The Preface Paradox stems from a common enough source — that bit of text found in the prefaces of many academic books along the lines of, “the errors that are found herein are mine alone,” absolving advisors and other editors of any blame.

While this may seem nice enough, with this single bit of text the author has asked us to accept two mutually incompatable beliefs:

1. Such an author has written a book that contains many assertions, and has factually checked each one carefully, submitted it to reviewers for comment, etc. Thus, he has reason to believe that each assertion he has made is true.

2. However, he knows, having learned from experience, that, in spite of his best efforts, there are very likely undetected errors in his book. So he also has good reason to believe that there is at least one assertion in his book is not true.

Somehow, despite these paradoxical facts, we know to trust the author, and trust that his or her mistakes, if any, will be few and far between, a wayward needle or two in the haystack of facts. The paradox is an epistemic one, related to how we know what we know, and, because of its relationship between what’s likely (the facts are correct) and what’s not (the facts are errors), classed with another paradox involving the lottery.

To my emerging collection of paradoxes, I now add another: the joyfully alliterative Parrondo’s Paradox, which states that “Given two games, each with a higher probability of losing than winning, it is possible to construct a winning strategy by playing the games alternately.” The paradox was discovered in 1999.

This article from the New York Times written shortly afterward describes one of Parrondo’s experiments with two games involving weighted (non random) coins: “when a person plays either game A or game B 100 times, all money taken to the gambling table is lost. But when the games are alternated — playing A twice and B twice for 100 times — money is not lost. It accumulates into big winnings. Even more surprising, he said, when game A and B are played randomly, with no order in the alternating sequence, winnings also go up and up.”

When visualized, these games take on a rachet-like shape — a shape central to the explanation of trivial phenomena, like the Brazil Nut Effect, and more fundamental matters, like the design of enzymes and proteins.

As promised, here’s a list of all paradoxes discussed by Peter Cave in this episode of Philosophy Bites: Liar’s Paradox, Zeno’s Paradox, a paradox of murder more legal than philosophical (but nonetheless reminiscent of the third story from one of the greatest opening sequences of any movie ever which recalls this 1994 Internet meme), Russell’s Paradox (or the Barber Paradox), Poisoned Chalice (or Kavka’s Toxin Puzzle), the Surprise Exam (or Unexpected Hanging Paradox or Bottle Imp Paradox), and Buridan’s Ass. Cave’s book, This Sentence Is False: An Introduction to Philosophical Paradoxes, comes out later this year. In the meantime, a lengthy list of paradoxes is available here on Wikipedia.

From a short story by Robert Lewis Stevenson written in 1891, we get the “bottle imp paradox,” a paradox that shares some qualities with both the Sorites paradox and the unexpected hanging paradox. As the story goes, the protagonist is offered the opportunity to buy, for whatever price he wishes, a bottle containing a genie who will fulfill his every desire. The only catch is that the bottle must thereafter be resold for a price smaller than what he paid for it, or he will be condemned to live out the rest of his days in excruciating torment. Obviously, no one would buy the bottle for 1¢ since he would have to give the bottle away, but no one would accept the bottle knowing he would be unable to get rid of it. Similarly, no one would buy it for 2¢, and so on. However, for some reasonably large amount, it will always be possible to find a next buyer, so the bottle will be bought. But where’s the limit? Makes me wish I could take Prof. Laurence Goldstein’s PHIL 2511 course on “Paradoxes.” Fascinating.

“My view is, roughly speaking, that the Free Rider problem is, simply, the Sorites Paradox applied to human action or human decision-making.” Richard Tuck discusses the concept of Free Riding on Philosophy Bites. More on the Sorites Paradox here. More on Free Riders here.

A Sorites paradox is a key issue in the philosophy of language, with the word “Sorites” translating in Greek to “heap.” The paradox can be stated in two premises, 1) A heap of sand is comprised of a large collection of grains, and 2) A heap of sand minus one grain is still a heap. The problem is that if you continue executing premise 2, you’ll soon have neither heap nor even one sand grain left. The many possible resolutions to this problem suggest a cache of ways philosophers try to deal with problems of vagueness. Some resolutions trivialize the problem, casting “heap” as a basically meaningless term. Others try to set limits, either fixed by number (although what’s the difference between 9,999 grains and 10,001?) or positional (a heap has sand grains supporting other sand grains off the ground, multiple heaps cannot also belong to a single heap, etc.). So-called “fuzzy” logic allows not just for on and off positions (“heap” and “not-heap”) but also a third position, “unsure,” which may be subdivided into “mostly-heap,” “partly-heap,” etc. Other solutions may be as simple as group consensus (the majority of people would call it a “heap”) or as complex as hysteresis, which suggests certain systems make it impossible to predict an output from a given input and instead it must be taken into account whether the heap in question started out as a heap, a desert, or a single grain of sand. More on the wonderful podcast Philosophy Bites.