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)
(A Δ2)
(A Δ3)
(A Δ4)
(A Δ5)
BLΔ中的推理规则为MP规则和Δ规则, MP规则为从
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A和B, 有Γ, A⊢B当且仅当Γ⊢ΔA→B.
SBL是BL在增加了公理
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) ~~A→A;
(~2)
(~3)
在SBL~系统中, 令
(SBLΔ)SBLΔ的公理系统;
(~1) ~~A→A;
(~3)
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
定义2.1.设
定义2.2. Goguen命题逻辑系统也称为乘积系统, 记∏.设
$\forall x,y\in {{\Pi }_{\sim ,\Delta }},\sim x=1-x,\Delta x=\left\{ \begin{align} & 1,x=1 \\ & 0,x<1 \\ \end{align} \right.,\\ x\vee y=\max \left\{ x,y \right\},x\wedge y=\min \left\{ x,y \right\},$ |
$x \to y = \left\{ \begin{gathered} 1,{\text{ }}x = 0 \hfill \\ \frac{y}{x} \wedge 1,{\text{ }}x > 0 \hfill \\ \end{gathered} \right. = \left\{ \begin{gathered} 1,{\text{ }}x \leqslant y \hfill \\ \frac{y}{x},{\text{ }}x > y \hfill \\ \end{gathered} \right.,$ |
称
注:
定义2.3.设
定义2.4.在
接下来我们在
设
以下几点若在文中无特别说明, 则均不发生变化.
(1) 在
(2)
(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.在
${{\tau }_{n,\Gamma }}(kA)=\left\{ \begin{align} & 1,\text{ }N(\Gamma )=\varnothing \\ & \text{ }\frac{1}{|N(\Gamma )|}\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}),\text{ }N(\Gamma )\ne \varnothing } \\ \end{align} \right.,$ |
其中,
定理2.1.设
${{\tau }_{n,\Gamma }}(kA)=\left\{ \begin{align} & 1,\text{ }{{N}^{*}}(\Gamma )=\varnothing \\ & \frac{1}{|{{N}^{*}}(\Gamma )|}\sum\limits_{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in {{N}^{*}}(\Gamma )}{k\overline{A}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})},\text{ }{{N}^{*}}(\Gamma )\ne \varnothing \\ \end{align} \right.,$ |
其中,
证明:因为
由定义2.4可知,
有
所以, 当
当
$\sum\limits_{({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})\in \Pi _{\sim ,\Delta }^{m+l}}{k\overline{A}({{x}_{1}},...,{{x}_{m}},{{x}_{m+1}}...,{{x}_{m+l}})}=\\ \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}\times {{n}^{l}}}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}=\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in \Pi _{\sim ,\Delta }^{m}}{k\overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\times {{n}^{l}}}.$ |
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
为方便表述, 将
定理2.2.设
(ⅰ)若ΓA, 则
(ⅱ)若Γ~A, 则
证明: (ⅰ)若ΓA, 则
结合Δ连接词的运算性质可得,
结合~连接词的运算性质可得,
由定义2.5可得,
(ⅱ)若Γ~A,
结合~连接词的运算性质可得,
结合Δ连接词的运算性质可得,
由定义2.5可得,
定理2.3.设
证明:因为
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.设
证明:由于
当
从而有
即
因此,
引理2.1.设
证明:
(1) 当
当
(2) 当
当
定理2.5.设
(ⅰ)
(ⅱ)
证明:设A, B含有相同的原子公式
(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,
(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
再由定义2.5可知,
引理2.2.设
证明:首先令
1) 当
2) 当
定理2.6.设
证明:设A, B含有相同的原子公式
$\lambda \overline A ({x_1},{x_2},...,{x_m}) \vee \mu \overline B ({x_1},{x_2},...,{x_m}){\text{ = }}\lambda \overline A ({x_1},{x_2},...,{x_m}) +\\ \mu \overline B ({x_1},{x_2},...,{x_m}) - (\lambda \overline A ({x_1},{x_2},...,{x_m}) \wedge \mu \overline B ({x_1},{x_2},..,{x_m})),$ |
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,
引理2.3.设
证明:首先令
1) 当
2) 当
综上, 可得
定理2.7(Γ-k真度的MP规则).设
证明:设A, B含有相同的原子公式
$\mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\ge \lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})+(\lambda \overline{A}({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\to \mu \overline{B}({{x}_{1}},{{x}_{2}},...,{{x}_{m}}))-1,$ |
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,
推论2.1.设
引理2.4.设
证明:首先令
1) 当
1.1) 当
1.2) 当
1.2.1) 当
1.2.2) 当
2) 当
2.1) 当
2.2) 当
2.2.1) 当
2.2.2) 当
综上可得
定理2.8(Γ-k真度的HS规则). 设
证明:设A, B, C含有相同的原子公式
$(\lambda \overline{A}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\ge (\lambda \overline{A}\to \mu \overline{B})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})+(\mu \overline{B}\to \eta \overline{C})({{x}_{1}},{{x}_{2}},...,{{x}_{m}})-1,$ |
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,
推论2.2.设
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
解:根据定义2.5来计算
$\begin{align} & \overline{A}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{A}(x,y)=(\sim x\vee \Delta y)\to y, \\ & \overline{B}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{B}(x,y)=(\sim x\to \sim y)\to x, \\ & \overline{C}(x,y):\Pi _{\sim ,\Delta }^{2}\to [0,1],\overline{C}(x,y)=(\Delta x\to \sim y)\to \sim x. \\ \end{align}$ |
为了方便理解, 特做出如下图表.
