On misconceptions in modern logic

The article discusses the inconsistency of three "indisputable" principles in modern logic: the inconsistency of the concept of "set," the absolute necessity of axioms in logic, and the infallibility of syllogistic reasoning. To address the first misconception, the authors propose incorporating the algebra of sets into the foundations of logic, following the approach outlined in R. Courant and G. Robbins' book What is Mathematics? The second misconception is addressed by deriving known laws of algebra of sets, corresponding to classical logic, through the enumeration method. The third misconception is resolved by developing a mathematical model of polysyllogistic reasoning based on the laws of algebra of sets. The novelty of this proposed reasoning model lies in introducing restrictions alongside premises, with any violation of these restrictions signaling errors in reasoning. This model enhances the analytical capabilities of logical analysis, enabling the detection of errors in traditional syllogistic reasoning, including instances where some correct inferences are classified as "incorrect" modes. Furthermore, new laws of algebra of sets are formulated and justified: the law of paradox, the condition of non-empty intersection, and the law of existence.