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.     

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.     

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.     

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.     

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.     

Decidability of the interpolation problem and of related properties in tabular logics

Propositional modal and positive logics are considered as well as extensions of Johansson’s minimal logic. It is proved that basic versions of the interpolation property and of the Beth definability property, and also the Hallden property, are decidable on the class of tabular logics, i.e., logics given by finitely many finite algebras. Algorithms are described for constructing counterexamples to each of the properties mentioned in handling cases where the logic under consideration does not possess the required property.     

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.     

Fuzzy constructive logic

We introduce a logical system in which the principles of fuzzy logic are interpreted from the point of view of the constructive approach The language of predicate formulas without functional symbols and symbols of constants is considered. The notion of identically trae predicate formula in the framework of the introduced logic is defined; two variants of this definition are given. Theorems concerning identically true predicate formulas are proved. Some connections between the introduced logic and the constructive (intuitionistic) predicate calculus are established. Bibliography: 40 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 130–152.     

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.     

Advances in the ŁΠ and logics

 The ŁΠ and logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics, including the three mainly investigated ones (G?del, product and Łukasiewicz logic). The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both logics. Then we will see many other logics contained in the ŁΠ family of logics (namely logics induced by the continuous finitely constructed t-norms and Takeuti and Titani's fuzzy predicate logic). Received: 1 October 2000 / Revised version: 27 March 2002 / Published online: 5 November 2002 Partial support of the grant No. A103004/00 of the Grant agency of the Academy of Sciences of the Czech Republic is acknowledged. Key words or phrases: Fuzzy logic – Łukasiewicz logic – Product logic     

On squares of modal logics with additional connectives

The paper gives an overview of new results on two-dimensional modal logics of special type, “Segerberg squares.” They are defined as usual squares of modal logics with additional connectives corresponding to the diagonal symmetry and two projections onto the diagonal. In many cases these logics are finitely axiomatizable, complete and have the finite model property. Segerberg squares are interpreted in the classical predicate logic.     

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.     

Three ways of recognizing inessential formulas in sequents

The first method is based on the familiar method of lowering thinnings downwards and is a further development of the lemmas on weeding of [1]. The second method is based on the use of sufficiently wide classes of sequents, for which derivability in the intuitionistic predicate calculus coincides with derivability in the classical predicate calculus and the familiar property of disjunction is true. By this method one can get, e.g., a syntactic proof of the following assertion. If the positive formula A is derivable in the theory of groups under additional assumptions of the form then A is also derivable in the theory of groups without these assumptions. As the third method there is proposed a syntactically formulated test for the conservativeness of extensions of intuitionistic axiomatic theories. With the help of this test one can get, for example, a syntactic proof of the hereditary undecidability of the intuitionistic theory of equality, with additional axioms which are the formula, all formulas of the form and all negations of formulas derivable in the classical predicate calculus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 163–175, 1979.     

Hypersequent calculi for intuitionistic logic with classical atoms

We discuss a propositional logic which combines classical reasoning with constructive reasoning, i.e., intuitionistic logic augmented with a class of propositional variables for which we postulate the decidability property. We call it intuitionistic logic with classical atoms. We introduce two hypersequent calculi for this logic. Our main results presented here are cut-elimination with the subformula property for the calculi. As corollaries, we show decidability, an extended form of the disjunction property, the existence of embedding into an intuitionistic modal logic and a partial form of interpolation.     

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.     

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.     

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.