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.     

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.     

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.     

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 method of proving interpolation in paraconsistent extensions of the minimal logic

The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess _CIP, nor even the restricted 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. 5, pp. 627–648, September–October, 2007.     

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.     

Table admissible inference rules

A recursive basis of inference rules is described which are instantaneously admissible in all table (residually finite) logics extending one of the logics Int and Grz. A rather simple semantic criterion is derived to determine whether a given inference rule is admissible in all table superintuitionistic logics, and the relationship is established between admissibility of a rule in all table (residually finite) superintuitionistic logics and its truth values in Int. Translated from Algebra i Logika, Vol. 48, No. 3, pp. 400–414, May–June, 2009.     

Algebraic semantics for superintuitionistic predicate logics

We construct a unified algebraic semantics for superintuitionistic predicate logics. Assigned to each predicate logic is some deductive system of a propositional language which is kept fixed throughout all predicate superintuitionistic ones. Given that system, we build up a variety of algebras w.r.t. which a given logic is proved to be strongly complete. Supported by the Russian Arts Foundation (RAF), grant No. 97-03-04089a. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 68–95, January–February, 1999.     

Projective beth properties in modal and superintuitionistic logics

Projective Beth properties in superintuitionistic and normal modal logics are considered. Their interrelations and connections with interpolation properties of the logics are established. Algebraic counterparts for the projective Beth properties are found out. Supported by the Russian Humanitarian Science Foundation, grant No. 97-03-04089. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 316–333, May–June, 1999.     

Bases of admissible rules for <Emphasis Type="Italic">K</Emphasis>-saturated logics

Admissible inference rules for table modal and superintuitionistic logics are investigated. K-saturated logics are defined semantically. Such logics are proved to have finite bases for admissible inference rules in finitely many variables. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 750–761, November–December, 2008.     

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.     

Independent bases for admissible rules in pretable logics

Independent bases of admissible inference rules are studied; namely, we treat inference rules in pretable modal logics over S4, and in pretable superintuitionistic logics. The Maksimova-Esakia-Meskhi theorem holds that there exist exactly five pretable S4-logics and precisely three pretable superintuitionistic ones. We argue that all pretable modal logics and all pretable super-intuitionistic logics have independent bases for admissible inference rules. Supported by RFFR, and Rybakov’s part, by the Turkish Scientific Technical Research Council (TUBITAK, Ankara). Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 206–226, March–April, 2000.     

Barwise's information frames and modal logics

 The paper studies Barwise's information frames and answers the John Barwise question: to find axiomatizations for the modal logics generated by information frames. We find axiomatic systems for (i) the modal logic of all complete information frames, (ii) the logic of all sound and complete information frames, (iii) the logic of all hereditary and complete information frames, (iv) the logic of all complete, sound and hereditary information frames, and (v) the logic of all consistent and complete information frames. The notion of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4 respectively. To develop general theory, we prove that (i) any Kripke complete modal logic is the modal logic of a certain class of information frames and that (ii) the modal logic generated by any given class of complete, rarefied and fully classified information frames is Kripke complete. This paper is dedicated to the memory of talented mathematician John Barwise. Received: 7 May 2000 Published online: 10 October 2002 Key words or phrases: Knowledge presentation – Information – Information flow – Information frames – Modal logic-Kripke model     

Separability of normalizable superintuitionistic propositional logics

The problem of separability of superintuitionistic propositional logics that are extensions of the intuitionistic propositional logic is studied. A criterion of separability of normal superintuitionistic propositional logics, as well as results concerning the completeness of their subcalculi is obtained. This criterion makes it possible to determine whether a normalizable superintuitionistic propositional logic is separable. By means of these results, the mistakes discovered by the author in the proofs of certain statements by McKay and Hosoi are corrected.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 606–615, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-00944.     

The modal logic of the countable random frame

 We study the modal logic M L ^r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L ^r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. Received: 1 May 2000 / Revised version: 29 July 2001 / Published online: 2 September 2002 Mathematics Subject Classification (2000): 03B45, 03B70, 03C99 Key words or phrases: Modal logic – Random frames – Almost sure frame validity – Countable random frame – Axiomatization – Completeness     

The Lyndon property and uniform interpolation over the Grzegorczyk logic

We consider versions of the interpolation property stronger than the Craig interpolation property and prove the Lyndon interpolation property for the Grzegorczyk logic and some of its extensions. We also establish the Lyndon interpolation property for most extensions of the intuitionistic logic with Craig interpolation property. For all modal logics over the Grzegorczyk logic as well as for all superintuitionistic logics, the uniform interpolation property is equivalent to Craig’s property.     

Joint consistency in extensions of the minimal logic

Analogs of Robinson’s theorem on joint consistency are found which are equivalent to the weak interpolation property (WIP) in extensions of Johansson’s minimal logic J. Although all propositional superintuitionistic logics possess this property, there are J-logics without WIP. It is proved that the problem of the validity of WIP in J-logics can be reduced to the same problem over the logic Gl obtained from J by adding the tertium non datur. Some algebraic criteria for validity of WIP over J and Gl are found.     

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 relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.