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July 25th, 2026 (Permalink)

100% MORE!

…[T]he problem with ads isn't so much what is said or demonstrated as what we infer from what is said or demonstrated, and the best way to induce false inferences is to sketch alluring pictures and leave out crucial bits of information. … A survey shows that this medication works more quickly. Than what does it work more quickly?1

I recently saw an advertisement for an expensive electric toothbrush that made me wonder whether I should purchase one. My current toothbrush has the advantage that it's cheap, but how good does it clean my teeth? Should I pay over ten times as much for a replacement?

The ad claimed, in large print, that the electric toothbrush would remove "100% more plaque". More than what? This is a dangling comparative2, that is, a comparison missing one of the objects compared. What is the electric toothbrush compared to: does it remove more plaque than the previous model, or than a competing brand, or what?

I should have written that it is a semi-dangling comparative since the ad completed the comparison in smaller print: "vs. a regular manual toothbrush". So, it's being compared to the kind of toothbrush I currently use. Why was the rest of the comparison in small print? The ad writer probably didn't want to draw attention to the fact that the ad was comparing an electric toothbrush to a manual one. Perhaps any electric toothbrush would remove "100% more" plaque than a manual one, in which case I should purchase the cheapest electric one I can find instead of the expensive one advertised.

Now, that explains the semi-dangling comparative and the small print, but what about "100% more": what does that mean? 100% of something is, of course, all of it, so adding 100% more to something increases it by the same amount. Thus, if the electric toothbrush removes 100% more plaque than a manual one, it removes twice as much plaque. Why didn't the ad just say that instead of using a potentially confusing percentage?

Percentages, which are a type of fraction, are generally harder to understand than multiplication, since many people find division and fractions difficult. Unlike percentages, we have frequent experience with multiplication by two, which makes "100% more" more confusing than "twice as much" or "double", though perhaps not 100% more confusing. This is an advantage for the ad writer whose goal is not to write clearly but to sell toothbrushes.

My guess is that the writer wanted to trick the casual peruser of the ad into thinking that "100%" referred to all of the plaque, that is, that the toothbrush would remove 100% of it. The use of the percentage together with the semi-dangling comparative could lead an unwary reader astray, yet the fine print would legally protect the company from a charge of false advertising. It's not a literally false ad, but it's a misleading one.

An important fact about percentages is that they always have a base, that is, a percentage is always a fraction of something, which is called its base. If I tell you that I will have 100% more toothbrushes tomorrow than I have today, how many will I have tomorrow? You can't answer this question without knowing the base, that is, how many brushes I have today. So, when confronted by any claim involving a percentage, one should always ask: what is the base? What is the base of the ad's claim that the electric brush removes 100% more plaque than a manual one?

The base, in this case, is the amount of plaque removed by a manual toothbrush, but the ad doesn't tell you what that is. Suppose that a manual toothbrush removes 1% of the plaque on your teeth, then how much would the electric one remove, according to the ad? 2%, which is double the manual one, that is, 100% more. This is an example of a feature of percentages to keep in mind: a large percentage―such as 100%―can represent a small amount if its base is small. 100% of nothing is still nothing, and 100% of very little is not much.

In addition to the base, there's a lot of other information missing from this ad that is needed to evaluate the strength of its argument for purchasing the toothbrush. For instance, how were the two types of toothbrush compared? How much plaque is removed by brushing surely must depend on how long one brushes. Was the amount of plaque removed by the two toothbrushes measured after the same length of time spent brushing? It would seem that this would have to be the case for a fair test, but perhaps an electric toothbrush is just faster than a manual one. If so, you might be able to remove "100% more plaque" with a non-electric toothbrush by brushing twice as long. Would the savings in time be worth the much higher price of the electric brush?

In any case, this is not a review of the electric toothbrush or a recommendation about what to use to clean your teeth―dammit, Jim, I'm a logician, not a dentist. I'm just using the ad's claims as an example of what you need to be on the lookout for in advertising. Ads are trying to sell you something; if not a toothbrush then something else, such as a political candidate. Watch out for dangling comparisons, what's hidden in the fine print, and missing information.

