How Completeness and Correspondence Theory Got Married

Abstract

It has been said that modal logic consists of three main disciplines: duality theory, completeness theory and correspondence theory; and that they are the pillars on which this edifice called modal logic rests. This seems to be true if one looks at the history of modal logic, for all three disciplines have been explicitly defined around the same time, namely in the mid-seventies. While it is certainly true that modal logic can be divided in this way, such a division creates the danger that the subareas are developed merely in their own right, disregarding the obvious connections between them. Moreover, such historically grown divisions always run a risk of enshrining certain errors that have accidentally been made and subsequently hindered the development. One such error is the idea that Kripke-frames are the natural or fundamental semantics for modal logic. Although I agree that Kripke-models are the most intuitive models and that they are in many cases indeed the intended models, I cannot go along with the claim that they are in any stronger sense fundamental. Philosophically as well as mathematically, to start with Kripke-frames is to start at the wrong end; philosophically, because nothing warrants the belief that possible worlds exist — in fact, for my ears this is a contradictio in adiectu — and indeed it is much more plausible to say that possible worlds are philosophical fiction. And mathematically, because Kripke-frames are deficient in some respects and these deficiencies do not apply to modal algebras and also because the former can be obtained canonically from the latter. It is surprising how long it was possible to ignore Stone’s representation theory for boolean algebras and also Tarski’s work together with Jónsson in which Kripke-semantics appeared long before Kripke himself came to discover it.