Categorical logic – hair length and tail length for “Gangles” Given: All spotted Gangles have long tails. Short-haired Gangles always have short tails. Question: If those two facts are true, is the statement “Long-tailed Gangles never have short hair” true, false, or uncertain?

Introduction / Context:
This is a syllogistic inference using implication and its contrapositive. We translate descriptive phrases into logical conditionals to see whether the third statement follows from the earlier ones.

Given Data / Assumptions:

• S1: All spotted ⇒ long-tailed.
• S2: Short-haired ⇒ short-tailed.
• Claim to evaluate: Long-tailed ⇒ not short-haired.

Concept / Approach:
From S2 (Short-haired ⇒ short-tailed), take the contrapositive: not short-tailed ⇒ not short-haired, which is equivalent to long-tailed ⇒ not short-haired. Hence, S3 is directly implied by S2 alone. S1 is not even needed to prove S3, though it is consistent.

Step-by-Step Solution:
1) Write S2 as SH ⇒ ST (short-haired implies short tail).2) Take contrapositive: not ST ⇒ not SH.3) “Not short tail” is “long tail.” Therefore: LT ⇒ not SH.4) This matches the statement “Long-tailed Gangles never have short hair” ⇒ True.

Verification / Alternative check:
Create a compatible model: Short-haired Gangles always short-tailed; any long-tailed Gangle cannot be short-haired without violating S2. The result holds universally.

Why Other Options Are Wrong:

• False / Uncertain: The contrapositive of S2 compels the conclusion.
• Both true and false: Not applicable under consistent logic.

Common Pitfalls:
Misusing converse (short-tailed ⇒ short-haired), which is invalid; only the contrapositive preserves truth.

Final Answer:
True