Remark

Under the hypotheses of C5, if the theory $\mathscr{T}$ turns out to be contradictory, the same will be true of $\mathscr{T}'$. For if $A$ and "not $A$" are theorems in $\mathscr{T}$, then $(T_1|a_1) \ldots (T_h|a_h)A$ and $\mathrm{not}(T_1|a_1) \ldots (T_h|a_h)A$ are theorems in $\mathscr{T}'$. \* For example, if the theory of groups were contradictory, the theory of sets would also be contradictory. \*