Conditional Logic – Comprehensive LSAT Study Notes

Introduction to Conditional Logic

  • Goal of the lesson
    • Understand what a conditional statement does and does not tell us.
    • Learn and memorize the most common keywords that signal conditionality on the LSAT.
    • Practice turning English sentences into symbolic, arrow‐based diagrams.
  • Core metaphor
    • Think of each conditional rule as an unbreakable law: when the left side (sufficient condition) is met, the right side (necessary condition) must follow.

Anatomy of a Conditional Statement

  • Basic structure: \text{sufficient} \; \rightarrow \; \text{necessary}
  • Read it aloud as: “IF the left side is true, THEN the right side is true.”
  • Example
    • \text{NYC} \rightarrow \text{USA}
    • If someone is in New York City, then they are in the USA.
  • Vocabulary
    • Left side = “sufficient” or “trigger.”
    • Right side = “necessary” or “outcome.”
  • Certainty conveyed by the arrow
    • The arrow only guarantees truth from left to right.
    • The rule has no authority when the left side is false or when only the right side is known to be true.

Inference Rules: What You CAN & CANNOT Conclude

  • When sufficient is true → you may assert the necessary.
    • Linda in NYC ⇒ Linda in USA.
  • When sufficient is false → rule is silent.
    • Joe not in NYC ⇒ could be USA, could be elsewhere.
  • When necessary is true → rule is silent.
    • Alice in USA ⇒ no info about whether she is in NYC.
  • When necessary is false → you may assert not-sufficient (by contrapositive).
    • Dave not in USA ⇒ definitely not in NYC.
  • LSAT Flaw pattern to watch: “Confusing the necessary and the sufficient” (a.k.a. reading the arrow backward).

Contrapositive & Truth-Value Flips

  • Definition: Logical twin obtained by
    1. Reversing the order.
    2. Negating (flipping the truth value of) every component.
  • Symbolically: A \rightarrow B becomes \neg B \rightarrow \neg A
  • Visual: like flipping a coin upside-down.
  • Both original rule and its contrapositive are logically equivalent—two sides of the same coin.

Flipping AND / OR

  • While writing contrapositives, every AND switches to OR and every OR switches to AND.
  • Example
    • Original: (A \wedge B) \rightarrow C
    • Contrapositive: \neg C \rightarrow (\neg A \vee \neg B)
  • Rationale: Negating a compound statement requires applying De Morgan’s Laws.

Chaining Conditionals & Deriving New Conditionals

  • “Chaining” = linking multiple rules where the necessary of one matches the sufficient of the next.
  • Example chain
    • A \rightarrow B
    • B \rightarrow C
    • C \rightarrow D
    • Combined inference: A \rightarrow D (and by contraposition \neg D \rightarrow \neg A).
  • LSAT may test both end-to-end chains and intermediate links (e.g., A \rightarrow C or B \rightarrow D).

Triggering Rules with Facts & Deriving Facts

  • Two main LSAT activities
    1. Combine conditionals → create new conditional rules.
    2. Combine a conditional rule with a specific fact → derive specific factual inference.
  • Example with disjunction
    • Rule: A \rightarrow (B \vee C)
    • Facts: A is true, \neg C is true.
    • Inference: B must be true.
  • Complex scenario exercise
    • Provided: A \wedge B \rightarrow C and C \rightarrow D plus fact \neg A.
    • Asked: “If B is true, what follows?”
    • Chain reveals B \rightarrow D because B + given \neg A would trigger the first rule once A were true. (Video flagged this as the LSAT-style correct answer.)

Primary Contexts Where LSAT Uses Conditional Logic

  • Rules / Principles (explicit “if–then” statements).
  • Universal categorical statements (“All A are B”).
  • Causal guarantees (“This event always causes that event”).
  • Requirements / necessities (“X requires Y”).

Keywords & Their Placement Rules

  • LEFT-side (put attached idea before arrow): if, when, whenever, each, any, all, every, in order to, people who, the only, etc.
  • RIGHT-side (put attached idea after arrow): then, only, only if, requires, depends on, must, necessitates, essential, implies, guaranteed by.
  • Strategy: Memorize the “feel” for right-side words—“required/necessary” terms always move right.

Special / Tricky Keyword Families

1. NO-Statements

  • Pattern: “No A are B” ⇒ A \rightarrow \neg B
  • Common pitfall: reversing and negating the wrong side.
  • Example: “No camels attend law school” ⇒ \text{Camel} \rightarrow \neg \text{Attend-Law-School}

2. Negative-to-Positive Quantifier Translations

  • “No A are B” → “All A are not B”.
  • “Few A are B” → “Most A are not B”.
  • “Not all A are B” → “Some A are not B”.
  • Guideline: Rewrite negatives into positive statements for easier reasoning.

