In Lewis Carroll's story "What the Tortoise Said to Achilles," the swiftfooted warrior has caught up with the plodding tortoise, defying Zeno's paradox in which any head start given to the tortoise should rilake him uncatchable. (In the time it would take for Achilles to close the gap, the tortoise would have progressed a small amount; in the time it took to close that gap, the tortoise would have moved a bit farther, ad infinitum.) The tortoise offers Achilles a similar paradox from logic. Achilles pulls an enormous notebook and a pencil from his helmet, and the tortoise dictates Euclid's First Proposition:
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
The tortoise gets Achilles to agree that anyone who accepts A and B and "If A and B then Z" must also accept Z. But now the tortoise disagrees with Achilles' logic.
He says he is entitled to reject conclusion Z, because no one ever wrote down the if-then rule on the list of premises he must accept. He challenges Achilles to force him to conclude Z. Achilles replies by adding C to the list in his notebook:
(C) If A and B are true, Z must be true. The tortoise replies that he fails to see why he should assume that just because A and B and C are true, Z is true. Achilles adds one more statement—
(D) If A and B and C are true, Z must be true.
—and declares that "Logic [must] take you by the throat, and force you" to accept Z. The tortoise replies, Thinking Machines 99
"Whatever Logic is good enough to tell me is worth writing down. So enter it in your book, please. We will call it
(E) If A and B and C and D are true, Z must be true."
"I see," said Achilles; and there was a touch of sadness in his tone. Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring tortoise, and was writing in his notebook, which appeared to be nearly full.
I don't get it A, B does imply Z. Why need the third rule? C that A and B implies Z, and after that D that, A, B, and C implies Z and zo on?
I got this from pinker Stephen's How the Mind's work.