Photo by AMagill

The Matrix trilogy presents a number of yin yang pairs, one of which is seen in the Oracle and the Merovingian. Although they take on the form of humans in the matrix, they’re actually computer programs. And they’ve been designed to specialize in different kinds of logic.

The Oracle is gifted with foresight based on inductive reasoning. Although she’s not actually psychic, she was specifically created for the purpose of understanding humans, and this gives her amazing powers of prediction. She can effectively see into the future, up to the point where free will presents a choice. She says nobody can see past a choice they don’t understand, thus showing her limitations.

The Merovingian is gifted with hindsight based on deductive reasoning. Believing that everything is determined by cause and effect, he thinks that someone’s power is based on their understanding of why events unfolded the way they did, and he understands this quite well. But his ability makes him overconfident, because he really doesn’t know everything. For example, he didn’t know that his wife would turn against him, because he saw no cause that would create that effect.

The Merovingian and the Oracle are opposites in this regard, and he dismisses her as a silly fortune teller. He laughs at Neo and his friends for visiting him just because the Oracle advised them to, when they didn’t really know why they were there or what they expected to happen. They were just blindly following orders, without knowing the answer to that all-important question: why?

But despite his mocking, the Merovingian secretly desires the Oracle’s powers of induction to complement his powers of deduction. And understandably so, when you consider what one could do with both of them.

This post is fairly long and complicated, but it’s virtually guaranteed to boost your reasoning skills. Read it once, then enjoy the benefits of improved logic for a lifetime.

Inductive reasoning

The Oracle inductively knew that Neo was coming, and she was ready with cookies in The Matrix (1999).

With inductive reasoning, you reach a conclusion that is believed to be true but not guaranteed. Specifically, you use observations of particular cases to make a generalization. While this may not seem logical, we do it all the time.

Here’s a common example:

“The stock market has averaged 10% annual returns in the past, so it’s reasonable to expect that it will continue to do so.”

Statements like this are often followed with an admission that 10% returns are not guaranteed. As they say, past performance does not guarantee future results.

However, predicting 10% returns based on available data seems preferable to ignoring the past, and deciding that returns of 10%, -100%, or 1,000,000% are equally probable. In other words, we naturally want to make a prediction, so we should make one that fits the pattern.

The card game Mao is based on inductive reasoning. New players are not told what the rules are, because the point of the game is to figure out the rules. Players have no choice but to observe the game, make many mistakes, and slowly piece together the rules by induction.

Here are some examples of so-called strong induction:

“Mary always hates horror movies, so she’ll hate this horror movie.”

“I’ve never seen a green canary, so your canary is probably not green.”

“Technology has changed a lot in the last 100 years, so it will change a lot in the next 100 years.”

“White eggs have a hard shell, so brown eggs must have a hard shell.”

“Five channels are showing static, so the cable is out.”

“Penicillin killed these bacteria, so it will kill other bacteria.”

“Pi does not terminate or repeat after the first million digits, so it never terminates or repeats.”

This is called strong induction because the conclusion is likely to be true, assuming the premise is true.

However, these statements could be stronger if the wording were more specific. What does it mean that “Mary always hates horror movies?” Which ones, and how similar are they to the one you’re predicting she’ll hate? You could also say she’ll probably hate the movie, to acknowledge the possibility that she might not.

Here are some examples of what’s called weak induction:

“I could run fast 60 years ago, so I can run fast now.”

“I always sleep until noon, therefore everyone always sleeps until noon.”

“I made a wish and it came true, so all wishes come true.”

This is weak induction because the arguments aren’t very convincing at all. There’s a very weak link between the premise and the conclusion.

However, what one person considers weak induction, someone else may consider strong induction. For example, Isaac Newton induced his theory of gravity from observing the motions of planets and falling apples. This theory was undoubtedly met with varying degrees of resistance, depending on how strong someone considered the induction to be.

Any induction, particularly weak induction, carries the risk of overgeneralization, which can lead to prejudice and delusion.

Rejecting inductive reasoning

On the other hand, if you don’t generalize at all, that’s a problem too. If you know how to drive a Camry, it would be crazy to say that you don’t know how to drive a Corolla because it’s a different model. We have to generalize to survive.

