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September 19th, 2021 (Permalink)

What's New? The Fallacy of the Sheep is!

While writing the previous entry I was somewhat surprised to discover that it was the fourth example over the last ten years of the fallacy of the sheep, yet I didn't have a separate entry for that fallacy. So, now I've filled that lacuna. You can access the new entry from either the Taxonomy on the Main Menu, or the drop-down menu of the alphabetized fallacy entries, both available in the navigational frame to your left. Check it out!

September 16th, 2021 (Permalink)

"Breakthrough" Infections and the Fallacy of the Sheep

…[T]he operator of a diner…offered rabbit sandwiches at a remarkably low price. When questioned about it, he admitted that he used some horse meat to keep his costs down. "But I mix 'em fifty-fifty," he avowed. "One horse to one rabbit."1

In the last few months we've started hearing a lot about "breakthrough" infections. Google Trends shows that the number of searches for the phrase "breakthrough infection" was almost none before April of this year, then peaked in the first week of last month, has declined some since, but remains much higher than before April2. What exactly is a "breakthrough" infection?

A breakthrough infection is simply one in which a vaccinated person becomes infected with the virus the vaccine protects against; it does not mean that the person has any symptoms of the disease3. Since no vaccine is 100% effective in preventing infection, there will always be some breakthrough infections.

Nonetheless, recent events have led to some alarm about coronavirus breakthrough infections: in late June, it was reported that nearly half of new cases in Israel were in people who were vaccinated4, and 75% of those infected in Provincetown, Massachusetts a month later were also vaccinated5.

Some people have misinterpreted these statistics to mean that the vaccinated are just as likely, or even more likely, to become infected than the unvaccinated6. Doing so, however, commits the fallacy of the sheep7.

You may have heard the old riddle: Why do white sheep eat more than black sheep? Because there are more of them. It's not a very good riddle, but it illustrates an important point. When comparing two or more classes of things, you should take into consideration the size of the classes compared. Unless the two classes are the same size, comparing the absolute numbers of a characteristic, such as an infection, between the two classes will be misleading8.

In the case of Israel noted above, the fact that half of those infected were vaccinated and half were unvaccinated, does not mean that the former were just as likely to be infected as the latter. Israel has a high rate of vaccination, with about two-thirds of Israelis having received at least one dose of a vaccine, and about 90% of those being fully vaccinated, at the time of the report9. This means that there were at least twice as many vaccinated Israelis than unvaccinated, yet only half of the new cases were in those vaccinated. Similarly, close to 95% of the population of Provincetown was vaccinated, yet only three-quarters of those infected10. So, there were more white sheep than black ones.

It's likely that we will see more breakthrough infections in coming months as the rate of vaccination continues to increase. So, when you read reports about breakthrough infections in a particular area, always look for the rate of vaccination there. The higher that rate, the higher the percentage of all infections will be breakthrough ones.


  1. Darrel Huff, The Complete How to Figure it: Using Math in Everyday Life (1996), p. 391.
  2. "breakthrough infection", Google Trends, accessed: 9/16/2021.
  3. See:
  4. Marianne Guenot, "Israel says the Delta variant is infecting vaccinated people, representing as many as 50% of new cases. But they're less severe.", Business Insider, 6/24/2021.
  5. Kristina Fiore, "No, Most COVID Infections Are Not Occurring in Vaccinated People", MedPage Today, 8/6/2021.
  6. Jessica McDonald, "Posts Misinterpret CDC's Provincetown COVID-19 Outbreak Report", Fact Check, 8/6/2021.
  7. Stephen K. Campbell, Flaws and Fallacies in Statistical Thinking (1974), pp. 101-104. For other examples of the fallacy, see:
  8. Katelyn Jetelina calls the mistake "base rate bias" or "the base rate fallacy"―see the entry for the latter from the drop-down menu to your left. I'm not sure what "base rate bias" refers to, but I don't think that this is an example of the fallacy of the same name. The base rate fallacy and the fallacy of the sheep are similar, but the latter is a better match for the mistake committed here. See: Katelyn Jetelina, "Israel, 50% of infected are vaccinated, and base rate bias", Your Local Epidemiologist, 6/27/2021.
  9. Edouard Mathieu & Hannah Ritchie, "Vaccinations and COVID-19―Data for Israel", Our World in Data, 2/8/2021. See the charts; accessed: 9/16/2021.
  10. Ellen Barry & Beth Treffeisen, "'It's Nowhere Near Over': A Beach Town's Gust of Freedom, Then a U-turn", The New York Times, 7/31/2021. The outbreak took place due to an influx of visitors to the town, and 80% of the infections due to the event were in such visitors, but I haven't been able to find out what percentage of those visitors were vaccinated. Presumably that percentage was less than 95%, so that the overall percentage of vaccination among those in the town at the time of the outbreak would also be less, but how much less I don't know.

The March Hare, the Dormouse, and the Mad Hatter
September 11th, 2021 (Permalink)

Play with Your Mind, Part 2

Note: Here's an example of the second of the two types of logic puzzle found in this month's new book, Games for Your Mind1, namely, a "knights and knaves" puzzle. Specifically, it's from the master of this type of puzzle, Raymond Smullyan:

…Alice [and] the Duchess…had the following remarkable conversation.

