Achilles wins the race easily. Can he ever force the tortoise to admit he's right?
In Lewis Carroll's 1895 dialogue, Achilles gets the tortoise to accept two premises: 'A: things equal to the same are equal to each other' and 'B: the two sides of this triangle are things equal to the same.' The tortoise still refuses to accept the conclusion, 'Z: the two sides of this triangle are equal to each other', until Achilles gives one more reason. What goes wrong?
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Every time Achilles adds the missing logical rule as a new premise ('if A and B are true, then Z must be true'), the tortoise simply asks why that rule should make him accept Z either, demanding it be added as yet another premise, forever. Carroll's point: logical inference can't be reduced to just more premises, because you eventually need the rule of inference itself, not another fact, to make the jump.
— Lewis Carroll, What the Tortoise Said to Achilles — Mind, Vol. 4, No. 14, 1895