IN 1895 Lewis Carroll published in Mind 1 [ New Series, vol. IV, pp. 278-80.] a brief dialogue, `What the Tortoise said to Achilles'. The intention of the story is, plainly enough, to raise a difficulty about the idea of valid arguments, a difficulty similar, or so Carroll implies, to Zeno's difficulty about getting to the end of a race-course. Different writers have said different things, usually briefly, about what the difficulty is. Let us first consider just what happens in the story and then try to see what problems it raises.