There are many deductions with different flavor to show that is logical to think proposition $A \implies B$ is true when A is false. For example:
All of above responses directly or indirectly, use this statement:
“p implies q” means that if p is true, then q must also be true.
My question is not in about "Why that may be formally logical". I want to know what is the type of above statement about the meaning of "$\implies$". Is that a definition? An axiom? A theorem? or ...? Really I want to find the root and reliable source reference which presented that meaning. I think there must be a famous book or reference in about logic which is the root of that meaning and all mathematicians accept that. So I can rely to that for similar questions.
In WIKIPEDIA, you can see:
Definitions of the material conditional:
Logicians have many different views on the nature of material implication and approaches to explain its sense ...
As a truth function
In classical logic, the compound p→q is logically equivalent to the negative compound: not both p and not q.
As a formal connective
... Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated.
If above vision be true, then
“p implies q” means that ... will be a
definition. Is this true?