When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

“This would vitiate all science” is what W. V. Quine, one of the preeminent logicians of the twentieth century, would say: “Any conjunction of the form p and not-p logically implies every sentence whatever; therefore acceptance of one sentence and its negation as true would commit us to accepting every sentence as true, and thus forfeiting all distinction between true and false.” To see what Quine means, suppose you believe both p and not-p. Because you believe p, you must also believe p or q, where q is any arbitrary proposition. But from p or q and not-p, it obviously follows that q. Hence any arbitrary proposition is true. (The Latin phrase for this is ex contradictione quodlibet.) The idea that a contradiction is bad because absolutely anything follows from it might seem strange to a non-logician. Bertrand Russell was once trying to get this very point across at a public lecture when a heckler interrupted him. “So prove to me that if two plus two is five, I’m the Pope,” the heckler said. “Very well,” Russell replied. “From ‘two plus two equals five’ it follows, subtracting three from each side, that two equals one. You and the Pope are two, therefore you are one.”