Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.

Artificial Intelligence, both as a hope of making substantial progress, and a fear of the unknown and unimaginable, has its roots in human dreams. These dreams are materialized by means of rational intellectual efforts. We see the beginnings of such a process in Lullus’s fancies. Many scholars and enthusiasts participated in the development of Lullus’s art, ars combinatoria. Amongst them, Athanasius Kircher distinguished himself. Gottfried Leibniz ended the period in which the idea of artificial intelligence was shaped, and started the (...) new period, in which artificial intelligence could be considered part of science, by today’s standards. (shrink)

Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.

We ask which ideas of cognitive science have their roots in traditional logic, grammar and rhetoric.We also emphasize the presence of cognitive science in the pages of Studies in Logic, Grammar and Rhetoric since its very beginning.

Leibniz may be considered as the first computer scientist. He made major contributions to engineering and information science. He invented the binary system, fundamental for virtually all modern computer architectures. He built a decimal based machine that executed all four arithmetical operations and outlined a binary computer. The concepts of lingua characteristica (formal language, programming language) and calculus ratiocinator (formal inference engine or computer program) are the base of the modern logic and information science. Leibniz was groping towards hardware and (...) software concepts worked out much later by Charles Babbage and Ada Lovelace. He anticipated the universal Turing machine. (shrink)

The well known Master Argument of ancient Stoic logician Diodorus Cronus is an argument in favour of the philosophical doctrine of fatalism. Perhaps in antiquity this argument was a subject of the most celebrated controversy about temporal truth and modality. This argument is a subject of logical analysis, especially in connection with temporal logic, also today. 1 The most elegant tense-logical formulation of the Master Argument has been given by A. N. Prior. Discreteness and irreflexivity of time are semantical assumptions (...) of Prior’s formulation of the argument. The assumption of discreness is troublesome. 2 The problem of this assumption is a subject of M. J. White’s paper “The Necessity of the Past and ModalTense Logic Incompleteness”. 3. (shrink)

The position of Polish informatics, as well in research as in didactic, has its roots in achievements of Polish mathematicians of Warsaw School and logicians of Lvov-Warsaw School. Jan Lukasiewicz is considered in the world of computer science as the most famous Polish logician. The parenthesis-free notation, invented by him, is known as PN (Polish Notation) and RPN (Reverse Polish Notation). Lukasiewicz created many-valued logic as a separate subject. The idea of multi-valueness is applied to hardware design (many-valued or fuzzy (...) switching, analog computer). Many-valued approach to vague notions and commonsense reasoning is the method of expert systems, databases and knowledge-based systems. Stanis3aw Jaokowski's system of natural deduction is the base of systems of automatic deduction and theorem proving. He created a system of paraconsistent logic. Such logics are used in AI. Kazimierz Ajdukiewicz with his categorial grammar participated in the development of formal grammars, the field significant for programming languages. Andrzej Grzegorczyk had an important contribution to the development of the theory of recursiveness. (shrink)

Scientific knowledge is acquired according to some paradigm. Galileo wrote that the “book of nature” was written in mathematical language and could not be understood unless one first understood the language and recognized the characters with which it was written. It is argued that Turing planted the seeds of a new paradigm. According to the Turing Paradigm, the “book of nature” is written in algorithmic language, and science aims to learn how the algorithms change the physical, social, and human universe. (...) Some sources of the Turing Paradigm are pointed out, and a few examples of the application of the Turing Paradigm are discussed. (shrink)

Leibniz zamierzał stworzyć lingua characteristica universalis – język, w którym dałaby się zapisać wszelka wiedza, oraz calculus ratiocinator – metodę umożliwiającą rachunkowe określenie prawdziwości dowolnego zdania tego języka. Taka idea leży u podstaw współczesnej logiki. Taka idea w odniesieniu do poprawności sprzętu i programów leży u podstaw metod logicznych weryfikacji systemów informatycznych.Dziś widzimy, że dalszy rozwój informatyki zależy istotnie od postępu badań logicznych, a logika – ta starożytna dyscyplina – znalazła nowe bogate pole dociekań, zyskała nowe perspektywy badań. Jej zastosowania (...) mogą mieć tak znaczący wymiar praktyczny, o jakim nie śniło się filozofom. (shrink)

In the introduction to the volume on negation, first the source ways of understanding it from antiquity to modern times are presented, as well as the basic points of contention connected with it. Subsequently, the works contained in this volume are briefly presented in the order in which they appeared.

Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for (...) example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications. Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration. (shrink)

Building on our diverse research traditions in the study of reasoning, language and communication, the Polish School of Argumentation integrates various disciplines and institutions across Poland in which scholars are dedicated to understanding the phenomenon of the force of argument. Our primary goal is to craft a methodological programme and establish organisational infrastructure: this is the first key step in facilitating and fostering our research movement, which joins people with a common research focus, complementary skills and an enthusiasm to work (...) together. This statement—the Manifesto—lays the foundations for the research programme of the Polish School of Argumentation. (shrink)

Artykuł jest próbą wyodrębnienia i ogólnego opisu aspektów filozoficznych, zawartych w teorii duchowości ekologicznej papieża Franciszka oraz porównania ich z myślą tomistyczną, rozwijaną przez jej polskich przedstawicieli w czasach współczesnych. Aspekty te dostrzeżono na gruncie wielu dyscyplin filozoficznych, ponieważ w tego typu duchowości chodzi o stosunek człowieka do najogólniej pojętej rzeczywistości realnej, czyli do samego siebie, przyrody, kultury i religii. W toku analizy obu stanowisk zauważono liczne analogie, a także pewne rozbieżności w sposobie widzenia przyrody, osoby ludzkiej, kultury oraz religii, (...) które przynajmniej częściowo można tłumaczyć niejednoznacznością niektórych sformułowań papieskich. -------------- Zgłoszono: 19/08/2020. Zrecenzowano: 26/09/2020. Zaakceptowano do publikacji: 18/10/2020. (shrink)

W artykule poruszono problem perspektyw rozwoju, a tym samym przyszłości filozofii tomistycznej na podstawie sytuacji w Polsce. Otóż taką szansę widzi się dla tej filozofii w badaniach bardziej partykularnych, znajdujących się poza głównym obiektem zainteresowań metafizyki ogólnej. Każdorazowo przybierają one postać metafizyk szczegółowych i stosowanych. Pierwsza z nazw nawiązuje do zawężonego pola badań, druga natomiast jest uzasadniona korzystaniem z dorobku innych nauk filozoficznych, bardziej podstawowych, włącznie z metafizyką ogólną. Metafizyki tego rodzaju stanowią szeroko pojęte studium tego, co jest najbardziej fundamentalne (...) nie dla bytu jako takiego, ale dla wszelkich podstawowych typów przedmiotów w sferze natury i kultury. Nie stanowią one jednak alternatywy dla metafizyki ogólnej, która bada rzeczywistość w wymiarze transcendentalnym, ponieważ korzystając z jej dorobku, zachowują z nią łączność oraz stanowią jej konieczne dopełnienie. Jako zaś typowe dyscypliny filozoficzne, dające możliwość rozwijania owych metafizyk, podano w artykule filozofię przyrody, antropologię filozoficzną oraz dwie formy filozofii ludzkiego działania, czyli religię i moralność. W ich przypadku wskazywano zarówno na tematy już podjęte przez wielu przedstawicieli myśli tomistycznej w Polsce oraz na te, jakie są możliwe do podjęcia. Realizacja tych form metafizyki odpowiada nie tylko strukturze rzeczywistości pluralistycznej i jednocześnie powiązanej ze sobą szeregiem zależności, ale też problemom aktualnym, nowym, wymagającym ujęcia gruntownego i całościowego, jakie stwarza dopiero perspektywa metafizyczna, rozpatrywana z pozycji filozofii tomistycznej. (shrink)

In the paper some varieties E s n of BCK-algebras with condition defined by W. H. Cornish [1] are considered. A characterization of simple algebras in E s 2 is given and some properties of simple algebras in E s 3 are indicated.

