IMI/Publicaţii/BASM/Ediţii/BASM n.2 (8), 1992/

On the cardinality of formula bases and absence of finite approximation with respect to completeness in propositional provability logic. (Russian)

Authors: Rusu Andrei


The probability logic $G$ permits to immerse the intuitionistic logic into Peano Arithmetic. Its formulas are built by connectives $\Cal E$, $\vee$, $\supset$, $\neg$, $\bigtriangleup$. A system $\Sigma$ of formulas is called complete (with respect to expressibility) in the logic $L$ if all formulas are expressible via $\Sigma$ in $L$. A system of formulas is reffered to as formula basis in the logic $L$, if it is complete in $L$, but every proper subsystem of it is incomplete in $L$. Theorem 1. For every integer positive $k$ there are formula bases of power $k$ in the logic $G$ and in its extentions, which are not local tabular. A logic $L$ is called finitely approximable with respect to completeness, if for every finite incomplete in $L$ system $\Sigma$ there exists an extension of the logic $L$ in which $\Sigma$ is incomplete too. Theorem 3. The logic $G$ and its extensions, which are not local tabular, are not finitely approximable with respect to completeness. Theorem 5. An extension of logic $G$ is finitely approximable with respect to completeness if and only if it is local tabular.