We know that given any Turing machine m and any input n, we can construct a finite set of sentences ∆ and a corresponding halting sentence H such that ∆ ⊨ H iff TMm eventually halts on input n. Thus if there were an effective method for determining that ∆ ⊭ H then we could solve the halting problem and thereby refute the Church-Turing Thesis.
However, given that there is an effective positive test for first-order entailment, why cant we solve the halting problem by considering the negation of the halting sentence, viz. ∆ ⊨ ¬H?