In the paper we consider some principal notions of non-well-founded proof theory in argumentation. This theory is based on the assumption of Anti-Foundation Axiom that every graph tree has a unique decoration. A decoration of a graph is an assignment of a derivable formula to each node of the graph in such a way that the premisses of the root-derivable formula assigned to a node are the derivable formulas assigned to the children of that node. According to Anti-Foundation Axiom in proof theory, cyclic graph and infinite graph trees have a decoration too. This means that there are cyclic and infinite proof trees. The natural interpretation of cyclic proofs in argumentation is their consideration as confirmation procedure, where premisses are compatible with a derivable statement, but they do not prove this in the standard meaning. As model example we use the case of excluding Boris Pasternak from the Association of Writers of the USSR.