Let me begin by saying that the answer to part of this question has been previously addressed HERE: Prove/Disprove: if $x^2 = a^2$, then $x = a$
However, I am hoping to add something to this inquiry that I think is deserving of a new post all together.
The first part of this question is to determine whether the following statement is true or false for all groups with property of $\ \ x^2=a^2$:
$x^2=a^2 \rightarrow x=a$
Starting with what is assumed:
$x\circ x = a \circ a \ \ \ \ \ \ \ $Then, performing some manipulation...
$ x\circ x \circ x^{-1} = a \circ a \circ x^{-1} $
$x \circ e = a \circ a \circ x^{-1}$
$x = a \circ a \circ x^{-1}$
Now, the only way to show that $x = a$ would be if $a \circ x^{-1} = e$
This would mean that $ a = (x^{-1})^{-1}$ but $(x^{-1})^{-1}$ is just another way of writing $x$.
As such, we need $a = x$ in order to prove $a=x$
Said differently, proving the claim requires the assertion of the claim.
I know a little bit of formal logic to understand that this cannot be..."correct".
Is there a particular name for this type of logical fallacy? And, more importantly, if during proof solving (for any given proof) I encounter this situation, can I immediately conclude that the statement is always false no matter how I choose to tackle it?
Said differently, I know that proofs can be attacked using several different methods (contrapositive, contradiction, induction, etc). If I choose any single proof method and the following phenomenon arises, (i.e. proving the claim requires the assertion of the claim), do I know for certain that there will be NO WAY of ever finding a proof for the claim regardless of if I try a different method?