April 2018 – Law and Logic

Using Aristotle’s Tools

Aristotle worked out almost all of the rules of categorical reasoning more than 2300 years ago, and logic students for the last two millenia have been learning how to apply them. Since the Law School Aptitude Test is primarily a test of logical reasoning, these rules may be able to help law students, too. This post is for students who know the basics of categorical reasoning and want to learn how to use Aristotle’s tools to do better on the LSAT.

There are four types of categorical statements:

Universal Affirmative (“A”): All subjects are predicates
Universal Negative (“E”): No subjects are predicates
Particular Affirmative (“I”): Some subjects are predicates
Particular Negative (“O”): Some subjects are not predicates

These four types of statements “distribute” their terms in different ways. A term is “distributed” if the categorical statement tells you something about every member of that category. Here is how the terms are distributed in the four types of statements:

A (“All subjects are predicates”): subject is distributed, predicate is not
E (“No subjects are predicates”): subject is distributed, predicate is distributed
I (“Some subjects are predicates”): subject is not distributed, predicate is not distributed
O (“Some subjects are not predicates”): subject is not distributed, predicate is distributed. (This last category is much less intuitive than the other three! If you say, “Some presidents are not Caucasian,” then you may assert that every Caucasian is not that president–i.e., not Barack Obama.)

There is a standard form for categorical syllogisms:

Major premise: Major term and Middle term
Minor premise: Minor term and Middle term
Conclusion: Minor term (subject) and Major term (predicate)

There are several rules for categorical statements:

There must be exactly three terms, each of which is used in the same sense.
The middle term must be distributed in at least one premise.
If a term is distributed in the conclusion, it must be distributed in at least one premise.
No argument may have two negative premises.
If an argument has a negative premise, it must have a negative conclusion.
If an argument has two universal premises, it cannot have a particular conclusion.

With these tools in hand, let us see how they work on some “sufficient assumption” questions. (All of the following examples are from the June 2007 LSAT, which the LSAC has placed in the public domain.)

Section 2, Question 6

An undergraduate degree is necessary for appointment
to the executive board. Further, no one with a felony
conviction can be appointed to the board. Thus,
Murray, an accountant with both a bachelor’s and a
master’s degree, cannot be accepted for the position of
Executive Administrator, since he has a felony
conviction.

The conclusion of this argument is “Murray… cannot be accepted for the position of Executive Administrator.” If we treat “Murray” as the “category of all Murrays” we can rewrite this as “No Murray is an Executive Administrator.” “Murray” is the subject of this conclusion (which makes “Murray” the “minor term”) and “Executive Administrator” is the predicate (which makes that the “major term”). We can sketch out the “standard form” categorical syllogism as follows:

Major premise: Executive Administrator AND Middle Term
Minor premise: Murray and Middle Term
Conclusion: No Murray is an Executive Administrator

Applying our rules to what we have so far, one of our premises must be positive and the other must be negative. The conclusion is an “E” type statement (“No S is P”), which means that both the major and minor terms are distributed in the conclusion. That means they must also be distributed in at least one premise. The middle term must also be distributed. Thus, every term must be distributed.

The only way to get that many distributed terms with one positive premise is to have one “A” type premise (“All S is P”) and one “E” type premise (“No S” is “P”). That means we must either have an “All Murray is (Middle term)” or “No Murray is (Middle term).”

Looking at the stimulus, we see an “A” type statement about Murray. “Murray has a felony conviction.” Let’s plug that new information into our standard form, and make the other premise an “E” type statement that uses “felony conviction” as the middle term:

Major premise: No Executive Administrator has a felony conviction
Minor premise: (All) Murray has a felony conviction
Conclusion: No Murray is an Executive Administrator

Do we have an answer that says, “No Executive Administrator has a felony conviction”? Unfortunately, no. That doesn’t mean we’re wrong in our analysis–just that there’s more information in the stimulus.

The stimulus tells us that “no one with a felony conviction can be appointed to the executive board.” That gets us very close to the answer we need! Answer Choice B says, “Only candidates eligible for appointment to the executive board can be accepted for the position of Executive Administrator. When we combine that with “no one with a felony conviction can be appointed to the executive board,” we get “No Executive Administrator has a felony conviction.” Answer Choice B is the correct answer.

Section 2, Question 15

A new government policy has been developed to avoid
many serious cases of influenza. This goal will be
accomplished by the annual vaccination of high-risk
individuals: everyone 65 and older as well as anyone
with a chronic disease that might cause them to
experience complications from the influenza virus.
Each year’s vaccination will protect only against the
strain of the influenza virus deemed most likely to be
prevalent that year, so every year it will be necessary
for all high-risk individuals to receive a vaccine for a
different strain of the virus.

