# Thread: IS this proof valid or invalid?

1. ## IS this proof valid or invalid?

I attached a picture of the proof. Any help is appreciated!

2. ## Re: IS this proof valid or invalid?

Originally Posted by Crayola

I attached a picture of the proof. Any help is appreciated!
Maybe it just my browser, but I see absolutely no proof in that link.

In any case this is true: $\neg \left( {\forall x} \right)\left[ {P(x)} \right] \equiv \left( {\exists x} \right)\left[ {\neg P(x)} \right]$

4. ## Re: IS this proof valid or invalid?

Originally Posted by Plato
Maybe it just my browser, but I see absolutely no proof in that link.

In any case this is true: $\neg \left( {\forall x} \right)\left[ {P(x)} \right] \equiv \left( {\exists x} \right)\left[ {\neg P(x)} \right]$
Right, sorry I meant statement. So why wouldn't the given be valid?

5. ## Re: IS this proof valid or invalid?

Originally Posted by Crayola
Right, sorry I meant statement. So why wouldn't the given be valid?
On the set of all real numbers, let $P(x)$ mean “x is positive”.

$\neg \left( {\forall x} \right)\left[ {P(x)} \right]$ means “Some real number is not positive”. Literally, “It is false that all real numbers are positive”.

Whereas, $\left( {\forall x} \right)\left[ {\neg P(x)} \right]$ means “No real number is positive”. Literally, “All real numbers are not positive”.

Now can you answer the question?

6. ## Re: IS this proof valid or invalid?

Okay, thank you!