Multiple-conclusion logic

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.

This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley 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).