How To Really Understand The Raven Paradox?

The raven paradox knocks at the door of one of the most fundamental aspects of science. At its core, it stems from the following question:

“What is evidence?”

Logician and philosopher Carl Gustav Hempel formulated a playful-sounding logic puzzle in the 1940s to demonstrate how this paradox occurs. He starts with the following statement:

“All ravens are black.”

This statement is also where the paradox’s name comes from. Following this, let us say that you are tasked with figuring out whether this statement is true or not.

On your adventure, you encounter a black raven first. You would intuitively consider this as evidence confirming the original statement. Since it is just one instance of ‘a’ black raven, it is not outright proof of the original statement, but just a piece of evidence that confirms the statement.

Next, you encounter a green apple. Would you consider this as evidence in favour of or against the original statement? Intuitively, you are likely to treat this piece of information as irrelevant to the topic at hand. But what if I told you that this is indeed evidence in favour of the original statement?

It sounds counter-intuitive and false. Well, this is the challenge that the raven paradox presents us with.

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The Background

The bigger question that we are trying to answer here is:

“Can we come up with a logical system that assesses evidence in favour or against any given statement?”

In this essay, I will be breaking down how the paradox occurs using an intuitive approach that puts three major puzzle pieces together. We will be avoiding complex formal expressions to make our lives as easy as possible.

To begin understanding the challenge better, we will need to cover the four fundamental concepts listed below first. But before you proceed, just know that these are just a bunch of fancy-sounding words that describe simple concepts of how human beings think.

1. The concept of confirmation.

2. Instance confirmation

3. The equivalence condition.

4. The contrapositive statement.

Again, don’t let the fancy-sounding words intimidate you. I assure you that they are simple concepts. Let us start with confirmation.

The Concept of Confirmation

To be frank with you, we have already covered the concept of confirmation in this essay. It just happens to be so that I haven’t explicitly defined it.

Consider the analogy of a detective story. Let us say that the detective is tasked with solving a crime. She discovers that the butler was at the crime scene on the day the crime happened.

The detective considers this as evidence to the hypothesis (not proof!) that the butler committed the crime. This is what we define as confirmation. If the statement were “The butler committed the crime.”, the fact that he was at the crime scene on the crime day is considered as confirmation of the hypothesis.

Later on, it becomes apparent that the butler was not at the crime scene at the exact time of the crime. The detective considers this as evidence against the statement that the butler committed the crime. This is what we define as disconfirmation.

Apart from this, the detective learns that there was a sandstorm on Mars at the same time as the crime. The detective considers this statement as irrelevant for her case. This is what we define as a neutral statement. It is neither confirmation nor disconfirmation of the hypothesis statement.

Now that we have covered the concept of confirmation, let us move on to instance confirmation.

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Instance Confirmation

Tounderstand instance confirmation, let us go back to the original statement:

“All ravens are black.”

This is what we would call a general hypothesis. The notion of instance confirmation says that if we encounter a single instance of the general hypothesis, it counts to some degree as evidence confirming the general hypothesis.

In the case of our original statement, the fact that you encountered a black raven would count as evidence to some degree confirming the general hypothesis that all ravens are black.

Right, that’s all there is to instance confirmation. Let us now move on to the equivalence condition.

The Equivalence Condition

Let us say that we have two statements that logically convey precisely the same meaning, even if they use different wording/grammatical structures. This pair of statements is what we would define as equivalent statements.

Let us consider the following example:

Statement A: Frozen water is completely made of ice.

Statement B: Ice is what frozen water is completely made of.

Even though these two statements use different kinds of sentence structures, they convey the same meaning. Consequently, they are equivalent statements.

The equivalence condition states that any two equivalent statements are either both true or both false. There cannot come a situation where one of them is true and the other is not. Such a situation in logic is what is known as a contradiction.

Right, so far, so good. Onward to the next piece of the puzzle we go.

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The Contrapositive Statement

Remember that I told you that all of these concepts are simple. Well, I lied; I apologise! The contrapositive statement is probably the only concept that is slightly advanced.

We almost never use it in real life (explicitly). It is a concept that is reserved for scientific usage in mathematics, logic, and philosophy.

The contrapositive statement is a way of expressing logical equivalence that uses negation. It is probably far easier to directly look at an example. Consider the following two statements.

Statement A: If it is raining, then the grass is wet.

Statement B: If the grass is not wet, then it is not raining.

Statement B sounds starkly different to statement A, but in the language of logic, both these statements are saying exactly the same thing!

In a general sense, the contrapositive statement takes the following form:

Years ago, when I first encountered this concept as part of discrete mathematics, it took me quite a while to really understand and digest this concept. So, don’t be alarmed if this is confusing for you. It is okay to be confused.

At the same time, it is very important that you understand how the contrapositive statement works. Without this key piece of the puzzle, the raven paradox will likely remain out of reach. So, please take your time to grasp the contrapositive statement before you proceed with reading this essay.

If it is any motivation, understanding the contrapositive principle will help you a great deal in weeding out fake news statements and other misinformation cases in real life. It would strengthen your sense of logic.

Right, if you have grasped the notion of the contrapositive statement, we are finally ready to deal with the main challenge.

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The Raven Paradox — Revealed

We are now going to use every piece of the puzzle we have picked up until this point to understand the raven paradox. Let us again consider the original statement. But this time, let’s try to formulate its contrapositive:

The contrapositive statement appears to be a really bad way of writing the original statement. But given that these two statements convey the same meaning, they are equivalent statements.

Now, let us consider the scenario where you encounter a green apple. If you consider the original statement, this encounter appears intuitively irrelevant. But if you consider the contrapositive of the original statement, the encounter of a green apple serves as evidence in favour of the original hypothesis (that all ravens are black).

The green apple is a non-black entity and it is a non-raven. The logic checks out. Here is a summary of what we just did:

1. We first formulated our original statement in its contrapositive form.

2. We then established that the original statement and its contrapositive form are equivalent statements.

3. By looking at an entity that has nothing to do with ravens (in the form of a green apple), we were able to make an ‘instance confirmation’ of the contrapositive statement.

4. Since the contrapositive statement and the original statement are equivalent, the green apple provides ‘instance confirmation’ (evidence) of the original statement.

In short, we could confirm a hypothesis about ravens simply by looking at a green apple or a blue car, etc. This sounds intuitively absurd. But it makes logical sense. This reveals a contradiction between inductive logic and intuition — the raven paradox!

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Final Thoughts

Inthe end, we used logically consistent ingredients in the form of ‘instance confirmation’, the equivalence condition, and the contrapositive statement to arrive at a logically inconsistent conclusion.

To make sense of this paradox, we need to make one of the following choices:

1. One of the following three concepts is false: instance confirmation, the equivalence condition, or the contrapositive statement.

2. The conclusion cannot follow the logically consistent ingredients we put together.

3. The conclusion must be true, even though it intuitively appears to be false.

My question to you: How would you proceed? While you ponder upon the meaning and implications of this paradox, rest assured that you are not alone.

The scientific community and deep thinkers alike have been trying to crack this challenge for decades now. In a follow-up essay, I will cover how one could approach tackling the raven paradox!

Update: I have now published the follow-up essay on How To Really Tackle The Raven Paradox?

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References and credit: Carl Gustav Hempel (scientific paper) and Marc Lange (presentation).

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Further reading that might interest you: The Thrilling Story Of Calculus  and The Strong Law Of Small Numbers.