Syllogistic adequate to Yu.V. Ivlev’s interpretation of categorical propositions in first-order logic

Abstract

The article examines the system of pure positive syllogistic, which is adequate to the Yu. V. Ivlev’s translation of categorical propositions into the language of first-order logic. According to this translation, particular propositions are true and general propositions are false if their subjects are empty. In syllogistic language we formulate the axiomatic calculus and prove that this calculus is recursively equivalent to the system of fundamental syllogistic. For this purpose, we define two translations: one of them embeds Ivlev’s syllogistic into fundamental, and the other embeds the second system into the first. We further proved a metatheorem that the axiomatic syllogistic calculus is embedded into the classical first-order logic through Ivlev's translation. The deductive features of the Ivlev’s syllogistic are investigated. It has been established that 21 modes of simple categorical syllogism, the laws of diagonals and the laws of subordination of the logical square, the conversion with restriction for general statements, the law of syllogistic identity for particular affirmative propositions are valid in it. The three modes of the syllogism of the fourth figure, pure conversions and the law of syllogistic identity for universal affirmative statements, are found to be invalid.