If the proposition ‘All thieves are poor’ is false, which of the following…
2016
If the proposition ‘All thieves are poor’ is false, which of the following propositions can be claimed certainly to be true?
Propositions :
- A.
Some thieves are poor.
- B.
Some thieves are not poor.
- C.
No thief is poor.
- D.
No poor person is a thief.
Show answer & explanation
Correct answer: B
Concept: A categorical proposition takes one of four forms — A (universal affirmative: "All S are P"), E (universal negative: "No S are P"), I (particular affirmative: "Some S are P"), or O (particular negative: "Some S are not P"). The traditional Square of Opposition fixes how these relate:
Contradictories (A and O, or E and I): always have opposite truth values — if one is false, the other is certainly true, and vice versa.
Contraries (A and E): cannot both be true, but can both be false at the same time.
Subcontraries (I and O): cannot both be false, but can both be true at the same time.
Subalternation (A to I, E to O): the universal implies the particular, but this inference runs only in that one direction.
Application: "All thieves are poor" is an A-type proposition (S = thieves, P = poor), and the stem states it is false. A and O are contradictories, so a false A makes its O-form certainly true.
Identify the given proposition: "All thieves are poor" = A-type ("All S are P").
The O-form for the same subject-predicate pair is "Some thieves are not poor".
By the contradictory relation, exactly one of A and O is true and the other false.
Since A is given false, O = "Some thieves are not poor" must certainly be true.
Cross-check: the remaining propositions stand in weaker relations to A, so A being false does not fix their truth value either way.
Proposition | Logical form | Relation to "All thieves are poor" | Fixed by A being false? |
|---|---|---|---|
Some thieves are poor | I-type (particular affirmative) | Subalternate of A | No — subalternation only runs from A being true to I being true, not from A being false. |
No thief is poor | E-type (universal negative) | Contrary of A | No — contraries can both be false at once, so E is not forced true. |
No poor person is a thief | E-type (universal negative, predicate-led) | Contrary-type relation to A | No — the same indeterminacy applies; A being false does not force this statement true. |
Result: only the contradictory (A-O) relation guarantees a fixed truth value when A is false. So "Some thieves are not poor" is the one proposition that can be claimed certainly true.