Conventional Wisdom Archives - The Silent Knight

A Better Solution to the Blue-eyed Monk Problem

The Problem

There are variations to this problem, but we will go by the XKCD version discussed in www.xkcd.com/blue_eyes.html. It goes:

“A group of people with assorted eye colors live on an island. They are all perfect logicians — if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.”

“On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.”

“The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following: ‘I can see someone who has blue eyes.’”

“Who leaves the island, and on what night?”

The Traditional Solution

The traditional/standard/irrefutable solution is described everywhere, including in xkcd.com/solution.html. Basically, if there were only one blue-eyed monk (BEM), then he wouldn’t see any. He would realize he must be it, and leave that night. If there were two, they would each see only one and would assume the other guy would leave that night. When they saw each other the next day, they would both leave. And the people who were worried that they might be Number 3 would all breathe a sigh of relief. And so on. For the case described above, all 100 would leave on Night 100.

Disclaimer (and Personal History)

There is a reason why I was never invited to Logical Island. When I first read this problem, decades ago, I didn’t even figure it out on my own. I cheated and read the solution. But then I thought, “That’s pretty slick”! When I ran across the problem again, a decade or two later, I remembered the answer and still thought it was pretty cool. But I thought it was too bad that there wasn’t a quicker solution. The third time I ran across it, another decade or two later, I became obsessed with finding that solution. It wasn’t going well, but a few nights later it occurred to me that the answer might involve modular arithmetic^Explained. A test using Mod 6 was effective.

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In the Mod 6 Method, the “proof” proceeds normally up through five BEMs. Those seeing four leave on Night 5. But for the case of six BEMs, those seeing five (you know what that means) would await their findings on the morning of Day 5 to decide their fate. Those seeing six BEMs, however, would leave the first night since six is 0 (mod 6). Their first clue as they approached the dock would be the dozens of other monks at the dock instead of the six others they were expecting. Even more important, perhaps, is that none of them had blue eyes. They would all have a good laugh (probably against the rules), and then quietly go back to bed. Those who originally saw five (a.k.a. 5 mod(6)) were sleeping when all of this happened and wouldn’t know anything about it. The problem would take care of itself four days later.

The next logical question would be “Why is six so special?” If all of the monks can’t agree on the same base/modulus without communicating, the scheme fails. I introduced my idea on a different forum. (I posted a couple of blog articles here as the plan developed, but they have been superseded by this one.) One commenter quipped (sarcastically), ‘if that scheme is even legal, what’s to stop you from just having everyone who sees an even number of BEMs go to the ferry on Night One?’ I thought that was way too extreme, but as I was typing “ridicul . . .”, a bell went off in my head. Of course. That’s it! Mod 2. But that commenter wasn’t buying it.

The Fastest Answer

In my plan, everyone who sees an even number of BEMs (remember, zero is an even number) goes to the dock the first night. If when they got to the ferry, everyone else there had blue eyes, they would all board the ferry. Otherwise, having non-blue eyes, they would all quietly return home. If the problem isn’t solved that night, everyone that sees an odd number of BEM’s (everyone else) leaves on the second night.

Let’s say there were 99 BEMs (or 101). They would all see 98 (or 100) BEMs, and they would (as in the case of one BEM) get on the ferry the first night. That would be that. But if there were 100 BEMs, those seeing all 100 would all go to the ferry the first night. They see nothing but non-BEMs and all go home quietly. The next morning, those seeing 99 and hoping that those seeing 98 acted accordingly, would realize that wasn’t the case. They would all leave that second night (as in the two BEM case). End of story.

Advantages

If there are 200 monks between the ages of, say, 19 and 85, with minimal turnover, you can expect three deaths each year. That means there is about a 50% chance that the number of BEMs will mysteriously change while the 100-day clock is ticking. (I’ve heard no discussion about how the traditional method handled such things. But a similar situation is addressed in the next-to-last paragraph [Go]). My new method virtually eliminates that possibility. Even if nobody dies, there is over three months of meticulous counting that can be avoided.

Other Concerns and Considerations

More About the Technique

As we’ve discussed, this method is actually based on the traditional method, but folds the number/time line into the smallest possible segments using modular arithmetic. For a monk that sees 100 BEMs, there are only two possibilities. Either there are 100 BEMs (which all see 99) and he is among the 100 others, or he is one of 101 BEMs. Nobody seeing 100 thinks there might be 102 or 99. If you see 100 and think you might be one of 101, then you don’t care what the non-existent monks that see 102 BEMs think. For those seeing 100 BEMs, this scheme folds the other monks from both possibilities – those seeing 99 and those seeing 101 – into heading to the ferry the same day.

