FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2001, VOLUME 7, NUMBER 1, PAGES 47-69
E. E. Zolin
Abstract
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This paper introduces the notion of \emph{modality}
as an operator $\nabla_\psi$ ,
defined on the set of propositional modal formulas by the equality
$\nabla_\psi(F)=\psi(F)$ , where $\psi(p)$ is a formula of one variable $p$ .
Defining the logic $L(\nabla)$ of modality $\nabla$ over logic $L$ as
the set
of all provable in $L$ formulas of the propositional language extended by
the operator $\nabla$ , the notion of
\emph{exact interpretability} ($\hookrightarrow$)
of a logic $L_1$ in a logic $L_2$ can be formalized as follows:
$L_1 \hookrightarrow L_2$ iff $L_1=L_2(\nabla)$ for some modality $\nabla$ .
The question about the number of logics, which are exactly interpretable
in some fixed logic, is considered in this paper.
Answers to this question
are obtained for the following family of known modal logics:
logics of boolean
modalities, normal logics K ,
K4 ,
T ,
S4 ,
S5 ,
GL ,
Grz ,
logics of provability. A number of results concerning the absence of exact
interpretability of some logics of this family in others are offered as well.
All articles are published in Russian.
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Last modified: May 10, 2001