Logical Reasoning – Beyond the Scope?

A common thread I’ve seen among students focusing on the Logical Reasoning section is that they often miss questions where the correct answer is one that introduces material not found in the passage – that is to say, it goes “beyond the scope” of the initial reading.

Invariably, these students have done some good prep on their own, or taken a class, and they have learned (correctly) that it’s very common for a wrong answer to introduce new material.  In fact, that’s often a good way to eliminate answer choices.  But it appears that the sources that have taught students about that concept have taken it a bit too far – there are questions in which the right answer DOES introduce new material.  In fact, there are times where a right answer almost HAS TO introduce new material. 

It all comes down to the question type.  The most common question type in which the advice to avoid new material is spot-on is the “find the conclusion” question.  Many, many LSAT questions are of this type, and the conclusion will flow from the premises – which will be given in the passage – so an answer choice that presents new material will be wrong.  But there are other question types.  For instance, a “find the flaw” question is a GREAT candidate for a correct answer that introduces new material.  Consider – one of the more common “flaws” is that the speaker fails to consider an alternative explanation.  So of course the “flaw” won’t appear in the passage – the reason the passage is flawed is because it’s not in there!  I’m going to update this entry and include some examples from past LSATs so you can see what I mean in practice.  But for now, be sure to ask yourself…Is this the kind of question where it makes sense that new material might be a correct answer?

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Contrapositive 401

Last add (for a while, anyway) about the contrapositive.  If you’ve read the earlier posts (and if you haven’t, you should) on the topic, this should be a review for a short while:

Alex and Bob are two of a group of students, all of whom must take either a math class or an English class.  The condition you’re given is, if Alex takes English, the Bob takes math, or, symbolically:  A(e) -> B(m).  The contrapositive can be diagrammed in the affirmative:  If Bob takes English (i.e. “not math”), then Alex takes math (i.e. “not English”):  B(e) -> A(m).  So:

A(e) ->B(m)

B(e) -> A(m)

That’s the review.  Many students mistakenly infer from this that Alex and Bob must be in different classes.  But that’s not correct.  Notice that there are two conditional relationships – one attaches if Alex takes English, and the other attaches if Bob takes English.  There’s nothing there that says they can’t both take math.  There will usually be a question or two that will trip you up if you’re not ready for this one.  They can’t both be in English, but they could be in math together, or they could be in different classes.  Make sure you understand this one; it’s important.

Logical Reasoning: What’s with the adverb?!

How are you at identifying parts of speech?  If you’re looking for a logical reasoning tip, and you know your parts of speech, I’ve got one for you.  And it’s all about adverbs.

Quick grammar review:  Adverbs are similar to adjectives, except that adjectives modify nouns, and adverbs modify verbs, adjectives, and other adverbs.  For instance (adverb in CAPS):

He drove QUICKLY.

Porsches are USUALLY fast.

etc.

Adverbs are used to provide nuance.  For instance, in a novel, you might be told that someone said something “pointedly,” or said something “loudly,” which might express anger, for instance.  But they’re not critical to telling you exactly what happened.  Hemingway said that from Tolstoy, he learned to “distrust” adverbs.  The “meat” of the sentence consists of nouns & verbs. 

This is especially important in the logical reasoning section of the LSAT.  Logic and argument are about nouns & verbs.  Who did what?  Nuances like adjectives are out of place.  And that’s important, because they should stand out like a red thumb.  Here’s an example…look at these two arguments:

A1:  All cats have four legs.

A2: Spot is a cat.

A3: Therefore, Spot has four legs.

Perfectly valid syllogism.  If the premises (A1 & A2) are true, the conclusion (A3) MUST be true.  You can “Venn diagram” this (Google search if you don’t know Venn Diagrams).  The little circle, cats, is completely enclosed by the big circle, things with 4 legs.  Now look at this:

B1.  Mammals usually cannot fly.

B2.  Spot is a mammal.

B3.  Therefore, Spot cannot fly.

