软件学报  2017, Vol. 28 Issue (7): 1629-1639   PDF    
Goguen命题逻辑系统公理化扩张的Γ-k真度理论及性质
高晓莉, 惠小静, 朱乃调    
延安大学 数学与计算机科学学院, 陕西 延安 716000
摘要: 首先对n值Goguen命题逻辑进行公理化扩张,Goguen~,Δ,记为∏~,Δ.利用公式的诱导函数给出公式在kk任取~或Δ)连接词下相对于局部有限理论Γ的Γ-k真度的定义;讨论了∏~,Δ中Γ-k真度的MP规则、HS规则等相关性质;最后,在Γ-k中定义了两公式间的Γ-k相似度与Γ-k伪距离,得到了公式在k连接词下相对于局部有限理论Γ-k的相似度与Γ-k伪距离所具有的一些良好性质.
关键词: Goguen命题逻辑系统     Γ-k真度     Γ-k相似度     Γ-k伪距离    
Theory and Property of Γ-k Truth Degree on Axiomatic Extension of Goguen Propositional Logic System
GAO Xiao-Li, HUI Xiao-Jing, ZHU Nai-Diao    
College of Mathematics and Computer Science, Yan'an University, Yan'an 716000, China
Foundation item: National Natural Science Foundation of China (11471007);Natural Science Foundation of Shaanxi Province of China (2014JM1020);Graduate Innovation Fund of Yan'an University (YCX201612)
Abstract: Axiomatic extensions of n-valued Goguen propositional logic system denoted as ∏~, Δ is first studied in this paper. Using induced function, the definition of Γ-k truth degree of formula relative to local finite theory Γ under the k conjunction is given. The MP rule, HS rule, and some correlation properties are also discussed. Finally, the definition of Γ-k similarity degree and Γ-k pseudo-metric in ∏~, Δ between two formulas is presented, and some good properties about Γ-k similarity degree and Γ-k pseudo-metric relative to local finite theory Γ under the k conjunction are simultaneously obtained.
Key words: Goguen propositional logic system     Γ-k truth degree     Γ-k similarity degree     Γ-k pseudo-metric    

关于命题逻辑结论程度化的思想被Pavelka在20世纪70年代提出以来[1], 大量学者投入到了这一研究领域, 并取得了丰富的成果[2-18].其中, 文献[2-6]从逻辑概念程度化入手, 给出了命题逻辑系统中公式的真度理论.文献[7-10]利用赋值集的随机化方法, 在命题逻辑系统中给出了公式的随机真度概念, 实现了计量逻辑学与概率逻辑学的融合.文献[11, 12]对Łukasiewic命题逻辑系统和R0Ł3n+1命题逻辑系统中公式相对于局部有限理论Γ的真度进行了研究, 提出了公式的相对Γ-真度, 把一般真度作为相对真度的特例, 拓展了真度理论的应用范围.文献[13, 14]通过视赋值集为通常乘积拓扑空间, 利用其上的Borel概率测度在命题逻辑系统中引入了Borel概率真度的概念, 从而使计量逻辑学中命题的真度概念成为所研究工作的一个特例.

然而, 在目前广泛受到大家关注的命题逻辑系统中, G del命题逻辑系统和Goguen命题逻辑系统中的否定过强而使相关研究受到了阻碍.文献[15, 16]引入了基本连接词对合否定~.文献[17]引入连接词Δ, 并提出了基本逻辑系统BL的公理化扩张BLΔ系统, 同时与对合否定相结合建立了SBL~系统, 在该系统中, Δ演绎定理和强完备性定理都成立, 从而使得在G del命题逻辑系统和Goguen命题逻辑系统中的研究得以顺利展开.文献[18]便是在SBL~系统中以推理中命题的真值为基础, 运用Δ转换词建立了推理中前提与结论的真值关系定理, 实现了Δ模糊逻辑系统的计量化.

本文以Goguen命题逻辑系统为例, 拟在SBL公理化扩张中展开计量化研究.首先在n值Goguen命题逻辑系统中添加了两类算子, 即对合否定和连接词Δ, 将其作为SBL~系统的公理化扩张, 记为Goguen~, Δ或∏~, Δ.然后利用公式的诱导函数给出公式在k(k任取~或Δ)连接词下相对于局部有限理论Γ的Γ-k真度的定义; 讨论了∏~, Δ中Γ-k真度的MP规则、HS规则等相关性质; 最后, 在∏~, Δ中定义了两公式间的Γ-k相似度与Γ-k伪距离, 得到了公式在k连接词下相对于局部有限理论Γ的Γ-k相似度与Γ-k伪距离所具有的一些良好性质.

1 预备知识

定义1.1[18]. BLΔ的公理系统如下.

(BL)BL的公理系统;

(A Δ1) $\Delta A\vee \neg \Delta A;$

(A Δ2) $\Delta (A\vee B)\to (\Delta A\vee \Delta B);$

(A Δ3) $\Delta A\to A;$

(A Δ4) $\Delta A\to \Delta \Delta A;$

(A Δ5) $\Delta (A\to B)\to (\Delta A\to \Delta B).$

BLΔ中的推理规则为MP规则和Δ规则, MP规则为从$A,A\to B$推得B.Δ规则为$A\to \Delta A.$

如果£ 是BL的公理化扩张, 那么把£ Δ记为£ 的扩张, 其方式正如BL扩张为BLΔ一样, BLΔ系统中下面的Δ演绎定理成立.

定理1.1(Δ演绎定理)[18].令£ 是BLΔ的公理化扩张, 那么对任意理论Γ, 公式AB, 有Γ, AB当且仅当Γ⊢ΔAB.

