The Conditional Logic and Quantifier Guide

Conditional logic still shows up in Inference, Principle, and Sufficient Assumption questions. This illustrated guide covers it in eleven parts.

By Germaine Washington ·

With Logic Games gone, formal logic matters less on the LSAT than it once did, and prep has understandably shifted with the test. However, that has also made it easier to underprepare for the question types where conditional logic still shows up: Inference, Principle, Sufficient Assumption, Parallel Reasoning, and others.

That creates an opportunity for students to gain a competitive advantage by familiarizing themselves with the test’s logic. Learn the rules, practice the translations, and you’ll find that improvement here is much more consistent than in many other areas of the LSAT. What follows is a quick reference guide to help you with that practice.

Part 1: Conditional relationships

Identify the sufficient and necessary conditions first. Then translate the relationship into a diagram or plain English.

Conditional and "all" statements

A conditional is a relationship in which one condition guarantees another. Write it A guarantees B and read it as "if A, then B."

  • A is sufficient: if A occurs, B follows.
  • B is necessary: B is required for A.
  • One direction only: A guarantees B, but B does not guarantee A.
A map of France with Paris marked by a filled dot inside the country.
You are in Paris. If you are in Paris, then you are in France. Therefore, you are in France. Paris guarantees France

The contrapositive

For A guarantees B, the absence of B guarantees the absence of A. Two steps get you there:

  1. Reverse — place the conditions in the opposite order.
  2. Negate — negate both conditions.

So A guarantees B becomes not B guarantees not A. The contrapositive has the same meaning as the original conditional relationship.

A map of France with the marker placed outside the country, in the Atlantic, and Paris drawn as an empty circle.
You are not in France. If you are not in France, then you are not in Paris. Therefore, you are not in Paris. not France guarantees not Paris

Part 2: Indicators

Which condition does the indicator introduce?

Sufficient condition indicators

The arrow points away from the term being modified by the indicator.

  • Invariably / Inevitably / Ensures / Guarantees / Is

    A guarantees B

    A valid ticket ensures entry.Ticket guarantees Entry

  • Will enable [someone] to / Allows

    Condition guarantees Outcome

    A key allows you to open the door.Key guarantees Can Open Door

  • All / Every / Any / When(ever) / Provided that / As long as

    A guarantees B

    Any valid contract is binding.Valid Contract guarantees Binding

Necessary condition indicators

The arrow points toward the subject of the indicator.

  • Can be explained only by

    Phenomenon guarantees Cause

    A crash can be explained only by pilot error.Crash Happened guarantees Pilot Error

  • Only / Only if / Only by / Only through

    A guarantees B

    The car starts only if there is gas.Starts guarantees Gas

  • Requires / Must / Essential / Presupposes / Precondition

    A guarantees B

    Evolution requires time.Evolution guarantees Time

  • Depends on / Is contingent on / Is governed by

    A guarantees B

    Survival depends on adaptation.Survive guarantees Adapt

  • Impossible without / In the absence of / But for

    Result guarantees Requirement

    Victory is impossible without sacrifice.Victory guarantees Sacrifice

Part 3: Quantifiers

How much of one group overlaps another?

  • All

    12 of 12

  • Most

    9 of 12, over half

  • Some

    2 of 12, at least one

  • None

    0 of 12

  • Few

    3 of 12, under half

Filled circles show how many members also belong to the second group.

  • "Most" (more than 50%). More than half of the members of one group are in another group. "Most basketball players are tall" is Basketball Players most are Tall People and tells you that over half of basketball players are tall.
  • "Some" (at least one). At least one member of a group is in another group. "Some basketball players are short" is Basketball Players some are Short People and tells you at least one basketball player is short.
  • Many is a vague quantity. Treat it as some.
  • Few establishes two relationships at once. "Few politicians are independent" gives you both Politician some are Independent and Politician most are not Independent.

Hierarchy (all > most > some). A stronger quantifier can also imply a weaker one. For example, "All dogs are mammals" also tells you that some dogs are mammals.

Quantifier indicators

  • Some / Many / Several / Often / At least one

    Some

    Many birds migrate.Birds some are Migrate

  • Most / Majority / Usually / Typically / Almost all

    Most

    Most businesses fail.Businesses most are Fail

  • Few / Rarely / Seldom

    Most of A are not B

    Few people live to 100.People most are not Live to 100

  • Not all / Not every

    A some are not B

    Not all that glitters is gold.Glitters some are not Gold

  • No / None

    A guarantees not B

    No square is a circle.Square guarantees not Circle

Part 4: Valid conclusions

The rule: an "all" conditional carries forward whatever quantifier reaches it.

All-to-all chain

All poodles are dogs, and all dogs are mammals. Every poodle therefore belongs to the larger group of mammals.

