逻辑系统MTL中任意量词的代数研究

Algebraic Study of Universal Quantifier in Monoidal T-Norm Based Logic

一元MV-代数(L,?)由MV-代数连同其上的一个一元算子?构成,其中?是一元多值谓词逻辑中任意量词的代数解释.考虑到MTL-代数是一类范围最广的基于三角模的逻辑代数,为了给一元三角模谓词逻辑中任意量词提供统一的代数语义,文章引入了一元MTL-代数,研究其基本的代数性质以及对应的滤子,证明了一元MTL-代数是一元MV-代数和一元布尔代数的自然推广.其次,基于一元滤子刻画了可表示一元MTL-代数,证明了半单一元MTL-代数的次直积表示定理,揭示了强一元MTL-代数和可表示一元MTL-代数是等价的.最后,引入了一元MTL-代数簇对应的模态逻辑系统,证明了其完备性定理,并得到了其成为半线性逻辑的充分必要条件.

The concept of monadic MV-algebras as MV-algebras endowed with an inner operator,which are algebraic counterpart of the logical notion of universal quantifiers "for any" in its original meaning.In this paper,for obtaining the most general algebraic counterpart of universal quantifiers on the t-norm based fuzzy logics,we introduce the notion of monadic MTL-algebras,or more precisely,MTL-algebras with universal quantifiers,and study some related algebraic properties.Firstly,we study some basic properties of monadic MTL-algebras and their corresponding monadic filters.Secondly,we start a systematic study of the main subvarieties of monadic MTL-algebras,some of which constitute well known algebras:Monadic MV-algebras and monadic Boolean algebras.In addition,we give some representations of monadic MTL-algebras based on monadic filters.In particular,we characterize representable monadic MTL-algebras and prove the subdirect representation theorem of semisimple monadic MTL-algebras.Finally,we establish the modal logic of monadic MTLalgebras and prove its(chain) completeness,and obtain a necessary and sufficient condition for this modal logic to be semilinear.