Exercise: Supposing that an electric toothbrush really did remove 100% more plaque than a manual one, what is the largest percentage of plaque that the manual brush could remove?

Notes:

  1. ↑ John Allen Paulos, A Mathematician Reads the Newspaper (1995), p. 86.
  2. ↑ See: Dangling Comparative, 10/16/2023.
New Book
July 18th, 2026 (Permalink)

Sorry, Wrong Number!

Quote: An individual study is a brick, and it only proves itself when incorporated into a useful structure. Bad bricks either get rejected as not fitting into the structure, or exposed when things built on them collapse. But the Wrong Numbers you'll find in this book do not try to fit into structures, they are used to throw through windows instead. They go direct to the public, claiming the authority of science. But it's cargo-cult science1. It looks like science, with equations and data and references and peer review and jargon, but it was never exposed to critical scrutiny, never cross-checked with related work, never used to build structures of understanding.2

Title: Wrong Number

Subtitle: How to Extract Truth From a Blizzard of Quantitative Disinformation

Author: Aaron Brown

Comment: Brown is a writer who has had an interesting career, including professional poker playing, financial risk management―whatever that is―and teaching mathematics in college3. He's the author or co-author of four previous books, including ones on risk and chance. He also has done a series of short videos with the same title as the new book4. According to Brown:

Almost half the chapters in this book are based on studies I examined for…videos, and include some phrasing and research contributed by the team. Bill Fallon, my editor…, noticed the attention generated by the videos and suggested that Wrong Number was a good theme for a book.5

So, if you like the videos then you may like the book, and vice versa.

Date: 2026

Summary: Judging from the Table of Contents and the first, introductory chapter, the chapters appear to be mostly case studies of bad scientific studies, like the short videos Brown has done. Based on their titles, some of the chapters appear to cover the same topics as do some of the videos, though presumably in greater depth.

The Blurbs: The book is positively blurbed by Philip Tetlock, author of Expert Political Judgment6, despite the fact that there is actually a chapter titled "Philip Tetlock"7! Given that I haven't read the chapter, I don't know whether it is positive or negative about Tetlock, though I'd wager that it's positive given the endorsement. While I don't doubt Tetlock's sincerity, it's an obvious conflict of interest to have a subject of the book endorse it in a blurb, especially without revealing that conflict.

Disclaimer: I haven't read this book yet and, therefore, can neither review nor recommend it, but it looks interesting enough to read and I thought others might also be interested. I may review the book in the near future if I have anything new or interesting to say about it.

Notes:

  1. ↑ If you're unfamiliar with "cargo cult science", see: Richard P. Feynman, "Cargo Cult Science", Caltech commencement address, 1974.
  2. ↑ And the planes don't land. P. 8. Subsequent citations to only page and chapter numbers are to the New Book.
  3. ↑ All information on Brown from: "About the Author", Wiley, accessed: 7/13/2016.
  4. ↑ For a list of episodes, see: "Wrong Number with Aaron Brown", YouTube, accessed: 7/14/2016.
  5. ↑ P. 9.
  6. ↑ Philip Tetlock, Expert Political Judgment: How Good is It? How Can We Know? (2005).
  7. ↑ Ch. 24.
July 8th, 2026 (Permalink)

Sow and Sew

Grandfather William sewed the seeds of this particular tragedy when he left a will that turned the estate into a trust because he wanted to cut out his wastrel first born son….1

The words "so", "sew", and "sow", in descending order of familiarity, are homophones, that is, they are pronounced identically. "So" is a common and highly ambiguous word with too many meanings to list, but what is important for our purposes is that in most of those meanings it's an adverb2, whereas both of the other homophones are verbs. So, it's more likely that the verbs will be confused with each other than with the adverb.

"To sew" means to use needle and thread in making or repairing clothes or other items3. It used to be a common word, but in 21st century America I don't think many people still make or mend their own clothes. "To sow", meaning to scatter or plant seeds, is the least familiar of the three4.

