What is the logical flaw in this reasoning? Abusing $T \equiv T \vee F$. This is not a "Haha gotchu mathematicians!" question. I'm seriously trying to learn. Please don't take my question to be insincere.
IMPORTANT EDIT: I've noticed that people are focussing too much on my examples that they're distracted by my real question which is logic based. I use various examples to explain my "logic based" confusion, I'm not confused about the examples. I do not require explanations for my examples.

This question stems from a question I asked long long long time ago and someone answered that it is incorrect to write $|x|=±x$ and I took their word for it because well... I wasn't experienced enough to ask the right questions.
Since then my instinct to deal with $|x|$ has been to use: $$ |x|= \begin{cases} x, \ x≥0 \\\\ -x, \ x <0 \end{cases}$$
because well... that is the definition of $|x|$.
However I was going through my questions yesterday, when I realised, "Wait hold on, why is the equation $|x|=±x$ incorrect? Because "$±$" means "plus OR minus", you're not insisting that $|x|$ is $x$ AND $-x$. You're only saying it is either $x$ OR $-x$.

But hold on, there's more. With that argument in mind, you can always write $\sqrt{9}=±3$ even though it's just $3$. You can even go more bonkers with this logic by writing $$\sqrt{9} = 3 \text{ or } -3 \text{ or } -193e^2$$ as long as one of them is true. You get the point, right?
You can keep adding on nonsense using the fact that $T \equiv T \vee F$ like so:
$$( \sin x = 0) \equiv (\sin x =0 \text{ or }  \cos x = 0) $$ and then get $x \in \{\frac{nπ}{2} : n \in \mathbb{Z}\}$ as the solution which is absurd.

Question: Where is the logical error here?
 A: *

*$$f(x)=|x|\tag1$$ can be translated as $$\Big(x<0\implies f(x)=-x\Big)\quad\text{and}\quad \Big(x\geq 0\implies f(x)=x\Big),$$
whereas
$$f(x)=\pm x\tag2$$ can be translated as $$ x\in\mathbb R\implies\Big(f(x)=-x\quad\text{or}\quad f(x)=x\Big).$$ Statement $(1)$ fully describes the function (in which each input has a certain output) $f,$ whereas statement $(2)$ conveys only partial information about it.


*\begin{align} &\forall x{\in}\mathbb R&\bigg(\,f(x)=|x|&\implies f(x)=\pm x\,\bigg)\\ &\forall x{\in}\mathbb R{\setminus}\{0\}&\bigg(\,f(x)=\pm x&\;\kern.6em\not\kern-.6em\implies f(x)=|x|\,\bigg)\\ &\forall (x,y){\in}\mathbb R^2 &\bigg(|x|=|y|&\iff x=\pm y\bigg) \end{align}


*Therefore, \begin{align}|x| &:= \begin{cases}-x &\text{ if }x<0; \\x &\text{ if }x\geq0\tag A\end{cases}
\\\\ |x| &:\not=\;\pm x
\\\\ |x|&=\;\pm x\tag B
\\\\ \pm x&=\;|x|.\tag C\end{align}
Definition $(\text{A})$ fully specifies $|x|,$ while statement $(\text{B})$ means $$|x|=-x\quad\text{or}\quad |x|=x.$$ So, $\pm x$ is less informative than $|x|.$ Thus, $$|x|=\pm x$$ is not an identity!
Notice that, unlike statement $(\text{B}),$ statement $(\text{C})$ never gets conflated with definition $(\text{A}).$ This is because the statement feel less definitive when the properties/possibilities of the subject $|x|$ is displayed before the subject itself.


*Compare:

*

*$$\begin{align}&\lvert2x\rvert=x-1\\\iff&\bigg(x<0 \;\text{ and }-2x=x-1\bigg) \:\text{ or }\: \bigg(x\geq0 \;\text{ and }\; 2x=x-1\bigg)\\\iff&\bigg(x<0 \;\text{ and }\; x=\frac13\bigg) \:\text{ or }\: \bigg(x\geq0 \;\text{ and }\; x=-1\bigg)\\\iff& x\in\emptyset\end{align}$$

*$$\begin{align}&\lvert2x\rvert=x-1\\\iff&\pm2x=x-1 \;\text{ and }\; x-1\geq0\\\iff& x\in\emptyset\end{align}$$

*$$\begin{align}&\color{red}{\lvert2x\rvert}=x-1\\\color{red}{\implies}&\color{red}{\pm2x}=x-1\\\iff& x\in\left\{-1,\frac13\right\}.\end{align}$$ (Here, both solutions being extraneous is due to the equation being inconsistent.)



