Statements: I. Some cats are dogs. II. No dog is a toy. Conclusions: I. Some dogs are cats. II. Some toys are cats. III. Some cats are not toys. IV. All toys are cats. Choose the option that must follow.

Introduction / Context:
This question examines conversion of a particular affirmative and its interaction with a universal negative. We must determine which conclusions are logically necessary in all models consistent with the premises.

Given Data / Assumptions:

• Premise 1: Some Cats are Dogs (∃x: x ∈ Cat ∩ Dog).
• Premise 2: No Dog is a Toy (Dog ∩ Toy = ∅).

Concept / Approach:
“Some A are B” entails the symmetric “Some B are A.” A universal negative about Dogs and Toys guarantees that any Dog cannot be a Toy. Combine these to deduce about the Cats that are Dogs.

Step-by-Step Solution:
1) From “Some Cats are Dogs,” infer by symmetry: Some Dogs are Cats (Conclusion I is true).2) Those same cats that are dogs cannot be toys (from “No Dog is a Toy”). Therefore, at least some Cats are not Toys (Conclusion III is true).3) Conclusion II (“Some Toys are Cats”) is unsupported; the premises speak only about Dogs and Toys being disjoint, not about Toys intersecting Cats.4) Conclusion IV (“All Toys are Cats”) is far stronger than anything given and is not entailed.

Verification / Alternative check:
Model: Let Dogs = {d1}, Cats = {d1, c2}, Toys = {t1}. Premise 1 holds via d1; Premise 2 holds since Toys do not contain d1. Here, I and III hold; II and IV fail.

Why Other Options Are Wrong:

• II and III: II is not forced.
• I and II: II is not forced.
• Only I: Misses III, which necessarily follows.

Common Pitfalls:
Assuming that Cats and Toys must overlap simply because Dogs are excluded from Toys. The premises do not impose any relation between Cats (that are not Dogs) and Toys.

Final Answer:
Only Conclusions I and III follow.