命题泛逻辑的演算理论及推理研究
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摘要
本文研究课题源于国家自然科学基金项目(No．60273087)“经验知识推理理论研究”与北京市自然科学基金项目(No．4032009)“不精确推理理论研究”。
如何处理各种不确定性和演化已成为当前人工智能深入发展的关键。已十分完善的经典数理逻辑是刚性逻辑，只能解决确定性问题。如何使经典数理逻辑柔性化，以包容各种不确定性和演化，是逻辑学研究面临的新挑战。在此背景下，各种非标准逻辑和现代逻辑大量涌现。何华灿教授在研究各种逻辑规律的基础上提出了能包容各种逻辑形态和推理模式的泛逻辑学理论框架，为研究复杂系统中的不确定性和演化过程奠定了理论基础。
本文属于泛逻辑的基础理论研究，围绕“命题泛逻辑的演算理论”这个主线，对命题泛逻辑的语义、语构和推理进行了深入研究。主要研究成果和创新点如下：
1．将广义重言式理论引入命题泛逻辑，对h，k取一些固定值时的广义重言式理论进行了刻画，得到一些重要结论：当c∈[0.75，1]时，关于I_(h=c)而言，F(S)中只有3种不同的广义重言式，即，可达0-重言式、0~+-重言式和重言式：当h=1，k=0.5时，关于I_(h=1，k=0.5)而言，F(S)中只有5种不同的广义重言式，即可达0-重言式、0~+-重言式、可达(1／2)-重言式、(1／2)~+-重言式和重言式：当h=0.75，k=0.5时，关于I_(h=0.75，k=0.5)而言，F(S)中对每一有理数均存在可达广义重言式，且是类类互异的。
2．当h∈(0，1]时，以零级泛与运算模型为逻辑“与”的解释，以零级泛蕴涵运算模型为逻辑“蕴涵”的解释，建立命题泛逻辑演绎系统UL_(h∈(1，1])，并证明其可靠性和完备性。
3．当h∈(0，1]，k∈(0，1)时，以一级泛与运算模型为逻辑“与”、一级泛蕴涵运算模型为逻辑“蕴涵”、一级泛非运算模型为逻辑“非”为背景，引入一种新的代数系统L∏G~-，以此代数为语义建立命题演绎系统UL_(h∈(0，1])~-，并证明其可靠性和完备性。
4．提出了基于泛逻辑且针对常见模糊推理模型的推理规则和泛蕴涵推理机；用含有泛蕴涵推理机的模糊系统对一元、二元函数的图形进行描绘；针对三种模糊系统进行了实验数据比较。结果表明，在相同的规则下，含有泛蕴涵推理机的系统误筹最小。
上述研究成果为最终解决命题泛逻辑的标准完备性奠定了理论基础，为进一步建立谓词泛逻辑提供了理论保障。
How to deal with various uncertainties and evolution become the key problems for further development of AI. The well-developed classical mathematical logic is the rigid logic and can only solve problems of certainty. How to make classical mathematical logic more flexible to contain various uncertainties and evolution is a new challenge that logics faces. Under this situation, all kinds of non-classical and modern logic are much proposed. Based on studies of general law of logics, Professor Huacan He proposed a universal logic principle frame which can contain many logical modalities and reasoning modes. It established a theoretical basis for the studies of complex uncertainty problems and evolution process.
This thesis is a theoretical research on universal logic. Centered on the calculus of propositional universal logic, its semantics, syntax and reasoning are deeply studied. The main results and main innovations in this thesis are as following:
1. The generalized tautologies theory is introduced into the propositional universal logic, and that theories for some fixed value of h and k are described, hence, some important results are obtained: when c∈[0.75, 1], if the semantic is explained by I_(h=c), there are only three different generalized tautologies in F(S), that is, accessible 0-tautology、 0~+-tautology and tautology; when h=1, k=0.5, if the semantic is explained by I_(h=1,k=0.5), there are only five different generalized tautologies in F(S), that is, accessible 0-tautology、 0~+-tautology、accessible (1/2)-tautology、 (1/2)~+-tautology and tautology in F(S); when h=0.75, k=0.5, if the semantic is explained by I_(h=0.75,k=0.5), the accessible tautologies exist for each rational number in F(S), and they are difference each other.
2. When h∈ (0,1], the propositional calculus deductive system UL_(h∈(0,1])) of universal logic is built up based on the 0-level universal AND operators as logic conjunction and 0-level universal IMPLICATION as logic implication. And its soundness and completeness are proved.