  1. Poodle guarantees DogPremise
  2. Dog guarantees MammalPremise
  3. Poodle guarantees MammalConclusion
Every Poodles is a Mammals. MammalsDogsPoodles

Most followed by all

Over 50% of US citizens live in the continental U.S., and everyone in the continental U.S. lives in North America. So over 50% of US citizens live in North America.

  1. US Citizens most are Continental USPremise
  2. Continental US guarantees North AmericaPremise
  3. US Citizens most are North AmericaConclusion
Every U.S. citizens is a North America. North AmericaContinental U.S.U.S. citizens

Some-to-all chain

Some sea creatures are predators. Since every predator consumes other animals, those same sea creatures must consume other animals.

  1. Sea Creatures some are PredatorsPremise
  2. Predators guarantees Consume Other AnimalsPremise
  3. Sea Creatures some are Consume Other AnimalsConclusion
Something can belong to sea creatures and predators at once. Consume other animals Sea creatures Predators Some parrotfishsurgeonfishsharktigerwolfvulturecarrion fly

Two majorities within one group

Eight of the twelve students study biology and seven study Spanish: fifteen enrollments among twelve students. Since each group is more than half the class, at least one student must study both. In this case, three do.

  1. Students most are Study BiologyPremise
  2. Students most are Study SpanishPremise
  3. Study Biology some are Study SpanishConclusion
Eight study biology and seven study Spanish, so three of the twelve must study both. Biology 8both 3Spanish 7

No valid conclusion

Be very careful in dealing with a mixture of "all" and "some" conditionals, as you can see in this example. Even though every single person born in Alabama may be American, a strong relationship, and some Americans are born in Nebraska, it is clearly the case that no one (not some) born in Alabama is born in Nebraska. And so you can never confidently conclude anything from an "all" into "some" relationship.

  1. Born in Alabama guarantees AmericansPremise
  2. Americans some are Born in NebraskaPremise
  3. Born in Alabama some are Born in NebraskaDoes not follow
A map of the United States with Alabama shaded solid and Nebraska shaded with diagonal stripes, two states that do not touch.

Part 5: Compound statements

  • And / Both

    A + B

    He is rich and famous.Rich + Famous

  • Or / Either … or

    A or B

    We can walk or drive.Walk or Drive

  • Neither … nor

    not A + not B

    Neither rain nor snow stops the mail.not Rain + not Snow

"And" and "or" inside a conditional

  • "And" in the necessary condition — A guarantees (B and C)

    Splits into two relationships: A guarantees B and A guarantees C.

  • "And" in the sufficient condition — (A and B) guarantees C

    Both sufficient conditions are required together. Do not split.

  • "Or" in the necessary condition — A guarantees (B or C)

    At least one of the necessary conditions is required. Do not split.

  • "Or" in the sufficient condition — (A or B) guarantees C

    Splits into two relationships: A guarantees C and B guarantees C.

Minimums, maximums, and nested conditions

  • At least one (minimum of one)

    "You must choose A or B." Refuse one and you must take the other.

    not A guarantees Bnot B guarantees A

  • Not both (maximum of one)

    "You cannot choose both A and B." Take one and the other is out.

    A guarantees not BB guarantees not A

  • Nested conditional

    "If A happens, then if B happens, C happens." Read it as an "and" sufficient condition.

    (A and B) guarantees C

Part 6: Negation

Negating a statement means identifying its exact logical opposite.

Negating "and" and "or": De Morgan's laws

When you take the contrapositive of a statement containing "and" or "or," change "and" to "or" and "or" to "and."

Original

A guarantees (B and C)

If you get promoted, you get a raise and an office.

Contrapositive

(not B or not C) guarantees not A

If you don't get a raise or you don't get an office, you aren't promoted.

Original

(A or B) guarantees C

If it rains or snows, we stay inside.

Contrapositive

not C guarantees (not A and not B)

If we don't stay inside, it isn't raining and it isn't snowing.

Negating a quantifier

  • All

    "All A are B" A guarantees B negates to "some A are not B" A some are not B.

  • Most

    "Most A are B" (over 50%) negates to "half or less" (0–50%).

  • Some

    "Some A are B" (at least one) negates to "none" A guarantees not B.

Part 7: Exclusion, and the "unless" family

To translate a statement using "unless," identify the idea that follows "unless," negate that idea, and make the negated form sufficient. The other part of the sentence goes on the necessary side of the arrow.

  1. Start with the sentence [B] unless [A] — "I will go to the movie unless it rains."
  2. Negate [A] and make it sufficient "If it does not rain, then I will go to the movie."
  3. Convert to a diagram not Rains guarantees Go

Use the same method for "except", "without", "until", and "or else".

