# 模态逻辑

$\Diamond p = \lnot\, \Box\, \lnot\, p.$
$\Box p = \lnot\, \Diamond\, \lnot\, p.$

## 真势模态

• 可能的如果它“可能”为真（不管实际上是真是假）;
• 必然的如果它“不可能”为假;
• 偶然的如果它“不是”必然为真，就是说，可能为真可能为假。偶然的真理是“实际上”为真，但“可能曾经不是”的真理。

## 公理系統

"X非必然的"等价于"非X可能的"。
"X非可能的"等价于"非X必然的"。

• $\Box p$（必然的p）等价于$\neg \Diamond \neg p$（非可能的非p）
• $\Diamond p$（可能的p）等价于$\neg \Box \neg p$（非必然的非p）

• 必然性规则：如果p是 K的定理，则$\Box p$也是。
• 分配律公理：如果$\Box (p \rightarrow q)$$(\Box p \rightarrow \Box q)$（这也叫做公理K）

• $\Box p \rightarrow p$（如果p是必然的，则p是事实）

• 4: $\Box p \rightarrow \Box \Box p$
• B: $p \rightarrow \Box \Diamond p$
• D: $\Box p \rightarrow \Diamond p$
• E: $\Diamond p \rightarrow \Box \Diamond p.$

• K := K + N
• T := K + T
• S4 := T + 4
• S5 := S4 + BT + E
• D := K + D.

KS5形成了嵌套的系统层级，建造了正规模态逻辑的核心。D主要对探索模态逻辑的道义解释的人有价值。

## 引用

• M. Fitting and R.L. Mendelsohn (1998) First Order Modal Logic. Kluwer Academic Publishers.
• James Garson (2003) Modal logic. Entry in the Stanford Encyclopedia of Philosophy.
• Rod Girle (2000) Modal Logics and Philosophy. Acumen (UK). The proof theory employs refutation trees (semantic tableaux). A good introduction to the varied interpretations of modal logic.
• Robert Goldblatt (1992) "Logics of Time and Computation", CSLI Lecture Notes No. 7, Centre for the Study of Language and Information, Stanford University, 2nd ed. (distributed by University of Chicago Press).
• Robert Goldblatt (1993) "Mathematics of Modality", CSLI Lecture Notes No. 43, Centre for the Study of Language and Information, Stanford University. (distributed by University of Chicago Press).
• G.E. Hughes and M.J. Cresswell (1968) An Introduction to Modal Logic, Methuen.
• G.E. Hughes and M.J. Cresswell (1984) A Companion to Modal Logic, Medhuen.
• G.E. Hughes and M.J. Cresswell (1996) A New Introduction to Modal Logic, Routledge.
• E.J. Lemmon (with Dana Scott), 1977, An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, no. 11 (ed. by Krister Segerberg), Basil Blackwell, Oxford.
• J. Jay Zeeman (1973) Modal Logic. D. Reidel Publishing Company.