模态逻辑(modal logic)

包含诸如必要性、可能性、不可能性、偶然性、精确的关联性以及某些其他紧密相关的概念模式的规范系统。建构一组模态逻辑最直截了当的方法就是在某些非模态逻辑的系统中增加一个新的原始运算素，设计成其中一种模式的表现，以此来定义其他的模态运算素，并增加包含那些模态运算素的公理与／或转换规则。举例来说，某人可以在古典命题的运算法中增加L这个符号，意为「那是必要的」；以此方式，Lp就读作「p是必要的」。可能性的运算素M（「那是有可能的」）可以用L来定义为Mp=@8(logicNon.jpgL@8(logicNon.jpgp（译按：「p是有可能的＝非p是非必要的」，其中@8(logicNon.jpg表示「非」）。除了古典命题逻辑的公理与推论规则之外，有的系统可能有自己的二组公理与一组推论规则。模态逻辑中某些独特的公理如：(A1)Lpp（若p是必要的，则p）与(A2)L(pq)(LpLq)（若（若p则q）是必要的，则（若p是必要的，则q是必要的））。在这个系统中，新的推论规则是「必要性规则」：若p为系统的定理，则Lp也是。藉由增加的公理可以获得更强的模态逻辑系统。有人增加了LpLLp（若p是必要的，则「p是必要的」是必要的）的公理，其他的人则增加了MpLMp（若p是有可能的，则「p是有可能的」是必要的）的公理。

English version：

modal logic

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 as 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.