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October 27th, 2005 (Permalink)

The Scariest Puzzle Imaginable

The Boojum is a scary monster. In fact, it is the scariest monster of them all. Even worse, it is the scariest monster imaginable! It is also mysterious: no one knows what it looks like, because everyone who sees it dies immediately from fright. The only thing known for sure about the Boojum is that it is the scariest monster that you can imagine.

You might think or hope that the Boojum is only a figment of the imagination. However, if the Boojum were only imaginary, then it would be possible to imagine a scarier monster, namely, a real Boojum. Here's where it gets really scary! There are only two possibilities:

However, a real Boojum that doesn't exist is self-contradictory. Therefore, there is only one possibility left:


If you can find a flaw in this scary proof of the Boojum's existence, send your explanation to the fallacist by Halloween.

Update (11/1): Apparently, this puzzle was scarier than I imagined! Everyone was too frightened to solve it, though Eric Williams did recognize that the puzzle is a variant of Anselm's "proof" of the existence of God―more precisely, the part before it gets really scary. However, the really scary part is the most important part of the puzzle. So, I am extending the deadline until the beginning of next week: send in your solution by 11/6 or a real Boojum may pay you a visit!


October 26th, 2005 (Permalink)

Logic Check: Dueling Comparatives

I've pointed out frequently how politicians like to use dangling comparatives to mislead voters. For instance, a politician may claim to have lowered something that his constituents would like to see lower, such as taxes, crime, or unemployment. But the comparative word "lower" is left dangling, that is, only the first term of the comparison is supplied, while the other is left out. "Lower than what?" is the question to ask. Similarly, politicians will claim that something is higher that voters would like to see higher, such as social security benefits, wages, or economic growth. "Higher than what?" should be asked.

Annenberg Political Fact Check has put out a report about dueling television ads in the New Jersey race for governor. In an ad, the Republican challenger has accused the Democratic incumbent of voting to raise taxes, while the incumbent has responded with an ad claiming to have voted to lower taxes. In some cases, the ads are talking about the very same votes!

Fact Check suggests that this means that they are contradicting one another, but not necessarily. Both are using dangling comparatives. For instance, in one case, the incumbent voted for a bill that would have lowered taxes from their current level, but would have left them higher than a competing bill. This is no more a contradiction than that John is both taller than Jim and shorter than Joe. However, both candidates are being deceptive by dropping the object of the comparison.

Source: James Ficaro, "Taxing the Truth in New Jersey Ad War", Annenberg Political Fact Check, 10/24/2005


October 21st, 2005 (Permalink)

Barnum Statements

"Analysts long ago noticed that many horoscopes contained a large number of very vague statements. Examples include such waffling as: 'You have considerable hidden talent that you have not yet used to your advantage,' and, 'While you have some personality weaknesses, you are generally able to compensate for them.' Critics of astrology have dubbed such declarations 'Barnum statements' after the American showman who coined the phrase, 'There's a sucker born every minute.'

"Most strikingly, surveys have shown that when a sample of people are shown Barnum statements from horoscopes, ninety percent of them believe the statement applies to them, and can link what is said in even the most crass tabloid horoscope to events in their lives or their hopes and aspirations. The fact that on occasion these horoscopes have been either deliberately fabricated or written by hard-pressed journalists makes no difference at all. One research scientist, Geoffrey Dean, has also noted that when horoscopes contain succinct but slightly more specific personality references, such as 'you have a good imagination,' they are seen as less relevant to individuals reading them than the horoscopes full of Barnum statements. The reason for this is clear: with a Barnum statement, anyone can make of it what they will."

Source: Michael White, Weird Science (1999),

Resource: Robert Todd Carroll, "Barnum Effect", Skeptic's Dictionary

October 20th, 2005 (Permalink)

Name That Fallacy!

"Astrology would be considered a scientific theory if judged by the same criteria used by a well-known advocate of Intelligent Design to justify his claim that ID is science, a landmark US trial heard on Tuesday. Under cross examination, ID proponent Michael Behe, a biochemist at Lehigh University in Bethlehem, Pennsylvania, admitted his definition of 'theory' was so broad it would also include astrology."


Source: Celeste Biever, "Astrology is Scientific Theory, Courtroom Told", New Scientist, 10/19/2005

October 18th, 2005 (Permalink)

"Sorry"s and "Thank You"s

October 14th, 2005 (Permalink)

Poll Watch

Mark "Mystery Pollster" Blumenthal has a post about a recent poll result that seems to show President Bush's approval rating among black Americans at only 2%. Bush's rating has been less than 50% among the general populace recently, and doubtless it is even lower among blacks, but 2% is implausibly low. The pollsters wisely did not draw attention to this number, but some reporters have pounced upon it. However, the sample of blacks was less than 100, and it is hasty generalizing to conclude that the poll's result accurately represents black public opinion.