x | y | ||||||
0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 1 | ||
0 | 1 | 0 | 1 | 1 | 1 | ||
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | |||
1 | 1 | 1 | 1 | ||||
1 | 1 | 1 | 1 | ||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | |
0 | 1 | 0 | 1 | 1 | |||
1 | 1 | 1 | 1 | ||||
1 | 1 | 1 | 1 | ||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | |
1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
从表中可以看出,
$\begin{align} & \quad {{\tau }_{4,\Gamma }}((\Delta A\wedge \sim B)\to \sim C)=\frac{1}{13}\sum\limits_{(x,y)\in N(\Gamma )}{13\times 1}, \\ & {{\tau }_{4,\Gamma }}((\Delta A\to \sim C)\vee (\sim B\to \sim C))=\frac{1}{13}\sum\limits_{(x,y)\in N(\Gamma )}{13\times 1}, \\ \end{align}$ |
因此,
定义3.1.设
${{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\wedge (\mu B\to \lambda A)),$ |
称
定理3.1.设
证明:设A, B含有相同的原子公式
$\begin{align} & {{\xi }_{n,\Gamma }}(\lambda A,\mu B)={{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\wedge (\mu B\to \lambda A)) \\ & \text{ }={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-{{\tau }_{n,\Gamma }}((\lambda A\to \mu B)\vee (\mu B\to \lambda A)) \\ & \text{ }={{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+{{\tau }_{n,\Gamma }}(\mu B\to \lambda A)-1. \\ \end{align}$ |
定理3.2.设
(ⅰ)
(ⅱ)
(ⅲ)
证明:设A, B含有相同的原子公式
(ⅰ)
所以有,
从而有,
则
$\sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\lambda \overline{A}\to \mu \overline{B})\wedge (\mu \overline{B}\to \lambda \overline{A}))({{x}_{1}},{{x}_{2}}},...,{{x}_{m}})=\\ \sum\limits_{({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\in N(\Gamma )}{((\mu \overline{B}\to \lambda \overline{A})\wedge (\lambda \overline{A}\to \mu \overline{B}))({{x}_{1}},{{x}_{2}},...,{{x}_{m}})}.$ |
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,
因此
(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.设
证明:设A, B, C含有相同的原子公式
$\begin{array}{l} {\xi _{n,\Gamma }}(\lambda A,\mu B) + {\xi _{n,\Gamma }}(\mu B,\eta C) - 1 = ({\tau _{n,\Gamma }}(\lambda A \to \mu B) + {\tau _{n,\Gamma }}(\mu B \to \lambda A) - 1) + \\ ({\tau _{n,\Gamma }}(\eta C \to \mu B) + {\tau _{n,\Gamma }}(\mu B \to \eta C) - 1) - 1\\ \le {\tau _{n,\Gamma }}(\lambda A \to \eta C) + {\tau _{n,\Gamma }}(\eta C \to \lambda A) - 1\\ = {\xi _{n,\Gamma }}(\lambda A,\eta C). \end{array}$ |
定义3.2.设
${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\xi }_{n,\Gamma }}(\lambda A,\mu B),$ |
称
定理3.4.设
${{\rho }_{n,\Gamma }}(\lambda A,\mu B)=1-{{\tau }_{n,\Gamma }}(\lambda A\to \mu B)+1-{{\tau }_{n,\Gamma }}(\mu B\to \lambda A).$ |
证明:由定理3.1可知,
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. 设
(ⅰ)
(ⅱ)
(ⅲ)
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
本文对n值Goguen命题逻辑进行了公理化扩张
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