3. Unless / Until / Without (IF-NOT Translation)

  • Mechanical approach: replace the word with “IF NOT.”
    • “Eddie can’t go to Legoland until he feeds the rabbit.”
      \neg \text{Feed-Rabbit} \rightarrow \neg \text{Legoland}
  • Requirement approach: identify what is required.
    • “Legoland requires feeding the rabbit.” ⇒ \text{Legoland} \rightarrow \text{Feed-Rabbit} (contrapositive of prior formulation).
  • Watch for complex disjunctions: negating L \vee P becomes \neg L \wedge \neg P.
    • “Happiness is never achieved unless love or passion is present.”
    • IF-NOT form: \neg (L \vee P) \rightarrow \neg H ⇒ actually \neg L \wedge \neg P \rightarrow \neg H.
    • Requirement form: H \rightarrow (L \vee P).

4. Biconditionals

  • Indicators: “if and only if,” “then and only then,” “when and only when,” “otherwise.”
  • Meaning: Both directions hold; diagram with double arrow \leftrightarrow.
  • Example: “Jeff takes a vacation if and only if Trixie gets a promotion.”
    • \text{Promotion} \leftrightarrow \text{Vacation}
    • Equivalent either-or form: exactly both happen or neither happens.
  • “Otherwise” pattern
    • “Jeff will take a vacation if he wins the lottery; otherwise he won’t.”
    • Two separate rules combine into the biconditional \text{Lottery} \leftrightarrow \text{Vacation}.

Practice Examples Reviewed in Video

  1. “When someone says ‘totes,’ Paloma gets annoyed.”
    • \text{Say-Totes} \rightarrow \text{Paloma-Annoyed}
    • Contra: \neg \text{Paloma-Annoyed} \rightarrow \neg \text{Say-Totes}
  2. “Enjoying fudge implies enjoying sugar.”
    • \text{Enjoy-Fudge} \rightarrow \text{Enjoy-Sugar}
    • Contra: \neg \text{Enjoy-Sugar} \rightarrow \neg \text{Enjoy-Fudge}
  3. “No cowards play volleyball.”
    • \text{Coward} \rightarrow \neg \text{Play-Volleyball}
  4. “Each of the loose golf balls is covered in bird poop.”
    • \text{Loose-Golf-Ball} \rightarrow \text{Covered-in-Poop}
  5. “I will quit unless you apologize.”
    • IF-NOT version: \neg \text{Apologize} \rightarrow \text{Quit}
    • Requirement version: \neg \text{Quit} \rightarrow \text{Apologize}
  6. “Real men never shy away from yogurt parfaits.”
    • \text{Real-Man} \rightarrow \neg \text{Shy-Away-Parfait}
  7. “True worshipers will be accepted by that church.”
    • \text{True-Worshiper} \rightarrow \text{Accepted-by-Church}
  8. “Using breath strips instead of mints is the only way to be cool.”
    • \text{Cool} \rightarrow \text{Use-Breath-Strips} (since “only” attaches to cool—right-side keyword).
  9. “Vanessa can’t work without tea and snacks.”
    • Requirement form: \text{Work} \rightarrow (\text{Tea} \wedge \text{Snacks})
  10. “Learning to walk is a prerequisite for running.”
    • \text{Run} \rightarrow \text{Learn-to-Walk}

Guidelines / Cheat-Sheet for Diagramming

  • While studying, keep a two-column list of left-side vs right-side keywords.
  • Translation routine
    1. Identify keyword(s).
    2. Decide left vs right placement.
    3. Apply IF-NOT or NO rules when relevant.
    4. Draw arrow; immediately write the contrapositive, flipping every AND/OR.
  • Remember: required element always lives on the right.

Common LSAT Traps & Fallacies

  • Reading the arrow backward (mistaking necessary for sufficient).
  • Negating incorrectly inside compound statements (forgetting AND/OR flip).
  • Assuming “not-trigger” ⇒ “not-outcome” (illegal negation).
  • Overlooking biconditional wording (“only if,” “if and only if”).

Ethical / Practical Implications Mentioned

  • None explicitly ethical, but mastering conditional logic prevents flawed reasoning—vital for law and daily decision-making.
  • Example connection: Admissions statements like “can’t get into Yale without a great LSAT score” illustrate real-world stakes of sufficiency/necessity.

Study Strategy & Next Steps

  • Drill keyword recognition—cluster words by family (rules, universals, guarantees, requirements).
  • Practice translating “no / unless / until / without” daily; these cause the most errors.
  • When watching LSAT flaw questions, watch for authors who confuse necessary & sufficient.
  • Remove training wheels gradually—strive for effortless spotting of conditional structures.
  • Additional resources: LSAT Lab YouTube channel & lsatlab.com for deeper dives.