Some help in this regard arrived in the form of the concept of a matrix, essentially a set of truth-values (usually but not necessarily finitely many) plus a truth-value table for each connective.
This idea, which was due to Jan ukasiewicz, was then generalized into the notion of an algebra (essentially a set with operators) and taken into modal logic by Alfred Tarski and his collaborators.
Till this day, the area of modal propositional logic is more definitive than the relatively more unsettled area of modal predicate logic.
The possible-worlds semantics, introduced by Kripke in the early 1960s, may be cast in the following form (which differs from Kripke's original formulation in terminology and, to some extent, in substance).
Furthermore, more recently, other formal languages have been suggested, which, although not modal logic in a strict sense, are closely related to it, such as dynamic logic.
Modern modal logic began in 1912 when Clarence Irving Lewis published a paper in Mind, in which he recommended that the logic of Principia Mathematica be supplemented with what he called intensional connectives.
The set of all normal logics, ordered by set inclusion, forms a lattice of immense complexity, as do sets of more inclusive classes of nonnormal modal logics.
The efforts to explore these structures continue but are increasingly a concern for mathematicians rather than for philosophers.
It would seem that to take a stand in such matters is to rely on implicit semantic ideas, however sketchy.
It was accordingly an important step when at last, thanks to Kripke and others, formal semantics were articulated.