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It's funny my original ambiguous question might be considered a simple mathematical question. Yet when I try to be more precise it falls into philosophy. :(
John: Yeah you might be right. However many philosophical problems later became formalized in mathematical language. And surely asking a philosopher this question will only result in more philosophy!
Oh very sorry some guy on the street :( Yeah maybe instead of possible worlds... we can think of them as accessible worlds. For instance a world that is like ours but some of the facts about it are changed. So maybe instead of wearing a blue shirt today I wear a white one. However if "2+2=4" was not true this world would be vastly different and inaccessible (impossible?). This perhaps makes it more clear what contingent and necessary mean.
There is no truth-value assignment for P where P or not P is not true. This is a tautology and necessarily true. But consider a more complex statement, for example the fundamental theorem of algebra. This does not seem to be a formal tautology as there exists truth-value assignments where the statement is false. But surely it seems to be necessarily true. Could the fundamental theorem of algebra be false in a world where it exists?
Define 'necessarily true' as a the statement that is true in all possible worlds (where such a statement could exist). Maybe the questions is rather is all mathematical theorems necessarily true?
