The modern, 20th-century history of modal logic is told from a mathematical and algebraic point of view in this paper, which also describes the most important results in the field. It argues that, similar to classical propositional logic, modal logic has benefited from algebraic treatment.

Section 1 introduces the paper. Section 2 is dedicated to the early syntactic development of modal logic, and Gödel’s provability interpretation of modality and relationship with intuitionistic logic. Section 3 examines modal algebras, the first type of semantics for modal logic, and the topological interpretation of modality. Section 4 surveys the development of relational semantics in the 1950s, with references to the work of Kripke, Carnap, Hintikka, Bayart, Kanger, and Montague, and gives credit to Kripke for his attractive exposition of the topic. Sections 5 and 6 present the main outcomes in the field in the 1960s and 1970s resulting from the semantic analysis of modal logic, and their relationships with modal algebras. Section 7 illustrates some mathematical applications of modal logic, including dynamic logic, temporal logic of concurrency, &mgr;-calculus, and the arithmetic and topological interpretations of modal operators.

The only downside of the paper is that it is not possible to expose, in a survey like this, all the relevant mathematical constructions (and proofs) in a step-by-step manner. A reader not fully conversant with algebraic methods might therefore find it hard to follow the line of some arguments, and consequently, might not understand the importance of the issues at hand there. On the other hand, the paper exposes the main idea in clear and crisp language, and refers the reader to the original works for the full details. The bibliography is impressive, with close to 300 records; any reader seriously interested in a deeper understanding of the topics discussed in this paper can find many good pointers here to the main results and ideas in the mathematical treatment of modal logic.