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).

See also

• Sequent calculus