= 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.
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