The Grue Paradox: Induction Does Not Really Exist.

The grue paradox deals with the scientific notion of induction and challenges its objective validity. I have already covered the mathematical notion of induction and the logical notion of confirmation (the raven paradox) in separate essays. A sound understanding of these two topics will, in my opinion, help you appreciate the depth of the grue paradox.

Interestingly, the grue paradox and the raven paradox are closely related; more on that later. The grue paradox makes a solid argument against induction. To even begin appreciating this argument, it makes sense to visit/revisit why induction is relevant to science.

So, I will be starting this essay there. From thereon, I will be touching upon some of the challenges that the philosophers of science have faced with induction over the years.

Following this, we will finally be arriving at the grue paradox. Once we have covered the grue paradox, I will be exploring a few paradoxical states of confirmation. Without any further ado, let us begin.

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Why is Induction Relevant to Science?

If you think about it, the entire business of science hinges upon the following process:

Step 1: Make empirical observations and measure/gather data about a phenomenon.

Step 2: Form a hypothesis about what causes the observed phenomenon.

Step 3: Repeat experiments that confirm the cause for the phenomenon, and use inductive logic to generalise the cause in the form of a scientific law to predict future occurrences of the phenomenon.

In short, science confirms a hypothesis using observations/measurements from experiments/real world and uses inductive logic to establish the hypothesis as a scientific law. This scientific law, then, predicts/describes this phenomenon in the future (or past/present).

The transition from hypothesis/theory to law is not a simple one though. I have covered this in detail in this essay. If you are interested, check it out. But for the purposes of this essay, this level of understanding of the relationship between induction and science should suffice.

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The Philosophy of Science: Induction Does Not Really Exist

One of the biggest challenges of science is that we don’t know why it works the way it does. Every scientific law ever created is a model of how a phenomenon occurs. Scientists create such laws based on what they observed in the past using induction.

If this pattern changes, then the law would not hold anymore. So, the more confirming instances there are, the stronger our trust or belief in scientific laws becomes. However, if there is only one instance that contradicts a scientific law, then our belief in it gets shattered.

In essence, induction works, until it doesn’t! So, should we give up on it? Well, John Stuart Mill made a good case for induction by arguing that nature has regularities, and it is only by induction we know that it has regularities.

It is easy to note that this logic is circular; he knew this as well. But it kind of does the job and numerous contemporary philosophers such as Max Black agreed with him.

A German philosopher named Hans Reichenbach took this argument even further in what is now famous as a pragmatic justification of induction. He argued that if there is any way for us to guess what the unexamined parts of nature look like, it has to be by induction. In essence, he meant that if induction does not work, nothing else in science would work either.

Bertrand Russell and Rudolf Carnap seemed to agree with Reichenbach, although Russell did not find the argument satisfying enough. Carnap, on the other hand, tried to construct a probability-based inductive logic system from scratch in his work titled “Logical Foundations of Probability” (1950). Notable philosophers such as Karl Popper and Thomas Kuhn thought that this approach was misguided though.

Then came Carl Gustav Hempel with his raven paradox. If you have read my essay on this paradox, you know how notorious it can be. Following this, an American philosopher named Nelson Goodman came up with the grue paradox in his book titled “Fact, Fiction, and Forecast”.

Many respectable philosophers and scientists of the time regarded this paradox to be equally notorious as the raven paradox. Now, it is our turn to experience what this paradox is all about.

The Grue Paradox Explained

We use the word “emerald” to describe a green gemstone; the colour is due to trace amounts of chromium or sometimes vanadium. Based on this, expressions such as “emerald green” have become commonplace.

Let us say that the law here is “all emeralds are green”. But one fine day, a prolific and established scientist comes along and claims that “all emeralds are grue.”

People don’t really understand what “grue” means. So, they demand an explanation from the famous scientist. This is the explanation he provides:

All emeralds will be green until 31.12.3022.

But from 01.12.3023 onwards, all emeralds will be blue.

To capture both the properties of all emeralds, I propose the predicate “grue” (green-blue).

If we are to entertain this scientist, every passing day that we observe “green” emeralds confirms this scientist’s hypothesis. So, it should increase our belief in this hypothesis.

But let’s be honest. Does it really? The answer is no. This is, in essence, the grue paradox. If this confirmation does not increase our trust or belief in the scientist’s hypothesis, why should any confirmation of any law gain our trust or belief? While you ponder upon that question, let me present you with more complications.

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The Paradoxical Nature of Confirmations

It is clear that the grue paradox (as well as the raven paradox) deals with the notion of confirmation. One of the philosophical questions seems to be how and why does confirmation increase our trust or belief?

This discussion becomes more complicated when we consider the fact that under certain circumstances, confirmations can actually discredit or even falsify a hypothesis. Here is an example from Paul Berent:

Let us say that the basic hypothesis is, “All men are less than 100 feet tall”. All of a sudden, we discover a man who is 99 feet tall. While this observation should logically confirm the hypothesis, you and I both know that our confidence in the hypothesis would be significantly shaken.

Here is another one: suppose that we have ten cards whose values range from ace all the way up to 10. Now, I shuffle the cards and deal them face down in a row. Your hypothesis is “no card with value n is in the nth position from the left.”

To your amazement, you confirm your hypothesis 9 times in a row as you turn the first nine cards over. However, this should not increase your confidence in your hypothesis. If none of the turned cards is 10, then the tenth one has to be the 10, thus falsifying your hypothesis.

On a related note note, Richard K. Guy once wrote a hilarious yet ingenious paper titled “The Strong Law of Small Numbers”. I have covered this paper and its implications in this essay. If you are interested, check it out. Turning back to induction, where does all of this leave us?

The Grue Paradox: Induction Does Not Really Exist

Paradoxes like the grue paradox, the raven paradox, and the related family of confirmation-based veridical paradoxes help us come to terms with the following truth:

We don’t even have a qualitative understanding of inductive logic, let alone a quantitative one.

Outside of mathematics, the notion of induction loses its deterministic promise. However, it still works, and what’s worse, it’s the best we’ve got!

As we fumble around to make sense of our universe and nature, our challenges with induction might very well reflect the limitations of our very nature as human beings.

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Reference and credit: Martin Gardner.

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Further reading that might interest you:

• How To Really Solve The Kissing Circles Puzzle?
• Randomised Statistical Trials Are Not Always The Best Option.
• 3 Reasons Why Deep Space Travel Is Really Challenging

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