In the Venn-diagram test of the categorical syllogism AEE in figure IV, which…
2024
In the Venn-diagram test of the categorical syllogism AEE in figure IV, which regions must be shaded to represent the first premise?
Answer: C. The regions S ∩ P ∩ M̄ and S̄ ∩ P ∩ M̄ — ConceptA universal affirmative proposition ‘All A are B’ means there are no members in A but outside B. In a three-circle Venn diagram, every cell of A ∩ B̄…
- A.
The region S ∩ P ∩ M
- B.
The region S ∩ P̄ ∩ M
- C.
The regions S ∩ P ∩ M̄ and S̄ ∩ P ∩ M̄
- D.
The regions S̄ ∩ P̄ ∩ M and S ∩ P ∩ M
Show answer & explanation
Correct answer: C
Concept
A universal affirmative proposition ‘All A are B’ means there are no members in A but outside B. In a three-circle Venn diagram, every cell of A ∩ B̄ is therefore shaded; the third class only splits that empty part into two cells.
Application
In figure IV, the mood AEE has the major (first) premise ‘All P are M.’
Translate it as P ∩ M̄ = ∅.
The S circle divides P ∩ M̄ into S ∩ P ∩ M̄ and S̄ ∩ P ∩ M̄.
Shade both cells, because their union is exactly P ∩ M̄.
Cross-check and contrast
S ∩ P ∩ M lies inside M, so it is not part of P outside M.
S ∩ P̄ ∩ M lies outside P, so it cannot express the emptiness of P outside M.
The mixed pair S̄ ∩ P̄ ∩ M and S ∩ P ∩ M does not cover the two cells of P ∩ M̄.
Hence the required regions are S ∩ P ∩ M̄ and S̄ ∩ P ∩ M̄.