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LSAT PrepTest 41, LR2, #25

A Guide to Combining Quantified Statements

Trevor McKinnon, Unsplash

Imagine that you're working through a logical reasoning section and feeling pretty good. You're reading slowly, you're anticipating, you're scoffing at the pathetic trap answers. And then you get a stimulus like this: 

The vast majority of animals with a cloaca are birds. However, all members of the mammalian order called monotremes possess a cloaca. Some monotremes have developed venomous spikes near their legs to help protect against predators, but most monotremes do not have predators. Although ancestors of monotremes were were aquatic, today, almost all members of the monotreme order live on land. The most well-known monotreme, however, is primarily aquatic.

Welcome to one of the trickiest issues in LSAT Logical Reasoning: making inferences from multiple quantified statements. In this post, I provide a comprehensive guide (with examples) to the valid and invalid inferences that the LSAT expects us to know.

At the end, I summarize with just three easy points to keep in mind that can help make sense of the broader set of rules. 

ALL + ALL 

1. All + All: When the SUFFICIENT conditions of BOTH conditional statements match, you can validly conclude a "some" relationship between the necessary conditions. 

Everyone in New York is in the United States.

Everyone in New York loves the Mets.

VALID INFERENCE: Some people in the US love the Mets. (There is at least one person in the US who loves the Mets -- the people in New York.)

 2. All + All: When the SUFFICIENT of one statement matches the NECESSARY of the other, this allows us to connect the conditional statements using the TRANSITIVE PROPERTY. 

Everyone in Manhattan lives with roommates.

Everyone who lives with roommates has a dirty kitchen.

VALID INFERENCE: Everyone in Manhattan has a dirty kitchen.

 3. All + All: When both NECESSARY CONDITIONS of the statements match, there is NO VALID INFERENCE from combining the two statements. 

Everyone in New York is in the United States.

Everyone who wears cowboy boots is in the United States.

INVALID: Some people in NY wear cowboy boots. (Doesn't have to be true because maybe the people who wear cowboy boots are all in Texas or some other state).  

ALL + MOST 

1. All + Most: When the SUFFICIENT of the "all" statement matches the LEFT SIDE OF THE "MOST" statement, you can validly conclude a "some" relationship between the other terms. 

Everyone in New York is in the United States.

Most people in New York use public transportation regularly.

VALID INFERENCE: Some people in the United States use public transportation regularly. (At least one person in the US uses public transportation regularly -- the people in New York)

 2. All + Most: When the SUFFICIENT side of the "all" statement matches the RIGHT SIDE OF THE 'MOST" statement, you can validly conclude a "most" relationship between the other terms. 

Everyone who uses public transportation regularly has been late to work.

Most people in New York use public transportation regularly.

VALID INFERENCE: Most people in New York have been late to work. (If most New Yorkers are regular users of public transportation, and all of those regular users have been late, then most New Yorkers have been late)

 3. All + Most: When the NECESSARY CONDITION of the "all" statement matches ANY TERM OF THE "MOST" statement, there is NO VALID INFERENCE from combining the statements. 

Every Mafia family leader lives in New York.

Most people in New York use public transportation regularly.

INVALID: Some Mafia family leaders use public transportation regularly. (Doesn't have to be true because maybe the Mafia leaders all use private cars and chauffeurs -- they don't have to be part of the "most people in NY" who use public transportation regularly.)

Every Mafia family leader lives in New York.

Most Yankees players live in New York.

INVALID: Some Mafia family leaders are Yankees players. (Doesn't have to be true because maybe the Mafia leaders and Yankees players are two entirely different groups of people who each happen to live in New York)

 ALL + SOME 

1. All + Some: When the SUFFICIENT CONDITION of the "all" statement matches ANY OF THE TERMS IN THE "SOME" statement, you can validly conclude a "some" relationship between the other terms. 

Everyone in New York is in the United States.

Some people in New York celebrate Jewish holidays. (Or, some people who celebrate Jewish holidays are in New York.)

VALID INFERENCE: Some people in the United States celebrate Jewish holidays. Or, Some people who celebrate Jewish holidays are in the United States. (We know at least one person in the US celebrates Jewish holidays -- at least one person in New York.)

  2. All + Some: When the NECESSARY CONDITION of the "all" statement matches ANY OF THE TERMS IN THE "SOME", there is NO VALID INFERENCE from the combination. 