As mentioned, the solution must be so logical as to be adopted unquestionably by all monks without discussion. First, remember that they were able to come up with the original method without discussion (something I couldn’t do). Why would they stop at the first answer they came up with when they can clearly see the disadvantages to that approach. This answer adds another simple mathematical concept for the next logical step. But if you know of another logical solution candidate, even completely unrelated to these techniques, speak right up.

I mentioned that there are variations to this puzzle. This technique would not work in variations that require the monks who discover their eye color to quietly commit suicide in their homes. And it is the ferry part, not my new technique, that would minimize the damage of an oracle mistakenly (or maliciously) claiming he saw a BEM when there were actually none.

Questions Answered

Even if based on the original method, doesn’t the mod 2 math allow two possible answers? Why wouldn’t they have those seeing an odd number of BEMs go first? Because the “let odd go first” plan is not the direct result of simple mod 2 math. And the first two cases (1 and 2 BEM’s) in my technique are exactly the same as the original method.

I’ve heard that the inductive method proves that the original technique is the only one possible. Nonsense! All mathematicians know there are commonly more than one way to solve or prove anything. Just a couple years ago, two teenage girls found a new trigonometric proof to the Pythagorean Theorem^Article.

I’ve heard that these monks are illegally communicating information. No, they are giving the same information at night as they were giving during the day – no more information than in the original approach. In either case, when the lone BEM disappears the first night, he is telling every other monk (that was worried that he might be Number 2) that they can stand down.

But what would happen if some of the monks stuck with the old technique? Well, I guess all monks weren’t perfectly logical. But if there were an odd number of BEMs, for example, and only some of those seeing an even number go to the ferry, as my plan dictates, then they would see other BEMs and leave as required. The next day, those who chose the traditional solution would notice that there were far fewer BEMs. Regardless of whether the new number of remaining BEMs was odd or even, they would realize what happened and leave the next night.

(If they still couldn’t figure it out, they could at least proceed with the traditional plan using the new BEM count at the original starting point. [Return to Advantages]

Would they be allowed to take action without knowing their eye color? Nothing in the rules restricts their access to the ferry dock or prevents them from noticing the eye color of those they meet. I would expect logical monks to be curious and willing to conduct simple experiments that could benefit themselves and the group.

Any questions?

Will 2020 See The Extinction Of The Red-eyed Monks?

For those of you not familiar with what has been called “the hardest logic puzzle in the world”^Link, The Blue-eyed Monk problem or Blue-eyed Islander problem (Another version), check out the above reference links. For what it’s worth, that “hardest” title is probably an exaggeration. But my new version, inspired by recent events, might be harder.

In this version of the famous logic problem, there are a number of monks on this remote island. All of them are “perfect logicians”. Every monk sees every other monk every day, but they are rugged individualists who never talk to each other or communicate in any way. They also take full responsibility for their actions and obey all rules and protocols. (Yes, this problem is entirely fictional.) Strangely, there are no mirrors or reflective surfaces on the island. The ferry visits the island every night to drop off supplies. It would take the monks to the mainland, if necessary. But none of the islanders wants to leave and no strangers are allowed on the island. In this version of the problem, all of the monks have red eyes.

One day, all but one of the monks noticed that one of their peers had blue eyes. But they all continue to go about their lives. Many days later, almost all of them notice that two of the monks now have blue eyes. Still, life goes on. This trend continues with a slowly increasing frequency until many, many days later, the first monk to have developed blue eyes is found dead. Doctors on the mainland soon discover that some as-yet-unknown, but contagious pathogen hit the island. The number of monks infected has been growing 2% every day for quite some time. The only visible symptom is the changing of eye color from red to blue. The disease is not contagious until the eyes go blue but then remains contagious until the infected person dies. How it spreads is still unknown, but obviously, direct contact or even close proximity is not required.

By the time the doctors figure all of this out, red-eyed monks have started changing eye color at the rate of one every other day and the second blue-eyed monk has just died. If a blue-eyed monk gets to the mainland in time, they can be cured. but the ferry does have a limited capacity. The monks are pacifists, so there will be no shooting of blue-eyed monks (yes, I might have a personal stake in this directive). All of this information was left on a large sign at the ferry dock the next day.

