  This counterexample can also be understood in terms of behaviour of premise combination. Here when we say \(X,A \vdash B\) is true, we are not just saying that \(B\) holds in any circumstance in which \(X\) and \(A\) both hold.  If we are after a genuine entailment A \(\rightarrow\) B, then we want \(B\) to be true in any (related) circumstance in which \(A\) is true. So, \(X,A \vdash B\) says that in any possibility in which \(A\) is true, so is \(B\). These possibilities might not satisfy all of \(X\). (In classical theories of entailment, the possibilities are those in which all that is taken as necessary in \(X\) are true.)