Everyone in New York is in the United States.

Some people in the United States call a drinking fountain a "bubbler".

INVALID: Some people in New York call a drinking fountain a "bubbler". (Doesn't have to be true because maybe the people who call a drinking fountain a "bubbler" don't live in New York -- maybe they live in Massachusetts or Wisconsin.)

 MOST + MOST 

1. Most + Most: When the LEFT SIDES OF BOTH "MOST" STATEMENTS match, you can validly conclude a "some" relationship between the right sides. 

Most people in Manhattan live with roommates.

Most people in Manhattan are paying too much for housing.

VALID INFERENCE: Some people who live with roommates are paying too much for housing. (There must be at least one person in the world who lives with roommates and pays too much for housing -- at least one person in Manhattan. Remember, "most" means over half, or in other words, at least 51 out of 100. If over half of people in Manhattan have roommates and over half of people in Manhattan are paying too much for housing, then even if you tried to separate the roommate group and paying-too-much group from each other as much as possible, there would still be 1 person who both has a roommate and pays too much.)

 2. Most + Most: When the LEFT SIDES of the "most" statements DO NOT MATCH, there is NO VALID INFERENCE from the combination of the two. 

Most members of the Knicks live in New York.

Most people who live in New York are terrible basketball players.

INVALID: Some members of the Knicks are terrible basketball players. (Doesn't have to be true because maybe they are excellent basketball players even though most NY people are terrible.)

Most members of the Knicks live in New York. 

Most members of the Yankees live in New York.

INVALID: Some members of the Knicks are members of the Yankees. (Doesn't have to be true because maybe the Knicks and Yankees are two entirely different groups of people)

 MOST + SOME / SOME + SOME  

NO VALID INFERENCE from the combinations of these statements.  

Most (or some) actors live in California.

Some actors live in London.

INVALID: Some people who live in California live in London. (That doesn't make sense - the actors who are in California could just be an entirely different group from the actors who live in London.)

Most (or some) actors live in California.

Some people who live in California astrophysicists.

INVALID: Some actors are astrophysicists. (The two groups could just be entirely separate - sure, some of each group lives in California, but the actors and astrophysicists could just be different groups of people.)

So What's the Takeaway? Three Key Points

That was a lot of information, I know. To help you apply these rules, let me boil it down to three key points that I use when facing a problem involving inferences from quantified statements.

• Any combination involving an "All" statement must have the SUFFICIENT CONDITION match one of the terms in the other statement. Look back on the valid inferences for All + All, All + Most, All + Some—they only happen when the sufficient condition is the matching term, NOT the necessary condition. (Except for when two conditional statements combine using the transitive property: A -> B -> C. In this case the sufficient condition of one matches the necessary of the other.)

• The only possible combination that does not involve an "All" statement is when the LEFT SIDES of two "Most" statements match. Most + Some and Some + Some will never result in an inference.

• There are only two situations that allow you to conclude something stronger than "Some". The only two combinations that allow you to conclude something stronger than "Some" are

  • (i) All + All when you're using the transitive property (A -> B -> C = All A are C)

  • (ii) Most + All when the right side of the "Most" statement matches the sufficient of the "All" (A -most-> B -> C = Most A are C).

Now see if you can figure out what we can infer from the monotreme problem at the beginning of this post:

The vast majority of animals with a cloaca are birds. However, all members of the mammalian order called monotremes possess a cloaca. Some monotremes have developed venomous spikes near their legs to help protect against predators, but most monotremes do not have predators. Although ancestors of monotremes were were aquatic, today, almost all members of the monotreme order live on land. The most well-known monotreme, however, is primarily aquatic.

Future LSAT Reading Comp Passages

This section of the post collects interesting online articles that you might see edited down to an actual LSAT RC passage in the distant future. Read these recommendations every week and I can guarantee that your RC score will improve or you'll learn something interesting about the world, or both.

🧠  The Brain Processes Speech in Parallel With Other Sounds | Quanta Magazine

🧑🏽‍⚖️  Grand corruption as a systemic parasite upon society | Aeon Essays

🧑‍🎨 How Bosch Experienced his Own Kind of Hell | hyperallergic.com

🎲  Dew-Becker v. Wu: Daily Fantasy Sports as Gambling | NULR Of Note