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We are always solving problems, some of which have instantly obvious solutions and others that do not. In the latter case, the word puzzle comes to mind instead. A problem can be defined therefore as a question that presents enough information from which we can envision a solution or answer; a puzzle, on the other hand, seems to lack the relevant information for envisioning a solution.
If told that in a right-angled triangle, two of the sides are 4 and 5 units, and asked to determine the length of the third side, we can easily come up with the answer, if we know the Pythagorean theorem (answer: 3). This problem has all the information we need to arrive at an answer (given background knowledge). Now, let's consider the following:
Two people want to cross a river. The only way to get across is with a boat that they find on one side; but that boat can only take one person at a time. The boat cannot return on its own, and there are no ropes to haul it back, yet both persons manage to cross using the boat. How did they do it?
The solution is that the two individuals were on opposite sides of the river. This is a “gotcha” type of puzzle, as the late Martin Gardner called such puzzles in his marvelous book, Gotcha (1982). There was no mention in the statement of the puzzle that the two individuals were on the same side. It cannot be assumed that they were.
So far, so good. Now, consider the following, from Henry Dudeney’s classic puzzle anthology, The Canterbury Puzzles (1907):
A child asked, “Can God do everything?” On receiving an affirmative reply, she at once said: “Then can He make a stone so heavy that He can’t lift it?”
What do we make of it? It appears to have no answer at all. In fact, it is a strange type of problem, called a logical paradox, with no solution. It is difficult to pinpoint who came up with the first paradoxes, but two Greek philosophers, Eubulides of Miletus (c. fourth century BCE) and Zeno of Elea (c. 490-430 BCE), appear to be among the first to have devised them. Equivalents have been found, however, in ancient China, from the Warring States era (479-221 BCE), a fact which suggests that paradoxical thinking is likely universal.
The study of paradoxes has been of enormous value to philosophy, logic, mathematics, science, and other fields, leading to paradigm shifts within them. Zeno’s paradoxes, for example, led over time to the concept of limits which, in turn, was the basis for the calculus. Various philosophical treatments of paradoxes have been put forth, such as the one by philosopher W. V. Quine, in his classic book, The Ways of Paradox, and Other Essays (1966). But for the present purposes, a paradox can be envisioned as an experiment in non-linear thinking. Consider the classic question posed by Plutarch in the first century CE: “Which came first, the chicken or the egg?” If we assume that it was the chicken, then the question becomes: How can the chicken precede the egg from which it is born? If we assume that it was the egg, then the question becomes: How can the egg precede the chicken, since the chicken is required to produce it? Clearly, we cannot resolve it with cause-and-effect linear thinking. It is an example of infinite regress (or circular) thinking, which is more common than one might think.
Interestingly, Zeno’s paradoxes are recounted by Aristotle in his Physics. Although he dismissed them as examples of specious reasoning, the very fact that he recorded them is indirect evidence that he understood their enormous implications for grasping the nature of the human mind.
The paradoxes provided here are 10 classic ones. Even if well-known, they are worthwhile revisiting from time to time, since they shed light on how human thinking is at times, literally, paradoxical, revealing that the imagination sometimes manifests itself as an inner trickster, always prepared to contrive some paradox that will mischievously undermine our most elaborate logical structures.
Paradoxes
1. The Liar Paradox
This is truly the most "classic of all classic" logical paradoxes. It is a perfect example of circular or self-referential reasoning. Some attribute it to Eubilides, others to Protagoras (c. 190-420 BCE). Its most famous articulation has been attributed, however, to the Cretan poet Epimenides in the sixth century BCE:
Epimenides, a Cretan, made the following statement: “All Cretans are liars.” Did Epimenedes speak the truth or not?
2. Jourdain’s Paradox
This is British logician Philip Jourdain’s (1879-1919) version of the Liar Paradox, which he devised in 1913:
The following is printed on the front side of a card: “The sentence on the back side of this card is true.” But on the card’s back side the statement reads: “The sentence on the front side of this card is false.” What do you make of the card?
3. The Barber Paradox
Apparently distressed by the challenge to logical reasoning that the Liar Paradox posed, the English philosopher Bertrand Russell (1872-1970) formulated the following version to examine the nature of self-referential reasoning more precisely:
The village barber shaves all and only those villagers who do not shave themselves. So, shall he shave himself?
4. Grelling’s Paradox
This is yet another paradox exemplifying how self-reference works. It was formulated by Kurt Grelling and Leonard Nelson in 1907—hence it is also known as the Grelling-Nelson Paradox:
Is the word “heterological,” meaning “not applicable to itself,” a heterological word?
5. The Dichotomy Paradox
This is one of Zeno’s paradoxes of motion, which he formulated to support the views of his own teacher, Parmenides, that motion was (logically) impossible. It is also known as the Race Course Paradox. A version goes like this:
The great Achilles, who is fleet of foot, starts racing to the end line. However, he will never get there, even though he will get infinitely close to it. How so?
