摘要
本论文将语言真值格蕴涵代数引入到格值一阶逻辑中,对语言真值格蕴涵代数的一些代数结构、基于语言真值格值一阶逻辑的不确定性推理的理论与方法、基于语言真值分层格值一阶逻辑的不确定性推理的理论与方法进行了研究,并取得如下四个方面的研究成果：
一、关于格蕴涵代数的研究
1.在格蕴涵多项式的基础上,重新给出了格蕴涵代数不等式的定义,提出了几类格蕴涵代数不等式解集的结构特征,并得到了有些解集可构成滤子和理想的结构；
2.提出了WLI-理想的定义,给出了WLI-理想以及由WLI-理想所生成拓扑空间的性质,得到了满足第二可数性公理的充分必要条件；
3.提出了赋范格H蕴涵代数、蕴涵距离d→、V-距离d∨、∧-距离d∧的定义,给出了赋范蕴涵满射、赋范格H蕴涵同态、赋范格H蕴涵同构以及赋范同构的性质,并证明了收敛数列的有界性和蕴涵距离是有界的结论.
二、关于语言真值格蕴涵代数的研究
1.给出了语言真值格蕴涵代数的若干性质,得到了(aj,bn)→(ai,bm)=(ap,bh)→(ai,bm)(?)aj=ap且bn=bh(?)(ai,bm)→(aj,bn)=(ai,bm)→(ap,bh)；
2.给出了语言真值格蕴涵代数中对偶分子的推理性质,证明了对偶分子在一定程度上对算子∨、∧、→具有闭性；
3.提出了语言真值格蕴涵代数中蕴涵不可约元素的定义,给出了它的推理性质.
三、关于语言真值格语言真值格值一阶逻辑系统Lv(n×2)F(X)上的不确定性推理研究
1.提出了基于语言真值格值一阶逻辑系统Lv(n×2)F(X)的不确定性推理理论与方法；
2.基于语言真值格值一阶逻辑系统Lv(n×2)F(X),给出了带广义量词的推理规则,并证明了这些推理规则的合理性；
3.提出了两种推理模型中的不确定性推理理论与方法.
四、关于基于语言真值分层格值一阶逻辑系统Lv(n×2)fl的不确定性推理研究
1.基于语言真值分层格值一阶逻辑系统LV(n×2)fl,给出了推理规则在(α0,β0)≤∧θ∈Lv(n×2)(θ∨θ')(θ≠(an,t))水平下的不确定性推理理论与方法,并证明了这些推理方法的合理性；
2.提出了基于语言真值分层格值一阶逻辑系统LV(n×2)fl的不确定性推理模型的正则条件以及基于语言真值分层格值一阶逻辑系统LV(n×2)fl在蕴涵不可约元素集下的不确定性推理理论与方法.
The idea of linguistic truth-valued lattice implication algebra is introduced into the lattice-valued first-order logic in this thesis, the study presented in this dissertation concentrates on some structures of linguistic truth-valued lattice implication algebra, theories and methods of uncertainty reasoning based on linguistic truth-valued lattice-valued first-order logic Lv(n×2)F(X) and linguistic truth-valued gradational lattice-valued first-order logic Lv(n×2)fl·The innovation and main results are summarized as follows:
Part One. The study of lattice implication algebra
1. According to the lattice implication polynomial, the notion of inequality in lattice implication algebra was redefined. The several characterizations of solution set in unary lattice implication algebra inequalities are investigated, and the constructions of filters and ideals by some solution sets are obtained.
2. The concept of weak Li-ideals is defined, weak Li-ideals and topological space properties based on weak Li-ideals of lattice implication algebra are investigated, and obtained the necessary and sufficient condition of second countable axiom in (L,TW(L)).
3. The notion of normed lattice H implication algebras, implication distance d→v—distance and∧-distance are introduced. The properties of normed implication epimorphism, normed lattice H implication homomorphism. normed lattice H implication isomorphism and normed isomorphism are given. The boundedness of convergence sequence and sequences operations (i.e.,(?),(?),∨,∧,→) to implication distance are proved.
Part Two. The study on linguistic truth-valued lattice implication algebra
1. The properties of linguistic truth-valued lattice implication algebra are given, and obtained(aj,bn)→(aj,bm)= (aP,bh)→(ai,,bm)(?)aj= aP and bn= bh(?) (ai,bm)→(aj,bn) = (ai,bm)→(aP,bh).
2. The reasoning properties of dual molecule in linguistic truth-valued lattice implication algebra are investigated, and it is proved that operators∨、∧、→have some degree closed.
3. The concept of implicative irreducible element in linguistic truth-valued lattice implication algebra is introduced, and its reasoning properties are obtained.
Part Three. The study of uncertainty reasoning based on linguistic truth-valued lattice-valued first-order logic Lv(n×2)F(X)
1. The uncertainty reasoning theory and approach based upon linguistic truth-valued lattice-valued first-order logic Lv(n×2)F(X) are proposed.
2. The inference rules with generalized quantifiers based upon linguistic truth-valued lattice-valued first-order logic Lv(n×2)F(X) are given, and the reasonableness of these inference rules are proved.
3. The uncertainty reasoning theories and approaches under two inference models based upon linguistic truth-valued lattice-valued first-order logic Lv(n×2)F(X) are proposed.
Part Four. The study of uncertainty reasoning based on linguistic truth-valued gradational lattice-valued first-order logic Lv(n×2)fl
1. The uncertainty reasoning theory and approach of the inference rules under the level (α0,β0)≤∧(θ∨θ')(θ≠(an,t)) based upon linguistic truth-valued gradational lattice-valued first-order logic system Lv(n×2)fl is proposed, and the reasonableness of these methods are proved.
2. The regular conditons of uncertainty inference model, the uncertainty reasoning theory and approach under the set of implicative irreducible elements based upon linguistic truth-valued gradational lattice-valued first-order logic system Lv(n×2)fl are discussed.
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