SBL是BL在增加了公理$\neg \neg A\vee \neg A$之后的公理化扩张.SBLΔ也为SBL的公理化扩张.SBL~系统是在SBL系统中增加了对合否定连接词~后形成的逻辑系统.

定义1.2[17].作为SBL的公理化扩张, SBL~的公理系统如下.

(SBL)SBL的公理系统;

(~1) ~~AA;

(~2) $\neg A\to \sim A;$

(~3) $\Delta (A\to B)\to \Delta (\sim B\to \sim B).$

在SBL~系统中, 令$\Delta A=\neg \sim A$, 便可以建立SBLΔ系统与SBL~系统之间的关系.即SBL~有如下的等价公理系统.

(SBLΔ)SBLΔ的公理系统;

(~1) ~~AA;

(~3) $\Delta (A\to B)\to \Delta (\sim B\to \sim B).$

SBL~中的推理规则也为MP规则和Δ规则.如果£ 是SBL的公理化扩张, 那么把£ ˷记为£ 的扩张, 其方式正如SBL扩张为SBL~一样, 而且G del˷和∏~是SBL~公理化扩张的两个基本类型.由于SBL~也是BLΔ的公理化扩张, 因此SBL~中的Δ演绎定理也成立.

定理1.2(强完备性定理)[17].令£ 是SBL~的公理化扩张, 那么对理论Γ和公式A, 下面条件等价.

(ⅰ) Γ⊢A;

(ⅱ)对每个£ 代数和理论Γ的每个模型e, 均有$e(A)=1.$

2 Γ-k真度的定义及性质

定义2.1.$S=\left\{ {{p}_{1}},{{p}_{2}},... \right\}$是可数集, ~, Δ是S上的一元运算, $\vee ,\wedge ,\to $S上的二元运算, $F(S)$是由S生成的$(\sim ,\Delta ,\vee ,\wedge ,\to )$型自由代数, 称$F(S)$中的元为命题或合式公式, 称S中的元为原子公式.

定义2.2. Goguen命题逻辑系统也称为乘积系统, 记∏.设${{\Pi }_{\sim ,\Delta }}=\left\{ 0,\frac{1}{n-1},...,\frac{n-2}{n-1},1 \right\},$${{\Pi }_{\sim ,\Delta }}$中规定

$\forall x,y\in {{\Pi }_{\sim ,\Delta }},\sim x=1-x,\Delta x=\left\{ \begin{align} & 1,x=1 \\ & 0,x<1 \\ \end{align} \right.,\\ x\vee y=\max \left\{ x,y \right\},x\wedge y=\min \left\{ x,y \right\},$
$x \to y = \left\{ \begin{gathered} 1,{\text{ }}x = 0 \hfill \\ \frac{y}{x} \wedge 1,{\text{ }}x > 0 \hfill \\ \end{gathered} \right. = \left\{ \begin{gathered} 1,{\text{ }}x \leqslant y \hfill \\ \frac{y}{x},{\text{ }}x > y \hfill \\ \end{gathered} \right.,$

$\text{Gogue}{{\text{n}}_{\sim ,\Delta }}$n值乘积命题逻辑系统的扩张, 简记为${{\Pi }_{\sim ,\Delta }}.$

注:${{\Pi }_{\sim ,\Delta }}$作为n值乘积系统的公理化扩张, 是在n值乘积系统的基础上增加了对合否定和连接词Δ两类算子, 由于乘积系统是SBL系统, 因此${{\Pi }_{\sim ,\Delta }}$$\text{SB}{{\text{L}}_{\sim }}$的公理化扩张, 满足$\text{SB}{{\text{L}}_{\sim }}$的公理系统及定理1.1和定理1.2.

定义2.3.$A=A({{p}_{1}},{{p}_{2}},...,{{p}_{m}})\in F(S),$A对应一个nm元函数$\overline{A}$.在$\Pi _{\sim ,\Delta }^{m}$中, $\overline{A}:{{\Pi }_{\sim ,\Delta }}\to [0,1],$这里$\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})$是由运算符号$\sim ,\Delta ,\vee ,\wedge ,\to $${{x}_{1}},{{x}_{2}},...,{{x}_{m}}$连接而成, 其方式恰如$A=A({{p}_{1}},{{p}_{2}},...,{{p}_{m}})\in F(S)$由连接词$\sim ,\Delta ,\vee ,\wedge ,\to $将原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}}$连接而成那样, 称$\overline{A}$是公式A所诱导的函数.

定义2.4.$\Pi _{\sim ,\Delta }^{m}$中, 设$\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})$$F(S)$中命题公式$A({{p}_{1}},{{p}_{2}},...,{{p}_{m}})$所诱导的函数.定义:$l\ge 0,\forall ({{x}_{1}},$$..,{{x}_{m}},...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l},$${{\overline{A}}^{l}}:\Pi _{\sim ,\Delta }^{m+l}$$\to [0,1],$${{\overline{A}}^{l}}({{x}_{1}},...,{{x}_{m}},...,{{x}_{m+l}})=\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$${{\overline{A}}^{l}}$为函数$\overline{A}$的直到第l元的扩张.

接下来我们在$\text{Gogue}{{\text{n}}_{\sim ,\Delta }}$命题逻辑系统中, 利用诱导函数给出公式在k连接词下相对于局部有限理论Γ的Γ-k真度的定义, 并讨论Γ-k真度的相关性质.

$\Gamma \subseteq F(S),A\in F(S),$本文规定${{S}_{\Gamma }}=\{p\in S|\exists B\in \Gamma ,$p是构成B的原子命题}, ${{S}_{A}}=\{p\in S|p$A中出现}, 当${{S}_{\Gamma }}$有限时, 称Γ为$\text{Gogue}{{\text{n}}_{\sim ,\Delta }}$命题逻辑系统的局部有限理论.

