Logical syllogism — determine which conclusions follow beyond doubt. Statements: 1) All men are vertebrates. 2) Some mammals are vertebrates. Conclusions: (1) All men are mammals. (2) All mammals are men. (3) Some vertebrates are mammals. (4) All vertebrates are men.

Introduction / Context:
This problem checks your ability to recognize necessary conclusions from two categorical premises involving sets: men, mammals, and vertebrates.

Given Data / Assumptions:

• All men ⊆ vertebrates.
• Some mammals ⊆ vertebrates.
• We must accept only what must be true in every model that satisfies the statements.

Concept / Approach:
From “Some mammals are vertebrates,” we can immediately convert to “Some vertebrates are mammals.” Universals (“All …”) do not allow converses (e.g., “All vertebrates are men”) without evidence. Also, nothing connects all men to mammals directly.

Step-by-Step Solution:
Check (3): Some vertebrates are mammals — directly supported by Statement 2 via simple conversion; hence it follows. Check (1): All men are mammals — not implied; men could be vertebrates that are not mammals. Check (2): All mammals are men — no support; mammals could include many non-human species. Check (4): All vertebrates are men — unjustified and clearly false in general.

Verification / Alternative check:
Construct an example: let vertebrates include fish and men; let mammals include dogs (a subset of vertebrates). Then “Some vertebrates are mammals” is true (dogs), but (1), (2), (4) are not forced. This validates only (3).

Why Other Options Are Wrong:
(1) and (2) over-generalize; (4) is an extreme converse/generalization without basis. Only (3) is logically compelled.

Common Pitfalls:
Mistaking subset relations for equality, and assuming converses of universal statements are true.

Final Answer:
Only (3)