Mathematical Implication

The way it ties into philosophical implication can be confusing.

This cleared it up for me - forget math! Just remember what you thought when you first heard about implication.

In common sense implication (let’s think of A => B as if A, then B) B must be true if A is true.

Think about it. I tell you the sky will explode if birds chirp.

A => B form is If birds chirp, then the sky will explode.

A = birds chirp

B = sky explodes

If birds do indeed chirp, then the sky MUST EXPLODE.

In other words, if you told me that birds chirped and the sky did not explode (True => False), I would tell you that you are lying.

This explains…

T | F | F

Back to the math. Common sense wise it is impossible for A to be true without B being true. It’s a lie if that’s the case. Always. So, in math, we say that statement is logically FALSE.

But what about the others?

T | T | T

Is easy. It follows the common sense rules. If A is true, and B is true, then you listened to me, and I was right. Everything checks out!

But

F | T | T

??? This says birds did not chirp, but the sky did explode! How is this true?

Common sense - we simply say that when birds chirp, the sky will explode. We don’t say that birds chirping is the ONLY thing that causes the sky to explode. A nuclear bomb could hit the moon for all we know! I could not say you are lying.

So, mathematicians say ‘that could be true, so it’s true’.

And finally,

F | F | T

If you told me birds did not chirp and the sky did not explode, I couldn’t tell you that you are lying either, so that checks out!

Hopefully that clears it up. This revelation cleared it right up for me.

 
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