(X & Y) = ~ (~X + ~Y)

(X & ~X) = ~(~X + X)

X & ~X =FALSE

~( X + ~X)= FALSE

I'm beginning to feel like in a quarrel of the deaf.

Syntax: The rules guiding proper and improper sentences in a formal language.
Semantics: meaning given to sentences in a formal language.
Truth-table semantics is one semantics applicable to the propositional calculus (->, ~, V ^,..).

One can create a proof system over a formal language. This will include axioms, and methods by which one can manipulate and combine existing theorems into new ones. A sentence will be considered a theorem, or provable, if it can be derived by those methods.

By the way, I don't follow your manipulation - the operations you are using are only applicable to datums which are either true or false, according to truth-table semantics. There is no "neither true nor false" value in truth-table semantics.

If you're using a new semantics, then you'll have to explain what you mean when you say that a sentence is true or false or neither under that semantics.

Then you should refer to my first paragraph. The "man who killed his grandfather" paradox is merely an extreme of the basic paradoc, which is that by going to the past, you alter it, thus supposedly making it impossible for that same you who went to the past to have existed in the first place. Another thing is, that any transportation to the past would have to be intertwined with a transportation to a different location, since all the bodies in the universe, including the one we're living on, constantly move around, so that if you stay in the same place, and go to the past, you are as likely to find yourself in space instead of on any cellestial body, not to mention Earth. Then again, how do you define your current location without referring to movement?

By the way, I realize that the notion of "timelines" is simplistic, but there is a valid interpretation of Quantum Mechanics, called the "Many Worlds Interpretation." You will have to refer to a Physicist on any further discussions, though, as I don't know much about it, or indeed, about Quantum Mechanics.