General logic-systems and finite consequence operators (Q874755)

A nonempty language \(L\) is non-organized if all other properties that can be defined for \(L\) are ignored when a consequence operator property is determined. The author discusses the significance of using the general logic-system concept and finite consequence operators defined on non-organized languages. It is analyzed how properties of finite consequence operators are independent from language organization and how, in some particular cases, they depend only upon one simple language characteristic. It is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language that cannot be generated from any finite logic-system.