Logical Link Control

While humans are much smarter than other creatures, our intelligence varies greatly from person to person. Some of us are geniuses, most are average and some are living with a mental disability. While most of us identify academic or general intelligence as the primary indicator of mental aptitude, there are other ways we can be smart. According to developmental psychologist Howard Gardner, there are actually nine ways of measuring intelligence:

  1. Musical
  2. Visual
  3. Verbal
  4. Logical
  5. Bodily
  6. Interpersonal
  7. Intrapersonal
  8. Naturalistic
  9. Existential

Of course, expanding the term to include these categories is highly subjective, and it erodes the traditional understanding of intelligence as the capacity for reasoning and understanding. And if intelligence can be narrowed to such specific abilities as music or physical ability, then why not include additional classifications for programmers, powerlifters, comedians, cashiers, magicians, memorizers, sharpshooters and competitive eaters? Just as redefining art makes everyone an artist, a more inclusive understanding of intelligence means everyone is highly intelligent. This would also mean that many animals, and even machines, are more intelligent than humans, which is something that we know isn’t true. While general intelligence may be difficult to define, opening the door to alternate meaning only weakens its meaning.

One of the abilities associated with general intelligence is logical comprehension, which is the ability to analyze, comprehend, abstract and navigate the layers of a causal system. A wonderful example of this occurs in the classic 1987 film The Princess Bride, when the then-masked Westley challenges a Sicilian named Vizzini to a battle of wits. In a lively and captivating back-and-forth between the two men, Vizzini attempts to discover, by pure reason, which of two goblets of wine has been poisoned by Westley with a fictitious toxin known as iocane powder. Let’s see if we can follow the layers of reasoning.

  1. The first line of reasoning is easy to comprehend. In it, Vizzini he asserts that “only a fool would drink reach for what he was given.”
  2. He then claims that Westley must know that he isn’t a fool, stating that he “can clearly not choose the wine in front of [Westley].”
  3. Vizzini goes on to accuse his opponent of knowing that he isn’t a fool, which means that he shouldn’t drink his own wine.
  4. He then deduces that because the poison originates in Australia, a land “entirely populated by criminals,” which implies that Westley would anticipate not being trusted, meaning that Vizzini should not drink the wine in front of Westley.
  5. Aware that Westley must have predicted his ability to determine the poison’s origin, Vizzini reverses his position again.
  6. Vizzini then accuses his opponent of poisoning his own goblet and planning to trust his physical strength to withstand the poison.
  7. He then points to Westley’s education, and therefore knowledge of his own mortality, as the reason why Westley would “put the poison as far from [himself] as possible.”
  8. In a final attempt to dupe his enemy, Vizzini distracts Westley and switches the goblets before both men drink.

While this depth of reasoning is impressive, the layers don’t actually require placement in a precise order. After all, Vizzini could have pointed to the poison’s Australian origin at the beginning, and arrived at the same conclusion. Logical comprehension isn’t just about understanding complexities, but also following a path of thought. Here’s a more structured example that becomes increasingly complex as layers are added. With each statement, a negation is added, inverting the meaning of the sentence.

  1. I will be going to the party.
  2. I won’t be going to the party.
  3. I won’t be not going to the party.
  4. It’s a lie that I won’t be not going to the party.
  5. It’s not a lie that I won’t be not going to the party.
  6. It’s false that it’s a not a lie that I won’t be not going to the party.
  7. It’s not false that it’s not a lie that I won’t be not going to the party.

Most people lose the ability to comprehend the meaning of these sentences somewhere between levels 2 and 4. However, many realize that it’s not necessary to understand the entire sentence at all. If we merely count the number of negatives, we can determine that the person will be attending the party if number is even, and they won’t be attending if the number is odd. Now let’s consider another example that illustrates a more complex logical thought process.

