This note is a summary of the talk given on June 10, 2010 at the University of Opole during a meeting of The Group of Logic, Language and Information. We limit ourselves to some major points of the talk, skipping all the minor details. The main hero of this note is the notion of the intended model of a theory. Some emphasis is put on the role of extremal axioms in the characterization of such models. The notion of the intended model seems to be of some interest for linguists, too. This is one of the reasons for which this paper is published in a linguistic journal.