Exercise 5, § 3

Let $\mathscr{T}$ be a logical theory with explicit axioms $A_1$, $A_2$, ..., $A_n$. Then $A_n$ is independent of the other axioms of $\mathscr{T}$ (§ 2, Exercise) if and only if the theory whose signs and schemes are the same as those of $\mathscr{T}$, and whose explicit axioms are $A_1$, $A_2$, ..., $A_{n-1}$, "not $A_n$", is not contradictory.