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March 31st, 2015 (Permalink)

The Puzzle of the Mount Rushmore Four

Bank robbers seem to like wearing masks of presidents in order to conceal their identities―see the Resources, below, for a couple of previous examples. Perhaps it's because of all those pictures of presidents on the currency. Now, a new gang of four has robbed several banks while wearing masks representing presidents Washington, Jefferson, Theodore Roosevelt, and Lincoln. As a result, the police have nicknamed them "the Mount Rushmore gang".

After the gang's most recent robbery the police brought in four men―one of whom was named Cross―whom they suspect of being the Mount Rushmore thieves. Unfortunately, no physical evidence was found directly linking any of the four suspects with the robberies. They immediately lawyered up and have since refused to say a word.

The police put the four suspects in a line-up for witnesses from the bank, but because of the masks the witnesses had only been able to see the robbers' eyes. Luckily, each of the suspects had a different color of eyes―one had dark brown eyes.

Putting together all of the evidence they were able to gather, the police had only the following six clues:

  1. The suspect named Dawson had not worn the Roosevelt mask.
  2. The fourth man in the line-up was not the suspect named Ambrose.
  3. During the robbery, the suspect named Ballard entered the bank first, followed by the suspect who stood third in the line-up, then came the man who had worn the Jefferson mask, and finally the blue-eyed robber.
  4. The suspect with hazel eyes stood last in the line-up.
  5. In the line-up, Dawson stood to the immediate left of the man who had worn the Washington mask.
  6. When the second man in the line-up entered the bank he was immediately followed by the green-eyed robber.

Of course, the police are baffled by this evidence. Can you help them determine the masks each suspect wore, their eye colors, the order in which each entered the bank during the robbery, and the order in which they stood during the line-up?



March 27th, 2015 (Permalink)

Conspiracy Theorists and Other Bad Thinkers

…[N]one of us can deny that intellectual vices of one sort or another are at play in at least some of our thinking. Being alive to this possibility is the mark of a healthy mind.

That's philosopher Quassim Cassam from a very interesting article at Aeon magazine applying the theory of intellectual virtues and vices to conspiracy theorists (CTists)―see Source 2, below. If you're not familiar with intellectual virtues and vices, here's an introductory sketch―for a longer, more challenging introduction, see the Resource, below:

The theory of intellectual virtues and vices is an outgrowth of an approach to ethics that goes back as far as Plato and whose most influential exponent was Aristotle. Ethical virtues are such things as courage, honesty, self-control, and so on, while the vices are cowardice, dishonesty, self-indulgence, and the like. The intellectual virtues and vices are special types of the ethical ones related to intellectual matters having to do with the formation and maintenance of beliefs, evaluation of evidence, willingness to change beliefs, etc. Such virtues include intellectual humility, curiosity, and open-mindedness; corresponding vices are intellectual arrogance, willful ignorance, and closed-mindedness.

Here's why Cassam thinks that it's worthwhile applying the theory of intellectual virtues to conspiracy theorists:

The problem with conspiracy theorists is not…that they have little relevant information. The key to what they end up believing is how they interpret and respond to the vast quantities of relevant information at their disposal.

You can see in this short quote that Cassam's approach is to reject the idea that CTists are ignorant in favor of the notion that there is something wrong with the way they think. I agree with him that the problem with CTists is usually not a lack of information per se―typically, they have loads of information supporting their pet conspiracy theory (CT), though much of it will be false―but it may be that such false information isn't the kind of relevant information in question. Rather, what the CTists need is the information that would refute their favored CT. In my experience, CTists are frequently ignorant of exactly this kind of information, despite the masses of pseudo-evidence they can spew forth in support of their CT.

Moreover, the evidence refuting most CTs is readily available―"The truth is out there!" to quote an infamous conspiracy-mongering TV show―but CTists don't believe it. The CTist pays close attention to the evidence that supports his favorite CT, but ignores or downplays that which refutes it. He exercises little or no skepticism about evidence favorable to his theory, which is why he ends up believing so many falsehoods. In contrast, his skepticism immediately goes into overdrive when forced to confront counter-evidence, which he simply dismisses.

