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Proof: show that negation of converse is true?

  1. Feb 18, 2012 #1
    Hi everyone,

    I was thinking about logic and proofs and I concluded that "proving the negation of the converse of an implication to be true" proves "the implication to be true". But strangely I can't find any information about this proof method, so I doubt if I am correct.

    Just to be clear, here is an example:
    Implication: "I am human" implies that "I am an animal".
    Negation of the converse: "I am an animal" does not imply that "I am human".

    So, is my reasoning flawed here?
     
  2. jcsd
  3. Feb 18, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi bentley4! :smile:
    But "I am not an animal" implies that "I am not human".

    I don't follow the rest of what you're saying. :confused:
     
  4. Feb 18, 2012 #3

    jedishrfu

    Staff: Mentor

    By negative of converse I think you mean the contrapositive:

    P implies Q

    Is equivalent to:

    not Q implies not P

    You can use truth tables to prove it.

    See Wikipedia search on: p implies q
     
  5. Feb 18, 2012 #4
    Dear jedishrfu,

    Nope. I know that when the contrapositive is true, the implication must be true as well. But this is not what I am asking. Thnx for the response though.
     
  6. Feb 18, 2012 #5
    Hey Tiny-tim : ),

    You are just saying that if the implication is true, than the contrapositive must be true. I know, but my question is just if the negation of the converse must also be true if the implication is true.

    Using the example:
    (1) Implication: "I am human" implies that "I am an animal". (True)
    (2) Negation (of the implication): "I am human" does not imply that "I am an animal". (False)
    (3) Converse: "I am an animal" implies that "I am human". (False)
    (4) Negation of the converse: "I am an animal" does not imply that "I am human". (True)
    (5) Contrapositive: "I am not an animal" implies that "I am not human". (True)

    So what I am saying is that if (1) or (5) is true, (4) must also be true.
    Can anyone prove that the negation of the converse is false if the implication is true?
     
  7. Feb 18, 2012 #6
    Consider A => A. That's true.

    The converse is A => A.

    The negation of the converse is not(A => A). That's false.
     
  8. Feb 18, 2012 #7

    jedishrfu

    Staff: Mentor

    But you can still prove/disprove your assertion via truth tables and then you have an answer to your question.
     
  9. Feb 18, 2012 #8

    jedishrfu

    Staff: Mentor

    P___q__ p->q___ q->p___ ~(q->p)___ ~q____~p____~q->~p

    t___t____t_______t_______f_______f_____f_______t
    t___f____f_______t_______f_______t_____f_______f
    f___t____t_______f_______t_______f_____t_______t
    f___f____t_______t_______f_______t_____t_______t

    ((sorry cant get formatting right web form keeps changing uppercase to lower case))
     
    Last edited: Feb 18, 2012
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