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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?

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    The fact that all of "Pythagorean triangle is right" angled is undoubtable! – Sancho Panza Nov 1 '15 at 1:23
  • I would agree with Sancho Panza. What about deriving the right angleness of a Pythagorean triangle from the law of cosines? Whether one doubts a mathematical statement or not, depends on how much one knows about the domain in question. – Jo Wehler Nov 1 '15 at 2:15
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    I can also doubt that I doubt any material thing exists, therefore I'm a material thing. Or I can also doubt about the doubt itself... – sure Nov 1 '15 at 15:05
  • @Sancho Panza Even if that were the case truth of premises has no bearing on validity of arguments. There were several centuries between Thales and Euclid (who is believed to have proved the converse Pythagorean theorem), so plenty of time to doubt. Not to mention that both rely on the parallel postulate, which also proved to be doubtable. – Conifold Nov 1 '15 at 18:18
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It seems to me that Arnauld's "syllogism" does not invalidate Descartes' argument.

Descartes model of knowledge is based on two pillars : intution and deduction.

The knowledge must start from the intuition of "first principles", like the axioms of geometry : form them we deduce theorems that are true because they are "derived" from true principles by way of "elementary" inference steps that are so simple that we perceive them clearly and distinctly, and thus valid, i.e. indubitable.

Arnauld seems to agree on the fact that :

"if we clearly and distinctly perceive a conceptual distinction, by "truthworthiness" of God, we can conclude with the reality of that distinction".

Now for Arnauld's counterexample :

  • we clearly and distictly perceive [clare et distincte percipio] that a triangle is rectangle;

  • we doubt about Pythagorean theorem : we do not have an intuition about its truth, but we have only a proof of it; thus, whitout a proof that transfer certainty from (geometrical) first principles by way of elementary clear and distinct inference steps, we cannot have certainty about it;

  • thus, we have to conclude that we can conceive of a triangle being rectangle without "being forced" to conceive it as satisfying also P's theorem.

Conclusion :

"we can clearly and distictly perceive a conceptual distinction, without being licensed to conclude that it "reflects" a real distinction.


It seems to me that, as pointed at by Conifold, we have here a difficulty with "epistemic modalities".

If the principle Arnauld agrees on is that :

"if we clearly and distinctly perceive a conceptual distinction, then we cannot doubt about it; and if we cannot doubt it, by "truthworthiness" of God, the distinction is grounded in a real distinction",

we are not entitled to "invert" it [the Inverse of p → q is ¬ p → ¬ q], i.e. to conclude that :

if we can doubt it, then it is not a real distinction.

What we can say is that :

"if we doubt it, then we do not perceive it clearly and distinctly";

thus, from the fact that we can doubt about being a rectangular triangle must implies the fact that it have to satisfy P's theorem, we have to conclude that we have no intuition about the truth of P's theorem.

Only after having proved it from (geometric) first principles we attain the certainty of its truth.

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    But doesn't this mean that Arnauld succeeds? If doubting means that we do not perceive something clearly and distinctly then from doubting that any material thing exists Descartes only gets that he has no intuition of its existence. Arnauld's example of two necessary truths, one intuited clearly and distinctly, the other not, would then be exactly what is needed to invalidate him. Could it be that Arnauld reads Descartes as using epistemic modalities while Descartes has something more de re in mind? – Conifold Nov 1 '15 at 20:33
  • @Conifold - you are right : "from doubting that any material thing exists Descartes only gets that he has no intuition of its existence"; D needs the existence of God and its "truthworthiness" in order to deduce the existence of the external world. – Mauro ALLEGRANZA Nov 2 '15 at 9:18

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