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June 11th, 2017 (Permalink)

Cuckoo for Cocoa Puffery

There are three sets of words to be on the lookout for in media reports of statistical studies of medicine or nutrition:

  1. "Could", "may", "possible", "suggest", and other words indicating that the results of a study are weak.
  2. "Link", "association", "connection", "relationship", and other words that mean that two variables are related statistically, but not necessarily causally.
  3. Other words, such as "help", that indicate that any possible effect is probably small.

With that in mind, I have highlighted every occurrence of these words in the following excerpt from a recent news story1. They first make an appearance in the headline:

Eating chocolate may help prevent a fairly common heart problem

Eating chocolate has been linked with a reduced risk of heart disease and stroke. Now a study from Denmark suggests that regular consumption of the treat may help to prevent the development of atrial fibrillation, a type of irregular heartbeat.

In the study…researchers found that people who ate chocolate at least once a month had rates of atrial fibrillation that were 10 to 20 percent lower than those who ate chocolate less often. …

When researchers took into consideration other factors that might influence development of atrial fibrillation, such as alcohol intake, smoking, obesity, high blood pressure and cholesterol levels, the study showed an association between people with a moderate intake of chocolate and a lower risk of developing atrial fibrillation.

The study does not prove a cause-and-effect relationship. And although the exact mechanism of how chocolate may prevent atrial fibrillation is not known, itís possible that compounds in chocolate called flavonoids may play a role, the researchers said.

Flavonoids have antioxidant and anti-inflammatory properties…. They may limit the inflammatory process in the body, reducing the stickiness of the blood and leading to less scarring of connective tissue. All of these factors may help prevent the electrical remodeling of the heart that leads to atrial fibrillation….

The findings showed that for women, the strongest association was seen in those who ate a one-ounce serving of chocolate once a week: This level of consumption was linked to a 21 percent lower risk of atrial fibrillation compared with those who ate less chocolate. For men, the strongest association was seen in those who ate two to six one-ounce servings of chocolate weekly. These men had a 23 percent lower risk of atrial fibrillation. …

All in all, the findings suggest that compared with some other snack choices, a moderate intake of chocolate may be a heart-healthy snack….

If only The Washington Post and other news media would publish health articles with these three types of word highlighted, as above, then it would be obvious how weak the studies often are. I don't pick on the above story on the grounds that it is especially bad; rather, it's a typical example of this type of reporting.

It's also typical in its first-the-good-news-then-the-bad-news structure: in the first half, we're given the good news: chocolate is good for you! In the second half, we're told the bad news: never mind! This is why, as I've noted before2, you should read these articles all the way to the end. Unless, of course, all you want to read is the fake news.


  1. Cari Nierenberg, "Eating chocolate may help prevent a fairly common heart problem", The Washington Post, 5/27/2017
  2. Caveat Lector, 9/3/2011

June 9th, 2017 (Permalink)

Puzzle: Your Mileage May Differ

A family owns two cars: a sport utility vehicle that gets 20 miles per gallon and a compact car that gets 40 MPG. Both vehicles are getting old and the family has decided to trade one in on a replacement that gets better mileage on a tank of gasoline. The possibilities are switching the SUV for one that gets 25 MPG or replacing the compact with one that gets 50 MPG. They drive each car about the same amount. Assuming that the only relevant consideration is how much they can save on gas, which vehicle should they trade in?


June 8th, 2017 (Permalink)

Amphibolous Headline

Teen accused of killing Uber driver with machete in one hand, knife in the other1

My first thought when I saw this headline was: if an Uber driver came at me with a machete in one hand and a knife in the other, I guess I'd kill him, too. However, if you read the story beneath the headline, it was the teen who is accused of using a machete and knife to kill the driver.

This headline isn't funny and neither is the story beneath it. However, it is ripe for misinterpretation since the headline is ambiguous; more specifically, it is amphibolous2 because of a misplaced modifier3. Grammatically, the modifying phrase "with machete in one hand, knife in the other" should modify "Uber driver", which it immediately follows, and not "teen" or "killing". Using brackets to indicate the scope of the modifying phrase, three distinct meanings of the headline can be disambiguated:

  1. Teen accused of killing [Uber driver with machete in one hand, knife in the other]

    The phrase acts as an adjective, modifying "Uber driver", which means that the driver had the weapons.

  2. [Teen with machete in one hand, knife in the other] accused of killing Uber driver

    Again, the phrase acts as an adjective, but now modifies "teen". The teen had the weapons, but when accused, not necessarily while killing.

  3. Teen accused of [killing, with machete in one hand, knife in the other] Uber driver

    The phrase now acts as an adverb, modifying the verb "killing" rather than a noun. This version accurately represents what the news story reported the teen was accused of doing.

