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Letters from alumni about Thinking about thinking, cover story January 29.


April 5, 2003

I have read with great interest, if not full comprehension, the ongoing discussion of the logical problem presented by the waiter who said, "You can have the fish or else not both the soup and salad." My concern is that one must also recognize the difference of these words when spoken, as here, or written.

If the customer cannot understand the spoken sentence, a likelihood in this case, he will, of course, ask what its speaker means, with immediate clarification — feedback. Put if the sentence were written (on a menu?) this is not possible. Thus its author in a process called (self-) editing must make take care that his meaning can be easily understood by his reader. If this reader is a trained logician (unlikely), the quoted form will suit; otherwise it must be rephrased so as to be understood by the reader to whom its author addresses it because immediate feedback is not possible.

But suppose the writer's purpose is obfuscation. Then a writer will offer a sentence just like the obscure one, though it requires trained skill to do this. Perhaps President Bush has developed this skill, but I rather think that his obfuscations come naturally.

William B. Hunter '37
Greensboro, N.C.

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April 9, 2003

In Professor Johnson-Laird's response to William H. Weigel (March 26, 2003), he considers the following problem: "You get a grade of zero if you miss a class without an excuse, or else you get the mean grade of your other classes if you have an excuse. You missed a class without an excuse. What follows?"

It appears to me that the problem, as expressed by Professor Johnson-Laird is ambiguous because there are at least two nonequivalent interpretations of the first sentence in the problem. The first interpretation, which has been selected by the professor, relies upon the fact that the words "or else" are often used to denote what is referred to in logic as an "exclusive or," meaning that one but not both of the following two statements must be true: "You get a grade of zero if you miss a class without an excuse," or "You get the mean grade of your other classes if you have an excuse." The professor states that this is consistent with a case where a student has no excuse but still doesn't get a zero grade, and in fact it is, because in such a scenario the first statement would be false and the second might very well be true.

However in the English language, when "or else" is preceded by a statement of the form "If A, then B," it may also be interpreted (and I would suggest that it is usually interpreted) to mean "but otherwise." In this interpretation of the first sentence, it could be rewritten as, "If you miss a class without an excuse, then you get a grade of zero, but otherwise (i.e., if you have an excuse) you get the mean grade of your other classes." This differs from the original statement only in that (1) "or else" has been replaced by "but otherwise," (2) the "if" clauses have been moved in front of their consequences, and (3) parentheses and the abbreviation "i.e." have been used to indicate that the second "if" clause is actually redundant. Stated in this way, I think just about any reader would say that a person who misses a class without an excuse will absolutely get a zero grade, which is exactly what "nearly everyone, experts and nonexperts alike, conclude," according to the
professor.

There appears to be a rule in the English language under which "or else" preceded by "if A then B" means "but otherwise", while "or else" not preceded by such a conditional means "exclusive or". The strange word order ("B if A" rather than "if A then B") and the redundant second condition makes it less certain which interpretation of "or else" is correct in this case, so equally rational people might very well differ in the interpretation that they choose, but in any event the professor's interpretation is by his own admission only shared by a small minority. Thus, to the extent that usage determines the interpretation where logic is not able to do so, one would have to conclude that it is in fact the professor who has something to learn (and marvel at) in this particular case.

Zachariah Cobrinik ’81
New York, N.Y.

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March 16, 2003

I appreciated the example of thinking outside the box given by Canice Lawler *91 in response to the footbridge dilemma described in your excellent article "Thinking about Thinking" (January 29). I'd like to offer another example.

Professor Kahneman applies Occam's Razor to the system 1/system 2 distinction: "[It] is going to be tied to the brain, or it will vanish." I suggest that the "or" he uses is an inclusive or, not an exclusive or.

As the article shows, there is increasing evidence of different types of thinking: automatic and reflective (or believer and skeptic, for the religiously inclined). It seems likely that these two two types will be "tied to the brain" but that wouldn't preclude the vanishing of the system 1/system 2 theory. What if there are more than two types of thinking?

If we divide all thinking into "images" versus "words" (a pop version of the automatic and reflective types), we're confronted with the fact that neither explains what goes on when a child is learning an imitative task such as tying shoes. Few, if any, children learn to tie from hearing a verbal explanation, and few would learn from seeing a sequence of snapshots.