The conclusion here is “every year it will be necessary for all high-risk individuals to receive a vaccine for a different strain of the virus.” That is an A-type statement, “All years’ vaccines are vaccines for a different strain of the virus,” which we can simplify to “All Years’ Vaccines are Different Vaccines.” Putting that in standard form:

Major premise: Different Vaccines AND Middle term
Minor premise: Year’s Vaccines AND Middle term
Conclusion: All Year’s Vaccines are Different Vaccines

The conclusion is affirmative, which means we cannot have any negative premises. The minor term is distributed, which means we must have a premise which distributes that term. That can only be “All Years’ Vaccines are (Middle term).” The middle term is not distributed in that statement, but it must be distributed somewhere, and there is only one way to do that using an affirmative statement: “All (Middle term) Different Vaccines.” We can write out our standard form as:

Major premise: All Middle Term are Different Vaccines
Minor premise: All Year’s Vaccines are Middle Term
Conclusion: All Year’s Vaccines are Different Vaccines

Looking at our stimulus, we see that “each year’s vaccination will protect only against the strain of the influenza virus deemed most likely to be prevalent that year.” Let’s plug that in as our middle term.

Major premise: All vaccinations against the strain most likely to be prevalent are Different Vaccines
Minor premise: All Year’s Vaccines are vaccinations against the strain most likely to be prevalent
Conclusion: All Year’s Vaccines are Different Vaccines

Looking through the answers for “all vaccinations against the strain most likely to be prevalent are Different Vaccines,” we see “Each year the strain of influenza virus deemed most likely to be prevalent will be one that had not previously been deemed most likely to be prevalent.” That is the answer!

Section 2, Question 23

Philosopher: An action is morally right if it would be
reasonably expected to increase the aggregate
well-being of the people affected by it. An action
is morally wrong if and only if it would be
reasonably expected to reduce the aggregate wellbeing
of the people affected by it. Thus, actions
that would be reasonably expected to leave
unchanged the aggregate well-being of the people
affected by them are also right.

The conclusion is “actions that would be reasonably expected to leave unchanged the aggregate well-being of the people affected by then are (also) right,” which can be simplified as “All Actions that don’t affect aggregate well-being are Right actions.”

Major premise: Right actions and Middle term
Minor premise: Actions that don’t affect aggregate well-being and Middle term
Conclusion: All Actions that don’t affect aggregate well-being are Right actions

The conclusion is affirmative, and the minor term is distributed. There is only way to write an affirmative statement that distributes a term: “all actions that don’t affect aggregate well-being are Middle term.” That leaves the middle term undistributed, which means it must be distributed in the major term, which can only be written one way: “All Middle term are right actions.” We can write out the standard form this way:

Major premise: All Middle term are right actions.
Minor premise: All actions that don’t affect aggregate well-being are Middle term.
Conclusion: All Actions that don’t affect aggregate well-being are Right actions

We have an affirmative conclusion, so we must have two affirmative premises. One statement in the stimulus says, “An action is morally wrong if and only if it would be reasonably expected to reduce the aggregate well-being of the people affected by it.” That is a double negative, of sorts–all actions that are NOT right actions are NOT actions that don’t affect aggregate well-being, and vice versa.

Aristotle’s rules aren’t getting us very far here. Let’s peek at some answer choices and see if any of them might line up with either our major premise or our minor premise. Answer choice C is the only one that fits the pattern at all. Let’s plug it in to our standard form:

Major premise: All actions that are not morally wrong are right actions.
Minor premise: All actions that don’t affect aggregate well-being are actions that are not morally wrong.
Conclusion: All Actions that don’t affect aggregate well-being are Right actions

Sure enough, that matches–the “if and only if” statement in the stimulus is logically equivalent to “all actions that don’t affect aggregate well-being are actions that are not morally wrong.”

Section 3, Question 5

Atrens: An early entomologist observed ants carrying
particles to neighboring ant colonies and inferred
that the ants were bringing food to their
neighbors. Further research, however, revealed
that the ants were emptying their own colony’s
dumping site. Thus, the early entomologist was
wrong.

