On the expressibility of propositions

In possible world semantics propositions are defined as sets of possible worlds.A proposition P is said to be expressible in a formal language L if there exists a set of formulas r of L such that the possible worlds satisfying r are precisely those contained in P. It is well known that not every proposition is expressible in a given language L; in other words there exists a gap of expressibility between propositions and formulas. As shown herein, this gap can cause problems in modelling belief dynamics. Motivated by these problems in this article we study the expressibility of propositions. More precisely we investigate conditions under which the expressibility gap reaches its lower bound, and we show that even under these conditions there exist infinitely many propositions that are not expressible in a given language L, unless L contains only finitely many logical equivalence classes.