Modal Tree-Sequents

We develop cut-free calculi of sequents for normal modal logics by using treesequents, which are trees of sequences of formulas. We introduce modal operators corresponding to the ways we move formulas along the branches of such trees, only considering fixed distance movements. Finally, we exhibit syntactic cut-elimination theorems for all the main normal modal logics. Mathematics Subject Classification: 03B45, 03F05.     

A New Modal Lindström Theorem

We prove new Lindstr?m theorems for the basic modal propositional language, and for some related fragments of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. In addition we discuss new connections with interpolation properties, and the modal invariance theorem. Mathematics Subject Classification (2000): Primary 03B45; Secondary 03C95     

La connaissance commune en logique modale

The problem of Common Knowledge will be considered in two classes of models: a class K.* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models. MSC: 03B45, 68T25.     

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     

An Inheritance Criterion for the Admissible Inference Rules of K4

We give a criterion for inheritance of the admissible inference rules of the modal logic K4 by the finitely approximable modal logics extending K4. We give a negative solution to the question of inheritance of the admissible rules of K4 by tabular logics. We exhibit a series of examples of modal logics which inherit or fail to inherit the admissible inference rules of K4.     

Classically complete modal relevant logics

A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.     

A formalization of Sambins's normalization for GL

Sambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut-free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.     

Structuralist Logic: Implications, Inferences, and Consequences

On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B20, 03B42, 03B60     

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.     

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.     

The Fields of Real and Complex Numbers with a Small Multiplicative Group

We consider the model theory of the real and complex fieldswith a multiplicative group having the Mann property. Amongthese groups are the finitely generated multiplicative groupsin these fields. As a by-product we obtain some results on groupswith the Mann property in rings of Witt vectors and in fieldsof positive characteristic.k 2000 Mathematics Subject Classification03C10, 03C35, 03C60, 03C64, 03C98, 13K05.     

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.     

Barwise's Information Frames and Modal Logics

The article studies Barwise's information frames and settles the problem of Barwise dealing in finding 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; (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. Toward a general theory, we prove that any Kripke complete modal logic is a modal logic of a certain class of information frames, and that every modal logic generated by any given class of complete, rarefied, and fully classified information frames is Kripke complete.     

Definability in Normal Extensions of S4

A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

Jacobi transplantation revisited

A transplantation theorem for Jacobi series proved by Muckenhoupt is reinvestigated by means of a suitable variant of Calderón–Zygmund operator theory. An essential novelty of our paper is weak type (1,1) estimate for the Jacobi transplantation operator, located in a fairly general weighted setting. Moreover, L ^p estimates are proved for a class of weights that contains the class admitted in Muckenhoupt’s theorem. Research of ó. Ciaurri and K. Stempak was supported by the grant MTM2006-13000-C03-03 of the DGI. Research of A. Nowak and K. Stempak was supported by MNiSW Grant N201 054 3214285.     

Modal positive operators

For every modal positive operator, there exists a formula defining a least fixed point for that operator in partially ordered Kripke models with the property of being cofinal for every infinite ascending chain. A similar result obtains also for strict partially ordered Kripke models with the same property. Supported by the Russian Humanitarian Science Foundation (RHSF), grant No. 97-03-04089, and by the Siberian Branch of the Russian Academy of Science through PSO RAN No. 473, grant No. 3. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 585–597, September–October, 1999.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Prefinitely axiomatizable modal and intermediate logics

A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.