The conclusion here is “the early entomologist was wrong,” but that needs to be unpacked a bit. The entomologist concluded that “ants were bringing food,” so the opposite of that is “ants were not bringing food.” We can make that a categorical statement by saying, “No particles that ants were carrying were food particles.” Putting that in standard form, we have:

Major premise: food particles and Middle Term
Minor premise: particles that ants were carrying and Middle Term
Conclusion: No particles that ants were carrying were food particles

The conclusion is negative and distributes both terms. Since the middle term must also be distributed and we must have one and only one negative premise, we must distribute every term with one affirmative premise and one negative premise. The only way to do that is to have one A-type statement and one E-type statement.

There aren’t a lot of choices for a middle term–the only additional information in the stimulus is about a dumping site, and that information is affirmative and relates to the particles the ants were carrying. We can pencil that in as our middle term:

Major premise: no food particles are particles from the dumping site
Minor premise: all particles that ants were carrying were from the dumping site
Conclusion: No particles that ants were carrying were food particles

Answer choice C is a match for our major premise. QED!

Section 3, Question 24

Sociologist: Romantics who claim that people are not
born evil but may be made evil by the imperfect
institutions that they form cannot be right, for
they misunderstand the causal relationship
between people and their institutions. After all,
institutions are merely collections of people.

The sociologist concludes that romantics can’t be right when they claim that people are not born evil but may be made evil by the imperfect institutions that they form. The logical opposite of “people may be made evil by institutions” would be “no people are made evil by institutions.” This can be put in standard form:

Major premise: beings made evil by institutions and middle term
Minor premise: people and middle term
Conclusion: No people are beings made evil by institutions

The passive “things made evil by institutions” is questionable. We don’t have to write our conclusion in the passive voice. We could say, instead, that “no institutions are things that make people evil,” in which case our standard form would be:

Major premise: things that make people evil and middle term
Minor premise: institutions and middle term
Conclusion: No institutions are things that make people evil

As we have seen twice before, the conclusion is a universal negative, which means that we must distribute the major term, minor term, and middle term using one affirmative premise and one negative premise. This requires one A-type statement and one E-type statement, with the middle term undistributed in the A-type. If our first formulation is correct, one statement must read either “All people are middle term” or “All beings made evil by institutions are middle term.” If our second formulation is correct, one statement must read either “All institutions are middle term” or “All things that make people evil are middle term.”

The stimulus states that “institutions are merely collections of people.” If we zero in on that as our middle term, we get:

Major premise: No collections of people are things that make people evil
Minor premise: All institutions are collections of people
Conclusion: No institutions are things that make people evil

Answer choice E is broad enough to include our major premise. That answer says, “The whole does not determine the properties of the things that comprise it.” That is certainly consistent with “No collections of people are things that make people evil,” and none of the other answers are anywhere close. That makes answer choice E our best bet.

Conclusion

If you can do categorical syllogisms in your sleep, using Aristotle’s rules would appear to provide a fairly reliable way to analyze sufficient assumption questions. That doesn’t mean that this is faster or better than other techniques!

Playing with a Full Deck

Countless successful students prove that people can see significant and sustained improvement on the LSAT over time. That’s great for them, but how can you get such results? Here is how I use index cards and the “blind review” process to help my students get the score they want.

There are lots of LSAT prep materials (PowerScore, Kaplan, Princeton Review, etc.). which range from “brief but useless” to “overlong and overwhelming.” You can buy a 900 page book that is “guaranteed” to raise your score, but the book won’t actually help you until something in that book enables you to eliminate a wrong answer or choose a right answer. If you’re lucky, those 900 pages might contain thirty concrete and specific insights or techniques that consistently affect your score.

I never require my students to buy any specific prep materials. I want them to make the most of whatever they start with, whether they come to me with the LSAC “SuperPrep” or the entire PowerScore LSAT Bible Trilogy. Instead, I help them figure out generally-applicable techniques that enable them to eliminate a particular wrong answer or identify a particular right answer, and write it out on my tutoring whiteboard in a format that would fit on an index card. At the end of each session, I expect them to save those whiteboards and copy out any new cards and add them to their personal card deck.

Once you have some cards in your personal deck, I will ask you to do one logical reasoning section under time pressure. For students who are already scoring at or above the national average (152), that means under 35 minutes. I don’t put a specific time limit on students who are still scoring under 152, but I do have them time each individual question with a stopwatch.

Once you complete that logical reasoning section under time pressure, don’t score it! You have completed the “testing” phase of the process, but you haven’t even started the “learning” phase. The next step is to “blind review” each question, using your personal card deck. Let’s walk through that process, using Section 2 of the June 2007 LSAT (which is the only test that the LSAC makes generally available).

Question 1: Which one of the following most accurately expresses
the main conclusion of the economist’s argument?