So, I generally agree with Cassam that CTists are bad thinkers. Nonetheless, such bad intellectual traits can lead to CTists lacking relevant information because they fail to seek it out (a lack of curiosity), fear the consequences to their pet beliefs (intellectual cowardice), and actively resist being educated (willful ignorance). For this reason, simply making correct information available will probably have little effect on CTists, since they are likely to dismiss or downplay it if they come across it. As Cassam puts it:

…[T]here remains the problem of what to do about such people as [the CTist]. If he is genuinely closed-minded then his mind will presumably be closed to the idea that he is closed-minded. Closed-mindedness is one of the toughest intellectual vices to tackle because it is in its nature to be concealed from those who have it. … What if [he] is too far gone and canít change his ways even if he wanted to? Like other bad habits, intellectual bad habits can be too deeply entrenched to change. This means living with their consequences. Trying to reason with people who are obstinately closed-minded, dogmatic or prejudiced is unlikely to be effective.

The philosopher Stephen Law has used the apt phrase "intellectual black hole" to refer to such things as CTs. Once someone has been sucked into the intellectual black hole of a CT, he may be forever unreachable. This is a discouraging fact, but it should also encourage us to do what we can to prevent people from falling into such holes. One thing we can do is to put up intellectual signposts along the way: "Caution: intellectual black hole ahead!", "Dead end", "No exit".

You might wonder what the value is of condemning CTists, pseudoscientists, and others as "bad thinkers", especially if the conclusion follows that there is little if anything we can do to make their thinking better. A negative point is that it may be a waste of time and effort to directly argue with CTists since they are often not receptive to evidence against their pet CTs. A positive point is that if we want to reduce the prevalence of conspiracy theories and pseudoscience, we need to educate people in the intellectual virtues. As Cassam writes: "If we care about the truth then we should care about equipping people with the intellectual means to arrive at the truth and avoid falsehood."


  1. "Make your own Road Construction Sign", Atom Smasher
  2. Quassim Cassam, "Bad Thinkers", Aeon, 3/13/2015

Resource: Rosalind Hursthouse, "Virtue Ethics", Stanford Encyclopedia of Philosophy

March 22nd, 2015 (Permalink)

What's New?

I've added a new contextomy to the "Familiar Contextomies" page―see the Source, below. This is another misleading quote used by 9/11 conspiracy theorists to suggest that something other than an airliner crashed into the Pentagon on September 11th, 2001.

Source: Familiar Contextomies: Danielle O'Brien

March 16th, 2015 (Permalink)

Wikipedia Watch

In previous watches, I've mentioned hoaxes that have been played on Wikipedia―see the list below. Unfortunately, hoaxing Wikipedia is not a new pastime, and some have even done it for college credit! The longest-lived hoax that I was aware of lasted for five years before it was discovered, but now a decade-long hoax has been uncovered―see the Source, below, for the details. This is just the latest record-holder, and it's likely that even longer-lived ones will eventually come to light. This is why I argued, in the most recent watch listed below, that we can't really know how fast hoaxes or other types of misinformation are exposed.

It's hard to imagine something like this happening to the Encyclopaedia Britannica, or to any other traditional encyclopedia for that matter. The fact that anyone can add something to Wikipedia would seem to make hoaxing unavoidable. This is one reason why it's not a trustworthy source of information; at the very least, you need to verify what it says with at least one independent source.

Source: Mason, "Jared Owens, God of Wikipedia", Wikipediocracy , 3/15/2015

Previous Wikipedia Watches: 5/16/2012, 1/9/2013, 12/21/2014

Reader Response (3/23/2015): Pat Heil writes:

Encyclopaedia Britannica (EB) and other "reliable" sources are at the mercy of the tendentious, the poor scholar, and the latter day scholar who accepts as authorities people whose material is out of date. I feel quite sure that editions of EB published between 1912 and 1953 treated Piltdown Man seriously as a forebear of humanity. So all authorities deserve to be questioned, not just Wikipedia.

I expect you're right about the Piltdown hoax, but that's not the sort of hoax that's been played repeatedly on Wikipedia. It is possible that a traditional encyclopedia such as the Britannica might run a hoax article, but it's far less likely to happen than to Wikipedia. I agree that it's a good idea to double-check every source, and to trust none completely, but some are more trustworthy than others. No source―not even Britannica―is 100% reliable, but some are more reliable than others, and many are more reliable than Wikipedia.

March 14th, 2015 (Permalink)

Logical Literacy: "You can't prove a negative."

I have some good news and some bad news. I'll save the good news till later. Here's the bad news: despite what you may have heard, you can too prove a negative!

