In my talk, I will discuss several “paradoxes of material implication”. In general, these “paradoxes” are inference patterns that involve implication and are classified as valid by classical logic despite their highly unintuitive character. I will focus mainly on inference patterns, in which implication interacts with disjunction and negation. In these cases, classical logic seems to give particularly unsatisfactory “predictions” about validity.  However, the common theories that avoided these unwanted features of classical logic usually removed also some important parts of classical logic that are intuitively acceptable (e.g. the equivalence between A or B and if not-A, then B). I will propose a semantic theory (and a corresponding deductive cal culus) that has the ambition to avoid the most serious paradoxes and, at the same time, to preserve these acceptable parts of classical logic.