Martin Schell '74
Klaten, Central Java  

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March 15, 2003

Billy Goodman's article about thinking in the January 29 issue presents several logic problems, and the analysis of the waiter’s offer (page 31, answer 1) seems to fail on two counts. First, if the waiter says “you may have the fish or else not the meat”, the customer has only two choices to state: “fish” or “not meat.” If he says “not meat,” the waiter can bring him anything which meets that criterion. It might be lobster thermidor, sautéed hummingbird tongues, a peanut butter sandwich, or nothing at all. The customer doesn’t get to specify how he wishes the “not meat” order to be fulfilled.

Secondly, most of us who use the English language understand that not “meat and fish” is a subset of not “meat”, in just the same way that “feminist bank teller” is a subset of “bank teller” on page 29. If your cardiologist tells you to “eat not meat”, it is clear that he means that surf and turf is a no-no. When the waiter says you may have “not meat” it may very well be that the restaurant has run out of meat that day or is a Kosher dairy establishment. In any event the diner cannot assume that an order of “fish and meat” will or can be fulfilled.

John M. Matsen ‘57
Annandale, N.J.

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March 12, 2003

Regarding Tom Brueckner’s letter anent the card hand problem, did he miss something, or did the originator miss something, or did I? Tony’s letter doesn’t state the size of the hand and, unless limited to just a few cards, one might assume the “hand” could be up to 48 (full pinochle deck) or even 52 (full bridge deck). If so, then the problem hand could obviously contain not only four aces but ALSO 4 Kings, 4 Queens, etc. Was this some sort of “trick” question, or has my reasoning been “trumped?”

Regarding the bridge-over-train problem there appears to be NO “moral” answer! If pushing the other person off to stop the train, murdering him/her to save the others is immoral, so is the alternative of committing suicide by jumping off oneself. What does appear to be moral is the option of doing nothing. Some would disagree with this, I’m sure, but one is not under any moral obligation to sacrifice one’s self, or anyone, to save the others, is one?

Yet another alternative would be to procrastinate; weighing alternative choices such as “Should I push him/her off?” or “The other person’s a woman, so perhaps I should sacrifice myself (or vice versa)” or “This other guy is so big, if I try to shove him off he'll more likely toss me off ... !” etc., in the process failing to act at all, so that by the time one has decided the train has already sped past and killed the others.

The real lesson from this is that while on one hand the Philosopher can comfortably take all the time desired to arrive at whatever conclusion, no matter how moral or immoral, no matter whether right or wrong in fact, in "real life" the Sergeant in battle, the top executive of some firm, the cop on the beat, the Secretary of Defense facing terrorist emergency, must often make quick decisions despite limited information due to the limits of time and urgency of the situation. The true Philosophy to be derived is that real men will make such decisions and act in time, in good faith, and manfully accept the consequences, "right or wrong". And presuming "good faith" underlay the decision, other good men, including Philosophers, should empathize with whomever wound up in error despite acting in good faith and not just feed him/her to the fish.

Much as I value the philosophic side of a Princeton Education, it becomes meaningless unless the student is also taught this principle, that one must often have the courage to make a timely decision based on less than complete information and manfully accept the responsibility and consequences. Short of that, the education is just a waste, a philosophical exercise, isn't it?! Sometimes the need for decisive action outweighs taking time to moralize. Ask your nearest USMC Officer or NCO.

John J. Auld Jr. ’50
Chesterfield, Mo.

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February 18, 2003

Re: The footbridge dilemma

Thank you for making the point that the moral choice of jumping off the bridge rather than pushing a stranger was omitted from the problem and for the reassurance that comes from knowing that I'm not the only one to notice the omission.

Stanley Kalemaris '64
Melville, NY

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February 18, 2003

I like "brain teasers". (Certainly in part because they're entertaining, but also because as a lawyer I spend a lot of time trying to render life's vicissitudes into "precise" words, and vice versa.) However, I must say that in reading Mr. Goodman's account of the respective research being done by Professors Cohen, Kahneman, and Johnson-Laird, among others, I was uneasy with the conclusions they drew from their use of the "thought problems" described in the article.

