The article discusses logic traditions, which had been shaped in the VI century in the Eastern and Western parts the Roman Empire. Using the Philoponius's comments to treaties of Aristotle, to the treaty “On Hypothetical Syllogism” by Boethius, to the works of Peripatetics, Neo-Platonists and Stoics as examples, the article considers the Greek and Latin traditions in the theory of hypothetical syllogisms, draws conclusions about the specificities of those traditions, and about their mutual influence.