If the statement ‘some men are cruel’ is false, which of the following…
2011
If the statement ‘some men are cruel’ is false, which of the following statements/statement are/is true ?
(i) All men are cruel.
(ii) No men are cruel.
(iii) Some men are not cruel.
Answer: C. (ii) and (iii) — Concept — the traditional square of opposition. Categorical statements come in four standard forms: A = “All S are P” (universal affirmative), E = “No S is P”…
- A.
(i) and (iii)
- B.
(i) and (ii)
- C.
(ii) and (iii)
- D.
(iii) only
Show answer & explanation
Correct answer: C
Concept — the traditional square of opposition. Categorical statements come in four standard forms: A = “All S are P” (universal affirmative), E = “No S is P” (universal negative), I = “Some S are P” (particular affirmative) and O = “Some S are not P” (particular negative). Two relations of the square settle any item of this kind. Contradiction: I and E are contradictories, so exactly one of that pair is true and the other false. Subalternation: a true universal carries the particular beneath it (A carries I, E carries O), and therefore a false particular forces its own universal to be false.
Application. Here the subject class S is “men” and the predicate P is “cruel”, and the datum is that the I proposition “Some men are cruel” is false.
The I proposition is false. Its contradictory is the E proposition “No men are cruel”, and contradictories always take opposite truth values, so E comes out true. Statement (ii) holds.
E is now established as true. By subalternation a true universal negative carries the particular negative beneath it, so the O proposition “Some men are not cruel” comes out true. Statement (iii) holds.
The A proposition “All men are cruel” would, by subalternation, carry the I proposition “Some men are cruel”. I is false, so A cannot be true either. Statement (i) does not hold.
Cross-check — read the same datum as sets. “Some men are cruel” being false means the class of cruel men is empty.
Statement | Standard form | Truth value |
|---|---|---|
(i) All men are cruel. | A — universal affirmative | False |
(ii) No men are cruel. | E — universal negative | True |
(iii) Some men are not cruel. | O — particular negative | True |
With no cruel men at all, “No men are cruel” is plainly satisfied, and every man lies outside the cruel class, so “Some men are not cruel” is satisfied too, while “All men are cruel” would need at least one cruel man and so fails. The step from E to O relies on the traditional assumption of existential import — the class “men” is taken to be non-empty — which is the convention the square of opposition assumes. In modern predicate logic without that assumption, E would not carry O.
Result. The true statements are (ii) “No men are cruel” and (iii) “Some men are not cruel”.