Paradox Lost covers ten of philosophy's most fascinating paradoxes, in which seemingly compelling reasoning leads to absurd conclusions. The following paradoxes are included:
The Liar Paradox, in which a sentence says of itself that it is false. Is the sentence true or false?
The Sorites Paradox, in which we imagine removing grains of sand one at a time from a heap of sand. Is there a particular grain whose removal converts the heap to a non-heap?
The Puzzle of the Self-Torturer, in which a series of seemingly rational choices has us accepting a life of excruciating pain, in exchange for millions of dollars.
Newcomb's Problem, in which we seemingly maximize our expected profit by taking an unknown sum of money, rather than taking the same sum plus $1000.
The Surprise Quiz Paradox, in which a professor finds that it is impossible to give a surprise quiz on any particular day of the week... but also that if this is so, then a surprise quiz can be given on any day.
The Two Envelope Paradox, in which we are asked to choose between two indistinguishable envelopes, and it is seemingly shown that each envelope is preferable to the other.
The Ravens Paradox, in which observing a purple shoe provides evidence that all ravens are black.
The Shooting Room Paradox, in which a deadly game kills 90% of all who play, yet each individual's survival turns on the flip of a fair coin.
Each paradox is clearly described, common mistakes are explored, and a clear, logical solution offered. Paradox Lost will appeal to professional philosophers, students of philosophy, and all who love intellectual puzzles.
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Michael HuemerParadox Losthttps://doi.org/10.1007/978-3-319-90490-0_1
Begin Abstract
1. Introduction
Michael Huemer1
(1)
Philosophy Department, University of Colorado Boulder, Boulder, CO, USA
Michael Huemer
End Abstract
1.1 What Is a Paradox?
First, some words about what a paradoxisnât. Some people understand the word âparadoxâ to refer to a case in which reality is contradictory, that is, a situation that you would correctly describe by contradicting yourself. I do not use the word this way, because I find it inconvenient. If we use âparadoxâ to denote a situation containing a true contradiction, then we will have to say that, by definition, there are no paradoxes, since contradictions are necessarily false â thus apparently depriving this book of its subject matter. I should then have to say that this is a book about âapparent paradoxesâ, that the next chapter is about âthe Liar Pseudo-paradoxâ, and so on. This would be tedious. So I wonât understand âparadoxâ that way.
Some people use âparadoxâ simply to refer to a contradictory statement, or apparently contradictory statement, such as âNobody goes to that restaurant anymore because it is too crowded.â That also is not what I mean by âparadoxâ. Such statements are either false, or simply have an alternate meaning that is different from the most superficial interpretation (as in the statement, âI am nobodyâ, which really just means âI am unimportantâ). In either case, there is no real puzzle.
I understand a paradox, roughly, as a situation in which we have seemingly compelling reasoning for a contradictory or otherwise absurd conclusion.1 We feel that we cannot accept the conclusion, but nor can we readily identify a flaw in the reasoning. For example, consider Zenoâs famous paradox of motion (figure 1.1):
Fig. 1.1
Zenoâs paradox
In order for an object to move from point A to point B, the object must first travel half the distance. Then it will have to travel half the remaining distance. Then half the remaining distance again. And so on. This is an infiniteseries. An infiniteseries has no end; hence, it is impossible to complete an infinite series. Therefore, it is impossible for the object to reach point B. Thus, no object can move anywhere.
The conclusion is absurd, so the reasoning must be wrong. Nevertheless, the reasoning has a certain obvious, intuitive force, and it is difficult to say exactly what is wrong with it.
To count as âparadoxicalâ, the reasoning for the absurd conclusion must have widespread appeal â that is, the reasoning must be of a sort that would seem compelling to typical human reasoners; an idiosyncratic error that I personally canât seem to shake does not qualify. Thus, if I make a calculation error in multiplying a string of numbers, which results in my deriving a logically impossible conclusion, this will not count as a paradox â not even if I personally cannot find the error after many tries.
To count as âparadoxicalâ, a piece of erroneous reasoning must also have a certain sort of robustness: paradoxes bear extended contemplation and discussion. Paradoxes can have solutions and attempted solutions, but the correctness of a given solution will be a matter of debate, at least for some time, even among the experts. A paradox is not merely a problem whose solution, though known to experts, is unknown to most non-experts.
