Let's consider these two statements.
Everything is either complicated or not complicated
Either everything is complicated or nothing is complicated.
Why would one of these statements be a tautology, and the other one be invalid?
Here's my attempt.
Say C(x) represents it's complicated
1st statement is ∀x [ C(x) v ¬C(x) ], is this right?
2nd statement is [∀xC(x)] v [¬∀xC(x)]
I'm not sure if it's correct though, also I can justify why the 1st statement is always true, as it's a disjunction of C(x) and ¬C(x), but I don't seem to be able to justify why the 2nd statement is false. Please help!
Your symbolizations are correct!
The second statement is not necessarily false, but could be false ... if some things are complicated and some other things are not. This is why the second sentence is not a tautology.
The fact that a sentence is not a tautology does not mean that it is false, but that it can be false.