Public Enterprise Korean Language Logic Ch0. Bricks of Logic — Categorical Propositions and Quantifiers (All / Some)

Chapter 0. Bricks of Logic: Categorical Propositions and Quantifiers

The first wall you hit when solving logical reasoning problems is deciding how to turn a sentence into “symbols.” In particular, categorical propositions containing “All” and “Some” split cleanly into those you can rewrite as a conditional (A→B) and those you cannot. If this foundation isn’t solid, you can’t even begin the harder puzzles.

1. The four standard forms of categorical propositions

A categorical proposition defines the relationship between subject and predicate in terms of “Quantity” and “Quality.”

The 4 types of categorical propositions (AEIO)
Type	Standard statement	Symbolization	Valid transformation
A (Universal Affirmative)	All S are P	S → P	Contrapositive (~P → ~S)
E (Universal Negative)	No S is P	S → ~P	Conversion (P → ~S)
I (Particular Affirmative)	Some S is P	S ∩ P ≠ ∅	Conversion (Some P is S)
O (Particular Negative)	Some S is not P	S ∩ ~P ≠ ∅	No valid conversion

★ Absolute rule: If you write “Some S is P” as S → P, you are eliminated on the spot.

All: An arrow (→) is allowed.
Some: No arrow is allowed. Read it as “the intersection (∩) is non-empty.”

2. The logical definition of “Some”

In everyday language, “Some student was late today” tends to conjure up a single person, but in logic, “Some” means “at least one.”

Some S is P: There exists at least one thing that is both S and P.
Negation: The negation of “Some S is P” is “No S is P” (a universal negative).

3. Visualization strategy with Venn diagrams

Don’t try to solve complex categorical syllogisms in your head — draw them.

Categorical proposition visualization algorithm

Check the premises

Separate the 'All' propositions from the 'Some' propositions.

Shade the excluded region

For 'All S are P' → shade out (delete) the region that is S but not P.

Mark existence

For 'Some S is P' → mark an 'X' in the overlap between S and P.

Check the conclusion

Look at the diagram and verify whether the conclusion is necessarily true.

4. Evolving problem practice (Step 1: Basics)

Question 0: Which of the following is logically equivalent to “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]

Symbolize the sentence: Philosopher (S) → Logician (P)
An “All” proposition is a conditional, so its contrapositive holds.
Contrapositive: ~P → ~S (“If you are not a logician, you are not a philosopher”)

Correct answer: 2

5. Evolving problem practice (Step 2: Advanced)

Question 1: Given that 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.
Anyone who is not a certain politician is not a lawyer.
Some politicians are legal experts.
No one is both a politician and not a legal expert.

[Thought process: solving without a Venn diagram]

Premise 1: Lawyer → Legal expert
Premise 2: Some lawyer = politician (i.e., there exists someone who is both a lawyer and a politician)
That “some lawyer” from premise 2 is, by premise 1, also necessarily a “legal expert.”
Therefore, there exists at least one person who is both a politician and a legal expert.

Correct answer: 3

🚀 Insight from the premium course

When “Some” shows up on the actual test, approach it from the angle of existence. If you’re chaining arrows together and “Some” interrupts the chain, from that point on you can no longer connect a plain arrow — you can only rely on the “intersection” property.

In the next lesson, we’ll build practical skills around the conversion rule for ~(A→B) and eliminating disjunctions.