A: I will focus on your problem with $T\iff T\lor F$.
Sure, under assumption that some formula $A(x)$ depending on a parameter $x$ is true (independent of the value for $x$), you may use that to deduce that $A(x)\iff A(x)\lor B(x)$ (since $T\iff T$).
However, in your examples, the formula $A(x)$ only holds true for some special values of $x$. It could thus happen that for some values of $x$, $A(x)$ fails to hold, while $B(x)$ is true. In this case, you can clearly not write $A(x)\iff A(x)\lor B(x)$ (since $F$ is not equivalent to $T$).
A: The term $\pm$ (or sometimes $\mp$), like any notation, carries a lot of meaning and connotations, because we see certain notations in certain places. To use $\pm$ is to say that the choice of either plus or minus should make sense, and perhaps depending on further context one is preferred over the other. To say $|x|$ is to unambiguously say that the quantity is positive and equal in absolute value to $x$.
$|x|\neq\pm x$ because the left hand side is a uniquely determined positive value, whereas $\pm$ is an ambiguous statement suggesting both states of plus or minus are valid unless further context to the question says otherwise. If you are doing algebraic manipulation, and you write $\pm x$ instead of $|x|$, you will find yourself in a nightmare of superimposed states, where you must deal with both cases of plus and minus and as such it is an inferior notation to $|x|$. There is neither reason nor motivation to write $|x|$ as $\pm x$, and $|x|$ is always only one value, is always unique; $\pm x$ is not.
You say one can go bonkers with this logic, saying $\sqrt{9}=3\vee -193e^2$. There is a good quote from somewhere, I don't remember exactly where, saying that a good notation frees the mind to focus on the problem at hand. $\sqrt{9}=3\vee -193e^2$ is not a good use of the $\vee$ notation, and is indeed bonkers as you say. I don't think it is so much a logical error but more a semantic error: logical conjunctions like "or" generally signal that either state is possible, and in any further working or proof we must account for all the states. You can chain "or"s and other conjunctions, and the point of doing this is to logically determine one or more solutions to whatever problem you're facing - introducing absurdities serves absolutely no purpose. It is not correct to write $3\vee-193e^2$ because to take mathematical notation to such a highly pedantic level is to undermine the purpose of notation in the first place, and it will trip you up to write like that if you ever write a proof or work on a harder problem, because littering the working with absurdities and ambiguous notation is not how we do maths.
A: The reason $\lvert x \rvert \neq \pm x$ is that the left hand side, $\lvert x \rvert$ is a function, therefore it produces only a single value for a given argument. On the other hand, $\pm x$ is a $set$ of possible values. Single-valued and multivalued maps are fundamentally different kinds of quantities, so equating them is meaningless.
Here's what I mean, graphically. This is a plot of $y= \lvert x \rvert$,

