Insufficient Logic Does Not Compute (Properly, In Any Case)

Sometimes, insufficient logic can lead to the wrong result. That is, either less or more analysis would derive the right answer, but with an intermediate level the wrong answer results. Occasionally the wrong answer gets enshrined in folklore. Often the wrong answer is deliberately induced by clever manipulation of the presentation.

One example that came up recently is a really old puzzle that I remember from my childhood - long before steamboats. The puzzle, like most, can be told in several guises and elaborated in numerous ways, but the underlying mistake is always the same. The protagonist in every version I’ve heard is a young boy, so I will stick with that detail:

An agile youth comes to a long bridge that is arched up toward the middle as most bridges are. The bridge has a sign saying that it can safely carry only 100 pounds. The boy weighs 96, so that is okay, but he is carrying three 2-pound balls for a total of 102 pounds. The bridge is too long to throw a ball across it and the arch is too high to permit rolling a ball across. Yet after a few minutes of concentration, the boy sets off with the balls and safely crosses the bridge. How does he do it?

We can overlook the obvious absurdity of the bridge only carrying a small load. Similarly we can accept with a grain of salt the possibility that the safety margin was zero and that the load limit was that well known. And the process of overlooking obvious scene-setting for a logical problem is the sucker trap that lures people astray. We are used to thinking that in such an obviously artificial situation, this is a logical puzzle requiring some clever stepping outside the box to solve. Therefore instead of thinking as though the boy were in the real world with real balls crossing a real bridge, we are lulled into thinking of a mathematical bridge, boy, and balls.

With that mental set, we cleverly think that the boy can carry two balls, but suppose the other one is in the air? That’s it! He simply juggles the balls as he crosses the bridge. At any time one is in the air and he never exceeds the load limit of the bridge. After all, we are also told he is agile, so he must be able to juggle.

I’ve actually seen this answer given in print as the correct one. Yet it is obviously bogus. A plot of the force exerted on the bridge as a stationary boy juggles would oscillate up and down by several pounds as he throws the balls, but the average would always be 102 pounds. When he throws a ball up, the force downward increases, and when he stops the falling ball, the force increases. He could accomplish much the same effect by simply hopping. When he is in the air, he isn’t exerting any force on the bridge at all. And that brings up another problem. When the boy is simply walking across the bridge without any balls, he could exceed the weight limit momentarily depending on how he walks or runs, and if he should stumble and fall, the bridge would collapse!

It’s easy to dismiss this obvious case as a childhood puzzle, but the psychological tricking us into using the wrong mental model to solve problems is not always so obvious. We are constantly bombarded with advertisements, religious warnings, and political pronouncements that sound like good logical presentations and lead us to a conclusion that might not be in our best interests.

Since I don’t want to offend any manufacturers, preachers, or politicians and get sued, I will use a safe example involving sex. I read part of a presentation by an abstinence-only anti-sexual activity speech. The presenter was trying to make the point the STDs are extremely dangerous and normal contraceptive methods are inadequate protection. “Would you trust your life to this?” he exhorted as he held up a flimsy piece of thin latex. Think about the logic and misdirection of that! It would do a parlor magician proud. Another fallacy he used in the same presentation was to say that abstinence in the only 100% effective way to avoid unplanned births. Sounds good - you don’t do it, and you don’t get pregnant. Except that when a method of birth control is proposed, it must be tested, not as an ideal logical thing, but as ground truth in the field, and when that is done, abstinence is probably the least effective of the common methods of birth control. Its failure rate as a method is greater than the failure rate of other techniques. This is totally independent of moral or health issues.

Maybe in a later article I will take on some of the religious and political misdirections that are designed to impede accurate analysis. Readers with pet peeves are welcome to suggest targets.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.