An Essay In Modal Logic

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Furthermore, more recently, other formal languages have been suggested, which, although not modal logic in a strict sense, are closely related to it, such as dynamic logic.

Modern modal logic began in 1912 when Clarence Irving Lewis published a paper in Mind, in which he recommended that the logic of Principia Mathematica be supplemented with what he called intensional connectives.

Much to his surprise he later found himself confronted by a veritable embarrassment of riches: an ever increasing number of modal logicsnot only his own famous quintuple of systems S1, S2, S3, S4, and S5 (his own tentative favorite was S3, the so-called Survey system) but also many, in fact, infinitely many others.

Since he never translated his semantic intuitions into a formal structure, the differentiation between different proposals became a problem.

Historically, even though for a long time it is modal predicate logic that has been of particular interest to philosophers, propositional modal logic has received much more attention from formal logicians, probably because agreement on what constitutes a generally accepted conceptual framework for research was reached much earlier in the latter area.

To the set of the usual truth-functional connectives, add two new connectives: a box operator del/Feys/von Wright logic, and KT4 and KT45 as the logics S4 and S5, respectively.The development of modal logic, both material and formal, preceded in steps.Propositional logics were studied extensively before predicate logicians had been able to work out a generally accepted common ground.The logics KD, KD4, and KD45, of special interest to deontic and doxastic logic, are sometimes called weak T, weak S4, and weak S5, respectively.The logics KT4G and KT4H are better known as S4.2 and S4.3, respectively, and the logic K4W as the Gb logic GL.Modal predicate logic does not exhibit the relative orderliness or maturity of its propositional relative.Philosophical questions such as the proper treatment of individuals persist.Till this day, the area of modal propositional logic is more definitive than the relatively more unsettled area of modal predicate logic.The possible-worlds semantics, introduced by Kripke in the early 1960s, may be cast in the following form (which differs from Kripke's original formulation in terminology and, to some extent, in substance).The set of all normal logics, ordered by set inclusion, forms a lattice of immense complexity, as do sets of more inclusive classes of nonnormal modal logics.The efforts to explore these structures continue but are increasingly a concern for mathematicians rather than for philosophers.

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