Statements: All men are vertebrates. Some mammals are vertebrates.…

Concept: In a syllogism, a conclusion only follows if it can be validly derived from the given statements. A universal affirmative statement ('All X are Y') cannot be reversed into 'All Y are X' — that reversal is an invalid conversion. A particular affirmative statement ('Some X are Y'), however, always converts validly into 'Some Y are X', because asserting an overlap between two classes is symmetric.

Application: Premise 1 places men entirely inside vertebrates ('All men are vertebrates'). Premise 2 asserts that mammals and vertebrates overlap ('Some mammals are vertebrates'). Checking each conclusion against these:

1. All men are mammals — no premise relates men to mammals directly, so this cannot be derived.

2. All mammals are men — the same issue in the other direction; nothing connects mammals to men.

3. Some vertebrates are mammals — this is exactly the valid conversion of 'Some mammals are vertebrates', since a particular affirmative converts symmetrically. This follows.

4. All vertebrates are men — this would require reversing the universal statement 'All men are vertebrates', which is an invalid conversion; other vertebrates need not be men.

Cross-check: Drawing the classes as circles confirms this: vertebrates contains men entirely, and mammals overlaps vertebrates in some region — but that region need not touch the men-circle at all. The only relationship guaranteed in every such diagram is the overlap between vertebrates and mammals stated in reverse, i.e., conclusion (3).

Result: Only conclusion (3) follows.