Andrzej Grzegorczyk (1922-2014)
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On the Minimal Non-Fregean Grzegorczyk Logic
Joanna Golinska-Pilarek´ On the Minimal Non-Fregean Grzegorczyk Logic To the Memory of Andrzej Grzegorczyk Abstract. The paper concerns Grzegorczyk’s non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, MGL, which satisfies Grzegorczyk’s fundamental requirements. We present a sound and complete semantics for MGL and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of MGL, which makes it a generic non-Fregean logic. Keywords: Non-Fregean logic, Descriptive equivalence, Equimeaning, Extensionality. Introduction There are plenty of logical systems discussed in the literature which are fo- cused on different aspects of formal representation of real world problems. Over the years, the question as to which logic is adequate has been a matter of lively discussion. One of the fundamental controversies is the philosoph- ical status of classical two-valued logic, and in particular the paradoxes of material implication and equivalence. In order to remedy the dissatisfactory aspects of classical logic, many logical systems have been introduced, such as modal, intuitionistic, relevant, non-Fregean, and paraconsistent logics, among others. In [3] Andrzej Grzegorczyk introduced yet another logical system in- tended to cover the fundamental properties of descriptive equivalence of sentences. His main assumption is that the human language is used pri- marily to create descriptions of reality and logical connectives are the main tool to represent them formally. In order to be able to speak about inter- actions between descriptions—as Grzegorczyk argues—a logical language Presented by Heinrich Wansing; Received February 21, 2015 Studia Logica (2016) 104: 209–234 DOI: 10.1007/s11225-015-9635-y c The Author(s) 2015. -
L. Maligranda REVIEW of the BOOK by MARIUSZ URBANEK
Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 51 L. Maligranda REVIEW OF THE BOOK BY MARIUSZ URBANEK, “GENIALNI – LWOWSKA SZKOL A MATEMATYCZNA” (POLISH) [GENIUSES – THE LVOV SCHOOL OF MATHEMATICS] L. Maligranda. Review of the book by Mariusz Urbanek, “Genialni – Lwowska Szko la Matema- tyczna” (Polish) [Geniuses – the Lvov school of mathematics], Wydawnictwo Iskry, Warsaw 2014, 283 pp. ISBN: 978-83-244-0381-3 , Mat. Stud. 50 (2018), 105–112. This review is an extended version of my short review of Urbanek's book that was published in MathSciNet. Here it is written about his book in greater detail, which was not possible in the short review. I will present facts described in the book as well as some false information there. My short review of Urbanek’s book was published in MathSciNet [24]. Here I write about his book in greater detail. Mariusz Urbanek, writer and journalist, author of many books devoted to poets, politicians and other figures of public life, decided to delve also in the world of mathematicians. He has written a book on the phenomenon in the history of Polish science called the Lvov School of Mathematics. Let us add that at the same time there were also the Warsaw School of Mathematics and the Krakow School of Mathematics, and the three formed together the Polish School of Mathematics. The Lvov School of Mathematics was a group of mathematicians in the Polish city of Lvov (Lw´ow,in Polish; now the city is in Ukraine) in the period 1920–1945 under the leadership of Stefan Banach and Hugo Steinhaus, who worked together and often came to the Scottish Caf´e (Kawiarnia Szkocka) to discuss mathematical problems. -
New Publications Offered by The
New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to http://www.ams.org/bookstore-email. Algebra and Algebraic Contemporary Mathematics, Volume 544 June 2011, 159 pages, Softcover, ISBN: 978-0-8218-5259-0, LC Geometry 2011007612, 2010 Mathematics Subject Classification: 17A70, 16T05, 05C12, 43A90, 43A35, 43A75, 22E27, AMS members US$47.20, List US$59, Order code CONM/544 New Developments in Lie Theory and Its On Systems of Applications Equations over Free Carina Boyallian, Esther Galina, Partially Commutative and Linda Saal, Universidad Groups Nacional de Córdoba, Argentina, Montserrat Casals-Ruiz and Ilya Editors Kazachkov, McGill University, This volume contains the proceedings of Montreal, QC, Canada the Seventh Workshop in Lie Theory and Its Applications, which was held November 27–December 1, 2009 at Contents: Introduction; Preliminaries; the Universidad Nacional de Córdoba, in Córdoba, Argentina. The Reducing systems of equations over workshop was preceded by a special event, “Encuentro de teoria de G to constrained generalised equations over F; The process: Lie”, held November 23–26, 2009, in honor of the sixtieth birthday Construction of the tree T ; Minimal solutions; Periodic structures; of Jorge A. Vargas, who greatly contributed to the development of The finite tree T0(Ω) and minimal solutions; From the coordinate ∗ T Lie theory in Córdoba. group GR(Ω ) to proper quotients: The decomposition tree dec and the extension tree Text; The solution tree Tsol(Ω) and the main This volume focuses on representation theory, harmonic analysis in theorem; Bibliography; Index; Glossary of notation. Lie groups, and mathematical physics related to Lie theory. -
L. Maligranda REVIEW of the BOOK by ROMAN
Математичнi Студiї. Т.46, №2 Matematychni Studii. V.46, No.2 УДК 51 L. Maligranda REVIEW OF THE BOOK BY ROMAN DUDA, “PEARLS FROM A LOST CITY. THE LVOV SCHOOL OF MATHEMATICS” L. Maligranda. Review of the book by Roman Duda, “Pearls from a lost city. The Lvov school of mathematics”, Mat. Stud. 46 (2016), 203–216. This review is an extended version of my two short reviews of Duda's book that were published in MathSciNet and Mathematical Intelligencer. Here it is written about the Lvov School of Mathematics in greater detail, which I could not do in the short reviews. There are facts described in the book as well as some information the books lacks as, for instance, the information about the planned print in Mathematical Monographs of the second volume of Banach's book and also books by Mazur, Schauder and Tarski. My two short reviews of Duda’s book were published in MathSciNet [16] and Mathematical Intelligencer [17]. Here I write about the Lvov School of Mathematics in greater detail, which was not possible in the short reviews. I will present the facts described in the book as well as some information the books lacks as, for instance, the information about the planned print in Mathematical Monographs of the second volume of Banach’s book and also books by Mazur, Schauder and Tarski. So let us start with a discussion about Duda’s book. In 1795 Poland was partioned among Austria, Russia and Prussia (Germany was not yet unified) and at the end of 1918 Poland became an independent country. -
Polish Mathematicians and Mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire)
Science in Poland Stanisław Domoradzki ORCID 0000-0002-6511-0812 Faculty of Mathematics and Natural Sciences, University of Rzeszów (Rzeszów, Poland) [email protected] Małgorzata Stawiska ORCID 0000-0001-5704-7270 Mathematical Reviews (Ann Arbor, USA) [email protected] Polish mathematicians and mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire) Abstract In this article we present diverse experiences of Polish math- ematicians (in a broad sense) who during World War I fought for freedom of their homeland or conducted their research and teaching in difficult wartime circumstances. We discuss not only individual fates, but also organizational efforts of many kinds (teaching at the academic level outside traditional institutions, Polish scientific societies, publishing activities) in order to illus- trate the formation of modern Polish mathematical community. PUBLICATION e-ISSN 2543-702X INFO ISSN 2451-3202 DIAMOND OPEN ACCESS CITATION Domoradzki, Stanisław; Stawiska, Małgorzata 2018: Polish mathematicians and mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire. Studia Historiae Scientiarum 17, pp. 23–49. Available online: https://doi.org/10.4467/2543702XSHS.18.003.9323. ARCHIVE RECEIVED: 2.02.2018 LICENSE POLICY ACCEPTED: 22.10.2018 Green SHERPA / PUBLISHED ONLINE: 12.12.2018 RoMEO Colour WWW http://www.ejournals.eu/sj/index.php/SHS/; http://pau.krakow.pl/Studia-Historiae-Scientiarum/ Stanisław Domoradzki, Małgorzata Stawiska Polish mathematicians and mathematics in World War I ... In Part I we focus on mathematicians affiliated with the ex- isting Polish institutions of higher education: Universities in Lwów in Kraków and the Polytechnical School in Lwów, within the Austro-Hungarian empire. -
Stefan Banach
