You have suggested that mirror-particles of one another, i.e. particles which look like the mirror image of one another, are actually commutative if we introduce additional dimensions. For instance, a 2D spiral with right orientation can be turned upsidedown in 3D to produce a 2D spiral with left orientation (clockwise rotation of the spiral) (1). I asked whether the existence of zillions of chiral molecules, i.e. molecules which are mirror images of one another, hints that no extra spatial dimensions exist in molecular scales.
Moreover, string theory suggests there are several additional spatial dimensions which are 'comapctified' - like a closed string with a small but finite radius. Thus, chiral particles smaller than the radius of these dimensions should be able to switch to their mirror-particles and back, i.e. their wavefunction should commute with the mirroring operator.

(1) Riddle of the Day - Why is the counterclockwise direction defined as the positive mathematical direction?

I'm not deeply familiar with string theory, but I can imagine a small closed loop, let's say a circle, which looks from a distance like a dot. I can also imagine a cylinder with a finite small radius which looks from a distance like a line. That cylinder has one infinite dimension running along it, which is visible at any scale, and another compact dimension which is visible only on a scale small enough. This compact dimension must be quantized, as phi(r,z) should be equal to phi(r+2pi*n*R,z) for any r, where R is the radius of the comapct dimension.

The mirroring operator (let's tag it P) simply flips x with -x, like a mirror. which means Px=-x (that's an eigenvalue equation, -1 is the eigenvalue of P). Please warn me if I'm relating to stuff you haven't previously encountered, when I say for instance that x^2 is conserved under mirroring, thus x^2 commutes with P. Px^2=x^2P, thus [P,x^2]=0 - the commutator is 0. Any symmetric function surely commutes with P.
If a particle has an extra dimension through which to flip upsidedown to its mirror image, then it should be conserved under mirroring. Its wavefunction should commute with P.

Now, since we know chiral molecules do not spontaniously switch to one another, then I assume we can deduce they have no extra dimension to 'flip through'. If string theory allows particles to 'flip through' those compact dimensions, which is something I honestly don't know, then we can limit the radius of these dimensions to the lowest scale at which we can tell that a particle doesn't flip to its image. Spin-up electrons do flip to spin-down states, but molecules don't flip to their mirror-molecules.

(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.

What does consistency mean in a world without the law of contradiction?