One of the earliest proofs of a negative is what is sometimes referred to as "Euclid's Second Theorem", which is that there are an infinite number of prime numbers. Why is this a negative? Take a close look at that word "infinite". Do you see the "in" on the front? "In-finite", that is, not finite. Are you still not convinced? Specifically, what Euclid proved is that there is no greatest prime number; that there must be an infinite number of primes follows from this fact, because if there were a finite number of primes then there would be a greatest one.

How did Euclid prove that there is no greatest prime number? Without going into the details, he first assumed that there is a greatest prime, and then showed how to find an even bigger prime. This, of course, is a form of reductio ad absurdum (RAA), in which you prove something false by assuming that it's true and showing that a contradiction follows from that assumption, which is a common form of proof in logic and mathematics.

Euclid's Second Theorem is just one of many negative theorems in logic and math. Other famous negative theorems are Fermat's Last Theorem and Gödel's Incompleteness Theorems―note the word "incompleteness": in-completeness, that is, not complete.

Moreover, as philosopher Steven Hales points out―see the Source, below―"you can't prove a negative" is itself a negative statement, so if it's true then you can't prove it! So, how would you know that "you can't prove a negative" is true?

Let's move on to the good news: while it's not true as a general matter that you can't prove a negative, there is something to the idea. However, getting at what that something amounts to is not easy, but that's what I'll try to do in the remainder of this entry.

First of all, I was speaking of "proof" in the strict, logical sense of the word, above. Unfortunately, in this strong sense of the word, the only things you can "prove" are propositions of logic and mathematics. However, there is a weaker, common use of the word "prove" to mean something like "establish beyond a reasonable doubt". This is the sort of "proof" standard used in criminal cases in American courts. In this weaker sense, it's possible to "prove" things that are not part of logic or math, such as that the defendent is guilty. For the rest of this entry, I'll use the word "establish" for this weaker sense of "prove".

So, instead of "you can't prove a negative" I'll be discussing "you can't establish a negative", instead. However, it's still not true that you can't establish a negative. Consider the claim that all swans are white. It's easy to establish the negation of this claim, at least if you live in Australia: just produce a non-white swan, which has in fact been done since there are black swans in Australia.

In order to find any truth in the claim that you can't establish a negative we have to focus upon a specific type of negative statement, namely, a negative existential statement. An existential statement―or, more specifically, an affirmative existential statement―is just a claim that some particular thing or type of thing exists. For instance, "there is a Loch Ness monster" is an affirmative existential statement; "there are bigfeet [bigfoots?]" is another. A negative existential statement is just the negation of an affirmative one, so "there is no Loch Ness monster" and "there are no bigfeet" are both negative existentials.

Affirmative existential statements are comparatively easy to establish; at least, that is, if they are true. Consider the claim: "a black swan exists". To establish this statement, all that is necessary is to produce a black swan, which is easy if you live in Australia. In contrast, the negation of "a black swan exists"―namely, "it is not the case that a black swan exists" or "there are no black swans"―would be much harder to establish even if it were true. To establish that there were no black swans, you would have to examine every swan and see that none is black. While not completely impossible, this is for all practical purposes undoable. Thus, it can be much more difficult to establish a negative existential claim than an affirmative one.

Thus, it is reasonable to place the burden of proof on those who make affirmative existential claims rather than on those who deny them. This is one reason why some skeptics have made the "you can't prove a negative" claim: the burden of proof is not on the skeptic to disprove the existence of the Loch Ness monster, bigfeet, flying saucers, etc., but on those who claim they exist.

To sum up, you can prove a negative, but it's much harder to establish a negative existential statement than an affirmative one. As a result, the burden of proof is on those who make affirmative existential claims rather than on those who deny them.



March 3rd, 2015 (Permalink)

Sobriety Check, Part 2

In part one, we saw that the statistical claim that underage drinkers spent $22.5 billion on alcoholic beverages in the United States in 2001 was implausible―see the Resource, below. In order for this to be true, underage drinkers would have had to spend an average of over $600 apiece. However, though this is an implausibly high amount, it isn't impossible.

To perform this statistical check, all that we needed was the information included in a short New York Times article that reported the claim―see Source 2, below―as well as the statistical benchmark that approximately four million babies are born each year in the U.S. Unfortunately, the information contained in the Times report, together with statistical benchmarks, does not appear to be sufficient to show that the original paper that reported this statistic must be in error. In this sequel, we will turn to the paper itself―see Source 1, below―and use a different technique to check it.