I can sympathise with the editor who wrote the headline, since it's difficult to pack so much information into a brief headline unambiguously. However, here's a suggested revision:

Teen accused of using a machete in one hand and a knife in the other to kill Uber driver

This is four words longer but three times clearer.


  1. Samantha Schmidt, "Teen accused of killing Uber driver with machete in one hand, knife in the other", The Washington Post, 6/1/2017
  2. See Amphiboly.
  3. See Robert J. Gula, Precision: A Reference Handbook for Writers (1980), sections 10B & 10C
May 30th, 2017 (Permalink)


Q: You have pages on Improper Transposition (IT) and Denying the Antecedent (DA), but I don't think the distinction between them is clear. Indeed, you use the same counterexample for both fallacies. Would it be fair to say that DA and Affirming the Consequent (AC) are two types of IT?―Bill Adlam

A: No, it wouldn't be quite fair to say that, though those three fallacies are very similar and frequently confused. The subtle difference between them is perhaps best explained by examples, so here's a simple example of IT:

If it's raining then it's cloudy.
Therefore, if it's not raining then it's not cloudy.

Notice, that there is only one premiss1, which is a conditional statement―"if it's raining then it's cloudy." Similarly, the conclusion is a conditional statement. Now, compare that to a corresponding example of DA, that is, one that has the same content:

If it's raining then it's cloudy.
It's not raining.
Therefore, it's not cloudy.

This argument has the same first premiss as the example of IT, but it has one additional premiss, namely, "it's not raining." This may seem to be an insignificant difference but it's not. For instance, suppose that two people, A and B, asserted the example of IT and the example of DA, respectively. Then B would be claiming that it's not raining, since that's the second premiss of B's argument. Moreover, B is asserting in the conclusion that it's not cloudy. In contrast, A is only asserting that if it's raining then it's cloudy and if it's not raining then it's not cloudy. Unlike B, A never asserts that it's not raining or not cloudy. So, A and B are not asserting the same things when they make these arguments, which means that these particular arguments are not identical, even though they concern the same content and have similar forms.

For this reason, the two counterexamples you mentioned, while containing the same content, are different arguments. The same points apply, mutatis mutandis2, to the relation between the other form of IT and AC.


  1. Or "premise", if you prefer.
  2. With the corresponding changes.

May 27th, 2017 (Permalink)

New Book: Open to Debate

The new book with the above title, by Heather Hendershot, is subtitled: "How William F. Buckley Put Liberal America on the Firing Line"1. It appears to be an historical examination of Buckley's long-running television show Firing Line, which I've mentioned previously in my review of the documentary Best of Enemies2 about Buckley's contretemps with Gore Vidal.

Readers younger than thirty are unlikely to remember either Buckley or the show, which left the air in 1999. Hendershot seems to be interested in it because it often featured reasonably civil debates between the conservative Buckley and his liberal guests, hence the title. Moreover, this fact is of interest now when politics seems to be particularly polarized and uncivil; for example, just the other day a politician reportedly "body-slammed" a journalist for asking an unwanted question3.

However, the Buckley-Vidal incident, chronicled in Best of Enemies, was an infamous case of incivility that occurred on television in 1968, though not on Buckley's show and there was no "body-slamming" involved. So, polarization and incivility are certainly not new in politics. Here's Hendershot on the book's timeliness:

Looking back from today's world of shrieking pundits―a world in which TV talking heads deliberately strike up personality-based feuds with their ideological foes―the Buckley approach seems positively mind-blowing. How have we allowed our political discourse to become so coarsened and polarized? Frankly, there could not be a better time to revisit Firing Line. Even if―like me―you disagree with most of Buckley's political positions, it is impossible not to find value in a program that offers such a compelling model of gracious and rigorous political engagement.4

I may review the book in the near future.


  1. Heather Hendershot, "Open to Debate". The author's website for the book, it contains links to reviews, interviews, and excerpts.
  2. Movie Review: Best of Enemies, 2/23/2016
  3. Chronicle Staff, "Gianforte charged with election-eve assault", Bozeman Daily Chronicle, 5/25/2017
  4. P. xxxiv.

May 25th, 2017 (Permalink)

WHO said that?

Here are some headlines:

WHO says halving sugar target has extra benefit1

Bad Day For Bacon: Processed Meats Cause Cancer, WHO Says2

WHO says Ebola outbreak continues to spread in West Africa3

I don't know, WHO?


