Chapter 0. Bricks of Logic: Categorical Propositions and Quantifiers

논리 추론 문제를 풀 때 가장 먼저 만나는 벽은 문장을 어떻게 ‘기호’로 바꿀 것인가입니다. 특히 “모든”과 “어떤”이 포함된 정언명제는 조건문(A→B)으로 바꿀 수 있는 것과 없는 것이 명확히 구분됩니다. If this foundation is shaken, difficult quizzes cannot even begin.

1. Four standard forms of categorical propositions

A categorical proposition defines the relationship between subject and predicate as ‘Quantity’ and ‘Quality’.

4 types of categorical propositions (AEIO)
type	standard statement	symbolization	valid conversion
A (uniform positive)	All S are P	S → P	Treatment (~P → ~S)
E (universal negation)	Not all S are P	S → ~P	Replacement (P → ~S)
I (specific positive)	Some S is P	S ∩ P ≠ ∅	Replace (some P is S)
O (specific negation)	Some S is not P	S ∩ ~P ≠ ∅	Not convertible

★ Absolute caution: If you write “Some S is P” as S → P, you are immediately eliminated.

All: Arrows (→) can be used.
Some: Arrows cannot be used. It should be understood as ‘there is an intersection (∩)’.

2. Logical definition of ‘something’

일상 언어에서 “오늘 어떤 학생이 지각했다”라고 하면 ‘한 명’만 떠올리기 쉽지만, 논리학에서 **‘어떤(Some)‘은 ‘적어도 한 명 이상(At least one)‘**을 의미합니다.

Some S is P: There is at least one being that is both S and P at the same time.
Negation: The negation of “Some S are P” is “Not all S are P” (universal negation).

3. Visualization strategy through Venn diagram

Do not solve complex categorical propositions and syllogisms in your head, but draw them.

Categorical proposition visualization algorithm

Check premises

Distinguish between 'all' propositions and 'some' propositions.

Marking areas

'All S are P' → Shade (delete) areas that are S but are not P.

Existence indication

'Some S is P' → Mark 'X' at the intersection area of S and P.

Verification of conclusion

Look at the picture and check whether the conclusion is necessarily true.

4. Solving evolving problems (Step 1: Basics)

Question 0: Which of the following has the logical equivalent of “All philosophers are logicians”?

Some philosophers are logicians.
If you are not a logician, you are not a philosopher.
Some logicians are philosophers.
If you are not a philosopher, you are not a logician.

[Thought Process]

Sentence symbolization: Philosopher (S) → Logician (P)
Since ‘all’ propositions are ‘conditional sentences’, treatment is established.
Treatment: ~P → ~S (If you are not a logician, you are not a philosopher)

Correct answer: 2

5. Solving evolving problems (Step 2: Advanced)

Problem 1: If the following two premises are true, which conclusion is necessarily true?

Premise 1: All lawyers are legal experts.

Premise 2: Some lawyers are politicians.

Not all politicians are legal experts.
Unless you are a certain politician, you are not a lawyer.
Some politicians are legal experts.
There is no one who is both a politician and a legal expert.

[Thinking Process: Solving Without a Venn Diagram]

Premise 1: Lawyer → Legal Professional
Premise 2: Some lawyer = politician (i.e., there exists someone who is both a lawyer and a politician)
The ‘some lawyer’ in premise 2 is naturally also a ‘legal expert’ according to premise 1.
Therefore, there is at least one person who is both a politician and a legal expert.

Correct answer: 3

🚀 Insight from the 200,000 won lecture

When “something” comes up in the actual test, approach it from the perspective of ‘Exist’. 화살표 사슬을 연결하다가 중간에 “어떤”이 끼어들면, 그 지점에서 더 이상 일반적인 화살표 연결은 불가능하고 오직 ‘교집합’ 성질만 이용할 수 있습니다.

In the next enrichment class, we will learn the conversion principle of ~(A→B) and practical skills through declarative elimination.

Public Enterprise Korean Language Logic Ch0. Bricks of logic — categorical propositions and quant...