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I was wondering lately whether Descartes argument for the existence of undoubtable truth could be formalized. I tried to formalize his argument in FOL, but only his light version proving that there does not exists relation doubt(x,x), x that doubts x, doubt that doubts itself, there exists some undoubtable truth. Here is my formalization -->

Cogito Ego sum formally

Is this a reliable formalization of it?

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  • One of the most "reasonable" reading of Cogito is that it is not an inference but an intuition. Commented Nov 26, 2021 at 8:29
  • The cartesian formulation is based on an axiom that cannot be proven. Conclusions of Godels incompleteness theorem are that whenever a system tries to make a statement about itself, it fails. Commented Nov 26, 2021 at 11:18
  • Logically, an undoubtable truth is just part of common truth, then it can be just as simple as T(i)->E(i). Descartes' problem is empirical, exceeds logic, and can't be proven, because such very existence sustains the structure of truth.
    – RodolfoAP
    Commented Nov 26, 2021 at 17:20
  • Cogito itself is easy to formalize: I think → I am (major premise); I think (minor premise) ⊨ I am (conclusion). What you are looking for, I am guessing, is formalizing Descartes's argument for the minor premise ("we cannot doubt of our existence while we doubt"). Something like: ¬X → I doubt X, doubting → thinking ⊨ I think. Here X = "I think" and you'll need modal logic to formalize that because doubting here is a propositional attitude (operator on propositions). See logic of beliefs for similar setup.
    – Conifold
    Commented Nov 29, 2021 at 7:28

2 Answers 2

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There is a fundamental flaw in "Cogito, ergo sum." To state that "I think" already assumes that the "I" exists, and therefore, already assumes the conclusion.

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    That's a misreading of it, & it's purpose. It could be framed 'thinking happens, therefore existence is happening in some sense'. It is that minimal a claim, but is then used as the basis for 'clear and distinct' perceptions to count as evidence.
    – CriglCragl
    Commented Nov 26, 2021 at 9:46
  • I agree that "Thinking happens, therefore existence is happening" avoids the contradiction that I mention. But Descartes wrote "Je pense, donc je suis" on one occasion and "Cogito, ergo sum" on another, so in both cases he used the first person singular. I can judge only what he wrote. Also, I said nothing about its purpose. Commented Nov 26, 2021 at 20:17
  • "already assumes the conclusion" You take it proving he exists is the purpose, as a conclusion.
    – CriglCragl
    Commented Nov 27, 2021 at 13:50
  • You are assuming facts not in evidence. I am not assuming anything, but basing my point on what Descartes actually wrote: the portion of "Je pense, donc je suis" that comes after "donc", and the portion of "Cogito, ergto sum" that comes after "ergo". What are you basing your point on? Commented Nov 28, 2021 at 16:59
  • The whole of The Meditations On First Philosophy. There is obviously substantial literature on this plato.stanford.edu/entries/descartes-epistemology/#CogiErgoSum As I see it the assertion is of a subjectivity, not a self, which comes later.
    – CriglCragl
    Commented Nov 28, 2021 at 17:45
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Long comment

One of the most "reasonable" reading of Cogito is that it is not an inference but an intuition.

Having said that, your formalization is not very clear to me...

Are the first four lines the premises of the argument? If so, where do you use 2nd premise?

Does doubt(x,x) means that x doubt about itself? The first x is a "subject" formulating a thought, the second one is an "object of thought".

How this relates to 4th premise: "if doubt(x,x), then not doubtful(x)"? If my reading is correct, it sounds wrong.

In conclusion, IMO 3rd and 4th ones seem wrong.

But the real issue is with 2nd premise:

"if doubtful(x), then there is some y such that doubt(y,x)".

But this IS the Cogito!

And thus the Cogito is the premise of the argument; in conclusion, the argument is circular.

Way out: the "argument" is not an argument at all but the "elucidation" of an ungrounded intuition: my "experience" of doubt is the evidence that there is a subject formulating that thought, and this subject is "myself".

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  • The four lines are the premises. I did not use the 2nd one, just let it be there so that I see what do I mean by sth being doubtable (definition). Like you say, by doubt(x,x) I meant that x is a doubt of x, or, x doubts x. Premise 4 is there to "create" a contradiction, so that I may come to conclusion that there does not exists any x such that doubt(x,x). The third premise, is the negative of what I am trying to prove.
    – Eauriel
    Commented Nov 26, 2021 at 9:27

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