The Puzzle of the Dead Presidents
After your first success in helping the police crack a bank robbery (see the Resource below), they turn to you for help with another case: A gang of five men wearing masks of five former presidents have committed a series of armored car robberies.
During one of the crimes, the driver of the armored car was shot and killed by a robber. There are no witnesses to the shooting other than the five criminals involved. The gang was caught while still wearing their masks and have refused to reveal their identities, so the police know them only by the name of the president each impersonated. The police determine that each member of the gang had a specific job: the mastermind, the lookout, the wheelman, the safecracker, and the triggerman. Unfortunately, the police aren't able to figure out which role each "president" played. The police are especially interested in who killed the armored car driver, namely, the triggerman.
The members of the gang were questioned individually while attached to a lie detector, and each criminal made only two statements. Assuming that the lie detector is accurate, each criminal made one true statement and one false statement, but the operator of the detector couldn't tell which was which. As usual, the police are baffled.
Can you determine who the triggerman was?
Resource: The Puzzle of the Masked Men, 6/28/2009
Name that Fallacy!
Doctors commonly prescribe treatments, even when they know theyíre not effective, because in the face of assertive patients, demanding a specific treatment, many will choose an easy life. … Most sore throats are caused by viruses. Doctors usually avoid antibiotics…. Patients were randomly given either 10 days of antibiotics, or advice to watch and wait, or a delayed prescription, which they could use if things hadnít settled in a few days. Patients got better in each group at pretty much the same rate, as we would expect, and got back to work or school at the same pace. More interesting, though, is that many more patients given antibiotics came away with the view that antibiotics were effective (87% vs 55%), and many more intended going back to the doctor if they had another sore throat (79% vs 54%). So while prescribing antibiotics had marginal benefits on the sore throat at best, it hugely enhanced belief in antibiotics, and intention to go back to the GP. Then, one year later, the researchers returned to the same patients for a follow up study: and they found, looking back, that patients who had been prescribed antibiotics originally were 39% more likely to go back to the GP when they had a sore throat.
Source: Ben Goldacre, "Doing Nothing", Bad Science, 3/19/2010
I've added a new contextomy to the "Familiar Contextomies" page.
Resource: Familiar Contextomies: James Madison, Fallacy Watch
"Intensional fallacy" is the name of a new fallacy, or perhaps a new name for an old fallacy. I came across the fallacy under this name for the first time in a long, fascinating book review by philosophers Ned Block and Philip Kitcher. I'll let them introduce the book and its authors:
In their controversial new book, What Darwin Got Wrong, Jerry Fodor and Massimo Piattelli-Palmarini set out to dismantle [the Darwinian] framework. They argue that standard evolutionary thinking―what they call Darwinism―is guilty of a basic logical error, not a mistake in biology but an "intensional fallacy." That fallacy, they say, undermines the entire enterprise.
I've never heard the term "intensional fallacy" before. That fount of all human wisdom, Wikipedia, has an entry for it, but the description is of the Masked Man Fallacy. More importantly, the Internet Encyclopedia of Philosophy has a brief entry in its article on fallacies, but it also describes the Masked Man Fallacy.
"Intensional fallacy" is an unfortunate name as it is pronounced and spelled almost exactly the same as the more familiar "intentional fallacy". The intentional―with a "t"―fallacy is not a logical fallacy, but is the supposed mistake of judging a work of art based on the intentions of the artist. Here's Block and Kitcher's description of the intensional―with an "s"―fallacy:
[Fodor and Piattelli-Palmarini] allege that Darwinism is guilty of an "intensional fallacy." … There are some sentences in which, if you substitute one name for another, and both are names for the same thing or person, you always go from a true sentence to a true sentence, or from a false sentence to a false sentence. "Madonna" and "Louise Ciccone" name the same person. The sentence "Madonna is a woman" is true. If you substitute "Louise Ciccone" for "Madonna," you obtain the sentence "Louise Ciccone is a woman," which is also true. Not all sentences work this way. Our world is full of people who do not know that Madonna is Louise Ciccone. If Bert is one of these people, then the sentence "Bert believes that Madonna is a star" may well be true, even though "Bert believes that Louise Ciccone is a star" is false. … There are some contexts, such as "is a woman," in which substitution of names that name the same entity preserves truth (or falsehood); these contexts are said to be extensional. Other contexts, such as "Bert thinks that is a star," allow for changes from truth to falsehood under similar substitutions; these are intensional.