Source: Mark Blumenthal, "2% Among African-Americans?", Mystery Pollster, 10/14/2005

October 13th, 2005 (Permalink)


Q: I was sorry to see that you don't have an entry for the pathetic fallacy. Any particular reason?―Ruth Logie, Johannesburg, South Africa

A: The word "fallacy" in general means "error"; more specifically, it means "common error". So, there are books with "fallacy" in the title which deal with common false beliefs. The Fallacy Files is concerned with logical errors, as opposed to factual errors, or error in general.

There is some controversy among logicians about what a logical fallacy is. I take the narrow view that a logical fallacy is a type of error in reasoning, so that factual errors are to be distinguished from logical fallacies. One can picture reasoning as a process which takes inputs―premisses―and produces an output―the conclusion. Errors in the premisses or conclusion are not the business of logic; rather, a logical fallacy is an error in the process―an argument―that produces the conclusion from the premisses. Furthermore, a mistake in the reasoning process must be an instance of a common type of error to count as a fallacy.

In contrast, some logicians take the expansive view that a logical fallacy is any error which occurs in the course of reasoning, even if it is only a false premiss. I think that this view is too broad, since any factual error may become the premiss of some argument. Sciences other than logic are concerned with correcting factual errors; if the expansive view were correct, logic would tend to expand to cover all of the other sciences. As D. H. Fischer said: "Logic is not everything. But it is something…."

The "pathetic" fallacy is the name given by the art critic John Ruskin to the error of attributing emotions and other psychological states to the natural world. Ruskin gives the example of a poem with the lines:

They rowed her in across the rolling foam―
The cruel, crawling foam.

Since the foam cannot be cruel, the poem is literally false. This is the sort of error that philosophers call a category mistake, that is, attributing a characteristic to an object which is the wrong type of thing for that characteristic. Only animals can be cruel, and foam is not an animal. Thus, the pathetic fallacy appears to be a type of conceptual error, rather than an error in the process of reasoning.

It's possible that such examples might be logical boobytraps, that is, linguistic snares―like vagueness and ambiguity―that might cause the unwary to commit a logical fallacy. However, I can't think of any examples, so it would seem not to be a common type of logical error. Also, the pathetic fallacy seems to occur primarily in poetry―all of Ruskin's examples are from poems―and I think that most people do not take poetry literally, but understand that poetic language is metaphorical. However, thanks to your question, Ruth, I'll keep my eyes open for examples of the pathetic fallacy as a logical boobytrap. If any readers have such an example, please send it to the Fallacy Files!

Source: John Ruskin, "Of The Pathetic Fallacy"

October 12th, 2005 (Permalink)

Check it Out

Continuing the theme of causation/correlation confusion is Carl "The Numbers Guy" Bialik's latest column:

Local governments and a national public-service ad campaign headlined by Jamie Lee Curtis and Barbara Bush told parents to eat dinner with their children…based on a study that showed frequent family dining reduced the risk of substance abuse in kids by 50%. …[T]he claim that frequent dinners reduce risk by 50% doesn't account for age―a key failing. You might be unsurprised to learn that 17-year-olds are more likely to use drugs than 12-year-olds. Older teens are also the ones most likely to eat dinner away from their families.

This also connects with the previous entry on the ubiquity of appeals to celebrity. Celebrities such as Curtis and former first lady Bush are well-meaning, but they don't know what they're talking about.

I would have linked to this earlier, but the Wall Street Journal is now providing it to subscribers only, whereas the "Numbers Guy" columns used to be available to nonsubscribers. Fortunately, nonsubscribers can now read this column thanks to the Canadian Foundation for Drug Policy.

Source: Carl Bialik, "The Link Between Dinner and Drugs", The Numbers Guy, 10/7/2005

October 11th, 2005 (Permalink)

The Cult of Celebrity

October 10th, 2005 (Permalink)

Baggini's Back

After a lengthy hiatus, Julian Baggini has posted a new Bad Moves column. This one concerns the type of ad hominem fallacy in which people give a dismissive psychological explanation of their opponents' arguments. For example:

There is…The Lady Doth Protest Too Much manoeuvre―which states that the more someone denies something the more likely it is to be true. As every schoolboy knows, someone who makes a big point of denying he is gay, for example, is probably gay himself.

