Description logics refer to a family of formalisms concentrated around concepts, roles and individuals. They belong to the most frequently used knowledge representation formalisms and provide a logical basis to a variety of well known paradigms. The main reasoning tasks considered in the area of description logics are those reducible to subsumption. On the other hand, any knowledge representation system should be equipped with a more advanced reasoning machinery. Therefore in the current paper we make a step towards integrating description (...) logics with second-order reasoning. One of the important motivations behind introducing second-order formalism follows from the fact that many forms of commonsense and nonmonotonic reasoning used in AI can be modelled within the second-order logic. To achieve our goal we first extend description logics with a possibility to quantify over concepts. Since one of the main criticisms against the use of second-order formalisms is their complexity, we next propose second-order quantifier elimination techniques applicable to a large class of description logic formulas. Finally we show applications of the techniques, in particular in reasoning with circumscribed concepts and approximated terminological formulas. (shrink)

In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication (set inclusion) and are not applicable to the first-order case. We make a shift to arbitrary relations “ordering” the underlying universe. (...) This allows us to incorporate background theories into higher-order quantifier elimination methods which, up to now, has not been achieved. The technique we propose subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the “ordering” relations are interpreted as implication and reveals the common principle behind these approaches. (shrink)

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. (...) Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties.In the current paper we:investigate properties of well-supported models of 4QLprove the correctness of the algorithm for computing well-supported modelsshow that 4QL has PTIME data complexity and captures PTIME. (shrink)

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotation theories^ can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, (...) specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21]. (shrink)

Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning. In the current paper we first generalize the result of Nonnengart and Szałas [17] by allowing second-order variables to appear within higher-order contexts. Then we focus on a semantical analysis of conditionals, using the introduced technique and Gabbay’s semantics provided in [10] and substantially using a (...) third-order accessibility relation. The analysis is done via finding correspondences between axioms involving conditionals and properties of the underlying third-order relation. (shrink)

This paper describes an experimental platform for approximate knowledge databases called the Approximate Knowledge Database, based on a semantics inspired by rough sets. The implementation is based upon the use of a standard SQL database to store logical facts, augmented with several query interface layers implemented in JAVA through which extensional, intensional and local closed world nonmonotonic queries in the form of crisp or approximate logical formulas can be evaluated tractably. A graphical database design user interface is also provided which (...) simplifies the design of databases, the entering of data and the construction of queries. The theory and semantics for AKDBs is presented in addition to application examples and details concerning the database implementation. (shrink)

Grammar logics were introduced by Fariñas del Cerro and Penttonen in 1988 and have been widely studied. In this paper we consider regular grammar logics with converse ( REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various (...) optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG c logics is coNP-complete. (shrink)

Mathematical theory of voting and social choice has attracted much attention. In the general setting one can view social choice as a method of aggregating individual, often conflicting preferences and making a choice that is the best compromise. How preferences are expressed and what is the “best compromise” varies and heavily depends on a particular situation. The method we propose in this paper depends on expressing individual preferences of voters and specifying properties of the resulting ranking by means of first-order (...) formulas. Then, as a technical tool, we use methods of second-order quantifier elimination to analyze and compute results of voting. We show how to specify voting, how to compute resulting rankings and how to verify voting protocols. (shrink)

Rough Sets were introduced by Z. Pawlak in the year 1982 with the intention to address knowledge representation and data processing from the angle of computation and decision making. The main idea...

In this paper we study automated reasoning in the modal logic CPDLreg which is a combination of CPDL (Propositional Dynamic Logic with Converse) and REGc (Regular Grammar Logic with Converse). The logic CPDL is widely used in many areas, including program verification, theory of action and change, and knowledge representation. On the other hand, the logic REGc is applicable in reasoning about epistemic states and ontologies (via Description Logics). The modal logic CPDLreg can serve as a technical foundation for reasoning (...) about agency. Even very rich multi-agent logics can be embedded into CPDLreg via a suitable translation. Therefore, CPDLreg can serve as a test bed to implement and possibly verify new ideas without providing specific automated reasoning techniques for the logic in question. This process can to a large extent be automated. The idea of embedding various logics into CPDLreg is illustrated on a rather advanced logic TEAMLOGK designed to specify teamwork in multi-agent systems. Apart from defining informational and motivational attitudes of groups of agents, TEAMLOGK allows for grading agents' beliefs, goals and intentions. The current paper is a companion to our paper (Dunin-Kęplicz et al., 2010a). The main contribution are proofs of soundness and completeness of the tableau calculus for CPDLreg provided in (Dunin-Kęplicz et al., 2010a). (shrink)

This paper is concerned with n-localization property introduced by Newelski and Rosłanowski in [10] and getting it for CS iterations of forcing notions.

