Formal systems incorporating modalities such as necessity, possibility, impossibility, contingency, strict implication, and certain other closely related concepts. The most straightforward way of constructing a modal logic is to add to some standard nonmodal logical system a new primitive operator intended to represent one of the modalities, to define other modal operators in terms of it, and to add axioms and/or transformation rules involving those modal operators. For example, one may add the symbol L, which means It is necessary that, to classical propositional calculus; thus, Lp is read as It is necessary that p. The possibility operator M (It is possible that) may be defined in terms of L as Mp = ¬L¬p (where ¬ means not). In addition to the axioms and rules of inference of classical propositional logic, such a system might have two axioms and one rule of inference of its own. Some characteristic axioms of modal logic are: (A1) Lp p and (A2) L(p q) (Lp Lq). The new rule of inference in this system is the Rule of Necessitation: If p is a theorem of the system, then so is Lp. Stronger systems of modal logic can be obtained by adding additional axioms. Some add the axiom Lp LLp; others add the axiom Mp LMp.
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