So what do the monks do? Will they go extinct? Will 2020 see the end of all logical thought on this planet? Only you can answer that question. Good luck! Leave your answers in the comment section below.

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In the interest of full disclosure and transparency, I have addressed the original monk problem before. After much fumbling, I did come up with a better (faster) solution, which was widely regarded as cheating (see A Better Solution to the Blue-eyed Monk Problem.

A Pair Of Perspectives On Pearls

Several Sundays ago, our pastor began his sermon by talking about the world’s largest pearl.

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This pearl, which was just recently revealed after being hidden for ten years, is 26 inches long and 12″ wide, weighing 75 pounds. That’s over 2½ times longer and five times heavier than the previous record-holder, which was found in another giant clam near the same Philippine island in 1934 (both the diver and the clam lost their lives in the earlier case).

Filipino Fisherman Reveals 75-Pound Pearl He Kept Hidden For A Decade.

Our preacher used the pearl as an example of how God can take an irritant and help you turn it into a treasure. I believe that is a common way of looking at this from a human perspective. But for some reason, I saw the issue differently.

Another Possible Moral To This Story

Turning trash into treasure – is that what the clam was actually trying to do? Does the clam even know that the old irritant now has such great value (estimated to be over a hundred million dollars)? Was this 75-pound object, which started as a gain of sand and had been growing for over a hundred years^A, actually less irritating to the clam than the original grain of sand? I suspect not!

The preacher could have used this as an allegory, showing how man, because of pride, will try to solve a problem by himself but fail, despite putting a great deal of effort and time into it. And, as is often the case, he could even make the situation worse, not better. “But look how pretty I made it.” Can’t you just see that giant clam trying to sing Frank Sinatra’s “My Way”^video, but failing, of course, because it has a 75-pound lump of calcium in its mouth?

Is There A Third Possibility?

One of the beauties of life is that some situations can share many lessons. Reality is like that. Even as I was writing this, a third perspective began taking shape in my mind. Of course, I ignored it. But if you found another pearl of wisdom in this parable, please share in the comment section below. Thank you, and thanks for listening.

Not Quite Clear On The Concept “Innocent Until Proven Guilty”

Listening to discussions on Facebook about the Brett Kavanaugh nomination, I was surprised and disappointed to see the “innocent until proven guilty” principle so often misapplied (and in two different ways). It became clear to me that a lot of people just don’t understand the concept.

A Larger Doctrine

When something is being awarded to somebody, whether it is good, bad, large, or small, most people would like to think the recipient deserved the award. It is the presenter’s responsibility to make sure that’s the case. The more extreme the action, whether reward or punishment, the more effort the presenters should take to see that the award has been earned.

The Bad

When the award is a punishment, this doctrine takes the form of “innocent until proven guilty”. If the death penalty is under consideration, for example, we need to go to great lengths to be sure we aren’t making a mistake. I’ll save the discussion of the two types of possible errors – letting a murderer go free vs. hanging an innocent person – for another day.

The Good

When the action under consideration is a reward, one would expect some law of symmetry to apply, and it does. In this case, the slogan “innocent until proven guilty” has no place. Whether it is the Mega Millions jackpot or the Nobel Prize, one does not assume a prospect is ‘innocent’, or deserving, until proven otherwise. It is up to the claimant to prove they deserve the award. For the Mega Millions jackpot, that would be by showing the winning ticket and some form of identification. The Nobel Prize has even more stringent requirements. Republicans have no trouble applying this principle to welfare recipients but seem to get tripped up when it comes to Presidents and Supreme Court nominees.

The Ugly

In either of the above cases, it is well understood, as stated above, that the candidate will be fully vetted. And the more significant the award, the more serious the investigation. For someone to insist that a candidate is “innocent until proven guilty”, especially for a reward, and then refuse to hold a meaningful investigation into any evidence of guilt is the height of duplicity. But that seems to be the current state of the Republican Party. It hasn’t always been this way.

Anyone with a logical alternate interpretation of the facts is welcome to share. You can be sure the civility of this discussion will be maintained.

Please Help! The Special Theory Of Relativity Is Haunting (or is it Taunting) Me!

