Where am I thinking wrong in the "only if" part of "If and only if" statements?

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• 2 Likes

  David Wayne Free Trial Member

  June 2014 edited June 2014 571 karma

  I'm not 100% sure as to what you're question is, but i do know that you may not be entirely clear on how contrapositives work. In your latter statement, you've written
  
  H will go before J only if it is after M
  
  This means that H-J -> M-H.
  
  This doesn't mean anything unless our sufficient condition has kicked or our necessary condition has been failed. If the sufficient condition kicks, we'd need to have M-H-J. If our necessary condition was failed, meaning H-M, then it must be the case that J is before H. So we'd have J-H-M.
  
  When you have a biconditional, there are to possible worlds: Either A and B obtain, or ~A and ~B obtain. (obtain means happens, by the way)
  
  But if you just have a regular conditional statement, A-B, there are three possibilities:
  A obtains, therefore B obtains
  A does not obtain, B obtains
  A does not obtain, B does not obtain.
  
  So your second conditional statement provides for three possible worlds:
  H-J, M-H
  J-H, M-H
  J-H, H-M
  
  Possibilities 1 and 3 depend on whether the sufficient condition kicks, or whether the necessary condition fails, respectively. Possibility 2 is the situation that occurs when the sufficient condition has been failed, thereby making the rule irrelevant. Does this clear things up? In sum, you are not wrong to infer J - H - M from your second statement; it is the result of the necessary condition being failed. Similarly, in your first example, you've
  concluded M - H - J. But that only accounts for satisfying the sufficient condition. If you failed the necessary condition, you'd have J-H-M. Of course, the third possibility would be that the sufficient condition failed and the necessary condition was satisfied. So whereas you've stated one possible world for your first conditional statement, you've neglected 2 others, and in your second conditional statement, you've similarly neglected two others.
  
  Also, by the way, this entire discussion assumes we're dealing with a one to one mapping game in which the pieces cannot stack in the same slot. The contrapositive of A-B is the negation of {A-B}, meaning either A and B are in the same slot, or B is before A. If your logic game only permits for one piece per slot, then we must necessarily employ the latter definition exclusively.
  
  The recent LSAT's employ these concepts viciously on the logic games section, so you should make sure you know this perfectly. JY has picked up on this trend (he calls it a 'new LSAT invention') and he explains it quite well in the logic games videos from the recent exams. So review the 50's and 60's games well if you haven't already. If you haven't gotten to those exams yet, fear not, JY will make everything clear when you watch the explanations.
  
  
• psbrathwaite Member

  June 2014 207 karma

  Michael, That was a thorough reply. Thank you. I apologize for the confusing with my questioning.
  
  Essentially, all I am trying to do is notate " H will go before J if and only if it is after M."
  
  
• 2 Likes

  David Wayne Free Trial Member

  June 2014 edited June 2014 571 karma

  H-J <--> M-H
  Possibility one: Sufficient and Necessary satisfied. M-H-J
  Possibility two: Sufficient Failed, Necessary failed. J-H-M
  
  I can't do a better job of explaining this than the "Advanced Logic" section of the 7sage syllabus. It may be necessary to give each video more than 2 or even 3 views in order to master the topics.
• psbrathwaite Member

  June 2014 207 karma

  I suppose I just feel as if this statement would have been the same if it read
  
  " H will go before J only if it is after M"
  
  Wouldn't I be know that M - H - J is a possibility, be cause it would happen if it was after M
  
  and then that J - H - M is the other?
  
  The first "if" in " if and only if "...I just feel like it didn't do anything in this situation?
  
  It feels like a biconditional without it
• 3 Likes

  David Wayne Free Trial Member

  June 2014 571 karma

  If you just have a regular conditional statement, A->B, there are three possibilities:
  A happens, therefore B happens
  A does not happen, B happens
  A does not happen, B does not happen.
  
  If you have a bicondtional statement, A <--> B, then the 2nd possibility above wouldn't be possible. You could only have
  
  A happens, B happens
  A does not happen, B does not happen
  
  So yes, the conditional relationship encompasses the two possibilities that a biconditional relationship can produce, but it also necessarily includes a possibility that a biconditonal expressly forbids, namely one element being negated, and the other being not negated. Its either both negated, or both not negated, whereas a conditional relationship allows for one to be negated and the other not to.
• psbrathwaite Member

  June 2014 207 karma

  That last paragraph hit the nail on the head.
• David Wayne Free Trial Member

  June 2014 571 karma

  You finally deemed a post as 'like-worthy'. It was all worth it.
• psbrathwaite Member

  June 2014 edited June 2014 207 karma

  Man, thanks.
• 1 Like

  psbrathwaite Member

  June 2014 207 karma

  If you just have a regular conditional statement, A->B, there are three possibilities:
  A happens, therefore B happens -- # 1
  A does not happen, B happens --- # 2
  A does not happen, B does not happen. -- # 3
  
  If it rains, you get wet --# 1
  you can get wet even if didnt rain -- #2
  it didnt rain, so you didnt get wet --#3
  
  Did I do it right, finally? lol
  
  I will hit the bed now.
  
  For # 2, , A doesn't cause B to happen,
  
  We're just saying B can happen without A.
  right?