Multiple-conclusion logic

A multiple-conclusion logic is one in which logical consequence is a relation, ${\displaystyle \vdash }$, between two sets of sentences (or propositions). ${\displaystyle \Gamma \vdash \Delta }$ is typically interpreted as meaning that whenever each element of ${\displaystyle \Gamma }$ is true, some element of ${\displaystyle \Delta }$ is true; and whenever each element of ${\displaystyle \Delta }$ is false, some element of ${\displaystyle \Gamma }$ is false. Such a reading is related to Gerhard Gentzen's interpretation of the multiple-succedent sequent calculus LK, though Gentzen interprets his sequents ${\displaystyle \Gamma \vdash \Delta }$ as formulae ${\displaystyle (\bigwedge \Gamma )\supset (\bigvee \Delta )}$.¹

This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley² but has not been widely adopted.

Some logicians (for example, Greg Restall³) favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).