# 武卡谢维奇逻辑

## 实数值语义

• ${\displaystyle w(\theta \rightarrow \phi )=F_{\rightarrow }(\theta ,\phi )}$
• ${\displaystyle w(\neg \theta )=F_{\neg }(\theta )}$
• ${\displaystyle w(\theta \wedge \phi )=F_{\wedge }(\theta ,\phi )}$
• ${\displaystyle w(\theta \vee \phi )=F_{\vee }(\theta ,\phi )}$

${\displaystyle F_{\wedge }}$, ${\displaystyle F_{\vee }}$, ${\displaystyle F_{\neg }}$${\displaystyle F_{\rightarrow }}$ 的值明确给出自:

• ${\displaystyle F_{\wedge }(x,y)=Max\{0,x+y-1\}}$
• ${\displaystyle F_{\vee }(x,y)=Min\{1,x+y\}}$
• ${\displaystyle F_{\neg }(x)=1-x}$
• ${\displaystyle F_{\rightarrow }(x,y)=Min\{1,1-x+y\}}$

### 求值的性质

${\displaystyle F_{\wedge }}$${\displaystyle F_{\vee }}$ 满足

• ${\displaystyle F_{\wedge }(0,0)=F_{\wedge }(0,1)=F_{\wedge }(1,0)=0}$${\displaystyle F_{\wedge }(1,1)=1}$
• ${\displaystyle F_{\vee }(0,0)=0}$${\displaystyle F_{\vee }(0,1)=F_{\vee }(1,0)=F_{\vee }(1,1)=1}$
• ${\displaystyle F_{\wedge }}$${\displaystyle F_{\vee }}$连续性的。
• ${\displaystyle F_{\wedge }}$${\displaystyle F_{\vee }}$ 在每个构成上是严格递增的。
• ${\displaystyle F_{\wedge }}$${\displaystyle F_{\vee }}$ 在如下意义上是结合性的: ${\displaystyle F(a,F(b,c))=F(F(a,b),c)}$ 对于每个 ${\displaystyle F\in \{F_{\wedge },F_{\vee }\}}$

• ${\displaystyle F_{\neg }(0)=1}$${\displaystyle F_{\neg }(1)=0}$
• ${\displaystyle F_{\neg }}$ 是连续的。

## 引用

1. ^ Łukasiewicz J., 1920, O logice trojwartosciowej (Polish, On three-valued logic). Ruch filozoficzny 5:170–171.
2. ^ Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86.
3. ^ Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
4. ^ Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.