Syllogism — Rats, Cats, Dogs, and Cows Statements: • Some rats are cats. • Some cats are dogs. • No dog is a cow. Conclusions: (1) No cow is a cat. (2) No dog is a rat. (3) Some cats are rats.

Introduction / Context:
This syllogism mixes “some” and a categorical negative (“No dog is a cow”). The task is to check which conclusions must follow without overreaching beyond what is guaranteed by the premises.

Given Data / Assumptions:

• Some Rats (R) are Cats (C): R∩C ≠ ∅.
• Some Cats (C) are Dogs (D): C∩D ≠ ∅.
• No Dog (D) is a Cow (W): D∩W = ∅.
• Conclusions to test: (1) No Cow is a Cat. (2) No Dog is a Rat. (3) Some Cats are Rats.

Concept / Approach:
Only conclusions that hold for all possible diagrams are valid. A single counterexample diagram that satisfies premises but falsifies a conclusion is enough to reject it.

Step-by-Step Solution:

Verification / Alternative check:
Construct a diagram where one element lies in R∩C∩D (allowed by the first two premises) and D and W remain disjoint. All premises hold; (2) becomes false, confirming it does not follow. Similarly, allow some C to overlap W (not prohibited). This falsifies (1) while satisfying premises.

Why Other Options Are Wrong:

• Any option containing (1) or (2) overstates what is given.
• Only (3) is supported with certainty.

Common Pitfalls:
Assuming that if some cats are dogs and no dog is a cow, then no cat is a cow. That generalization is invalid because “some cats” does not describe all cats.

Final Answer:
Only (3)