武卡谢维奇逻辑

无穷多值 Łukasiewicz 逻辑是实数值逻辑，其中来自命题演算的句子被指派上在 0 到 1 之间的任意精度的真值。求值有如下递归定义:

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

${\displaystyle F_{\wedge }}$ , ${\displaystyle F_{\vee }}$ , ${\displaystyle F_{\neg }}$  和 ${\displaystyle F_{\rightarrow }}$  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

在这个定义下，求值满足如下条件:

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

• ${\displaystyle F_{\wedge }(0,0)=F_{\wedge }(0,1)=F_{\wedge }(1,0)=0}$  和 ${\displaystyle F_{\wedge }(1,1)=1}$ 。
• ${\displaystyle F_{\vee }(0,0)=0}$  和 ${\displaystyle F_{\vee }(0,1)=F_{\vee }(1,0)=F_{\vee }(1,1)=1}$ 。

• ${\displaystyle F_{\wedge }}$  和 ${\displaystyle F_{\vee }}$  是连续性的。
• ${\displaystyle F_{\wedge }}$  和 ${\displaystyle F_{\vee }}$  在每个构成上是严格递增的。
• ${\displaystyle F_{\wedge }}$  和 ${\displaystyle F_{\vee }}$  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 ${\displaystyle F\in \{F_{\wedge },F_{\vee }\}}$ 。

所以 ${\displaystyle F_{\wedge }}$  和 ${\displaystyle F_{\vee }}$  都是连续t-规范的。

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

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.