Brain Games: 8 Philosophical Puzzles and Paradoxes

Suppose someone tells you “I am lying.” If what she tells you is true, then she is lying, in which case what she tells you is false. On the other hand, if what she tells you is false, then she is not lying, in which case what she tells you is true. In short: if “I am lying” is true then it is false, and if it is false then it is true. The paradox arises for any sentence that says or implies of itself that it is false (the simplest example being “This sentence is false”). It is attributed to the ancient Greek seer Epimenides (fl. c. 6th century BCE), an inhabitant of Crete, who famously declared that “All Cretans are liars” (consider what follows if the declaration is true).

The liar paradox is important in part because it creates severe difficulties for logically rigorous theories of truth; it was not adequately addressed (which is not to say solved) until the 20th century.

In the 5th century BCE, Zeno of Elea devised a number of paradoxes designed to show that reality is single (there is only one thing) and motionless, as his friend Parmenides had claimed. The paradoxes take the form of arguments in which the assumption of plurality (the existence of more than one thing) or motion are shown to lead to contradictions or absurdity. Here are two of the arguments:

Against plurality:

(A) Suppose that reality is plural. Then the number of things there are is only as many as the number of things there are (the number of things there are is neither more nor less than the number of things there are). If the number of things there are is only as many as the number of things there are, then the number of things there are is finite.

(B) Suppose that reality is plural. Then there are at least two distinct things. Two things can be distinct only if there is a third thing between them (even if it is only air). It follows that there is a third thing that is distinct from the other two. But if the third thing is distinct, then there must be a fourth thing between it and the second (or first) thing. And so on to infinity.

(C) Therefore, if reality is plural, it is finite and not finite, infinite and not infinite, a contradiction.

Against motion:

Suppose that there is motion. Suppose in particular that Achilles and a tortoise are moving around a track in a foot race, in which the tortoise has been given a modest lead. Naturally, Achilles is running faster than the tortoise. If Achilles is at point A and the tortoise at point B, then in order to catch the tortoise Achilles will have to traverse the interval AB. But in the time it takes Achilles to arrive at point B, the tortoise will have moved on (however slowly) to point C. Then in order to catch the tortoise, Achilles will have to traverse the interval BC. But in the time it takes him to arrive at point C, the tortoise will have moved on to point D, and so on for an infinite number of intervals. It follows that Achilles can never catch the tortoise, which is absurd.

Zeno’s paradoxes have posed a serious challenge to theories of space, time, and infinity for more than 2,400 years, and for many of them there is still no general agreement about how they should be solved.

Also called “the heap,” this paradox arises for any predicate (e.g., “… is a heap”, “… is bald”) whose application is, for whatever reason, not precisely defined. Consider a single grain of rice, which is not a heap. Adding one grain of rice to it will not create a heap. Likewise adding one grain of rice to two grains or three grains or four grains. In general, for any number N, if N grains does not constitute a heap, then N+1 grains also does not constitute a heap. (Similarly, if N grains does constitute a heap, then N-1 grains also constitutes a heap.) It follows that one can never create a heap of rice from something that is not a heap of rice by adding one grain at a time. But that is absurd.

Among modern perspectives on the sorites paradox, one holds that we simply haven’t gotten around to deciding exactly what a heap is (the “lazy solution”); another asserts that such predicates are inherently vague, so any attempt to define them precisely is wrongheaded.

Although it bears his name, the medieval philosopher Jean Buridan did not invent this paradox, which probably originated as a parody of his theory of free will, according to which human freedom consists in the ability to defer for further consideration a choice between two apparently equally good alternatives (the will is otherwise compelled to choose what appears to be the best).

Imagine a hungry donkey who is placed between two equidistant and identical bales of hay. Assume that the surrounding environments on both sides are also identical. The donkey cannot choose between the two bales and so dies of hunger, which is absurd.

The paradox was later thought to constitute a counterexample to Leibniz’s principle of sufficient reason, one version of which states that there is an explanation (in the sense of a reason or cause) for every contingent event. Whether the donkey chooses one bale or the other is a contingent event, but there is apparently no reason or cause to determine the donkey’s choice. Yet the donkey will not starve. Leibniz, for what it is worth, vehemently rejected the paradox, claiming that it was unrealistic.