The objective of the paper is a presentation and an interpretation of Władysław Stróżewski’s views on aesthetics. Built upon the foundation provided by classical metaphysics, Stróżewski’s aesthetics is simultaneously a continuation of the tradition of phenomenological interpretation of art. Stróżewski extends Roman Ingarden’s aesthetic theory by including the idea of numinous concretion, largely inspired by the works of Rudolf Otto. As a result, this new phenomenological perspective transcends the narrowly understood ‘aesthetics’ of the work of art towards the inherent dimension (...) of the sacred. Moreover, Stróżewski’s scrutiny of the aspect of art that goes beyond the aesthetic reveals an important relationship obtaining between his insights and the theological aesthetics of Hans Urs von Balthasar. Thus Stróżewski’s ideas turn out to be of significance to contemporary aesthetic and theological debates. In particular, he shows the movement of beauty from aesthetic to transcendental, and ultimately towards the sacred, where numinosum is manifested as Mystery. (shrink)

The paper consists of two parts, outlined in the title.I. In the historical science time appears as an element of the historian’s workshop. The historian collects source information, evaluates them and assigns respective dates. Only on the ground of thus “processed” sources may he reproduce the past: events and longer development processes, setting them in time. This dated time is understood colloquially as something objective, which runs one way and may be measured.II. A historian who studies the past reality depends (...) on the extent to which those sources are preserved. As a result of their interpretation, with the use of increasingly excellent research methods, he reproduces that reality. He must carry out an ongoing selection of historical facts found. This exposes him to temptations of such selection of those facts so as to adjust the historical knowledge to serve well the national interests, his own political opinions, etc. In this way the historian exerts influence on social awareness, which may consists of various myths and stereotypes, which sometimes lead to negative social behaviors: aggression, chauvinism, etc. Therefore, for researcher honesty’s sake and in the sense of moral responsibility, he must be careful to be as impartial as possible in his work. (shrink)

Natural Deduction Natural Deduction is a common name for the class of proof systems composed of simple and self-evident inference rules based upon methods of proof and traditional ways of reasoning that have been applied since antiquity in deductive practice. The first formal ND systems were independently constructed in the 1930s by G. Gentzen and S. Jaśkowski and … Continue reading Natural Deduction →.

The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with intuitively correct (...) principles. (shrink)

The aim of the paper is to examine the nature of moral and legal norms in a broader context: first, taking into account logical and methodological assumptions, second, in the perspective of psychology of emotions and legal policy. The basic subject of the research carried out by Leon Petrażycki was represented by law. Originally, it had a psychological character, not an objective, eternal, and unchanging one. To fully understand the genesis and nature of morality and law, Petrażycki addressed the study (...) of mental phenomena, especially emotional experiences. First, however, he developed appropriate rules of logic and scientific methodology. Then he developed a new classification of mental phenomena, among which the fundamental role is played by bilateral emotions. At some stage, emotions begin to cooperate with cognitive processes, first of all with imaginations. Imaginations of acts, such as theft, betrayal, murder, can cause repulsive emotions, and type imaginations, such as truthfulness, charity, justice can evoke apulsive emotions. On the basis of such associations, judgments are created over time, the content of which becomes a basis for fundamental rules of conduct, that is, for norms. There are two fundamentally different types of norms: moral norms and legal norms. The norms of the first type are imperative and represent the nature of validity, while the norms of the second type are imperative-attributive and they also always entitle someone to something, i.e. they give someone a right. This division determines a fundamental difference between morality and law. (shrink)

Książka stanowi wprowadzenie do twórczości Richarda Rorty'ego, a także próbę osadzenia jej na szerszym tle filozofii współczesnej, ze szczególnym uwzględnieniem filozofii postmodernistycznej. Obecne są w niej szersze analizy stanu kultury współczesnej, a także relacje z istotnych koncepcji obecnych we współczesnej humanistyce, bez których nie można zrozumieć ewolucji współczesnej humanistyki i specjalnej roli, jaką odegrała w niej filozofia Richarda Rorty'ego.

