Substitutional Quantification in Truth-Theories for Modal Languages

If we wish to formulate an axiomatic truth-theory interpreting a modal language and treat the symbol of necessity as a sentential operator and not as a quantifier over possible worlds, there arise various problems. These are due partly to the fact that words could have meant something other than what they actually mean and partly to certain principles of modal metaphysics. One of those principles is existentialism about propositions: a proposition that is expressed in a sentence containing a non-empty name could not exist if the referent of the name did not exist. The paper explains how the problems arise. It also explains how we can avoid them using substitutional quantification in the metalanguage. The truth-theory we can construct in that way is compositional and homophonic, and its theorems interpreting the various sentences of the modal language are derived in a straightforward way. The paper develops one such theory for a propositional modal language and one for a first-order modal language. The first-order case involves a number of intricacies that are discussed.