Quantification of Modal Propositions in Traditional Logic

Abstract

Abstract: Modal systems did not emerge until recently with Lewis Clarence, who raised the issue of strict implication. Although modal systems have evolved and diversified significantly since then, they remain loyal to his frameworks, as they are considered foundational in modal logic. In 1940, the American logician Ruth Barcan Marcus (1921-2012) attempted to introduce the theory of quantification to modal logic to expand it by adding quantifiers to all fields of contemporary modal logic. Since most historians of contemporary logic attribute the beginning of the quantification of modal logic to her, this might suggest that traditional modal propositions, whether in Aristotle's or Islamic logicians' works, were not quantified. Therefore, in this research, we addressed traditional modal propositions and explained how Aristotle and Muslim logicians used quantifiers in modal propositions. We also highlighted the role these quantifiers play, particularly in inferential structures, whether direct or indirect inferences, and identified some differences between them and the inferences related to categorical propositions. Keywords: Proposition; Quantifier; Modality; Necessity; Possible.