A long time ago (when I was in the sixth or seventh grade) in a galaxy far, far away (namely Southern California), I was introduced to Einstein’s Special Theory of Relativity by way of a story about two astronauts on different spacecraft watching a bouncing-light-beam clock, and I was really impressed. But upon further review as I was chewing my cud (See definitions of “ruminate”), as I was wont to do walking home from school, it didn’t seem to make as much sense. I developed some questions but didn’t know where to get answers, and as life pressed on my attention wandered elsewhere, and everyone lived happily ever after. . .

Until recently. In the last year, the subject has come up several times, the questions seem to be the same, and I still don’t know where to turn.

The Story

Although not exactly as I remember it, www.dummies.com describes a similar thought experiment in the second section, “Unifying space and time”, with a spacecraft traveling at ½ the speed of light, but doesn’t give much explanation. The Star Garden gives a more detailed explanation.

The Problem

“Time Dilation”, Section 7.2.2 of Reference 2 concludes

“The time between heartbeats is also slower, and so from the perspective of a stationary person, a moving person appears to be living their life at a slower rate. Conversely, from the perspective of the moving person, the stationary person seems to live their life as if it is being fast-forwarded. If they travel fast enough, then they could see the stationary person age before their eyes.”

My problem with that conclusion is that based on the second paragraph of Section 7.1 at the beginning the article, which states

“there’s no such thing as absolute speed or velocity, and something can only be said to be moving at a constant velocity relative to something else. In the same way, something can only be said to be stationary relative to something else”,

how do we really know which astronaut is supposed to be aging before our eyes? What if we put a bouncing-light clock on each spacecraft? Would it explode?

The authors of Reference 2 seem to address this issue at the bottom of the next section, 7.2.3, where they say

“The twin paradox

The twin paradox asks why the astronaut can consider themselves to be moving and the Earth to be stationary, when Galileo’s relativity shows that there’s no such thing as absolute velocity. Why can’t the astronaut consider themselves to be stationary while the Earth moves away at tremendous speeds?

The answer is acceleration. Galileo’s relativity applies to inertial – that is non-accelerating – reference frames. The fact that the astronaut must have accelerated before getting to such high speed means that they know they are the one that is moving.”

To me, this sounds bogus; any acceleration before or after the experiment should be immaterial. Let’s have three observers; one person remains on Earth while two astronauts board different spacecraft, each leaving the Earth in opposite directions and reaching similar stable speeds well in excess of ½ the speed of light (meaning their relative speed would exceed the speed of light in a non-relativistic world). Each of the observers has their own bouncing-light clock. If you start counting after their speed stabilizes, exactly how do each of the three observers see the ages change for the other two?

One Last Question

A question that one might ask in each of these scenarios is “how does the state of the bouncing light in one spacecraft become known to the other observers?” Reference 1 states that Amber, on a different spacecraft, would see the bouncing light travel further between bounces, as if Amber had super X-ray vision and/or was otherwise experiencing the light beam in real time. How does that work? If she had to wait for reflected light rays from the event to reach her eyes, would that affect the apparent outcome in any way?

One phenomenon that may or may not have anything to do with the solution to this problem involves ocean waves. In deep water, a wave’s speed is nearly proportional to the square root of its wavelength^A,

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$S \cong 1.251 \sqrt{W}$

where S is the wave’s speed (measured in meters per second) and W is its wavelength (in meters).
For shallow water waves, the speed is proportional to the square root of the depth.

$S \cong 3.1 \sqrt{d}$

where d is the water’s depth (in meters).

but in all cases, it is much less than the speed of light. If an observer were to watch the crest of a wave as it moved along a seawall, or along any imaginary line that wasn’t along the wave’s direction of travel (directly away from a point source, or in the direction of the wind, or perpendicular to the wavefront, etc.), then the apparent speed would be greater than the calculated or expected speed, and as the angle of that reference line approached 90° to the direction of travel, the apparent speed would approach infinity.

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$A = \frac{S}{\cos \theta}$

where A is the apparent speed, S is the expected speed, and θ is the angle between the reference line and the direction of travel.

which is well above the speed of light. As far as I know, this has no implications or gives no reason for hope for wannabe time travelers.

In Summary

So now you see my dilemma. To repeat the title plea, please help me understand. A crucial early step in solving any problem may be asking the right questions. Finding those should be as important, and in some cases may be as difficult as answering them. So let’s get started. Thank you for your help. If I do figure it out without your help, I’ll let you know.

We Can Simplify Our Student Grading System

In the works, I have two different questions for you:

‘Do Medium-sized Egos Really Exist?’, and
‘Should Law Enforcement Officers Be Allowed To Use The “I was afraid for my life” Defense?’