  1. "WHO says halving sugar target has extra benefit", NHS Choices, 3/6/2014
  2. Allison Aubrey, "Bad Day For Bacon: Processed Meats Cause Cancer, WHO Says", NPR, 10/26/2015
  3. Liz Szabo, "WHO says Ebola outbreak continues to spread in West Africa", USA Today, 10/24/2014

May 22nd, 2017 (Permalink)

Lesson in Logic 18: Categorical Syllogisms

Finally, what I've been leading up to and you've been waiting for is here: how to use Venn diagrams to evaluate categorical syllogisms. In the previous lesson1, you learned how to diagram the premisses of such an argument on a pretzel2, so you're almost there. All that's left is to learn how to evaluate such a diagram for validity.

First, a definition: a categorical syllogism is an argument with two premisses and a conclusion, all of which are categorical statements. Furthermore, the statements making up a categorical syllogism have exactly three categorical terms among them. Also, one of the three categorical terms3 occurs in both premisses but not in the conclusion. The other two terms occur once each in the conclusion and once in one of the premisses. Here's an example of a categorical syllogism:

  1. All woodpeckers are birds.
  2. All sapsuckers are woodpeckers.
  3. Therefore, all sapsuckers are birds.

In this example, 1 and 2 are the premisses and 3 is the conclusion, as indicated by the word "therefore". The three terms are: woodpeckers, birds, and sapsuckers. "Woodpeckers" is the middle term because it occurs in both premisses but not in the conclusion. "Birds" occurs once in the conclusion and once in the first premiss, and "sapsuckers" also occurs once in the conclusion and once in the second premiss. So, this argument fits the definition of a categorical syllogism.

To test an argument of this type for validity, do the following:

  1. Diagram the premisses on a pretzel.
  2. Do not diagram the conclusion.
  3. Answer the question: Does the diagram show the conclusion of the syllogism to be true?
    • If so, then the syllogism is valid.
    • If not, then the syllogism is invalid.
Categorical syllogism

Here's how this works on the example, above. The diagram to the right shows the first premiss in red and the second in blue. Does the diagram show the conclusion to be true? In order for the conclusion to be shown true, the area of the diagram representing all sapsuckers that are not birds, which is outlined in yellow, would have to be empty. The diagram does show that area as empty. Therefore, the argument is valid, as should be intuitively obvious.

The only tricky part of evaluating syllogisms with Venn diagrams comes when a premiss and the conclusion of the argument are particular statements4. It may happen that the diagram of a syllogism of this type does not look exactly the way you would diagram the conclusion. In particular, the "X" from the particular premiss may not be on a line. However, the point to remember is that if the diagram of the premisses shows the conclusion to be true, then the argument is valid. This is best shown with an example: Categorical syllogism

  1. All sapsuckers are woodpeckers.
  2. Some birds are not woodpeckers.
  3. Therefore, some birds are not sapsuckers.

If you diagram the first premiss―shown in red―before the second, as you learned to do in the previous lesson1, then you will not put the "X" on a line, since the area outlined in yellow is empty. Given the first premiss, birds that are not woodpeckers must be in the area marked by the "X".

To evaluate the example for validity, look at the diagram to see whether it shows the conclusion to be true. In this case, look to see whether the diagram shows that there are birds which are not sapsuckers. The diagram does indeed show that there is at least one bird that is outside of the circle representing sapsuckers. Therefore, the argument is valid.

You'll learn more from practice than just by reading, so try the following exercises.

Exercises: Use a Venn diagram to evaluate the following arguments. Indicate whether each is valid or invalid.

  1. No woodpeckers are raptors. All sapsuckers are woodpeckers. Therefore, no sapsuckers are raptors.
  2. All flickers are woodpeckers. All sapsuckers are woodpeckers. Therefore, all sapsuckers are flickers.
  3. No sapsuckers are raptors. Some woodpeckers are sapsuckers. Therefore, some woodpeckers are not raptors.
  4. No woodpeckers are raptors. All sapsuckers are raptors. Therefore, no sapsuckers are raptors.
  5. Some woodpeckers are sapsuckers. All sapsuckers are raptors. Therefore, some raptors are not woodpeckers.

Answers to the Exercises

Next Lesson: Sorites!


  1. Lesson in Logic 17: Pretzel Logic, 4/28/2017
  2. A three-circle Venn diagram.
  3. Called "the middle term".
  4. That is, I and O statements. If both premisses are particular, then the syllogism will be invalid. Similarly, if only one premiss is particular, but the conclusion is not. For now, take my word for it, but when you've become skilled in testing syllogisms for validity you can prove these facts for yourself.

