Philosophy:Multiple-conclusion logic

Short description: Mathematical logic

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

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

Some logicians (for example, Greg Restall) 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