ABSTRACT
The route from purely truth-functional semantics of propositional logic to a denotational one of predicate logic led via the accommodation of quantification and the engagement of the universe of individuals. But there is also a completely different possible route, the route we take when we want to move to modal propositional logic; it leads via the accommodation of modalities and via the engagement of the “universe” of possible worlds. In this way we reach the intensional variety of formal semantics, which, according to philosophers like Carnap, is necessary to account for the meanings of empirical expressions. The idea is that the truth value of a sentence may be relative to something (“circumstances”), and hence that its meaning is not a truth value but the way in which its truth value depends on this something. This variety of semantics can be again generalized all the way to lambda-categorial grammar. Moreover, it is suitable to accommodate the kind of logic avoiding variables, for lambda-categorial grammar can be transformed into variable-free combinatory logic.
