De re – de dicto dichotomy and apodeictic syllogistic

Abstract

Aristotle's syllogistic is a modal deductive system, and his assertoric syllogistic is only a narrow fragment of it. This modal logical theory drew objections from Aristotle's ancient and medieval successors and commentators. Aristotle considered some “mixed” syllogisms with one apodeictic premise, one assertoric premise and apodeictic conclusion to be valid. His pupils Theophrastus and Eudemus introduced the principle that the conclusion always has the same modal character as the weaker of the premises, thereby they rejected all mixed modal syllogisms.

In medieval logic, a distinction was made between $\textit{de dicto}$ and $\textit{de re}$ modalities. It was demonstrated that propositions with $\textit{de dicto}$ and $\textit{de re}$ modalities have different deductive characteristics. Aristotle's apodeictic syllogistic contains both: reasonings valid only under $\textit{de dicto}$-interpretation of modalities (e.g. the law of $i^\square$-conversion) and reasonings valid only under$\textit{de re}$-interpetation (e.g. modus $Ba^\square rbara^\square$). When we accept the “principle of the weakest premise”, apodeictic syllogistic can be naturally interpreted as containing \textit{de dicto} modalities.

The eminent Polish logician Jan _ukasiewicz suggested that both modal syllogistic versions were incorrect. In his opinion all mixed modi formed from the valid categorical syllogisms (e.g. $Barba^\square ra^\square$ rejected by Aristotle) are also valid. _ukasiewicz justified these modi by means of his positive assertoric syllogistic and four-valued modal logic, which contains some theorems unprovable in normal modal calculi.

We set out two translations of apodeictic and assertoric propositions into the modal first-order logic with equality (G.E. Mints' modal system $\textbf{T}$): the first provides the validity of all the laws of Aristotle's apodeictic syllogistic, the second one preserves the validity of all apodeictic syllogisms accepted by _ukasiewicz. So, the apparatus of modern quantified modal logic can be used for the “rehabilitation” of apodeitic fragments of Aristotle's syllogistic as well as _ukasiewicz' syllogistic. DOI: 10.21146/2074-1472-2018-24-2-108-115