Eliminating Wrong Answers

I’m always looking for objective ways to eliminate wrong answers. Here’s a tip from “bswise2″ I just read on the 7Sage.com forum:

In an “assumption” question (whether necessary or sufficient), read the conclusion and look for any new terms that do not appear elsewhere in the stimulus. The right answer MUST contain that term. Eliminate all that do not.

Logic Games: Must Be Easy

I have been struggling to explain “must be” questions to my logic games students. Sometimes they are super easy. Sometimes they are super hard. In the first case, you can read the correct answer right off your sketch. In the second case, the only way to solve one “must be true” question is to find four “could be false” solutions–and my students who are having trouble with logic games in general get tied up in knots trying to figure that out.

i have also been struggling with helping students know when to stop working out initial inferences. They know they are supposed to draw a sketch, write down the rules, and start looking for the obvious implications of those rules, but how do you know when you’re done?

As things would happen, it turns out that there is one simple answer to both of these difficult problems. I call it the “must be easy” rule.

Most logic game questions fall into one of two categories: “must be” and “could be” questions. “Which of the following must be true” is an obvious “must be” question, and so is “which of the following cannot be true.” In theory, the answer to “must be” question is something you might add to your logic game sketch. Thus, if question 3 asks, “Which of the following cannot occur on Tuesday,” you should be able to pencil in a “Not X” underneath Tuesday on your sketch.

A “must be easy” question is one where the answer is already penciled into your sketch. The only way to tell whether a “must be” question is a “must be easy” question is to look at it. It should take about three seconds per answer to decide whether you have already deduced the answer to this question. If you have, you’re done. That was fast–and easy!

If you haven’t already found the answer to your “must be” question, take another look at it. Is it a “focused” question? Logic game questions that ask about  a specific entity or slot are “focused,” a “must be focused” question is practically shouting at you, “Hey! There’s something about this slot or entity that you haven’t figured out yet.” If so, now is the time to think it through and add it to your sketch.

If it’s not easy, and it’s not focused, it may still be important–especially if it comes early in the game. I have seen this on several grouping games that have “/A->B” rules on grouping games.  (I have written about the importance of such rules here.) Take a moment to ask whether this is a “must be important” question.

If it isn’t easy, and it isn’t focused, and you can’t quickly think of any less-common but very-important deductions you should be making, it “must be hard.” That is a subject for another day. (Add hyperlink HERE when that day comes…)

Here are some examples from “10 Actual, Official LSAT Preptests Volume V.”

Preptest 62, Game 1, Question 4: if you already noticed that both gas and satellite must both fall on the last three days, Question 4 is a “must be easy” question. If not, now’s the time to ask, “What do I know about these days?”

Preptest 62, Game 2, Question 8: if you realized that “/P->O” means you have to have either purple or orange in every window, this “must be easy.” If not, this is neither easy nor focused–but it’s a critical inference. If you don’t understand that every window must have either purple or orange in it, this is one of the two hardest logic games in recent years. If you do see that the “/P->O” rule means “P or O” in every window, then it’s pretty straightforward.

Preptest 63, Game 1, Question 2: if you realized that “/H->P” means “H or P” on each court, including the appellate court which only has three slots and one is already filled, then this “must be easy.” If not, this is neither easy nor focused–but it is important! Question 3 and 4 both depend on exactly the same inference.

Preptest 63, Game 2, Question 7: if you noticed that W can’t come last and T comes before W so T can’t come next to last, then this was easy. If you notice that “at least two of the members dive after so-and-so,” then this is focused. The question is asking, “Who can’t dive last or next-to-last?”

Preptest 63, Game 3, Question 12: if you always knock out a whole staircase of entities at either end of a sequence that includes an ordered chain (“A…B…C” means B and C are out on the first day, and C is out on the second day, and A and B are out on the last day, and A is out on the next-to-last day), then this question “must be easy.” If you don’t, then it focuses your attention on Thursday.

No More Double-Nots

As I wrote recently, I have been trying to find a better way to explain the important features of a “/A->B” rule in a grouping game. The “/A->B” and “A->/B” rules can be literal “game changers” in any in/out, sorting, or matching game, but they are extraordinarily hard to explain. A student has to understand conditional reasoning thoroughly and have an almost intuitive grasp of how the necessary and sufficient terms work to be able to read either of those rules and see the implications in a grouping game. Most people who are that good at conditional logic don’t need me to tutor them.