Google's Ngram Viewer, which graphically displays the frequency of words and phrases in books, shows the unsurprising result that "so" was far more common in the last century than "sew" and "sow"5. As a result, the difference between "sew" and "sow" is dwarfed by their distance from "so". Examining the latter two words separately reveals that "sow" has been more common most of the time than "sew", which is not what I expected. Less surprisingly is that, as the population became less agricultural and more urban in the first part of the last century, the frequency of "sow" declined, though it appears to have levelled off in the past fifty years. More surprisingly, "sew" has actually been used slightly more frequently in that time frame6. While a century ago, "sow" was several times more common than "sew", the two words are recently used at about the same frequency.

"Sow" is now seldom encountered in its literal meaning, but more often in such idioms as "sowing wild oats", "reaping what you sow", and "sowing the seeds of" something, such as tragedy in the sentence quoted above. In that example, grandpa Bill obviously sowed the seeds of that particular tragedy, whatever it was, rather than "sewed" them: what would "sewing" seeds mean?

Though I've seen it before, confusing these two words may not be a very common error since none of the reference books I usually consult discuss it. Perhaps it only occurs in the familiar idioms mentioned above. Bryan Garner does mention the mistake of "sewing" wild oats, and provides three examples from newspapers7. This is the same substitution shown in the example sentence that inspired this entry, but in a different though related idiom: to sow wild oats is a metaphorical sowing of a particular type of seed.

As with many of the other confusions we've seen in these entries, this one seems to go only one way, namely, "sew" is used where "sow" should be. I don't recall ever seeing it go the other way, which may be because city dwellers are more familiar with sewing than with sowing and, thus, don't know how the latter is spelled.

My fast and filthy tip to avoid confusing "sow" and "sew" is not to use tired idioms8 such as "sowing wild oats", "reaping what you sow", and "sowing the seeds of" something, especially if you're unsure what such phrases literally mean. Unless you write about farming or gardening, you'll probably never need the word "sow" in its literal meaning. Similarly, if you don't write about needlework or surgery, you probably won't need "sew".

So, don't "sew" the seeds of confusion or you will reap what you "sew".

Notes:

  1. ↑ Robert Thorogood, Death Knocks Twice (2017), p. 319
  2. ↑ "So", The Britannica Dictionary, accessed: 7/7/2026.
  3. ↑ "Sew", Ibid., accessed: 7/7/2026.
  4. ↑ "Sow", Ibid., accessed: 7/7/2026.
  5. ↑ "so,sew,sow", Google Books Ngram Viewer, accessed: 7/8/2026.
  6. ↑ "sew,sow", Google Books Ngram Viewer, accessed: 7/8/2026.
  7. ↑ Bryan Garner, Garner's Modern English Usage (2016), under "Sow".
  8. ↑ See: George Orwell, "Politics and the English Language", Horizon, Vol. 13, No. 76, 4/1946, p. 255, "Dying metaphors".
Puzzle
July 4th, 2026 (Permalink)

Independence Day at the Logicians' Club

Those fun-loving logicians of the Logicians' Club (LC) celebrate Independence Day each year with a patriotic puzzle party, and what a wild party it is! This year's party was even wilder than usual since it celebrated the 250th anniversary of American independence. Here's how it went:

Every member wore a hat that was either red, white, or blue. The LC is, as I'm sure you know, a rather exclusive organization, so there were only six hats available: three red, two blue, and one white. The white hat was reserved for the president of the club, so all the regular members received either a red or blue hat. All of the club members were aware of these facts.

The president of the club was the first to arrive at the venue chosen for the party, and the first thing he did was make sure that the room had no mirrors or other shiny surfaces. When the first three members arrived, it was time to begin the festivities. Carrying the six hats in an opaque bag, the president approached each member from behind and placed a hat on the member's head in such a way that the member could not see its color. However, each member was able to see the color of the other two members' hats.

After the three members had each been issued a hat, the president asked one―Ms. X―what color her hat was.

"I don't know," Ms. X said, and the other two logicians heard her answer. The president turned to the second member, Mrs. Y, and asked the same question.

"I don't know," Mrs. Y said, and again the other two logicians heard her answer. The president turned to the final member, Professor Z, and asked the same question.

Prof. Z was just about to reply in the same way as the previous two logicians when he suddenly stopped and correctly gave the color of his hat.

What was the color of Prof. Z's hat? Are you LC material? If you can solve this problem, you are!

Previous Independence Day puzzles:

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