FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 1, PAGES 47-69

**Relative interpretability of modal logics**

E. E. Zolin

Abstract

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This paper introduces the notion of *modality* as an
operator $$Ñ_{y}, defined on
the set of propositional modal formulas by the equality
$$Ñ_{y}(F)=y(F), where
$$y(p) is a formula of one
variable $p$.
Defining the logic $L($Ñ) of
modality $$Ñ over
logic $L$ as
the set of all provable in $L$ formulas of the
propositional language extended by the operator $$Ñ, the notion of
*exact interpretability* ($\$\; \backslash hookrightarrow\; \$$) of a logic $L$_{1} in
a logic $L$_{2} can be
formalized as follows: $\$\; L\_1\; \backslash hookrightarrow\; L\_2\; \$$ iff $L$_{1}=L_{2}(Ñ) for some
modality $$Ñ.
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.

Location: http://mech.math.msu.su/~fpm/eng/k01/k011/k01104h.htm

Last modified: May 10, 2001