What is the meaning of "$\pm$"? Can we write $1=\pm1$? or $x^2=4\iff \pm x=\pm 2$? Does "$\pm$" mean "positive or negative"?
Is it allowed to write: $1 = \pm1\;$?
Is it allowed to write: $X^2 = 4 \iff \pm X = \pm2\;$?
 A: Beside the other answer(s), care must be taken when used more than once in the same expression/equation or used along with “$\mp$”; usually that means that the same sign is understood throughout/respectively or that the signs are supposed to be arbitrary irrespective of multiplicity of appearance. The context will make it clear, or a diligent author will make this explicit prior.
So, on the one hand, for instance
$$(a^2-b^2)(c^2-d^2)=(ac\pm bd)^2-(ad\pm bc)^2$$
means either\both $$(a^2-b^2)(c^2-d^2)=(ac+bd)^2-(ad+bc)^2$$ or\and $$(a^2-b^2)(c^2-d^2)=(ac-bd)^2-(ad- bc)^2$$ is\are correct; whereas
$$(a^2+b^2)(c^2+d^2)=(ac\pm bd)^2+(ad\mp bc)^2$$ means  either\both $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$$ or\and $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$$ is\are correct.
However, on the other hand, the inequality
$$0\ne\pm 1\pm1\pm1$$
will usually mean any combination of the signs regardless of its double appearance is meant——that is,
$0\ne \pm 3$ and $0\ne\pm 1$.
A: In order to answer this question we look at first at examples where the symbols $\pm$ and $\mp$ are sometimes used.

Quadratic equation:
\begin{align*}
x^2+px+q=0\qquad\qquad\qquad\qquad x_{1,2}=-\frac{p}{2}\color{blue}{\pm}\sqrt{\frac{p^2}{4}-q}\tag{1}
\end{align*}
Geometric series:
\begin{align*}
\frac{1}{1+x}=1-x+x^2-x^3\color{blue}{\pm}\cdots\tag{2}
\end{align*}
Trigonometric addition formula:
\begin{align*}
\cos\left(\alpha\color{blue}{\pm}\beta\right)=\cos\left(\alpha\right)\cos\left(\beta\right)\color{blue}{\mp}\sin\left(\alpha\right)\sin\left(\beta\right)\tag{3}
\end{align*}

Comments:

*

*In the first example (1) the symbol $\pm$ is used to denote the two roots $x_1$ and $x_2$ of the quadratic equation at the left hand side of (1). This is just a compact notation for
\begin{align*}
x_1&=\frac{p}{2}\color{blue}{+}\sqrt{\frac{p^2}{4}-q}\\
x_2&=\frac{p}{2}\color{blue}{-}\sqrt{\frac{p^2}{4}-q}\\
\end{align*}


*In the second example (2) the symbol $\pm$ is used to indicate how the terms represented by the three dots are added resp. subtracted, namely starting with $+$ followed by $-$, then followed by $+$, etc. Here we do not have any choice to select either $+$ or $-$. Here the order is relevant and we have to continue the series as described. We could also write
\begin{align*}
\frac{1}{1+x}=1-x+x^2-x^3+x^4\color{blue}{\mp}\cdots
\end{align*}
which indicates the next term $x^5$ is to subtract, the following term $x^6$ is to add, and so on.


*The third example (3) gives a compact notation for two different identities, namely
\begin{align*}
\cos\left(\alpha\color{blue}{+}\beta\right)&=\cos\left(\alpha\right)\cos\left(\beta\right)\color{blue}{-}\sin\left(\alpha\right)\sin\left(\beta\right)\\
\cos\left(\alpha\color{blue}{-}\beta\right)&=\cos\left(\alpha\right)\cos\left(\beta\right)\color{blue}{+}\sin\left(\alpha\right)\sin\left(\beta\right)
\end{align*}
Note, that the usage $\pm$ at the left-hand side of (3) and $\mp$ at the right-hand side of (3) is crucial. Here we do not use $\pm$ to provide alternatives, but instead to provide two different identities in compact notation.
Conclusion: When looking at (1) - (3) what can we say about the usage of $\pm$?

*

*The symbols $\pm$ (and similarly $\mp$) have different meanings depending on the context. Sometimes it means two valid alternatives as in (1), sometimes it means one and only one extension as in (2) and sometimes it is just as compact notation for two different things as in (3).


*The examples also follow some useful mathemactical writing guidelines:

*

*readability: They provide information which is easy to read, easy to grasp.


*conciseness: They provide useful information in a concise manner

With the examples above in mind let's consider the question:
Is it allowed to write: $1 = \pm1$?
Although this might be syntactically allowed, it is clearly a misusage of $\pm$ and should not be used. The right-hand side provides no additional information than already given by the left-hand side. Worse it has the drawback of being less concise than the left-hand side.
Is it allowed to write: $X^2 = 4 \Leftrightarrow \pm X = \pm2$?
To indicate the two solutions of $X^2=4$ it is recommended to write
\begin{align*}
X_{1,2}=\pm 2
\end{align*}
as it is stated in (1). A prefixing of $X$ with $\pm$ here is also some kind of misuse. It does not provide any more information and is rather confusing, since the reader has to think about how plus and minus at both sides of $\pm X=\pm2$ are coupled in order to give a valid statement. This clearly violates readability and conciseness.
A: $±$ means both the positive and the negative value. For example, $±1 = -1, 1$.
$X^2 = 4 \iff ±X = ±2$ is technically correct, but it is redundant, as $+X = +2$ is the same as $-X = -2$, and similarly $+X = -2 \iff -X = +2$. It is usually written as just $X = ±2$.
To remove ambiguity, it is perfectly fine to write $x \in \{-2, 2\}$.
A: In logic language, $1 = \pm 1$ is understood as "1 is equal to 1 or 1 is equal to -1". So it is true.
To answer your question on $x^{2} = 4$, first we get
\begin{equation}
\forall x \in \mathbb{C},\left(x = 2 \longrightarrow x^{2} = 4\right) \wedge \left(x = -2 \longrightarrow x^{2} = 4\right)
\end{equation}
Using logic rules, it leads to
\begin{equation}
\forall x \in \mathbb{C}, \left(x = 2 \vee x = -2\right) \longrightarrow x^{2} = 4
\end{equation}
At the same time, we have
\begin{equation}
\forall x \in \mathbb{C}, \textrm{there exist at most two distinct solutions for $ax^{2} + bx + c = 0$}.
\end{equation}
As a result,
\begin{equation}
\forall x \in \mathbb{C}, x^{2} = 4 \longleftrightarrow \left(x = 2 \vee x = -2\right),
\end{equation}
or, in brevity, we get $x = \pm 2$.
PS. who can help me write "there exist at most two distinct solutions for $ax^{2} + bx + c = 0$" in logic language?
A: As suggested in comments (and the other answer), the expression $\pm$ generally refers to a set; in this way, it's very similar to another piece of notation, the so-called 'big O' notation, for representing a set of functions with a certain rate of growth. We often write $a=\pm b$ when talking about the set of possible solutions of an equation; but this is an equality among sets, not numbers. So, $1=\pm 1$ would generally not be considered accurate, whereas both $x^2=4\iff \pm x=\pm 2$ and $x^2=4\iff x=\pm 2$ would be correct (though the latter is less confusing). Also, it may be worth noting that multiple instances of $\pm$ 'multiply' combinatorially; $\pm a\pm b$ usually refers to the set $\{a+b, a-b, -a+b, -a-b\}$.
