Killgrace Books

Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic). This month we are moving onto a traditional Egyptian puzzle.

The Problem: A woman is walking by a river with her child when her child is snatched from her by a crocodile. The crocodile tells her that it will return the child if she can correctly answer the question “What do you intend to do with the child?”

The Answer: The mother is traditionally supposed to reply “You intend to keep it.” to guarantee the return of her child.

 With this one it is not the problem that breaks, it is the solution. The crocodile could equally well answer false, and eat the child on the spot. Eating something is not the same as keeping it. It could also drop the child in the river so neither of them have it. The error is in that assuming the crocodile has only two options – to keep the child or hand it back. It also has options to destroy the child or dispose of it so no one has it.

Even “You intend to keep it” does not guarantee the return of the child or its condition when it is returned. Worse if the crocodile is, as many in myth were, a sage akin to an Eastern Dragon, the answer might actually be false. Its intent may be simply to test the mother’s intelligence.

So what could she say? The actual answer could be a negative. “You do not intend to return the child safely to me.” However there is no room in the question for a negative. Likewise the conditional, “You intend to return it if I can answer your riddle” may be disqualified although it is undoubtedly true.

Or as a mathematician friend put it: “You intend to use it to make me answer a stupid bloody question.”

Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic).

The problem: If a man has a full head of hair, removing one hair will not make him bald. Therefore the act of removing one hair cannot make a man bald.

The Answer:
This is a well-known paradox by Eubulides, a greek philosopher.
However the standard answer is incorrect. The definition of bald is “possessing no hair on the scalp”. If hair still remains, the individual is balding – that is, in the process of going bald. Therefore if there is no more hair to remove, the individual is bald.

This type of problem is the sort of thing programmers have to solve all the time, and if it was a true paradox then most coding would be impossible. Viewed programmatically, the paradox fails very easily:

$h = 100,000; //total amount of hair on head
$bald = false; //person is not bald
for (h, h=0, h–) {
   //remove one hair at a time until none remain.
If h ==0 {
bald = true;// person is now bald.
}
}

You could even take this a stage further. “Balding” is described as in the process of going bald – having lost the majority of hair. Majority is defined as more than 50% – so…

$h = 100,000; //total amount of hair on head
$total-hair = //original amount of hair on head
$status = “not bald”; //person is not bald
for (h, h=0, h–) {
  //removes one hair at a time until none remain.
If (h ==0) {
$status = “bald”;// person is now bald.
}
elseif (h<=( ($total-hair/2)-1) ){
$status = “balding”; //person is balding
}
}

Ignoring my rather pedantic programming digression, the correct answer is simple.

When no further hair remains to be removed, the individual is bald.

Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic).

The Problem: The Athenians preserved Theseus’ ship for posterity. This meant replacing damaged parts as they wore out over time – the sails, the timbers, the decorations. At what point is it no longer Theseus ship?

The Answer: This is a curious one because there isn’t any true answer, but it is an interesting debate.

Some people would argue that it stops being Theseus’ ship when the last original part is replaced. This overlooks one thing – that the ship is not just its physical existence, it is its design. A question that must be asked is whether the original sailors would recognise it. If Theseus could see it today and immediately identified it as his ship, would anyone have the right to gainsay him even if all the timbers had been replaced?

I’ll go back to programming for an example. In Object Oriented Programming you have a class, which is a type of object and an instance, which is a specific member of that class. In this case we have a class: Ship and an instance: Theseus’ ship.

To make it easier to understand, let’s take a more radical change. Suppose we had class Password and an instance Mary’s password. Mary forgets her password, so she changes it to something completely new with no link to the old one. The new password is still Mary’s password. It still does everything the old password did, it unlocks the account, it needs to be remembered etc.

It isn’t Mary’s password any less because the letters have changed. It had better not be, since if it isn’t Mary’s password the login program won’t work and Mary’s locked out!

So going back to class: Ship and instance: Theseus’ ship, is changing the timber enough to make it not his ship? Or is it the point where the program – in this case the population of Athens – no longer recognises that instance as Theseus’ ship no matter how much it may have changed?

There is an extended problem, where the rotting planks are used to construct a vessel at another location so there are now two Ships of Theseus, and the issue is determining which is the true ship. This is more easily disposed of by anyone familiar with how quickly wood rots and sails break down. The second constructed ship will never at any one time have the entire body of Theseus’ ship present at its location. The fact the ship may have timbers from the original does not make it the original.

As a side note, this issue has actually been encountered in aircraft restoration – if parts of two spitfires are used to restore a third, which aircraft does the restoration count as in terms of flying hours, etc? It counts as the aircraft originally registered, not the ones used to restore it regardless of parts. The Iron Duke steam engine was broken down to save space in 1919 only to have a replica rebuilt at a new location using many of the salvaged parts less than a year later. The new engine still counts as a replica, not the original.

Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic).

The problem: If the chance of someone in the population being hit by lightning is 650,000 a year, and Kelly is hit by lightning in 2009 what are the chances she will be hit by lightning again in 2010?

The Answer: The standard answer is one in 650,000 as years do not affect each other. However this is incorrect. Kelly is no longer only part of the group “general population”, she is also part of a smaller group: people who have already been hit by lightning. The chances of that group being hit may not be the same as the general population.

This is because people who have already been hit by lightning include a subset, and in some years a majority, of people who engage in behaviours more likely to get them struck by lightning – e.g. steeplejacks, deep sea anglers, high altitude construction workers, mountaineers, and depending on your definition of lightning, electricians who immediately skew the odds.

Their chances of being struck by lightning are higher. The average (mean) chance across a group is the total number of probabilities divided by the number of members in the group. As there are more people in the group with higher risks, the average chance of being struck by lightning again is higher. By being struck by lightning, Kelly therefore enters a higher risk group.

However, even this is incorrect. What we do not know is how Kelly reacted to the first strike. If she changed jobs, decided she was not at risk, or even if she survived the strike. This is why mathematical models must be treated with caution.

The correct answer is two-part:

1. i) given no changes in circumstance (ceteris paribus – all things being equal) her chance of being struck by lightning should be the same as the year before.
2. ii) without knowing more about her behaviour and circumstances, that probability is impossible to determine.

To replace the videos, which I proved to be dreadful at keeping up with, I’m starting a new series for 2015.

Pedants Paradox was inspired by a christmas gift that had me tearing my hair out: a book of paradoxes. The problem is that most of them aren’t paradoxes, those that are are presented badly, and a statistician of my acquaintance did not think much of the math. Therefore, I’m picking them apart in the hope it will provoke a bit of a debate on the blog.

I’ve got one a month coming out on the fourth Monday of each month, and several already written and queued up. Feel free to argue it out in the comments or message board.

(I’ve also finally figured out how to syndicate this, so tumblr can start getting some traffic.)