My overall concern is that their work may say more in a given situation about how the brain uses or relies on language than about how (or when) it uses or relies on logic. The fact is that many if not all of these "thought problems" — for example, the trolley/footbridge dilemma, the "Linda problem," and the ball/bat arithmetic — are, like any good brain teaser, designed to confuse or confound. I have to wonder then if employing them and thereby tricking some test subjects into thinking "the wrong way" proves much more than that we humans can sometimes "fool" one another. (Indeed, who's fooling whom in some of these cases seems an apt question.)

Consider the "Linda problem." Its aim, as I understand the article, is to evaluate whether a given subject "sees" that one of the eight possibilities — Linda as a "bank teller" — includes within it one of the other possibilities — Linda as a "bank teller and feminist" —  which would mean that the latter cannot, as a matter of logic, be more likely than the former. But if in the test, the eight possibilities are simply, and equally, listed 1-8, then it would be quite natural to interpret the one possibility of "bank teller" to mean "bank teller [who is not a feminist]." That does not qualify as a "snap judgment." So all that "Linda" would really prove in this event is the incommensurability of the tester and test subject, with each logically seeing the other as having committed a category-mistake. (No wonder Professor Kahneman wonders that so many test subjects are "blind"!)

Now imagine going to a store to buy a ball and bat. You pay 10 cents for the ball and then ask the price of the bat. The clerk says, "A dollar more." Would you actually say at that point "What do you mean? More than what? The ball's cost? The 10 cents already paid? What?!" Of course not. You'd simply hand over another dollar because you'd know you were being asked for only a dollar. That's just how language in real life gets used, and more to the point how a person's brain is "trained" (virtually hardwired?) to reason in like situations. So what is really happening when a test subject is presented with the same scenario as a purely abstract proposition and misses its "obvious" (and quite opposite) logic? Is it really the case the person is being careless and "not checking the math"? Or does the phrase "a dollar more . . ." trigger the person's brain to use the "obvious" (and quite common) logic of the real life situation? Here again, the problem's wording may interfere with "getting a good read" of the test subject's rationality.

The trolley/footbridge dilemma similarly may demonstrate the dichotomy of logic in abstraction and real life, but in a more complex fashion. What is most curious about the footbridge dilemma (to me) is not simply that it tends to evoke an emotional response, but that it should do so despite being an abstraction. It may be, then, that the brain cannot help but take such problems seriously, close or at least closer to how it would if the dilemma were real. But if that is what is happening, then all bets are off — both on whether the brain "in the back of its mind" is accepting of the absurd lack of contingency in the two stated problems (hitting the switch will lead to the one person being killed, pushing the person onto the tracks will stop the train — yeah, right, that must be true) and on what the emotional response itself in the second problem is signifying.

Surely any kind of "principled" conflict can generate an emotional response — for example, the "right" in preventing harm to others versus the "wrong" in killing another. (And what if the test subject happens to be a "large" person too? His or her brain could essentially overrule the problem as posed and, through pure logic, infer an alternative, and "more troubling," solution, which would then only further complicate the dilemma.) Moreover, must it be true that a subject is "fighting" the emotional reaction that occurs in order to "think straight"? Perhaps the emotional response actually works to reinforce rationality, serving as a kind of alarm when either "something important" or, more important, "something sacred" is at stake and so "attention must be paid" lest a "bad" decision gets made.

In sum, it strikes me that this research, to be credible, must further account for how language (and experience) can "inform" logic (and conceptualization) in the brain. And the use of "brain teasers" to understand this sort of convergence appears to present as "tricky" a proposition for the testers as for the test subjects.

Bruce K. Adler ’79
Budd Lake, N.J.

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February 8, 2003

The English language does not express logical distinctions as clearly and unambiguously as one might hope. It seems to me that a subject's failure to come up with the expected answers for a couple of the examples cited in "Thinking about Thinking" (January 29 feature) could indicate, not faulty logic, but different understanding of the meanings of the questions.

Regarding the statement, "You can have the fish, or else not both the soup and the salad," to get the "right" answer one must interpret "You can have the fish" to mean that you can have fish and perhaps some other items as well, rather than that fish would constitute your entire meal. Another ambiguity is that, if A is true, then "A or else B" might (exclusive "or") or might not (inclusive "or") imply that B is necessarily false.

The "Linda problem" is a misleading trick question. Many subjects, expecting it to be an inquiry about Linda, rather than an exercise in pure logic, will not take the option "bank teller" literally but will treat it as not overlapping with "actively feminist bank teller". Their technically incorrect ranking of the latter above the former reflects their correct belief that, if Linda were a bank teller, then she more likely than not would be an actively feminist one.

