A Kripke Semantics for the Logic of Gelfand Quantales
A Kripke Semantics for the Logic of Gelfand Quantales
复制标题
Gelfand Quantales逻辑的Kripke语义
DOI：
10.1023/a:1012495106338
复制
发表时间：
2001
期刊：
影响因子：
0.7
通讯作者：
W. MacCaull
中科院分区：
文献类型：
--
作者：
G. Allwein;W. MacCaull
gelfand量子是完整的单位量化， *如果所有b，则满足任何元素a的属性，然后给出a * a * a * a * a = a = a = a = a。命题逻辑，称为Gelfand逻辑，与Gelfand量相对于Gellipke语义，它显示了Gelfand逻辑的声音和完整性。 rasiowa/sikorski样式的语义图表系统还介绍了属性，如果封闭了图表的所有分支，那么所讨论的公式是Gelfand逻辑的定理。公式无效；因此它不是Gelfand逻辑的定理。
Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a ⊙ b ≤ a for all b, then a ⊙ a* ⊙ a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic.