Consider a little girl with a genie – the magical genie. The genie tells the little girl, “wish for anything and I shall make it come true”. Excited, the girl thinks hard and says, “genie, I wish that wishes could never come true“. Would that wish ever come true? If the genie made the little girl’s wish true, then the little girl’s wish would not come true, simply because the little girl wished that wishes would not come true, right? Now, that is outright controversy! That is an example of a paradox, but what are paradoxes really? Let us dive right into demystifying paradoxes but, be ready to think hard and make sure your tummy is full – don’t get hungry thinking too much now.
A paradox is a statement that leads to a self-contradictory or a logically unacceptable conclusion despite logical reasoning from a true assumption(s). In other words, our minds believe they are being tricked when they look into a statement. It is important to note that a paradox is different from a dilemma and by way of thought, you can say that the dilemma gives you one of two choices, a paradox does not. For a paradox to be a paradox, it has to contain some kind of contradiction be is logical, visual or otherwise.
There are three kinds of paradoxes according to one W. V. Quine, an American philosopher of the 20th century. The three categories are; veridical, falsidical and antinomy. I shall cover each one of them and give an example.
Veridical : This is a paradox that produces a result that appears to be absurd but is proven true nonetheless. In simpler terms, I could tell you that 1 + 1 = 10 but you would dispute that. However upon research, you would find that it is true because I was doing the calculation in binary, not decimal. That example is not a paradox at all and that is why I give you this: the Birthday Paradox.
Falsidical : This kind of a paradox produces a result that not only appears false but is indeed false when the underlying fallacy is identified. A popular paradox in this category is the Achilles and the Tortoise Paradox of a Greek philosopher, Zeno. In his argument, before the invention of mathematical concepts to prove the fallacy, the philosopher noted that an athlete like Achilles could never catch a normal, slow moving tortoise if the tortoise was given a head start and continued to move ahead. We know that is a fallacy but wait, why did he say so?
According to Zeno, there would always be a distance between the tortoise and the athlete because the tortoise is always moving. In this sense, if a head start of 100 meters is given to the tortoise and it moved at a speed of 1 meter every 5 seconds, while Achilles moves 100 meters in 10 seconds then by the time Achilles got to the 100th meter mark, the tortoise would have moved two meters and so forth. There shall always be a distance between the two no matter how small. Duh? He will still catch up and overtake the tortoise.
That paradox bugged minds for centuries before mathematics came to the rescue. Aristotle crushed this paradox and the easiest way to understand why it is a fallacy is to consider a mathematical sequence. An infinite number of steps of a finite distance shall always total up to the length of that finite distance provided that nothing is done to the distance – that is; distance is not changed. Take into consideration a whole number, 1. Divide it into two, you get two halves. Divide one of those halves by two, you get two quarters. Divide one of the quarters by two, you get two eighths. If you add two-eighths to the quarter and then to the half, you get a whole, 1. This paradox can be easily observed in mortgage loans. Moving on…
Antinomy : This is a paradox that reaches a self-contradictory result even after following proper and accepted reasoning. In other words, no one has an answer really or, it does not make sense at all. Consider a time traveler who goes back in time to kill his grandfather or the genie paradox we had at the start of this article, what is the solution there? This is an incompatibility of two laws. Let us use a very simple example to put the point across.
I am lying. Is that statement true or false? If I am truly lying, how would you tell for sure that I am, furthermore how would you tell that that statement is true if I have already told you I was lying? For that statement to be true, it must be false and vice versa. Are you lost there? That is what a paradox is. We may not know the truth at the moment, and we may never come to know the truth after all.
There are other categorizations of paradoxes from various authors and each is justified. But for now, you are able to understand what a paradox is and the types there are. If you have a paradox that you think is unsolvable, be sure to leave it in the comments.
Now you know.