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!