In Terence Tao's Analysis I, he says that "the statement '$x=3$ if and only if $x^2=9$'" is false. However, isn't it more accurate to say that this statement is not always true? Isn't this statement true sometimes (i.e. when $x\neq-3)$ and false other times (i.e. when $x=-3$). Perhaps the $\forall x$ quanitifer had been left implicit, and what he really means is that the following statement is false:
$$
\forall x:x=3 \leftrightarrow x^2=9 \, .
$$
However, at this point in the book he hadn't introduced quantifiers, and so I'm unsure what is intended.
To add to my confusion, in this post, it states that the meaning of "If $A$, then $B$" is "whenever $A$ is true, $B$ is true". According to this definition, the statement "$x=3$ if and only if $x^2=9$" is plainly false. However, the negation of $A \leftrightarrow B$ is $A\leftrightarrow \neg B$. So the negation of "$x=3$ if and only if $x^2=9$" is "$x=3$ if and only if $x^2\neq9$", which is also false according to this definition in the linked post. A statement and its negation can't both be false, so I'm unsure what's gone wrong.
Best Answer
Indeed, in the statement "$x = 3$ if and only if $x^2 = 9$", the universal quantifier ("for all $x$ it is true that...") has been left implicit. As a logical statement, $x = 3 \leftrightarrow x^2 = 9$ does not have a truth value, as $x$ is still a free variable.
This explains the problem with negation: although the negation of $A \leftrightarrow B$ is $A \leftrightarrow \neg B$, the negation of $\forall x (A(x) \leftrightarrow B(x))$ is $\exists x (A(x) \leftrightarrow \neg B(x))$. And indeed there is an $x$ such that $x = 3$ is false but $x^2 = 9$ is true.