I understand that we just have to fill the last two raws in the truth table with any value, and that we randomly chose True, and that the value True makes matters easier sometimes (I don't know an example of that, but I read that somewhere). But the question is, since mathematics is tied to the real world, how does the random truth value not cause any problems? I read the following, in a reply in some thread on the forums:- "Note that we're essentially selecting axioms here; we're choosing the rules that govern the logical deductions we'll be making in the future (i.e. we're setting up a logical system, not making inferences based on that system)." It sounds to be the type of answer I'm looking for, but for the lack of my knowledge, I still can't understand/imagine the full picture. Moreover, the explanation still doesn't explain how the theorems we make still apply in the real world. So, can you help me with this please? Particularly, can you explain how choosing the rules of inference randomly still lets theorems correspond to the real world?