Public Enterprise Korean Language Logic Ch1. Conditionals and Disjunctions — ~(A→B) and Logical Transformation
Chapter 1. The Fundamentals of Conditionals and Disjunctions
The single most fatal mistake in Korean-language logic problems is negating a conditional statement. It happens because everyday intuition and the formal rules of logic pull in opposite directions. In today’s lesson, we’ll resolve that conflict for good and give you the ability to mechanically convert any sentence.
0. Truth table simulation
Work through the truth tables for the conditional (A→B), the disjunction (A∨B), and the conjunction (A∧B) yourself to see how each connective behaves.
| A | B | A→B | A∨B | A∧B |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | T | F |
| F | F | T | F | F |
1. Negating a conditional: ~(A → B) ≡ A ∧ ~B
This is the single most important formula. What is the only situation that breaks (negates) the promise “If it rains, the ground gets wet”?
- Key point: It is not “If it doesn’t rain, the ground doesn’t get wet”!
- Formula: Leave the antecedent (A) untouched, negate only the consequent (B), and join them with “and” (∧).
Test-takers’ #1 mistake: The negation of “If you pass, I’ll buy you a car” is not “If you don’t pass, I won’t buy you a car” — it’s “a situation where you passed, yet no car was bought.”
2. The conditional-disjunction interchange formula
Logic problems only fall into place once you can freely move between conditionals (→) and disjunctions (∨).
- Application: “I eat an apple or a pear (A ∨ B)” → “If ~A, then B (if I don’t eat an apple, I eat a pear)”
- Practical tip: The instant you see an “or (∨)” sentence, convert it into a conditional with a single arrow.
3. Evolving problem practice (Step 3: Logical transformation)
Question 2: When the proposition “If Cheolsu is the culprit, then Younghee is innocent” is false, which of the following must be true?
- Either Cheolsu is not the culprit, or Younghee is innocent.
- If Cheolsu is not the culprit, Younghee is not innocent.
- Cheolsu is the culprit and Younghee is not innocent.
- Either Cheolsu is not the culprit, or Younghee is not innocent.
[Step-by-step algorithm]
- Symbolize the sentence: Cheolsu is the culprit (A) → Younghee is innocent (B)
- What the question asks: find the situation where ~(A → B) is “true” (since the original proposition is “false”).
- Apply the formula: ~(A → B) ≡ A ∧ ~B
- Interpretation: “Cheolsu is the culprit (A), and Younghee is not innocent (~B).”
Correct answer: 3
4. Evolving problem practice (Step 4: Compound conditionals)
Question 3: Given that all three of the following conditions are true, which one must be true?
- Condition 1: Either A goes forward or B goes forward.
- Condition 2: If B goes forward, C does not go forward.
- Condition 3: C goes forward.
- Both A and B go forward.
- B goes forward and A does not.
- A goes forward and B does not.
- Nothing goes forward.
[Thought process: building a chain of arrows]
- Convert condition 1 (a disjunction): ~A → B (equivalently, ~B → A)
- Condition 2: B → ~C
- Condition 3: C (a confirmed fact)
- Start reasoning backward:
- Since C is true, the contrapositive of condition 2 (~(~C) → ~B, i.e. C → ~B) confirms ~B (B does not go forward).
- Since ~B is confirmed, the converted form of condition 1 (~B → A) confirms A (A goes forward).
- Conclusion: A goes forward and B does not.
Correct answer: 3
🚀 Insight from the premium course
For complex problems, the standard approach is to start from a “confirmed fact” (here, C) and work backward along the reverse direction of the arrow (the contrapositive). The moment you spot “or (∨),” don’t hesitate — convert it into an arrow. The path forward will reveal itself.
In the next lesson, we’ll use De Morgan’s laws in depth and reductio ad absurdum (proof by contradiction) to break down seemingly unbeatable puzzles.
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