以下几点若在文中无特别说明, 则均不发生变化.

(1) 在$\Pi _{\sim ,\Delta }^{m}$中讨论.

(2) $k,\lambda ,\mu ,\eta $任取Δ, ~.

(3) 真值函数的上划线不包括kA前的k.

(4) 基本语法、语义概念如定理、逻辑等价、重言式、矛盾式等均与经典命题逻辑一样.

定义2.5.$\Pi _{\sim ,\Delta }^{m}$中, 设$\Gamma \subseteq F(S),{{S}_{\Gamma }}$有限, $A\in F(S),S={{S}_{\Gamma }}\cup {{S}_{A}}=\{{{p}_{1}},{{p}_{2}},...,{{p}_{m}}\},$

${{\tau }_{n,\Gamma }}(kA)=\left\{ \begin{align} & 1,\text{ }N(\Gamma )=\varnothing \\ & \text{ }\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),\text{ }N(\Gamma )\ne \varnothing } \\ \end{align} \right.,$

其中, $N(\Gamma )=\{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}|\forall B\in \Gamma ,\overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1\},$${{\tau }_{n,\Gamma }}(kA)$为公式Ak连接词下相对于局部有限理论Γ的Γ-k真度, 简称Γ-k真度.

定理2.1.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限, $S={{S}_{\Gamma }}\cup {{S}_{A}}=\{{{p}_{1}},{{p}_{2}},...,{{p}_{m}}\},{{S}^{*}}=\{{{p}_{1}},{{p}_{2}},...,{{p}_{m}},{{p}_{m+1}},...,{{p}_{m+l}}\}\subseteq S,$

${{\tau }_{n,\Gamma }}(kA)=\left\{ \begin{align} & 1,\text{ }{{N}^{*}}(\Gamma )=\varnothing \\ & \frac{1}{|{{N}^{*}}(\Gamma )|}\sum\limits_{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in {{N}^{*}}(\Gamma )}{k\overline{A}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})},\text{ }{{N}^{*}}(\Gamma )\ne \varnothing \\ \end{align} \right.,$

其中, ${{N}^{*}}(\Gamma )=\{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l}|\forall B\in \Gamma ,{{\overline{B}}^{l}}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})=1\}.$

证明:因为$N(\Gamma )=\{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}|\forall B\in \Gamma ,\overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1\},$

${{N}^{*}}(\Gamma )=\{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l}|\forall B\in \Gamma ,{{\overline{B}}^{l}}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})=1\},$

由定义2.4可知, $\forall ({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l},{{\overline{B}}^{l}}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})=\overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$

${{N}^{*}}(\Gamma )|=|N(\Gamma )|\times {{n}^{l}},$

所以, 当$\text{ }{{N}^{*}}(\Gamma )=\varnothing $时, $N(\Gamma )=\varnothing ,$${{\tau }_{n,\Gamma }}(kA)=1.$

$\text{ }{{N}^{*}}(\Gamma )\ne \varnothing $时, $N(\Gamma )\ne \varnothing ,$由于${{\overline{A}}^{l}}({{x}_{1}},...,{{x}_{m}},...,{{x}_{m+l}})=\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$得到

$\sum\limits_{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l}}{k\overline{A}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})}=\\ \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}\times {{n}^{l}}}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\times {{n}^{l}}}.$

同时,

$\begin{array}{l} \frac{1}{{|{N^*}(\Gamma )|}}\sum\limits_{({x_1},...,{x_m},{x_{m + 1}}...,{x_{m + l}}) \in {N^*}(\Gamma )} {k\bar A({x_1},...,{x_m},{x_{m + 1}},...,{x_{m + l}})} \\ = \frac{1}{{|N(\Gamma )| \times {n^l}}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} {k\bar A({x_1},{x_2},...,{x_m}) \times {n^l}} \\ = \frac{1}{{|N(\Gamma )|}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} {k\bar A({x_1},{x_2},...,{x_m})} . \end{array}$

从而有${{\tau }_{n,\Gamma }}(kA)=\frac{1}{|{{N}^{*}}(\Gamma )|}\sum\limits_{({{x}_{1}},...,{{x}_{m+1}},...,{{x}_{m+l}})\in {{N}^{*}}(\Gamma )}{k\overline{A}({{x}_{1}},...,{{x}_{m+1}},...},{{x}_{m+l}}).$

为方便表述, 将${{N}^{*}}(\Gamma ),\sum\limits_{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in {{N}^{*}}(\Gamma )}{k\overline{A}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}},...,{{x}_{m+l}})}$仍记作$N(\Gamma ),\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}.$

定理2.2.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限,

(ⅰ)若Γ‘A, 则${{\tau }_{n,\Gamma }}(\Delta A)=1,{{\tau }_{n,\Gamma }}(\sim A)=0;$

(ⅱ)若Γ‘~A, 则${{\tau }_{n,\Gamma }}(\Delta A)=0,{{\tau }_{n,\Gamma }}(\sim A)=1.$

证明: (ⅰ)若Γ‘A, 则$\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$$\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1,$

结合Δ连接词的运算性质可得, $\Delta \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1,|N(\Gamma )|=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\Delta \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})};$

结合~连接词的运算性质可得, $\sim \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=0,\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\sim \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=0.$

由定义2.5可得, ${{\tau }_{n,\Gamma }}(\Delta A)=1,{{\tau }_{n,\Gamma }}(\sim A)=0.$

(ⅱ)若Γ‘~A, $\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),\sim \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1,$

