#32) Prove It, Mel!

Hello, and welcome back to "Always Be Better" with Mel Windham.

Today we're going to talk about logic. If you haven't seen the first five videos, don't worry. You can pick up the rest after watching this one, as today I'm going to discuss something important I've left out previously.

Now, think about the last time you've been in a political discussion. I bet someone said, "Prove it." What does this mean? Assuming the person really wants an explanation, they are asking to see some kind of logic that may help to explain or persuade. And I bet that was the next thing you heard was: "Because of this" or "Because of that."

Sometimes the logic works, but more often than not, another person is likely to say, "Your logic is faulty because of this or that." And so on. Which can sometimes be fun -- but ... we'll get there.

For today we'll stay simple. Let's look at our very first example from lesson 1:

If Sam is human, he has a face.	A -> B
Sam is human.	A
Therefore, Sam has a face.	∴ B

This is the simplest type of proof. It makes sense to us. If the first two sentences are true, then so is the third.

We can reduce the logic into symbols to the right. And see those three dots at the bottom? Those are the gist of the proof -- the Eureka moment. It lets the world know that we figured this out -- using LOGIC!

But is it really that simple? How do we know that the first sentence is TRUE? I decided to ask Google and was not surprised to see some interesting counterexamples. I'd show it here, but the images are a bit disturbing. If you feel up to it, try it yourselves.

And what about the second statement? How do we know that Sam is human? Again, we can turn to Google to see some robots that look surprisingly human. Can we rely on our eyes alone? And wouldn't you know it? I now have an urge to watch Blade Runner!

But sure -- if the first two sentences are TRUE, then the third has to be TRUE. So, issues only exist with the first two sentences. This may be the main challenge with logic. If you don't have something to start with, it's just impossible to use all the logical tools I've shown before. We wouldn't be able to prove anything, and we'd all be dumbfounded.

In other words, if we wish to make any progress, we must first agree on the starting points. Once these are set, then we can use logical tools to figure out whatever we wish -- step by step -- proof after proof. This is what I left out before, and now I will show you the two types of starting points.

#1) We have DEFINITIONS. We must agree as to what words mean. Here's an example ...

Can you prove that 3 is more than 2? Well, what is 3? And what is 2? We must first define counting numbers. The number 1 starts us off. Then 2 comes next, and then 3, and then 4, and so on. Someone had to come up with this, and everyone had to accept, solely because that was how it was defined.

And what about the words "more than"? Well, that has to be defined, too. A number is "more than" another if it comes "after" it. So is 3 more than 2? We defined 3 to come after 2. So, we know that 3 is more than 2. Eureka!! Proof by definitions!

Of course, if we can't accept the definitions, we can't accept the proof.

And #2) We have AXIOMS. These are concepts that are considered to be unprovable, but generally accepted. 

For example, let's take parallel lines. If we draw out two parallel lines on a piece of paper, they don't look like they're going to meet. It just makes sense. A lot of classical Euclidean geometry depends on this fact.

But wait -- what if we draw these two parallel lines on a globe? The curvature can cause them to meet -- just like how longitudinal lines meet at the poles. If we change the axiom to be "parallel lines can meet," it creates a whole new branch of math, called non-Euclidean geometry. 

When it comes to politics, you're probably familiar with the most famous example of verbal axioms from the beginning of the Declaration of Independence.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

The words "to be self-evident" is the exact definition of an axiom. It does not purport to prove these concepts, but asks us to accept them as being TRUE, as -- isn't it obvious?

Often, I see arguments blow up because the two individuals can't come to agree on DEFINITIONS. Or they refuse to accept AXIOMS proposed by the other party. Without these starting blocks, all logic fails on a sandy foundation.

To review, all proofs require DEFINITIONS, AXIOMS, and then LOGIC to come up with conclusions.

The Challenge

As you go about your business the next couple of weeks, I'd like you to listen to everyone around you. Look for DEFINITIONS and AXIOMS. How well are people using the tools of logic? Do their proofs have solid foundations? 

And if you haven't already done so, I invite you to go back and watch the first five videos on logic. And while you're at it you can make sure to LIKE and SUBSCRIBE. And tell all of your friends about this series. If you feel so inclined, you can even begin your own series on how to be better.

And I'll see you in a couple of weeks when I give an introduction to Politics. Yep -- it'll be fun -- right here on "Always Be Better."