Gonzalo Llach

Keynote presentation delivered at the 9th World Congress on the Square of Opposition (Square 9), Université du Québec à Montréal, June 2026. This presentation introduces the Hexagon of Indeterminate Propositions and the Logic of Determination. It shows that the traditional Square of Opposition emerges as a theorem of the system at the third layer of opposition (O3). Do not hesitate to contact me if you want me to deliver this talk at your university online. Just write me to the email at the end of the presentation.

This article is dedicated to constructing the Hexagon of Indeterminate Propositions and providing its theoretical foundation in Aristotle’s work and the commentary tradition. It is both a logical and philosophical inquiry, as the centuries-long discussion about indeterminate propositions is deeply rooted in the intellectual heritage of Platonism. The discussion emerges from a philosophical motif whose resolution aims to offer—hopefully—a fresh perspective for engaging with Aristotle’s works, while also contributing to a better understanding of the coherence of his logical writings. Surprisingly, modern scholars have not paid as much attention to this topic as ancient and medieval commentators did. The central question is how and why indeterminate propositions relate to the traditional propositions that form the Square of Opposition (A-E-I-O), which Aristotle refers to as determinate propositions.

The hypothesis of a theoretical unity between On Sophistical Refutations and Prior Analytics presents a major challenge to scholars attempting to unify the criteria of analysis. This paper examines this problem and proposes a middle ground between the perspectives of Woods and Boger to address this crucial question: If a unitary and coherent theory of deduction exists, why does not the technical apparatus of syllogistic modes for analyzing fallacies appear in SE? This paper makes useful contributions to the discussion on the internal unity of Aristotle's logical writings on around categorical syllogistics and the doctrine of sophistical refutations.