结合~连接词的运算性质可得, $\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=0,|N(\Gamma )|=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\sim \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})},$

结合Δ连接词的运算性质可得, $\Delta \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=0,\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\Delta \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=0,$

由定义2.5可得, ${{\tau }_{n,\Gamma }}(\Delta A)=0,{{\tau }_{n,\Gamma }}(\sim A)=1.$

定理2.3.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限, 若$N(\Gamma )\ne \varnothing ,$${{\tau }_{n,\Gamma }}(\sim kA)=1-{{\tau }_{n,\Gamma }}(kA).$

证明:因为$N(\Gamma )=\{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in {{\Pi }_{\sim ,\Delta }}|\forall B\in \Gamma ,\overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1\},$$N(\Gamma )\ne \varnothing ,$

所以,

$\begin{align} & {{\tau }_{n,\Gamma }}(\sim kA)=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\sim k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})} \\ & \text{ }=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(1-k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}))} \\ & \text{ }=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{1}-\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{k\overline{A}({{x}_{1}},}{{x}_{2}},...,{{x}_{m}}) \\ & \text{ }=1-\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})} \\ & \text{ }=1-{{\tau }_{n,\Gamma }}(kA). \\ \end{align}$

定理2.4.${{\Gamma }_{1}}\subseteq {{\Gamma }_{2}}\subseteq F(S),A\in F(S),{{S}_{{{\Gamma }_{2}}}}$有限, 若${{\tau }_{n,{{\Gamma }_{1}}}}(kA)=1,$${{\tau }_{n,{{\Gamma }_{2}}}}(kA)=1.$

证明:由于${{\Gamma }_{1}}\subseteq {{\Gamma }_{2}}$, 则$N({{\Gamma }_{2}})\subseteq N({{\Gamma }_{1}}),$$N({{\Gamma }_{2}})=\varnothing $时, ${{\tau }_{n,{{\Gamma }_{2}}}}(kA)=1,$

$N({{\Gamma }_{2}})\ne \varnothing $时, 可知$N({{\Gamma }_{1}})\ne \varnothing ,$因为${{\tau }_{n,{{\Gamma }_{1}}}}(kA)=1,$所以$\frac{1}{|N({{\Gamma }_{1}})|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N({{\Gamma }_{1}})}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=1,$

从而有$|N({{\Gamma }_{1}})|=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N({{\Gamma }_{1}})}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}.$

$\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N({{\Gamma }_{1}}),$$k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1;$$\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N({{\Gamma }_{2}}),$$k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1.$

因此, $|N({{\Gamma }_{2}})|=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N({{\Gamma }_{2}})}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})},$${{\tau }_{n,{{\Gamma }_{2}}}}(kA)=1.$

引理2.1.$\forall a,b\in \Pi _{\sim ,\Delta }^{m},$则(1) $1\to \mu b=\mu b;$(2) $\lambda a\to \mu b\ge \mu b.$

证明:

(1) 当$\mu b=1$时, $1\to \mu b=1\to 1=1=\mu b;$

$\mu b<1$时, $1\to \mu b=\frac{\mu b}{1}=\mu b.$

(2) 当$\lambda a\ge \mu b$时, $\lambda a\to \mu b=\frac{\mu b}{\lambda a}\ge \mu b;$

$\lambda a<\mu b$时, $\lambda a\to \mu b=1\ge \mu b.$

定理2.5.$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 若Γ⊢λA, 则

(ⅰ) ${{\tau }_{n,\Gamma }}(\lambda A\to \mu B)={{\tau }_{n,\Gamma }}(\lambda A\wedge \mu B)={{\tau }_{n,\Gamma }}(\mu B);$

(ⅱ) ${{\tau }_{n,\Gamma }}(\mu B\to \lambda A)=1.$

证明:设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},$若Γ⊢λA, $\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$$\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1.$

(ⅰ)由引理2.1(1) 可知,

$(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\to \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\\ 1\to \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$
$(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\wedge \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\\ 1\wedge \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$

所以,

$\begin{align} & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})} \\ & \text{ }=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}, \\ \end{align}$

则有

$\begin{align} & \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})} \\ & \text{ }=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}. \\ \end{align}$

由定义2.5可得, ${{\tau }_{n,\Gamma }}(\lambda A\to \mu B)={{\tau }_{n,\Gamma }}(\lambda A\wedge \mu B)={{\tau }_{n,\Gamma }}(\mu B).$

(ⅱ)由引理2.1(2) 可知,

$(\mu \overline{B}\to \lambda \overline{A})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\to \lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\ge \lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=1.$

类似于定理2.5(ⅰ), 得到$\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\mu \overline{B}\to \lambda \overline{A})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=1.$

再由定义2.5可知, ${{\tau }_{n,\Gamma }}(\mu B\to \lambda A)=1.$

引理2.2.$\forall a,b\in \Pi _{\sim ,\Delta }^{m},$$\lambda a\vee \mu b=\lambda a+\mu b-(\lambda a\wedge \mu b).$

证明:首先令${{*}_{1}}\text{=(}\lambda a\vee \mu b)-\lambda a-\mu b+(\lambda a\wedge \mu b),$再分两种情况进行讨论.

