>As an example of such statements, I present self-contradictory ones. There is no need to give any further examples,
Sorry, that won't cut it. A self contradictory statement is essentially nonsense, and has no meaning. To prove that something can be both true and false (or neither true nor false) you're going to have to show me a real example.
>You will have to give a very tricky definition for "proposition" in order for this to be true
No, I believe the standard definition will suffice. Any statement about something is either true or false (nonsensical self-contradicting statements notwithstanding, because they essentially have no meaning.) Can this apply to statements about the future? Absolutely. Example:
Socrates is mortal.
Therefore, either Socrates is mortal, or he is not. (Either he will die, or he won't die. Statements about the future.) Socrates cannot be both mortal and immortal at the same time, nor can he be neither (the two conditions are mutually exclusive, and collectively exhaustive.)
>We can see that something happens, but we can't tell exactly what at any time
That doesn't matter, the fact that it is happening at all is proof that there were absolute initial conditions, and absolute current conditions, regardless of whether or not they can be perfectly determined by us.