Because MΓ∪{A} is not assumed to be a subset of MΓ, the implication from Γ to C need not extend to supersets of Γ.

The consequence relations of classical logics are monotonic. That is, if a set Γ of formulas implies a consequence C then a larger set Γ ∪ A will also imply C. A logic is nonmonotonic if its consequence relation lacks this property. Preferred models provide a general way to induce a nonmonotonic consequence relation. Invoke a function that for each Γ produces a subset MΓ of the models of Γ; in general, we will expect MΓ to be a proper subset of these models. We then say that Γ implies C if C i