Relevance logics, also called relevant logics, are non-classical logics developed as attempts to avoid the paradoxes of material and strict implication.
Relevance logics, also called relevant logics, are non-classical logics developed as attempts to avoid the paradoxes of material and strict implication. These so-called paradoxes are valid conclusions that follow from the definitions of material and strict implication but are seen, by some, as problematic.
For example, the material implication $p \rightarrow q$ is true whenever $p$ is false or $q$ is true, i.e. when $\neg p \lor q$ is true. So if $p$ is true, then the material implication is true when $q$ is true. Among the paradoxes of material implication are the following: $$ p \rightarrow (q \rightarrow p) $$ $$ \neg p \rightarrow (p \rightarrow q) $$ $$ ( p \rightarrow q ) \lor ( q \rightarrow r) $$ The first asserts that every proposition implies a true one; the second that a false proposition implies every proposition, and the third that for any three propositions, either the first implies the second or the second implies the third.
Similarly, the strict implication $p \rightarrow q$ is true whenever it is not possible that $p$ is true and $q$ is false, i.e. when $\neg\Diamond(p \land \neg q)$ is true. Among the paradoxes of strict implication are the following: $$ (p \land \neg p) \rightarrow q $$ $$ p \rightarrow (q \rightarrow q) $$ $$ p \rightarrow (q \lor \neg q) $$ The first asserts that a contradiction strictly implies every proposition; the second and third imply that every proposition strictly implies a tautology.
