A Relevant Logic may be broadly defined as a Sociative Logic in which Disjunctive Syllogism fails. That is, it is any logic in which: 1. There is no theorem of the form $A \rightarrow B$ in which the antecedent and consequent do not share at least one sentential variable. 2. $(A \wedge (\neg A \lor B)) \rightarrow B$ is not a theorem. Many (though not all) Relevant Logics are also Paraconsistent, i.e. the rule form of Explosion fails in them $A \wedge \neg A \vdash B$. It is often useful to adopt Routley’s term “Pararelevant Logics” for logics which are both Relevant and Paraconsistent.
There are a huge number of relevant logics and any attempt at a taxonomy presents enormous difficulties. But for now we will focus on those systems that are in some sense based upon $FDE$ (what we might call the ‘mainstream’ systems). At some point in the future we will also discuss other related systems and additionally will discuss the so-called ‘Semi-Relevant Logics’.
Axiom Systems