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:

  1. There must be exactly three terms, each of which is used in the same sense.
  2. The middle term must be distributed in at least one premise.
  3. If a term is distributed in the conclusion, it must be distributed in at least one premise.
  4. No argument may have two negative premises.
  5. If an argument has a negative premise, it must have a negative conclusion.
  6. 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!

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