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In the paper we consider the classical logicism program restricted to first-order logic.The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), thedoi:10.21146/2074-1472-2016-22-1-125-135 fatcat:mhi4a3ccdza45bznq5xu6cjgtq