Thus, for example, I do not consider the Monty Hall Problem to be a paradox. The Monty Hall Problem goes like this:
You are a contestant on the game show Letâs Make a Deal, with host Monty Hall. You know how the game works: at a certain point in the game, Monty shows the contestant three closed doors. One of the doors has a nice prize behind it (say, a new car); the other two have goats behind them (assume that no one wants a goat). The contestant is allowed to choose one of the doors, and will be allowed to have whatever is behind it. After the contestant chooses, but before he reveals what is behind the chosen door, Monty opens one of the other two doors and shows the contestant a goat.2 He never opens the door with the car behind it; he always shows the contestant a goat.3 Monty then asks if the contestant would like to change their choice, that is, to switch to the other closed door. Thus, suppose you initially choose door #1. Monty then opens, say, door #3 and shows you a goat behind it. He then asks if you would like to change your choice from door #1 to door #2. Should you switch?
Most people have a strong intuition that it doesnât matter whether you switch to door #2 or stick with door #1; thatâs because most people think that the prize is now 50% likely to be behind door #1 and 50% likely to be behind door #2. The correct answer, however, is that you should definitely switch to door #2: door #1 has a 1/3 probability of having the real prize behind it, and door #2 now (after you saw the goat behind door #3) has a 2/3 probability of having the real prize.
It can be difficult to convince people of this. In fact, almost everyone, on first hearing the problem, gives the wrong answer, and persists in that answer until bludgeoned for a while with probability calculations or experiments.4 In this case, there are compelling arguments (discussed below and in fn. 3) for a highly counter-intuitive answer. Nevertheless, I do not consider this a paradox. One reason is that this problem is not robust enough to bear debate among experts. The Monty Hall Problem has a well-known, objectively correct solution that can be shown to be so in a fairly brief span of time; it does not, for example, bear years of reflection.
Why does the prize have a 2/3 probability of being behind door B? This is beside my present point (which, remember, was just to define âparadoxâ); however, in case you canât sleep until you know, an explanation follows. (You can also do a calculation employing Bayesâ Theorem, but the following is probably going to be more satisfying.)
Suppose Monty runs the game 300 times. Each time, the location of the good prize is randomly selected from among the three doors. We would expect that in about 100 of these games, the contestantâs initial guess is correct, that is, the first door they pick has the prize behind it. The other 200 times, the initial guess is wrong. Therefore, if the contestants always stick with their initial guess, then 100 of the 300 will win the real prize, and 200 will receive goats. (The 200 who initially selected a goat door canât possibly improve their result by sticking with that choice!)
Now, on the other hand, suppose that the contestants always switch doors. Then the 100 contestants who initially picked the correct door will lose, as they give up that door. But the other 200, the ones who initially picked wrong, will all switch doors. And they will all switch to the correct door, since the correct door will be the only remaining door, after rejecting the door they initially picked and the goat door that Monty just opened.
So the âswitch doorsâ strategy wins 2/3 of the time, whereas the âstick with your doorâ strategy wins only 1/3 of the time.
Now, back to our main point: a paradox is a piece of reasoning, or a situation about which such reasoning is constructed, that has widespread and robust appeal, but that leads to a contradictory or absurd conclusion, where even experts have difficulty identifying the error in the reasoning. This account of paradoxicality makes it species-relative: perhaps a superintelligent alien species would find our âparadoxesâ so easy to see through that the aliens would not consider these puzzles paradoxical at all. Nevertheless, for humans at the present time, there are many paradoxical situations and pieces of reasoning.
1.2 What Is a Solution?
Paradoxes trade on intellectual illusions: cases in which something that is actually wrong seems correct when considered intellectually. It is characteristic of illusions that what seems to be the case is not necessarily what we believe â often, in fact, we know that things cannot be as they appear, yet this does not stop them from appearing that way. We know, for instance, that the reasoning in Zenoâs Paradox is wrong, since we know that objects move. But this does not stop the reasoning from exerting its intellectual pull: each step in the reasoning still seems correct and the steps seemingly lead to the conclusion that nothing moves.
Intellectual illusions, however, are at least somewhat more tractable than sensory illusions. Sensory illusions (for example, the illusion whereby a straight stick appears bent when half-submerged in water) virtually never respond to reflection or additional evidence-gathering. No matter how much you understand about the actual situation, the half-submerged stick still looks bent. Intellectual illusions are not that way. Admittedly, as noted, they do not respond immediately to the knowledge that the conclusion of a piece of reasoning is wrong â the mere fact that we know that objects move does not dispel the force of Zenoâs reasoning. But usually, there is some degree of understanding, of the right points, that dispels the illusion. It is often possible for someone to identify the particular point at which the fallacious reasoning goes wrong, and to explain wh...
