Statements: Some hills are rivers. Some rivers are deserts. All deserts are…

Concept: When two statements are chained through a common (middle) term in the same direction — the first statement's predicate matching the second statement's subject — a definite conclusion follows only in specific patterns: two particular ('some') statements chained this way never yield any definite conclusion; a particular statement chained with a universal ('all') statement in that order yields a particular conclusion linking the two outer terms, dropping the shared middle term; and any valid 'Some A are B' conclusion converts simply to 'Some B are A'.

1. 'Some hills are rivers' (hills→rivers) and 'Some rivers are deserts' (rivers→deserts) chain through the shared term 'rivers', but both are particular — chaining two particular statements never yields a definite conclusion linking hills and deserts.

2. 'Some rivers are deserts' (particular, rivers→deserts) chained with 'All deserts are roads' (universal, deserts→roads) shares the term 'deserts' in the required chain order — particular chained with universal yields a particular conclusion dropping the shared term, so 'Some rivers are roads' follows.

3. 'Some rivers are roads' converts simply to 'Some roads are rivers' — this matches the claim linking roads and rivers, so that claim holds on its own.

4. Chaining 'Some hills are rivers' (particular, hills→rivers) with the derived 'Some rivers are roads' (particular, rivers→roads) again gives two particular statements sharing 'rivers' — so no definite conclusion links hills and roads either.

Cross-check with a concrete model: let hills = {h1, h2}, rivers = {h2, r1}, deserts = {r1, d1}, and roads = {r1, d1} exactly (every desert is a road, and nothing else is). Here h2 is both a hill and a river; r1 is both a river and a desert. Since r1 is a river and, being a desert, also a road, 'some roads are rivers' is true in this model. But h2 is not a road, so 'some roads are hills' is false; and neither desert (r1, d1) is a hill, so 'some deserts are hills' is false too — confirming that only the roads-rivers link is forced, while the other two can fail.

Only the claim 'Some roads are rivers' is forced true in every model that satisfies the three statements, while the claims linking roads to hills and deserts to hills can fail — so only that one conclusion follows.