Conceivability, possibility and essence: Arnauld's revenge

Abstract In this paper, I examine two connected strands of the 4th Set of Objections and Replies to Descartes' Meditations on First Philosophy . On the one hand, I look at Arnauld's defence of the view that real distinction proofs require adequate knowledge and Descartes' view that they only require complete understanding. On the other hand, I engage Arnauld's famous right‐angled triangle, T , and Pythagorean property, P , counterexample to Descartes that vivid and clear thought of separability entails knowledge of separability by God. Following a comment in Almog, I extend out the exchange over the counterexample by considering non‐Euclidean geometries. I argue that while at first it seems that Descartes has a plausible response to Arnauld's counterexample by considering T in a non‐Euclidean space where P does not hold. However, Arnauld actually has a powerful response when we consider what is the essence of T across geometries, Euclidean or otherwise.