Whereas this is $y = \pm x$,

(Notice how 'or' translates to superimposition of the different possibilities).
A: The logical flaw in your thought, is that you thought that $T \equiv T \lor F$ implies that $\forall x [A(x) \equiv A(x) \lor B(x)]$; and this is incorrect. This flaw made you think that the set for solutions of $A(x)$ which is $\{x: A(x)\}$ is the set union of itself with $\{x: B(x)\}$ and of course caused the absurdity you referred to.
The correct implication is:
$(T\equiv T \lor F) \implies \forall x [A(x) \implies (A(x) \iff A(x) \lor B(x))]$
This shows that the set of solutions for $A(x)$ is $\{x:A(x)\}$ because any object substituting $x$ would only be a solution if it satisfies $A(x)$, now if it satisfies $B(x)$ but not satisfy $A(x)$ then we'll have $F \equiv F \lor T$ which is not true and so not a solution, and of course if it neither satisfy $A(x)$ nor $B(x)$ then it is not a solution. Of course if $x$ satisfies $A(x)$ then we get $T \equiv T \lor F$ always, so it is always a solution despite whether $B(x)$ is true or not.   So logically speaking we have the set of solutions equal to $$\{x: A(x)\} \cap \{x: B(x)\} \cup \{x:A(x)\}$$ which is $\{x:A(x)\}$ itself.
I think this is the answer to your logical question which is basically Zuy's answer.
However, your question is also interesting in another sense, that of proper symbolism of relations and functions. And although this was not your question, but its related to it in some sense. To answer that aspect I'd say that everything depends on how do we define the expressions we write. For instance $F(X)= \pm Y$ is a doubious expression, it's confusing, because generally the expression $F(X)=Y$ is reserved for when $F$ is a one place function symbol, so it assigns ONE value $Y$ to each argument $X$, while $\pm Y$ is usually taken to denote two distinct values that are $+Y$ and $-Y$, so this will supply the impression that the expression $F(X)= \pm Y$ means that $F$ is some relation symbol that sends $X$ to $+Y$ and also sends $X$ to $-Y$, i.e. a One-to-Many relation, but this is confusing because of the use of $=$ which entails that $F$ must be a function. To properly write matters in a logical language one must first see how to write formulas using $\sf One-to-Many$ relation symbols, one better avoid using the equality $=$ symbol, so we better write $F(X,Y)$, so doing that we can for example define:
$ F(X, \pm Y) \iff [F(X,+Y) \land F(X,-Y)]$
So for example if we intend to use the symbol $\sqrt \ $ to designate the converse relation of the square function, then we better write it this way:
$\sqrt \ \ {}(X, \pm Y) \iff [(+Y)^2=X \land (-Y)^2=X] $
However, we can abuse notation and insert the equality symbol and define:
$\sqrt x = \pm y \iff [(+y)^2=x \land (-y)^2=x] $
Which is a wrong way of writting matters because logically it would be read as $\sqrt x = + y \land \sqrt x = -y$ and this clearly leads to $+y = -y$ by identity theory, which is contradictory! But unfortunately this is the intended meaning many times.
Now if we intended for example for the square rooting to be a function, so we can fix one value to each argument, like by arranging it to be the positive value only, so this way $\sqrt 9 =3 \land \sqrt 9 \neq -3$, but if we stick to this interpretation, then $\sqrt 9 = \pm 3$ is a FALSE statement!.
The OP is under the impression that $F(X) = \pm Y$ is a disjunctive expression, to mean that $F(X)=+Y \lor F(X)= - Y$, and this is a wrong capture, the reason is because it would lead to the explosion he is alluding to, so I can for example write $\sqrt 9= \pm 3; 8^2 = \pm 64 , 6/2 = \pm 3, 1+3=\pm 4, |3|=\pm 3,...$ and all would indeed be logically valid under that interpretation, and clearly this is a redundant symbolism, and so it is not the correct capture of what's going on. Yet, if you insist on interpreting it this way, then there is no logical flaw with it, as long as one of the values is true, then you can add as much other values you want and the result remains logically valid, but the possible expressions under that interpretation would of course be REDUNDANT. So you can abuse $T \equiv T \lor F$ as much as you want, there is no logical flaw about it, but when you use this to define expressions, then you'ed expect an explosion of redundant expressions. [Be aware that this abuse must not lead you to the logical flaw about the set of solutions that was pointed out in the begining of this answer]
The reality of the matter is that $F(X)= \pm Y$ is an incorrect symbolism, it is an abuse of notation, and should be just taken in an informal sense to mean that $F(X,+Y)$ is true and $F(X,-Y)$ is true also. The proper way of writing it is as $F(X, \pm Y)$, and the proper definition of it was given above.
A: Writing $x=\pm3$ does not simply mean "$x=+3$ or $x=-3$". It usually means that both alternatives are possible. So writing $x=\pm3$ for the solutions to $x^2=9$ is correct, but not for the solution to $x+2=5.$
Also, the symbols $\pm$ and $\mp$ are primarily used when there are two expressions that only differ by a sign. One can then treat them both at the same time by using $\pm$ or $\mp$ where the signs differ.
Examples:

*

*$(x\pm y)^2=x^2+y^2\pm 2xy$ instead of $(x+y)^2=x^2+y^2+2xy$ and $(x-y)^2=x^2+y^2-2xy$,

*$-(x\pm y)=-x\mp y$ instead of $-(x+y)=-x-y$ and $-(x-y)=-x+y$,

*$x_\pm=-\frac{p}{2}\pm\sqrt{\frac{p^2}{4}-q}$ instead of $x_+=-\frac{p}{2}+\sqrt{\frac{p^2}{4}-q}$ and $x_-=-\frac{p}{2}-\sqrt{\frac{p^2}{4}-q}$.

