From Page 45 of Magnus/Button's book, ForAllx, the author(s) write
is equivalent to stating that the sentence $A$ is a contradiction.
The given definition of "$⊨$", a tautological entailment, is the following: The sentences $A_1, A_2,... ,A_n$ tautologically entail the sentence $C$ if there is no valuation of the atomic sentences which makes all of $A_1, A_2,... ,A_n$ true and $C$ false.
I am not sure how "$A ⊨$" can make sense, unless we admit that a statement of nothing is actually a sentence, but if we do, it seems that such a sentence must be both true and false, which we cannot admit in truth functional logic.
On the other hand "$⊨ C$" does make sense; it means that $C$ is a tautology. This is because (all the sentences on the right side are true) is a true statement, and if C could be false, the entailment does not hold. Therefore $C$ must be true for all valuations.