"I am lying." True of False

We can look at this from two perspectives: One if that "I" refers to I as in me and stuff. Another perspective is that "I" refers to someone whose name is "I". We shall now look at this issue from the first perspective.

If one says that oneself is lying, it is impossible. Why? Allow me to explain. If one is lying and says, "I am lying", that means that one is not lying. However, if that person is telling the truth, then he/she is lying. This loop will go on forever, thus a paradox occurs, you can never tell whether the person is telling the truth or lying. If that's the case, this situation is impossible and cannot be solved- ever. We conclude that the first scenario is completely impossible.

Now, the second scenario is slightly different. Person A is calling Person I a liar. Given that person A is telling the truth, Person I is lying. This scenario is completely and absolutely possible, as there is no conflict. However, what if Person A is lying? If so, person I would be telling the truth and Person A is lying, no conflict is present here either. Thus, the truth of the sentence differs based on whether the person speaking is telling the truth. Thus if the speaker is telling the truth, the sentence is true. Otherwise, the sentence is consider false. However, this applies only to this scenario.

What, then, is the conclusion? Is the sentence true or false? How are we to know? One cannot tell if another is lying, and one cannot solve a paradox, what does this leave us with, then? Simple, this sentence remains undefined in terms of its truth.