Both of these require some preparation/research, but I hope to have them ready before too long. For now, I’ve chosen a lighter topic about a scheme that, because it’s not being implemented as designed, could well be simplified.

Some Background

When I started school, we got one grade, from A to F (I never learned why E was left out), to represent our mastery of the subject. Then, at some point, they introduced a separate grade for effort (from 1 to 3) and another for conduct (also A through F, also without the E). These were promoted as independent variables that could give more insight into the performance of one’s child. I soon had reason to question the independence of these variables.

What’s The Best Grade You Can Get

Conventional wisdom tells us that the highest grade one can get would now be an A1A. I’m not here to discuss the merits of bad behavior, so we will focus only on the first two symbols. To me, it was obvious that an A3 would be more desirable. Here’s why –

Suppose it’s a leap year and you are betting on track events at the Summer Olympics. In the first heat, the first place runner comes in with a time of, say, 4:00.00. At the end she is visibly spent (lying on the ground, breathing heavily, and sweating profusely). Her grade would clearly be an A1. In the next heat, the winner has the exact same time but isn’t even breathing hard. I would give her an A3. Keep in mind that it is not uncommon for runners in the early heats of big events to pace themselves to save some effort for later heats, if they can afford to.

Of course, both runners advance to the finals with the best times. Again, conventional wisdom gives the higher grade to the first runner. But tell the truth – which one are you betting your hard-earned money on in the finals?

My Experience

So you can see what grade I was trying for. But the truth is teachers don’t give A3 grades, even if you never turn in your homework. This isn’t a case of political correctness (whereby we fashion our remarks based on the possible objections of imaginary people with hyper thin skins or real fools priding themselves on how easily offended they can be). It is another common problem in the political arena whereby people refuse to let facts get in the way of their idea of the way things should work in their perfect (but grossly oversimplified) world. In their view, the very fact that you got an A proves that you were trying really hard because hard honest work is what made America great.

The problem is that once you link those previously independent variables (effort and results), you are really only working in a one-dimensional world. You don’t need two grades to adequately describe it.

Looking From The Other Side

But you may be saying to yourself “Silent, you are the anomaly! Only the very rare person who can find a task at which they can succeed without unbelievable effort would have the luxury of taking your position on this topic”. If you really think failure is the norm, then answer this:

Do you really think someone who, for whatever reason, didn’t meet the minimum requirements for success in this class, would prefer an F1 over an F3? From what I’ve observed, the opposite has usually been the case. If you give him an F1 you are saying “bless his little heart, he gave it his best shot but is just too stupid to make the grade”. That may even be true, but giving him an F3 gives him an alibi (or more accurately, reinforces the excuses he’s been giving even without your blessing) that he’s really very, very intelligent, but just didn’t put forth the effort.

I’m not saying to give all failures a “3” – you could just make the variable independent, open your eyes, and give the student whatever they truly earned.

Conclusion

There are actually two ways to cure this problem: we could start treating effort and results as the independent variable they are (which is probably too agonizing a task for most teachers). Or we could just stop giving the effort grade. I propose the latter. What do you think?

When Sailors Should Split Tacks

American inventor Thomas Edison once said “Genius is one percent inspiration, ninety-nine percent perspiration”^A. From my experience, when you are competing, whether for business or pleasure, or trying to solve a problem, or just trying to get something done, you can usually do pretty well without that stroke of genius if during the remaining 99% of the time you can just keep from screwing up.

A Sailing Example

One summer I had the opportunity to race sailboats by donating the perspiration needed to handle the sails. Since sailboats can’t sail directly into the wind, which is often exactly where you need to go, you must regularly choose which side of the wind is best, or which side of the course is best, or . . . the bottom line is that in sailboat racing, as in life, there are plenty of decision-making opportunities.

If we happened to get behind early in the race, by the time we got to a point that needed a decision, our competition had already gotten to that point and had already made their decision. Our skipper, reasoning that we would never catch up if we did everything our competition had done, invariably would make the opposite choice at that point (hence, splitting tacks, or sailing on the opposite side of the wind as our competition). More often than not we would get further behind. As it turned out, we didn’t win many races that summer. Now I will use just a little math to show you why not.