3. When h∈(0,1], k∈ (0,1), based on the 1-level universal AND as logic conjunction,
如何处理各种不确定性和演化已成为当前人工智能深入发展的关键。已十分完善的经典数理逻辑是刚性逻辑，只能解决确定性问题。如何使经典数理逻辑柔性化，以包容各种不确定性和演化，是逻辑学研究面临的新挑战。在此背景下，各种非标准逻辑和现代逻辑大量涌现。何华灿教授在研究各种逻辑规律的基础上提出了能包容各种逻辑形态和推理模式的泛逻辑学理论框架，为研究复杂系统中的不确定性和演化过程奠定了理论基础。
本文属于泛逻辑的基础理论研究，围绕“命题泛逻辑的演算理论”这个主线，对命题泛逻辑的语义、语构和推理进行了深入研究。主要研究成果和创新点如下：
1．将广义重言式理论引入命题泛逻辑，对h，k取一些固定值时的广义重言式理论进行了刻画，得到一些重要结论：当c∈[0.75，1]时，关于I_(h=c)而言，F(S)中只有3种不同的广义重言式，即，可达0-重言式、0~+-重言式和重言式：当h=1，k=0.5时，关于I_(h=1，k=0.5)而言，F(S)中只有5种不同的广义重言式，即可达0-重言式、0~+-重言式、可达(1／2)-重言式、(1／2)~+-重言式和重言式：当h=0.75，k=0.5时，关于I_(h=0.75，k=0.5)而言，F(S)中对每一有理数均存在可达广义重言式，且是类类互异的。
2．当h∈(0，1]时，以零级泛与运算模型为逻辑“与”的解释，以零级泛蕴涵运算模型为逻辑“蕴涵”的解释，建立命题泛逻辑演绎系统UL_(h∈(1，1])，并证明其可靠性和完备性。
3．当h∈(0，1]，k∈(0，1)时，以一级泛与运算模型为逻辑“与”、一级泛蕴涵运算模型为逻辑“蕴涵”、一级泛非运算模型为逻辑“非”为背景，引入一种新的代数系统L∏G~-，以此代数为语义建立命题演绎系统UL_(h∈(0，1])~-，并证明其可靠性和完备性。
4．提出了基于泛逻辑且针对常见模糊推理模型的推理规则和泛蕴涵推理机；用含有泛蕴涵推理机的模糊系统对一元、二元函数的图形进行描绘；针对三种模糊系统进行了实验数据比较。结果表明，在相同的规则下，含有泛蕴涵推理机的系统误筹最小。
上述研究成果为最终解决命题泛逻辑的标准完备性奠定了理论基础，为进一步建立谓词泛逻辑提供了理论保障。
How to deal with various uncertainties and evolution become the key problems for further development of AI. The well-developed classical mathematical logic is the rigid logic and can only solve problems of certainty. How to make classical mathematical logic more flexible to contain various uncertainties and evolution is a new challenge that logics faces. Under this situation, all kinds of non-classical and modern logic are much proposed. Based on studies of general law of logics, Professor Huacan He proposed a universal logic principle frame which can contain many logical modalities and reasoning modes. It established a theoretical basis for the studies of complex uncertainty problems and evolution process.
This thesis is a theoretical research on universal logic. Centered on the calculus of propositional universal logic, its semantics, syntax and reasoning are deeply studied. The main results and main innovations in this thesis are as following:
1. The generalized tautologies theory is introduced into the propositional universal logic, and that theories for some fixed value of h and k are described, hence, some important results are obtained: when c∈[0.75, 1], if the semantic is explained by I_(h=c), there are only three different generalized tautologies in F(S), that is, accessible 0-tautology、 0~+-tautology and tautology; when h=1, k=0.5, if the semantic is explained by I_(h=1,k=0.5), there are only five different generalized tautologies in F(S), that is, accessible 0-tautology、 0~+-tautology、accessible (1/2)-tautology、 (1/2)~+-tautology and tautology in F(S); when h=0.75, k=0.5, if the semantic is explained by I_(h=0.75,k=0.5), the accessible tautologies exist for each rational number in F(S), and they are difference each other.
2. When h∈ (0,1], the propositional calculus deductive system UL_(h∈(0,1])) of universal logic is built up based on the 0-level universal AND operators as logic conjunction and 0-level universal IMPLICATION as logic implication. And its soundness and completeness are proved.
3. When h∈(0,1], k∈ (0,1), based on the 1-level universal AND as logic conjunction,
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