Statements: All roads are poles. No pole is a house. Conclusions: Some roads…

Concept: In syllogisms, if either premise is negative (an "is not"/"no" statement), the conclusion must also be negative — an affirmative conclusion (a "some"/"all" statement) can never validly follow from a negative premise. Also, a universal negative statement of the form "No A is B" converts validly to "No B is A" (simple conversion).

Application: Here the first premise, "All roads are poles," places roads entirely inside poles. The second premise, "No pole is a house," is negative and, by simple conversion, also gives "No house is a pole" — poles and houses share no members in either direction. Since roads sit wholly inside poles, and poles share nothing with houses, roads and houses cannot share anything either.

• "Some roads are houses" claims roads and houses overlap. But roads lie entirely within poles, and poles have zero overlap with houses, so roads and houses have zero overlap too — this claim cannot be derived.

• "Some houses are poles" claims houses and poles overlap. The negative premise (and its valid conversion) establishes exactly the opposite — zero overlap between houses and poles — so this claim cannot be derived either.

Cross-check: Draw the poles circle, place the roads circle fully inside it (from "All roads are poles"), and draw the houses circle completely separate from the poles circle (from "No pole is a house"). Every diagram satisfying both statements keeps the houses circle away from both the poles and the roads circles, so no diagram supports either proposed conclusion — confirming there is no valid overlap to draw either conclusion from.

Result: Since one premise is negative, no affirmative conclusion can be drawn, and the diagram check confirms roads and houses never overlap and houses and poles never overlap. So neither of the two given conclusions follows.