The issue of the emergence of formal axiomatic logical systems due to the emergence of logical antinomies in formal axiomatic systems, specifically the issue of developing formal logical axiomatics in the calculus of considerations was investigated in the considered research. At the same time, in order to determine the characteristics of the implementation of the logical-methodological principles and provisions of the deductive reasoning obviously, conceptual-logical foundations of the calculus of considerations was studied and the main propositions of the calculus of (...) considerations and the analysis of the initial logical operations on them were given in the study. The basic laws of logic and the expression of one logical act with another one were also explained in the research work by giving the concepst of proportional form and tautology in the calculus of considerations. At the same time, main properties and the procedure of setting of the formal deductive theory in the calculus of considerations were studied in the article. In order to provide an adequate characterization of the conceptual and methodological bases of the calculus of considerations in the research work, the basic logical laws of the calculus of considerations developed by the German mathematician and thinker P. Hilbert were also given. In addition, the main principles and methodological provisions of the deductive method were investigated, and the rules of deriving new conclusions from the axioms were interpreted. At the end of the reviewed article, the main principles and methods of the formal axiomatic mathematical systems built on a deductive basis, specifically the calculus of considerations, were studied and a logical-methodological analysis of the calculus of considerations was carried out on this basis. In this framework, the syntactic and semantic analysis of formalized language in formal axiomatic logical systems was carried out separately, the constituent parts of those systems, including the construction scheme, were given. (shrink)