Q:

Clarification on deductive consequence

The Standford Encyclopedia on Philosophy page on Classical Logic has the following theorem:
Theorem 12. A set Γ is consistent if and only if there is a formula θ such that it is not the case that Γ ⊢ θ.
Can someone explain the negation at the end: "such that it is not the case that Γ ⊢ θ". I was expecting this to be in the affirmative, "such that it is the case...". Does it have anything to do with Γ being contingent: satisfiable and falsifiable?
Thanks

A:

No, a consistent set need not be falsifiable.  The intuition behind the definition is that consistency means that not too much should be deducible from $\Gamma$.  If absolutely every statement were deducible, then that would include various statements that contradict each other.
Often people define consistency to mean that it should not be possible to deduce two explicitly contradictory statements, i.e., statements of the form $\theta$ and not-$\theta$.  You should check (if your reference didn't do it for you) that (for classical logic) this is equivalent to the definition you quoted.

