The topic of conditionals is ancient, not as ancient as Aristotle and logic itself, but damn near: hard thinking on this topic began with the Dialectical School which featured such worthies as Philo the Logician and Diodorus Cronus, circa late 4th to mid-3rd centuries B.C. In nuce, those gentlemen had wrapped their minds around what much later came to be called material and strict implication, Philo around the former, Diodorus around the latter. The topic of conditionals is also deep and fascinating. But then no topic in philosophy lacks for fascination. The mansion of philosophy has countless rooms, each a labyrinth. Be sure to secure your thread of Ariadne before plunging on . . . .
The other day it occurred to me that 'material' in 'material implication' is best thought of as an alienans adjective. Normally, an FG is a G. Thus a nagging wife is a wife, a female duck is a duck, cow's leather is leather, and a contingent truth is a truth. But if 'F' is alienans, then either an FG is not a G, or it does not follow from x's being an FG that x is a G. For example, your former wife is not your wife, your quondam lover is not your lover, a decoy duck is not a duck, artificial leather is not leather, negative growth is not growth, and a relative truth is not a truth. Is an apparent heart attack a heart attack? It may or may not be. One cannot infer from 'Jones had an apparent heart attack' to 'Jones had a heart attack.' So 'apparent' in 'apparent heart attack' is alienans.
Now if p materially implies q, does it follow that p implies q? Obviously not. I am breathing materially implies 7 + 5 = 12, but the first does not imply the second. Material implication is no more a kind or species of implication than former wives are a kind of wives, or artificial leather is a kind or species of leather. Just as 'artificial' shifts or alienates the sense of 'leather,' 'material' shifts or alienates the sense of 'implication.'
Material implication is rather a necessary condition any implication must satisfy if it is to be what it is, namely, a genuine implication. For all will agree that in no case does p imply q if p is true and q false. Thus material implication does capture something essential to every genuine implication. But if X is essential to Y, it does not follow that X is a kind of Y.
Once we appreciate that 'material' in 'material implication' is an alienans adjective, and that material implication is not a kind of implication, we are in a position to see that that the 'paradoxes' of material implication are not paradoxes strictly speaking, but arise from foisting the ordinary sense of 'implication' upon 'material implication.'