Topic: GENERAL LOGIC
MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
By Robert Goldblatt
Over a period of three decades or so from the early 1930’s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures often called Kripke models, whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related the subsets of a relational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Techniques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms particularly first and second order monadic predicate logic.
The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic and this has at least three dimensions. To begin with there are the new mathematical ideas, when and why they were introduced, and how they interacted and evolved. Then there is the use of method and results from other areas of mathematical logic, algebra and topology in the analysis of modal systems. Finally, there is the application of modal syntax and semantics to study notions of mathematical and computational interest.
Appeared in: Handbook of the History of Logic. Volume 6 Dov M. Gabbay and John Woods (Editors)