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The Aristotelian square of opposition is a formal structure representing precise logic relations between four types of propositions: universal affirmative, particular affirmative, universal negative, particular negative. In this article we present a formal translation in modern terms of the Aristotelian square, representing each type of proposition in the language of set theory and the first-order predicate logic. Starting from this formalization, we can derive one of the less intuitive theses affirmed by Aristotle: the sub-contrary propositions of the square can be both true, but they cannot be both false. Finally, we note that the logical-formal basis of Hume’s problem of empirical induction and that of Popper’s theory of falsification can be both derived from the logical relations defined in the Aristotelian square. [ABSTRACT FROM AUTHOR]