INvalid.  What’s the difference?  “usually.”  The adverb acts as a qualifier.  In this case, it tells you that a small portion of mammals are NOT in the “things that cann0t fly” category.  The argument fails, because even though most mammals can’t fly, Spot MIGHT be a bat.

This focus on the adverb comes up in all sorts of Logical Reasoning question types, and in many different ways.  In argument B, above, it might be “Find the Flaw” (the speaker over generalizes that something that is usually true must be true in a particular case).   Or in an “Unspoken Assumption” question (the speaker assumes that Spot is not one of the group of mammals that can fly).  Or in a “strengthen the argument” question (Which of the following would strengthen the author’s argument?  ‘Spot is not a bat’).

Adverbs don’t need to be in arguments.  So when they ARE in your LSAT argument, very, very often it’s because they’re relevant to the right answer.  An easy way to identify most adjectives is that many end in “LY.”  Not all of them, but a lot of them.  Train yourself to spot them.  Somehow, some way, they’re relevant to the right answer.  Maybe the right answer is the only one that addresses something the adverb brings up.  Maybe the “qualifying” character of the adverb eliminates a wrong answer, when you’ve got it down to two choices.  The bottom line, though, is that they serve a purpose, and that purpose usually isn’t necessary to the speaker’s argument — it’s examsmanship.  It’s necessary to creating right or wrong answer choices.

Study Tip: Specifying your weaknesses

I met with a new student last weekend.  She’s going to be taking the September 26th LSAT, so I had limited time to work with her, but I had some concerns about how beneficial our session would be.  I was pleasantly surprised, though — she not only knew what section she needed help with (Logical Reasoning), she also knew which type of questions gave her trouble.  Specifically, the “identify the unspoken assumption” type of question. 

Because she had done such a great job of pinpointing her problem area, I was able to focus our session to give her some helpful tips, and we were able to work through a variety of practice test questions specific to her needs. 

When you’re first learning about the LSAT, there’s room for general improvement, as it’s all new to you.  After you’re familiar with the test, though, and you have a “baseline” — an approximate score that you’re generally around — then your best bet for improvement is focusing on particular weak points.  Odds are, you’re leaving a disproportionate number of points on the table in one of the three sections, or in one of a few specific subtypes of questions.  There are only so many types of logic games.  A solid majority of Logical Reasoning questions are one of four types.  Don’t just do practice tests – analyze your results.  Where are those extra 5-10 points you’re looking for going to come from?  If you can answer that question, you’re a lot more likely to find them.

Logic Games: Contrapositive (Part 3)

So, in the two previous points, I’ve highlighted a few of the most important things (for LSAT Logic Games purposes) about the contrapositive.  To recap:

1) The basic format is, “If not Q, then not P.”

2) It’s the only valid inference from a conditional (If P then Q)

3) When there are only two possibilities, “If not Q, then not P” can be expressed affirmatively (see post below).

One other thing you have to be aware of…when the Q has an “and” in it, it becomes an “or” when you’re writing “not Q.”  That sounds gobbledygookish, so let’s look at some actual statements. 

First, let’s take it in general terms.  For instance: “If Spot is a cat, then Spot has four legs and meows.”  We have a classic conditional (if P then Q) set-up, so we know that “If not Q, then not P” is valid.  But to negate “Spot has four legs and meows,” we only need to negate EITHER possibility.  So, for instance, if Spot has 4 legs but doesn’t meow, he’s not a cat; maybe he has 4 legs and barks, and he’s a dog.  (Yes, he could be a cat with damaged vocal cords, but we assume the premise is true as given).

For diagramming purposes, it’s important to recognize the difference; seeing the “or” that isn’t in the initial statement actually gives you more to diagram.  For instance:

“If Bob takes English, then Ken takes Math and Science.”  This can be diagrammed (among other ways) as:  B(E) –> K(M+S)

The way to diagram the contrapositive is with TWO statements:

K(~M) –> B(~E)  AND

K(~S) –> B(~E).