PowerScore uses a special “double not arrow” (“<-|->”) to represent one of these rules, but I find this arrow hard to use and impossible to teach.  I have searched and searched for a better way to teach this concept, without any success, so finally decided I would make up new symbols for the “/A->B” and “A->/B” rules. Within a few hours of posting my idea, somebody suggested something even better (which proves that you can’t find everything you need to know by searching for it).

This helpful commenter reminded me that any conditional statement (“A->B”) can be written as an “or” statement (“/A or B”). This is something I teach all my students to help them understand why “unless,” “except,” “without,” and “until” all have a “not” in them. (“You won’t pass unless you understand logic can be written either as “/P or L” or as “P->L.”) That means that “/A->B” can be rewritten as “//A->B,” or, if you take out the double negative, “A or B.”

“A or B” is really easy to work with. In a logic game, I just write down A/B and I know exactly what to do with it. Each slot must have an A, or a B, or both. (You must always remember, of course, that “A or B” means “A or B or both.”)

What about the other rule, “A->/B”? The same rewriting approach gives us “/A or /B.” That works out to be the same as “NOT (A and B).” On a logic game, I write that as an [AB] block with a slash through it. I can’t have A and B together. If I’m working on an in/out game, that means at least one of those has to be “out,” so I can write “A/B” over in my “out” column.

If you can remember that these two rules produce “A/B” or “/[AB],” you’re already way ahead of most other test-takers. If you can remember which is which, you have a significant strategic advantage. Most grouping games with one of these rules in them have one or more questions that directly depend on them. I have seen grouping games with three questions that can be answered in 15 seconds or less if you start with “A/B” in the right slot.

So–how do you remember which is which? “/A->B” means that if you don’t have A, you must have B. That’s exactly what an “or” statement says. If you can remember that much, you can instantly rewrite “/A->B” as “A/B” every time. If you can’t you have to ask, “So what happens if A is ‘in’? I guess B can do whatever it wants. So  A and B can both be ‘in,’ so that means ‘A/B.'”

Better Than a Double-Not

I teach my students that as soon as they realize a logic game is a grouping game they should start tapping one foot and saying “count… count… count” in the back of their minds. Grouping games are all about counting. The instant you can fill up any group, you have usually solved the problem.

This priority on counting makes information about slots that have to be full (or empty) especially valuable. Two common conditional rules provide just that kind of information. They are:

  • A->/B
  • /A->B

The “/A->B” rule is so important for grouping games that PowerScore uses a special symbol (“A<-|->B”, or “the double-not arrow”) to note it. As a person who understands how important this is for grouping games, I think the “double-not arrow” is brilliant. As a tutor who wants to explain it to my students, I think it is both frustrating and confusing. That’s why I have come up with two arrows of my own.

Before I unveil my new arrows, let’s see why it is so important to spot the “double-not arrow” situation and so confusing to use it. Let’s walk through what “/A->B” really means. In a grouping game, we tend to think of items as “in” or “out” rather than “true” or “false,” so we’ll use in/out terminology for this discussion.

  • If A is OUT, B is IN means:
    • Either A or B must be IN
    • A and B cannot both be OUT
    • A and B can both be IN

This means that any in/out game with a “/A->B” rule will always have either A or B “in.” By the same logic, and in/out game with the other rule (“A->/B”) will always have either A or B “out.” That is essential information! But how do you teach that to a student who is still trying to figure out the basics of conditional reasoning?

I have searched the Internet looking for clues. This is a PowerScore symbol, so I figured they must have a way to teach it. If they do, they haven’t printed it or posted it yet. Instead, they have forum discussions where they try to help people untangle themselves after they get it all confused–which is what I have been doing. Up until now.

What we need here are some simple symbols that make this easy and obvious. Fortunately, not only can we come up with such symbols, we can write them out with a keyboard. Note how the slash comes first in the “/A->B” situation, but comes second in the “A->/B” case. Let’s turn those slashes into pictures. If we put the forward slash first, we can make a “/\” picture. If we put it second, we get a “\/” picture.

