Chapter 1. Standards of conditional sentences and declarations

The most fatal mistake in public enterprise logic writing is negation of conditional statement. This is because the intuition of everyday language and the formulas of logic clash. In today’s class, you will completely resolve that conflict and gain the ability to mechanically convert any sentence.

1. Negation of conditional statement: ~(A → B) ≡ A ∧ ~B

This is the most important formula. What is the only situation that breaks (denies) the promise, “When it rains, the ground gets wet”?

Key point: It’s not “if it doesn’t rain, the ground doesn’t get wet”!
Formula: Leave the antecedent (A) as is, negate only the consequent (B) and connect it with ‘and (∧)’.

Negation of Conditional

~(A → B) ≡ A ∧ ~B

Only 'A but not B' makes the conditional false.

No. 1 mistake made by test takers: The negation of “I will buy you a car if you pass” is not “They won’t buy you a car if you don’t pass”, but “A situation where they won’t buy you a car even if you pass”.

2. Exchange formula for conditional and declarative statements

Logic problems can be solved by freely switching between conditional statements (→) and declarative statements (∨).

Conditional statement-declaration conversion (Arrow to Or)

A → B ≡ ~A ∨ B

'If not A, or at least B' is logically equivalent to 'If A, then B'.

Use: “Eat an apple or a pear (A ∨ B)” → “If A, then B (If you don’t eat an apple, eat a pear)”
Practical tip: Immediately convert all ‘or(∨)’ sentences into conditional sentences with a single arrow drawn.

3. Solving evolving problems (Step 3: Logical transformation)

Problem 2: Which of the following is necessarily true when the proposition “If Cheol-soo is the culprit, then Young-hee is innocent” is False?

Either Cheolsu is not the culprit or Younghee is innocent.
If Cheolsu is not the culprit, Younghee is not innocent.
Cheolsu is the criminal and Younghee is not innocent.
Cheolsu is not the culprit and Younghee is not innocent.

[Step-by-Step Algorithm]

Sentence encoding: Cheolsu (A) → Younghee’s innocence (B)
Problem requirement: Find a situation where ~(A → B) is ‘true’. (Because the original proposition is ‘false’)
Formula substitution: ~(A → B) ≡ A ∧ ~B
Interpretation: “Cheolsu is the criminal (A), and Younghee is not innocent (~B).”

Correct answer: 3

4. Solving evolving problems (Step 4: Compound conditional statements)

Problem 3: When all three conditions below are true, which one is necessarily true?

Condition 1: Either A is promoted or B is promoted.

Condition 2: If B is promoted, C is not promoted.

Condition 3: C is promoted.

Both A and B are promoted.
B is promoted and A is not promoted.
A is promoted and B is not promoted.
Nothing is promoted.

[Thought Process: Creating a Chain of Arrows]

Condition 1 (Declaration) Transformation: ~A → B (or ~B → A)
Condition 2: B → ~C
Condition 3: C (Confirmed Information)
Start reverse inference:

Since C is true, ~B (B not pursued) is confirmed by the treatment of condition 2 (~(~C) → ~B, that is, C → ~B.
Since ~B is confirmed, A (promotion of A) is confirmed by the conversion of condition 1 (~B → A**).**

Conclusion: A is promoted and B is not promoted.

Correct answer: 3

🚀 Insight from the 200,000 won lecture

The rule of thumb for complex problems is to start with an ‘established fact’ (C in this case) and work backwards in the direction of the arrow. If you see “or (∨)”, don’t worry, just change it to an arrow. You will see a path.

In the next supplementary lesson, we will overcome the unassailable quiz with Advanced practical use of De Morgan’s law and Reductio ad (contradiction proof).

Public Enterprise Korean Language Logic Ch1. The basics of conditional statements and declaration...