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Kindly comments on Proofs, Implications, and Models - Less Wrong
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What? For any function f, if x=y, then f(x) = f(y). Squaring is a function. Do you mean something else by truth-preserving?
Squaring can introduce truth into a falsehood. For example, if we write -5 = 5, that's false, but we square both sides and get 25=25, and that's true. Furthermore, squaring doesn't preserve the truth of an inequality: -5 < 3, but 25 > 9.
Ah- if you don't define the (principle) square root to be the inverse of squaring, the apparent contradiction goes away.
I concluded that you wanted to be able to preserve falsehood as well. Squaring preserves falsehood in the domain of the nonnegative reals, exactly like multiplication and division by positive values does.