Syllogism – Multiple “some” statements about mountains (avoid unjustified negatives) Statements: • Some mountains are hillocks. • Some mountains are rivers. • Some mountains are valleys. Conclusions to test: I. All mountains are either hillocks or rivers or valleys. II. No valley is a river. III. Some rivers are valleys.

Introduction / Context:
When all premises are “some”-type statements, they assert the existence of overlaps with a single set (“mountains”) but do not provide relationships among the non-mountain categories themselves. Any universal claim or cross-category relation beyond mountains must not be assumed.

Given Data / Assumptions:

• M ∩ H ≠ ∅ (some mountains are hillocks).
• M ∩ R ≠ ∅ (some mountains are rivers).
• M ∩ V ≠ ∅ (some mountains are valleys).

Concept / Approach:
“Some” indicates at least one element exists in the intersection described. However, it does not cover all of M, nor does it establish any R–V, H–R, or H–V relations outside of M. Therefore, universal claims like “all mountains are …” or “no valley is a river” cannot be justified without additional data.

Step-by-Step Solution:
I: “All mountains are either hillocks or rivers or valleys.” The premises give three non-empty intersections but do not say anything about the remaining mountains. Some mountains could be none of the three. I does not follow.II: “No valley is river.” There is zero information about how V relates to R. II does not follow.III: “Some rivers are valleys.” Again, no given overlap between R and V is specified. III does not follow.

Verification / Alternative check:
Construct a model with three distinct mountain subgroups: M∩H, M∩R, and M∩V, all disjoint. All premises remain true while II and III fail and a portion of M lies outside H∪R∪V, refuting I.

Why Other Options Are Wrong:

• Any option claiming one or more conclusions follow assumes extra relations not present.
• “Only either II or III” suggests an exclusive inference, which is unsupported.

Common Pitfalls:
Assuming that multiple “some” overlaps imply an overlap among the non-mountain categories; turning existential statements into universal coverage.

Final Answer:
None follows