is used to describe this state of dispute. Three types of fixed points of the dispute are considered: minimal, internal and maximal. Each fixed point can be complete or incomplete. Using fixed points I give definitions of solvability, partial solvability and unsolvability of a dispute. In particular, the dispute may turn out to be solvable only in epistemic aspects that is, it may finish in a change of the attitudes of the agents, but not solvable in the aspects of interaction or influence. Fixed points allow characterizing such a concept as “endless dispute”, as well as establishing the relationship between the results of the dispute, which are expressed by various aspects of its assessments. It is concluded that a fixed point as a logical concept is the basis of the metaphor of a computational procedure or “processing”, which remains the leading one in our understanding of argumentative interactions.