Last Update: 8 February 2009
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Some of you have said that the "Addition" rule of inference, which says:
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".)
For more information on this, see my Web pages: