Cogito and logic

peace,
I’m currently studying the logic of Rene Descartes, and our teacher argued in class that, among other things, Descartes also undermines logic and that the cogito argument is independent of logic.
This reminded me of your claim about the inability to assert valid things without logic and raised a question for me.
As I understand it, classical logic can be challenged in four ways –

1. To challenge the fact that truth and falsehood are the only truth values ​​for any claim (I will not address this challenge)
2. Challenge how logical relationships work (e.g., argue that both false and true are true)
3. Challenge the way to determine the validity of an argument (and propose a different way than when the premises are true then the conclusion is necessarily true).
4. To argue that there is no such thing as logical connections at all and a way to infer validity (and I would like to suggest that perhaps in this way it would be possible to claim meaningful things about the world).

Regarding those who challenge the way logical relations work and the way to determine validity, I understand your claim about the lack of value in their arguments. For example, a person who claims that P and neither P is true, claims everything and claims nothing. But this person does not challenge the existence of logic, but only what the truth table of the logical relation of negation or of both looks like (and perhaps the truth tables of other logical relations).
What about a person who does not believe at all in the existence of logical connections and the validity of arguments, and all his claims will be independent intuitions/observations that are not related to other claims?
Such a person, for example, will claim that the truth value of – the sun is green – is false. And if you ask him – then, then the truth value of the sun is not green is also true? He will ask you this and also? And this and not? Because he does not use logical connections at all. He does not claim P and not P, this person’s argument is admittedly more limited, but there is no contradiction in his case and his argument is not nonsense, he simply does not use logic at all (does not change it, but does not use it).
If so, it would be possible to argue that Descartes’ argument in his Logic – I (the thinker) exist is truly illogical. It is simply an intuition/observation that I (the thinker) exist. That is, when you completely give up on logic (and do not change it), Is it possible to claim things about the world without them being nonsense?
I hope I was clear,
And thanks a lot in advance 🙂