The addition rule (often abbreviated “ADD”) is perhaps the most counter-intuitive rule in propositional logic natural deduction. The addition rule is defined formally as follows:
Premise: p
Conclusion: p v q
The addition rule (ADD) tells you that if you have any statement p on a proof line, then you can conclude the disjunction p v q on a subsequent proof line, where q refers to any simple or compound statement.
At first, the addition rule may seem implausible because the statement q in the conclusion does not appear in the premise. So why are you justified in concluding “either p or q” when you are given only “p” as a premise?
Remember that a disjunction tells you only that at least one of the two disjuncts is true. Therefore, the statement p v q tells you only that one of the two statements, literally either p or q, is true. Well, if you know that statement p is true (from the premise), then the statement p v q (either p or q) must be true, regardless of what the statement letter q refers to, and regardless of whether q is actually true or false. In other words, if p is true, then it must necessarily follow that the weaker claim “either p or q” is true, because at least one of the two disjuncts (in this case, p) would always be true.
The addition rule (ADD) can be used to introduce statement letters into a proof that do not appear in the given premises. If you are attempting a logic proof problem in which the conclusion contains a letter not appearing in the premises, look for a way to use the addition rule to introduce the missing letter so that you can derive the conclusion.