Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes).
In order to achieve its special goal, logic was forced to develop its own formal tools, most notably its own grammar, detached from simply making direct use of the underlying natural language.
In case of a so-called extensional functor we can in a sense abstract from the "material" part of its inputs and output, and regard the functor as a function turning directly the extension of its input(s) into the extension of its output.
The attempts for such deep logical analysis have a long past: authors as early as Aristotle had already studied modal syllogisms.
As mentioned, motivations for settling problems that belong today to intensional logic have a long past.
The exact semantic content of these assertions therefore depends crucially on the nature of the accessibility relation.
The answer to this question characterizes the precise nature of the system, and many exist, answering moral and temporal questions (in a temporal system, the accessibility relation relates states or 'instants' and only the future is accessible from a given moment.
Richard Montague could preserve the most important advantages of Church's intensional calculus in his system.
Unlike its forerunner, Montague grammar was built in a purely semantical way: a simpler treatment became possible, thank to the new formal tools invented since Church's work.