Imagine you’re engaging in a game of rock-paper-scissors with someone. But before you begin, your opponent tells you that he will choose rock. Is he telling the truth? How would you deduce what to choose next? Well you might think something like this (see how many layers you can follow):

  1. I know that he won’t actually choose rock, like he said, so I’ll choose scissors to cut his paper or tie with his scissors.
  2. He knows that I know he won’t choose rock, so he’ll choose rock to crush my scissors, since he knows that scissors are best counter to someone not choosing rock.
  3. I know that he knows that I know he won’t choose rock, so I’ll choose paper to cover the rock he chose to crush the scissors I chose in response to his rockless strategy.
  4. He knows that I know that he knows that I know he won’t choose rock, so he’ll choose scissors to cut the paper I chose to cover the rock he chose to counter my scissors, which are perfect against an opponent without a rock.
  5. I know that he knows that I know that he knows that I know he won’t choose rock, so I’ll choose rock to crush the scissors he chose to cut the paper I chose to cover the rock he chose to counter the scissors I used against his non-rock.
  6. He knows that I know that he knows that I know that he knows that I know he won’t choose rock, so he’ll choose paper…

At this point the cycle would continue indefinitely. assuming that either of us could actually fathom such strategies. It is possible, however, to simply count the occurrences of the word know in order to formulate the solution (just as we do with multiple negatives), but again, this circumvents actual understanding. Let’s take a look at an example without a pattern that can easily explain its meaning.

Imagine a sprinter who only changes speed in increments of 1 m/s. After completing each one-second interval, he will travel a distance equal to his velocity at the last interval, and his velocity will increase by his acceleration at the previous interval. Let’s also imagine that our sprinter continues to accelerate his acceleration during the experiment.

Time 00:00 00:01 00:02 00:03 00:04 00:05 00:06 00:07 00:08 00:09 00:10
Position (m)  0  0 1 3 7 14 26 46 79 133 220
Velocity (m/s)  0 1 2 4 7 12 20 33 54 87 137
Acceleration (m/s^2)  1 1 2 3 5 8 13 21 33 50 73
m/s^3  0  1 1 2 3 5 8 12 17 23 30
m/s^4  0 0 1 1 2 3 4 5 6 7 8
m/s^5  0 0 0 1 1 1 1 1 1 1 1

It’s a little difficult to determine what’s happening here, but our sprinter is basically adding another layer of acceleration after each interval. Let’s describe each of the characteristics of the sprinter.

  1. Velocity (m/s) : change in position per second
  2. Acceleration (m/s^2): change in velocity per second
  3. m/s^3: change in the rate of acceleration per second or changing how fast we accelerate
  4. m/s^4: change in the speed of the rate of acceleration per second or changing how fast we’re changing our acceleration
  5. m/s^5: change in degree of the speed of the rate of acceleration per second or changing how fast we’re changing the rate at which we change our acceleration

If it isn’t already clear by now, things tend to get complicated really quickly after the third step. Part of the problem is that our brains are less able to recognize patterns among similar symbols. But even when we vary the terminology, the process is extremely difficult to follow. Here are four more examples of logical processes that can be hard to follow:

  1. Nested loops
  2. Layered arguments
  3. Recursion
  4. Paradoxes

It’s difficult to pinpoint the exact issue, but it seems to be related to the limits of our mind’s working memory described by cognitive psychologist and Mad Max director, George Miller. Miller observed that humans can only keep track of 3 to 7 things at once. It seems that keeping track of a logical process is more difficult than independent parts. The difference here is that layers seem to compound the complexities as they’re added. Let’s imagine a system for using our working memory to keep track of both independent and layered thoughts. Here’s how it might work:

  1. We start with 3 to 7 memory units to spend on thoughts.
  2. Each independent thought costs 1 unit.
  3. Each layered thought after the first costs 2 units.

If our first layered thought counts as an independent thought, with a cost of 1, and each following thought costs 2, then our results confirm what we observed:

  1. We can track 3 to 7 independent thoughts.
  2. We can track 2 to 4 layered thoughts.

So what can we learn from all of this? For starters, we can confirm that measuring intelligence should be done in terms of reasoning and comprehension, not words or emotions. We also know that the complexities of some processes can be circumvented by simply counting the occurrence of a word or phrase. In addition, we know that tracking a logical process has roughly twice the mental cost of tracking independent thoughts. So next time you encounter a logical process, do the following:

  1. See how many layers deep you can go.