If Γ ⊨ φ, then Γ ∪ {¬φ} has no model and is therefore, by Henkin's theorem, contradictory.

1 Deductive Calculi It is common in model theory to regard a logic as comprising at least three different things: a class of structures, a formal language to describe these structures, and a satisfaction relation that determines when a formula of the language is true with respect to a given structure. A deductive calculus might be added. In fact, any calculus for one-sorted first-order logic can be easily extended to a many-sorted one; the only rules which need to be adapted are the ones deali