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Fun and mind-boggling paradoxes that will pick at your brain

Do you like thinking about stuff? Or maybe, solving puzzles? Whatever the case is, here are some interesting paradoxes so you can impress your friends with your infallible logic!

Paradoxes are one of the most interesting products of the human imagination. They are and have been omnipresent throughout essentially all of human history. While the average Joe would normally shrug the idea off, paradoxes actually have far-reaching implications on the way humans approach thinking. In summary, it teaches us that even when things may appear to be right, they are wrong. But before we dive into the mystical and contradictory world of paradoxes, we must first understand what is meant by the term.

The definition of the word “paradox” is:

a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.

Interesting, right? If that weren’t a mouthful of a sentence enough, philosopher Willard van Orman Quine separated paradoxes into three categories: falsidical, veridical, and antinomies. These may be difficult to explain, so let’s take a look at some examples.

WARNING: MATH INCOMING!!!

This is a falsidical paradox. It revolves around the claim that all horses have the same color, so if we take a set with n number of horses, all of them will be the same color. This may be notated as P(n).

If there is one horse in the set, n will be equal to 1, and therefore the claim holds; that one horse is the same color as itself.

Now, let’s assume that n = k. Therefore, P(n) becomes P(k). Hence, in a set with k number of horses, they will all be the same color. In order to prove that all horses are the same color, we need to show that P(k+1) is true as well.

To do this, we can number the horses in both sets:

P(k): {h1, h2, …, hk}

P(k+1): {h2, h3, …, hk+1}

Since both sets have the same number of elements, there are k horses in each set. By our hypothesis, all horses in both sets are the same color.

When we “combine” both sets, we get:

T = {h2, h3, …, hk}

Since this new set contains all horses from P(k) and P(k+1), this new set contains horses of the same color.

Therefore, because P(k) is true, then P(k+1) is also true.

However, we see that the new set T is two less than the other sets because of the omission of h1 and hk+1. Therefore, if we determine that P(k+1) has only one element, the set T would have -1 elements, which is absurd. To make it regular, we would have to prove this hypothesis for k being greater than 2. However, this makes the base case (when we proved that the statement was true for n = 1) false.

This is known as proof by induction. Although many times it is a valid method of mathematical proof, it often leads to absurd conclusions such as this. We know that horses are not all the same color, which goes to show how math can be pretty misleading, even if it sounds legitimate.

In the words of Quine himself, a falsidical paradox “packs a surprise, but is seen as a false alarm when we solve the underlying fallacy.” In other words, the problem may appear to be true and reasonable until we expose its soft and false underbelly.

The infinite monkey theorem

This theorem can be considered a veridical paradox because its premise sounds false, but is true.

Imagine this: a monkey is placed in front of a typewriter and is allowed to type random keys for an infinite amount of time.

Now, let me ask you a question: what is the probability that the monkey types the entire script of Hamlet, by William Shakespeare? One may be inclined to believe that it is 0, and if I didn’t know any better, I would say so too. But is this really the case?

This problem has largely to do with probabilities, so let’s recap on that. Let’s say that on any given day, there is a 30% chance of a job promotion (we would usually express this as a decimal: 0.3). Assume that, for any given day, there is also a 5% (0.05) chance of a car accident happening. Hence, the chance of them happening at the same time would be 0.30.05, which is 0.015, or 1.5%. These are called statistically independent events; they don’t depend on each other to happen.

The same applies for our good monkey friend. If the typewriter has 30 keys (this is a pretty limited typewriter), and each key is pressed at random, it holds that each key has an equal chance of being pressed. If you paid attention to your math class, then you would know that the probability of any key being pressed is 1/30. That being said, the probability of the monkey writing “hello” would be (1/30)5, since you are multiplying 1/30 by itself five times (one for each letter). As the number of characters the monkey types (we’ll call this Cn) increases, the power n of (1/30)n increases, and the probability of the word or phrase containing n characters being written decreases. Chances are looking pretty slim for the monkey, it appears.

However, this is the premise of a veridical paradox. If I tell you that the chance of the monkey writing the whole Hamlet play is 1, or 100%, would you believe me? Probably not, because it sounds absurd. But consider the following perspective: what if we were to calculate the chances of the monkey NOT typing a word?

You might remember that if, say, I have a 1 in 6 chance of getting a 45 in the IB, there is a 1 - ⅙ chance I don’t. In other words, I have a 5 in 6 chance of getting a grade below 45. If we apply this reasoning to the monkey theorem, the probability of the monkey not typing “hello” (following our previous example) would be:

P = [1 - (1/30)^5]^x

With x being the number of blocks (groups) of 5 letters. As x increases, the chances of the monkey not typing “hello” decrease exponentially, reaching 0 at infinity. Voilá! As x approaches infinity, the chances of the monkey typing a word of your choice is 1.

The omnipotence paradox and the grandfather paradox

Ok, enough math. Let's break down the word “antinomy”. “Anti-”, as I am sure most of you know, means “against”. “-Nomy” is derived from the Greek word “nomos”, which means “law”. “Against law” is the literal translation of the word “antinomy”. But that doesn’t give us a whole lot of information. To better understand this type of paradox, take a look at the following statement:

“This sentence is a lie”

There is obviously something wrong here. By claiming that the above sentence is a lie is to automatically assume that it is true. In turn, by doing that, you admit that the sentence is a lie.

But what does this tell us about antinomies? Based on the reasoning above, antinomies are propositions that can neither be true nor false.

An example of this is the omnipotence paradox. Omnipotence is the state of being able to do anything and everything, or having unlimited power over things. If an omnipotent being exists, can they create a stone which they cannot lift? If so, then they would not be omnipotent, and they would not have created the stone in the first place.

Some people use this as an argument to disprove the existence of (a) God. However, it is perfectly reasonable to contest that such an entity would not abide by humanity’s notions of logic; it would transcend the idea of paradox.

The grandfather paradox works in a similar way. Say you build a time machine, go back in time and kill your grandfather. If you killed your grandfather, you never would’ve existed. Hence, you wouldn’t have created the time machine, effectively disabling you from performing such a heinous crime.

This is a theoretical reason why one wouldn’t be able to travel back in time, apart from all the other "sciency" stuff.

Congratulations, you have made it this far! We come to the end of our journey through the mystical land of paradoxes and logic. Hopefully you gained some insight into the different types of paradoxes and can now distinguish one from the other, and now have a better idea of how to think critically and look deep into the arguments and ideas you are presented to in life. This was a hard journey, but one thing’s for sure: expect seeing this exact same article somewhere else.

Further reading and sources