 Lewis was dissatisfied with the extensional truth functional logic Principia Mathematica, and with its understanding of implication as material implication, according to which the truth of the if-then conditional \(p \supset q\) expressing the material implication of \(q\) by \(p\) is a function just of the truth or falsity of \(p\) and \(q. p \supset q\) is equivalent to \({\sim}(p \amp{\sim}q)\) and is true just in case it isn’t the case both that \(p\) is true and \(q\) is false. As a result, among the theses of Principia Mathematica are \(p \supset(q \supset p)\) and \({\sim}p \supset(p \supset q)\). In other words, a true proposition, whatever it happens to be, is implied by any proposition whatsoever, true or false, and a false proposition, whatever it happens to be, implies any proposition whatsoever, true or false. Lewis didn’t deny these theses, properly understood relative to the definition of material implication. However, he did think that these so-called “paradoxes of material implication” meant that material implication doesn’t provide a proper understanding of any ordinary notion of implication, according to which one proposition implies another just in case the latter logically follows from and is deducible from the former.