  • /A->B turns into A<-/\->B
  • A->/B turns into A<-\/->B

Pictures are helpful if they mean something, so let’s call the “/\” picture an “erupting volcano.” The “erupting volcano arrow” means that something is erupting, so that something must be in your slot. The “\/” looks like a “leaky funnel,” which means something is leaking, which means something must be out.

If you can remember that “slash comes first” means “/\,” and “/\” means “erupting volcano,” and “erupting volcano” means something must be in, you can turn a “/A->B” rule into a full slot within seconds. And if you can remember what a “leaky funnel” does, you’ll fill an out slot just as fast.

And… if you’re tapping your foot, saying, “count… count… count” in the back of your  head, that full or empty slot just made the game much easier!

“Some People Say” in “Find the Conclusion” Questions

I have previously noted the value of spotting the “some people say” formula [hereafter, “SPS”] in an LSAT Logical Reasoning stimulus here and here. In this post, I document how frequently and consistently the SPS formula appears in just one type of question.

There are 27 “Main Point” questions in 10 Actual, Official LSAT Preptests, Volume V [hereafter, “Volume V”], and 14 of them have an SPS.

Note: I consider stimuli with a blank line at the end and a question stem that reads, “Which one of the following most logically concludes the argument” to be “inference-like” questions, not “Main Point” questions. None of these questions in Volume V include an SPS.

Here is the breakdown:

  • PT 62, Section 4, Q1. The SPS is, “One suggestion is that….” Answer C states, “The suggestion that… is mistaken.”
  • PT 62, Section 4, Q12. The SPS is followed by, “Such criticism, however, is never sincere.” Answer D states, “A politician criticizing… is being insincere.”
  • PT 63, Section 1, Q8. This is a double-decker SPS. “Your article was unjustified in criticizing environmentalists for claiming….” Answer E says the evidence “does not warrant the article’s criticism of the environmentalists’ claim.”
  • PT 63, Section 3, Q10: The SPS is followed by, “However, this accusation rests on a fuzzy distinction.” Answer A is “the claim… rests on a fuzzy distinction.”
  • PT 65, Section 1, Q9: Answer C is “the claim… is likely to be incorrect.”
  • PT 65, Section 4, Q14: The SPS is global warming “would cause more frequent and intense tropical storms.” Answer E is “Global warming probably will not produce more frequent and intense tropical storms.”
  • PT 66, Section 4, Q5: The SPS is “A famous artist once claimed that all great art imitates nature.” Answer B is “Either the artist’s claim is incorrect, or most great music is not art.”
  • PT 66, Section 4, Q9: The SPS is followed by, “Surely he is mistaken.” Answer A is “Terrence Gurney is mistaken when he suggests…”
  • PT 66, Section 4, Q26: The SPS takes up two long  sentences, and is followed by “This general line of argument may be reasonable, but… humans did not evolve from chimpanzees.” Answer B is “The assumption that something like human language must exist in some species from which humans evolved has no clearcut linguistic implications for chimpanzees.”
  • PT 68, Section 3, Q8: The SPS is followed by “However, there is little evidence to support this belief.” Answer B says, “There is as yet little reason to accept….”
  • PT 68, Section 3, Q11: The SPS is, “Many people assume that personal conflicts are inevitable.” Answer D is “Personal conflicts are not inevitable.”
  • PT 69, Section 4, Q1: The SPS is followed by a clause that says, “but they need to reassess that view.” Answer E is “Scientists need to reconsider the  belief that….”
  • PT 70, Section 4, Q16: The SPS is “Some heartburn-medication advertisements imply that unrelieved heartburn is likely to cause  esophageal cancer.” This is followed by “This is simply false.” Answer C is “Unrelieved heartburn is not likely to cause esophageal cancer.”
  • PT 71, Section 1, Q7: The SPS is followed by “Gillette’s argument is not persuasive, however, because  he fails to consider…” Answer E is “Gillette’s argument is unconvincing because it ignores…”

While I have yet to research this pattern in older tests, the most recent tests demonstrate that the SPS pattern is well worth recognizing. SPS appeared in just over half the “Main Point” questions in Volume V and was a reliable indicator of the correct answer in every case.

