Carroll’s Paradox

CARROLL’S PARADOX

Lewis Carroll is probably best known for being the author of Alice’s Adventures in Wonderland but he was in fact a mathematician called Charles Lutwidge Dodgson who worked at the University of Oxford. He wrote about a logical paradox in an article entitled “What the Tortoise Said to Achilles” for the philosophical journal Mind. The article is written in the form of a dialogue, in which a tortoise argues Achilles into a corner, or rather, an infinite abyss. His use of the tortoise and Achilles characters is a nod to Zeno’s paradoxes.

In Carroll’s paradox, he explores how logical arguments are built up by logical implications. Using logic requires rules of inference; adding further premises cannot replace the act of applying a rule. Carroll’s paradox is about what happens if we don’t do that – we just never get there, endlessly filling in more logical steps and never reaching a conclusion. It’s like trying to draw a genuinely infinite fractal tree, rather than one that just looks filled in enough – you can never finish in a lifetime.

Carroll’s tortoise asks Achilles to compare two sides of a triangle to see if they are the same length. You might pick up a ruler and measure both sides and discover they are both 5 cm and so they’re the same as each other.

This involves the statements:

A: Both sides of the triangle equal the length of 5 cm.

Z: Both sides of the triangle equal each other.

The tortoise asks Achilles if Z follows from A, and Achilles says yes of course. Perhaps you agree. But the tortoise says this is only true if we know that A implies Z. So we have another statement mediating between them:

A: Both sides of the triangle equal the length of 5cm.

B: A implies Z.

Z: Both sides of the triangle equal each other.

Now the tortoise asks Achilles if A and B together imply Z. But that is itself a new statement:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

Z: Both sides of the triangle equal each other.

Now the tortoise asks if A, B and C imply Z. But that is a new statement:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

D: A, B and C together imply Z.

Z: Both sides of the triangle equal each other.

The story ends with the tortoise still sitting there torturing Achilles by making him write down all these intermediate statements; it becomes clear that this will go on forever.

So how do we ever deduce anything from anything else? Or have we in fact never correctly deduced anything from anything else? The answer is that we have to use the rule of inference, modus ponens (mentioned in Chapter 4), which we must assume to be valid in order to get anywhere at all. This paradox warns us that there is always a level of meta-logic controlling our logic, and we can only understand this by keeping those levels separate.

This is somewhat like attempts to find all the factors that caused something to happen, as we discussed in Chapter 5:

A: I dropped the glass.

B: A implies Z because the glass was too fragile.

C: A and B imply Z because the floor was too hard.

D: A, B and C together imply Z because gravity intervened.

E: A, B, C and D together imply Z because I didn’t catch the glass.

F: A, B, C, D and E together imply Z because nobody else caught the glass.

G: A, B, C, D, E and F together imply Z because... . . .

Z: The glass broke.

If we don’t decide enough is enough, we will never be able to conclude anything.