A teacher announces to her class that there will be a surprise test sometime during the following week. The students begin to speculate about when it might occur, until one of them announces that there is no reason to worry, because a surprise test is impossible. The test cannot be given on Friday, she says, because by the end of the day on Thursday we would know that the test must be given the next day. Nor can the test be given on Thursday, she continues, because, given that we know that the test cannot be given on Friday, by the end of the day on Wednesday we would know that the test must be given the next day. And likewise for Wednesday, Tuesday, and Monday. The students spend a restful weekend not studying for the test, and they are all surprised when it is given on Wednesday. How could this happen? (There are various versions of the paradox; one of them, called the Hangman, concerns a condemned prisoner who is clever but ultimately overconfident.)

The implications of the paradox are as yet unclear, and there is virtually no agreement about how it should be solved.

You buy a lottery ticket, for no good reason. Indeed, you know that the chance that your ticket will win is at least 10 million to one, since at least 10 million tickets have been sold, as you learn later on the evening news, before the drawing (assume that the lottery is fair and that a winning ticket exists). So you are rationally justified in believing that your ticket will lose—in fact, you’d be crazy to believe that your ticket will win. Likewise, you are justified in believing that your friend Jane’s ticket will lose, that your uncle Harvey’s ticket will lose, that your dog Ralph’s ticket will lose, that the ticket bought by the guy ahead of you in line at the convenience store will lose, and so on for each ticket bought by anyone you know or don’t know. In general, for each ticket sold in the lottery, you are justified in believing: “That ticket will lose.” It follows that you are justified in believing that all tickets will lose, or (equivalently) that no ticket will win. But, of course, you know that one ticket will win. So you’re justified in believing what you know to be false (that no ticket will win). How can that be?

The lottery constitutes an apparent counterexample to one version of a principle known as the deductive closure of justification:

If one is justified in believing P and justified in believing Q, then one is justified in believing any proposition that follows deductively (necessarily) from P and Q.

For example, if I am justified in believing that my lottery ticket is in the envelope (because I put it there), and if I am justified in believing that the envelope is in the paper shredder (because I put it there), then I am justified in believing that my lottery ticket is in the paper shredder.

Since its introduction in the early 1960s, the lottery paradox has provoked much discussion of possible alternatives to the closure principle, as well as new theories of knowledge and belief that would retain the principle while avoiding its paradoxical consequences.

This ancient paradox is named for a character in Plato’s eponymous dialogue. Socrates and Meno are engaged in a conversation about the nature of virtue. Meno offers a series of suggestions, each of which Socrates shows to be inadequate. Socrates himself professes not to know what virtue is. How then, asks Meno, would you recognize it, if you ever encounter it? How would you see that a certain answer to the the question “What is virtue?” is correct, unless you already knew the correct answer? It seems to follow that no one ever learns anything by asking questions, which is implausible, if not absurd.

Socrates’ solution is to suggest that basic elements of knowledge, enough to recognize a correct answer, can be “recollected” from a previous life, given the right kind of encouragement. As proof he shows how a slave boy can be prompted to solve geometrical problems, though he has never had instruction in geometry.

Although the recollection theory is no longer a live option (almost no philosophers believe in reincarnation), Socrates’ assertion that knowledge is latent in each individual is now widely (though not universally) accepted, at least for some kinds of knowledge. It constitutes an answer to the modern form of Meno’s problem, which is: how do people successfully acquire certain rich systems of knowledge on the basis of little or no evidence or instruction? The paradigm case of such “learning” (there is debate about whether “learning” is the correct term) is first-language acquisition, in which very young (normal) children manage to acquire complex grammatical systems effortlessly, despite evidence that is completely inadequate and often downright misleading (the ungrammatical speech and erroneous instruction of adults). In this case, the answer, originally proposed by Noam Chomsky in the 1950s, is that the basic elements of the grammars of all human languages are innate, ultimately a genetic endowment reflecting the cognitive evolution of the human species.

Suppose you are sitting in a windowless room. It begins to rain outside. You have not heard a weather report, so you don’t know that it’s raining. So you don’t believe that it’s raining. Thus your friend McGillicuddy, who knows your situation, can say truly of you, “It’s raining, but MacIntosh doesn’t believe it is.” But if you, MacIntosh, were to say exactly the same thing to McGillicuddy—“It’s raining, but I don’t believe it is”—your friend would rightly think you’d lost your mind. Why, then, is the second sentence absurd? As G.E. Moore put it, “Why is it absurd for me to say something true about myself?”

The problem Moore identified turned out to be profound. It helped to stimulate Wittgenstein’s later work on the nature of knowledge and certainty, and it even helped to give birth (in the 1950s) to a new field of philosophically inspired language study, pragmatics.

I’ll leave you to ponder a solution.