Process: Do I have an index card for this? No, I don’t, because I usually get main point questions right. Looking at my test paper, I see that I originally underlined the phrase “not all efforts to increase productivity are beneficial” because it looked like the conclusion to me. I know that the conclusion is the phrase that everything else in the paragraph supports, and that’s true here. The answer I chose was B, which says,”Some measures to increase productivity fail to be beneficial.” I’m pretty sure that “Not all are” is logically identical to “Some are not,” so I’ll stick with my original answer. Since I double-checked my reasoning, I’m going to take my ball-point pen and draw a blue circle around the black pencil circle I started with.

Question 2: Which one of the following uses flawed reasoning that
most closely resembles the flawed reasoning used in
the argument above?

Process: Parallel flaw–let’s see, I do have an index card for this one.

Parallel flaw: read the stimulus and see if you can spot the flaw.
If you can see it now, you can do it now.
If you can’t see the flaw, this will take longer than any other question and you’re still likely to get it wrong. Save it for last!
If you still haven’t seen the flaw, read the answers CAREFULLY. One of them has the hint you need.

Well, that’s good–because I couldn’t see the flaw with the clock ticking and I still can’t see it. Let’s look at the answers for hints. Nothing in A… B is interesting. If I mix something extremely toxic (like arsenic) with something non-toxic (like water), I’ll bet it’s still extremely toxic, not “moderately toxic.” That’s sort of like the stimulus–breeding a dog that barks a lot with a dog that doesn’t bark won’t necessarily give you a moderate barker. Let’s circle B with the blue pen. Since this card seems to have actually helped me here, let’s put a little tick mark in the top right corner of the card.

Question 3: Which one of the following most logically completes
the argument?

Process: This looks like some kind of inference. I have a card for that.

“CAN’T BE FALSE” FAMILY:
1. Pick an answer.
2. Could the opposite of that answer ever happen?
3. Reread the stimulus–if the answer you picked CAN’T BE FALSE, you’re right!
4. Avoid answers that COULD BE TRUE, answers which reference something you have no information about, and answers which are EXPLANATIONS of the stimulus rather than DEDUCTIONS from it.

I picked answer A, “people at a century’s end reminisce about their own lives.” Could that ever be false? Maybe. I mean, some people may look back on their own lives but others wouldn’t. A might be wrong. Let me look at the others. B looks wrong–no information about fearing. C talks about looking forward to the next century. That COULD be true, but it doesn’t have to be true. D talks about the history of the century just ending. Could that be false? People don’t care about the century just ending, but the stimulus says that people behave toward the century much as a person behaves towards his or her own life. If we look back at our lives at the end of a century, I guess people at the end of a century must look back on the century. OK, let’s circle D in blue, and add a tick mark to the card. (Hmmm– that’s the fifth tick mark on this card. I guess I should be thinking about this while the clock is running–I want to get this question right the first time, not just on blind review!)

Question 5: The reasoning in the consumer’s argument is most
vulnerable to criticism on the grounds that the
argument…

Process: Flaw! OK, there’s a card for every different answer to a flaw question. Let’s look at these answers. A talks about bias. I have a card on this.

Flaw: Bias.
This is a specific form of the more general ad hominem fallacy, which attacks the person rather than the argument.
Bias may show that a witness is unreliable, but it does not prove that the witness is lying.
Any conclusion that says the opposite of what the biased witness says is flawed.

Let’s see–answer choice A starts with the phrase, “treats evidence that there is an apparent bias.” Yes, the stimulus does do that. It goes on “as evidence that the Connorly Report’s claims are false. Yes, it does that, too. Looks like a perfect example of an ad hominem fallacy, which is what I though the first time. Keep the pencil circle around A–I’m so sure of this one that I’m not going to put a blue circle around it.

Question 6: The argument’s conclusion follows logically if which
one of the following is assumed?

Process: I hate these! OK, I know it’s an assumption question of some kind, but what kind? I have a card here somewhere…

NECESSARY ASSUMPTIONS v SUFFICIENT ASSUMPTIONS
C->A A->C
Does the question include the word “if”? Probably “sufficient.”
Does the question include words like “rely,” “depends”? Necessary.

OK, it does say “if” in the question. “If I assume, the conclusion follows.” Looks like it’s a sufficient assumption. (Add a tick mark to this card.) Now, I know I have another card about that.