As a general matter, not every dubious statistic that you come across in the news media can be detected by the use of statistical benchmarks. Sometimes the needed benchmarks won't be available, or a questionable number may survive a benchmark test. Luckily, there's an alternative test that sometimes works when the benchmark test fails.

Turning now to the paper itself, here's how the researchers arrived at the estimate of the amount spent by underage drinkers on alcoholic beverages:

  1. They used census data from 2000 to estimate the number of people between the ages of 12 and 20. Nowhere do they actually provide this figure, as far as I can tell, but based on the information they do give I estimate it as a conservative 40 million. See the Technical Appendix, below, if you want to know exactly how I arrived at this estimate.
  2. Using survey data, they estimated the proportion of those in this age group who are drinkers, and thus underage drinkers. Again, the precise number is not given, but based on the data I estimate it as approximately 19 million. Again, see the Technical Appendix, below, for the details of this estimate.
  3. Again using survey data, they estimated the mean number of drinks by underage drinkers in a month: 35.2.
  4. From the two previous numbers, they calculated the total number of drinks taken by underage drinkers in 2001, which they give as just over 20 billion.
  5. From this last number, together with data about the average price of alcoholic beverages, they estimated the amount of money spent on underage drinking.

Now, as I suggested above, it's unlikely that you can use benchmarks to check these statistics, but there's another way you can do it. You can check these numbers using only the information above. Moreover, you won't need any sophisticated math, though the use of a calculator would make the calculations less tedious. When you've done so, click on "Sobriety Check", below, to see the results of one such check.

Sobriety Check


  1. Susan E. Foster, Roger D. Vaughan, William H. Foster, Joseph A. Califano Jr, "Estimate of the Commercial Value of Underage Drinking and Adult Abusive and Dependent Drinking to the Alcohol Industry", Archives of Pediatrics & Adolescent Medicine, 5/2006
  2. Eric Nagourney, "Addiction: Sales Estimates Paint Portraits of Alcohol Abusers", The New York Times, 5/2/2006

Resource: Sobriety Check, 2/24/2015

Sobriety Check: The problem with these numbers occurs in the step from 2 and 3 to 4. First of all, since step 4 is expressed in drinks per year, we need to figure the mean number of drinks per year based on the number per month; this, of course, is simple: just multiply 35.2 by 12, to get about 423―to be conservative, I'm rounding up. Now, if you multiply the number of underage drinkers in step 2 by the average number of drinks in a year they supposedly drink, the result is a little over eight billion. This, of course, is less than half the number given in step 4. As a result, each of the following numbers will be no more than half of what is claimed by the study. Specifically, the total amount spent by underage drinkers will be a little less than $11 billion, instead of the $22.5 billion claimed by the researchers. As a percentage of the total amount spent on alcohol, this is about 8.5%, exactly half of the 17% they claimed. See the Technical Appendix, below, if you want to know exactly how I arrived at these estimates.

What happened in this study? How did the researchers go so wrong? I suggested in part one that they were drunk when they wrote it, but of course I wasn't serious. Joel Best, in his excellent book Stat-Spotting, from which this example is taken, suggests that it was simply an arithmetical error. However, even if this is true, why wasn't this error caught by the researchers before the paper was published? Also, the paper was published in a peer-reviewed scientific journal; how did it manage to survive the peer review process? Finally, the published paper generated a news report in The New York Times; how did it manage to get past the reporter, editors, and fact-checkers of that august publication?

What can we learn from this example? Here are the main points:


  1. "New Study in Archives of Pediatrics and Adolescent Medicine and CASA* Report Reveal", PR Newswire, 5/1. This is a CASA press release about the study. No year is given but it must have been 2006. The Times article appeared the next day and was probably based on this release.
  2. Christopher Shea, "In Drug-Policy Debates, a Center at Columbia U. Takes a Hard Line", The Chronicle of Higher Education, 10/3/1997 (PDF)

Technical Appendix: In the interest of showing my work, unlike the authors of this dubious paper, I will explain here how I estimated those numbers that were not provided by those authors. Below, I explain where the first two numbers that I used in the list, above, came from. If you're not interested in cross-checking my math, then you probably won't be interested in the following:

  1. To make this estimate, I used the fact that the population of the United States passed the 300 million mark in 2006, as mentioned by Joel Best (p. 7). Now, the authors of the paper actually based their estimates on the 2000 census, which would have had a somewhat lower figure, of course. I used the higher figure because it's easier to use―it's round and at hand―and it's a conservative choice because it tends to increase slightly the number of underage drinkers.

