# Logical fallacy in “Pyramids” by Terry Pratchett

Teppic meets Ephebian philosophers who are trying to prove that a tortoise can outrun an arrow. I understand this is because the tortoise moves, so when the arrow gets to where the tortoise was, it has moved on.

Then there is this bit:

"On the way to the tavern Xeno had explained to him, for example, why it was logically impossible to fall out of a tree."

Please can someone tell me what he is telling Teppic? How is it logically impossible to fall out of a tree? Does he mean that by the time you reach the ground, the Earth has moved?

(Sorry if I've used the wrong words here, Orcs are not strong on philosophy.)

• Perhaps because one's usually not actually in a tree? – Solemnity Mar 9 '13 at 9:39
• Paradox! That's the word I couldn't think of! (c: – NiceOrc Mar 11 '13 at 10:47

The paradox where the tortoise can outrun the arrow is the real life 'Achilles and the Tortoise', which you already seem to understand. Achilles (the arrow) can never catch the Tortoise because by the time he reaches the tortoise's position the tortoise has advanced a little - then when he has caught up to that position it has advanced a little more, and so on.

The 'impossible to fall out of a tree' is referred to in the article as "Dichotomy Paradox". In short: in order to fall from the tree to the ground, you must first fall halfway. To fall halfway you must first fall a quarter of the way, and before that an eighth and so on. In fact before travelling ANY DISTANCE AT ALL there is some distance you must travel before doing it. Hence you can never move at all, because there is always some other step (moving a smaller distance) that you must do first.

It's important to note that Zeno (apparrently unlike Xeno) didn't believe that movement was impossible, but was pointing it out as a philosophical problem. Finding the actual flaw is in the logic is actually quite challenging.

• You are overall correct, but let me point out the Arrow Paradox and Achilles and the Tortoise are two separate paradoxes :) – Andres F. Apr 2 '13 at 21:36
• It was later in the pub though that Xeno came to refute his own hypothesis. I believe the conversation went thusly "So if I try punch you in the face, my fist will never hit you?" Shortly there after Xeno got punched in the face. P.S. I just made that up, but honestly when someone tries to argue that reality isn't real I always offer to punch them in the face (or if they're particularly obnoxious and male, kick em in the plums) cause it all starts feeling very real then :) – Binary Worrier Apr 3 '13 at 7:59
• @BinaryWorrier In the Discworld novels, the whole point of said philosophers is that they will argue endlessly about anything. In Small Gods, some Ephebian philosophers "prove" the gods don't exist, after which they seek shelter to avoid the wrath of said gods. – Andres F. Apr 4 '13 at 5:32
• An old joke goes like this: A Mathematician, a Physicist, and an Engineer are at a school dance. All the boys are one one side of the gym, all the girls on the other. Every few seconds they close half the distance between them. When asked how long it will take them to meet, the Physicist says, "never," the Mathematician says, "an infinite amount of time," and the Engineer says, "in a couple of minutes they'll be close enough for all practical purposes." – John Bode Apr 4 '13 at 18:38
• Old question, I know, but I do wonder how they ever managed to get to the pub? – Steve Ives Jul 18 '16 at 12:33

Pratchett is referring to one of Zeno's paradoxes, outlined by the Greek philosopher that is the basis for the Ephebian philosopher.

Loosely paraphrased, the Arrow paradox talks about how taking an analog, continuous motion (the arrow flying towards the target, and in this case a man falling out of a tree) becomes impossible if you break down the motion into discrete steps. No movement occurs in any discrete step, but if the whole is comprised of the steps, how can it have movement?

• If I understand correctly, you cannot fall from a tree, only choose to experience moments of levitation? :) – Eureka Mar 9 '13 at 9:49
• It doesn't seem obvious to me that the "impossible to fall out of a tree" bit has to refer to the same idea explored in the Arrow Paradox before. – Michael Borgwardt Mar 9 '13 at 15:10
• being as it's Xeno talking, he probably does have it on his mind. – jwenting Mar 9 '13 at 16:45
• The Arrow paradox and the Dichotomy paradox, outlined in the link I gave, can both account for the "never reach the ground" statement in the original question. They are also two of the most famous paradoxes attributed to Zeno. The third paradox is the one with Achilles and the Tortoise, which is explicitly referenced (appropriately mangled, of course) in the same paragraph. – Avner Shahar-Kashtan Mar 9 '13 at 18:48

Not being aware of a variant of the Earthly Xeno's arguments that cover falling out of tree in particular I have always assumed that the author was simply giving us another example of the Disc's Xeno preferring pure logic over real world experience to the point of asserting things that are easily shown to be untrue.