Symbols and Synonyms

An LSAT logical reasoning stimulus always contains most of the major components of a logical argument. A complete argument always has at least three “statements”: one conclusion and at least two premises. All but the most trivial logical argument have at least two “terms.” After looking through a large number of published LSAT stimuli, I have yet to find any stimulus that does not include at least two of these “statements” and at least two “terms.” I have previously written about how to find a conclusion, and it is relatively simple to find a premise. In this post, I want to help students discover the terms in the argument.

We need to know what a “term” is before we can find one, however. A “term,” in a logical context, means any concept that can be precisely expressed in words. A term could be “the best third baseman in the 1960 World Series,” “the true meaning of life,” or “yellow.”

In a complete LSAT question (which consists of the stimulus as well as the correct answer), all key terms must appear at least twice.  The reason for this is that an argument must connect the concepts to form a conclusion. A term that only appears once is not really necessary to the argument, and is therefore not a “key” term.

Here are the most common argument structures in LSAT stimuli:

  • Modus Ponens: A→B, A, ∴B
  • Modus Tollens: A→B, ~B, ∴~A
  • Hypothetical Syllogism: A→B, B→C, ∴A→C
  • Disjunctive Syllogism: A v B, ~A, ∴ B
  • Conditional Counterexample: A, ~B, ∴ ~(A→B)

Note that each argument uses each term exactly twice. While it is possible to construct more elaborate arguments that use terms even more often, it is not possible to construct an argument that uses a key term less than twice. Even the simplest possible argument uses its one term twice:

  • Double Negative: A, ∴~(~A)

This means that every complete LSAT question (stimulus plus correct answer) must use each key term at least twice. Every published LSAT question has at least two key terms. Finding these terms is a foundational skill for LSAT success.

The English language makes this difficult. The LSAT writers intentionally use different words to express these key terms. They exploit negatives, double negatives, and contrapositives to make it hard to match up concepts.

Here is an example from the June 2007 PrepTest, the only test that LSAC has made freely available to all:

Although video game sales have increased steadily over
the past 3 years, we can expect a reversal of this trend
in the very near future. Historically, over three quarters
of video games sold have been purchased by people
from 13 to 16 years of age, and the number of people
in this age group is expected to decline steadily over
the next 10 years.

Which one of the following, if true, would most
seriously weaken the argument?

(E) Most of the people who have purchased video
games over the past 3 years are over the age
of 16.

The conclusion of the argument is “we can expect a reversal of this trend in the very near future.” That conclusion must contain at least one key term, but the presence of the word “this” means we are going to have to look  back  into the stimulus to figure out  what “this” refers to.

The phrases “video game sales,” “video games sold,” and “purchased video games” all refer to the same concept. The phrases “people from 13 to 16 years of age” and “people in this age group” refer to a different concept, and “over the age of 16” negates that term. The word “increase” provides us with a third term, while “decline” and “reversal of this  trend” are its negation. (Since two out of three of these instances are “decreasing,” let us call this key term “decrease” and treat “increase” as the negation.)

Since the key terms are expressed in different English words, we need some way to show that they mean the same thing. The easiest way to do this is with symbols. I will use “S” for “video game sales,” “D” for decline, and “T” for “teenager” (by which I mean the 13-16 year old customer). With these symbols, we can rewrite the argument as follows:

S→T (sales are to teenagers)
T→D (teenagers are decreasing)
∴ S→D (sales are decreasing)

The correct answer to this question is “Most of the people who have purchased video games over the past 3 years are over the age of 16.” Converting this into our symbols, we get “S→~T.” This doesn’t just weaken the above argument, it destroys it.

Learning how to spot these synonyms (and antonyms, like “increase” and “decrease”) takes work, but it is essential. If you don’t recognize the repeated terms in a stimulus, you don’t really understand the argument. If you don’t really understand the argument, you’re just guessing at the answers. If you’re guessing at the answers, your score is going to be a whole lot lower than you want it to be. So let’s start spotting synonyms!

Exploiting Antitheses

I have previously identified the “some people say” principle in an LSAT logical reasoning stimulus, and have engaged in some discussion of this phenomenon at 7Sage.com.  After some further thought and research, I have decided to revise and extend my earlier remarks.

