Logical Paradoxes

Achilles and the Tortoise

Achilles and the Tortoise is one of Zeno’s paradoxes. It appears to demonstrate that even the fastest of runners, Achilles, could not catch the slowest of creatures, the tortoise, if the tortoise were given a head start.

To get from one place to another takes time. If the distance between the two is very small, or the speed of travel is very fast, then it may take a very short amount of time, but it will take some time nevertheless. It is impossible to move from one point to another instantaneously.

To catch someone, you need to cross the distance between you and them, you need to move from where you are to where they are. If there is any distance between you at all, then this will take time.

Suppose that the person that you are trying to catch is moving away from you, then in the time that it takes you to get from where you are to where they are, they will have moved on. If you begin at point A, and they begin at point B, then by the time you reach point B they will be at point C. The person that you are trying to catch will no longer be at the point that you have reached.

To catch them, then, you will need to reach the point that they are now at, you will have to get from point B to point C. Doing so, though, will again take time. Point C is at a distance from point B, and so by the time you have reached point C, your target will have reached point D. This process can be repeated ad infinitum, without you ever catching your target.

Zeno illustrated this with the example of Achilles and the tortoise. If, in a race, the tortoise, who moves slowly, is given a head-start on Achilles, then no matter how quickly Achilles runs he will never catch the tortoise.

The Paradox of the Stone

God is all-powerful, or as theologians put it, “omnipotent”; there is nothing that he cannot do. This is part of the definition of “God”.

So can God create a stone that is so heavy that he cannot lift it? Either he can or he can’t.

If God can’t, then he isn’t all-powerful. If God can’t create a stone that he can’t lift, then there is something that he can’t do: create the stone.

If God can create a stone that is so heavy that he can’t lift it, though, then he also isn’t all-powerful. If God can create a stone that is so heavy that he can’t lift it, then there’s something that he can’t do: lift that stone.

There is, therefore, no way of answering the question above that preserves God’s omnipotence. If there is an omnipotent God, then he neither can nor can’t create a stone so heavy that he cannot lift it. This, though, is absurd; he must be either able or unable to perform this feat.

This is the paradox of omnipotence. Many critics of theism have used it to argue that the concept of omnipotence is self-contradictory, that there can be no omnipotent being, and so that God cannot exist.