How To Really Tackle The Raven Paradox?

In my previous essay on the raven paradox, I covered how the paradox occurs. It all started with the playful-sounding logic puzzle that logician and philosopher Carl Gustav Hempel formulated in the 1940s. He starts with the statement:

“All ravens are black.”

Given this statement, on the one hand, we consider the sighting of a black raven intuitively as evidence confirming the statement. On the other hand, we do not consider the sighting of a green apple intuitively as evidence confirming the original statement. However, the real challenge occurs when our scientific logic system does consider the sighting of a green apple as confirmational evidence as well.

We essentially apply the plausible concepts of ‘instance confirmation’, equivalence condition, and contrapositive statement to arrive at an implausible-looking conclusion. This is the paradox — it reveals a contradiction between intuition and inductive reasoning. For more details, please check out my original essay on this topic.

In this essay, I will be covering how one could choose to tackle this paradox. To begin, let us analyse the fundamental question underlying the challenge.

This essay is supported by Generatebg

The Challenge of the Raven Paradox

In all its complexity and minutiae, the raven paradox is asking a very fundamental question:

What is “evidence”?

It reveals to us our own inadequate definition of the word “evidence”. If we are somehow able to come to an agreement on what this word means, we would be able to tackle this paradox.

But alas! Some of the deepest challenges arise from the simplest of questions. The scientific community and deep thinkers alike have been at odds over the meaning of the word “evidence” over the past decades.

We will take a look at some of the most prominent approaches people have taken to tackle the raven paradox so far. Before we proceed, know that any attempt to tackle the paradox must also give a unique and consistent explanation as to why the paradox occurs.

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Acceptance of the Raven Paradox

Proponents of the “acceptance” approach choose to merely accept non-ravens as relevant evidence of the original statement. Hempel himself chose this approach to tackle the paradox. He argued that the result looks paradoxical only because we have prior knowledge like in the case of the raven example.

To explain his point further, consider the following statement:

“All sodium salts burn yellow.”

Its contrapositive statement is as follows:

“Whatever does not burn yellow is not a sodium salt.”

Now imagine a situation where you observe someone holding a colourless cube of some unknown material in a flame. You notice that the cube does not burn.

Based on this, you consider this as evidence that this material is not a sodium salt. Later on, the person holding the cube reveals to you that the substance is ice (made of water).

Had you known that it was ice all along, you might not have considered the event of it not burning as evidence of the original statement (or its contrapositive).

In short, supporters of this line of thought argue that the notion of evidence should be separated from prior knowledge. Evidence should be considered unbiased from what we already know or do not know. In Henpel’s own words:

“If we assume this additional information as given (that the substance is ice), then, of course, the outcome of the experiment can add no strength to the hypothesis under consideration. But if we are careful to avoid this tacit reference to additional knowledge … the paradoxes vanish.”

– Carl Gustav Hempel

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The Bayesian Approach to the Raven Paradox

The Bayesian faction of scientists and deep thinkers argues that prior knowledge should not be ignored and that evidence should be built on top of what we already know. Based on Bayesian probability, this approach naturally considers each instance of an event in relation to the total number of possible occurrences.

For instance, when you observe a black raven, you should also consider the total number of all ravens. That is likely a very large number, but still doable. However, when we consider the contrapositive condition, things change.

In the event that you observe a green apple, you should also consider the total number of all non-black entities. This is likely a significantly larger number than the total number of ravens.

This approach argues that the paradox occurs because we intuitively estimate the amount of evidence provided by the sighting of a green apple to be zero. In reality, it is likely non-zero but infinitesimally small.

At this point, we arrive at another challenge! The Bayesian faction is sub-divided into two more groups (human beings are truly special). One group argues for what we have just seen so far: the sighting of a black raven provides more evidence than the sighting of a green apple.

The other group argues that based on logical equivalence, both events must symmetrically provide the same amount of evidence, in that, they both reduce the entropy by the same amount.

We haven’t even started considering time dependencies. For example, when I say that you should consider all ravens, does this include all the ravens of the past, the present, and the future? How are you supposed to know how future ravens would look? As you see, things get really complicated really fast.

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Where Do We Go from Here?

Even today, the debate between the various approaches to tackle the raven paradox is still ongoing. I have just scratched the surface of the scientific discussion with this essay. There are other nuanced approaches that I haven’t even covered here (I just covered the most prominent ones).

At its core, the debate circles around two questions:

1. What is “evidence”?

2. How do we handle prior knowledge when treating evidence?

As human beings, we seem to be divided on answers to these questions. Here’s my challenge: every argument sounds convincing until you read or consider the other one. A part of me wants to accept them all; just not all at once.

I meditate often upon this topic. Over the years, I have come to believe that it might very well be impossible to arrive at a single “solution” for this paradox. The way we tackle this paradox remains very context-dependent.

Depending upon the requirements and boundary conditions we have, one approach may be more fruitful than the other. In situations where it makes sense to include and weigh prior knowledge, the Bayesian approach makes sense. On the other hand, when gauging events with independent probabilities, it makes sense to take the acceptance approach from Hempel and co.

Things get even more complicated when we start considering advanced phenomena such as ergodicity (a topic for another day). In short, my approach to tackling the raven paradox is: “it depends!“

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Reference: Carl Gustav Hempel (scientific paper).

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Further reading that might interest you: Logarithms: The Long Forgotten Story Of Scientific Progress and How To Make Working With Squares More Fun In Math?