武卡谢维奇逻辑在数学中,Łukasiewicz 逻辑是非经典、多值逻辑。它最初由扬·武卡谢维奇定义为叫做“三价逻辑”的三值逻辑[1];它后来被推广为 n 值(对于所有有限 n)和无限多值变体,命题和一阶都有[2]。它属于t-规范模糊逻辑[3] 和亚结构逻辑[4]类。 实数值语义 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: w ( θ → ϕ ) = F → ( θ , ϕ ) {\displaystyle w(\theta \rightarrow \phi )=F_{\rightarrow }(\theta ,\phi )} w ( ¬ θ ) = F ¬ ( θ ) {\displaystyle w(\neg \theta )=F_{\neg }(\theta )} w ( θ ∧ ϕ ) = F ∧ ( θ , ϕ ) {\displaystyle w(\theta \wedge \phi )=F_{\wedge }(\theta ,\phi )} w ( θ ∨ ϕ ) = F ∨ ( θ , ϕ ) {\displaystyle w(\theta \vee \phi )=F_{\vee }(\theta ,\phi )} F ∧ {\displaystyle F_{\wedge }} , F ∨ {\displaystyle F_{\vee }} , F ¬ {\displaystyle F_{\neg }} 和 F → {\displaystyle F_{\rightarrow }} 的值明确给出自: F ∧ ( x , y ) = M a x { 0 , x + y − 1 } {\displaystyle F_{\wedge }(x,y)=Max\{0,x+y-1\}} F ∨ ( x , y ) = M i n { 1 , x + y } {\displaystyle F_{\vee }(x,y)=Min\{1,x+y\}} F ¬ ( x ) = 1 − x {\displaystyle F_{\neg }(x)=1-x} F → ( x , y ) = M i n { 1 , 1 − x + y } {\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: F ∧ {\displaystyle F_{\wedge }} 和 F ∨ {\displaystyle F_{\vee }} 满足 F ∧ ( 0 , 0 ) = F ∧ ( 0 , 1 ) = F ∧ ( 1 , 0 ) = 0 {\displaystyle F_{\wedge }(0,0)=F_{\wedge }(0,1)=F_{\wedge }(1,0)=0} 和 F ∧ ( 1 , 1 ) = 1 {\displaystyle F_{\wedge }(1,1)=1} 。 F ∨ ( 0 , 0 ) = 0 {\displaystyle F_{\vee }(0,0)=0} 和 F ∨ ( 0 , 1 ) = F ∨ ( 1 , 0 ) = F ∨ ( 1 , 1 ) = 1 {\displaystyle F_{\vee }(0,1)=F_{\vee }(1,0)=F_{\vee }(1,1)=1} 。 F ∧ {\displaystyle F_{\wedge }} 和 F ∨ {\displaystyle F_{\vee }} 是连续性的。 F ∧ {\displaystyle F_{\wedge }} 和 F ∨ {\displaystyle F_{\vee }} 在每个构成上是严格递增的。 F ∧ {\displaystyle F_{\wedge }} 和 F ∨ {\displaystyle F_{\vee }} 在如下意义上是结合性的: F ( a , F ( b , c ) ) = F ( F ( a , b ) , c ) {\displaystyle F(a,F(b,c))=F(F(a,b),c)} 对于每个 F ∈ { F ∧ , F ∨ } {\displaystyle F\in \{F_{\wedge },F_{\vee }\}} 。所以 F ∧ {\displaystyle F_{\wedge }} 和 F ∨ {\displaystyle F_{\vee }} 都是连续t-规范的。 F ¬ ( 0 ) = 1 {\displaystyle F_{\neg }(0)=1} 和 F ¬ ( 1 ) = 0 {\displaystyle F_{\neg }(1)=0} 。 F ¬ {\displaystyle F_{\neg }} 是连续的。引用 ^ Łukasiewicz J., 1920, O logice trojwartosciowej (Polish, On three-valued logic). Ruch filozoficzny 5:170–171. ^ Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86. ^ Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. ^ 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.