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.     

Note on grammatical translations of logical calculi

It is shown that there exists no grammatical translation into classical (propositional) logic of the modal logics, nor of intuitionistic logic and of the relatedness and dependence logics, as defined in Richard L. Epstein's bookThe Semantic foundations of logic. In the book the result is proved for translations without parameters.Classical propositional logicPC can be translated into other logics. Usually the grammatical structure of propositions is preserved, in the sense of the following definition.     

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.     

Two Semantical Approaches to Paraconsistent Modalities

In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291–310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency, introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic. These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics. Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al. (Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and many-valued logics, defining new blends of logics whose relevance we intend to show.     

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.     

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     

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.     

On the interpolation property of some intuitionistic modal logics

LetL be one of the intuitionistic modal logics considered in [7] (or one of its extensions) and letM _L be the algebraic semantics ofL. In this paper we will extend toL the equivalence, proved in the classical case (see [6]), among he weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of varietyM _L. We will also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of varietyM _L. Then we obtain the Craig interpolation theorem and Robinson theorem for two intuitionistic modal logics, one ofS ₄-type and the other one ofS ₅-type, showing the super-amalgamation property of the corresponding algebraic semantics.     

Polymodal Lattices and Polymodal Logic

A polymodal lattice is a distributive lattice carrying an n-place operator preserving top elements and certain finite meets. After exploring some of the basic properties of such structures, we investigate their freely generated instances and apply the results to the corresponding logical systems — polymodal logics — which constitute natural generalizations of the usual systems of modal logic familiar from the literature. We conclude by formulating an extension of Kripke semantics to classical polymodal logic and proving soundness and completeness theorems. Mathematics Subject Classification: 03G10, 06D99, 03B45.     

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.     

Sequent calculi for branching time temporal logics of knowledge and beliefwith awareness: Completeness and decidability

In this paper, we consider branching time temporal logic CT L with epistemic modalities for knowledge (belief) and with awareness operators. These logics involve the discrete-time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. In addition, the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”). In a language of these logics, there are awareness operators. For these logics, we present sequent calculi with a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for these calculi are proved. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 328–340, July–September, 2007.     

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.     

Minimally Generated Abstract Logics

In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9–42, 1973)—nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ-versions (κ ≥ ω), introduce a hierarchy of κ-prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (κ-) compactness of the consequence relation together with the existence of a (κ-)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.     

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.     

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.     

Uniform interpolation and the existence of sequent calculi

This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus $\mathsf{G4ip}$ for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of $\mathsf{K}$ and $\mathsf{KD}$ (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.     

Modal operators on bounded commutative residuated <Emphasis Type="Italic">ℓ</Emphasis>-monoids

Bounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVá,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras. The first author was supported by the Council of Czech Government, MSM 6198959214.     

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.     

Finite bases with respect to admissibility for modal logics of width 2

It is proved that every finitely approximable and residually finite modal logic of depth 2 over K4 has a finite basis of admissible inference rules. This, in particular, implies that every finitely approximable residually finite modal logic of depth at most 2 is finitely based w.r.t. admissibility. (Among logics in a larger depth or width, there are logics which do not have a finite, or even independent, basis of admissible rules of inference.) Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 436–455, July–August 1999.     

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