Logic
A logical proof is a list of statements put in a specific order to prove a logical point.  Each statement in the proof is either an assumption made for the sake of argument, or has been proven to follow from earlier statements in the proof. All proofs must start with some assumptions, such as "men exist" in the above syllogism. A proof shows that one statement, the conclusion, follows from the starting assumptions. With a proof, we can prove that "Aristotle is mortal" logically follows from "Aristotle is a man" and "All men are mortal".