# Running, Logically

There’s no evidence that Charles Dodgson (who you probably know better as Lewis Carroll) was ever a runner, but he gets to the essence of competitive running in his book Through the Looking Glass, and What Alice Found There when he writes, “It takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”

Dodgson was also a mathematician. He was a serious scholar, but still found ways to indulge his taste for whimsy, a desire we here at Mathematical Runner can identify with. One way he did this was by creating sorites, puzzles made up of a series of statements (premises) that, when analyzed by the rules of logic, lead to a conclusion. Here’s an example from Dodgson’s book Symbolic Logic:

All babies are illogical.
Nobody is despised who can manage a crocodile.
Illogical persons are despised.

You can think of this as a series of “If->then” statements:

If it is a baby then it is not logical.
If it can manage a crocodile then it is not despised.
If it is not logical then it is despised.

…which you then chain together to find the conclusion.

Here, you also need to know that if a premise is true, so is its contrapositive. Thus, given “If it can manage a crocodile then it is not despised”, we know that “if it is despised then it cannot manage a crocodile”.

Therefore, the solution to the puzzle is “No baby can manage a crocodile.”

Here, we usually use math to look at running. But this time, we’re going to use sorites based on running to illustrate the math. For example:

My hats are the only things I have that are made of tin;
I find all my running shoes very useful;
None of my hats are of the slightest use.

Therefore: My running shoes are not made of tin.

Here’s a more challenging example:

All runners, except ultramarathoners, have a certain amount of common sense;
No one, who lives life to the fullest, can be anything but well-trained;
None but a hop-scotch player knows what real happiness is;
No one well-trained has a grain of common sense;
No couch-potato ever plays hop-scotch;
No ultramarathoner is ignorant of what true happiness is.

No couch-potato lives life to the fullest.

“What’s all this got to do with running?” you might ask. Well, as an internet user you might have noticed that there’s a lot of arguing going on these days, about running and any number of other topics. Maybe if people try applying formal logic to a topic they enjoy, they’ll be able to better apply those same principles to more serious topics.

Or you can stop worrying about it and just have a little fun.

Anyhow, here are a few more sorites for you to ponder. The answers are below (no cheating!).

1:

No runner is too slow
I usually finish races before the walkers
Anyone who enters a race is a runner

2:

My coach is well worth listening to about running;
No one can remember the Mexico City Olympics, unless he is very old;
Nobody is really worth listening to about running, unless he can remember the Mexico City Olympics.

3:

The only exercises, that my doctor allows me, are such as are not painful;
Nothing that makes me happy is inappropriate;
Injuries are always painful;
My doctor allows me all exercise that is appropriate.

4:

Every idea of mine, that cannot be expressed as a Syllogism, is really ridiculous;
None of my ideas about winning marathons are worth writing down;
No idea of mine, that fails to come true, can be expressed as a Syllogism;
I never have any really ridiculous idea, that I do not at once refer to my coach;
My dreams are all about winning marathons;
I never refer any idea of mine to my coach, unless it is worth writing down.