The “some people say” formulation occurs anytime an LSAT logical reasoning stimulus refers to the words, thoughts, or beliefs of persons other than the author of the stimulus. This usually takes the form of indirect discourse (i.e., “critics insist” or “paleontologists hypothesize”) but it can be more subtle. The functional test that I apply is, “Can I put somebody else’s claim in quotes?” If so, we can label it a “some people say” claim.

In my original post, I argued that every “some people say” claim is the negation of the conclusion. That goes too far. There are a handful of cases where “some people say” something that directly supports the conclusion. I think of this as “expert testimony.” There are other cases where “some people say” what they will do in the future (campaign promises, for example). Logicians distinguish “promises” from “logical statements” (for good reason). There are some other cases where “some people say” things that are not simply the negation of the conclusion–but not many.

Instead of insisting that every “some people say” statement is the negation of the conclusion, it is more accurate and helpful to say that most “some people say” statements have this characteristic. I use the term “antithesis” for such statements, and define it as follows:

An antithesis is a claim which an LSAT logical reasoning stimulus rejects.

There are significant advantages to identifying an antithesis. The first is that it makes complex stimuli simpler. “Some people say” statements can be involved, and arguments involving them can get complicated. If you extract the antithesis, what remains is often short and simple.

The second advantage is in identifying the conclusion. While this is second nature for some students, it is very hard work for others. Since an antithesis is (by definition) a claim which the stimulus rejects, it is easy to look for a part of the stimulus that says so. If there is such a statement, that is usually the conclusion.

The third advantage is in understanding the argument as a whole. If you see a “some people say” statement and suspect that it is an antithesis, you know where the argument is going before you read it. There are only so many ways to deny a  claim, and a person who is thoroughly familiar with basic argument types can anticipate them.

Thus, for example, if the antithesis is a conditional statement (“A->B”), then the simplest argument to refute it is a counterexample (“A and ~B”). If the antithesis is an affirmation (“A”), the simplest refutation is a modus tollens (“A→B, ~B, ∴ ~A”). If the antithesis is a denial (“~B”), the simplest refutation is a modus ponens (“A→B, A, ∴ B”). While these aren’t the only ways to reject such claims, they are the most common–and a person who is actively looking for such argument patterns will be able to find them faster if they really are present.

The bottom line is this: identifying an antithesis can cut a complicated stimulus in half, expose its conclusion, and reveal the argument structure quickly and accurately. While not every “some people say” statement is an antithesis, the large majority are.

How to Find the Conclusion

The first thing I teach new LSAT students to do is “find the conclusion.” That’s because every logical reasoning question on the LSAT is about a logical argument, and every logical argument combines premises into a conclusion.  The conclusion of an LSAT stimulus is what the whole paragraph is about.

Every word, clause, and sentence in a stimulus relates to the conclusion in some way. Typical sentences may introduce the topic to be discussed, provide evidence supporting the conclusion, acknowledge weak points, or describe competing views, but they all contribute to the argument in some way. Once you locate the conclusion, the rest of the paragraph generally falls into place.

So–how does one find this conclusion? More specifically, how does someone who is brand new to the LSAT and completely unfamiliar with logic find the conclusion? This post provides a toolbox of “find the conclusion” tips and techniques.

