TITLE: When can we use the implies arrow?
QUESTION [1 upvotes]: So I'm a complete beginner to logic and set theory and I've been trying to learn a bit about conditional statements and logic however I'm quite confused on if then statements.
For example if $p\implies q$
I saw that this means that if p is true then q must be true also
My confusion is when can we use this arrow can we only use it if the statement is true?
As an example if I had the statement if $x^2>0
$ then $x<0$ could I write $p\implies q$ where p=$x^2>0
$ and $q=
 x<0$
Or would it be incorrect to do so as this statement is false if this is not true when we see the implies arrow in theorems and proofs how am I to know that the statement is true or false?.
Thanks in advance.

REPLY [1 votes]: $p\implies q$ is considered to have a value of $true$ in one of these two cases

$p$ is false

or

$p$ is true and $q$ is true

So e.g. if one says this statement:
if the Earth is larger than the Sun, then 5 is even
this statement is true even though both $p$ and $q$ are false here.
But if one says:
if the Sun is larger than the Earth, then 5 is even
this statement is false.
Now to your question. You should write
$p\implies q$
when the implication as a whole is $true$
i.e. in one of the two cases mentioned above.
By the way before studying first-order logic it might be better to learn propositional logic.
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