A type of consequence relation that allows multiple conclusions by construing consequence as a relation between sets of formulae (rather than between a set of formulae and a formula). In some interpretations of multiple-conclusion logic, an inference $\text{Γ}\models \Delta$ is read as:

• • It is impossible that the formulae in $\text{Γ}$ are all true while some formula in $\Delta$ is false.

On others, an entailment in multiple-conclusion logic is read:

• • If all formulae in $\text{Γ}$ are true then some formula in $\Delta$ is true.

There is a close relationship between the consequence relation in multiple-conclusion logics and the standard interpretation of a sequent in a sequent calculus, which admits a similar reading.

A case in which multiple-conclusion logic is useful is the semantics of supervaluationism. Models (or precisifications) are collections of classical valuations and make a formula $\phi$ supertrue when $\phi$ is true according to every valuation in the collection. Validity corresponds to preservation of supertruth from premisses to conclusions. Employing a multiple-conclusion framework brings out subtle differences. Notably, the principle of excluded middle holds, that is:

• • $p\models q\vee \neg q$

However, this does not reflect that a form of the principle of bivalence fails, that is, it is possible that a formula is neither supertrue nor superfalse. To represent this feature, one needs a set of conclusions:

• • $p⊭q,\neg q$

Some notions must be reformulated in a multiple-conclusion logic. For example, the notion of a Tarskian consequence relation must be subtly generalized to the case of a Scott consequence relation when multiple conclusions are taken into account.