1) 当$\lambda a\ge \mu b$时, ${{*}_{1}}\text{=}\lambda a-\lambda a-\mu b+\mu b=0;$

2) 当$\lambda a<\mu b$时, ${{*}_{1}}\text{=}\mu b-\lambda a-\mu b+\lambda a=0.$

定理2.6.$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 则${{\tau }_{n,\Gamma }}(\lambda A\vee \mu B)={{\tau }_{n,\Gamma }}(\lambda A)+{{\tau }_{n,\Gamma }}(\mu B)-{{\tau }_{n,\Gamma }}(\lambda A\wedge \mu B).$

证明:设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$由引理2.2可知,

$\lambda \overline A ({x_1},{x_2},...,{x_m}) \vee \mu \overline B ({x_1},{x_2},...,{x_m}){\text{ = }}\lambda \overline A ({x_1},{x_2},...,{x_m}) +\\ \mu \overline B ({x_1},{x_2},...,{x_m}) - (\lambda \overline A ({x_1},{x_2},...,{x_m}) \wedge \mu \overline B ({x_1},{x_2},..,{x_m})),$

其中,

$\lambda \overline A ({x_1},{x_2},...,{x_m}) \vee \mu \overline B ({x_1},{x_2},...,{x_m}){\rm{ = }}\lambda \overline A ({x_1},{x_2},...,{x_m}) +\\ \mu \overline B ({x_1},{x_2},...,{x_m}) - (\lambda \overline A ({x_1},{x_2},...,{x_m}) \wedge \mu \overline B ({x_1},{x_2},..,{x_m})),$
$\lambda \overline A ({x_1},{x_2},...,{x_m}) \wedge \mu \overline B ({x_1},{x_2},...,{x_m}) = (\lambda \overline A \wedge \mu \overline B )({x_1},{x_2},...,{x_m}),$

那么,

$(\lambda \overline{A}\vee \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})+\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})-(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$

因此,

$\begin{align} & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\vee \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}\text{=}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}+ \\ & \text{ }\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}- \\ & \text{ }\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}. \\ \end{align}$

同时,

$\begin{align} & \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\vee \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}+ \\ & \text{ }\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}- \\ & \text{ }\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\wedge \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}. \\ \end{align}$

由定义2.5可得, ${{\tau }_{n,\Gamma }}(\lambda A\vee \mu B)={{\tau }_{n,\Gamma }}(\lambda A)+{{\tau }_{n,\Gamma }}(\mu B)-{{\tau }_{n,\Gamma }}(\lambda A\wedge \mu B).$

引理2.3.$\forall a,b\in \Pi _{\sim ,\Delta }^{m},$$\mu b\ge \lambda a+(\lambda a\to \mu b)-1.$

证明:首先令${{*}_{2}}\text{=}\mu b-\lambda a-(\lambda a\to \mu b)+1,$再分两种情况进行讨论.

1) 当$\lambda a\le \mu b$时, ${{*}_{2}}\text{=}\mu b-\lambda a\ge 0;$

2) 当$\lambda a>\mu b$时, ${{*}_{2}}\text{=}\mu b-\lambda a-\frac{\mu b}{\lambda a}+1\text{=}\frac{\mu b(\lambda a-1)}{\lambda a}-\frac{\lambda a(\lambda a-1)}{\lambda a}\text{=}\frac{(\mu b-\lambda a)(\lambda a-1)}{\lambda a}\ge 0.$

综上, 可得$\mu b\ge \lambda a+(\lambda a\to \mu b)-1.$

定理2.7(Γ-k真度的MP规则).设$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 若${{\tau }_{n,\Gamma }}(\lambda A)\ge \alpha ,{{\tau }_{n,\Gamma }}(\lambda A\to \mu B)\ge \beta ,$${{\tau }_{n,\Gamma }}(\mu B)\ge \alpha +\beta -1.$

证明:设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$由引理2.3可知,

$\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\ge \lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})+(\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\to \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}))-1,$

因此,

$\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}\ge \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{\lambda \overline{A}({{x}_{1}},}{{x}_{2}},...,{{x}_{m}})+$
$\begin{align} & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}- \\ & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{1}, \\ \end{align}$

所以,

$\eqalign{ & \frac{1}{{|N(\Gamma )|}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} {\mu \overline B ({x_1},{x_2},...,{x_m})} \geqslant \frac{1}{{|N(\Gamma )|}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} {\lambda \overline A ({x_1},{x_2},...,{x_m})} + \cr & {\text{ }}\frac{1}{{|N(\Gamma )|}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} {(\lambda \overline A \to \mu \overline B )({x_1},{x_2},...,{x_m})} - \cr & {\text{ }}\frac{1}{{|N(\Gamma )|}}\sum\limits_{({x_1},{x_2},...,{x_m}) \in N(\Gamma )} 1 . \cr} $

结合定义2.5可得, ${{\tau }_{n,\Gamma }}(\mu B)\ge \alpha +\beta -1.$

推论2.1.$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 若${{\tau }_{n,\Gamma }}(\lambda A)=1,{{\tau }_{n,\Gamma }}(\lambda A\to \mu B)=1,$${{\tau }_{n,\Gamma }}(\mu B)=1.$

引理2.4.$\forall a,b,c\in \Pi _{\sim ,\Delta }^{m},$$(\lambda a\to \eta c)\ge (\lambda a\to \mu b)+(\mu b\to \eta c)-1.$

证明:首先令${{*}_{3}}\text{=}(\lambda a\to \eta c)-(\lambda a\to \mu b)-(\mu b\to \eta c)+1,$再分以下几种情况进行讨论.

