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)

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.

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.

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...) various time orderings. (shrink)

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.

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.

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.

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.

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.

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)

This book presents the refereed proceedings of the Sixth European Workshop on Logics in Artificial Intelligence, JELIA '96, held in Evora, Portugal in September/October 1996. The 25 revised full papers included together with three invited papers were selected from 57 submissions. Many relevant aspects of AI logics are addressed. The papers are organized in sections on automated reasoning, modal logics, applications, nonmonotonic reasoning, default logics, logic programming, temporal and spatial logics, and belief revision and paraconsistency.