Why do we use "If p,then q" instead of "Not p or q"? I read almost all posts on material implication and vacuous truths on the site.
I understood that it was introduced for mathematical convienience use and it does not have to perfectly align with our natural language intuition of the expression. I understood all the "breaking promise" analogy and the "subset" analogy.
What I don't understand is the reason why we choose the "if p, then q" construct insead of sticking to the "Not p or q"?
To me it seems that the second one is far more intuitive and resolves pretty much all the paradoxes that material implication arise.
Let's examine an example:  

If pigs can't fly, then I can't walk on water

What would you answer if someone was to ask you, "is the above statement true or false"?
I honestly would answer, "I don't know, it really doesn't seem true"
But what about...

Either pigs can fly or I can't walk on water

This seems right, seems true to me in an intuitive way.
Another example can be:  

If 2+2=5, then 2+2=6

False?True? What would you choose? Doesn't seem easy!
Look it this way...

It's not true that 2+2=5 or it's true that 2+2=6

Seems intuitive, clear.
The only reason I can think about is that with the "If p, then q" construct you underline a causal relationship between the antecedent and the consequent, which altough is useful in mathematical's contexts ( It's more immediate), it gives rise to vacuous truths and some paradoxes where there is no causal relationship. 
 I will soon provide other examples for what  I mean, if it's not clear!
 A: I think it's because the formulation $\phi\rightarrow\psi$ is more natural when it comes to conditional proofs and modus ponens. In a conditional proof you assume $\phi$ and from there prove $\psi$ then when discharging the assumption you conclude $\phi\rightarrow\psi$ - after all you have actually proven that $\psi$ follows from $\phi$.
In the case of modus ponens you have that $\phi$ and $\phi\rightarrow\psi$ and conclude $\psi$. Here in effect you're using that $\phi$ implies $\psi$. However some persons would be fine with modus ponens in the alternate form, that is we know that $\phi$ and $\neg\phi\lor\psi$ and from that conclude $\psi$.
Of course there are merits in the other formulation too. For example it might be easier to manipulate logical formulae if you only have two or three operators (that is $\neg$, $\land$ and $\lor$) to bother about. You need it to achieve disjunctive and conjuctive normal forms.
A: In english, the meaning of or can be two: 
- to link alternative ORRRR
- to use as a conjunction.
hence, in "not p or q", people might get confused easily that the "or" is an alternative link, and the meaning will goes wrong.
So, I guess, in convention people will just use "if p then q".
Yes, I'm one of those who get confused until I see your question when googled it.
Thanks to your question for clearing my mind lol
A: Psychologically, it's just easier to think about $A\implies B$ than it is to think about not $\neg A \lor B$. 
As for any apparent paradoxes when we have both $A\implies B$ and $\neg A$, I just reason that we cannot really infer anything about $B$ from these two statements alone. Without any possibility of inconsistency, B could be true or $B$ could be false, as in the truth table for $A \implies B$.
Example: We know that if it is rainy, then it must be cloudy. Based on this knowledge alone, if it is not raining, it may or may not be cloudy.
A: "What I don't understand is the reason why we choose the "if p, then q" construct insead of sticking to the "Not p or q"?"
Well doing such would cause (another) controversy among logicians.  I mean, it should get noted that their exist logical systems, such as Lukasiewicz 3-valued logic, where "if p, then q" does NOT mean the same thing as "not p or q".  Also, detachment with $\rightarrow$ makes for the least controversial of logical rules.  Furthermore, symbol usage ends up minimized with $\rightarrow$ instead of $\lnot$ and $\lor$.   
Additionally, I will note that all sound and complete systems strong enough to have a propositional calculus have vacuous truths.
That all said, there exist plenty of systems where $\lnot$ and $\lor$ get taken as primitive instead of $\rightarrow$.  So, it isn't like you have to use $\rightarrow$, so if you don't want to use $\rightarrow$, then don't do so.
A: I know that there are many sources that read implication like this: "if p then q", but it's only part of the story of how to address the expression coloquially, because you could be using the longhand version of which "if, then" are part of. Here's one way to frame the whole truth table in language:

Whether P is true or false, Q is true, because otherwise, if Q is false,
then P must be false for the implication to be true. P being true does
not imply that Q is false. Only that it's true.

The last two sentences refer to the implication itself. "Does not imply" means the negation of "implies" and therefore its value. "Only that it's true" is shorthand of "the antecedent being true implies that the consequent must be too for the implication to be true as well". Let's try to replace P and Q in the expression:

Whether you like this design or not, we're going to use it, because otherwise, if we don't use it, then you must have hated it. Liking this design does not imply that we're not using it. Only that we will.

It boils down to "The designer liking his work implies that his client will use it".
Reading implication with "if, else" sets the antecedent as a condition and the consequent as the consequence, but they're not the same thing. Think of consequent as the "agent of consequence", which means that it's condition necessary to verify whether the antecedent is true or false.
I hope that I've helped. If not, then check whether there is a mistake in what I said or not.