Autor porusza zagadnienie struktury Peryta, która posiada tę właściwość, iż środkiem świata jest miejsce założone przez obserwatora. W ten sposób zmienia się charakter zjawisk geometrycznych, w tym perspektywy.

The contemporary development of socialist economy is featured not only by high dynamics, but also by essential structural and qualitative changes. The economic policy and the socialist planning in particular tendtowards optimalization of economic growth, then to engage for its proper directions, structure and rythmicality. The every year growing national product ought to be, as it is possible, most suitable and useful for implementation of threefold objective: 1° the better satisfaction with the necessities of population, 2° the quicker modernization of (...) the national economy, and 3° the effective development of the foreign trade as well as the international cooperation.The superiority of a set of social criteria is becoming the general assumption of economic policy. A vision of the future human being and the mature society is determining prospective objectives and aims of the development as well is indicating ways and means for their realization by economic policy.The economic growth is not objective by itself, but has to serve for a comprehensive social advancement. At the same time the factors of social advancement advantageously impact the labour productivity and otherwisethey perform a role of additional stimulatives of the economic growth.The objectives of social advancement should be realized gradually basing on staging patterns of systematic raise of the living standards of the population. A hierarchy and a sequence of satisfaction with the both individual and collective necessities demand a skillful application of the criteria of the economic and social choice. We are using here the principle of social and economic rationality.Hence the factors of social advancement, involving the scientific and technical progress, belang to the fundumatel elements of the present economic and social policy. At the same time objectives and means of modern economic and social advancement are here strictly interlaced. (shrink)

This essay explores social forces which contributed to regaining independence by the Polish people and sovereignty by the Polish state after 45 years of Soviet domination There were four major factors or forces of historical change: workers' resistance (big-industry working class); intellectual opposition (dissidents); grass-roots movement (families, households and their microeconmuc activity); and the Catholic Church (in the late phase of the Polish People's Republic). The preliminary thesis is that Poland succeeded in transcending communism and Soviet domination as quickly as (...) it did thanks to its civil society traditions. In particular, universities and their intellectual influence on the young generations of Poles helped nurture the political opposition Equally, the grass-roots movement of Polish family households undermined the unrealistic, strange system of national (planned) economy, which otherwise could have lasted much longer in Poland, as it did in all of Eastem Europe. The conclusion follows that the historically formed cultural capital of the Polish people was the decisive factor in the nation's liberation from totalitarian rule. (shrink)

When asked in 1962 on what he was working Kazimierz Ajdukiewicz replied: Several years ago Polish Scientific Publishers suggested that I pre pare a new edition of The Logical Foundations of Teaching, which I wrote 1 before 1939 as a contribution to The Encyclopaedia of Education. It was a small booklet covering elementary information about logical semantics and scientific methodology, information which in my opinion was necessary as a foundation of teaching and as an element of the education of (...) any teacher. When I recently set to preparing the new edition, I rewrote practically everything, and a booklet of some 100 pages swelled into a bulky volume almost five times bigger. The issues have remained practically the same, but they are now analysed much more thoroughly and the threshold of difficulty is much higher now. The main stress has been laid on the methods used in the empirical sciences, and within that field, on the theory of measurement and the methods of statistical inference. I am now working on the last chapter of the book, concerned with explanation procedures and theory construction in the empirical sciences. When that book, which I intend to entitle Pragmatic Logic, is com pleted I intend to prepare for the press Vol. 2 of my minor writings, 2 Language and Cognition, which will cover some of my post-war pa pers. (shrink)