A simplified theory of boolean semantic types

The theory of semantic types in Keenan and Faltz (I985) is insufficiently constrained in the sense that it requires denumerable categories to be interpreted under certain conditions via nondenumerable algebras. An ontologically more austere version of the theory is proposed in which expressions are always interpreted in terms of finite algebras and it is shown how it is nonetheless possible to treat an infinite language by providing an inductively defined hierarchy of such algebras, each representing a stage of an expanding knowledge base. Some apparent obstacles are considered and disposed of and some advantages discussed, having to do with the alethic modalities and referential opacity induced by predicates of propositional attitude. Finally, it is shown that a weaker version of Keenan and Faltz's central mathematical result, the Justification Theorem, suffices for the revised system and a simple, intuitive proof for it is given.