A consequence relation $⊢$ (either syntactic or semantic) such that for all sets of formulae $\text{Γ}$ and all formulae $\phi$,

1. 1 $\phi ⊢\phi$

2. 2 If $\text{Γ}⊢\phi$ then $\text{Γ},\psi ⊢\phi$ for all formulae $\psi$

3. 3 If $\text{Γ},\psi ⊢\phi$ and $\text{Γ}⊢\psi$, then $\text{Γ}⊢\phi$

In multiple-conclusion logic, the analogous notion is that of a Scott consequence relation, in which clauses 2 and 3 are replaced with the following:

1. 2′ If $\text{Γ}⊢\Delta$ then $\text{Γ},\phi ⊢\Delta ,\psi$ for all formulae $\phi$ and $\psi$

2. 3′ If $\text{Γ},\psi ⊢\Delta$ and $\text{Γ}⊢\Delta ,\psi$, then $\text{Γ}⊢\Delta$