Each information system leads to a hierarchy of binary relations on the object set in a natural way; these relational systems can serve as frames for the semantics of modal logics. While relations of indiscernibility and their logics have been frequently studied, the situation in the case of relations which distinguish objects is much less clear. In this paper, we present complete logical systems for relations of complementarity derived from information systems.

ABSTRACT In this paper we introduce and investigate various classes of multimodal logics based on frames with relative accessibility relations. We discuss their applicability to representation and analysis of incomplete information. We provide axiom systems for these logics and we prove their completeness.

A method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

One of the important issues in research on knowledge based computer systems is development of methods for reasoning about knowledge. In the present paper semantics for knowledge operators is introduced. The underlying logic is developed with epistemic operators relative to indiscernibility. Facts about knowledge expressible in the logic are discussed, in particular common knowledge and joint knowledge of n group of agents. Some paradoxes of epistemic logic are shown to be eliminated in the given system. A formal logical analysis of (...) reasoning about knowledge is a subject of investigations both in logic and computer science , and several epistemic systems have been proposed to formalize the operator ‘an agent knows’. In the present paper we propose a formalization based on a semantic treatment of knowledge within the framework of rough set theory . The inspiration for the underlying epistemic logic came from the analysis of knowledge transfer in distributed systems developed in Orlowska and Sanders and from the author’s earlier work on indiscernibility and relative accessibility semantics. (shrink)

A method of defining semantics of logics based on not necessarily distributive lattices is presented. The key elements of the method are representation theorems for lattices and duality between classes of lattices and classes of some relational systems . We suggest a type of duality referred to as a duality via truth which leads to Kripke-style semantics and three-valued semantics in the style of Allwein-Dunn. We develop two new representation theorems for lattices which, together with the existing theorems by Urquhart (...) and Bimbo-Dunn, constitute a complete, in a sense, representation theory for lattices. As observed by Dunn and Hardegree, variations of Urquhart's duality arise by varying his disjointness assumption on the canonical frame. Four possible assumptions – disjoint, exhaustive, non-disjoint and non-exhaustive – are discussed in the paper. Each of the four corresponding representation theorems is expanded to a duality via truth. Based on these dualities we suggest four corresponding types of semantics for lattice-based logics. We also discuss a new topological representation of lattices. (shrink)

This article provides an overview of development of Kripke semantics for logics determined by information systems. The proposals are made to extend the standard Kripke structures to the structures based on information systems. The underlying logics are defined and problems of their axiomatization are discussed. Several open problems connected with the logics are formulated. Logical aspects of incompleteness of information provided by information systems are considered.

In the paper we define a class of languages for representation o knowledge in those application areas when a complete information about a domain is not available. In the languages we introduce modal operators determined by accessibility relations depending on parameters.

In the paper ordering relations for comparison of verisimilitude of theories are introduced and discussed. The relations refer to semantic analysis of the results of theories, in particular to analysis of concepts the theories deal with.

ABSTRACT In this paper it is shown that a broad class of propositional logics can be interpreted in an equational logic based on fork algebras. This interpetability enables us to develop a fork-algebraic formalization of these logics and, as a consequence, to simulate non-classical means of reasoning with equational theories algebras.

In this chapter the life, education, scientific path, and research of Ewa Orłowska are presented. Information on her service for the logic community, in particular on activities in scientific organisations, councils, and committees, on membership of editorial boards, and on participation in national and international projects is also mentioned.

ABSTRACT Propositional dynamic logic with converse and test, is enriched with complement, intersection and relational operations of weakest prespecification and weakest postspecification. Relational deduction system for the logic is given based on its interpretation in the relational calculus. Relational interpretation of the operators ?repeat? and ?loop? is given.

A survey of some classes of Post algebras is given including the class of plain semi-Post algebras, Post algebras of order m, m>1, as its particular instance, Post algebras of order ω+, and Post algebras of order ω + ω∗. Representation theorems for each of the classes are given. Some examples of the algebras in the classes are constructed.

The book presents logical foundations of dual tableaux together with a number of their applications both to logics traditionally dealt with in mathematics and philosophy (such as modal, intuitionistic, relevant, and many-valued logics) and to various applied theories of computational logic (such as temporal reasoning, spatial reasoning, fuzzy-set-based reasoning, rough-set-based reasoning, order-of magnitude reasoning, reasoning about programs, threshold logics, logics of conditional decisions). The distinguishing feature of most of these applications is that the corresponding dual tableaux are built in a (...) relational language which provides useful means of presentation of the theories. In this way modularity of dual tableaux is ensured. We do not need to develop and implement each dual tableau from scratch, we should only extend the relational core common to many theories with the rules specific for a particular theory. (shrink)

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...) are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)

Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation.

We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system.

We present several classes of logics for reasoning with information stored in information systems. The logics enable us to cope with the phenomena of incompleteness of information and uncertainty of knowledge derived from such an information. Relational inference systems for these logics are developed in the style of dual tableaux.

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.

We investigate complex algebras of the form arising from a frame where, and exhibit their abstract algebraic and logical counterparts.

Following the theory of Boolean algebras with modal operators , in this paper we investigate Boolean algebras with sufficiency operators and mixed operators . We present results concerning representability, generation by finite members, first order axiomatisability, possession of a discriminator term etc. We generalise the classes BAO, SUA, and MIA to classes of algebras with the families of relative operators. We present examples of the discussed classes of algebras that arise in connection with reasoning with incomplete information.

Based on Alasdair Urquhart’s representation of not necessarily distributive bounded lattices we exhibit several discrete dualities in the spirit of the “duality via truth” concept by Orłowska and Rewitzky. We also exhibit a discrete duality for Urquhart’s relevant algebras and their frames.

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of ♦-formulae, i.e. each ♦-formula has a ℒ-model iff it has a finite ℒ-model. Roughly speaking, a ♦-formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.

We investigate the join semilattice of modal operators on a Boolean algebra B. Furthermore, we consider pairs ⟨f,g⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle f,g \rangle $$\end{document} of modal operators whose supremum is the unary discriminator on B, and study the associated bi-modal algebras.

We consider fragments of the relational logic RL(1) obtained by posing various constraints on the relational terms involving the operator of composition of relations. These fragments allow to express several non classical logics including modal and description logics. We show how relational dual tableaux can be employed to provide decision procedures for each of them.