1) 当$\lambda a\le \eta c$

1.1) 当$\mu b\ge \eta c$时, ${{*}_{3}}\text{=1}-(\lambda a\to \mu b)-(\mu b\to \eta c)+1=\text{1}-(\lambda a\to \mu b)+1-\frac{\eta c}{\mu b}\ge 0.$

1.2) 当$\mu b<\eta c$

 1.2.1) 当$\lambda a\ge \mu b$时, ${{*}_{3}}\text{=}1-\frac{\mu b}{\lambda a}\ge 0;$

 1.2.2) 当$\lambda a<\mu b$时, ${{*}_{3}}\text{=0}\text{.}$

2) 当$\lambda a>\eta c$

2.1) 当$\mu b<\eta c$时, ${{*}_{3}}\text{=}\frac{\eta c}{\lambda a}-\frac{\mu b}{\lambda a}-1+1=\frac{\eta c-\mu b}{\lambda a}\ge 0.$

2.2) 当$\mu b\ge \eta c$

 2.2.1) 当$\lambda a>\mu b$时, ${{*}_{3}}\text{=}\frac{\eta c}{\lambda a}-\frac{\mu b}{\lambda a}-\frac{\eta c}{\mu b}+1\text{=}\frac{\eta c-\mu b}{\lambda a}-\frac{\eta c-\mu b}{\mu b}\text{=(}\eta c-\mu b)\left( \frac{1}{\lambda a}-\frac{1}{\mu b} \right)\ge 0;$

 2.2.2) 当$\lambda a\le \mu b$时, ${{*}_{3}}\text{=}\frac{\eta c}{\lambda a}-1-\frac{\eta c}{\mu b}+1\text{=}\frac{\eta c}{\lambda a}-\frac{\eta c}{\mu b}\ge 0.$

综上可得$(\lambda a\to \eta c)\ge (\lambda a\to \mu b)+(\mu b\to \eta c)-1.$

定理2.8(Γ-k真度的HS规则).$\Gamma \subseteq F(S),A,B,C\in F(S),{{S}_{\Gamma }}$有限, 若${{\tau }_{n,\Gamma }}(\lambda A\to \mu B)\ge \alpha ,{{\tau }_{n,\Gamma }}(\mu B\to \eta C)\ge \beta ,$${{\tau }_{n,\Gamma }}(\lambda A\to \eta C)\ge \alpha +\beta -1.$

证明:设A, B, C含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$由引理2.4可知,

$(\lambda \overline{A}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\ge (\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})+(\mu \overline{B}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})-1,$

因此,

$\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}\ge \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}+$
$\begin{align} & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\mu \overline{B}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}- \\ & \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{1}. \\ \end{align}$

所以,

$\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}\ge \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}+$
$\begin{align} & \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{(\mu \overline{B}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}- \\ & \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{1}. \\ \end{align}$

结合定义2.5可得, ${{\tau }_{n,\Gamma }}(\lambda A\to \eta C)\ge \alpha +\beta -1.$

推论2.2.$\Gamma \subseteq F(S),A,B,C\in F(S),{{S}_{\Gamma }}$有限, 若${{\tau }_{n,\Gamma }}(\lambda A\to \mu B)=1,{{\tau }_{n,\Gamma }}(\mu B\to \eta C)=1,$${{\tau }_{n,\Gamma }}(\lambda A\to \eta C)=1.$

下面将随机举出其中一个定理的例子来加以计算.

例2.1:在${{\Pi }_{\sim ,\Delta }}$二元四值中, 设$\Gamma =(\sim {{p}_{1}}\to \Delta {{p}_{2}})\to {{p}_{2}},A=(\sim {{p}_{1}}\vee \Delta {{p}_{2}})\to {{p}_{2}},B=(\sim {{p}_{1}}\to \sim {{p}_{2}})\to {{p}_{1}},$ $\to \sim {{p}_{2}})\to \sim {{p}_{1}},$试计算${{\tau }_{4,\Gamma }}((\Delta A\wedge \sim B)\to \sim C)={{\tau }_{4,\Gamma }}((\Delta A\to \sim C)\vee (\sim B\to \sim C)).$

解:根据定义2.5来计算${{S}_{\Gamma }}=\{{{p}_{1}},{{p}_{2}}\},{{S}_{A}}=\{{{p}_{1}},{{p}_{2}}\},{{S}_{\Gamma }}\cup {{S}_{A}}=\{{{p}_{1}},{{p}_{2}}\},$公式A, B, C所诱导的函数分别为

$\begin{align} & \overline{A}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{A}(x,y)=(\sim x\vee \Delta y)\to y, \\ & \overline{B}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{B}(x,y)=(\sim x\to \sim y)\to x, \\ & \overline{C}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{C}(x,y)=(\Delta x\to \sim y)\to \sim x. \\ \end{align}$

$\Gamma =(\sim {{p}_{1}}\to \Delta {{p}_{2}})\to {{p}_{2}}$可以写成诱导函数的形式为$\Gamma =(\sim x\to \Delta y)\to y.$

为了方便理解, 特做出如下图表.

