# How to Use the Addition Rule (ADD)

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.

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