I think the core of this kind of question is in what's meant by True or False, and the nature of definitions. I wanted to come at this in a more concrete way and in laymans terms, though some of the existing answers (@Bumble, @KristianBerry, @Lowri) touch on it.
Important notes to start:
"All models are wrong; some are useful."
You cannot boil a complete valid system down to a pure set of axioms. Every definition depends on other definitions, and there's no escape or bottom, though we can wrap it in higher- and higher-level formalism. In other words:
"It's turtles all the way down".
I'd argue that definitions are inherently arbitrary, and cannot be True or False in any Real sense. They are each a symbol arbitrarily linked to a concept; nothing more or less.
We often speak of them as such though, or as correct or incorrect, right or wrong; this is merely a shorthand for one or more of the underlying true traits of a definition: whether it's widely-shared and to a lesser degree useful or intuitive / consistent.
This is complicated by the common conflation of statements of fact and definitions.
Note that this argument is itself a definition, of 'definition'. I don't know if it's the most widespread one, but I'll argue it's a better model (that is, more useful) than the alternative I've seen argued - that every word has some Real fixed Aristolean meaning, decipherable with enough study or divine enlightenment*.
* This isn't an idle quip.
At the simplest level: If I tell someone "My dog is a fligabert - a poodle crossed with a wolf", they're liable to respond "That's not a real word". Of course, what differentiates 'Fligabert' from 'Labrador' but widespread use? If in 20 years this wolf-poodle breed becomes extremely popular, and widely known as a Fligabert, 'Fligabert' would naturally become the 'correct' word. For now it's 'incorrect', but only because there's only one person holding to the definition. If however I tell them it's a 'woodle', they're much more likely to accept it as a word (albeit perhaps with some rolled eyes) because despite not having heard it before, it matches the pattern of how cross-breeds are named.
If I designate a new mathematical operation φφ like f(x) = xφφ + y - z, and define it to mean "Cube every other positive term in this equation, skipping the one bearing the φφ", I can certainly do that.. But it'd probably not be a useful definition. In addition to not being shared by anyone else, it's also inconsistent with the way mathematical operations are usually styled.
All of those meanings vary in usefullness, but what meaning could there possibly be to calling any of them True or False on any deeper level?
Sufficiently-intuitive words can communicate their meaning perfectly well even if not previously widely-shared. Similarly, widely-shared words can become useless if their meaning becomes sufficiently counter-intuitive. See "Inflammable".
An important distiction here needs to be made between a definition and a statement of fact.
On their own, meant as definitions, there is no difference between
x = 2 and
4 = x + 2; one's simply cleaner. I think some of the other answers have mis-dissected it, calling it a statement of fact. As an equation, it could be read as "I define
X as the value that satisfies the equation
4 = x + 2. Such a value exists given the common definitions of all the symbols in that equation, but to call the equation True or False without a pre-determined
x is meaningless. The equation can only be True or False if all the values it depends on come in with prior constraints, and is therefore presented as a statement of fact.
Simiarly, if someone says "A unicorn is a species of maize that grows in isolated stalks", it's a False statement only if the speaker knows what the rest of us see as the meaning of "unicorn", and is claiming an equivalence between the underlying concepts - a horned horse and a stalk of corn. If the speaker honestly means it as a definition though, then we can still call it 'wrong', but only in the sense that it differs from the definition the rest of the english-speaking world agrees on for that word. The 'wrongness' goes no deeper. Humanity made up the word unicorn, and so the only 'true' meaning it can possibly have is whatever the group as a community believes it to mean.
Most of this doesn't matter much in everyday life; words are 'correct' when used the way everyone else uses them, or perhaps if their meaning is intuitive even if not well-known, and 'incorrect' otherwise. The model described above becomes incredibly important though when a group lacks a mutually-shared set of definitions.
Some concrete examples:
Subgroups disagree on the meaning of a particular word or symbol
The statement "The X historical flag flying above my house means Y" actually boils down to some combination of "The X flag means Y to me/my group" and "When this X flag was historically flown, it meant Y to the people flying it". The first statement is a definition, and cannot possibly be true or false - though it may certainly be useful or useless at communicating the intended meaning. The second is a provable statement of fact. Using terms like "true", "real", or "incorrect" as a shorthand here without breaking it down into pieces, substantially adds to the perceived disconnect between groups.
A common trend is for an older group to call the slang of the younger generation "made-up / incorrect" or "confusing"; This dispute is clarified by translating those complaints to 'not widespread among their community' or 'unintuitive'.
A document a group believes or defines to be true (like a Holy Book or Constitution, respectively) contains words or phrases which no longer have a commonly-understood meaning, and so one has to be defensibly determined.
- This need not overlap with the above, though it often will.
- A fundamentalist with a Holy Book has the fall-back of believing that every word there does in fact have some Real True meaning - whatever the Writer intended, which can in fact theoretically be discovered through sufficient study or divine enlightenment. Even here though, the True thing is the meaning itself, not the link between that word and that meaning - i.e. you may believe that the phrase 'dragon' has a very specific meaning in a particular chapter of a particular Holy Book, but it'd be fallacious to believe that every use of 'dragon' in secular media actually refers to that same specific meaning.
- With legal documents, we're in much more of a bind. Humanity's worked hard to shape an objective framework out of subjective words, but we can't escape the core nature of language; it means whatever everyone agrees it means. The best we can do, is make sure everyone shares the same definitions, and that all those definitions are consistent. If someone comes up and says "This phrase of this law means Z, not Y", the only place the court can turn is "What has that phrase meant in other parts of the law, the judicial record, or common parlance" (i.e., is any particular definition shared among its peers), and "Of any viable interpretations which one will yield a more reasonable result, both in this case and in future cases?" (i.e., is this or that definition useful, consistent, and intuitive). Even if a phrase does have a firmly established definition which clearly applies to a case, that doesn't save us from having to make arbitrary judgements; the definition itself is made of words which require definitions, and it's turtles all the way down. At no point in that chain will you hit an inarguable definition, merely (hopefully) eventual limited common agreement.