The Importance of the Addition Rule

1.     P & ~P
2.     P          1, simp
3.     P v Q    2, add
4.     ~P        1, simp
5.     Q         3, 4, DS

The addition rule is useful for a number of reasons. First, it simply follows from the truth-table for ‘or’ and the rules of validity. But as you can see here it is really helpful in showing why contradictions cannot be tolerated. If we allow contradictions into our beliefs or premises, then we can prove literally any proposition. This is what logicians call explosion. From one single contradiction, we can show that every statement, no matter what it is, is true.  Of course, it is absurd that every statement is true. So contradictions cannot be tolerated, and the addition rule nicely shows us why.

Applications: one of the reasons I love philosophy is that it helps us to locate all sorts of contradictions (or at least apparent contradictions) in our beliefs. Good philosophical thinking locates various tensions and then shows us that some claim or belief has to be given up lest we entertain contradictory beliefs. Do not let anyone get away with allowing contradictions. Love demands consistency, and contradictions are the chief enemy of consistency.


Here’s an apparent contradiction: God is one person.  God is three persons. From the claim that God is three persons, we can infer that God is not one person. So now we have a contradiction: God is one person & it is not the case that God is one person. Should we conclude that the doctrine of the trinity is contradictory? Or should we conclude something else? What's clear is that something needs to be done to remove the contradiction. But what? Extra credit for those who email me an attempted solution.

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