# Conditional proposition

I was looking for proof of conditional proposition, trying to understand why it is always true when it's hypothesis is false. While I was researching I found theories like material implication and objections of it's paradoxes and so on. And it was enough to convince myself. But the thing is that these arguments are all end up with it's counterargument. My question is: have the material implication arguments been proved?
If so, where can I find this proof in a mathematical, not philosophical way? If it just remains in not being proved, then how we can use the conditional operator in a mathematical proof?

• It is a definition (or convention): the conditional (or material implication) used in "standard" mathematical discourse is modelled with the well-known truth table. – Mauro ALLEGRANZA Mar 5 '17 at 10:44
• The key-feature of the conditional in mathematics is its use in modus ponens: when $P$ is true and $P \to Q$ is true we want to be licensed to infer $Q$, i.e. we want that $Q$ is true also. – Mauro ALLEGRANZA Mar 5 '17 at 10:46
• When $P$ is false... well, we are not interested into what we can infer from false premises. – Mauro ALLEGRANZA Mar 5 '17 at 10:47
• You can find many related posts on this site, searching for conditional or material implication. – Mauro ALLEGRANZA Mar 5 '17 at 10:49
• Thank you for your reply, one more thing, is the argument: P→Q is equivalent with ~p or q just argument or mathmatical law? – Jin Mar 5 '17 at 10:56