If Bob WERE in English, then Ken would be in BOTH math AND science.  So if Ken isn’t in science, Bob’s not in English.  Don’t fall into the trap of thinking that BOTH K(~M) AND K(~S) are necessary conditions to keep Bob out of English.  EITHER situation suffices.

If you’re not familiar with the tilde (~) in these diagrams, btw, see earlier post(s) of mine; it is used to designate (“not”) in logic, and I find it a handy LSAT abbreviation.  Feel free to use whatever works for you, though!

Contrapositive 201: Binary Games

Typically, when students get a handle on the contrapositive, they mistakenly view contrapositives as only able to be expressed in the negative (since they’re “If NOT Q, then NOT P”).  However, in some cases, the contrapositive can be expressed in the affirmative, as well.  This is important, because sometimes questions or answer choices are expressed in the affirmative, so for accuracy and speed, you have to accurately translate the given statement into its logical equivalent.

The key to when the contrapositive can be expressed in the affirmative is this: The question must be a binary variable; i.e. there must only be 2 options.  For instance, let’s say 6 students are each taking 1 class, and that class is either Math or Science.  If our statement is, “If Alex takes Math, then Bob takes Science.”  Sample diagram:

A(m) –> B(s).    Our hypothesis, P, is “Alex takes Math.”  Normally, our contrapositive is expressed as a negative.  If NOT Q, then NOT P.  But in this case, take a closer look at “NOT Q.”  There are only two classes, and everybody’s taking one.  “Not Q” = Bob doesn’t take science.  But since this is a binary game, “Bob doesn’t take science” = “Bob takes math.”  Similarly, “Not P” = “Alex doesn’t take Math,” which = “Alex takes science.”  So, our contrapositive, like the conditional, can be written in the affirmative:

B(m) –> A(s).   

Again, for this simple re-formulating to be accurate, we must be looking at a binary possibility – Alex takes Math, or Science.  Similarly, Bob takes Math, or Science.  If there were a third option, say, History, then we’d have to write the more normal expression:  ~B(s) –> ~A(m)  (If Bob doesn’t take science, Alex doesn’t take math).  In this case “doesn’t take science” isn’t equivalent to “takes math” because “takes history” is also an option.

Another situation in which it doesn’t work is in an incomplete grouping game.  Say we had 8 students altogether, but only 6 of them were taking classes.  Against, “Bob doesn’t take science” isn’t equivalent to “Bob takes math,” because there is a third option — Bob doesn’t take ANY class.

But for the binary possibilities, being able to immediate realize and formulate the contrapositive in the affirmative — 

B(m) –> A(s) — is potentially HUGE not necessarily in terms of accuracy (you’d have worked it out anyway, probably), but in terms of time.  Questions and answer choices are designed to see if you’ve made the connection, and they’ll be written that simply:  “If Bob takes Math which of the following must be true?”  And you won’t have a fact pattern that says “Bob takes math.”  But you WILL have one that says, “If Alex takes Math, then Bob takes Science.”  If you can look at that and diagram “If Bob takes Math, then Alex takes Science” in a few seconds, that’s golden.  And I promise, that concept will be on your LSAT, more than once.

Feel free to ask…

I’m open to taking specific questions about LSAT Prep, law school prep, the law school (particularly the 1L) experience (course selection, getting ready for finals, whatever), bar prep, finding a job, starting out as a new lawyer, etc.  Please feel free to drop a line.

That dreaded contrapositive (as relates to the LSAT), Part 1

Philosophy majors can probably skip this one, but for the other 99% of you…if you’re not thoroughly comfortable with the logical implications of conditional statements (“If P, then Q”), then stick around…they’re important on the LSAT, and particularly…the Logic Games.

Surprisingly, there are many levels that pertain to conditionals, so, we’re going to take it from Square 1. 

From the Statement Structure “If P, then Q” a few related statements can be constructed.  Only one of them, however, is accurate.  Let’s use actual statements with meaning, to make it easier to understand.  Let’s say I have a drink.  My conditional will be “If my drink is a vodka and tonic, then my drink contains alcohol.”  We can readily see that this follows the “If P, then Q” structure.   P = “my drink is a vodka and tonic”    Q = “my drink contains alcohol.”

