 Finite-variable dependence of a propositional rule base

Every finite collection of finite propositional formulas mentions only finitely many Boolean variables. Two Boolean valuations agreeing on those variables satisfy exactly the same formulas. They therefore have the same value under any weighted formula utility representation using that collection.

This observation limits representation over infinite sets. On $\mathbb{N}$, the function indicating whether the selected subset is infinite cannot be represented by a finite rule base: if $F$ is the finite set of mentioned indices, $\varnothing$ and $\mathbb{N}\setminus F$ agree on all those indices but require values zero and one. Completeness claims for finite propositional rule bases must therefore specify a finite underlying set or restrict the functions represented.