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Mathematical modal logic: A view of its evolution. (English) Zbl 1041.03015
This is a broad and detailed survey of the origins and development of the mathematical aspects of modal logic. It begins with a historical account of the emergence and early studies of formal modal logical systems, featuring first purely deductive, and later algebraic approaches. The theory of Boolean algebras with operators, developed by Jónsson and Tarski, is outlined, and other important precursors of Kripke semantics, including ideas and papers by McKinsey, Carnap, Meredith, Prior, Geach, Kanger, Montague and Hintikka, are discussed in detail. The seminal work by Kripke, introducing the relational semantics, and the following boom in the development of modal logic, beginning in the early 60s and still unabated, are then outlined. A good account is provided of the milestones of the major directions of the classical development of modal logic, viz. the completeness and correspondence theories, shaped in the works of Lemmon and Scott, Segerberg, Thomason, Sahlqvist, van Benthem, Fine and many others, as well as the duality theory between algebraic and Kripke semantics, much of which Goldblatt himself has developed, beginning with his dissertation. The last section contains a selection of the most important and well-studied mathematical interpretations of modality, including logics of programs and processes such as PDL, temporal logics of computations in reactive and concurrent systems, such as LTL and CTL, the modal mu-calculus, the provability logics, and Grothendieck topology as intuitionistic modality. The paper ends with a comprehensive bibliography comprising 275 entries.
Rich in both historical and logical content, and containing an encyclopedic amount of well-presented and expertly discussed classical knowledge on modal logic along with a wealth of little-known facts and details, the survey is a highly recommended, very interesting reading for a wide-ranging audience, from readers with general interest in logic and its history, to specialists in modal logic wishing to complete and consolidate their knowledge on the historical roots and developments of the subject.

MSC:
03B45 Modal logic (including the logic of norms)
03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
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