Paradoxes: Reasons and Contradictions at Once

Written by Emir Taha Macit

Paradoxes are self-contradictory statements that express two conflicting ideas. They appear logical but may also yield some unexpected results when examined. The contradictions may arise from various reasons, separating paradoxes into their specific types, such as logical and literary paradoxes. The types and instances of paradoxes have been well-researched throughout centuries and continue to be an interesting topic for scholars to cover. This article will analyze how paradoxes came to be and the major classes of paradoxes, while going over relevant examples for each.

Paradoxes are believed to have been first systematically formulated by the Greek philosopher Zeno of Elea in an attempt to defend his teacher, Parmenides’ philosophy of monism, an idea which posits that reality is singular and unchanging. To support his teacher, Zeno wrote a book containing numerous examples of paradoxes, a work that has not survived to this day. His argument with this book was that believing in the existence of more than one thing leads to a disarray of conclusions. Through retelling, some of his paradoxes were able to live on. The most famous of these were the paradox of Achilles and the tortoise and the dichotomy paradox.

The paradox of Achilles and the tortoise goes over an imaginative race between Achilles, a Greek warrior, and a tortoise. In the race, the tortoise gets a head start. When Achilles starts and, after a while, reaches where the tortoise once was, the tortoise has already gotten farther than it was. And as Achilles repeats this process on a continuously smaller scale, the same result is found. This paradox proposes that Achilles can never catch the tortoise, displaying the infinite subdivision of space and time, a topic that is also the foundation of the dichotomy paradox.

The dichotomy paradox supposes the Greek goddess Atalanta is constantly running towards the end of a path. To reach the end, she must first run the first half of the path. But once she reaches that point, she must now run half of the distance remaining until the end. As this halving process may go on indefinitely, this paradox is another showcase of infinite subdivision. Of course, both the dichotomy and Achilles’ paradoxes have their flaws. For example, about the dichotomy paradox, Zeno assumed that since the smaller segments of the path would keep getting divided, the time required to reach the end would be infinite. But, as later mathematics show, an infinite sum of decreasing values can still have a finite total. When taken into consideration, this completely undermines Zeno’s theory, but does not undermine the impact he has had on philosophy and mathematics overall.

Beyond Zeno’s paradoxes, with every passing century, new paradoxes continue to be formulated. These paradoxes can be grouped into many types depending on either their apparent outcome or the area of reasoning they are under. While the former contains veridical, falsidical, and antinomy paradoxes, the latter includes logical, mathematical, epistemic, and scientific paradoxes.

The first of the paradox types we will be going over is veridical. Veridical paradoxes are ones that may seem absurd at first, but turn out to be true when looked into. An example of this is the birthday paradox, which is about the fact that in a room of 23 people, there is a 50% chance of at least two of them sharing a birthday.

Falsidical paradoxes, on the other hand, are essentially the opposite of veridical ones. They may seem factual, but are actually untrue. Zeno’s paradoxes can be classified as falsidical, as they seemed true at first glance, but were later disproven.

Antinomies are paradoxes in which a contradiction somehow arises between two statements that are believed to be true, exposing a problem in the logic. A popular example of an antinomy is the Pinocchio paradox. This paradox questions that, if Pinocchio claims, “This sentence is a lie,” will his nose grow or not? Since he is lying by saying that it is a lie, will his nose grow, or will it stay the same size as he ended up telling the truth by claiming that his sentence was a lie? All antinomies are also logical paradoxes.

As we move on to the paradoxes classified under the area of reasoning, we should start with logical paradoxes. These paradoxes are closely related to antinomies, but are broader in their reasoning, including problems such as circular reasoning. A good example other than the Pinocchio paradox is the Epimenides paradox, in which Epimenides, a Cretan himself, claims that all Cretans are liars. If he was telling the truth, that means he was lying as a Cretan; but if he was lying, that means he was telling the truth.

The second type under this category is mathematical. As their type suggests, these are paradoxes arising from various systems in mathematics, often related to the concept of infinity. Other than Zeno’s paradoxes, the Monty Hall Problem is a good example of a mathematical paradox. The Monty Hall Problem is a scenario in which you are given an option to choose between three doors. Behind two of the doors, there is a goat, but behind one of them, there is a car that you can win if you choose correctly. You first choose a door, but then get the option to switch to the other after the host opens a door containing a goat. Are your chances higher if you switch, since you have only a 1/3 chance of having picked the correct door, while the others have a combined probability of 2/3; or are your chances unchangeable? Studies have concluded that switching is the better option, since the host will never open the door with the car, meaning the other door left may be the door with the car behind it.

Next, epistemic paradoxes. These paradoxes revolve around knowledge and involve various well-supported beliefs that somehow contradict each other, leading us to question our knowledge and correct our beliefs. The preface paradox can be given as an example. This instance involves an author who has written a book. The author has checked what she has written in the book and believes the facts to be true, but claims in the preface of the book that she has probably left some errors. The apparent contradiction in this paradox highlights the problems that lead to epistemic paradoxes.

The final type of paradoxes we will go over is scientific paradoxes. Instead of pure logic, they stem from scientific theories and laws. One example is the twin paradox. According to the paradox, there is a pair of twins, one staying on Earth and one traveling at near-light speed in space. It is suggested that the twin in space ages more slowly than the one on Earth due to Einstein’s theory of relativity. The age difference between twins, despite what we generally think of how twins age, is what leads to the paradox.

In conclusion, the varying types of paradoxes and how they work in general make them a complicated topic to talk about. Ever since the assumed first systematic formulation of a paradox by Zeno became a debate, paradoxes have been a hot topic. And as time goes by, paradoxes will likely continue to play a prevalent role in various areas, such as mathematics and philosophy.

References:

1. “What Is a Paradox? Easy Definition and Examples”, Oxbridge Editing, 2024, https://www.oxbridgeediting.co.uk/blog/what-is-a-paradox-easy-definition-and-examples

2. Fija Callaghan, “What is a Paradox? Definition, Types, and Examples”, Scribophile, (n.d.), https://www.scribophile.com/academy/what-is-a-paradox

3. “Zeno’s Paradoxes”, Stanford Encyclopedia of Philosophy Archive, 2002, https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno

4. Vashistha Patel, “Exploring paradoxes: Types of Paradoxes”, Medium, 2023, https://medium.com/@vashisthatpatel/exploring-paradoxes-types-of-paradoxes-d0c2b934bb3f

5. “Epistemic Paradoxes”, Stanford Encyclopedia of Philosophy Archive, 2006, https://plato.stanford.edu/archives/spr2021/entries/epistemic-paradoxes/#PrePar