This sounds like the beginning of a description of the Masked Man Fallacy, but never quite says what the fallacy is. In the Masked Man, one makes the mistake of substituting identicals into an intensional context, as if it were an extensional one. Using Block and Kitcher's example, to conclude that Bert believes that Louise Ciccone is a star from the fact that he believes Madonna is a star would be to commit the Masked Man Fallacy―or masked woman, in this case.
I'm not sure from the review what Fodor/Piattelli-Palmarini/Block/Kitcher mean by "intensional fallacy". It appears not to be the same as the Masked Man Fallacy, though clearly related. It may be either a more general fallacy, or perhaps the sibling fallacy of substituting co-extensional predicates within an intensional context. Perhaps the book is clearer.
I expect that I'll have more to say about the review soon, so read the whole thing and come back later. Be warned that it's a fairly philosophically sophisticated review, though not technical in terms of logic, and there's some mildly sophisticated discussion of evolution.
Update (3/7/2010): In reading the following, keep in mind that I haven't read the book yet, and so am relying upon the accuracy of Block and Kitcher (B&K)'s description of its contents.
The notion of "intension", and the related one of "extension", that seems most relevant here applies to properties such as "having a heart". The extension of "having a heart" is simply the class of everything that has a heart, whereas its intension is closer to what we think of as its meaning, namely, "having an organ that pumps blood" or whatever. Two properties that have different intensions may have the same extension, in which case they are called "co-extensive". For instance, "having a heart" and "having a kidney" are co-extensive.
Given that, I still can't figure out what the "intensional fallacy" is supposed to be. According to B&K, the problem is that evolution cannot make fine-grained intensional distinctions between properties, but only coarse extensional ones. That is, assuming that you have two intensionally different but co-extensive properties, there's no fact of the matter as to which one evolution selects for or against.
Natural selection and selective breeding differ in that the former can only differentiate extensionally distinct traits, whereas the latter can differentiate between intensionally distinct but co-extensive traits. For instance, a breeder of moths might select for a trait such as black wings, but it could be that black wings are co-extensive with nocturnal inactivity. Nonetheless, we can still say which trait is selected for, because the breeder can tell us. In contrast, we can't ask nature which it was selecting, so there is no fact of the matter whether evolution selected for black wings or nocturnal inactivity.
Where's the fallacy here? I don't even see the argument that's supposedly fallacious, which makes me wonder whether F&P are using the term "fallacy" in some unusual sense.
Keeping in mind that I'm no expert on evolution―and I haven't even read the book!―this strikes me as an interesting logical fact about natural selection, but one that doesn't undermine "Darwinism". In most cases where there are two intensionally-distinct but co-extensive traits―such as black wings and nocturnal inactivity―there are probably ways of telling which is selected for. For instance, we can take some nocturnally-active moths that have light-colored wings, paint their wings black, and see whether they are as successful at reproducing as the black-winged nocturnally-inactive moths. In other words, we would make the traits no longer co-extensive and see what happens.
In contrast, if two traits are linked in a way that cannot be causally teased apart, then it seems incorrect to say that evolution selected for one but not the other: it's either both or neither. But so what? I don't see how that undermines natural selection. It might undermine some hastily-drawn conclusions about what traits are naturally selected, but it wouldn't undermine the fact that some set of traits is, in fact, selected. It may complicate the story, but not show that it's fiction.
I may have more to say about this review in the future, but I'm afraid that I'm going to have to read the book itself if I want to get a better handle on the elusive "intensional fallacy".
Update (3/11/2010): There's an article by Fodor from a few years ago in which he foreshadowed the argument of the book, and which is available online (see the Source below). I'm not sure yet whether it's much help in understanding the/an intensional fallacy, but he claims that a more familiar fallacy is at work in the notion of "natural selection":
Itís a commonplace that Darwin constructed the theory of natural selection with an eye to what breeders do when they choose which creatures to encourage to reproduce. … Itís true, of course, that breeding, like evolution, can alter phenotypes over time, with consequent effects on phylogenetic relations. But, on the face of it, the mechanisms by which breeding and evolution operate could hardly be more different. … The present worry is that the explication of natural selection by appeal to selective breeding is seriously misleading, and that it thoroughly misled Darwin. Because breeders have minds, thereís a fact of the matter about what traits they breed for; if you want to know, just ask them. Natural selection, by contrast, is mindless; it acts without malice aforethought. That strains the analogy between natural selection and breeding, perhaps to the breaking point.
