Notes on Philosophical Paradoxes
A Zen master once told me: "Do the opposite of whatever I teach you." So I didn't.
~ Alan Wilson Watts
Achilles and the Tortoise
The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m. When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise always moving forwards while Achilles always plays catch up.
Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he can overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.
The Bootstrap Paradox
The Bootstrap Paradox is a paradox of time travel that questions how something that is taken from the future and placed in the past could ever come into being in the first place. It’s a common trope used by science fiction writers and has inspired plotlines in everything from Doctor Who to the Bill and Ted movies, but one of the most memorable and straightforward examples—by Professor David Toomey of the University of Massachusetts and used in his book The New Time Travelers—involves an author and his manuscript.
Imagine that a time traveler buys a copy of Hamlet from a bookstore, travels back in time to Elizabethan London, and hands the book to Shakespeare, who then copies it out and claims it as his own work. Over the centuries that follow, Hamlet is reprinted and reproduced countless times until finally a copy of it ends up back in the same original bookstore, where the time traveler finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?
The Boy or Girl Paradox
Imagine that a family has two children, one of whom we know to be a boy. What then is the probability that the other child is a boy? The obvious answer is to say that the probability is 1/2—after all, the other child can only be either a boy or a girl, and the chances of a baby being born a boy or a girl are (essentially) equal. In a two-child family, however, there are actually four possible combinations of children: two boys (MM), two girls (FF), an older boy and a younger girl (MF), and an older girl and a younger boy (FM). We already know that one of the children is a boy, meaning we can eliminate the combination FF, but that leaves us with three equally possible combinations of children in which at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child is a boy—MM—must be 1/3, not 1/2.
The Card Paradox
Imagine you’re holding a postcard in your hand, on one side of which is written, “The statement on the other side of this card is true.” We’ll call that Statement A. Turn the card over, and the opposite side reads, “The statement on the other side of this card is false” (Statement B). Trying to assign any truth to either Statement A or B, however, leads to a paradox: if A is true then B must be as well, but for B to be true, A has to be false. Oppositely, if A is false then B must be false too, which must ultimately make A true.
Invented by the British logician Philip Jourdain in the early 1900s, the Card Paradox is a simple variation of what is known as a “liar paradox,” in which assigning truth values to statements that purport to be either true or false produces a contradiction. An even more complicated variation of a liar paradox is the next entry on our list.
The Crocodile Paradox
A crocodile snatches a young boy from a riverbank. His mother pleads with the crocodile to return him, to which the crocodile replies that he will only return the boy safely if the mother can guess correctly whether or not he will indeed return the boy. There is no problem if the mother guesses that the crocodile will return him—if she is right, he is returned; if she is wrong, the crocodile keeps him. If she answers that the crocodile will not return him, however, we end up with a paradox: if she is right and the crocodile never intended to return her child, then the crocodile has to return him, but in doing so breaks his word and contradicts the mother’s answer. On the other hand, if she is wrong and the crocodile actually did intend to return the boy, the crocodile must then keep him even though he intended not to, thereby also breaking his word.
The Crocodile Paradox is such an ancient and enduring logic problem that in the Middle Ages the word "crocodilite" came to be used to refer to any similarly brain-twisting dilemma where you admit something that is later used against you, while "crocodility" is an equally ancient word for captious or fallacious reasoning
The Dichotomy Paradox
Imagine that you’re about to set off walking down a street. To reach the other end, you’d first have to walk half way there. And to walk half way there, you’d first have to walk a quarter of the way there. And to walk a quarter of the way there, you’d first have to walk an eighth of the way there. And before that a sixteenth of the way there, and then a thirty-second of the way there, a sixty-fourth of the way there, and so on.
Ultimately, in order to perform even the simplest of tasks like walking down a street, you’d have to perform an infinite number of smaller tasks—something that, by definition, is utterly impossible. Not only that, but no matter how small the first part of the journey is said to be, it can always be halved to create another task; the only way in which it cannot be halved would be to consider the first part of the journey to be of absolutely no distance whatsoever, and in order to complete the task of moving no distance whatsoever, you can’t even start your journey in the first place.
The Fletcher’s Paradox
Imagine a fletcher (i.e. an arrow-maker) has fired one of his arrows into the air. For the arrow to be considered to be moving, it has to be continually repositioning itself from the place where it is now to any place where it currently isn’t. The Fletcher’s Paradox, however, states that throughout its trajectory the arrow is actually not moving at all. At any given instant of no real duration (in other words, a snapshot in time) during its flight, the arrow cannot move to somewhere it isn’t because there isn’t time for it to do so. And it can’t move to where it is now, because it’s already there. So, for that instant in time, the arrow must be stationary. But because all time is comprised entirely of instants—in every one of which the arrow must also be stationary—then the arrow must in fact be stationary the entire time. Except, of course, it isn’t.
