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Suppose one were to prove that $\pi$ is irrational by claiming "because it is transcendental". This proof would be seen as circular. However, neither of "$\pi$ is irrational" and "$\pi$ is transcendental" is strictly stronger than the other, in fact, both are equally strong. So, what exactly is the formal definition of a circular proof?

user107952
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    I'd expect any proof involving $\pi$ to be circular. But what do you mean by saying "$\pi$ is irrational" and "$\pi$ is transcendental" are equally strong? In the sense that all true statements are equally strong, sure, but clearly "$x$ is transcendental" implies, and is not implied by, "$x$ is irrational". – Gerry Myerson Feb 13 '21 at 02:20
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    If one has a proof that $\pi$ is transcendental that does not rely on knowing that $\pi$ is irrational, and if has shown that every transcendental number is irrational, then there is nothing in the least circular about the proposed argument, never mind that $x$ is transcendental is strictly stronger than $x$ is irrational. – Brian M. Scott Feb 13 '21 at 02:30
  • Yes, "x is transcendental" is strictly stronger than "x is irrational". But "$\pi$ is transcendental" is equally as strong as "$\pi$ is irrational" – user107952 Feb 13 '21 at 18:31
  • Because $\pi$ is a specific constant. – user107952 Feb 13 '21 at 18:32
  • Are you familiar with the constant $\gamma$? It is a specific constant. Would you say "$\gamma$ is transcendental" is equally as strong as "$\gamma$ is irrational"? – Gerry Myerson Feb 14 '21 at 11:29
  • @GerryMyerson That depends on whether it is transcendental or not. – user107952 Feb 14 '21 at 15:54

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