6. Achilles and the Tortoise
This is another of Zeno’s famous paradoxes of motion. It goes like this:
Achilles decides to race against a tortoise. To make the race fairer, he allows the turtle to start at half the distance away from the finish line. In this way, Achilles will never surpass the turtle. Why?
7. The Arrow Paradox
This is one other of Zeno’s paradoxes. In this case:
Zeno asks us to imagine an arrow in flight. Can we prove that it is really moving?
8. The Socratic Paradox
This is Plato’s account of something that Socrates purportedly said to the Oracle at Delphi. How can we explain what it means?
All I know is that I know nothing.
9. Hilbert’s Paradox
This is a paradox that illustrates the counterintuitive properties of infinite sets, formulated by mathematician David Hilbert in 1924. It was presented as a puzzle by George Gamow in his 1947 book, One, Two, Three, Infinity.
Let us imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. “Sorry,” says the proprietor, “but all the rooms are occupied.” Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. “But of course!” exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on…. And the new customer receives room N1, which became free as the result of these transpositions. Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in and ask for rooms. “Certainly, gentlemen,” says the proprietor, “just wait a minute.” How does he now accommodate the guests?
10. Irresistible Force Paradox
This paradox is traced to a third-century BCE Chinese philosophical book, titled Han Feizi. Its most common version is the following:
What would happen if an irresistible moving body came into contact with an immovable body?
Answers
1. Let’s assume that Epimenides spoke the truth. Thus, his statement that “All Cretans are liars” is a true statement. However, from this, we must deduce that Epimenides, being a Cretan, is also a liar, as his true statement declares. But this is a contradiction: How can a liar speak the truth? Obviously, we must discard our assumption. Let’s assume the opposite, namely that Epimenides is in fact a liar. But, then, if he is a liar, the statement he made—“All Cretans are liars”—is true. But this is again a contradiction—liars do not make true statements. The inability to reach a conclusion produced by the Liar Paradox is because of self-reference—that is, it results because Epimenides includes himself in his statement; if he did not, no paradox would arise.
2. If the statement on the front side is true, then we would expect the statement on the back side to be true. But since the statement indicates that the front statement is false, we face a contradiction.
3. Let us assume that the barber decides to shave himself. He would end up being shaved, of course, but the person he would have shaved is himself. And that contravenes the requirement that the village barber should shave “all and only those villagers who do not shave themselves.” So, let us assume that the barber decides not to shave himself. But, then, he would end up being an unshaved villager. Again, this goes contrary to the stipulation that he, the barber, must shave “all and only those villagers who do not shave themselves”—including himself. It is not possible, therefore, for the barber to decide whether or not to shave himself. Russell argued that such undecidability arises because the barber is a member of the village. If the barber were from a different village, the paradox would not arise.
4. The paradox arises by defining “heterological” to mean “a word that does not describe itself.” So, the word “heterological” is heterological if it is not heterological—a contradiction, of course.
5. Achilles must first reach the halfway point in the racecourse; from there, he must go another half distance; from this new point, he must go yet another half distance; and so on ad infinitum. The distances thus form an infinite series with each term in it half of the previous one: {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …}. Zeno argued that the runner would logically never cross the end line, even though we know by experience that he actually does. This simple paradox has raised profound issues about time, space, and infinity.
6. This paradox, like the other paradoxes of Zeno, is an example of reductio ad absurdum (“reduction to the absurd”) or proof by contradiction, which may have started with Socrates. The argument goes somewhat like this. In order for Achilles to surpass the tortoise, he must first reach the halfway point, which is the tortoise’s starting point. But when he does, the tortoise will have moved forward another bit. Achilles must then reach this new point before attempting to surpass the tortoise. When he does, however, the tortoise has again moved a little bit forward, which Achilles must also reach again, and so on ad infinitum. In other words, although the distances between Achilles and the tortoise will continue to get smaller, Achilles will never surpass the tortoise, in logic. Of course, in reality, Achilles will do so, because motion is also a factor of time, not only of space.
7. At any given moment, the arrow has an exact location and so it is not moving. It cannot move to where it is not, because no time elapses for it to do so. It cannot move to where it is because it is already there. So, if it is motionless at every moment and is thus not moving.
8. How can we know that we know nothing? This is a contradiction, of course, but we do not read it in this way, but rather as connoting how little we really know, and thus we may not see it as a paradox because of how we interpret the words in it.
9. He moves the occupant of N1 into N2, the occupant of N2 into N4, and occupant of N3 into N6, and so on, and so on. Now the odd-numbered rooms became free and the infinite number of new guests can easily be accommodated in them. In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is smaller than the total number of rooms. However, in Hilbert’s Hotel, the quantity of odd-numbered rooms is not smaller than the total “number” of rooms, because infinite sets have the same number of elements.
10. In his Canterbury Puzzles, Dudeney made the following remark: “if there existed such a thing as an immovable body, there could not at the same time exist a moving body that nothing could resist.” His statement implies that there might actually be no paradox here. It is an example of a false dilemma in logic.