"The Cheshire Cat says that everyone here is mad," said Alice. "Is that really true?"

"Of course not," replied the Duchess. "If that were really true, then the Cat would also be mad, hence you could not rely on what it said."

This sounded perfectly logical to Alice.

"But I'll tell you a great secret, my dear," continued the Duchess. "Half the creatures around here are mad―totally mad!"

"That doesn't surprise me," said Alice, "many have seemed quite mad to me!"

"When I said totally mad," continued the Duchess, quite ignoring Alice's remark, "I meant exactly what I said: They are completely deluded! All their beliefs are wrong―not just some, but all. Everything true they believe to be false and everything false they believe to be true." …

"What about the sane people around here?… I guess most of their beliefs are right but some of them are wrong?"

"Oh, no, no!" said the Duchess most emphatically. "That may be true where you come from, but around here the sane people are one hundred percent accurate in their beliefs! Everything true they know to be true and everything false they know to be false."

Alice thought this over. "Which ones around here are sane and which ones are mad?" asked Alice. … "I've always wondered about the March Hare, the Hatter, and the Dormouse," said Alice. "The Hatter is called the Mad Hatter, but is he really mad? And what about the March Hare and the Dormouse?"

"Well," replied the Duchess, "the Hatter once expressed the belief that the March Hare does not believe that all three of them are sane. Also, the Dormouse believes that the March Hare is sane."2

Are the Mad Hatter, March Hare, and the Dormouse mad or sane?


  1. Playing with Your Mind, 9/7/2021
  2. Raymond Smullyan, Alice in Puzzle-Land: A Carollian Tale for Children Under Eighty (1982), pp. 20-22 & 24-25
  3. Ibid., pp. 146-147

September 8th, 2021 (Permalink)

Play with Your Mind, Part 1

Here's an example of one of the two types of logic puzzle found in this month's new book, Games for Your Mind1. This one was created by Lewis Carroll of Alice in Wonderland fame2. Consider the following clues:

  1. Every one who is sane can do Logic.
  2. No lunatics are fit to serve on a jury.
  3. None of your sons can do Logic.

Based on the above clues, are any of your sons fit to serve on a jury?


  1. Playing with Your Mind, 9/7/2021
  2. Lewis Carroll, Symbolic Logic and the Game of Logic (1958), pp. 112 & 132

New Book
September 7th, 2021 (Permalink)

Playing with Your Mind

Quote: "Among mathematical recreations generally, logic puzzles are the most ecumenical. All that is needed is a bit of patience and some clear thinking. No algebra, geometry, computational skill, or any other sort of specialized knowledge is required. The payoff for solving one, however, is the smile you wear on your face for the rest of the day."1

Title: Games for Your Mind

Subtitle: The History and Future of Logic Puzzles

Author: Jason Rosenhouse

Comment: Rosenhouse is a professor of mathematics who has written a previous book on the Monty Hall problem that I haven't read.

Date: 2020

Summary: Despite its subtitle, this book seems to be more an introduction to logic through puzzles than a discussion of their history or future. It's focused on two types of puzzle, rather than on all those that could be labelled "logic puzzles":

  1. Puzzles based on Aristotelian categorical logic. In particular, the book concentrates on puzzles created by Lewis Carroll, who is better known as the author of the Alice children's books than as a puzzle creator.
  2. What are traditionally called "knights and knaves" puzzles in which there are two types of character, "knights" that always tell the truth and "knaves" that always lie. The late Raymond Smullyan was the master of this type of puzzle, writing several books filled with them.

I've offered a number of puzzles of both types here on the weblog, including the recent "New Logicians' Club" puzzles, which are variations on the second type. Other common types of logic puzzle receive only passing mention in the book. You won't find much on Sudoku puzzles, despite the fact that they are currently the most popular type of logic puzzle. However, Rosenhouse has co-written an entire book on them2, so check that out if Sudoku seriously interests you.

Rather to my disappointment, another common type of puzzle that is largely missing is that which often goes by the name "logic puzzle"―there doesn't seem to be a more specific name for them. There are whole books and magazines devoted to this type of puzzle. Familiar examples are the Smith, Jones, and Robinson puzzle, and the Zebra puzzle―Rosenhouse does give a version of the latter as an example.

If you're interested in learning logic, but don't want to take a class or are daunted by the prospect of reading a text book, and if you enjoy puzzles, you might try this approach. You will, in fact, learn much more from working logic problems than just passively reading a book.

What little I've read of the book is clearly and accurately written, though perhaps aimed at a somewhat more sophisticated readership than I would have expected. For instance, there's a section of the first chapter on philosophy of logic3 that I would suggest skipping unless you're really interested in the topic. In fact, it wouldn't hurt to just skip the whole first chapter and get right to the puzzles in chapter 2.

Disclaimer: This book is from late last year, so it's not brand new, but I haven't read it in its entirety yet. My comments above are based on reading a short excerpt and studying its table of contents.


  1. Section 2.2
  2. Jason Rosenhouse & Laura Taalman, Taking Sudoku Seriously: The Math Behind the World's Most Popular Pencil Puzzle (2012)
  3. Section 1.2

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