Quantified Statements: All, Some, and None in Logic

So far in this series we have worked with propositional logic — statements that are simply true or false (P, Q, A, B…). But many real arguments involve claims about groups: all members of a set, some of them, or none. To handle these, we need predicate logic and its quantifiers.

This chapter covers everything you need for exam-level quantified reasoning.

The Two Quantifiers

Symbol	Name	Meaning	Example
∀x	Universal	”For all x…”	∀x: If x is a bird, x has wings
∃x	Existential	”There exists an x…”	∃x: x is a bird that cannot fly

In plain exam language:

Phrase	Quantifier
All, every, each, any	∀ (universal)
Some, at least one, there is	∃ (existential)
No, none, not any	∀… ¬ (universal negation)

Standard Statement Forms

Universal Affirmative: “All A are B”

Formal: ∀x: A(x) → B(x) Example: “All senators are politicians.”

Universal Negative: “No A are B”

Formal: ∀x: A(x) → ¬B(x) Example: “No senators are private citizens.”

Particular Affirmative: “Some A are B”

Formal: ∃x: A(x) ∧ B(x) Example: “Some politicians are honest.”

Particular Negative: “Some A are not B”

Formal: ∃x: A(x) ∧ ¬B(x) Example: “Some politicians are not honest.”

The Square of Opposition

These four forms are related in specific ways:

Universal Affirmative ←— Contrary —→ Universal Negative
"All A are B"                         "No A are B"
       |                                    |
   Subaltern                           Subaltern
       |                                    |
Particular Affirmative ←— Subcontrary —→ Particular Negative
"Some A are B"                          "Some A are not B"

Key relationships:

Contradictories (diagonals): If one is true, the other is false; if one is false, the other is true.
- “All A are B” ↔ contradicts ↔ “Some A are not B”
- “No A are B” ↔ contradicts ↔ “Some A are B”

Negating Quantified Statements

This is the most exam-tested skill:

Original	Negation
All A are B (∀: A → B)	Some A are not B (∃: A ∧ ¬B)
No A are B (∀: A → ¬B)	Some A are B (∃: A ∧ B)
Some A are B (∃: A ∧ B)	No A are B (∀: A → ¬B)
Some A are not B (∃: A ∧ ¬B)	All A are B (∀: A → B)

Rule: The negation of a universal is a particular (and vice versa), with the predicate flipped.

Example:

Original: “All civil servants are trustworthy.”
Negation: “Some civil servants are not trustworthy.” (You only need one counter-example to disprove a universal claim.)

Inference Rules: Categorical Syllogism

The classic three-line argument:

Barbara (most common exam form):

All A are B.
All B are C.
Therefore: All A are C. ✅

Celarent:

No A are B.
All C are A.
Therefore: No C are B. ✅

Darii:

All A are B.
Some C are A.
Therefore: Some C are B. ✅

Ferio:

No A are B.
Some C are A.
Therefore: Some C are not B. ✅

Common Invalid Inferences

Undistributed Middle:

All A are C.
All B are C.
Therefore: All A are B. ❌

Both A and B share property C, but that does not mean they share each other.

Example: “All dogs are animals. All cats are animals. Therefore all dogs are cats.” — Clearly invalid.

Illicit Conversion of “All”: “All A are B” does NOT imply “All B are A.” “All senators are politicians” does NOT imply “All politicians are senators.”

Existential Fallacy: From a universal statement alone, you cannot derive a particular about existence. “All dragons are fire-breathing” does not imply “Some dragons exist."

"Only” Statements

“Only A are B” means B → A, not A → B.

Example: “Only licensed professionals may prescribe medication.” Translation: Prescribes → Licensed. (Being licensed is necessary to prescribe.)

Example: “Only members may attend” = Attend → Member. Does this say members will attend? No. Does it say non-members definitely cannot attend? Yes.

Practice Problems

Problem 1: “All applicants must speak English.” An applicant does not speak English. What follows?

Solution

Universal affirmative: Applicant → SpeaksEnglish. Contrapositive: ¬SpeaksEnglish → ¬Applicant. The person who does not speak English is not (or cannot be) an applicant.

Problem 2: Negate “No department head failed the evaluation.”

Solution

“No A are B” negated = “Some A are B.” Negation: “Some department heads failed the evaluation.”

Problem 3:

All engineers know mathematics.
Some engineers work in government.
What follows?

Solution

From (1) and (2): Some engineers work in government AND know mathematics (applying Darii: All engineers know math + Some engineers work in government → Some government workers know math). Conclusion: Some government workers know mathematics.

Problem 4 (exam style): “Only those who score in the top 10% on the first round advance to interviews.”

Does advancing to an interview mean you scored in the top 10%?

Solution

“Only A advance” = Advance → Top10%. Yes: if you advanced to an interview, you must have scored in the top 10% (necessary condition).

Quantified logic is the bridge between the neat world of A → B and the messier world of real arguments about groups and categories. Master these four forms, their negations, and the basic syllogisms — and you will handle any categorical reasoning question the exam can produce.

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