Multiple-conclusion logic

Summary

A multiple-conclusion logic is one in which logical consequence is a relation, $\vdash$, between two sets of sentences (or propositions). $\Gamma \vdash \Delta$ is typically interpreted as meaning that whenever each element of $\Gamma$ is true, some element of $\Delta$ is true; and whenever each element of $\Delta$ is false, some element of $\Gamma$ is false. Such a reading is related to Gerhard Gentzen's interpretation of the multiple-succedent sequent calculus LK, though Gentzen interprets his sequents $\Gamma\vdash\Delta$ as formulae $(\bigwedge\Gamma)\supset(\bigvee\Delta)$.