x y $\begin{align} & \Gamma =(\sim x\to \\ & \Delta y)\to y \\ \end{align}$ $\begin{align} & \overline{A}(x,y)= \\ & (\sim x\vee \Delta y)\to y \\ \end{align}$ $\begin{align} & \overline{B}(x,y)= \\ & (\sim x\to \sim y)\to x \\ \end{align}$ $\begin{align} & \overline{C}(x,y)=(\Delta x \\ & \to \sim y)\to \sim x \\ \end{align}$ $\begin{align} & (\Delta A\wedge \sim B) \\ & \to \sim C \\ \end{align}$ $\begin{align} & (\Delta A\to \sim C)\vee \\ & (\sim B\to \sim C) \\ \end{align}$
0 0 1 0 0 1 1 1
0 $\frac{1}{3}$ 1 $\frac{1}{3}$ 0 1 1 1
0 $\frac{2}{3}$ 1 $\frac{2}{3}$ 0 1 1 1
0 1 1 1 1 1 1 1
$\frac{1}{3}$ 0 1 0 $\frac{1}{3}$ $\frac{2}{3}$ 1 1
$\frac{1}{3}$ $\frac{1}{3}$ 1 $\frac{1}{2}$ $\frac{1}{3}$ 1 1 1
$\frac{1}{3}$ $\frac{2}{3}$ 1 1 $\frac{2}{3}$ $\frac{2}{3}$ 1 1
$\frac{1}{3}$ 1 1 1 1 1 1 1
$\frac{2}{3}$ 0 1 0 $\frac{2}{3}$ $\frac{1}{3}$ 1 1
$\frac{2}{3}$ $\frac{1}{3}$ 1 1 $\frac{2}{3}$ $\frac{1}{2}$ 1 1
$\frac{2}{3}$ $\frac{2}{3}$ 1 1 $\frac{2}{3}$ $\frac{2}{3}$ 1 1
$\frac{2}{3}$ 1 1 1 1 1 1 1
1 0 0 1 1 0 1 1
1 $\frac{1}{3}$ $\frac{1}{3}$ 1 1 0 1 1
1 $\frac{2}{3}$ $\frac{2}{3}$ 1 1 0 1 1
1 1 1 1 1 0 1 1

从表中可以看出, $|N(\Gamma )|$为所有使$\Gamma =(\sim x\to \Delta y)\to y$的值为1元素的个数, 即$|N(\Gamma )|=13.$

$\begin{align} & \quad {{\tau }_{4,\Gamma }}((\Delta A\wedge \sim B)\to \sim C)=\frac{1}{13}\sum\limits_{(x,y)\in N(\Gamma )}{13\times 1}, \\ & {{\tau }_{4,\Gamma }}((\Delta A\to \sim C)\vee (\sim B\to \sim C))=\frac{1}{13}\sum\limits_{(x,y)\in N(\Gamma )}{13\times 1}, \\ \end{align}$

因此, ${{\tau }_{4,\Gamma }}((\Delta A\wedge \sim B)\to \sim C)={{\tau }_{4,\Gamma }}((\Delta A\to \sim C)\vee (\sim B\to \sim C)).$

3 Γ-k相似度、Γ-k伪距离的定义及性质

定义3.1.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限, 则有

${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\wedge (\mu B\to \lambda A)),$

${{\xi }_{n,\Gamma }}(\lambda A,\mu B)$为公式$A,B$$\lambda ,\mu $连接词下相对于局部有限理论Γ的Γ-k相似度, 简称Γ-k相似度.

定理3.1.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限, 则${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-1.$

证明:设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},$由定理2.6和定义3.1可知,

$\begin{align} & {{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\wedge (\mu B\to \lambda A)) \\ & \text{ }={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-{{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\vee (\mu B\to \lambda A)) \\ & \text{ }={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-1. \\ \end{align}$

定理3.2.$\Gamma \subseteq F(S),A\in F(S),{{S}_{\Gamma }}$有限, 则

(ⅰ) ${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\xi }_{n,\Gamma }}(\mu B,\lambda A);$

(ⅱ) ${{\xi }_{n,\Gamma }}(\lambda A\vee \mu B,\lambda A)={{\tau }_{n,\Gamma }}(\mu B\to \lambda A);$

(ⅲ) ${{\xi }_{n,\Gamma }}(\lambda A\wedge \mu B,\lambda A)={{\tau }_{n,\Gamma }}(\lambda A\to \mu B).$

证明:设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},\forall ({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma ),$

(ⅰ) $\forall a,b\in \Pi _{\sim ,\Delta }^{m},$显然有$(\lambda a\to \mu b)\wedge (\mu b\to \lambda a)=(\mu b\to \lambda a)\wedge (\lambda a\to \mu b).$

所以有, $(\lambda A\to \mu B)\wedge (\mu B\to \lambda A)=(\mu B\to \lambda A)\wedge (\lambda A\to \mu B).$

从而有, $((\lambda \overline{A}\to \mu \overline{B})\wedge (\mu \overline{B}\to \lambda \overline{A}))({{x}_{1}},{{x}_{2}},...,{{x}_{m}})=((\mu \overline{B}\to \lambda \overline{A})\wedge (\lambda \overline{A}\to \mu \overline{B}))({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),$

$\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\lambda \overline{A}\to \mu \overline{B})\wedge (\mu \overline{B}\to \lambda \overline{A}))({{x}_{1}},{{x}_{2}}},...,{{x}_{m}})=\\ \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\mu \overline{B}\to \lambda \overline{A})\wedge (\lambda \overline{A}\to \mu \overline{B}))({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}.$

同时,

$\begin{align} & \frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\lambda \overline{A}\to \mu \overline{B})\wedge (\mu \overline{B}\to \lambda \overline{A}))({{x}_{1}},}{{x}_{2}},...,{{x}_{m}}) \\ & =\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\mu \overline{B}\to \lambda \overline{A})\wedge (\lambda \overline{A}\to \mu \overline{B}))({{x}_{1}}},{{x}_{2}},...,{{x}_{m}}). \\ \end{align}$

由定义2.5可得, $\tau ((\lambda A\to \mu B)\wedge (\mu B\to \lambda A))=\tau ((\mu B\to \lambda A)\wedge (\lambda A\to \mu B)).$

因此${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\xi }_{n,\Gamma }}(\mu B,\lambda A)$

(ⅱ)

