Paradoxes of material implication

Summary

The paradoxes of material implication are a group of classically true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated with English words such as "implies" or "if ... then ...". They are sometimes phrased as arguments, since they are easily turned into arguments with modus ponens: if it is true that "if $P$ then $Q$" ($P \rightarrow Q$), then from that together with $P$, one may argue for $Q$. Among them are the following: