Describes a family of theorems of material implication and strict implication rejected by proponents of relevant logic as instances in which antecedents and consequents are irrelevant to one another. For example, where $\supset$ is material implication, the paradoxes of material implication include theorems of classical logic such as the following:

• • $\phi \supset \left(\psi \supset \phi \right)$

• • $\neg \phi \supset \left(\phi \supset \psi \right)$

• • $\left(\phi \supset \psi \right)\vee \left(\psi \supset \phi \right)$

The paradoxes of material implication led philosopher Clarence Irving Lewis (1883–1964) to develop logics of strict implication; when the conditional is the strict implication $\mathcal{⊰}$, then the foregoing formulae are not theorems of Lewis’ systems. However, strict implication may be thought to admit paradoxical theorems as well:

• • $\left(\phi \wedge \neg \phi \right)\mathcal{⊰}\psi$

• • $\phi \mathcal{⊰}\left(\psi \vee \neg \psi \right)$

• • $\phi \mathcal{⊰}\left(\psi \mathcal{⊰}\psi \right)$

In all these cases, there is a disconnect between the antecedent and consequent inasmuch as there is no need for the two to share subject-matter, i.e., these theorems fail to meet the variable sharing property. Hence, both paradoxes of material implication and paradoxes of strict implication are paradoxical in that they fail to enforce relevance between antecedent and consequent of a valid conditional.