${{\xi }_{n,\Gamma }}(\lambda A\vee \mu B,\lambda A)={{\tau }_{n,\Gamma }}(((\lambda A\vee \mu B)\to \lambda A)\wedge (\lambda A\to (\lambda A\vee \mu B)))$
$\begin{align} & ={{\tau }_{n,\Gamma }}(((\lambda A\to \lambda A)\wedge (\mu B\to \lambda A))\wedge ((\lambda A\to \lambda A)\vee (\lambda A\to \mu B))) \\ & ={{\tau }_{n,\Gamma }}((\mu B\to \lambda A)\wedge (\lambda A\to \lambda A)) \\ & ={{\tau }_{n,\Gamma }}(\mu B\to \lambda A). \\ \end{align}$

(ⅲ)

${{\xi }_{n,\Gamma }}(\lambda A\wedge \mu B,\lambda A)={{\tau }_{n,\Gamma }}(((\lambda A\wedge \mu B)\to \lambda A)\wedge (\lambda A\to (\lambda A\wedge \mu B)))$
$\begin{align} & ={{\tau }_{n,\Gamma }}(((\lambda A\to \lambda A)\vee (\mu B\to \lambda A))\wedge ((\lambda A\to \lambda A)\wedge (\lambda A\to \mu B))) \\ & ={{\tau }_{n,\Gamma }}((\lambda A\to \lambda A)\wedge (\lambda A\to \mu B)) \\ & ={{\tau }_{n,\Gamma }}(\lambda A\to \mu B). \\ \end{align}$

定理3.3.$\Gamma \subseteq F(S),A,B,C\in F(S),{{S}_{\Gamma }}$有限, 则${{\xi }_{n,\Gamma }}(\lambda A,\eta C)\ge {{\xi }_{n,\Gamma }}(\lambda A,\mu B)+{{\xi }_{n,\Gamma }}(\mu B,\eta C)-1.$

证明:设A, B, C含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},$由定理2.8和定理3.1可得

$\begin{array}{l} {\xi _{n,\Gamma }}(\lambda A,\mu B) + {\xi _{n,\Gamma }}(\mu B,\eta C) - 1 = ({\tau _{n,\Gamma }}(\lambda A \to \mu B) + {\tau _{n,\Gamma }}(\mu B \to \lambda A) - 1) + \\ ({\tau _{n,\Gamma }}(\eta C \to \mu B) + {\tau _{n,\Gamma }}(\mu B \to \eta C) - 1) - 1\\ \le {\tau _{n,\Gamma }}(\lambda A \to \eta C) + {\tau _{n,\Gamma }}(\eta C \to \lambda A) - 1\\ = {\xi _{n,\Gamma }}(\lambda A,\eta C). \end{array}$

定义3.2.$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 规定${{\rho }_{n,\Gamma }}:F(S)\times F(S)\to \left[ 0,1 \right],$

${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\xi }_{n,\Gamma }}(\lambda A,\mu B),$

${{\rho }_{n,\Gamma }}(\lambda A,\mu B)$为公式$A,B$$\lambda ,\mu $连接词下相对于局部有限理论Γ的Γ-k伪距离, 简称Γ-k伪距离, $(F(S),{{\rho }_{n,\Gamma }})$称为Γ-k逻辑度量空间.

定理3.4.$\Gamma \subseteq F(S),A,B,C\in F(S),{{S}_{\Gamma }}$有限, 则

${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+1-{{\tau }_{n,\Gamma }}(\mu B\to \lambda A).$

证明:由定理3.1可知, ${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-1,$

则有

$1-{{\xi }_{n,\Gamma }}(\lambda A,\mu B)=1-({{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-1),$

可得

${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+1-{{\tau }_{n,\Gamma }}(\mu B\to \lambda A).$

定理3.5.$\Gamma \subseteq F(S),A,B\in F(S),{{S}_{\Gamma }}$有限, 以下各结论成立.

(ⅰ) ${{\rho }_{n,\Gamma }}(\lambda A,\mu B)={{\rho }_{n,\Gamma }}(\mu B,\lambda A);$

(ⅱ) ${{\rho }_{n,\Gamma }}(\lambda A\vee \mu B,\lambda A)=1-{{\tau }_{n,\Gamma }}(\mu B\to \lambda A);$

(ⅲ) ${{\rho }_{n,\Gamma }}(\lambda A\wedge \mu B,\lambda A)=1-{{\tau }_{n,\Gamma }}(\lambda A\to \mu B).$

证明:在此只证明(ⅰ), 其他同理可证, 设A, B含有相同的原子公式${{p}_{1}},{{p}_{2}},...,{{p}_{m}},$由定理3.2(ⅰ)可知, 因为, ${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\xi }_{n,\Gamma }}(\mu B,\lambda A),$所以, ${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\xi }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\xi }_{n,\Gamma }}(\mu B,\lambda A)={{\rho }_{n,\Gamma }}(\mu B,\lambda A).$

4 总结

本文对n值Goguen命题逻辑进行了公理化扩张$\text{Gogue}{{\text{n}}_{\sim ,\Delta }}$(${{\Pi }_{\sim ,\Delta }}$), 并利用公式的诱导函数给出公式在k(k任取~或Δ)连接词下相对于局部有限理论Γ的Γ-k真度的定义; 讨论了${{\Pi }_{\sim ,\Delta }}$中Γ-k真度的MP规则、HS规则等相关性质; 定义了${{\Pi }_{\sim ,\Delta }}$中两公式间的Γ-k相似度与Γ-k伪距离, 得到了公式在k连接词下相对于局部有限理论Γ的Γ-k相似度与Γ-k伪距离所具有的一些良好性质.关于Γ-k真度、Γ-k相似度与Γ-k伪距离所具有的更多良好性质, 以及关于Γ-k真度的近似推理理论等, 我们将在另文中加以讨论.

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