In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.

Our society is in the middle of the AI revolution. We discuss several applications of AI, in particular medical causality, where deep-learning neural networks screen through big data bases, extracting associations between a patient’s condition and possible causes. While beneficial in medicine, several questionable AI trading strategies have emerged in finance. Though advantages in many aspects of our lives, serious threats of AI exist. We suggest several regulatory measures to reduce these threats. We further discuss whether ‘full AI robots’ should (...) be programmed with a virtual consciousness and conscience. While this would reduce AI threats via motivational control, other threats such as the desire for AI—human socioeconomic equality could prove detrimental. (shrink)

We present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$. This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely (...) model-theoretic proof of the compactness property for $\mathrm{InqML}$. Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$, which crucially uses $\omega $ -saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems. (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)

Recent years have seen the appearance of many English-language hand books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now (...) classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda tions but can discourage the beginner by their abstract form. A. G. (shrink)

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the Fidel-Vakarelov construction is (...) also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

This article focuses on the trajectory of life of the three generations of the Jankowski family in Siberia, Primorski Krai and Korea in the years 1863– 1945 in terms of their economic, cultural and scientific achievements. The founder of the Far Eastern branch of the family was Michał Jankowski. Exiled to Siberia for participation in the January Uprising of 1863, as a man of indefatigable energy and collaborator of Benedykt Dybowski, he undertook many initiatives, including hunting, wild ginseng collecting and (...) a gold mining administration. In 1880 he settled on the Sidiemi Peninsula near Vladivostok, where he established a farm specializing in horse breeding and ginseng gardening. In addition to farming, he conducted natural history and archaeological research and sent the acquired specimens and artifacts to Russian and European museums. The tradition of his father was continued in the second generation by Jerzy Mikhailovich Jankowski. Endowed with similar character and talent, he studied agriculture and husbandry in Texas and at the agricultural faculty of St. Louis University in Missouri. He expanded the farm to include commercial fishing and rice cultivation. The occupation of the Far East by the Red Army in 1922 forced the family to emigrate to Korea and Manchuria. The new family seat called Novina was built on the Ompo River near the town of Ch’ŏngjin, about 200 kilometers south along the Korean coastline from the mouth of the Tuman River. It was not only a modern farm with specialized animal husbandry and fruit farming, but was also a recreational, cultural, and spiritual enclave, with a school for children, a summer theater, an Orthodox church, and a resort for holidaymakers. The third and the most famous representative of the family was Walery Jurewicz Jankowski. In 1945, he joined the Soviet Army, where, due to his knowledge of Korean and Japanese, he served as an interpreter for the military intelligence at the headquarters of the 25th Army in P’yŏngyang. At the beginning of 1946, both Walery and his fathers were arrested and sent to the labor camps. The family was scattered around the world, from Australia to the North and Latin Americas and the farm was nationalized. The Jankowsky family achieved spectacular success in the Russian Far East and in Korea because they were able to apply the traditional Polish gentry ethos based on the extended structure of family and servants, which they were able to combine with a chivalrous fortitude and openness to hard life in the frontiers populated by various ethnic groups. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

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 →.

During the global financial crisis of 2007–2009 and the currentCovid-19 debacle, central banks acted quickly, boldly, and effectively. The paper argues that they did so thanks to the lessons learned from the past financial crises, which provided them with opportunities to reconsider their previous beliefs. A case in point is the banking crisis in the United States during the Great Depression of the 1930s that taught central banks to act rapidly and decisively in order to prevent an initial liquidity crisis (...) from escalating into a solvency crisis leading to bankruptcies of banks with potentially disastrous consequences for the entire economy. The paper revisits the consequences of the currency crises of the 1990s, which transformed exchange rate regimes across the world: the EMS one of 1992–1993 that accelerated the launch of the euro, and the Asian one of 1997–1998 resulted in the proliferation of floating exchange rates in emerging economies. (shrink)

A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance (...) with Dos̆en’s criteria for logical constants. The latter is a standard Gentzen’s sequent calculus and satisfies Hacking’s criteria for logicality, including cut elimination. It is also shown that provided rules are harmonious in the sense advocated by Gratzl and Orlandelli. (shrink)

The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is (...) proved in a constructive way for sequent calculi adequate for all logics under consideration. (shrink)

We examine some extensions of the constructive propositional logic with strong negation in the setting of varieties of $\mathcal{N}$ -lattices. The main aim of the paper is to give a description of all pretabular, primitive and preprimitive varieties of $\mathcal{N}$ -lattices.

