I admit it, I never liked possible worlds semantics. Montague grammar and any common relative uses it as part of the intensional semantics for natural language. The low value of possible worlds as a methodology was made very clear for me when I read Forster's essay "The Modal Aether," published in Reinhard Kahle's volume Intensionality (2005) in the Lecture Notes in Logic series. So I would like to see more linguistic semantics developed without this baroque and unfortunate construct. Yet, I also feel intensional logic is worth having in semantic theory, so I don't want to give up on it.
Doing away with possible worlds within intensional logic has been tried in at least two ways. Church's intensional type theory (published in Nous in 1973, 1974, and 1993) accomplishes the goal by assigning a special type for "intension" of an entity, and another for intension of a proposition. The Montague approach would assign as intension a function from possible worlds (or world-time indices). Church's approach further provides an infinite chain of intensions of intensions, represented as an infinite chain of new types. So, while Montague's scheme relies on an infinite function space, Church's scheme would require an infinite number of "primitive" types, which is a little uncomfortable.
Another way was explored by George Bealer, especially in his book Quality and Concept (1982). Bealer shows how intensional logic can be set up using an operator that he writes with , so that a formula [A] simply represents the intension of logical object A, which can be any of the usual things in a first-order system. Bealer takes the logical notion of intension as primitive, and does not attempt to define it as a function from such to such or whatever. My problem with this system is that it tries to be first-order, which means that you cannot organize a formula to reflect a decent linguistic composition of the subterms with respect to meaning composition. In other words, Bealer's system applied to natural language cannot be compositional (or so it seems to me), since it inherits the same problems for linguistic compositionality that have always plagued first-order logic.
But surely one of these systems or a relative could be applied to yield a decent intensional logic for natural language semantics that does not use possible worlds.