The Logic of the Illogical: Paradoxes

It’s a school night, and you’re faced with a choice: hit the books and study for the test tomorrow, scroll on Instagram for a quick distraction, or do nothing and hope for the best. Yet, none of these options feel fulfilling and leave you stuck in a state of indecision. This scenario illustrates a classic paradox of choice, in which too many options can actually lead to dissatisfaction. 

The allure of paradoxes lies in their ability to challenge the limitations of our thinking. They seemingly punch a hole into logic itself while simultaneously breaking and reinforcing the rules of reasoning. 

Paradoxes can be categorized into different types, each with its unique flavor.

Logical Contradiction

This is a situation in which simple logic contradicts itself; it is a statement that cannot be false but cannot be true. 

The Buttered Cat Paradox
As cats always land on their feet and buttered bread always lands butter-side down, what would happen if a piece of toast (butter side up) is tied to the feet of a cat and is then dropped? (For the sake of the paradox, we assume that these things always come true)

Semantic Paradoxes

These paradoxes stem from the limitations of language itself. Words or phrases used usually do not have set limitations or measurements on when they can be used.

Sorites (Heap) Paradox
Suppose you have a normal color gradient, as shown below. At what point does a color “change”? For example, at what point does the color orange cease to be orange and turn red?

Veridical Paradoxes

A veridical paradox is a true statement that appears to be false or contradictory at first glance. These paradoxes often involve counterintuitive or unexpected consequences of logical reasoning.

Zeno’s Paradox 

An archer fires an arrow at a target. To reach the target, the arrow must first travel halfway there. But before it can reach the halfway point, it must travel a quarter of the way. This process goes on infinitely. So the question is, how is motion possible if there are an infinite number of goals you must reach before getting to the end?

Falsidical Paradoxes

These are arguments that attempt to prove something that is obviously false, but contain a trick that is detectably responsible for the absurd conclusion.

The Infinite Chocolate Paradox
You may have heard of the viral trend in which if a bar of chocolate is cut in a certain way, an extra piece seemingly not part of the original bar is left. In truth, a gap is made when the pieces are put back together, and made to look like no change was made. 

So, how do we solve problems that have no solutions? Once we recognize a statement as a paradox, we have effectively solved the problem. Understanding that there is no logical solution is, in itself, the resolution. Paradoxes challenge our understanding of logic and language and push us to the limits of our cognitive abilities. While they do not have definitive answers, they provide us with valuable tools for intellectual growth.

Sources:

https://academic.oup.com/analysis/article/70/4/615/106991
https://www.youtube.com/watch?v=ppX7Qjbe6BM&t=1143s (examples)
https://cascadiaauthorservices.com/paradox-definition/

https://www.iangibsonillustration.com/blog/2017/8/19/cat-and-buttered-toast-theory

https://www.scienceabc.com/eyeopeners/infinite-chocolate-paradox.html