All this is well known to pollsters, who must expend considerable effort in trying to phrase questions so that they will not be misconstrued.

John G. Fletcher *59
Livermore, Calif.

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February 3, 2003

In "Thinking about Thinking" (PAW January 29), Johnson-Laird's logic problem, which trips up smart people, is described as follows: "In a hand of cards, only one of the following three assertions is true: There is a king or an ace or both in the hand. There is a queen or an ace or both in the hand. There is a jack or a ten or both in the hand. Is it possible that the hand contains an ace?" Smart people answer "Yes", and they are wrong. But the explanation for why the answer is "No" that PAW provides is at best unilluminating. Readers with middling logical abilities (such as myself) will be misdirected by the explanation: "If the first assertion is true, the second and third must be false — so there is no ace. The same reasoning applies if the second assertion is true." The proper explanation is different. First off, if the first assertion is true, it does not follow that the second is false. Indeed, if the first is true in virtue of there being an ace in the hand, it follows that the second is true. That's the problem for the "Yes" answer. If the ace possibility is realized by the truth of the first assertion, then this makes the second true as well, contradicting the initial assumption that only one of the three is true. The same reasoning applies to the supposition that only the second assertion is true in virtue of the hand containing an ace. That supposition leads, in the same way, to contradiction.

Tony Brueckner '74
Goleta, Calif.

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February 2, 2003

This is a response to Mr. Weigel and his worries about the waiter's words. He says: You can have the fish, or else not both the soup and the salad. It follows, surprisingly, that you can have all three dishes. The source of the surprise is that we normally think about what is true but not about what is false.

How do individuals understand "or else"?

Given the simple assertion: "He ate the fish or else the meat", most people list two possibilities:
Ate fish
Ate meat

So, you are right that the meaning of "or else" does not correspond to its inclusive meaning in logic, which is best paraphrased as: "fish or meat, or both". In other words, people treat an assertion of the form: A or else B, to mean that in one possibility A is true (and B is false), and in the other possibility B is true (and A is false). Given a simple assertion based on "not both", such as: "You ate not both the soup and the meat", individuals allow that there are several possibilities, but that the assertion does rule out the case in which you ate both the soup and the meat. We can put these two interpretations together to deal with what the waiter said: You can have the fish, or else not both the soup and the salad. "Or else" means that there are two cases to consider. In one, it is true that you can have the fish, and it is false that you can't have both the soup and the salad. If it's false that you can't have both the soup and the salad, then it is true that you can have both of them. Hence, in this first case, you can indeed have the fish, the soup, and the salad. The second case allows for various other meals, which need not detain us here. I hope that this account has convinced you that the analysis in PAW was correct. You worry about the passage giving the answers to the problems (on p. 31).

It argues that the phrase: You could have the fish or else not the meat, allows that you could have the fish and the meat. That claim is true. "Fish or else not meat" is compatible with two possibilities: In the first, you have fish and meat; in the second, you don't have fish and you don't have meat. But, as you say, it would not be permitted for the diner to have fish and to have not-meat. So, here, again there is no disagreement between us. Your final claim is that what is going on in these examples is, not a failure to apply logic, but a failure of language to be sufficiently precise, and that people are likely to recognize "the confusion and potential logical inconsistency embedded in the waiter's words." You may be right that people recognize the confusion in the waiter's words. In many studies, however, we have found that people slip into error with the greatest of confidence.

Consider problems of the following sort:
You get a grade of zero if you miss a class without an excuse, or else you get the mean grade of your other classes if you have an excuse. You missed a class without an excuse. What follows?

Nearly everyone, experts and nonexperts alike, concludes that you get a grade of zero. And they are highly confident that they are right. They have succumbed to an illusion.

"Or else" means that one of the assertions in the first sentence is false (and one of them is true). Suppose that it is false that you get a grade of zero if you miss a class without an excuse. In that case, even though you missed a class without an excuse, you need not get a grade of zero. Our propensity to neglect falsity here leads us to treat "or else" as though it meant "and." The moral of these studies is that when we think, we envisage possibilities and what is true in those possibilities. And so we get into trouble when it is crucial to think about what is false.

