### Book

## Leo Esakia on Duality in Modal and Intuitionistic Logics

This volume is dedicated to Leo Esakia’s contributions to the theory of modal and intuitionistic systems. Leo Esakia was one of the pioneers in developing duality theory for modal and intuitionistic logics, and masterfully utilizing it to obtain some major results in the area. The volume consists of 10 chapters, written by leading experts, that discuss Leo’s original contributions and consequent developments that have shaped the current state of the field.

This is a chapter in a book dedicated to Leo Esakia’s contributions to the theory of modal and intuitionistic systems. In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, T1 -spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, R^n (n ≥ 2). We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.

We study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, T1- spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, R^n (n>1). We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.

The paper draws attention to the epistemological obstacles that prevented Wittgenstein from acknowledging the modern view of modal logic, including the so-called propositional attitudes. Whilst suggesting a retrospective overview of the logic of epistemic modalities, it is argued that such obstacles primarily rely upon the nature of the logical space depicted in the *Tractatus Logico-Philosophicus* as well as the metaphysical status of the subject. Some relevant quotes are recalled to justify the essentially universal feature of logic according to the early Wittgenstein.

Recently some elaborations were made concerning the game theoretic semantic of Lℵ0 and its extension. In the paper this kind of semantics is developed for Dishkant’s quantum modal logic LQ which is also, in fact, the speciﬁc extension of Lℵ0 . As a starting point some game theoretic interpretation for the S L system (extending both Lukasiewicz logic Lℵ0 and modal logic S5) was exploited which has been proposed in 2006 by C.Ferm˝uller and R.Kosik . They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of Lℵ0 in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion.

*Advances in Modal Logic* is a bi-annual international conference and book series in Modal Logic. The aim of the conference series is to report on important new developments in pure and applied modal logic, and to do so at varying locations throughout the world. The book series is based on the conferences. Please consult thebackground pages for further details.

In this paper we introduce public announcements to Subset Space Logic (SSL). In order to do this we have to change the original semantics for SSL a little and consider a weaker version of SSL without the cross axiom. We present an axiomatization, prove completeness and show that this logic is PSPACE-complete. Finally, we add the arbitrary announcement modality which expresses ``true after any announcement'', prove several semantic results, and show completeness for a Hilbert-style axiomatization of this logic.

We consider modal logics of products of neighborhood frames and prove that for any pair L and L' of logics from set {S4, D4, D, T} modal logic of products of L-neighborhood frames and L'-neighborhood frames is the fusion of L and L'.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.