In other words, Fodor is claiming that natural selection involves a weak analogy with selective breeding. He's no doubt correct that the analogy of what happens in evolution with breeding breaks down at the notion of "selection". Natural "selection" is, indeed, a different process than artificial selection.
However, Fodor goes on to claim that the difference between breeding and evolution is that there's a fact of the matter of what traits a breeder selects for, but no fact of the matter as to what traits evolution "selects" for, because breeders have intentions but nature does not. This is where the/an intensional fallacy seems to enter in: that's "intensional" with an "s", though it seems that it could just as easily be "intentional" with a "t", since the difference between breeding and evolution is that the former has but the latter lacks intentions.
Confusingly enough, intentional contexts are also intensional contexts, that is, expressions of intent are propositional attitudes that create contexts into which you cannot substitute identicals (see the entry for the Masked Man Fallacy for more on "propositional attitudes"). For instance, Oedipus intended to marry Jocasta; Jocasta was Oedipus' mother; however, Oedipus did not intend to marry his mother.
As a result, the kind of selecting that breeders do is more fine-grained than what nature can do in "natural selection", that is, when two traits are co-extensive a breeder can intend to select for one of them, whereas nature either selects both or neither. However, I don't think that this means that there is never a fact of the matter about what traits are selected by evolution, which seems to be what Fodor is claiming―though I'm not too sure what he is claiming. I find his example perplexing:
The crucial test is whether oneís pet theory can distinguish between selection for trait A and selection for trait B when A and B are coextensive: were polar bears selected for being white or for matching their environment?
It seems obvious to me that nature selected for both, because in the environment of polar bears being white is matching the environment. Of course, if polar bears had been bred by a human breeder, that breeder might have bred for whiteness and not for matching the environment. But I still don't see why it's a problem that natural selection can't make that distinction. I wonder whether Fodor isn't falling victim to the same mistake that he accused Darwin of making, namely, taking the analogy between evolution and selective breeding too far.
Source: Jerry Fodor, "Why Pigs Donít Have Wings", London Review of Books, 10/18/2007
Untie the Nots, Part 4
In his famous essay, "Politics and the English Language", George Orwell gives what he calls "four specimens of the English language as it is now habitually written" as illustrations of "the mental vices from which we now suffer". The following sentence, by Harold Laski, is the first of these four:
I am not, indeed, sure whether it is not true to say that the Milton who once seemed not unlike a seventeenth-century Shelley had not become, out of an experience ever more bitter in each year, more alien to the founder of that Jesuit sect which nothing could induce him to tolerate.
This is practically unfathomable. The main problem is the piling up of multiple negatives, since the human mind appears to be limited in its ability to understand negation to about two or three per thought. Fortunately, double negatives cancel out, and formal logic provides techniques for reducing the number of negations.
What was Laski trying to say? In the essay, Orwell never explains the sentence, writing only that:
Professor Laski…uses five negatives in 53 words. One of these is superfluous, making nonsense of the whole passage, and in addition there is the slip alien for akin, making further nonsense, and several avoidable pieces of clumsiness which increase the general vagueness.
I'm unsure how Orwell came to a count of five negatives in the sentence, as I count at least six. There are only four "not"s, so I assume that he was counting either the "un" in "unlike" or "nothing" as an additional negation, but both should be counted.
I don't know how Orwell knew that "alien" was supposed to be "akin", but let's assume that he was right. Can you untie the "not"s in Laski's sentence, producing a translation that is understandable? Or was Orwell right that one of the negatives is "superfluous", rendering the sentence nonsensical?
Source: George Orwell, "Politics and the English Language", The New Republic, 6/17/1946.
If the lie detector is accurate, then each of the five criminals said one truth and one falsehood. So, let's start by assuming that Kennedy's first statement is true and his second is false, and see what happens.
If Kennedy's second statement is false, then either Reagan or Johnson is the lookout. However, given that his first statement is true, Johnson is the mastermind. Thus, Reagan must be the lookout. Since Johnson is the mastermind, it follows that Kennedy isn't, which means that Nixon's second statement is true. Hence, Nixon's first statement is false, but that means that Nixon is the lookout. This contradicts the previous conclusion that Reagan is the lookout. Therefore, our original assumption must be wrong, which means that Kennedy's first statement is false and his second true.