One possible conclusion to draw from “If P, then Q” is “If Q, then P.”  This is not a valid inference.  To see why, let’s just put our meaningful content into the formula.  If Q, then P translates to, “If my drink contains alcohol, then my drink is a vodka and tonic.”  Clearly, though, this does not follow.  My drink might be a rum and coke. 

Another possibility might be, “If not P, then not Q.”   Again, this is not a valid inference.  Let’s check it out:  “If my drink is not a vodka and tonic, then my drink does not contain alcohol.”  But, again, I could have a rum and coke.  “Not P” would be satisfied, but “not Q” wouldn’t be true – my rum and coke WOULD contain alcohol.

That leaves “If not Q, then not P.”  This is called the “contrapositive,” and it IS true.  Check it out: “If my drink does not contain alcohol, then my drink is not a vodka tonic.”  That one works. 

Very often, a Logic Games question will offer you incorrect answer choices based on drawing one of the faulty conclusions.  They want to see if you’ll jump on the wrong conclusions.  But the other thing they do is offer correct answer choices based on the contrapositive, and if you don’t spot it right away and allow for it in the diagram, you’ll either miss a correct answer, or spend far too long working it out.  Here’s how it plays out in LSAT land.  Let’s say we have a grouping game where 8 different students take either a Math class, a Science class, or an English class.  One of the clues might be:

“If Alex takes the English class, then Bob takes the Math class.”  You might diagram this a number of ways, for instance:  A(e) -> B(m)     Or  If A=e, B=m.    Something quick, visual, and understandable (to you).  Recognizing it as a conditional, though, you have to note that the contrapositive is true, and you have to get it diagrammed, as well.  There are various ways to diagram “not.”  I’m partial to the tilde (~), which is the symbol used in formal logic.  It’s fine to diagram it in other ways, though.  The important thing, though, is to get the contrapositive in your diagram, too, in this case:

~B(m) -> ~A(e),  or  If B~=m, A~=e.   In other words, if Bob doesn’t take the math class, then Art doesn’t take the English class.  This is the logical equivalent of the given statement.  The reason it’s important to write it out is that you’re guaranteed to get questions that tell you that Bob isn’t in the math class, and you have to be able to immediately rule out answer choices that put Alex in the English class.  The question might tell you straight out that Bob doesn’t take English, or it might say, “If Bob is in the science class, which of the following could be true.”  Then you have to make the connection:  Bob in science = Bob not in math = Alex not in English.  And you have to do it fast.  If you can lay out the contrapositives at a glance, and reflect them in your diagrams, you’ll be in good shape on a number of questions.  If you can’t, then you’re either going to come to faulty conclusions and get some answers wrong, or you’re going to use up valuable time figuring it out on the fly.  So learn to:

1) recognize conditional (“if P, then Q”) statements.

2) translate to the contrapositive (“if not Q, then not P”).

3. get the contrapositive diagrammed.

Caveat: Sometimes the Q comes before the P.  Just because the standard format is “if P, then Q,” don’t get lazy and assume that the first piece of information is the P, and the second is the Q.  It’s the “IF” that defines which part of the sentence is the hypothesis (P), and which is the conclusion (Q).  For instance:

“Carl takes Math if David takes Science.”  P = “David takes Science” (the part after the “IF”).  Q = “Carl takes Math.”  The contrapositive is:   “If Carl doesn’t take Math, then David doesn’t take science.”

Good practice Logic Game: Prep Test 33 (December 2000), section 4, game 2 (Questions 6-12).  Page 177 of “The Next 10 Actual, Official LSAT PrepTests.”

Unfortunately, this post only covers the first layer of thinking you need to have about the contrapositive.  Call it “Logic 101 for LSAT”  201 and 301 are more advanced, but they’re tremendously important.  Coming soon to an LSAT blog near you.

