Larisa Maksimova

Intuitionistic logic and implicit definability

It is proved that there are exactly 16 superintuitionistic propositional logics with the projective Beth property. These logics are finitely axiomatizable and have the finite model property. Simult ...

Definability and Interpolation in Non-Classical Logics

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra.In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johanssons minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi.

Definability in Normal Extensions of S4

A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.

Interpolation and implicit definability in extensions of the provability logic

The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog—the intuitionistic provability logic—and investigated these two logics and their extensions. In the present paper, different versions of interpolation and of the Beth property in normal extensions of the provability logic GL are considered. It is proved that in a large class of extensions of GL (including all finite slice logics over GL) almost all versions of interpolation and of the Beth property are equivalent. It follows that in finite slice logics over GL the three versions CIP, IPD and IPR of the interpolation property are equivalent. Also they are equivalent to the Beth properties B1, PB1 and PB2.