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Ive been reading online on this but everything has been vague so far, the most sensible explanation I've seen so far goes like this: All implications in logic are either true or false (and not both) so an implication of the form a => b (where a is false) is either true or false but the only case where such an implication is false is when a is true and b is false which is not the case therefore we must conclude that it is true.

Now this almost elucidates the issue for me but then it evokes another issue for me. If we can argue in the manner that is done above then we can essentially argue that implications between completely unrelated things are true. Consider for example the statement "If the derivative of sinx is cosx then quadratic equations have a general solution/formula". Now clearly this is either true or it is false, it clearly is not false as both (the derivative of sinx is cosx) and (quadratic equations have a general solution/formula) are true so therefore this implication is true. In general we can use such an argument to conclude that any implication a => b (where it is not true that a is true and b is false) is true. But this seems completely counterintuitive to me because there appears to be no casual link between the a and b in my concrete example and this idea suggests that implications can be true for statements that are completely unrelated and have no casual relationship between them which appears to contradict what I thought was the definition of an implication (namely that a entails b, or that a CAUSES b).

In fact what do we mean when we say then that an implication statement a => b is "true", do we mean that the truth of a actually necessarily causes the truth of b or that is possible for a to cause b even if we don't know there is a casual relationship?

By the way I haven't had any exposure to undergraduate Maths at all yet so please no heavy use of set theory or formal logic etc as I am just a beginner