A paradox is a statement which, although it seems to employ valid arguments and sound reasoning, it leads to a contradicting conclusion, a conclusion that goes contrary common sense or is self-contradictory. They are valuable in spotting weaknesses or deficiencies in philosophy, mathematics, physics, or other disciplines which employ logic and critical thinking as one of their main tools.

1. The ship of Theseus

Plutarch, the ancient Greek essayist and philosopher, devised this paradox by contemplating upon the question: “What makes a ship the ship that it is?”. Plutarch presented a case where one disassembles a ship into parts and take away all of its parts, replacing each and every part with a new part. Plutarch wondered whether the identity of the ship remained the same, despite having all of its part removed and replaced. If the answer is yes, then that would mean that what gives a thing its identity is not constituted upon its parts for Plutarch just revealed that an object can get all of its parts replaced and still hold the identity it already held.

The paradox was more recently discussed by Thomas Hobbes, the 17th century English philosopher, who set a new question asking: “If one takes all the disassembled parts and use them to build a new ship, would that ship be the ship of Theseus or an entirely different ship?

This paradox deals with the conception that the identity of an object rests on what it is composed of. This has led a number of philosophers to argue that an object’s identity goes beyond the matter which constitutes it.

2. This sentence is false

This paradoxical sentence serves as a contradiction of itself as it goes into an infinite regress of switching between the values of true and false. The paradox goes like this: If the sentence is false then the statement that ‘this sentence is false’ is true. But if the statement ‘this sentence is false’ is true then the sentence is false, which, in fact, makes the sentence true leading to a paradox and taking us back to where we started.

This paradox is also called the ‘liar paradox’. We can imagine someone show says “I’m lying”. If they are lying, then they are lying about lying, therefore they are telling the truth. But if they are telling the truth about lying, then they are lying, which leads us again to an infinite regress of the statement switching between values of true and false, contradicting itself.

This sentence is paradoxical because what determines the truth-value of the sentence (whether the sentence is true or false) is contained in the sentence itself. Therefore, the sentence puts itself to a self-referent loop.

This paradox goes back to Eubulides of Miletus who set two premises to present this paradox. First, when we take one million grains of sand, we refer to them as a heap of sand. Second, when we take out one grain of sand from the heap of sand, the heap still remains a heap. Now, repeating the second step and reapplying the rule will make us conclude that five or even one grains of sand constitute a heap of sand, which is false.

The problem appears as there is no clear limit at which the heap stops being a heap. Reapplying the rule, however, that “If we remove on grain of sand from the heap of sand, the heap of sand will still be a heap of sand.”

4. Zeno’s paradox of Achilles and the tortoise

Zeno of Elea, the 5th century ancient Greek philosopher, devised various paradoxes to show that motion as we perceive it in the world is, in fact, an illusion as change is not possible.

In order to support this, Zeno asked us to imagine a footrace between Achilles and a tortoise, at which the tortoise begins at a head start of 10 metres, for example. What Zeno wants to show is that it is impossible that Achilles will pass the tortoise, which goes contrary to real-world experience and intuition.

As the tortoise starts 100 metres ahead of Achilles, Achilles will take some time to cover these 100 metres and reach the point where the tortoise was at the start of the race. However, at the time it takes for Achilles to cover this distance, the tortoise will move at a distance of 10 metres, for example and will still be in front of Achilles. Now, in the time it will take Achilles to cover the extra 10 metres, the tortoise will move at a distance of 1 meter, let’s say.

The paradox lies on the fact that it will always take some time for Achilles to cover the distance that separates him and the tortoise. During that amount of time, however, the tortoise will always move forward, at a distance which Achilles also needs to cover, but where the tortoise will always move forward.

In this paradox, a prisoner is told by a judge that they will be executed in the following week, in a weekday, but that the day of execution is a surprise. The prisoner reasons that his day of execution will never come.

His reasoning is that, he cannot be hanged on Friday, because if the prisoner is still alive on Thursday, then they will know that they will be hanged on Friday, therefore the execution will not be a surprise. But if the prisoner cannot be executed on Friday, then they cannot be executed on Thursday either because, if they are alive on Wednesday, and they cannot be executed on Friday, then the execution cannot take place on Thursday for it will not be a surprise for the prisoner. This applies to the rest of the weekdays as well, ending up with the conclusion that the prisoner will not be executed.