Phil Johnson-Laird
Stuart Professor of Psychology
Princeton University
Princeton, N.J.

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January 31, 2003

This is a comment on the article "Thinking about Thinking" (January 29) in which there is reference to the "gambler's fallacy" as being "statistically nalve." There simply is no basis for this statement, statistical or otherwise. It does raise an interesting question, however, about possible causal relationships between supposedly independent statistical events, such as the toss of a coin or even the decay of radioactive atoms.

Here is a simple experiment that anyone can conduct: Toss a coin in any manner and record the sequence of heads and tails — no matter the time period or the manner of tossing. Do this for a fairly large number of tosses, say 100, and record the sequence of results. Interestingly, in the course of 100 tosses, you will likely record sequences of heads or tails that run 7 or 8 and even higher.

To calculate the probability of such a sequence under probability theory, it is assumed that the probability of a head or a tail is 1/2, and the probability of a sequence is the product of the probabilities of the separate events, considering them as independent. The probability of a run of 7 heads, for example, would be 1/2 to the 7th power or 1/128, which is a small probability but still within the realm of imagination.

Of more interest is to graph the trends in these tosses by assuming that the result of each toss represents a positive or negative displacement on the graph, starting with the first toss. The results are recorded as a cumulative sequence. The graphs are fascinating to examine, because they show definite trends in the sequences of heads or tails. The trends often prevail for many tosses and the number of heads and tails may not even out for the duration of the experiment. The trends are what gamblers refer to as runs of luck, and the question also arises as to whether the gambler can exert any influence over the results.

All of this brings up the question of causality and willpower as entering into statistical outcomes. This question is discussed in the accompanying article, which you might want to consider posting on the Internet for whatever value it might have for stimulating the thinking of the Princeton community.

Thomas V. Gillman '49
Eugene, Ore.

Further reading: Thomas Gillman's essay "Fallacy of the Assumption of Statistical Independence Between Successive Indeterminant Events".

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January 31, 2003

The January 29 cover story, Thinking about Thinking, poses a pair of philosophical questions, the "trolley dilemma" and the "footbridge dilemma," and suggests that they are morally equivalent. In the first, the reader must imagine that she can save five lives from a runaway trolley by hitting a switch that will turn the trolley on a side track, thereby killing one other unfortunate person. In the second, the reader imagines she is standing over the trolley track on a footbridge next to "a large stranger." With a timely push, she can shove the stranger onto the track, so that his body will prevent the train from reaching the other five.  People who have been posed these dilemmas are said to respond "yes" to the first, and "no" to the second, and philosophers are said to be puzzled as to why the responses should be different.

Perhaps the academic study of philosophy and morality have changed since my undergraduate days at Georgetown University. However, I suspect that morality has not changed so dramatically in people's minds. In the second problem, the morally correct answer is for the reader to jump onto the track herself. Pushing another person would indeed be murder. This option would surely occur to many hearers of the situation, and its absence in the formal problem shows that moral philosophers will forever be handicapped in their research if they negate the presence of love and self-sacrifice in the world. In contrast, it is clear in the first problem that you cannot offer your own life for that of the poor unsuspecting victims on the side track. No wonder those few people who agreed to murder in the second quesiton took a long time to think about it.

Canice Lawler *91
North Potomac, Md.

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January 29, 2003

"Thinking About Thinking" is a fascinating article, and reports some potentially significant work about what parts of the brain do what sorts of thinking. The (seemingly) purely emotional difference between the "tram" and "footbridge" problem is striking evidence of a physical explanation for an emotionally understandable difference in reaction to logically equivalent situations.

But the article also illustrates a serious problem about many such "logical" problems — the role of ambiguity in the question. As a litigator and sometime student of public opinion polls on difficult public policy issues, I have often been struck by the difficulty of asking truly clear questions about complicated issues, if the objective is genuinely to get meaningful answers. The "Linda" question, in particular, illustrates less about logic or the brain, than about the problem of imprecision in questions if you want meaningful answers. 

