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.     

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     

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.     

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.     

Some Multi-Conclusion Modal Paralogics

I give a systematic presentation of a fairly large family of multiple-conclusion modal logics that are paraconsistent and/or paracomplete. After providing motivation for studying such systems, I present semantics and tableau-style proof theories for them. The proof theories are shown to be sound and complete with respect to the semantics. I then show how the “standard” systems of classical, single-conclusion modal logics fit into the framework constructed.     

Axiomatization of modal logic squares with distinguished diagonal

Modal logics of squared Kripke frames with distinguished diagonal are considered. It is shown that many such logics, unlike ordinary two-dimensional products, cannot be axiomatized by formulas with finitely many variables. The method resembles that used to obtain a similar result for ≥ 3-dimensional products of modal logics. The proof uses, in particular, generalized Sahlquist formulas.     

Leibniz filters and the strong version of a protoalgebraic logic

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?⁺ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?⁺ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?⁺. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. Received: 30 October 1998 /?Published online: 15 June 2001     

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.     

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.     

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.     

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.     

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 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.     

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     

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.     

On Finite Model Property for Admissible Rules

Our investigation is concerned with the finite model property (fmp) with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem–K4 itself, S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp–the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width ≤ 2, there exists a zone for fmp w. r. t. admissibility. It is shown (Theorem 4.3) that all modal logics A of width ≤ 2 extending S4 which are not sub-logics of three special tabular logics (which is equipotent to all these λ extend a certain subframe logic defined over S4 by omission of four special frames) have fmp w.r.t. admissibility.     

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.     

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.     

CONSTRUCTING SELF-CONJUGATE SELF-ORTHOGONAL DIAGONAL LATIN SQUARES

1.IntroductionALatinsquareofordernisannxnarraysuchthateveryrowandeverycolumnisapermutationofann-setN.AtransversalinaLatinsquareisasetofpositions,oneperrowandonepercolumn,amongwhichthesymbolsoccurpreciselyonceeach.AdiagonalLatinsquareisaLatinsquarewhosemaindiagonalandbackdiagonalarebothtransversals.TwoLatinsquaresofordernareorthogonalifeachsymbolinthefirstsquaremeetseachsymbolinthesecondsquareexactlyoncewhentheyaresuperposed.ALatinsquareisself-orthogonalifitisorthogonaltoitstranspose.Inanea…     

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