Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
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Hájek 基本模糊逻辑和 Łukasiewicz 无限值逻辑
DOI：
10.1007/s001530200144
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发表时间：
2003
影响因子：
0.3
通讯作者：
A. Torrell
中科院分区：
文献类型：
--
作者：
R. Cignoli;A. Torrell
利用BL-代数理论，证明了命题公式ϕ在Ł-ukasiewicz无限值逻辑中可导的充要条件是它的双重否定˜˜ϕ在基本Fuzzy值逻辑中可导.如果sbl是基本逻辑按公理(φ&(φ→˜φ))→ψ)的扩展，则ϕ在经典逻辑中可导的充要条件是˜˜ϕ在sbl中可导。基本逻辑的公理扩张对应于BL-代数簇的子簇。证明了在具有相同自由生成元个数的MV-代数的子簇中，自由代数的正则元的MV-代数在相应的子簇中是自由的。对于由自由BL-代数的稠密元构成的广义BL-代数，也得到了类似的结果。
Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.