Defining Universal Logic as Composed by a Logical Structure of Two Kinds of Square of Opposition and Their Mutual Translations

. In order to define universal logic in the most general way, I start by observing some essential dualisms within Logic. First, I illustrate philosophical dualisms: the two meanings of the word “Truth”, the two argumentative faculties of the human mind, the two ways of conceiving Logic as a whole (calculus/language), the two reasoning faculties of human mind, the two kinds of the metaphysics of logic (ontology/henology). Then I observe the formal dualisms between classical logic and intuitionist logic: the two logical versions of a true proposition, the two kinds of a proof, the two kinds of a theoretical organization, the two kinds of square of opposition, the two main, mutual translations: the double negation translation KGG and its inverse translation PSR° (i.e. the application of the principle of sufficient reason suitably restricted). Incidentally, Béziau’s paradox is resolved as a consequence of this limitation in the application of PSR. As a comprehensive system of all previous dualisms but especially relying on the two kinds of theoretical organizations Universal Logic is defined as the following triadic structure: classical logic, intuitionist logic and their translations KGG and PSR°. In the light of this definition Béziau’s five questions about Universal Logic are successfully answered. In particular, the answer to the second question is obtained by taking into account that modal logic and Vasiliev’s original paraconsistent logic may be reduced to intuitionist logic; this result suggests an open question: whether only two main families of logics exist, i.e. the family of those reducible to classical logic and the family of those reducible to intuitionist logic. In order to answer Beziau’s first question the validity of Universal Logic for also Eastern thought is illustrated.