Korean Logic • Chapter 4 • 5 min read

Public Enterprise Korean Language Logic Ch4. A Master's Toolkit — Reductio ad Absurdum and Case Splitting

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Chapter 4. A Master’s Toolkit: Reductio ad Absurdum and Case Splitting

The contrapositive, syllogisms, and De Morgan’s laws you’ve learned so far are your “basic tools.” But what really rattles you on the actual test is “not knowing where to even start.” When not a single piece of information is confirmed, masters reach for two powerful weapons.

1. Reductio ad absurdum (proof by contradiction)

Reductio ad absurdum proves that an assumption is wrong by checking whether assuming it leads to a contradiction.

Reductio ad absurdum execution algorithm

1

Set up a hypothesis

Arbitrarily assume that one proposition of unknown truth value is 'true.'

2

Chain the inference

Based on that assumption, follow the chain of arrows to determine the truth value of the other propositions.

3

Look for a contradiction

Check whether the reasoning produces conflicting information (e.g., both A and ~A).

4

Draw the conclusion

If a contradiction shows up? → The initial assumption is 'false.' No contradiction? → The assumption is 'possible.'

2. Case splitting

Use this strategy when a conditional is loaded with “or (∨)” statements, or when there’s no confirmed information at all.

  • Method: Pick one variable, split into the [case where it’s true] and the [case where it’s false], and record the outcome of each in a table (matrix).
  • Tip: If both cases lead to a shared conclusion, that shared conclusion is exactly the “conclusion that must be true.”

3. Evolving problem practice (Step 5: reductio ad absurdum in action)

Question 4: The culprit is one of Person A, B, C, or D. Only one of the following four statements is true. Who is the culprit?

  • Person A: B is the culprit.
  • Person B: D is the culprit.
  • Person C: I am not the culprit.
  • Person D: B’s statement is false.

[Thought process: applying reductio ad absurdum]

  1. Assume A is the culprit

    • A (F), B (F), C (T), D (T) → 2 statements are true (contradiction! only one may be true)

  2. Assume B is the culprit

    • A (T), B (F), C (T), D (T) → 3 statements are true (contradiction!)

  3. Assume C is the culprit

    • A: “B is the culprit” is false (F)
    • B: “D is the culprit” is false (F)
    • C: since C is the culprit, “I am not the culprit” is false (F)
    • D: B’s statement really is false, so this is true (T)
    • Exactly 1 statement is true → condition satisfied!

  4. Assume D is the culprit

    • A (F), B (T), C (T), D (F) → 2 statements are true (contradiction!)

  5. Wait — spotting the contradictory pair first is faster. (a premium-course skill)

    • B and D stand in a contradictory relationship (D directly denies B’s statement → exactly one of them must be true, the other false).
    • So the single “true” statement has to belong to either B or D.
    • That means A and C must both be false (F).
    • Since C’s statement (“I am not the culprit”) is false → C is the culprit! (confirmed)
    • If C is the culprit, then B’s statement (“D is the culprit”) is false and D’s statement (“B’s statement is false”) is true — everything lines up.

Answer: The culprit is C, and D is the one who told the truth.

4. Evolving problem practice (Step 6: case splitting)

Question 5: The following conditions describe whether projects A, B, and C move forward. Which of these must be true?

  1. If A moves forward, B moves forward too.
  2. If B does not move forward, C does not move forward either.
  3. Either C moves forward, or A moves forward.

[Thought process: splitting into cases] Symbolize: (1) A→B, (2) ~B→~C (contrapositive: C→B), (3) C∨A

Case 1: A moves forward (A=T)

  • By (1), B=T
  • C is undetermined by the contrapositive of (2) (either T or F is possible)
  • (3) C∨A is satisfied (since A is T)

Case 2: A does not move forward (A=F)

  • By (3) C∨A, C=T is required
  • By the contrapositive of (2) (C→B), B=T is required

Putting it together: In both Case 1 and Case 2, B moves forward (T) without exception.

Answer: B is guaranteed to move forward.

🚀 Insight from the premium course

“Hard problems don’t stump you because you lack knowledge — they stump you because you lack the nerve to make an assumption.” If there’s no confirmed information, just grab any one variable and say “let’s assume it’s true.” Every formula of logic you’ve learned starts working the moment you do.

In the next lesson, we’ll take down the final bosses of logic puzzles using quantitative reasoning and precise case counting.

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