Let $A = \bigcup_{1\leq i \leq n} C_i$ be a sample space written as a union of (overlapping) events. Let $X$ be a separate "distinguished" event.
Let $S\subseteq \{1, 2, \ldots, n\}$. Define $$\mathbf{I}(S):=\bigcap_{i\in S}C_i$$
Suppose we know that $P(X| \mathbf{I}(S)) > 0.5$ for all subsets $S$. What can be said about $P(X)$?
I think it's probably true that $P(X) > 0.5$, but I don't know how to prove it.
Not necessary. Assume $A = \{1, 2, 3, 4, 5, 6\}$ with uniform measure, $C_i = \{i, 5, 6\}$ for $i = \overline{1,4}$ and $X = \{5, 6\}$.
Then $P(X | \textbf{I}(S)) = \frac{2}{3}$ if $|S| = 1$ and $P(X | \textbf{I}(S)) = 1$ for $|S| > 1$, but $P(X) = \frac{1}{3}$.
(that's assuming you don't allow $S = \varnothing$, if you do, then you get $\textbf{I}(\varnothing) = A$ and your statement is trivially true)