Logical entailment and intensional syllogistic

The article considers non-standard semantics for syllogistic systems. In these systems, the formulas of propositional logic are the meanings of general terms, and the values of categorical statements are determined by the logical relationship between these formulas. If formula A is the meaning of term S, and if formula B is the meaning of term P, then the formula “SaP” is valid if and only if A entails B. Otherwise, the formula of the form “SoP” is valid. The “SeP” formula is valid if and only if A entails -B, otherwise the “SiP” formula is valid. This semantics is based on the intensional approach to the construction of syllogistic theories that goes back to Leibniz’s ideas. Firstly, the entailment relation by which categorical statements are interpreted allows, as Voishvillo has shown, an intensional interpretation: A entails B if the logical content of B is a part of the logical content of A. Secondly, propositional formulas can be considered as abbreviations for quantifier-free first-degree formulas with a single free variable, and such formulas, according to Voishvillo, state the contents of concepts. Therefore, in this semantics, the value of a syllogistic formula is determined not by the relationship between the extensions of the subject and the predicate, but by the relationship between their intensions. This article shows how different types of entailment affect the class of universally valid syllogistic formulas. Theories that use the relations of classical entailment and FDE-entailment have already been constructed. The classical entailment can be defined on the set of Carnap-style (consistent and complete) state-descriptions, and the first-degree relevant entailment can be defined on the set of generalized state-descriptions (the latter can be either contradictory or incomplete). In the article, semantics using two other relations: the Hao Wang-style entailment logic (defined on the set of all consistent state-descriptions) and its dual logic entailment (defined on the set of all complete state-descriptions) were built. The deductive capabilities of four syllogistic systems that use different types of logical entailments were compared. The authors declare no conflicts of interests.

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