Galileo’s Paradox of the Infinite
In his final written work, Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638), the legendary Italian polymath Galileo Galilei proposed a mathematical paradox based on the relationships between different sets of numbers. On the one hand, he proposed, there are square numbers—like 1, 4, 9, 16, 25, 36, and so on. On the other, there are those numbers that are not squares—like 2, 3, 5, 6, 7, 8, 10, and so on. Put these two groups together, and surely there have to be more numbers in general than there are just square numbers—or, to put it another way, the total number of square numbers must be less than the total number of square and non-square numbers together. However, because every positive number has to have a corresponding square and every square number has to have a positive number as its square root, there cannot possibly be more of one than the other.
Confused? You’re not the only one. In his discussion of his paradox, Galileo was left with no alternative than to conclude that numerical concepts like more, less, or fewer can only be applied to finite sets of numbers, and as there are an infinite number of square and non-square numbers, these concepts simply cannot be used in this context.
The Potato Paradox
Imagine that a farmer has a sack containing 100 lbs of potatoes. The potatoes, he discovers, are comprised of 99% water and 1% solids, so he leaves them in the heat of the sun for a day to let the amount of water in them reduce to 98%. But when he returns to them the day after, he finds his 100 lb sack now weighs just 50 lbs. How can this be true? Well, if 99% of 100 lbs of potatoes is water then the water must weigh 99 lbs. The 1% of solids must ultimately weigh just 1 lb, giving a ratio of solids to liquids of 1:99. But if the potatoes are allowed to dehydrate to 98% water, the solids must now account for 2% of the weight—a ratio of 2:98, or 1:49—even though the solids must still only weigh 1lb. The water, ultimately, must now weigh 49lb, giving a total weight of 50lbs despite just a 1% reduction in water content. Or must it?
Although not a true paradox in the strictest sense, the counterintuitive Potato Paradox is a famous example of what is known as a veridical paradox, in which a basic theory is taken to a logical but apparently absurd conclusion.
The Raven Paradox
Also known as Hempel’s Paradox, for the German logician who proposed it in the mid-1940s, the Raven Paradox begins with the apparently straightforward and entirely true statement that “all ravens are black.” This is matched by a “logically contrapositive” (i.e. negative and contradictory) statement that “everything that is not black is not a raven”—which, despite seeming like a fairly unnecessary point to make, is also true given that we know “all ravens are black.” Hempel argues that whenever we see a black raven, this provides evidence to support the first statement. But by extension, whenever we see anything that is not black, like an apple, this too must be taken as evidence supporting the second statement—after all, an apple is not black, and nor is it a raven.
The paradox here is that Hempel has apparently proved that seeing an apple provides us with evidence, no matter how unrelated it may seem, that ravens are black. It’s the equivalent of saying that you live in New York is evidence that you don’t live in L.A., or that saying you are 30 years old is evidence that you are not 29. Just how much information can one statement actually imply anyway?
The Horn Paradox
The horn paradox rests on your answer to the unusual question: “have you lost your horns?” This being difficult to answer.
Intuitively, we would reply: “I have not,” but how are we supposed to understand that denial? That you still have horns? Or that you once did and don’t anymore? Or that you never have, and never will?
It becomes a question of what the not or negation means in your reply, to which there are multiple interpretations. The horned man paradox arises when someone adopts the view that “what you have not lost you still have.”
Put simply, it states:
Premise 1: If you have not lost something, you still have it.
Premise 2: You have not lost horns.
Conclusion: So you still have horns.
This argument follows the form of modus ponens, so has logical and valid reasoning. According to the assumptions we have made, premise 1 and 2 are true. If so, then the argument is sound and the conclusion true: we do have horns.
Of course, this conclusion is absurd. Humans do not have horns and never have. So where does this apparently sound argument go wrong?
The Solution
Most modern Philosophers regard this argument as fallacious. It commits the commonly known “fallacy of many questions.” To say you lost something implies that you had it to lose in the first place.
So if you never had horns, then premise 1 and 2 both make false presuppositions. Because of that, most believe the premises are false, or at least not true.
The Liar Paradox
This paradox is much more profound. It stems from claims made by Greek Philosopher-poet, Epimenides. He was a Cretan, yet claimed that all Cretans were liars.
Eubulides used this problematic claim to draw out a deeper Philosophical contradiction. The classic problem arises from the self-referential statement:
“This statement is false.”
The same problem also arises when you consider the claim: “I am lying.” Philosophical logicians have grappled with this problem for over 2,300 years. Here is the issue at hand:
Let’s call the classic liar sentence L. If L is true, then (according to the sentence), L is false. But the converse is also true: if we assume L is false, it follows that the liar sentence is true.
This shows that the liar sentence is true if and only if it is false. At any one time, the statment must be either true or false. Therefore, the statement is both true and false.
This is an obvious contradiction. According to classical logic, something must be either true or false but cannot be both.
The Solution
Some are inclined to argue the statement is meaningless and shouldn’t be discussed further — though this leaves the question “why is the liar sentence meaningless?” unanswered.
There are a variety of modern-day solutions to the problem. Some include adopting a different account of logic which allows statements to be both true or false, or introduces a new truth value that is neither.
