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Black Ravens And Green Apples!

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.

Note by Mursalin Habib
3 years ago

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This is a very interesting discussion, but I find myself wondering if the conclusion is valid. According to Wikipedia, "Nicod's criterion says that only observations of ravens should affect one's view as to whether all ravens are black. Observing more instances of black ravens should support the view, observing white or coloured ravens should contradict it, and observations of non-ravens should not have any influence."

So, I have to ask, what do you mean by evidence in this context? Why does probability come into play? It's not as though there's a limited amount of blackness in the universe; if there was, one could argue that seeing a black non-raven decreases the probability of ravens being black, since some blackness was wasted on the non-raven. One could also argue that seeing a green apple increases the probability of ravens being black, since no blackness was wasted on the apples. But here, there is no relationship between apples and ravens, which is why the result is so counter-intuitive, I suppose!

This segues into another admittedly philosophical question: what is the nature of knowledge? How does observation affect knowledge? Raj Magesh · 3 years ago

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@Raj Magesh

what do you mean by evidence in this context?

Basically what everyone means by evidence. Say that I claim that a certain statement is always true [for example: opposite charges attract]. If I observe certain phenomena that support that claim, I'd call those phenomena evidence for that statement. The more evidence I get, the more likely it is that my claim is true. In fact, this is what we do in physics. We observe the universe and come up with models, theories that describe how things happen. Sometimes these theories turn out to be wrong. We can never prove anything in physics, we can only disprove.

Why does probability come into play?

Probability does come into play. Here's how. Say that I claim that if you flip a coin, you'll always get a head. For evidence, I say that I flipped 3 coins and they all showed heads. Does that make my statement true? No. What if I flipped 1000 coins and got the same result every time? Now it is more likely that my claim is true. More evidence implies greater probability of something being right.

But here, there is no relationship between apples and ravens

This is actually not true. If you had only two non-black things in the entire universe, then looking at one of them ensuring that it's not a raven would increase the probability of all ravens being black by a lot. Technically if you can check out all the non-black things in the world and show that they are not ravens, you prove the claim that all ravens are black.

So why is it so counter-intuitive?

Here's what I think. The number of non-black things in the world are practically infinite. So, if you look at one of them and see that it's not a raven [for example a green apple], even though the probability increases, that increase is practically equal to zero. And our minds can't even comprehend such a small change. It has a preconceived notion about how much 'an increase' is. So the statement "a green apple increases the probability of all ravens being black" seems so counter-intuitive to us. Mursalin Habib · 3 years ago

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@Mursalin Habib @Raj Magesh , does this make sense? I was half-asleep when I wrote this last night. Mursalin Habib · 3 years ago

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@Mursalin Habib +1, for the clever reasoning on the last paragraph. Beakal Tiliksew · 3 years ago

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@Beakal Tiliksew Thanks! Mursalin Habib · 3 years ago

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@Mursalin Habib, do you also know about the 'Swedish Civil Defence Paradox' & the 'Barber's Paradox' & the 'St. Petersburg Paradox'? Ameya Salankar · 3 years ago

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@Mursalin Habib, when I first read about Hempel's Paradox, I was really baffled! But this paradox is easy to notice. Well, I don't understand why I didn't think of posting this in Brilliant. By the way, great job @Mursalin Habib.

Looking forward to your completed set! Ameya Salankar · 3 years ago

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Really nice post on something interesting.

Read the standard text by Copi-Cohen (same name: Logic) to get more information on this subject, plus things like fallacies (ways of making wrong arguments), deductive logic, inductive logic, breaking arguments down into their constituent parts, syllogisms, and much more. Star Light · 3 years ago

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cool:! Excuse me for my reasoning is usually impaired, but i can't help but to associate this paradox with the Schrödinger's cat. This i find bizarre. Reality is a malleable, the shear fact of your observation changes the reality of what you are seeing. Beakal Tiliksew · 3 years ago

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Really nice.... Didn't ever thot on this aspect before.... Great work sir Jatin Bhandari · 3 years ago

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@Jatin Bhandari Don't call me 'sir'! I'm just 16.

I'm glad you liked it! Mursalin Habib · 3 years ago

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