Discrete Structures

# The Addition Rule of Inference

 Last Update: 8 February 2009 Note: or material is highlighted

Some of you have said that the "Addition" rule of inference, which says:

From p
Infer (p ∨ q)

doesn't make any sense.

But that depends on what you mean by "sense" :-) It makes perfect logical sense; i.e., it is a truth-preserving move. On the other hand, perhaps you mean that "magically" bringing in q seems a bit...nonsensical. After all, q hadn't been mentioned before.

Moreover, this rule underlies what's called a "Paradox of the Material Conditional", namely, from a false statement, you can infer anything. This follows from the truth table for "→": If the antecedent is false, then the entire conditional is true, whether or not the consequent is true.

So, as Bertrand Russell, a famous atheist logician, once said: If you grant me any false statement, I can prove that I'm the Pope: Here's the proof:

 1 1+1=3 : premise (the "false statement" that we have granted Russell) 2 (1+1=3) ∨ "I, Bertrand Russell, am the Pope" : From 1, by Addition (!) 3 ¬(1+1=3) : axiom from arithmetic (clearly true) 4 "I, Bertrand Russell, am the Pope" : From 2,3 by Disjunctive Syllogism

This argument is perfectly valid. It is, of course, unsound, because premise 1 is false. Here's the same proof, without bringing in arithmetic or religion:

 1 (p ∧ ¬p) : premise: assumption of a false proposition 2 (p ∧ ¬p) ∨ q : from 1, by Addition 3 ¬(p ∧ ¬p) : tautology 4 q : from 2,3 by Disjunctive Syllogism

Any discomfort you feel about this is shared by many logicians. In fact, the rule of Addition is rather controversial for just those reasons. It is truth-functionally OK (because it's truth-preserving), but somehow seems to bring in an irrelevancy. (The same goes for Disjunctive Syllogism, as well as the truth-table for "→" and the interpretation of ordinary English "if...then" as "→".)

There are other systems of logic, called "relevance logics", that don't allow Addition, for just that reason. (In fact, our AI research group here uses such a logic for our knowledge representation and reasoning system, called "SNePS".)