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.

This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley^[1] but has not been widely adopted.

Some logicians 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).