Proof: show that negation of converse is true?

[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?

 

[tiny-tim]

hi bentley4!

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".

But "I am not an animal" implies that "I am not human".

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

 

[jedishrfu]

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

 

[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.

 

[bentley4]

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. ​

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?

 

[SteveL27]

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?

Consider A => A. That's true.

The converse is A => A.

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

 

[jedishrfu]

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.

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

 

[jedishrfu]

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 can't get formatting right web form keeps changing uppercase to lower case))