We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in ‘The World of Physics and the World of Sense’ (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's (...) second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology. (shrink)

Generational differences in societies are characteristics generally attributed to people’s age that constitute a sociocultural phenomenon. Divisions in the generations differ across nations and extend even to civilizations. Perception and recognition of the different characteristics of each generation affect the cooperation between people in social, political, and economic capacities, and subsequently extend to entities in the public, informal, commercial, and nongovernmental sectors. From the perspective of social justice, it is important to draw attention to how workplace management techniques are used (...) to promote equal opportunities among representatives of various generations. (shrink)

Spouses exhibit two kinds of behaviours: protective and transgressive. Protective acts are those aiming to overcome current problems, leading to preserving some balance. Transgressive acts are deliberately overstepping everyday marital reality and doing new things in new ways. They lead to changing the relation with the hope of improving it, but also create the risk of deterioration. The more transgressive behaviours spouses exhibit, the more chances they have to get to know each other and experience the joy of being part (...) of a union. Transgressive tendencies stem from a network personality structure and consist of five psychons: cognitive, instrumental, motivational, emotional, and personal. The success of a marriage is the effect of a specific form of transgressive behaviours in marriage exhibited by both spouses, which is recognizing difficulties as they appear, finding their sources, and taking steps together to overcome them. (shrink)

In terms of psychotransgressionism, personality is a network of five equipollent psychons, the content of which determines the personality’s functioning. The strength and power of the individual psychons underlies the tendency to undertake transgressive actions. In this study, we hypothesized that transgressive spouses are characterized by greater potential strength, greater power of cognitive, instrumental, motivational, emotional, and personal psychons than protective spouses. We operationalized all psychons, created the appropriate research tools, and then studied married couples. Using the Transgression Scale developed (...) by Studenski, we found a group of spouses with higher levels of transgression, and a group of spouses with lower levels of transgression. Transgressive wives are characterized by better knowledge about their husbands’ operational sphere, and are more aware of personal beliefs than protective wives. Similarly, transgressive husbands have greater knowledge of their wives’ operational sphere, stronger cognitive needs, and weaker personal needs than protective husbands. Transgressive husbands are characterized by a positive affective shift and have a greater awareness of personal beliefs than protective husbands. The potential brought into interpersonal relationships by transgressive spouses may create a climate conducive to building a satisfying marital relationship. (shrink)

Dowód pierwszej tezy Gorgiasza jest dedukcyjnym wnioskowaniem nie wprost, skon-struowanym przez wielokrotne użycie schematu modus tollens. Te z jego przesłanek, które przyjęte są bez dowodów, są łudząco podobne do niektórych prawd logicznych lub analitycznych. Gdyby rzeczywiście nimi były, w dowodzie musiałyby pojawić się sprzeczności. W czasach Gorgiasza dowód mógł uchodzić za poprawny; możliwość jasnego wskazania i opisania jego defektów poja-wiła się dopiero z wynalezieniem logiki formalnej. Gorgiasz mógł ogłosić swą tezę serio –jako tezę skrajnego nihilizmu metafizycznego –nie obawiając się kompromitacji. (...) Jeżeli dowód jest sofizma-tem, to wśród sofizmatów jest arcydziełem, najwybitniejszym osiągnięciem sofistów w tej dziedzi-nie. (shrink)

“Age of life” is one of the essential characteristics that differentiate people. Age perception is also associated with social justice. The concept of age is defined ambiguously. At the same time, the different age criteria also forms the basis of age differentiation and age discrimination. The population lead to distinctions of age groups, age categories, and generations. Differences between generations also lead to Study in the concepts of intergenerationality, intergenerational justice, and intergenerational policies.

In this paper, we want to present the genesis of Stanisław Leśniewski’s mereology. Although ‘mereology’ comes from theword ‘part’, mereology arose as a theory of collective classes. That is why we present the differences between the concepts of being a distributive class and being a collective class. Next, we present Leśniewski’s original mereology from 1927, but with a modern approach. Leśniewski was inspired to create his concept of classes and their elements by Russell’s antinomy. To face it, Leśniewski had to (...) define the concept of being an element of based on the concept of being part of. Leśniewski showed that in his theory, there is no equivalent to Russell’s antinomy. We will show that his solution has nothing to do with the original approach because, in both cases, we are talking about objects of a different kind. Russell’s original antinomy concerned distributive classes, and Leśsniewski’s considerations concerned collective classes. (shrink)