The following question has arisen when considering dependencies induced by probabilistic logic programs:
Let $A_1, \dots, A_n$ be proposition symbols, and let $\Omega_n$ be the set of all truth value assignments of $A_1, \dots, A_n$ (So $\Omega_n$ is the $2^n$-element Boolean algebra). Now consider $\Omega_n$ as a discrete probability space under the uniform probability measure. Then any propositional formula $\varphi$ in the proposition symbols $A_1, \dots, A_n$ can be seen as an event on $\Omega_n$.
Is the following statement true:
Let $\varphi$ and $\psi$ be positive propositional formulas (built from $\land$ and $\lor$, but no negation) in the proposition symbols $A_1, \dots, A_n$ such that $A_1$ occurs in every formula equivalent to $\varphi$ or $\psi$.
Then $\varphi$ and $\psi$ are dependent events in $\Omega_n$.
I have a proof for the statement when one of the formulas, say $\psi$, is a pure conjunction; then conditioning on that formulas will remove clauses in a minimal DNF for $\varphi$, which increases its probability. However, conditioning on a union of events is generally not well behaved, so I don't see how this generalises beyond conjunctions.