SUFFICIENT ASSUMPTIONS: POWERSCORE’S “MECHANISTIC APPROACH (TM)”
1. Find the conclusion. Look for a new term. It MUST BE in the correct answer.
2. Find any term that appears both in the conclusion and a premise. It should not be in the correct answer.
3. There should not be any brand new term in the correct answer–only something that has already appeared in at least one premise.

Right. So the conclusion here is “Murray cannot be accepted for the position of Executive Administrator. Good–Executive Administrator is a new term. Knock off all the answers that don’t include that term. Oh–that’s answer A, which is what I picked! Well, put a tick mark on this card–if I had remembered this I wouldn’t have chosen that. Moving on to step 2; “Murray” appears in the conclusion and the premises, so I guess I can knock out D and E. Rule 3 doesn’t help me choose between B and C, so I guess I’ve gone as far as the “Mechanistic Approach (TM)” will get me. Good thing I have another card here…

“CONCLUSION CAN’T BE FALSE” FAMILY
(MBT, MSS, MBF*, SA, PSA)
With MBT, MSS, MBF: the ANSWER can’t be false.
With SA, PSA: the CONCLUSION can’t be false if you pick the right answer.

So–the conclusion here is still, “Murray cannot be accepted for the position of Executive Administrator.” With answer choice B, Murray can’t be on the executive board (because he has a felony conviction), which means he can’t be Executive Administrator. That looks pretty good. With C, there’s nothing to keep Murray from being Executive Administrator. OK, C is out and B is in. Put a blue circle around B and add a tick mark to this card. (Hmm. That’s the first time this card has actually helped me!)

Questions 7-25… repeat this process until you finish the section.

When you (finally!) finish blind reviewing that last question, you’re ready to score the section. I encourage all my students to set up a free account at 7Sage.com (which may well be the single most fabulous LSAT resource on the Internet). 7Sage makes it easy to bubble in your blind review results–there’s a black bubble for your initial answer, and then, if you click on the question number, another line of blue bubbles pops up so that you can add your blind review result. When you save and score your answers, you’ll see your score for that section (and the test as a whole). In addition. 7Sage will sort the questions into “very low priority,” “low priority,” “high priority,” and “very high priority” questions.

The “very low priority” questions are those you got right the first time. For self-study purposes, these “very low” questions are unimportant. Since you already got it right, there’s nothing new to learn!

The “low priority” questions, by contrast, are very important. You got this question wrong under time pressure, but got it right when you worked through it with your card deck. Go over these “low priority” questions and take another look at the card that helped you get it right. Ask yourself, “What could I do different next time so that I get this right while the clock is running?” You already know how to get this question right–now you need to get it right the first time.

The “high priority” questions are also very important. If you have a card for this situation, it isn’t helping you yet. If you don’t have a card that helps you here, it’s time to get one! I ask my students to look over these high priority questions and flag the question on 7Sage if they can’t figure it out on their own. At our next tutoring session, we’ll use those flagged questions to figure out which generally-applicable insight or technique would help you get it right.

Over time, we will build up an index card deck that enables you to get a blind review score that is consistently ten or fifteen points higher than your target score. At that point, you’re officially “playing with a full deck.” Then you’ll keep doing practice tests, using those cards over and over again until they’re second nature to you. You won’t be shuffling through your card deck on test day–it will be second nature to you.

Can you see sustained and significant progress on the LSAT? Yes, you can!

What Am I Getting Myself Into?

Many LSAT courses claim that students can expect a ten-point gain if they attend the classes, read the book, and do the homework. That’s great for students who (only) want a ten-point gain, and who do well in classroom settings. I get the ones who want more than ten points, or need more than a lecture format. I get the gifted students and the ones with special needs.

Gaining ten points on the LSAT is good hard work, but it’s not that hard. It’s like running a five kilometer race (3.1 miles). Gaining twenty points on the LSAT is more like running a marathon.

Here’s what you need to go from couch potato to finishing a 5K race:

Commitment to the goal
A good pair of shoes and a water bottle
A realistic start–walk before you run!
Real rest between workouts
Feeling good about the progress you make each time you train

Here’s what you need to go from your first 5k to a marathon

12-20 more weeks of training
Three to five solid workouts per week
One long run every week
Work up to running 50 miles each week
Hydration and nutrition
Intervals and speed training

I don’t promise that you can improve 20 points if you spend 20 weeks preparing for the LSAT, but I firmly believe that the average young healthy human can finish a marathon if they really want to, and the average intelligent college graduate can score above average on the LSAT if they work at it the way marathon runners do.

I can’t raise your LSAT score. Only you can do that; you do it by working and thinking. I can tell you how to work and how to think to raise you score more effectively.