Propositional logic on the basis of algebraic system containing traditional syllogistics

The article explains the reasons to replace the multi-semantic basis of Aristotle in classical logic and traditional syllogistic with a mono-semantic basis, isomorphic to relationships “equivalent”, “entailing”, “independent”, which happen between terms of reasoning and random events in probability theory. Theoretical results and applications are discussed. The author identifies the drawbacks of the mathematical model which is the basis of classical logics. An advanced version of the mathematical model which is logic $\mathbf{S}_{L_1}$, based on non-degenerative Boolean algebra and an adjoint algebraic set-based system, is proposed. The article considers a non-classical interpretation of judgments in the orthogonal basis of syllogistics; it also describes the opportunities of effective computer validation of logical implication in semantics. A new method of solving logic equations is presented. The samples of solutions are presented.