Describes a deductive system that satisfies the variable sharing property, so that whenever $\phi \to \psi$ is a logical truth (or a theorem), $\phi$ and $\psi$ have at least one propositional variable in common. Motivated by the paradoxes of relevance, relevant logics were formulated by Alan Anderson (1925–1973) and Nuel Belnap (1930– ), drawing on the work of Wilhelm Ackermann (1896–1962) and Alonzo Church (1903–1995). There are several different semantics for relevant logics, but the most popular is the one developed by Richard Sylvan (né Routley) (1935–1996) and Robert Meyer (1932–2009). This is a world-semantics. It uses the Routley star to give the semantics for negation, and a ternary relation, $R$, to give the truth conditions for the conditional, as follows:

• • $\phi \to \psi$ is true at world $w$ iff for all worlds, $x$ and $y$ such that $\text{Rwxy}$, if $\phi$ is true at $x$, $\psi$ is true at $y$.

The philosophical meaning of the ternary relation is contested. As for the binary accessibility relation of modal logic, putting various constraints on the $R$ delivers a variety of different relevant logics.