The Negation of an Implication Statement?

[someperson05]

Hello,

So someone just asked me for assistance on a proof, and while I'm fairly certain you can't do what he did, I am not completely sure on the reasons.

To state it as formal logic,
If you have proposition A:
[tex]P \rightarrow Q[/tex]
And let's call proposition B
[tex]\neg (P \rightarrow Q)[/tex]
If you were to show B was false, then I think that does not imply A is true.

Am I right? And what logic is really going on above?

Thanks for any help you can provide.

EDIT:
I tried looking at the implication as,
[tex] P \rightarrow Q \equiv \neg P \vee Q[/tex]
which means that
[tex]\neg (P \rightarrow Q) \equiv P \wedge \neg Q[/tex]
which no longer seems to be really an implication statement.

 

[matiasmorant]

it is always true that

[tex]a\lor \neg a[/tex]


¬¬a is equivalent to a

 

[disregardthat]

Hello,

[tex]\neg (P \rightarrow Q) \equiv P \wedge \neg Q[/tex]
which no longer seems to be really an implication statement.

Why would it be, it's the negation of an implication statement.

 

[dkavlak]

I'm a little unclear on exactly what your question is and what system of logic you are talking about.

If you are talking about intuitionistic/constructive logic then a proof that B is false (i.e. a proof of the negation of B, which I believe in intuitionistic terms would be "a proof that B cannot be proven") would not be a proof of A, since intuitionistic logic does not have a double negation elimination rule. Intuitionistic logic also has a different definition of negation than classical logic (part of the reason there is no double negation rule).

However, in classical logic, a proof of the negation of B would be the double negation of A, which is equivalent to A via a rule of double negation elimination.

 

[blue_raver22]

Hello,

You are correct in your understanding that the negation of an implication statement does not necessarily imply the truth of the original statement. In fact, the negation of an implication statement is equivalent to the conjunction of the antecedent (P) and the negation of the consequent (\neg Q). This means that both P and \neg Q must be true in order for the negation of the implication to be true.

In terms of logic, the process of negating an implication statement can be seen as applying the De Morgan's Law, which states that the negation of a disjunction is equivalent to the conjunction of the negations of its individual components. So in this case, the negation of an implication statement is equivalent to the conjunction of the negation of the antecedent and the negation of the consequent.

I hope this helps clarify the logic behind the negation of an implication statement. Let me know if you have any further questions.