A logic studying the notions of necessity and possibility. Modal logic was of great importance historically, particularly in the light of various doctrines concerning the necessary properties of the deity, but was not a central topic of modern logic in its golden period at the beginning of the 20th century. It was, however, revived by C. I. Lewis, by adding to a propositional or predicate calculus two operators, · and ◊ (sometimes written N and M), meaning necessarily and possibly, respectively. Theses like p → ◊p and ·p → p will be wanted. Controversial theses include ·p→ ··p (if a proposition is necessary, it is necessarily necessary, characteristic of the system known as S4) and ◊p→ ·◊p (if a proposition is possible, it is necessarily possible, characteristic of the system known as S5). The classical model theory for modal logic, due to Kripke and the Swedish logician Stig Kanger, involves valuing propositions not as true or false simpliciter, but as true or false at possible worlds, with necessity then corresponding to truth in all worlds, and possibility to truth in some world. Various different systems of modal logic result from adjusting the accessibility relation between worlds. See Kripke model.