Mo, Tara and Dior

Mo went first, because his step was the simplest and the most foundational of the three: run the rule forward.

He wrote up a rule no one disputed — "If it rains, the ground gets wet" — and beneath it, a fact: "It is raining." Then he took the single honest step forward, and it came down on solid stone. "Therefore, the ground is wet."

"People dismiss this one as too obvious to bother with," Mo said, "which is exactly how you lose an argument. Watch how sturdy it is instead. If the rule genuinely holds, and the starting fact is genuinely true, then the conclusion cannot be false. It's not likely — it's forced. The ground must be wet. I'm not estimating and I'm not hoping. I'm walking a true rule forward by one honest step, and the ground catches me every single time." If P then Q; P; therefore Q.

Then Tara stepped up and ran the very same rule in the opposite direction — the move that reliably startles people.

"Same rule," she said. "If it rains, the ground gets wet." But she refused to start at the beginning. She started at the end, with a fact about the outcome: "The ground is not wet." And then she walked the rule backward.

"If rain always produces a wet ground," she reasoned, "then a dry ground is proof the rain never came — because if it had, the wetness would be here, and it isn't. Therefore, it did not rain." She smiled at the doubt on the students' faces. "It feels like a conjuring trick the first time, but there's nothing up my sleeve. It's the identical rule, read in reverse. No Q means no P — because P would have guaranteed Q. The dry ground testifies for itself. I never needed to glance at the sky." If P then Q; not Q; therefore not P.

And then came Dior, whose step worked in the harder case — when you had no clean rule at all, only a list of possibilities and a way to shrink it.

"The vase was knocked over," Dior said, laying out the facts like cards. "Either the cat did it, or the dog did it." Two candidates, and the truth had to be one of them. Then Dior located a single reliable fact: "The cat was shut outside the entire afternoon. It was not the cat."

"So I strike it from the list," Dior went on, drawing a slow line through the cat, "and with that branch gone, only one possibility is left standing — and whatever's left standing has to be true. Therefore, the dog." A shrug, and a quiet satisfaction. "I never caught the dog in the act. I didn't have to. I only had to be sure of the list — cat or dog — and sure of one thing it wasn't. Remove the false branch, and the true one is simply all that remains. The answer had been hiding inside the list the whole time; I only cleared away what it couldn't be." Either A or B; not A; therefore B.