Here's something incredibly interesting. I've been meaning to write about this for a really long time but somehow I didn't get the time. At last here I am with it. This post contains no math at all. So, anyone can join in.
So are you ready?
We're going to start with a really simple statement. It goes like this:
\(1.\) All ravens are black.
Seems simple enough, right? Now the contrapositive of statement \(1\) is something like:
\(2.\) Anything that is not black is not a raven.
If you read carefully, you'll quickly realize that 'contrapositive' is just a fancy term logicians use to mean that statements \(1\) and \(2\) are saying pretty much the same thing. They are logically equivalent. If \(1\) is true, so is \(2\).
Read both the statements again. Make sure you understand everything up to this point.
Have you done that?
Now consider the next statement:
\(3.\) I was looking out of the window and I saw a black raven.
Notice that statement \(3\) acts as evidence supporting statement \(1\). This is pretty intuitive as well. The more black ravens you see, the more likely it is for statement \(1\) to be true.
Now consider statement \(4\):
\(4.\) I see something green on my table [not black], and it's an apple [so it's not a raven].
Statement \(4\) is evidence for statement \(2\). This is obvious too. Because statement \(4\) describes an instance where something is not black and that something is not a raven.
Nothing up to this point is "incredibly interesting", but the fun stuff happens from now!
Do you remember that statements \(1\) and \(2\) were equivalent? So, anything that acts as evidence for statement \(2\) would also act as evidence for statement \(1\).
Statement \(4\) is evidence for statement \(2\). So, it should also be evidence for statement \(1\).
Now take a moment to realize what this implies.
This implies that looking at a green apple gives you evidence that all ravens are black! But that's crazy!
Can you really get information about the 'blackness' of ravens by looking at a green apple?
This paradoxical conclusion is called 'Hempel's Ravens'. This is a great example of how simple reasoning can sometimes be incredibly counter-intuitive. And you don't have to be restricted to green apples too. Anything non-black-non-raven would work as evidence for \(2\) and consequently for \(1\). So during the time you read this post, you have surely gathered a 'lot' of evidence supporting the notion that all ravens are black! How about that?
This note is going in my 'Counter-intuitive!' set. I'll be adding more stuff to it.