Interpolation and Definability in Extensions of the Minimal Logic

We study into the interpolation property and the projective Beth property in extensions of Johansson's minimal logic. A family of logics of some special form is considered. Effective criteria are specified which allow us to verify whether an arbitrary logic in this family has a given property. Supported by RFBR grant No. 03-06-80178, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1, and by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 726–750, November–December, 2005.     

The Projective Beth Property and Interpolation in Positive and Related Logics

We look at the interplay between the projective Beth property in non-classical logics and interpolation. Previously, we proved that in positive logics as well as in superintuitionistic and modal ones, the projective Beth property PB2 follows from Craig's interpolation property and implies the restricted interpolation property IPR. Here, we show that IPR and PB2 are equivalent in positive logics, and also in extensions of the superintuitionistic logic KC and of the modal logic Grz.2. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 85–113, January–February, 2006.     

Weak interpolation in extensions of the logics <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript> and <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>

Conditions are specified which are necessary and sufficient for a logic over K4 to possess the weak interpolation property. For this goal to be met, simple transitive modal algebras are described, and we establish a criterion for the class of such algebras to be amalgamable. For extensions of K4, the weak interpolation property is proved decidable. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 705–722, November–December, 2008.     

Intermediate predicate logic without the beth property

It is shown that a logic J _fd ^* characterized by all Kripke frames the domains of all nonmaximal worlds of which are finite lacks the Beth property. The logic is the first example of an intermediate superintuitionistic logic without the Beth property. The interpolation and the Beth properties are also proved missing in all predicate superintuitionistic logics which contain J _fd ^* and are contained in a logic characterized by frames of the form〈N _n , ≤,{Dk}k∈N n〉. Supported by the Russian Foundation for Humanities, grant No. 97-03-04089. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 107–117, January–February, 1998.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Fuzzy logics with modalities

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 731–757, November–December, 2006.     

Interpolation in superintuitionistic predicate logics with equality

It is shown that the Craig interpolation property and the Beth property are preserved under passage from a superintuitionistic predicate logic to its extension via standard axioms for equality, and under adding formulas of pure equality as new axioms. We find an infinite independent set of formulas which, though not equivalent to formulas of pure equality, may likewise be added as new axiom schemes without loss of the interpolation, or Beth, property. The formulas are used to construct a continuum of logics with equality, which are intermediate between the intuitionistic and classical ones, having the interpolation property. Moreover, an equality-free fragment of the logics constructed is an intuitionistic predicate logic, and formulas of pure equality satisfy all axioms of the classical predicate logic. Supported by RFFR grant No. 96-01-01552. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 543–561, September–October, 1997.     

A paraconsistent extension of Sylvan’s logic

We deal with Sylvan’s logic CC_ω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved. Supported by RFBR grant No. 06-01-00358 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 533–547, September–October, 2007.     

Predicate superintuitionistic logics without interpolation

Predicate superintuitionistic logics are considered. We prove that all such logics that contain a logic characterized by frames whose domains are all finite and are contained in the classical logic of finite domains do not have the interpolation and Beth properties. It is also established that the interpolation property is not shared by all predicate superintuitionistic logics which contain a logic characterized by frames whose domains of nonfinal worlds are all finite and which are contained in a logic characterized by all two-element frames with finite constant domains. Supported by the Competitive Basic Research Center of St. Petersburg State University, grant No. 93-1-88-12. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 105–117, January–February, 1996.     

A Modal Logic That is Complete with Respect to Strictly Linearly Ordered <Emphasis Type="Italic">A</Emphasis>-Models

An axiomatization is furnished for a polymodal logic of strictly linearly ordered A-frames: for frames of this kind, we consider a language of polymodal logic with two modal operators, □_< and □_≺. In the language, along with the operators, we introduce a constant β, which describes a basis subset. In the language with the two modal operators and constant β, an Lα-calculus is constructed. It is proved that such is complete w.r.t. the class of all strictly linearly ordered A-frames. Moreover, it turns out that the calculus in question possesses the finite-model property and, consequently, is decidable. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 560–582, September–October, 2005. Supported by RFBR grant No. 03-06-80178, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1, and by INTAS grant No. 04-77-7080.     

Restricted interpolation property in superintuitionistic logics

The restricted interpolation property IPR in modal and superintuitionistic logics is investigated. It is proved that in superintuitionistic logics of finite slices and in finite-slice extensions of the Grzegorczyk logic, the property IPR is equivalent to the projective Beth property PB2. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 54-89, January-February, 2009.     

Freedom from the interpolation property for tense calculi associated with Ershov spaces

We study into the question whether calculi associated with Ershov topological spaces possess Craig’s interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 745–762, November–December, 2007.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, September–October, 1997.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

We define the concepts of a triangular and a quasitriangular Jordan bialgebras. It is proved that a finite-dimensional Jordan algebra J over an algebraically closed field Φ admits the structure of a quasitriangular Jordan bialgebra with nonzero comultiplication, provided that J is not a direct sum of fields, algebras H(Φ₂) and H(Φ₃), null extensions of Φ, and of algebras with zero multiplication. Supported by RFFR grant No. 98-01-01142. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 40–67, January–February, 1999.     

Superintuitionistic logics and the projective beth property

It is proved that in superintuitionistic logics, the projective Beth property follows from the Craig interpolation property, but the converse does not hold. A criterion is found which allows us to reduce the problem asking whether the projective Beth property is valid in superintuitionistic logics to suitable properties of varieties of pseudoboolean algebras. It is shown that the principle of variable separation follows from the projective Beth property. On the other hand, the interpolation property in a logic L implies the projective Beth property in Δ(L). Supported by RFFR grants No. 96-01-01552 and No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 680–696, November–December, 1999.     

The Beth property and interpolation in lattice-based algebras and logics

We deal with logics based on lattices with an additional unary operation. Interrelations of different versions of interpolation, the Beth property, and amalgamation, as they bear on modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of implicative lattices, have been studied in many works. Sometimes these relations can and sometimes cannot be extended to the logics without implication considered in the paper. Supported by INTAS (grant No. 04-77-7080) and by RFBR (grant No. 06-01-00358). Supported by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 307–334, May–June, 2008.     

Automatic recognition of interpolation in modal calculi

We deal with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.3. Supported by RFBR grant No. 06-01-00358 and by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 103–119, January–February, 2007.