The Math

Without divine intervention or that long-awaited flash of inspiration, after a short time the leaders in this race will be the ones that make more correct moment-by-moment decisions. When our skipper got to his decision point, it is reasonable to assume that the competition ahead of him is batting above 500^D and already chose the short path. If the current leader has a success rate of, say 70%, then by blindly taking the other path, our skipper was limiting his success rate to 30%. This is NOT a winning strategy.

The more prudent leader would have chosen his battles; he would have evaluated every decision independently – more often than not this means he would have made the same choice as his competitor (assuming his own success rate is high enough to be competitive – certainly higher than 50%) – and he would bide his time while waiting for the competition to make their mistake. When his own evaluation led him to a different decision, he would quickly recheck his work (out of respect for his competitor’s 70% success rate) and then he would pounce.

A Non-sailing Example – Rush Hour Traffic

“Rush Hour”, referring to those busy couple of hours in the morning and another couple of hours in the afternoon when everybody is commuting to or from work at the same time and traffic is congested (as opposed to that time of day when Rush Limbaugh is delivering his political commentary), implies an urban environment, which implies a larger grid of streets and thus a richness of decision-making opportunities not completely unlike a fleet of sailboats tacking upwind, but familiar to a much larger segment of the population.

Many of you may have carpooled with somebody with the mentality of the skipper described above: either there is some sort of accident or s/he misjudged traffic again and finds him/herself behind schedule and facing the growing possibility that they will be late for work. Lacking patience or maturity, they assume the traffic must be better on one of the many alternative routes and blindly makes a turn (tacks) at the next intersection. When they discover that this path is also blocked, they immediately move to Plan C, then D, and so forth. Each maneuver has a small cost, which rapidly adds up, and then the path actually starts to get longer and they continue to dig themselves a deeper and deeper hole (oops, that’s not a sailing reference). The math is similar to that above.

To mix metaphors even more, compare this to the hitter swinging too hard for a home run. The problem is that in this game, after each errant swing the outfield fence is moved ten yards further away. Although still mathematically possible (at first), the odds of that game-winning home run drop with every swing. Those are the perils of panicking, shutting off your brain, closing your eyes, and trying to slug your way out of your problems.

The Moral

As you might have guessed, this article is not really about sailing, or traffic, or baseball. Blindly splitting tacks is a tactic of desperation. Desperation is often a result of one’s fears getting the best of them and may be one of the consequences of ignorance. It is never expedient to shut off your brain to save time (by the same token, except for specially trained pilots in specially designed aircraft, no self-respecting pilot would willingly turn off an airplane’s engines while still in the air), yet people try it every day. This is what happens when you panic. So get a grip! Just as in the sailing example, the prudent driver would carefully evaluate every decision (the more you practice, the easier it gets) instead of assuming the worst, bide your time, and make your bold move only when the conditions are right.

Simple English: The Problem With The “If” Statement

You are probably very proud of your grasp of English (unless you live in South Florida, in which case you may not give a damn). And yet I have seen plenty of people whose lack of understanding about basic structures like the “If” statement

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The next article in this series will discuss how misunderstandings about the conjunction “or” have caused so much trouble.

cause them to make terrible assumptions.

An Example

Suppose a young child is misbehaving to the point that the caregiving parent decrees “If you don’t knock that off, I’m going to paddle you” (This is an old example; I’m sure nobody would ever actually do that today 😉 ). As young children have been known to do, for whatever reason, the child continues with its behavior. The parent repeats the statement, with added emphasis. Nothing changes. The parent soon throws their hands up and says “wait until (your other parent) gets home”.

The parent’s first decree, like all “if” statements, had two parts; a condition and a consequence (joined by the conjunction “if”), with the understanding that if the condition is true, then the consequence will occur. It’s simple enough that even a young child can understand it. If the condition is met and the consequence is not accomplished, then the statement would be considered false. In short, the child knew that the parent was lying.

Now suppose the non-caregiving parent comes home, sees the objectionable behavior, makes a similar decree, and then the first parent points out that they had already made that decree to no avail. The child, for whatever reason, stops the objectionable behavior. To everybody’s surprise, the second parent paddles the child. Although the child and many of you listeners may think bad thoughts about this parent, one thing you can’t call him/her is a liar.

The Problem

As you can see here, the problem with the “if” statement is that is incomplete in the sense that it only addresses what happens when the condition is true, remaining completely silent to the possibility that the condition could be false. This allows most people to make the assumption that if the condition is false, the opposite of the consequence must occur. As the young child in our example learned, that assumption would be a mistake.

