Logical Reasoning — Quantifier Logic — Negation and Conjugation of “All” and “Any”

“All people die.” “Some birds cannot fly.” — These sentences contain quantifiers. A quantifier is a logical device that indicates “how many objects we are talking about.” Understanding quantifier logic lets you find the correct negation in complex reasoning problems and identify logical errors precisely.

Two Core Quantifiers

Universal Quantifier (∀)

Symbol: ∀ (“for all”)

∀x P(x): “P(x) holds for every x”

Natural-language expressions:

• “All ~ are ~”
• “~ is always ~”
• “No ~” (in a negative sentence)

Examples:

• ∀x (x is a person → x dies) = “All people die”
• “All prime numbers are greater than 1”

Existential Quantifier (∃)

Symbol: ∃ (“there exists,” some)

∃x P(x): “There is at least one x for which P(x) holds”

Natural-language expressions:

• “Some ~ are ~”
• “There is a ~ that is ~”
• “At least one ~ is ~”

Examples:

• ∃x (x is a bird ∧ x cannot fly) = “There is a bird that cannot fly”
• “Some integers are negative”

Negating Quantified Propositions: Core Rules

This is the point most often missed on exams.

Negating a Universal Proposition

$$\neg (\forall x P(x)) = \exists x \neg P(x)$$

The negation of “P(x) for every x” is “there exists an x for which P(x) is not true.”

Example:

• Original proposition: “Every student passed the exam”
• Negation: “There is a student who did not pass the exam” (at least one)

Incorrect negation: “No student passed the exam” → this is too strong a negation

Negating an Existential Proposition

$$\neg (\exists x P(x)) = \forall x \neg P(x)$$

The negation of “there exists an x for which P(x)” is “P(x) does not hold for every x.”

Example:

• Original proposition: “There is a bird that cannot fly”
• Negation: “All birds can fly”

Incorrect negation: “Some birds can fly” → this does not contradict the original proposition (both can be true)

Converting Quantifiers in Natural Language

Because exams use natural language instead of symbols, learn the following conversion table.

Universal Expressions

Natural language	Logical structure
“All A are B”	∀x (Ax → Bx)
“A is always B”	∀x (Ax → Bx)
“If A, then necessarily B”	∀x (Ax → Bx)
“There is no A that is not B”	∀x (Ax → Bx)

Existential Expressions

Natural language	Logical structure
“Some A are B”	∃x (Ax ∧ Bx)
“There is an A that is B”	∃x (Ax ∧ Bx)
“At least one A is B”	∃x (Ax ∧ Bx)

Note: “All A are B” is A → B (a conditional), whereas “Some A are B” is A ∧ B (a conjunction)!

Practical Negation Exercises

Question 1

Choose the correct negation of the following. “Every member paid the membership fee.”

① No member paid the membership fee ② There is a member who did not pay the membership fee ③ Some member paid the membership fee ④ There is a member who paid the membership fee

Solution: The negation of ∀x (member(x) → paid(x)) = ∃x (member(x) ∧ ¬paid(x)) = “There is a member who did not pay”

Answer: ②

Question 2

Choose the correct negation of the following. “Some employees worked overtime.”

① Every employee worked overtime ② No employee worked overtime ③ There is an employee who did not work overtime ④ Some employees did not work overtime

Solution: The negation of ∃x (employee(x) ∧ overtime(x)) = ∀x (employee(x) → ¬overtime(x)) = “No employee worked overtime”

Answer: ②

Syllogisms and Quantifiers

A classical syllogism using universal propositions:

Major premise: All A are B [∀x (Ax → Bx)] Minor premise: c is A [Ac] Conclusion: Therefore, c is B [Bc]

Example:

• Major premise: All mammals have hearts
• Minor premise: A whale is a mammal
• Conclusion: Therefore, a whale has a heart

Converse Syllogism Error (Invalid Syllogism)

The following is logically invalid:

Major premise: All A are B Minor premise: c is B Conclusion: Therefore, c is A ← Error! (different from affirming the antecedent)

Example:

• All dogs are animals
• A cat is an animal
• Therefore, a cat is a dog ← False!

This is the fallacy of affirming the consequent.

Quantifier Negation Summary Table

$$\begin{aligned} \neg (\text{All} A \text{are} B) &= \text{Some} A \text{are not} B (\text{there is an} A \text{that is not} B) \\ \neg (\text{Some} A \text{are} B) &= \text{No} A \text{are} B (\text{there is no} A \text{that is} B) \\ \neg (\text{If} A, \text{then} B) &= \text{There is} a \text{case where} A \text{but not} B \end{aligned}$$

LSAT-style tip: In questions asking for the correct negation of a proposition, the key is knowing the conversion between “all” and “some ~ are not.”

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Quantifier logic is a core tool in philosophy, mathematics, law (LSAT), and linguistics. Never forget that the negation of “all” is “some are not,” and the negation of “some” is “none.” In the next article (Ch7), we will cover proof by contradiction and mathematical induction — two powerful tools of proof.