Questions: Quantifiers: ALL, SOME, and NONE

Question 1

A study finds that some professional athletes use performance-enhancing drugs. A journalist writes the headline: 'Professional athletes use performance-enhancing drugs.' What quantifier error does the headline commit?

• A. Illicit conversion — the subject and predicate have been switched
• B. Overgeneralization — omitting 'some' implies a universal claim, making a conclusion with greater scope than the evidence warrants
• C. False dilemma — the headline suggests only two options exist
• D. No error — omitting a quantifier is standard journalistic practice that preserves the original meaning

Answer: (B)

Dropping 'some' produces a de facto universal claim: 'Professional athletes use drugs' implies all or most do. This is overgeneralization — the conclusion has greater quantifier scope than the evidence supports. The evidence warrants only 'Some professional athletes use performance-enhancing drugs.' This is one of the most common quantifier errors in public discourse.

Question 2

From the premise 'All senators are politicians,' what can we validly conclude?

• A. Nothing — universal statements don't imply particular ones
• B. Some senators are politicians — the universal entails the particular, assuming the class is non-empty
• C. Some politicians are senators — the subject and predicate can be freely swapped
• D. No non-senators are politicians — the contrapositive follows automatically

Answer: (B)

From 'All A are B' we can validly conclude 'Some A are B,' provided A is non-empty — the particular is weaker than the universal, so the universal implies it. What is NOT valid is the reverse: from 'Some A are B' you cannot conclude 'All A are B.' Option C is the error of illicit conversion: 'All senators are politicians' does NOT entail 'All politicians are senators.'

Question 3

In formal logic, 'Some A are B' is compatible with 'All A are B' — the word 'some' means at least one, not 'only some.'

• T. True
• F. False

Answer: True

Logically, 'some' sets a floor (at least one), not a ceiling. If all A are B, then certainly some A are B. The two statements are compatible. In everyday speech 'some' often pragmatically implies 'not all,' but in formal logic no such implication holds. This matters for evaluating arguments: 'Some students passed' does not conflict with 'All students passed' — it is consistent with it.

Question 4

'Most A are B' and 'No A are B' are contradictories — exactly one should be true.

• T. True
• F. False

Answer: False

'All A are B' and 'No A are B' are contraries, not contradictories. Contraries cannot both be true but can both be false: if some A are B and some are not, both universal claims are false simultaneously. Contradictories (such as 'All A are B' and 'Some A are not B') cannot both be true AND cannot both be false — exactly one must hold. The distinction between contrary and contradictory is essential for valid inference.

Question 5

Why can we not move from 'Some A are B' to 'All A are B' in logical reasoning?

Model answer: 'Some A are B' tells us only that at least one member of A has property B — it gives no information about the remaining members. Moving to 'All A are B' would claim that every member of A has property B, a scope the particular premise does not support. The conclusion reaches beyond the evidence: we observed a subset and generalized to the whole class, which is the fallacy of overgeneralization.

Formally: 'Some A are B' is existential (∃x(Ax ∧ Bx)), while 'All A are B' is universal (∀x(Ax → Bx)). An existential statement never entails a universal — no finite number of confirming instances proves a universal claim.