Since Kennedy's first statement is false, Johnson is not the mastermind, and since his second is true, neither Reagan nor Johnson is the lookout. Because Kennedy's second statement is true, Ford's second statement is false, which means that his first is true. Since Ford's first statement is true, Reagan's second statement is false, which means that his first is true. So, Nixon is not the lookout, and his second statement is false, meaning that Kennedy is the mastermind. Since Nixon is not the lookout, it follows that Johnson is not the wheelman, by Modus Tollens from Ford's first statement. So far, we have been able to conclude that none of the following is the lookout: Kennedy (since he's the mastermind), Reagan, Johnson, and Nixon; this leaves only Ford. Thus, Johnson's second statement is true and his first is false. The only way for Johnson's first statement to be false is for neither Nixon nor Johnson to be the safecracker. So we now know that Johnson is not the mastermind (Kennedy), the lookout (Ford), the wheelman, or the safecracker. Therefore, Johnson is the triggerman.
By continuing this reasoning, you can determine the remaining two positions in the gang, but I'll leave this as an exercise for the reader.
Solution to Untie the Nots, Part 4: First of all, notice that the overall structure of Laski's sentence is: "I am not, indeed, sure whether it is not true to say that p", where p is the proposition that Laski is not, indeed, sure whether it is not true to say. "Indeed" is an assuring term, similar to "of course", which doesn't change the meaning of the sentence, so let's drop it in the interest of simplicity.
"I am not sure whether it is not true to say that p" means the same as "I am not sure whether p", since it is true to say that p if and only if p is the case. So, we can eliminate one of the negations in Laski's statement, producing the following:
I am not sure whether the Milton who once seemed not unlike a seventeenth-century Shelley had not become, out of an experience ever more bitter in each year, more akin to the founder of that Jesuit sect which nothing could induce him to tolerate.
Logically, this is a complex proposition made up of the following four simple propositions about Milton:
We can eliminate another two negations in the first proposition by cancelling out "not un-". Usually, the "not un-" phrase is not a true double negation; for instance, to say that someone is "not unattractive" is not the same as saying that they are "attractive". This is because it's possible to be neither attractive nor unattractive. As a result, "F" and "not un-F" are usually contrary rather than contradictory. However, in this case it seems that Laski must have meant that Milton once seemed like a 17th-century Shelley, rather than that he was neither like nor unlike him: otherwise, why mention Shelley at all?
Logically, the four propositions are related to each other as conjuncts, so that the original sentence is a big conjunction. This can be seen by a simple example; consider the sentence: "Tom, who is Dick's brother, is best friends with Harry". Logically, this says that Tom is Dick's brother and best friends with Harry; grammatically, the first conjunct is a nonrestrictive clause.
One proviso is that the original passage claims that sentence 2 is related to sentence 4 causally: "Milton had become, out of an experience ever more bitter in each year, more akin to the founder of that Jesuit sect". That is, Laski was claiming that it was due to Milton's bitter experience that he became more akin to the intolerable founder. Therefore, in the translation, sentences 2 and 4 need to be related by causal language.
One way to diminish the number of negations in a sentence is to split it up into smaller sentences. Since a conjunction is logically equivalent to the set of its conjuncts, we can rewrite Laski's original sentence as a paragraph of four separate sentences:
Milton once seemed like a 17th-century Shelley. Nothing could induce him to tolerate the founder of that Jesuit sect. However, his experience was ever more bitter each year. I am not sure whether, as a result, he had not become more akin to that founder.
Here, there are only three negations in the entire paragraph, two of which are in the last sentence. It might be possible to reduce the number of negations in the final sentence, but only at the risk of changing its meaning, and it seems to be easily understandable.
There's still considerable residual obscurity in this paragraph, which is probably the result of being torn from its context. For instance, what Jesuit sect is Laski referring to, and who is its founder that Milton couldn't stand but became more like each year? Unfortunately, Orwell gave an insufficient citation for the source of the quote, and I haven't been able to find it―if anyone can find a copy of the essay that contains the original quote, please let me know. Given this lack of context, I can't tell whether the rewritten passage expresses what Laski intended, but it's not nonsense.