The “principle of explosion” explains why a single contradiction would destroy math
Unicorns roam free in fantasy novels and children’s stories, not so much in the real world, much less the cold, analytical ones of math and philosophy. But it turns out that these logical disciplines are only one misstep away from proving the existence of the long-adored mythic creatures—or proving any absurdity.
Contradictions keep math and philosophy on course through negative feedback. Like dead ends in a maze, they signal “this is not the way forward” and demand that you retrace your steps and choose a different path. Contradictions also underpin all paradoxes. Consider the infamous liar paradox: “This sentence is false.” If it’s true, then we should take it at face value: the sentence is false. If it’s false then it is not the case that the sentence is false, i.e., it’s true.
The argument that proves anything from a contradiction is remarkably straightforward. As a warm-up, suppose you know that the following statement is true.You know the above to be true. It doesn’t necessarily imply that Omar is married, nor does it imply that Maria is five feet tall. It only implies that at least one of those must be the case. Then you import an additional piece of knowledge. What can you conclude from this pair of assertions? We conclude that Maria must be five feet tall.
Some logicians find the principle of explosion so disturbing that they propose altering the rules of logic into a so-called paraconsistent logic, specifically designed to invalidate the arguments we’ve seen above. Proponents of this project argue that since unicorns have nothing to do with Omar’s marital status, we should not be able to learn anything about one from the other.
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