Logical Error Flashcards

[qwiz] [i]

Logical errors (more precisely known as “formal fallacies”) involve errors that can be reduced to symbolic logic, regardless of the actual content of the argument.  This quiz identifies all the formal fallacies found in 10 Actual Official LSAT Preptests Volume V, using Wikipedia’s name for each error.

PowerScore has trademarked their names for these errors, so the first few entries take PowerScore’s definitions and match them to the Wikipedia names.

[q multiple_choice=”true”] Take a conditional statement (A->B) and then negate both terms (/A=>/B). This logical error is known as:

[c] A trademarked PowerScore error

[f] Well, yes, but that doesn’t help us much here, does it? You can find what PowerScore calls this in Chapter 6 of the Logical Reasoning Bible.

[c] Denying the Consequent

[f] No, that one isn’t a fallacy. If the “consequent” (which the LSAT refers to as the “necessary term” is false, the “antecedent” (a/k/a the “sufficient term”) must also be false.

[c*] Denying the Antecedent

[f] Good! The “antecedent” is the first term in a conditional, which the LSAT routinely refers to as the “sufficient” term.

[q multiple_choice=”true”] Take a conditional statement (A->B), assume the second term is true, and conclude that the first term must be true as well.

[c] A trademarked PowerScore error

[f] Well, yes, but that doesn’t help us much here, does it? You can find what PowerScore calls this in Chapter 6 of the Logical Reasoning Bible.

[c*] Affirming the Consequent

[f] Good! The “consequent” is the second term in a conditional, which the LSAT routinely refers to as the “necessary” term.

[c] Denying the Consequent

[f] No, that one isn’t a fallacy. If the “consequent” (which the LSAT refers to as the “necessary term” is false, the “antecedent” (a/k/a the “sufficient term”) must also be false.

[q multiple_choice=”true”] Only people in group A were cured. Therefore if anyone wasn’t cured, he must not have been in group A.

[c*]  Denying the Antecedent

[f] Correct! You can’t get from C->A to /C->/A. That would be like going from “If I live in California then I live in America” to “If I don’t live in California I don’t live in America.”

[c] Appeal to probability

[f] No, that would mean mistaking a condition that MIGHT happen for  a condition that MUST happen.

[/qwiz]

 

Flaw Questions

“Flaw” type questions come in three basic forms–assumptions, logical errors, and fallacies. Doing well on flaw questions means doing well on each of these three very different challenges.

Assumptions

Assumptions are easiest for people with no special training in logic. It is easy to spot an assumption type answer, since  it tends to start with a phrase like “fails to consider that” or “takes for granted that.” In a typical assumption scenario, the answer choice will provide a new fact that would make a big difference to the argument. You don’t need to be a logic whiz to figure out how that new fact might change things. Since each assumption-type answer is unique to the facts in that particular argument, there is no easy way to train up to do better on assumption answers.

Logical Errors

Logical errors, by contrast, do not involve new facts. They are technically known as “formal fallacies,” which means they are wrong because of the “form” of the argument. Any “formal fallacy” can be reduced to symbolic logic so that the actual terms under discussion no longer matter. A stimulus that says “Albion is in Britain” can be rewritten as “A->B.” In a “formal fallacy,” it doesn’t matter whether “B” stands for “Britain” or “Botswana.” For example, if I say “I am in Britain, therefore I am in Albion,” I have committed the logical error that PowerScore refers to as a “mistaken reversal.” To do well on this type of flaw question, you need to do well on conditional logic as a whole. I am working on a flashcard deck for logical errors.

  • Logical error flashcards
Fallacies

The third type of flaw is a specialized version of logical errors that cannot be reduced to symbolic logic. These “informal fallacies” involve a host of tricks and traps for the unwary. Unscrupulous people have been using these fallacies to dupe people for so many centuries that most of them have Latin names. These Latin names never appear on the LSAT, which adds an unintended degree of difficulty to the test. The LSAT answer choices that describe these informal fallacies can be more bewildering than Latin, especially to people who have some familiarity with the traditional names. To address this problem, I am working on two sets of informal fallacy flash cards–one that identifies all the most common and/or recent fallacies by their Wikipedia names, and another which then connects those names to wording that mimics the LSAT answer choices.

  • List of fallacies
  • Fallacy flashcards
  • Flaw answer choices

Informal Fallacy Flashcards

[qwiz] [i]

Informal Fallacies: This quiz identifies all the informal fallacies in 10 Actual Official LSAT Preptests Volume V. Since many fallacies have more than one name, we have chosen the name used by Wikipedia for each fallacy.

[q multiple_choice=”true”] presents a situation in which only limited alternatives are considered, when in fact there is at least one additional option.

[c]  Sampling bias

[f] No, that would involve some kind of survey error.

[c] Begging the question

[f] No, that would mean assuming what you are trying to prove.

[c*] False dilemma

[f] Good! 

[q multiple_choice=”true”]  infers that something is true of the whole from the fact that it is true of some (or even every) part of the whole

[c*] Fallacy of Composition

[f] Good!

[c] Argument from ignorance

[f] No, that would mean using a lack of proof as a way of proving something.

[c] May/Must Fallacy

[f] Wikipedia doesn’t know about this one–it is the error of saying something must be true just because it might be true.

[/qwiz]

 

 

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!