Logical Analysis

The goal is to find which statement must be true given the premises and the special condition that "No P's are K's". The most relevant premises for this are (i) All G's are H's and (vi) All H's are J's or K's.

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  A. All P's are J's: Not necessarily true. We have no information that forces every P to be a J. A P could exist that is not an H, G, or J.
  


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  B. If any P is a G, it is a J: (Must be true)
  

  This is a valid deduction. Let's assume we have an object that is both a P and a G:

  
  
  
  
  1. Since it's a G, it must also be an H (from premise i).
  
  
  2. Since it's an H, it must be either a J or a K (from premise vi).
  
  
  3. Since it's a P, the special condition states it cannot be a K.
  
  
  4. Therefore, if it must be (J or K) but it cannot be K, it must be a J. The statement holds true.
  
  
  
  


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  C. No P is an H: Not necessarily true. An object could be a P, an H, and a J without violating any rules.
  


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  D. If any P is an H, it is a G: Not necessarily true. An H is not required to be a G (the converse of premise i is not given). An object could be a P and an H without being a G.
  

Final Answer: The correct option is (B) If any P is a G, it is a J.