I know a baby who rejects inductive reasoning. When he gets hungry, he cries. Most babies will stop crying when you start feeding them, but not him. He knows that just because baby food satisfied his hunger last time, there’s absolutely no guarantee that it will satisfy his hunger this time.

So he keeps crying, while mom shovels food into his mouth as fast as she can. He doesn’t stop crying until he actually feels full, and therefore has proof that his hunger was satisfied. And he’s a big baby, so this takes two jars.

You can imagine how his mom feels. She inductively reasons that since he’s done this every time, he’ll continue to do so for a while. Wouldn’t it be nice if her baby used inductive reasoning to determine what was likely to happen, instead of insisting on a guarantee?

Deductive reasoning

The Merovingian deductively believes that every cause and effect has already been determined, so he just sits back and enjoys himself in The Matrix Revolutions (2003).

With deductive reasoning, you apply known rules to given data to prove a conclusion. Unlike inductive reasoning, deductive reasoning lets you arrive at a guaranteed conclusion, as long as your reasoning is sound. This is what we usually think of as “logic.”

For example:

“The sum of the angles of every triangle is 180 degrees. In this triangle, two angles are 45 degrees, so the remaining angle must be 90 degrees.”

We like deductive reasoning for a couple of reasons. One, we’re so used to thinking of it as the definition of logic. Two, it’s a lot more certain than inductive reasoning, and people like certainty.

However, it also gives us false certainty. Suppose you’re looking at an animal, and want to prove that it can fly. So you use this logic:

“This animal is a bird. All birds can fly. Therefore, this animal can fly.”

There are a couple of problems here. Most importantly, the claim that all birds can fly is false. For example, penguins can’t fly. We call an argument like this valid because the deduction was logical, but not sound because it’s based on a false premise.

Also, how can you confirm that it’s a bird? On a math test, you’re given all the information you need. But in the real world, a problem won’t necessarily be set up so conveniently.

The problem with deductive reasoning is that it’s like an insurance policy that guarantees to pay off, but only if a particular set of conditions is met exactly. And you’d better read the fine print. It’s easy to be far more confident in the outcome than your logic warrants.

The deduction above is relatively simple, but verifying the preconditions is extremely difficult. How do you know it’s a bird? Because it looks like other birds you’ve seen? That’s inductive reasoning. What’s the definition of a bird anyway?

Is it true that all birds except penguins can fly? What about chickens? They can fly a little, or can they? What’s the definition of flight? What about injured birds? What about birds that are afraid to fly?

Math is logical, but filled with assumptions

Math makes heavy use of deductive reasoning, but it’s a lot less solid than we might think. Think back to your high school math. You learned how to deduce many things, but it was all based on a set of axioms that we just assumed to be true.

Some of these axioms seem obvious, like “all right angles are congruent.” Of course they are. How could they not be? But that’s an assumption that can’t be proven. Some axioms seem stupidly obvious, like “if A and B are true, then A is true,” or “x = x.” It hurts your head to even imagine them not being true, but we need to make these assumptions to support everything else.

In geometry, we even have three undefined terms: point, line, and plane. This goes beyond unproven – they’re not even defined!

For over 2,000 years, Euclid’s assumptions of geometry seemed so obvious that no one questioned them. Today, we need to specify “Euclidean geometry” when referring to the version that seems obvious to us, because there are different versions where these assumptions are violated.

Far from being an unquestionable universal truth, Einstein suggested that Euclidean geometry is a good model of physical reality only if the gravitational field is not too strong.

Yes, math is very logical, and you can say that A is definitely true, if you assume that B, C, D, E, F, G, and H are true. But in that case, what have you really proven?

An attempt at inductive reasoning in math

Say we have a function f, where

f(x) = x⁶ – 15x⁵ + 85x⁴ – 225x³ + 274x² – 120x

OK, but what is this, really? What’s f(0)? You plug it in and see that f(0) = 0. OK, what about f(1)? That’s also 0. You try f(2), f(3), and f(4), and they’re all 0 too. Looks like a pattern is emerging.

You try f(5) and get 0 again. Now you’re getting tired of this. Obviously, f(x) = 0 for all whole numbers x, right?

