If you read this post, then you will understand basic logic.

I want to talk about a strange math related topic.

The moment anyone starts seriously studying mathematics they immediately come across a few terms (all related to logic) that are very related. I specifically picked them because they are very applicable in even every day logic. They are:

1) Sufficiency and necessity.
2) logical implications (i.e. ⇒, ⇔, iff, etc.)

I'll start with the first.

Now, the first time someone studying mathematics comes across this is probably in mathematical optimization (necessary and sufficient conditions for maximization and minimization). Yet, it springs up constantly from there on. What does it mean?

Well, I'll start with an example.

Let's say that I want to prove that I'm a human being. Showing that I have blood is necessary, since human beings have blood, but not sufficient. Many different creatures have blood. Likewise, let's say that I want to prove I'm a male. Since my name is Brandon, that's sufficient, but not necessary in order to prove it. Certainly, there are many men who aren't named Brandon, but if you're named Brandon you're most certainly a male.

The distinction is very important. In these examples something that is necessary is exactly that, necessary, but that doesn't mean it completes the proof. Sufficiency, on the other hand, can prove it, but that doesn't mean it's necessary, since there may be examples where it's true that aren't related to it.

However, they can be taken together, and that's when it's the strongest. Let's say all males were named Brandon, and only males were named Brandon (no dogs, creatures, women, etc.). Well, showing that my name is Brandon is both necessary AND sufficient since it's sufficient to prove I'm a male, but at the same time for me to prove it it's necessary that my name is Brandon.

 This leads right to logical implication which is directly tied to this (and makes it much more intuitive). The first time you come across these would be in mathematical logic, which may be in early calculus, or as late as real analysis.

Here are the implication arrows. They are:

P⇒Q
Q⇒P
P⇔Q
 
Although the first two seem redundant, I'll use them with my previous examples. Let's say the statement P is "My name is Brandon" and Q is "I'm a male". The first says, essentially, "If my name is Brandon, then I'm a male." Which means the fact that my name is Brandon implies I'm a male. Let's switch the world around a bit, and say that all males are named Brandon, but so are some females. Then, the reverse statement would be true. "If I'm a male, then my name is Brandon", but then the previous statement would no longer be true since some females would be named Brandon. So now being named Brandon isn't enough to prove I'm male.

Notice how this works. With the first two implication arrows, you can take the previous statement and imply the latter, but you can't go back. My name is Brandon, therefore I'm a  male. However, if I'm a male that doesn't mean you can show my name is Brandon, since it's certainly not always true.

The last example follows necessity and sufficiency. I'll just restate my example of both necessity and sufficiency (the world where all men are named Brandon, and everyone named Brandon is a male), which is "If my name is Brandon, then I'm a male", and "If I'm a male, then my name is Brandon." In other words, I could take either fact and it would lead me to the other.

This is very simple, but incredibly powerful with logic. My name is Brandon⇒I'm a male. I'm a human being⇒I have blood. I'm from planet Earth⇔I'm from the third planet from the sun.

The first leads me to the second in all of those, but only in the last one can I go both directions.


Strange topic. Feels weird. I'll try something different next time.