This is an insidious type of poisoning the well, since the more you try to defend yourself against the charge, the more evidence you give of your guilt. Another instance is saying to someone who is arguing with you: "You talk too much!" or "You just like to argue!" This may shut the other person up, but it probably won't change anyone's mind, nor should it.

Source: Julian Baggini, "Motivation Speculation", Bad Moves, 10/10/2005

October 1st, 2005 (Permalink)


The last few weeks of this weblog seem to have been dedicated to examples of causal fallacies. It wasn't intentional, though; it's just a matter of what has caught my attention. I've just started reading economist Steven Levitt's new book, Freakonomics, and came across the following passage on the same theme:

[J]ust because two things are correlated does not mean that one causes the other. A correlation simply means that a relationship exists between two factors―let's call them X and Y―but it tells you nothing about the direction of that relationship. It's possible that X causes Y; it's also possible that Y causes X; and it may be that X and Y are both being caused by some other factor, Z.

He left out one possibility, though: it might be the case that X and Y are not causally related at all, and the correlation is simply coincidental. The following examples are good:

Think about this correlation: cities with a lot of murders also tend to have a lot of police officers. Consider now the police/murder correlation in a pair of real cities. Denver and Washington, D.C., have about the same population―but Washington has nearly three times as many police as Denver, and it also has eight times the number of murders. Unless you have more information, however, it's hard to say what's causing what. Someone who didn't know better might contemplate these figures and conclude that it is all those extra police in Washington who are causing the extra murders. Such wayward thinking, which has a long history, generally provokes a wayward response. Consider the folktale of the czar who learned that the most disease-ridden province in his empire was also the province with the most doctors. His solution? He promptly ordered all the doctors shot dead.

Source: Steven D. Levitt & Stephen J. Dubner, Freakonomics (2005), p. 10.

The Scariest Solution (11/7): Thanks to everyone who was brave enough to submit a solution to the scariest puzzle. This was probably the most difficult puzzle to appear in the Fallacy Files, since it's based on Anselm's "ontological proof" of God's existence, an argument that has bamboozled some smart philosophers, and caused others to spill a lot of ink attempting to explain what's wrong with it.

Some puzzlers adapted a version of the philosopher Immanuel Kant's celebrated criticism of Anselm's argument to apply to the puzzle. Now, there's definitely something to this point―though it's hard to identify exactly what it is―but I want to pass over it, because the first part of the puzzle was intended simply to motivate talking about a "real Boojum". Moreover, even if one grants the Kantian point, there's still a puzzle in the subsequent argument.

The first premiss of that argument is the disjunction: either a real Boojum exists or it doesn't. What does it mean to say that a real X exists? Nothing more nor less than to say that an X exists! The "real" is redundant: it's like saying an existing X exists. So, all the first disjunct says is: "A Boojum exists".

The second disjunct, however, is ambiguous, as it could mean either of two things:

  1. The negation has wide scope (for an explanation of the concept of scope, see the entry for scope fallacy), that is, it negates the first disjunct: "it is not the case that a real Boojum exists". Since the "real" is redundant, this means the same as: "it is not the case that a Boojum exists". Notice that this is not self-contradictory; in fact, it's true!
  2. The negation has narrow scope: "An existing Boojum is non-existent". This is the only sense in which the second disjunct is self-contradictory.

If we take the second disjunct in the first sense, then the argument fails because the second disjunct is true, not false. In contrast, if we take the second disjunct in the second sense, then the argument commits a black-or-white fallacy, since these are not the only two possibilities: there is the third possibility that the Boojum does not exist!

So, by introducing the redundant word "real", a potential scope ambiguity is introduced when the proposition is negated. Thus, the solution to the puzzle is that there is a negation scope fallacy in the argument for the Boojum's existence.

Technical Appendix: As if that wasn't scary enough, here is a more precise analysis of the argument using logical symbolism. Let "Bx" mean that x is a Boojum, and let "$y(x = y)" mean that x exists. "A real Boojum exists" can then be symbolized: $x(Bx & $y(x = y)), which is equivalent to $x(Bx). The second disjunct has two possible interpretations:

  1. Wide scope: ~$x(Bx & $y(x = y)), which is equivalent to:
  2. Narrow scope: $x(Bx & ~$y(x = y)), which is self-contradictory.

Source: The scariest puzzle was suggested by a puzzle concerning unicorns in Raymond Smullyan's book What is the Name of this Book? (Prentice-Hall, 1978), pp. 204-205.

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