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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.
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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.
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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 fromDoctor 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 Travellers—involves an author and his manuscript.
Imagine that a time traveller 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 traveller finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?
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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.
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