Tips for working on the LSAT’s Logical Reasoning section

It can seem difficult to efficiently work on improving your score on the logical reasoning section.  That section (like the reading comprehension section) feels a bit ‘messy’ — kind of hard to categorize.  The questions and concepts in the logical reasoning section, though, are fairly manageable.  It’s just harder to recognize them than in, say, Logic Games (Analytical Reasoning) where you can say, “Oh, that’s a linear game (or a grouping game, or whatever).”

One thing you definitely want to do is have a handle on the basic logical fallacies.  If that’s not something you studied as an undergrad (or only studied superficially), do an internet search.  I’d limit it to “basic” logical fallacies, because you don’t want a million obscure entries.  Philosophers (logicians) have been categorizing these things since the dawn of time.  What you need, though, are the basics.  There are only so many ways in which most arguments go astray.  A few of the common ones –

  • Improperly assuming that since B follows A, A must have caused B.
  • Improperly assuming that since A and B are positively correlated (occur together), then A caused B.  Maybe B caused A.  Maybe C caused A and B.
  • Improperly assuming that because A led to B this time, A must necessarily lead to B.

You don’t need to memorize what these fallacies are called (not even in English, let alone Latin!) but you do want to be familiar enough with them so that when you see examples of them, it clicks with you.  You can immediately say to yourself, “Hey, wait!  It doesn’t say that the crime was CAUSED by the poverty…”  You want to add this sort of pattern recognition to your internal toolbox. 

Being familiar with the fallacies helps on a number of question types.  You have questions that specifically ask for the problem in an argument, or ask for a similarly flawed argument.

The other thing you want to do, is get a really good grasp on how the arguments are structured.  If you’re working out of a 10 Actual LSATs book (and I strongly suggest that you do), pick out one test and, off the clock, just go through one entire Arguments section -question by question – and identify the function served by every single sentence or phrase in the prompt.  Don’t even worry about answering the questions; just focus on understanding the argument.  You should soon discover that there are only so many functions that pieces of information serve.  For instance, something might be a supporting example; something might be a counter-example to a position being challenged; something might be a piece of given information (to be supported or challenged); something might be an analogous illustration.  And so on. 

The purpose is to get in the habit of categorizing (and making sense of) the argument structure.  It’s easy to get caught in the trees and miss the forest in this section.  Don’t think so much about the content of the passages.  Just get used to seeing how the information is presented.  For instance, if you had a logic game about someone reading 6 books in order, you wouldn’t think of it as a “books” game; you’d think of it as a sequence game, and you’d think of it in the way you think of other sequence games.

If you have “The Next 10 Actual, Official LSAT Prep Tests,” (Tests 29-38), I can better illustrate when I’m suggesting you do.  Check out the passage that relates to questions 11-12 of Test 29 (October, 1999).  Go down the prompt, sentence by sentence.

The first sentence (species adapting) is presenting given information, up to the “but.”  It’s a given.  That information isn’t going to be the substance of an argument, because everyone already knows it.  But AFTER the “but,” we’re presented with an opinion – a belief (“only the most highly evolved…”)  We also have a preview that the author will disagree with the opinion “usually assumed.”

The next sentence is the author’s belief, prefaced with the tip-off “however.”

The next sentence is a example to support the author’s belief.

Everything after that is detail that continues the example, and is pretty much irrelevant to the structure.  So, we have –  1. Given   2. General belief.  3. Author’s contradictory belief.  4. Supporting counter-example to the general belief, to support the author’s belief.

Note that the general belief is a universal — something that is always true.  We can tell from the word “only” in line 2.  Therefore, any counterexample will refute the assertion.  For example, if someone’s assertion is that only non-mammals can fly, all you have to do is point out the bat – one counterexample that disproves the assertion.

Going through a section or Arguments in this way will help you in more readily making sense of the ways in which the arguments are constructed.  Every sentence in the prompt serves a purpose.  Illustration, analogy, etc.

Practicing Logic Games

…most people view their practice tests only in terms of accuracy — How many right? How many wrong? Let me give you an example of why there’s more to it than that.

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