The Solution

Don’t make stupid assumptions. As your lawyer would tell you, get it in writing. In the above example, since the second parent didn’t make any promises about what would happen if the behavior did stop, s/he can’t be accused of lying. If this example bothers you, I’m sure the second parent told the child afterward that the paddling was for not obeying the first parent, in which case we would be unable to judge the truthfulness of their claim until after the right set of conditions are met following some later episode of misbehavior (guesstimating any change in likelihood of that future misbehavior based on recent events will be left as an exercise for the reader). To lawyers, mathematicians, and the like, the parent’s explanation doesn’t matter to this case and is unnecessary.

P.S.

Logicians have named operators (or functions) fulfilling all sixteen patterns of truthfulness or falsehood of expressions based on the truthfulness or falsity of two variables, such as the condition and consequence of the “if” statement described above. Engineers call the statement that yields the results you thought the “if” statement provided the “exclusive nor” function, “nor” being short for not or, meaning “giving the opposite results than the ‘or’ function”. Some refer to it as logical equality. In English, it would be represented by a sentence including the phrase “If, and only if”, such as “I will ground you for the rest of your life if, and only if, you do not stop screaming this very second”. If that type of statement had been the norm in this household, the non-caregiving parent, upon hearing the lack of results achieved by the other parent, was still free to add other (most likely “or”) clauses (to be discussed in a later article) to his/her decree. If you are now totally confused, please do not sign any document containing more than six words before consulting an attorney, or at least a mathematician. On second thought, in cases like this, I would stick with the lawyer.

A Better Way To Handle The Harambe Incident

For those who haven’t heard about the gorilla named Harambe that was shot ten minutes after a toddler fell into his enclosure at the Cincinnati Zoo around 4 pm on Saturday, May 28, 2016, here is as good a source as any: Gorilla killed after 4-year-old falls into zoo enclosure. Apparently, authorities had both a tranquilizer gun and a rifle at their disposal, and chose the rifle to fatally shoot the gorilla even though the boy hadn’t yet been seriously injured because they were afraid that the tranquilizer wouldn’t act fast enough. They didn’t have to make that call. Here’s a better way.

Take both the tranquilizer and the rifle. The same person should not operate both.
Have other staff members make themselves immediately ready to rescue the child.
When both weapons are ready, shoot the gorilla with the tranquilizer.
Have the person with the rifle continuously evaluate the threat posed by the gorilla. If bodily harm from the gorilla is not immediately forthcoming, do not shoot.
If the parents get hysterical while you are evaluating the situation and behave in such a way as to adversely affect the behavior of the gorilla or the judgement of the zoo staff, shoot the parents. (So as not to make the same mistake as the Cincinnati Zoo staff did Saturday, I guess I should mention that you could use the tranquilizer gun for this if you had the forethought to bring the correct dose – even though at this point it wouldn’t be my weapon of choice. If you don’t have the correct dose, just pray that the staff isn’t acting under the same level of panic or incompetence as they exhibited with Harambe.)
Rescue the child as soon as practicable.

Although (admittedly based on limited information) I did not think the boy was in danger, and not all witnesses in Cincinnati felt the danger^A, those opinions don’t matter to the success of this plan. Since using the tranquilizer doesn’t prevent the use of the rifle, this plan could not have turned out worse for the child than the plan executed, and most likely would have turned out much better for all concerned. The zoo simply threw away options prematurely based solely on their worse fears instead of facts – that sounds like panic to me, and it sounds very unprofessional. If you feel differently, feel free to comment. If you see a reason that this plan would not work, feel free to comment.

How Large Is Your Universe?

We used to keep tropical saltwater fish. When getting a new fish, one important question that would inevitably come up would be how much space would it need. It amazed me to think that some fish, even in the wild, could be perfectly happy spending their whole lives patrolling one small rock. That was the extent of their universe. Each individual person, like each of those saltwater fish, lives in their own universe, each a different (but hopefully overlapping) subset of The Universe created by God.

In The Beginning

When you are born, your universe is very small – focused only on your mother’s breast. But it starts to grow immediately. Every experience gives you a new plank you can use to expand your universe. As a new experience comes to you, your mind stretches to make sense of that experience. In a later article we can discuss how important a strong imagination is to discovering the truth (this may seem ironic). This is important for the growth of your universe. But I don’t yet understand how strong imaginations are developed. Other attributes are also required.

