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.
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AOnly (1)
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BOnly (1) and (2)
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COnly (1) and (3)
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DOnly (2) and (3)
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EOnly (3)
Answer
Correct Answer: Only (3)
Explanation
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:
(3) Some cats are rats: Direct restatement of the first premise (“Some rats are cats”). Valid.(1) No cow is a cat: We only know that no dog is a cow and that some cats overlap with dogs. This says nothing about the remaining cats that are not dogs; they could overlap with cows or not. Hence (1) is not compelled.(2) No dog is a rat: From the premises, it is even possible that the “some cats that are dogs” are the same as the “some rats that are cats,” giving a dog that is also a rat. Therefore the universal negative (2) is not guaranteed.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)