Why doesn't $\pm a = \pm b \implies -a = b$? The problem arises from $a^2=b^2$. Since each variables could be positive or negative it feels like all permutations should be valid equalities?
 A: I think the problem lies in looking at $\pm a$ as some sort of object. $\pm$ is just a shorthand used. In reality, when you solve $a^2=b^2$, you say that $a$ and $b$ must have the same value squared, and that gives you your four solution pairs. There is no fundamental reason why a $\pm$ sign is used in equations like this. You just use it if the actual reasoning behind solving the equation permits it.
A: $$\pm a=\pm b\iff +a=+b\lor-a=+b\lor+a=-b\lor-a=-b$$
which can be simplified as
$$\pm a=\pm b\iff a=b\lor-a=b$$
(where $\lor$ means "or")
A: Simple answer comes from the number line. Plainly, A does not equal -A ( they are opposite values). Yet, squaring real numbers always results in a positive number. Therefore while $(-a)^2=(+a)^2$ the roots may be either equal OR opposite. This prevents your paradoxical conclusion. Thus  the correct conclusion must be -a=-b and +a=+b.
A: The point of the $\pm$ symbol is that it takes the same value ($+$ or $-$) anywhere it is used in one expression. Conversely, the $\mp$ sign is used for values that are $+$ when $\pm$ is $-$ and vice versa. So, for example,
$ \pm a \pm b $ can be either $a+b$ or $-a-b$, and
$ \pm a \mp b $ means either $-a+b$ or $a-b$.
(Note: if you only need one of them, it is conventional to use $\pm$.)
If you want to characterise the solutions to $a^2=b^2$, you can say $a=\pm b$, $a=\mp b$, or $\lvert a \rvert = \lvert b \rvert$.
Edit: the other thing to point out is that $a=-b \iff -a=b$, I suppose, so you cover all options with "$a=b$ or $a=-b$".