A: $|x|$ has one definite value for a given $x$. $\pm x$ has two values for a given $x$ (except when $x = 0$). Because of this, it's not clear what $|x| = \pm x$ should mean.
In mathematics, we define something called the pre-image of a function. Given a function $f : \mathbb{R} \to \mathbb{R}$ and a subset $A \subset \mathbb{R}$, we define
$$
f^{-1}(A) = \{x \in \mathbb{R} : f(x) \in A\}
$$
With this definition in hand, we can give some precise meaning to $|x| = \pm x$. Let $g(x) = |x|$. Then
$$
g^{-1}(\{x\}) = \{c \in \mathbb{R} : g(c)= x\} = \{x, -x\}
$$
If we think of $\pm x$ as the set $\{x, -x\}$, then we may (though no mathematician would) write $g^{-1}(x) = \pm x$.
A: You are right to state that symbols do not have meaning by themselves without context. When we write $\sqrt{9}=\pm 3$, what we are saying is all this: the equation $x^2-9=0$ has exactly two solutions $x=3$ and $x=-3$. So I would say that it is not exactly that $x=3$ or $x=-3$, that will depend on how $x$ is defined or computed elsewhere.
This is a different usage to writing $|x|=\pm x$ meaning $x$ or $-x$. In fact, I do think you could use it (even incorrect) to consider both possibilities in a single equation, for instance if the sign does not change the final result, e.g. $|x|^2=(\pm x)^2=x^2$. It does not seem very recommendable but I think everyone would understand it. Note that in this case the possibilities are either $x$ or $-x$ but not anything else, so it is not exactly a logical OR, just a shorthand for two equations.
A: What I think is being understood is the difference between a set and a statement with a truth value. Let me try and explain.
We'll take OP's example of $\sin x=0$ and run with it. What $\sin x=0$ actually tells us is the set of all values of $x$ (in a given domain) such that the statement holds. It has no inherent truth value, it's just a set (of solutions). But, once a $x$ is given, the statement $(\text{For the given }x,\ \sin x=0)$ indeed has a truth value. And here we see what the OP has a confusion in. OP's statement $$(\sin x=0)\equiv (\sin x=0\ \lor \cos x=0)$$ is trying to establish an equivalence between sets, and the logical or is actually just union, and the OP clearly establishes that this equivalence is not true. But what is true is that the following  statements can be compared as logical statements : Given an $x$ whether:
$$(\text{For the given }x,\ \sin x=0)\equiv (\text{For the given }x,(\ \sin x=0\ \lor\cos x=0))$$
(Thanks to Hans Lundmark for telling me that I wrote something wrong. I was thinking something else, and wrote something else haha)
Edit : Let me actually add another clarification to a problem OP mentioned in the question, and we'll see how both of them connects.
It has to do with the fact that OP writes $\sqrt 9=3\ \lor -3\ \lor -139e^2$. From a logical point of view, this is absolutely correct, because it's really 3 statements written together :
$$(\sqrt 9=3)\ \lor (\sqrt 9=-3)\ \lor (\sqrt 3=-139e^2)$$
and the first 2 statements are indeed true and hence makes the whole statement true. Being ruthless, we can literally add everything and say $$(\sqrt 9=3)\ \lor (\sqrt 9=-3)\ \lor (\sqrt 9=x,\ x\in\mathbb R\setminus\{3,-3\})$$ This statement is completely true because the first two components are true. This is a problem with "finding the correct solution given a set of statements connected by logical or". Because, if only 1 component is true, the whole statement becomes true, but that means there could be false components hidden in the statement.
An example where this happens is when we find solutions by squaring. Extraneous solutions creep in which doesn't solve the original equation, but the whole solution set as considered in the above fashion is indeed correct (try squaring both sides of $x^2=9$ and see that one has 2 extraneous solutions which doesn't solve this). Hence we are asked to check if indeed everything in the solution set is true, or there's imposters lurking in it (in our case, it would be everything that is not 3 or -3).
tl;dr : The statement OP wrote about $\sqrt 9$ is logically true, but since these are statements connected by logical or, one needs to check which of the solutions are actual solutions and which are not (because throwing in one solution makes the whole statement true).
In the case of the trigonometric statements, we again have 2 statements connected by logical or, and hence one needs to check which component is true and which is not for particular values of $x$.
Hope this helps in seeing where the problem lies, and the OP can transfer the arguments for the other cases as well.
