# Proof: show that negation of converse is true?

1. Feb 18, 2012

### bentley4

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. Feb 18, 2012

### tiny-tim

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

I don't follow the rest of what you're saying.

3. Feb 18, 2012

### 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

4. Feb 18, 2012

### bentley4

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.

5. Feb 18, 2012

### bentley4

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?

6. Feb 18, 2012

### SteveL27

Consider A => A. That's true.

The converse is A => A.

The negation of the converse is not(A => A). That's false.

7. Feb 18, 2012

### Staff: Mentor

But you can still prove/disprove your assertion via truth tables and then you have an answer to your question.

8. Feb 18, 2012

### 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