Nope, because f(6) = 720.

This is the problem with inductive reasoning. You can observe as many specific cases as you want, but you’ll never prove a generalization from observation unless you observe all possible cases.

However, our observations are still helpful, because we can apply deductive reasoning to them. The fact that f(0) = 0 means that x is a factor of f(x). The fact that f(1) = 0 means that x – 1 is a factor of f(x), and so on. By factoring everything out, we find that

f(x) = x(x – 1)(x – 2)(x – 3)(x – 4)(x – 5)

This form is much simpler, and now we can see exactly why f(x) = 0 only when x is 0, 1, 2, 3, 4, or 5.

So while our inductive reasoning failed, our observations turned out to be fuel for deductive reasoning.

Let’s try another one.

Mathematical induction

Let’s say you want to prove that 1 + 2 + 3 … + n = n(n + 1) / 2 for all natural numbers n. This seems far from obvious. How would you even get started?

Well, let’s plug in some numbers and check. It’s true for n = 1. Also 2, 3, 4, 5, 6, 7, 8, 9, 10…

But we saw in the last example that just because something is true for ten or a thousand or a million cases, that doesn’t guarantee that it’s true for all cases. So what can we do?

Think about when you set up a bunch of dominos to knock down in a chain reaction. It works like this:

If (1) all the dominos are set up in such a way that if one falls down, the next one falls down, and (2) the first domino falls down, then they all fall down, right?

Let’s use that same idea to prove that 1 + 2 + 3 … + n = n(n + 1) / 2 for all natural numbers n.

First, let’s check that it’s true for n = 1. Yes, it is.

Now, let’s check that if it’s true for some arbitrary number x, then it’s also true for x + 1, i.e., that 1 + 2 + 3 … + x + (x + 1) = (x + 1)(x + 2) / 2.

OK, if 1 + 2 + 3 … + x = x(x + 1) / 2, then 1 + 2 + 3 … + x + (x + 1) = x(x + 1) / 2 + (x + 1) = (x² + x + 2x + 2) / 2 = (x + 1)(x + 2) / 2. Done. And so the dominos all fall down.

This is called mathematical induction, yet it’s technically a form of deductive reasoning, because the conclusion is guaranteed if you do it right. However, it’s similar to inductive reasoning in that you’re taking a finite number of observations and generalizing them to an infinite number of cases.

But wait a minute. How do we know that the principle of mathematical induction, the “domino trick,” is true? Well, it’s usually taken as an assumption! It can only be proven if you make certain other assumptions, such as (1) the natural numbers are well-ordered, (2) every natural number is either 0 or the successor to another natural number, and (3) n + 1 > n for all natural numbers n.

Wow, we sure need to make a lot of assumptions in order to “prove” anything! Which brings up another point. Even if you manage to prove something, how can you prove the proof? Fermat’s Last Theorem was “proven” many times by reasoning that was ultimately revealed to be flawed.

Abductive reasoning

Deductive reasoning is often used in math, because they’re trying to prove things from a known starting point.

Inductive reasoning is often used in science, because they’re trying to discover things, not prove them.

And there’s another type of reasoning, called abductive reasoning, that’s often used by detective types, because they’re trying to explain things, not discover or prove them.

It works like this. A patient has certain symptoms, and goes to the doctor. The doctor knows that appendicitis will cause those particular symptoms. There are other possible causes, but appendicitis is far more likely, and therefore considered the best diagnosis based on the known information.

In some countries, doctors have been known to remove the appendix without actually testing for appendicitis. They thought it was better to occasionally be wrong than to consistently take the time and money to run a test to confirm a rather obvious diagnosis.

Now, using the method of your choice, possibly abductive reasoning, can you determine the point of this post?

• Slow And Steady Wins The Race
• When Logic And Intuition Fail
• Logic Test From My Interview
• The Illusion Of Certainty
• Einstein’s Riddle (The Zebra Puzzle)
• Why Beliefs Should Be Beyond Reason
• Outliers: The Story of Success
• The Case Of The Stolen Diamond
• My New Year’s Wish For 2009
• The Birthday That Never Arrives