  • Unless / Except / Without / Until / Or else

    not A guarantees B

    The alarm rings unless the code is entered.not Code guarantees Alarm

  • No / None / Never / Fail to be

    A guarantees not B

    No mammal breathes water.Mammal guarantees not Breathes Water

  • No [X] is both [Y] and [Z]

    X guarantees not (Y + Z)

    No number is both even and odd.Number guarantees not (Even + Odd)

  • Unless [A] or [B]

    not Trigger guarantees (A or B)

    The deal fails unless you pay cash or trade goods.Deal Succeeds guarantees (Pay or Trade)

  • Not [A], let alone [B]

    not A + not B

    He is not a mayor, let alone a president.not Mayor + not President

  • No one who is not [A] is [B]

    B guarantees A

    No one who is not a lawyer is a judge.Judge guarantees Lawyer

Part 8: Two-way and hypothetical statements

Biconditionals

A biconditional is a two-way conditional relationship in which both A guarantees B and B guarantees A are true. Having either A or B guarantees the other.

  • If and only if / Then, and only then / All and only / When and only when

    A if and only if B

    A number is even if and only if it is divisible by two.Even if and only if Divisible by 2

  • Equivalent to / Is the same as saying / Exactly when

    A if and only if B

    Having exactly 100 cents is the same as having one dollar.100 Cents if and only if One Dollar

Hypotheticals and complex statements

  • Otherwise

    not A guarantees B

    Wear a helmet, otherwise you risk injury.not Helmet guarantees Risk Injury

  • Had [A] occurred, [B] would have occurred

    A guarantees B

    Had the levee held, the city would have remained dry.Levee Held guarantees Dry

  • [B] would not have occurred had [A] not occurred

    not A guarantees not B

    The fire would not have started had the match not been struck.not Match Struck guarantees not Fire Started

  • To [verb] / In order to

    Goal guarantees Requirement

    In order to make an omelet, you must break eggs.Omelet guarantees Break Eggs

Part 9: Cause and obligation

These relationships can resemble conditional statements, but they describe causation, obligation, permission, or judgment rather than a guaranteed result.

Causal indicators

  • Promotes / Leads to

    Cause causes Effect

    Exercise promotes health.Exercise causes Health

  • As a result

    Cause causes Effect

    The dam broke; as a result, the valley flooded.Dam Broke causes Flooded

Value judgments, obligations, and rights

  • Should / Ought to / Is obliged to

    Condition guarantees Obligation

    Citizens are obliged to pay taxes.Citizen guarantees Duty to Pay

  • Justified in

    Condition guarantees Permissible

    A person is justified in acting in self-defense if attacked.Attacked guarantees Self-Defense Permissible

  • Permissible only if

    Permissible guarantees Condition

    Entry is permissible only if you have a badge.Entry Permitted guarantees Badge

  • Unethical / Wrong / Should not [action] except [exception]

    not Exception guarantees not Action

    Drivers shouldn't park here except when permitted.not Permitted guarantees not Park

Part 10: Probability and belief

  • A is likely / probably / we would expect B

    A most are B

    If you speed, you will likely get a ticket.Speed most are Ticket

  • A tends to B

    A most are B

    Power tends to corrupt.Power most are Corrupt

  • A is unlikely unless B

    A most are B

    Success is unlikely unless you plan.Success most are Plan

  • Can thus rule out

    A guarantees not B

    The alibi rules out the suspect.Alibi guarantees not Suspect

  • [Person] says [A], so they believe [B]

    Say(A) guarantees Believe(B)

    He says the bank is open, so he believes he can deposit checks.Says(Open) guarantees Believes(Deposit)

Part 11: Conditional errors

These errors take a narrow premise and read a broader relationship into it than the evidence supports.

Illegal reversal and illegal negation

  • Illegal reversal. "If A, then B" does not tell you "if B, then A."
  • Illegal negation. "If A, then B" does not tell you "if not A, then not B."
Every square is a rectangle, and most rectangles are not squares. Rectangles Squares
Squares sit inside rectangles: every square is a rectangle, but most rectangles are not squares.
  1. Square guarantees RectangleFollows
  2. Rectangle guarantees SquareDoes not follow
  3. not Square guarantees not RectangleDoes not follow
  4. not Rectangle guarantees not SquareFollows

The "most" error

You are not allowed to end a conditional chain in a "most" and draw a conclusion.

  1. NBA Players guarantees Tall PeoplePremise
  2. Tall People most are Non-AthletesPremise
  3. NBA Players some are Non-AthletesDoes not follow
Every NBA player can sit in the overlap, leaving no NBA player among the non-athletes. Athletes Tall people NBA players Not athletes
All of the NBA players could be among the tall people who are athletes. Nothing requires even one NBA player to be a non-athlete.

That is different from a most-most. In Two Majorities Within One Group the two majorities come from the same group rather than from a chain, so they have to overlap.

Back to Blog