When Growth Starts To Slow

Growth may start to slow, however, once your universe is large enough so that a new plank can fit entirely within your existing universe. Since you didn’t have to stretch your universe to accommodate that plank, you may feel that no more growth is necessary and discard that plank.

For example, in the story of the blind men and the elephant, which I embellished in “The Blind Men and the Elephant – The Full Story”, one scholar, “holding the tail, announced that an elephant was like a rope”. While their later behavior may lead you to question how scholarly they really were, a non-scholar would have been more likely to have declared that there are no elephants; what he was holding WAS a rope and he resented any efforts to try to fool him into believing otherwise.

In this man’s mind, his universe was already sufficient to describe what he had experienced, and so he threw out the new plank. Once this happens, it takes larger and larger planks to keep up any growth.

When You Have Reached Your Limit

At some point your mind may start subconsciously throwing out old planks to make room for new. In my first career, I was at a field unit (from which everyone starts). It was common to complain about how clueless the people in the district office were about what was going on in “the real world”, based on the decisions that were passed down to us. When someone in our unit was transferred to the district office, we took bets on how long it would take him/her to move to the dark side. The same thing happens when teachers with experience get transferred downtown, away from the classroom.

One could argue that it was the people in the field, who had experienced only one small piece of the puzzle (or elephant, if you will), were the ones with the smaller universes and thus were unqualified to pass judgement, while the transferee, with more experience in a larger world, was making decisions that would benefit the whole team. While that’s the way we would like to see it work, that doesn’t always happen.

My father, who had to join a union to learn his trade, could see only the benefits of the unions at the time and was a strong believer. Once he became a contractor and had to deal with unions “from the other side of the fence”, he could see only the negative. Apparently his universe was not capable of stretching to accommodate both views.

Sometimes the truth about elephants is too large to fit in somebody’s universe. When your universe stops stretching, it has reached its maximum capacity. Without new planks, it could actually start to deteriorate.

Then there are other people who are unwilling to stretch, and start throwing out new planks that don’t match or fit into a set of planks that they created themselves. We know those people are bigots. It’s when they grab everybody around them and try to force the others into their resultingly smaller universe that things could get ugly. I think it’s a bad idea to voluntarily throw planks out of your universe at any time. Here’s why –

The Descent

At some point as you age, your universe will start to become less resilient and will try to shrink. The process begins well before your universe has reached its largest size, but from then on it’s all downhill. I think I’ve already started the slide. If you don’t keep actively trying to add new planks to slow the process, it may act like shrink-wrap, too soon becoming so tight around your body that everybody will be able to see just how small those private parts really are that you had bragged so much about for so long. When they start laughing, you won’t care that the shrink-wrap is now too tight for you to breathe. I wrote this analogy specifically for men with bloated egos, (which may be my favorite target), but something analogous happens to all population groups.

To see the Note click here.To hide the Note click here.

“bloated egos”: Since this term is not in any medical journal or book on psychiatry, I will have to define it in a later post

Measuring Universes

While the size of one’s universe seems to be a far better metric^Defined by which to judge a person than more common and more superficial traits like size, sex, or hair quality, measuring this parameter is not as easy as it sounds. Since you can only measure something that is completely inside your universe, you can only judge people whose universe is small enough to be completely contained within yours, which usually means a child or imbecile. Maybe you are perfectly happy to always be comparing yourself to morons, but eventually your friends are going to correctly conclude that “it must take one to know one”.

If a person has any talent or experience that is not part of your universe, there is absolutely no way for you to tell how significant that talent is. If the common area between you and that other person is only a small part of your universe, it might be tempting to draw inferences unfavorable to that other person. But again, since you don’t know how large his/her unshared universe might be, your conclusions would be completely unsupported. It is entirely possible that their unshared universe could be larger than yours. But your unsupported conclusions would give evidence to any counterclaim that it is you who is the idiot.

A less common but more important question may be how to measure the size of your own universe. Sure, you can get from one side of your universe to the other, but what can you compare it to? Maybe you are like that little fish I mentioned at the beginning of this article, perfectly happy patrolling your own little rock while others swim in and out of your life on their way through. How do you know you are not missing out on something worthwhile just over that next rock? How do you know that something big is not soon coming along the path that will annihilate your universe and the universes of everyone around you? I don’t have answers to these questions, but clues might be found in the answer to two other questions: “How often do you discover a plank that isn’t yet part of your universe?” and “How hard are you really looking?”.