Summary

Modal logic has a remarkable history indeed. Originally conceived as the logic of necessity and possibility, its philosophical roots go back at least as far as Aristotle and the Stoic Diodorus Cronus.2 While Diodorus's ‘Master Argument’ could be considered an early example of modal propositional, more precisely temporal, reasoning,3 the Stagirite's investigations into modal syllogisms probably constitute the earliest serious essay on combining modality with quantification.4 Interest into modal distinctions kept very much alive during Mediaeval Scholastics, notably in the form of ontological arguments such as St. Anselm's,5 and only seems to have slowed down with the dawn of the Renaissance. While Leibniz, being an eminent contributor to the development of modern logic, is usually credited with the introduction of the notion of a ‘possible world’, the birth of modal logic as a technical discipline of mathematical logic is often considered to be Clarence Irving Lewis's investigations into the paradoxes of material implication in 19186, and, in the early thirties, Kurt Gödel's interpretation [Gödel, 1932; Gödel, 1933] of Heyting's version [Heyting, 1930] of Brouwer's intuitionistic propositional logic [Brouwer, 1907; Brouwer, 1908] in terms of the modality ‘it is provable’7. Despite Quine's early attempts at discrediting the intelligibility of modal discourse [Quine, 1947; Quine, 1953], its technical development went hand in hand with the rise of analytic philosophy, and both its technical sophistication as well as its philosophical significance virtually exploded in the 1970s with the unlikely success of Kripke's possible worlds semantics. In the early 21st century, modal logic is not only an important tool in philosophy and linguistics, it also provides, in its many disguises, key formalisms in artificial intelligence and knowledge representation.