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If I have proved a theorem $A$ on the basis of fact $B$, then, is it valid to prove $B$ by using the theorem $A$?

Frenzy Li
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CandidFlakes
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    Perhaps you may want to read this. – Roby5 Aug 04 '16 at 15:25
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    No, It is not. That amounts to saying that B is true because B is true. – Brian M. Scott Aug 04 '16 at 15:29
  • Term "fact" usually refers to some "true" statement. Such statement needs no proof (it is "already" true). If you did only prove that $B \implies A$ then: 1) not necessarily $B$ is true (counterexample: $x=y \implies |x|=|y|$, but $x=y$ doesn't hold for arbitrary $(x,y)$); 2) not necessarily $A \implies B$ is true (counterexample the same). – Abstraction Aug 04 '16 at 15:31

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That depends. If your goal is to show that the theorems are equivalent, then it is valid. If your goal is to show that the theorems are true, then it is not valid, but a case of circular reasoning.

As an example of how circular reasoning doesn't work to establish truth, consider the following:

Theorem A: $1=0$.

Theorem B: $2=0$.

I now prove A from B: Given $2=0$, I can multiply both sides of the equation with $\frac12$, and obtain $1=0$ $\square$

I also prove B from A: Given $1=0$, I can multiply both sides of the equation with $2$, and obtain $2=0$. $\square$

So if that sort of reasoning would establish truth, I would now have proven that $1=0$ and $2=0$.

However, what this reasoning did establish is that both claims are equivalent, so should I somehow manage to prove any of them, the other will also automatically be proved. And should I manage to disprove any of them, the other will also automatically be disproved.

celtschk
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