Formal system s 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 = 00AC; L 00AC; p (where 00AC; 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 2283; p and (A2) L(p 2283; q) 2283; (Lp 2283; 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 2283; LLp; others add the axiom Mp 2283; LMp.