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)

The Description Logic Handbook covers all aspects of the research in the field of knowledge representation. Written by some of the most prominent researchers in the field, and covering the basic technical material and implementational aspects, it is both a unique reference and a self-study guide.

The logical theory which is called thecalculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found (...) in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years. (shrink)

The purpose of the present paper is to show that modal propositional logics can be interpreted in a logic based on relational calculus. We consider languages with necessity operators [R], where R is an accessibility relation expression representing an element of the algebra of binary relations with operations −,∪,∩, −1 , ◦. The relational logic is based on relational calculus enriched by operations of weakest prespecification and weakest postspecification introduced in Hoare and He Jifeng and investigated in He Jifeng et (...) al . The logic is an extension of the system introduced in Or lowska and investigated in Buszkowski and Or lowska. (shrink)