A lot rides in the meaning of ‘to exist’.
Welcome to philosophy! 
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Interesting point. However, we say things like, “I will think about it.” or “I am not going to think about it at all.” which seems to suggest we have control over our thinking.
Doubting is a form of thinking. When one doubts about something, it doesn’t just stop there. It goes on to asking what caused the doubt, and also what could be the real case which is being doubted.
The reason that Descartes was able to think was he was existing before thinking. Hence “I think therefore I am” sounds like he was ignoring or forgetting the fact that he was existing before thinking.
There are cases for reasonable doubt where it is obvious that there is sufficient ground for one to doubt.
But there are also cases where there is no sufficient ground for doubt, which is to be seen as unreasonable doubt.
What was the ground for Descartes’ doubt for even his own existence? Was it a reasonable doubt? Or was it an unreasonable doubt?
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That’s true. 
In your threads on mathematics, you talk of the existence of numbers. I’m not sure whether Cartesian or Gotaman exitence is that controversial.
Buddhism is often mistaken for nihilism, due to the anatta doctrine. Descartes’ cogito is not that different from Christian/Kaccayana’s eternalism.
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I was listening to some Buddhists who were mulling over the Western myth which says that Buddhists deny the existence of the soul. Their conclusion was that it derives from Westerners who were hostile to Christianity and wanted a way to fully contradict a central Christian doctrine. They were also amused that, on occasion, the Western reinterpretation finds its way back into parts of the East. 
Something that functions as a self exists.
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When I studied Comparative Religion, we were told about the scholarly belief that the story of the life of Jesus was a direct influence on the formation of Mahāyāna Buddhism, and the idea that the bodhisattva would seek the enlightenment of the world before their own release. I don’t know if Buddhists would agree, but it seemed plausible to me.
That’s an interesting thesis. The break between Theravada and Mahayana always seemed a bit abrupt to me, but I have no idea whether it is true.
It developed over hundreds of years, with many factors in play. One of the major ones was the prohibition on monks carrying money, as it made it almost impossible for the travel required to propagate the teaching. Another was the need to accomodate lay disciples. There’s a famous Mahāyāna Sutra called the Vimalakirti, the chief character in which is a wealth silk merchant, who manages to best the Buddha’s most advanced disciples by virtue of his insight into śūnyatā.
It’s all quite well documented in e.g. Edward Conze - Buddhism its Essence and Development.
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There’s no doubt that Buddhism preaches anatta; it’s an incontrovertible fact. It’s simply a part of dependent origination.
It’s also a fact that Descartes’ thinking is solipsism; initially at least.
I’m only going by what is stated in Buddhism and in The Meditations.
What is the ground for no self in Buddhism? Why is it making sense to claim no self? Or is it just a religious claim?
Anatman is a dependent origin thing. I’ve already outlined the whole business in the OP. Yes, it is a Buddhist claim.
On the other hand we have Descartes who affirms his own existence as an I = soul = ego.
Your body is your self. You are now sitting in somewhere in the world perceiving and living. You see your self. You can point to your own self, and also to the world external to you. How can you fail to see or recognise your own self? How can one deny being oneself?
Chair is a device made with any material be it wood, metal or plastic to sit on. Chair has the shape which we can tell it is a chair, and surface for you to see on. How can you mistake a chair with not chair?
Conforms with ancient Judaic religion; they didn’t believe in a soul although they did make a big deal of breath (it seems they were ignorant about air). Modern Christianity however does have a doctrine on immaterial, imperishable souls. This doxa was already mature when Descartes penned his meditations. I haven’t a clue about Descartes’ religious beliefs though - his Deus Deceptor doesn’t shed light on the matter at all. It’s odd that Descartes should’ve doubted himself as his body. There are many people who don’t go beyond such an understanding of the self, but some do.
A chair is a chair. It’s just that it’s a composite thing - wood, nails, etc. It lacks inherent existence. The self is like that, allegedly.
This view in Buddhism seems to indicate that we lack a self. Hence anatta. It’s part of the 3 lingas of being - anicca, anatta, dukkha.
Ajita Kesakambali said the same thing: annihilationism (no soul, death is finito, everything is physical, life is for fun 
).
Pakudah Kaccayana swung to the opposite side, with his eternal soul.
I don’t know what Sanjaya Belathiputta would’ve said though.
P.S. I’ve made references to 3 of the 6 heretical teachers.
The Cogito was a result from the methodical doubt, they say. Descartes doubted everything, even his own existence. But he found it, by knowing the fact that he cannot doubt the fact he thinks, and doubts.
If he didn’t exist he couldn’t possibly doubt. But it was crystal clear that he was doubting, which means he was also existing.
However, Buddhist’s claim for no self still sounds unclear on their ground for the claim. There is no argument on the claim. Just saying no self doesn’t make much sense.
I believe I already spoke about this here and here
As for the anatta doctrine of Buddhism, it’s 20/20 clear that (inter)dependent origination has something to do with it. I can’t quite connect the dots though.
As I was saying, 3 of the 6 heretical teachers had pertinent opinions on the issue. Just as a side note, I find it incredible that a rich nastika tradition flourished in ancient India.
You need to explain your logic in plain English line by line.
The symbolic logic you laid out is unclear what it is trying to say.
And there is suddenly “y” in the logic, which is unclear what it stands for.
y is a variable. It’s not sudden and if it is that’s irrelevant to the logic. If something1 is something then there’s something2 that’s something1.
It’s saying that thinking things exist and dependently originated things don’t exist.
Variable is meaningful only if it contains a value of something.
Just placing an unknown variable in logical argument is meaningless with no indication of what type of data or content it could be, unless it is a math theorem which is to contain numbers.
It makes the point in the argument more unclear to decipher.
What are the premises, findings, assertions and conclusion in the argument?
What do you mean by thinking things? Are they any other existence apart from humans which thinks?
What is dependently originated things? Could you give some examples with explanation on what it means?