May 5th, 2017 (Permalink)

A Word Puzzle for This Friday

Consider the following list of words:

educated, ill-fated, notice, unite, decisive, branch

Which of the following words comes next in the list?

copper, truck, tenuous, neat, hum, explain


May 4th, 2017 (Permalink)

Poll Watch: No Margin for Error

Here's a fake news headline:

Hispanics give Trump higher approval rating than rest of U.S.1

According to the first two sentences of the article beneath the headline:

Hispanic support for Donald Trump has surged since Election Day, and now tops that of the president's overall approval rating. In its latest survey, Zogby Analytics said that Hispanic support has hit 45 percent, two points higher than the president's generic approval.1

Two points? The standard margin of error (MoE) for national polls is plus-or-minus (±) three percentage points, so this is within the MoE. Moreover, the statistic refers to the support of hispanics, a subgroup within the general population, and thus likely to be a subset of the sample taken for the poll. The smaller the sample, the wider the confidence interval that determines the MoE will be. So, the MoE for the hispanic subsample in this poll is likely to be greater than ±3 points2.

The reporting of this poll is worse than is usual for American newspapers, since most will at least give the MoE at the foot of the article even if they ignore it in the body. However, this Washington Examiner story doesn't even bother to mention it, which seems to violate American journalistic standards. At least the article links to Zogby's own report on the poll3, but the only thing Zogby has to say about Trump's support among hispanics is the following:

The biggest surprise in this new poll is Trump's approval among Hispanic voters, which is at 45% approval/51% disapproval. In February the numbers were less among Hispanics at 39% approval/53% disapproval.3

So, Zogby doesn't make a big deal about hispanic support for Trump exceeding support among the general population of likely voters, probably because they realize that it isn't statistically significant. Its report links to a tabulation of the results4, where you can finally find out that the MoE for the entire sample is ±3.3 points, so the alleged higher hispanic approval rating for Trump is within that margin. However, you can also read the following at the bottoms of the pages: "Subsets have a larger margin of error than the whole data set."4

In fact, the number of hispanics included in the poll was only 965, which means that the MoE at the 95% confidence level is ±10 percentage points6. So, not only is two percentage points far from significant, even the six point increase in hispanic approval of Trump is well within the MoE. As far you can tell from this poll, the supposed "surge" in Trump support among hispanics is just random sampling error.

That's why the Washington Examiner article is fake news: subtract the innumeracy and there's zero story left. Do hispanics support Trump more, less, or the same as the rest of us? Has hispanic approval of Trump increased in the last few months? Who knows? This poll certainly doesn't provide the answer.


  1. Paul Bedard, "Hispanics give Trump higher approval rating than rest of U.S.", Washington Examiner, 5/3/2017
  2. How to Read a Poll: Margin of Error Errors
  3. "Zogby Analytics Online Survey of Likely Voters", Zogby Analytics, 5/3/2017
  4. "The Zogby Poll: Trump overall approval down, but up among Hispanics", Zogby Analytics, 5/2/2017
  5. The last table, p. 3.
  6. Courtney Taylor, "How to Calculate the Margin of Error", Thought Co., 12/16/2014

Update (5/11/2017): In case you thought that the Washington Examiner's bad poll reporting was just a fluke, it's back a few days later with even worse reporting. Once again, the MoE is not reported, but that's the least of its problems. Here's how it originally began:

Couples are fighting over President Trump more than ever, and many are turning to divorce court to get out of their politically ravaged marriages. New data from Wakefield Research found that one in 10 couples, married and not, have ended their relationships in a battle over Trump. For younger millennials, it's 22 percent.1

Of course, the claim that ten percent of couples have broken up because of a difference of opinion over Trump is wildly implausible. Here's how Wakefield itself reported its results:

The results revealed that more than 1 in 10 Americans (11%) have ended a romantic relationship over political differences. This number jumps notably among the younger generation, with 22% of Millennials having broken up with someone over political differences.2

So, the 10% result refers to Americans who have had a romance end over politics, not just about Trump, and presumably occurring at any point in the person's lifetime, not just in the last year or two. The same applies to the 22% result for the so-called Millennials. These results really don't reveal anything about Trump's effect, if any, on romantic relationships, since it's possible that all of these break-ups happened before his foray into politics.

In addition to the poor reporting, the current version of the story on the Examiner's website3 has been silently edited to fix the worst misstatements of the poll's results. Of course, it's good that it's been corrected, but there ought to be some acknowledgment that changes were made. Otherwise, it looks as though the Examiner is trying to conceal its mistakes. However, Fox News reprinted the first half of the original article, which contained its worst misstatements, and still shows the unedited version so you can check it for yourself1.


  1. Paul Bedard, "Fights over Trump drive couples, especially millennials, to split up", Fox News, 5/8/2017
  2. "New Wakefield Research Study: The Trump Effect on American Relationships", Wakefield, 5/10/2017
  3. Paul Bedard, "Fights over Trump drive couples, especially millennials, to split up", Washington Examiner, 5/8/2017.

Via: Eugene Volokh, "No, a survey doesnít show that 1 in 10 couples have broken up over their views on Trump", The Volokh Conspiracy, 5/10/2017

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