Should it be allowed to apply classical logic to set theory? It is well known, that the generalized continuum hypothesis isn't provable from the standard axiom system ZFC.
GCH (generalized continuum hypothesis). For every infinite set A, there isn't a set M such that |A| < |M| < |P(A)| (where P(A) denotes the power set of A).
Using classical logic, one can deduce the following: 

GCH is either true or false.

But for me this statement does not make any sense because GCH is not true nor false, just because the concept of "set" is too ambiguous, in other words: the statement does not make any sense because "GCH is either true or false" isn't a proposition: it has no truth value.
How is one able to justify that one can say statements like

GCH is either true or false.

although this makes no sense for a non-platonist?
 A: One does not have to justify any philosophical statement in mathematics. Because philosophical statements represent some overarching drive as to where to move forward in mathematics.
You believe that something should be true, or something should be provable, means that you want to prove that this is the case. It is a light in the darkness which lies beyond the map. Where new research comes from.
But it is just a personal belief based on what you've seen so far, what you've proved, and how it all makes sense to you, in your mind.
What you said, however, is a bit misleading. Classical logic tells us that in a given model of $\sf ZFC$ a statement is either true or false, not to mention that if something is not provable does not mean that it is neither true nor false. It just means that you can't prove, from your current assumption which one it is. So if you are a Platonist, and you believe there is a true model of $\sf ZFC$, it just means that you have to decide for yourself - by the evidence that you see in mathematics - whether or not $\sf GCH$ should be true or false.
So if a Platonist (you or otherwise) feels that $\sf GCH$ is true, they might work towards formulating axioms that imply $\sf GCH$ (like Ultimate-L of Woodin) and towards pushing others into taking these axioms as canon. Or they might work to show that $\lnot\sf GCH$ has various undesirable consequences.
But at no point a Platonist is going to force you to accept an axiom. At best, they might argue that you're missing some bigger picture, or that all signs point at some direction. And if you're not a Platonist, like me for example, then you can just take that with a grain of salt, and focus on the mathematics instead.
In any case, without specifying a specific model there is no mathematical meaning to the statement "$\sf GCH$ is true". In some cases, like $\sf PA$, there is a standard model and "true" is a shorthand for "true in the standard model". But $\sf ZFC$ does not have this luxury (for better or worse) of a unique, well-specified standard model.
A: The viewpoint that the continuum hypothesis contains an intrinsic ambiguity ("is inherently vague") is a legitimate one, and has been pursued by the distinguished logician Sol Feferman; see e.g.,
Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, John R.
Does mathematics need new axioms? 
Bull. Symbolic Logic  6  (2000),  no. 4, 401–446 
where this viewpoint is developed in full.  Feferman's alternative is the predicative viewpoint inspired by Hermann Weyl.
