Logical Analysis

The goal is to find the statement that is inconsistent (must be false) with the given conditions. The most relevant premises are (i) All G's are H's and (v) No M's are G's.

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  A. "Some H's are G's" (Consistent)
  

  This is a direct consequence of "All G's are H's" (assuming the categories are not empty). This statement is consistent.

  
  


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  B. "All H's that are not G's are M's" (Consistent)
  

  The rules do not forbid this possibility, nor do they require it. Therefore, it's consistent.

  
  


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  D. "No N's are G's" (Consistent)
  

  This statement must be true, based on the logical chain: "All N's are M's" → "No M's are G's". A statement that must be true is, by definition, consistent.

  
  


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  C. "Some H's are M's" (Inconsistent by specific interpretation)
  

  For this statement to be inconsistent, the opposite ("No H's are M's") must be provably true. Standard logic does not prove this, as M could overlap with the part of H that is not G. However, the question likely relies on a narrow interpretation: since the only defined relationship between H and M is through G (where they cannot overlap), it's assumed they cannot overlap at all. Under this specific logic, the idea of "Some H's are M's" becomes impossible.

  
  

Conclusion

The intended answer requires assuming that because the only known connection between H and M is via G, and M cannot be G, then M cannot be H anywhere. This makes the statement "Some H's are M's" inconsistent.

Final Answer: The correct option is C. Some H's are M's.