Unfortunately, these solutions are still widely disputed, and the problem is considered unresolved.
Zeno of Elea
Zeno of Elea was a pre-socratic Philosopher (5th century BCE), who belonged to the Eleatic School (founded by Parmenides). He is most famous for his paradoxes which have been hailed by Bertrand Russel as “immeasurably subtle and profound.”
Everything we know of Zeno comes from the opening pages of Plato’s Parmenides. He wrote a book defending Parmenides, but it did not survive, and our knowledge of Zeno comes from the accounts of thinkers like Aristotle.
Zeno set out to defend Paremindes’ Philosophy against his critics. He rejected pluralism and change: for him, reality is made up of one indivisible, unchanging reality, and any apparent changes to reality are an illusion.
This view faced many criticisms, as it contradicts some of our basic beliefs of the world. But in defending it, Zeno aimed to show that a denial of Paremindes’ account results in absurd consequences.
Zeno wrote a variety of paradoxes and arguments against the existence of plurality and motion. Let’s talk through one of Zeno’s paradoxes of motion. After exploring this, I would recommend reading more of Zeno’s paradoxes.
The Dichotomy Paradox
This paradox appears in Aristotle's Physics (239b11). It adopts the assumption that Paremindes view is false, and aims to come to an absurd and contradictory conclusion. Doing so proves Paremindes view cannot be false, so must be true (this argumentative strategy is known as Reductio ad Absurdum).
If pluralism is false, then everything is infinitely divisible, including motion and movement. This paradox is known as a dichotomy because it involves repeated division into two. It is traditionally presented it as follows:
Suppose a fast runner, such as mythical Atalanta, needs to run a certain distance. Before she runs that distance, she must run 1/2 of that distance. But before she runs 1/2 of that distance, she must run 1/4 of that distance, and then 1/8, 1/16, and so on. This division can be made an infinite amount of times.
Therefore, movement of any distance requires us to travel an infinite amount of finite distances. In short, Atalanta must complete an infinite amount of tasks: she must run 1/2 of her distance, 1/4 of her distance, and so on forever.
Of course, as humans, we have a finite amount of time on earth. And because of that, it’s impossible for us to complete an infinite number of tasks.
A consequence of this is that movement (of any distance) is impossible. It’s even impossible for us to walk 1 meter. Here’s the argument in a valid form:
Premise 1: To walk 1 meter, you must complete an infinite number of tasks.
Premise 2: It’s impossible to complete an infinite amount of tasks in a finite time.
Premise 3: We have a finite amount of time (that is, we will die one day).
Conclusion: Therefore, it is impossible for us to walk 1 meter.
That conclusion is evidently false, but the difficulty is in determining where and why this seemingly valid argument fails. This question has been grappled and debated by Philosophers for centuries.
“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” — (Aristotle, Physics)
The Solution
Common responses to this problem remain inadequate. Simplicius of Cilicia (On Aristotle’s Physics, 1012.22) argues that the common experience of walking and standing proves that we can move 1 meter. But this did not impress Zeno: as a Parmenidean, he believed things were often not what they seemed — for appearances can be deceiving.
Acknowledging this, Aristotle believed he found a solution. As we divide the distances moved, perhaps we should also divide the time taken. It takes 1/2 the time to travel the remaining 1/2 distance, 1/4 of the time for the remaining 1/4, and so on for infinity. Thus, each fractional distance has just the right amount of finite time for us to complete it.
While Aristotle thought this response was satisfactory, he faced a fundamental problem. We are saying the time it takes to travel 1 meter is composed of an infinite number of finite pieces of time. Is that not just an infinite amount of time (to which we don’t have)?
This paradox and its solutions remain widely disputed even today.
The Paradox of the Stone
This problem is best characterized by theologian and Philosopher J.L Mackie, in his 1995 Evil and omnipotence. It focuses on the belief that the God of Christian thought is believed to be omnipotent — that is, he is all-powerful (or has unlimited power.)
The paradox asks a simple yet controversial question:
Could an omnipotent being create a stone too heavy for it to lift?
More generally, could an all-powerful being create something that even they cannot control?
Suppose that being could not create the stone, then it is not omnipotent, for there is something it cannot do. But if it can create the stone, then there is also something it cannot do (namely, lift the stone it created). The paradox arises because regardless of the answer: a seemingly omnipotent being is not omnipotent.
Wider Discussion on God
The stone paradox has been at the forefront of discussion when defining the parameters of what an omnipotent being can do. Though additional questions have arisen along the way. For example, could an omnipotent being create a circular square? Descartes answers yes, but western traditions since Aquinas have given the opposite answer.
The solution and answer to the paradox are still widely debated. But the view that an omnipotent being can do anything (no matter how logically absurd or contradictory,) is known as voluntarism.
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To go anywhere, you must go halfway first, and then you must go half of the remaining distance, and half of the remaining distance, and so forth to infinity: Thus, motion is impossible.
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