This goes back to Aristotle. It was picked up by Medieval and Arab philosophers (often associated with proofs of the existence of God) but I don't think it was taken very seriously by mathematicians until the 20th century. Kripke provided nice semantics for modal logics in terms of possible worlds in the 50s (I think) and since then the subject has blossomed. Nowadays modal logics are a commonplace tool in computer science.
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