  1. The “why” test. This is the least useful technique for beginners, but the most useful technique for advanced students. The conclusion is the statement in the stimulus that requires explanation. That makes sense, because every other word in the stimulus provides the explanation that is required.
  2. Conclusion Keywords. The easiest technique is to look for keywords like “therefore,” “thus,” “hence,” “so,” “it follow that,” “and so we see,” “clearly,” and so forth. Most LSAT prep materials provide a comprehensive list of typical conclusion keywords. Be careful about relying too heavily on these keywords, however, especially in a “main point” question where your only job is to find the conclusion. There can be more than one conclusion in an argument, and “main point” questions are notorious for packing them in!
  3. Eliminate the Evidence. Evidence is generally easier to spot than the conclusion. There are evidence keywords (“because,” “for,” “for example,” “since,” etc.) that always introduce a reason for something. (Remember: the conclusion is what makes you ask “why,” so the evidence provides the reasons.) Even without an evidence keyword, you can usually tell evidence because it seems non-controversial. Nobody asks “why” two plus two equals four. It just does. If something is presented as a given, it’s evidence. Once you exclude all the evidence in a stimulus, it’s much easier to pick the conclusion out of what is left.
  4. The position principle. The average writer either starts a paragraph with the point he or she is trying to make, or ends with it. The first and last sentences are emphatic, and most writers want to emphasize the point they are trying to make. Unless someone is trying to bury the conclusion (which LSAT writers often do!), it will typically come first or last.
  5. Sentence type. Kaplan Publishing notes that almost all LSAT conclusions fall into the following six categories: predictions, comparisons, recommendations, judgments, affirmations, or if/then statements. Affirmations (“the sky is blue”), comparisons (“Bob is taller than Sue”) and if/then statements (“If it rains, I get wet”) appear frequently in logical arguments, but the other three categories are rare. If you see a prediction, recommendation, or judgment in a stimulus, it is likely to be the conclusion.
  6. The “some people say” principle. I have written elsewhere about the fact that other people’s statements in an LSAT stimulus almost always point at the conclusion. The conclusion, in almost every case, denies what “some people say.” The conclusion may be as short as “But they are wrong” or as complex as “Romantics who claim that people are not born evil but may be made evil by the imperfect institutions that they form cannot be right.”

The “Should” Principle

The first skill I teach new students is to “find the conclusion.” This is essential but not easy. Until they develop an intuitive sense of what a conclusion is and how it relates to the argument as a whole, students need external cues to help them find the conclusion. One of those cues is what I call the “should” principle.

The “Should” Principle: if the word “should” appears in a stimulus, it will probably appear in the conclusion.

This is a “principle,” not a “rule, because it is not always and necessarily reliable. I haven’t gone through every PrepTest yet. I’m eager to find some counterexamples so that I can refine the principle a little. As of now, however, I have not found a single stimulus with the word “should” in it that doesn’t have “should” in the conclusion. (If you can find a counterexample, please identify it in the comments!)

There are two different reasons for this principle. The word “should” appears in moral arguments but not in factual arguments. One would expect the word “should” throughout a moral argument, especially in the conclusion.

We should obey the law.
The law says we should not drive faster than the speed limit.
THEREFORE, we should not drive faster than the speed limit.

Explicitly moral arguments are relatively rare, however. Arguments about facts are much more common. In real life and on the LSAT, however, a argument about the facts usually has a higher purpose. People may debate whether it will rain this afternoon, but not because they’re abstractly interested in the weather. What they really want to know is whether to cancel the picnic.

Kaplan’s LSAT Premier Chapter 9 lists six types of conclusions: value judgments, if/then conditionals, predictions, comparisons, assertions, and recommendations. When an argument about facts leads to a prediction, judgment, or recommendation, it may easily result in “should.”

The word “should” appears in six stimuli in the June 2007 PrepTest (the only test that is publicly accessible). All instances obey the “should” principle.

  • Section II, Question 10: “should” appears twice in this moral argument.
  • Section II, Question 16: “should” in the conclusion of “Sandra’s” short argument.
  • Section II, Question 17: “should” is the recommendation after a factual argument.
  • Section II, Question 20: “One should not assume” is the conclusion.
  • Section III, Question 14: “should” only appears  once  in this moral argument, in the conclusion.
  • Section III, Question 20: “should” is a recommendation.

A word of caution is in order–the “should” principle is a fast way to find the conclusion, but it is not a reliable indicator of the right answer. The LSAT is full of synonyms for “should.” For example, Section II, Question 10 is a “Main Point” question where the conclusion in the stimulus is “Double-blind techniques should be used whenever possible in scientific experiments.” The correct answer is, “It is advisable for scientists to use double-blind techniques in as high a proportion of their experiments as they can.”

Summing this up in LSAT terminology, “should” is a sufficient condition for a conclusion, but not a necessary condition.

That still doesn’t mean that every sentence with a “should” in it is a conclusion–the principle is narrower and more specific. If the word “should” only appears once in a stimulus, then it is in the conclusion. If “should” appears twice (or more) in a stimulus, then it will be in the conclusion as well as in a premise.