I can not doubt that I exist. I can doubt that any material thing exists. Therefore, I am not a material thing
This is a redux of Descartes's modal argument for dualism (e.g. in second and sixth meditations). Arnauld came up with a slick counter: I can not doubt that triangle inscribed in semicircle is right. I can doubt that Pythagorean triangle is right. Therefore, triangle inscribed in semicircle is not Pythagorean. A triangle is Pythagorean if the square on one side is equal to the sum of squares on the other two.
The statements after modalities are theorems of Euclidean geometry, Thales's and converse Pythagorean, the conclusion is necessarily false. At first glance Arnauld's substitution discredits Descartes's argument completely. But... Arnauld replaces existence with rightness. Existence, Kant tells us, is not a predicate, but rightness most certainly is. I am not sure if this makes a difference though. Next, Arnauld slaps double modality on (what is normally considered to be) necessary truths, while Descartes has it over contingents. By the way, Descartes's first premise surely doesn't mean that he can not imagine a world without himself, the double modality here is subtle.
I can doubt that Arnauld's substitution is legit. I can not doubt that Descartes gets his dualist conclusion too easily. Still, is there a reconstruction (in modal logic?) that makes Arnauld's argument invalid and Descartes's valid (if unsound)? Or at least makes them invalid for two different reasons? Are my doubts about Arnauld's substitution misplaced?