A bit new to modal logic.
I believe that it is the case if we say "It is possible that P is true in world w", then this statement is "true". This is not the same as saying P is true. Just that P is possibly true.
If we were to say "It is possible that P is true in world w and impossible that P is true in world w" this is a contradiction, so this statement is false.
Is that correct?
Wondering what the best way to formally state this would be. Can these types of statements be proven like in classical logic? I assume we can prove the truth of modal statements by constructing a proof and a theorem. However, in doing so, we don't actually prove whether P is true in a possible world and construct a truth table to show this is the case.
More specifically, to state the above formally, what are the truth values of the following statements? True, false, or do they not have a truth value?:
The second statement looks like a contradiction, so it should be false.