Chapter 4. Master’s Tools: Reductio and Number of Cases

The treatment, syllogism, and de Morgan that we have learned so far are the ‘basic tools’. But what frustrates us in real exams is **“not knowing where to start”****. When no information is confirmed, the masters unleash two powerful weapons.

1. Reductio ad absurdum

Reduction is a method of verifying whether a contradiction occurs when assuming a claim and proving that the assumption is wrong.

Reductio method execution algorithm

Hypothesis Setting

Assume one proposition whose true/false unknown is arbitrarily 'true'.

Chain inference

Based on that assumption, the truth/false of other propositions is determined along the arrow chain.

Contradiction discovery

Check whether conflicting information (such as A and ~A) appears during the inference process.

Drawing a conclusion

What if a contradiction occurs? → The initial assumption is 'false'. What if there is no contradiction? → The assumption is 'possible'.

2. Case Split

This is a strategy used when there is a lot of “or (∨)” in the conditional sentence or when there is no definitive information.

Method: Select one variable and divide it into [If A] and [If ~A] and record each result in a table (Matrix).
Tip: If there is a common conclusion in both cases, that is a “conclusion that is necessarily true”.

3. Solving evolving problems (Step 5: Practicing the method of reductio)

Problem 4: If only one of the following four statements is true, who is the culprit?

갑: 내가 범인이다.

Eul: Person A is not the culprit.

Byung: Eul is not the culprit.

Jeong: Illness is not the culprit.

[Thought Process: Applying the Reductio ad]

Assume A is true (culprit: A)

A (T), B (F), Byeong (T), Jeong (T) → There are 3 people who say it is true (contradiction! Only one person has to say it is true)

Assume B is true (culprit: not A)

What if the culprit is Eul?
A (F), Eul (T), Byeong (F), Jeong (T) → 2 true people (contradiction!)

Assume the illness is true (culprit: B)

A (F), Eul (F), Byeong (T), Jeong (T) → 2 true people (contradiction!)

Assume the statement is true (culprit: bottle)

A: The culprit is a disease, so ‘A is the culprit’ is false (F)
B: ‘A is not the culprit’ is true (T)
Bottle: ‘Eul is not the culprit’ is true (T)
Jeong: ‘Disease is not the culprit’ is false because ‘disease is the culprit’ (F)
2 true people (T, T) → Contradiction!

Wait! Find the contradiction first. (Paid lecture skill)

A vs Eul is a contradictory relationship (one must be true, one must be false).
Therefore, one person who is ‘true’ must be either A or E.
Class status must always be False (F).
Since the illness is false → Eul is the culprit. (Confirmed!)
Since the truth is false → The bottle is the culprit. (If there cannot be multiple culprits in one incident, check the problem setting)

Correct answer: The disease is the culprit, and B is the person who made the true statement.

4. Solving evolving problems (Step 6: Separating the number of cases)

Problem 5: There are the following conditions regarding whether to proceed with projects A, B, and C. What is necessarily true?

If A is promoted, B is also promoted.

If B is not promoted, C will not be promoted either.

Either C is promoted or A is promoted.

[Thinking Process: Divide by Number of Cases] Symbolization: (1) A→B, (2) ~B→~C (treatment: C→B), (3) C∨A

Case 1: When A is promoted (A=T)

by (1) B=T
C is unknown due to treatment in (2) (either T or F is possible)
(3) C∨A is satisfied (since A is T)

Case 2: When A is not promoted (A=F)

(3) C=T required by C∨A
By the treatment (C → B) in (2), B=T is required

Overall: Whether Case 1 or Case 2, In general, B is unconditionally promoted (T).

Correct answer: B will definitely be promoted.

🚀 Insight from the 200,000 won lecture

“Difficult problems are not solved because of lack of knowledge, but because of lack of courage to assume.” If you don’t have any definitive information, just grab one and say ‘let’s assume so’. All the formulas of logic will start working from that moment.

In the next supplementary lesson, we will defeat the final bosses of the logic quiz through mathematical logic and precise measurement of the number of cases.

Public Enterprise Korean Language Logic Ch4. Master's Tools — Reductio and Number of Cases