Nicolae Iorga jest postacią o dużym znaczeniu w historii i kulturze Rumunii. Bardzo często wypowiadał się o sprawach polskich w szerokim tego słowa znaczeniu, zarówno w kwestiach akademickich, jak i politycznych. Zaangażowanie Iorgi w Polskę jest widoczne niemal nieprzerwanie w trakcie jego kariery naukowej i politycznej. Od czasów szkolnych po jego tragiczną śmierć Polska i jej sprawy były obecne w jego działalności także za sprawą jego nauczycieli, profesorów czy późniejszych partnerów politycznych. Dzięki wykształceniu historycznemu Iorga postrzegał Polskę jako ważnego partnera (...) Rumunii na arenie międzynarodowej, jako gwarancję bezpieczeństwa politycznego, ale także ważnego partnera w kontekście rozwoju kulturalnego. Jako inicjator kursów języka i kultury rumuńskiej na Uniwersytecie Jagiellońskim, Nicolae Iorga otrzymał tytuł doktora honoris causa Uniwersytetu Wileńskiego w 1931 r. za ważną rolę w umacnianiu stosunków między Polską a Rumunią. (shrink)

Nicht erst die moderne Philosophie hat die fundamentale Doppelnatur des Körperlichen erkannt, das einerseits diejenigen Gegenstände auszeichnet, die wir mit größtmöglicher Distanz und Objektivität wissenschaftlich untersuchen, das sich uns andererseits und zuallererst aber auch in der Form des jeweils eigenen, in seiner Integrität stets bedrohten und letztlich dem Zerfall ausgelieferten Körpers aufdrängt als Bedingung, an der unsere eigene Existenz durch und durch hängt. Schon die Denker der Antike arbeiten sich an dieser Problematik ab; und es gilt, die gedanklichen Ressourcen nutzbar (...) zu machen, die sie für uns bereithalten. Die in diesem Band versammelten Beiträge spiegeln die antike Auseinandersetzung um den Körper und das Körperliche in all ihren Dimensionen und Schattierungen wieder. Gleichzeitig machen sie deutlich, dass diese Auseinandersetzungen nur geführt werden konnten vor dem Hintergrund einer althergebrachten und für lange Zeit sehr wirkmächtigen Tendenz, Organismen wie die Menschen als paradigmatische körperliche Wesen aufzufassen und dieses Paradigma zum Ausgangspunkt für Reflexionen über das Körperliche als solches zu nehmen. Im ersten Teil des Bandes geht es um das antike Nachdenken über den lebendigen, insbesondere menschlichen Körper. Im Mittelpunkt steht weniger die wissenschaftliche Beschäftigung mit diesem Körper als vielmehr die existenzielle Situation, in der sich der Mensch als ein körperliches Wesen befindet. Der zweite und dritte Teil behandeln die Frage nach den Möglichkeiten, Bedingungen und Grenzen der wissenschaftlichen Beschreibung von Körpern und ihrem Verhalten bei Platon und in der platonischen Tradition sowie bei Aristoteles und in der aristotelischen Tradition. (shrink)

Since the early Gestaltists there has been a strong interest in the question of how problem solvers get stuck in a mental impasse. A key idea is that the repeated activation of a successful strategy from the past results in a mental set (‘Einstellung’) which determines and constrains the option space to solve a problem. We propose that this phenomenon, which mostly was tested by fairly restricted experiments in the lab, could also be applied to more complex problem constellations and (...) naturalistic decision making. We aim at scrutinizing and reconstructing how a mental set determines the misinterpretation of facts in the field of political decision making and leads in consequence to wrong expectations and an ill-defined problem representation. We will exemplify this psychological mechanism considering a historical example, namely the unexpected stabilization of the Franco regime at the end of World War II and its survival thereafter. A specific focus will be drawn to the significant observation that erroneous expectations were taken as the basis for decisions. This is congruent with the notion that in case of discrepancy between preconceived notions and new information, the former prevails over the new findings. Based on these findings, we suggest a theoretical model for expectation violation in political decision making and develop novel approaches for cognitive empirical research on the mechanisms of expectation violation and its maintenance in political decision making processes. (shrink)

In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed successfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration.