If the proposition ‘No men are honest’ is taken to be false which of the…
2015
If the proposition ‘No men are honest’ is taken to be false which of the following proposition/propositions can be claimed certainly to be true ?
Answer: B. Some men are honest — Concept In Aristotelian logic the four categorical forms sit at the corners of the Square of Opposition: A ("All S are P"), E ("No S are P"), I ("Some S are…
- A.
All men are honest
- B.
Some men are honest
- C.
Some men are not honest
- D.
No honest person is man
Show answer & explanation
Correct answer: B
Concept
In Aristotelian logic the four categorical forms sit at the corners of the Square of Opposition: A ("All S are P"), E ("No S are P"), I ("Some S are P") and O ("Some S are not P"). A and O are contradictories, and E and I are contradictories — a proposition and its contradictory always carry opposite truth values, so fixing one of them as false fixes the other as true.
Two further relations decide the remaining cases. Contraries (A and E) can both be false together, so the falsity of one leaves the other undetermined; and a universal negative converts simply, which means "No S is P" and "No P is S" always carry the same truth value.
Application
The given proposition "No men are honest" is the E form, with subject term "men" and predicate term "honest persons". It is stated to be false.
Its contradictory is the I form "Some men are honest". Contradictories cannot both be false, so once the E form is false the I form is forced to be true — and it is the only one of the four propositions that the given falsity forces to be true.
"All men are honest" is the A form, the contrary of the given proposition. Contraries may both be false, so a false E leaves A undetermined: it could be either true or false.
"Some men are not honest" is the O form, the subaltern of the given E form. A false universal does not fix the truth value of the particular below it, so this one is undetermined as well.
"No honest person is man" is the simple converse of the given proposition, obtained by interchanging its subject and predicate terms. Because a universal negative converts simply, it carries the same truth value as the original — and the original is given as false.
Cross-check with two models
Let the class of men be {a, b, c}. Model 1: only a is honest. Model 2: all three are honest. Both models make "No men are honest" false, so both satisfy the condition in the question.
Proposition | Model 1 (only a honest) | Model 2 (all three honest) |
|---|---|---|
"No men are honest" (given) | False | False |
"All men are honest" | False | True |
"Some men are honest" | True | True |
"Some men are not honest" | True | False |
"No honest person is man" | False | False |
Between the two models the A form flips from false to true and the O form flips from true to false, while the converse stays false in both. Only "Some men are honest" holds in every model that makes the given proposition false, and that is exactly what "certainly true" demands.
Answer
Some men are honest.