The "Linda problem" concerns a bright, but also idealistic woman. Her stated career choices are "that she is active in the feminist movement, that she is a bank teller, or that she is a bank teller active in the feminist movement." (emphasis, as the lawyers say, supplied). We are asked to pick Linda's most likely choice. To most people, the proffered set of choices implies that the first two choices are mutually exclusive, i.e., that Linda chooses to be only a bank clerk with no broader cause that she seeks to advance or that she decides to be only an impractical activist with no visible means of support. (I leave it to others to judge that if "Linda" were "Larry" she would be stuck with the relatively low-status "real" job of "bank teller" and not, say, "investment banker." Moreover, "activist" is, for most actual activists, not a full-time alternative to a job, but a way of using leisure time — a further source of amb! iguity in the question. )

The supposed "logical" answer is "teller." The basis for this allegedly logically correct conclusion is that Linda could be a teller and at the same time also be an activist and therefore, "logically" it must be true that "teller" is more likely than "teller plus activist" since "teller" includes not only whatever chance there is of Linda being only a teller but also the combination of teller and activist. But this "correct" answer is not necessarily right even as matter of logic. It is equally true that the answer "activist" includes both the combined roles of teller from 9-5 and activist the rest of her waking hours, but also whatever chanced there is of Linda being just an activist. Nothing in the question tells us whether there are more people like "Linda" who, if they don't manage to combine practicality and ideals, chose only practicality — a necessary premise of the proposition that "only teller plus both" is mathematically m! ore likely than "only activist plus both." 

The real point is that if one understands the "pure" options ("teller" and "activist") as alternatives that excludebeing both gainfully employed and socially conscious, they are exposed as highly unlikely alternatives, compared to the "balanced" choice of working at a bank and seeking justice in her "spare" time. And most normal, and highly logical, people would so understand the problem. 

Walter B. Slocombe '63
Washington, D.C.

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January 29, 2003

Thank you for the cover story in the January 29 issue, which I found quite interesting. I want, however, to question the article's analysis of the "restaurant" puzzle and the "Linda" problem.

Regarding the restaurant puzzle, I think there is an internal inconsistency in your explanation. Your answer on page 31 states, "If you were told that you could have 'the fish or else the meat,' you could have one or the other, not both" (emphasis added). This seems to me a correct interpretation of the words "or else," particularly in a restaurant setting. (I note that "or else" does not have the same meaning as "or" in pure logic, because — at least in that realm — "or" does not imply "not both.")

Your explanation goes on to deduce that if the diner were offered "the fish or else not the meat," the diner could have both fish and meat. If "or else" means "not both," however, it would not be permitted for the diner both to have fish and to have non-meat. If the diner has fish, then he is having both fish and non-meat, which contradicts the meaning of "or else."

It seems to me as a layman that what is going on here is not a failure of the listener to apply logic in analyzing the question, but rather a failure of language (at least in these laconic phrases) to be sufficiently precise to express the nuances of choice that the waiter is offering. I suspect that many persons, upon hearing the waiter's offer, instinctively reject it, not because they are "befuddled" or incapable of understanding simple logic, but because they recognize the confusion and potential logical inconsistency embedded in the waiter's words.

In a somewhat similar vein, I question the analysis of the "Linda" problem. The test subject is offered, among others, the choices that (X) Linda is a bank teller active in the feminist movement and (Y) Linda is a bank teller. Given the juxtaposition of these two choices, an intelligent subject who is aware of the imprecise way in which most writers and speakers use language might very well assume that (Y) means "Linda is a bank teller who is not active in the feminist movement" (or, perhaps, "Linda is just a bank teller"). If, as I believe, that assumption is a reasonable (albeit not uniquely valid) interpretation of the words given, ranking (X) as more likely than (Y) is also quite reasonable. The "Linda" problem thus seems to me to reveal something about how the brain makes assumptions in interpreting the meaning of language. It does not seem to me to reveal, as your article suggests, that people "failed to reason proficiently" or that people are "blind."

Although the analogy is not perfect, the assumption above reminds me of experiments in which a baby who already knows the word "cup" hears the word "plastic" while being shown a plastic cup. Those experiments show that the baby assumes that the word "plastic" refers to the material rather than the cup itself, because he assumes that the new word means something different from the word "cup" that he already knows. In a somewhat similar light, the "Linda" subject appears to assume that (Y) is meant to be an opposite to (X), even though the test does not say that in so many words.
I would welcome any further discussion or clarification (or disagreement with my analysis). The topic is certainly fascinating and made for a intriguing article.

William H. Weigel ’71
Brooklyn, N.Y.

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