    The paper estimates the underage population as 15% of those aged twelve years or older (Table 2). So, I first estimated the size of the population no more than eleven years of age, using Best's benchmark figure of four million Americans of a given age (p. 7). Thus, approximately 44 million Americans are less than twelve, and subtracting this from the total population gives 256 million twelve and up. 15% of this figure is somewhat less than 40 million, so I again rounded up to be conservative. Joel Best went about this in a more direct way, using the benchmark figure to estimate the number of those aged twelve to twenty as 36 million (p. 25), which is comfortingly close to my figure.

  2. The estimate of the number of underage drinkers comes from the above estimate of those underage, together with the paper's statistic that 47.1% of underage Americans drink alcohol (Table 2): 47.1% of 40 million is about 19 million. Again, to be conservative, I have rounded up. From his estimate of the underage population, Best calculates this number as about 17 million underage drinkers (p. 25), which is again nicely close to my own estimate.

Two additional estimates, in the Sobriety Check above, are: first, the total amount of money spent on underage drinking; and, second, the percentage this amount represents of the total amount of money spent on drinking:

  1. According to the original paper, the average price for an alcoholic beverage ranged from a low of $1.05 for beer to a high of $1.34 for liquor (p. 475). To be conservative, I multiplied the highest price of $1.34 by the estimated total number of drinks by underage drinkers of eight billion, then rounded up.
  2. According to the paper, the total amount spent on alcoholic beverages in the United States in 2001 was $128.6 billion (p. 476), and $11 billion is about 8.5% of that amount.

At most steps in which I had to estimate a number, I rounded up, which was a conservative strategy because it gives the paper the benefit of the doubt. The goal here is not to get the most accurate estimate of the cost of underage drinking―I'm not in that business―but to give the paper's estimates the best chance of being correct. Despite my bending over backwards to give them a chance, the estimates still fail by a large margin, which shows just how far off they likely are.


Update (3/4/2015): I've changed a few infelicities in wording, above.

Solution to the Puzzle of the Mount Rushmore Four:

According to the third clue, the order in which the robbers entered the bank was: Ballard, the third man in the line-up, the man in the Jefferson mask, and the blue-eyed robber.

From the last clue, the robber with green eyes immediately followed into the bank the second man in the line-up. The second man in the line-up could not be the one who wore the Jefferson mask, since he was followed by the blue-eyed robber. Nor could he be the second to enter the bank since that was the third man in the line-up. Therefore, the second man in the line-up was the first to enter the bank, namely, Ballard. Also, the second to enter the bank, who was third in the line-up, has green eyes.

According to the fourth clue, the fourth man in the line-up had hazel eyes. We know that this man could not have entered the bank first since that was the second man in the line-up. Also, he could not have been the second or last into the bank, since those robbers had different colored eyes. Thus, he must have been the third into the bank, namely, Jefferson. By a process of elimination, we now know that Ballard has dark brown eyes, and that the first man in the line-up was the last to enter the bank.

From clue five, the suspect named Dawson stood to the immediate left in the line-up of the robber who wore the Washington mask. Dawson could not have been third in the line-up, which was to the immediate left of the man who wore the Jefferson mask. Nor could he have been second, since that was Ballard. Therefore, Dawson stood first in the line-up, and Ballard was the one who wore the mask of Washington.

By the first clue, the suspect named Dawson was not the robber in the Roosevelt mask. Thus, Dawson must have worn the Lincoln mask and the third man in the line-up wore the Roosevelt one.

Finally, according to the second clue, the suspect named Ambrose was not fourth in the line-up. Therefore, he must have been third, which leaves Cross in last position.

To sum up, the following table lists the robbers based on their position as they entered the bank:

Robbery Position: 1st 2nd 3rd 4th
Name: Ballard Ambrose Cross Dawson
Mask: Washington Roosevelt Jefferson Lincoln
Eye-Color:Dark Brown Green Hazel Blue
Line-up Position: 2nd 3rd 4th 1st

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