In Enumerative Combinatorics v. I, Stanley applies Möbius inversion to $B_n$, the poset of subsets of $[n]$, to show that $\forall f,g : B_n \to \mathbb{C}$,
$$ g(S) = \sum_{T \subset S} f(T) \, \, \forall S \subset [n] \\ \iff \\ f(S)= \sum_{T \subset S} (-1)^{|S-T|} g(T) \, \, \forall S \subset [n]. $$
How can I get the usual statement of the principal of inclusion-exclusion from this?