Putnam 2013 B3

Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that:

(i) if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and

(ii) if $S \in \mathcal{P}$ and $S \neq \emptyset$, then there is a subset $T \subset S$ such that $T \in \mathcal{P}$ and $T$ contains exactly one fewer element than $S$.

Suppose that $f: \mathcal{P} \to \mathbb{R}$ is a function such that $f(\emptyset) = 0$ and $$f(S \cup S') = f(S) + f(S') - f(S \cap S') \text{ for all $S,S' \in \mathcal{P}$.}$$ Must there exist real numbers $f_1,\dots,f_n$ such that $$f(S) = \sum_{i \in S} f_i$$ for every $S \in \mathcal{P}$?