Which of the following propositions are contradictory? (A) All squares are…
2024
Which of the following propositions are contradictory?
(A) All squares are rectangles
(B) No squares are rectangles
(C) Some squares are rectangles
(D) Some squares are not rectangles
Choose the correct answer from the options given below:
Answer: B. (A) and (D) Only — Concept: In traditional (Aristotelian) logic every categorical proposition has one of four standard forms: A, the universal affirmative "All S are P"; E, the…
- A.
(C) and (D) Only
- B.
(A) and (D) Only
- C.
(B) and (D) Only
- D.
(A) and (B) Only
Show answer & explanation
Correct answer: B
Concept: In traditional (Aristotelian) logic every categorical proposition has one of four standard forms: A, the universal affirmative "All S are P"; E, the universal negative "No S are P"; I, the particular affirmative "Some S are P"; and O, the particular negative "Some S are not P". Two propositions are contradictory when they can neither both be true nor both be false, so exactly one of them must be true for every possible subject and predicate. On the square of opposition contradiction runs along the two diagonals, A with O and E with I; the remaining relations are contrariety (A with E, which may both be false), subcontrariety (I with O, which may both be true) and subalternation (A with I, E with O).
Application
First classify each statement in the stem by its quantity (universal or particular) and its quality (affirmative or negative):
"All squares are rectangles" asserts the predicate of the whole subject class, so it is the A form (universal affirmative).
"No squares are rectangles" denies the predicate of the whole subject class, so it is the E form (universal negative).
"Some squares are rectangles" asserts the predicate of part of the subject class, so it is the I form (particular affirmative).
"Some squares are not rectangles" denies the predicate of part of the subject class, so it is the O form (particular negative).
Now read the diagonals of the square. A pairs with O and E pairs with I, so these four statements actually contain two contradictory pairs: "All squares are rectangles" with "Some squares are not rectangles" (A with O), and "No squares are rectangles" with "Some squares are rectangles" (E with I). Only one of those two pairs appears among the answer choices, and the stem directs you to choose from the options given below, so the pair to select here is (A) and (D). This also matches the ordinary meaning of denial: to deny "all squares are rectangles" is precisely to assert "some squares are not rectangles", nothing weaker and nothing stronger.
Contrast
The table below places every pairing that appears in this item, including the second contradictory pair that the answer choices do not offer:
Pair | Forms joined | Relation and why it is not contradiction |
|---|---|---|
(A) and (D) | A with O | Contradictories: exactly one of the two is true in every case. This is the contradictory pair that the answer choices offer. |
(A) and (B) | A with E | Contraries: they cannot both be true, but both are false when only part of the class has the predicate. |
(C) and (D) | I with O | Subcontraries: they cannot both be false, but both are true when part of the class has the predicate and part does not. |
(B) and (D) | E with O | Subalternation: an E that is true makes its O true, so the two can hold together. |
(B) and (C) | E with I | Contradictories as well, on the square’s other diagonal, but this pair is not offered among the answer choices. |
Cross-check
Contradiction is a relation between forms, so it must survive every substitution of subject and predicate, not just this one. Test with S = birds and P = creatures that fly:
A becomes "All birds fly" (false) and E becomes "No birds fly" (false). Both false at once, so A and E are contraries, not contradictories.
I becomes "Some birds fly" (true) and O becomes "Some birds do not fly" (true). Both true at once, so I and O are subcontraries, not contradictories.
E is false here while O is true; with S = squares and P = rectangles, E is false and O is false as well. Two propositions that are false together are not contradictories.
A is false here while O is true; with S = squares and P = rectangles, A is true while O is false. Exactly one holds in each case, which is the defining behaviour of contradictories.
Both diagonals behave this way: the E with I pair, "No squares are rectangles" and "Some squares are rectangles", also keeps opposite truth values under every substitution, so contradiction is not unique to A with O. Of the two contradictory pairs, only the A with O pair appears among the answer choices, and that is why "All squares are rectangles" together with "Some squares are not rectangles" is the pair to select here.