Stefan Banach Jeremy Noe March 17, 2005 1 Introduction 1 The years just before and after World War I were perhaps the most exciting and influential years in the history of science. In 1901, Henri Lebesgue formu- lated the theory of measure and in his famous paper Sur une g´en´eralisation de l’int´egrale definie, defining a generalization of the Riemann integral that had important ramifications in modern mathematics. Until that time, mathemat- ical analysis had been limited to continuous functions based on the Riemann integration method. Alfred Whitehead and Bertrand Russell published the Principia Mathematica in an attempt to place all of mathematics on a logical footing, and to attempt to illustrate to some degree how the various branches of mathematics are intertwined. New developments in physics during this time were beyond notable. Al- bert Einstein formulated his theory of Special Relativity in 1905, and in 1915 published the theory of General Relativity. Between the relativity the- ories, Einstein had laid the groundwork for the wave-particle duality of light, for which he was later awarded the Nobel Prize. Quantum mechanics was formulated. Advances in the physical sciences and mathematics seemed to be coming at lightning speed. The whole western world was buzzing with the achievements being made. Along with the rest of Europe, the Polish intellectual community flourished during this time. Prior to World War I, not many Polish scolastics pursued research related careers, focusing instead 1Some of this paragraph is paraphrased from Saint Andrews [11]. 1 on education if they went on to the university at all. -
A Philosophically Plausible Formal Interpretation of Intuitionistic Logic
MATHEMATICAL LOGIC A PHILOSOPHICALLY PLAUSIBLE FORMAL INTERPRETATION OF INTUITIONISTIC LOGIC BY ANDRZEJ GRZEGORCZYK (Communicated by Prof. A. HEYTING at the meeting of September 26, 1964} The notion of forcing introduced by P. J. CoHEN [2] in order to prove the independence of the axiom of choice, may be considered as a modi fication of the notion of fulfilment of a formula by a tree introduced by E. W. BETH [l] p. 447. But the word "forcing" used by Cohen suggests an interesting philosophical interpretation of intuitionistic logic. According to this interpretation, which shall be described in detail, the intuitionistic logic can be understood as the logic of scientific research (rather positivistically conceived) while on the other hand the classical logic is the logic of ontological thought. l. The formal description of scientific research Scientific research (e.g. an experimental investigation) consists of the successive enrichment of the set of data by new established facts obtained by means of our method of inquiry. When making inquiries we question Nature and offer her a set of possible answers. Nature chooses one of them. Hence scientific research may be conceived as a triple R=(J, o, P). The elements of R will be called: J - the information set, i.e. the set of all possible experimental data, o - the initial information, P - the function of possible prolongations of the informations. The set J may be finite or infinite. The elements of J are finite ordered collections of atomic sentences: Pt1(aJ), Pt2(aJ> ak), ... where Ptn are n-place predicates and a1, ak, .. -
I Topics in the Theory of Generative Grammar (The Hague, Paris: Mouton, 1969), P
NOTES I I Topics in the Theory of Generative Grammar (The Hague, Paris: Mouton, 1969), p. 11. 2 John Berryman, 77 Dream Songs (New York: Farrar, Straus and Giroux, 1965), p.5. 3 Time, Vol. 99, No.8, Feb. 21 (1972), p. 8. 4 See my Understanding Understanding (Ithaca: Cornell University Press, 1972), Chap. II for a discussion of "what is said". 5 See my Philosophic Turnings (Ithaca: Cornell University Press, 1966), Chaps. XI - XIII for a discussion of related matters. 6 D. B. Wyndham Lewis and Charles Lee (eds.), The Stuffed Owl: An Anthology of Bad Verse (New York: Capricorn Books, 1962), p. ix. This is the source of some of the verse cited in this essay. 7 The wild Barbary sheep also called an aoudad. 8 Peter Selz with texts by the artist, The Work ofJean Dubuffet (New York: The Museum of Modern Art, 1962), p. 97. 9 Op. cit .. p. 103. 10 John Skelton (1460-1529) wrote a funeral service for the sparrow Philip slain by a cat. I I All Souls College of Oxford University. Christopher Wren the room designer. 12 From Andrew Marvell's poem of the nymph complaining of the death of her faun. 13 Section XIX of the version edited by W. F. Stead. 14 In T. S. Eliot's Fragment of an Agon. IS See T. Roethke's The Meadow Mouse. in his The Far Field. 16 See M. Gell-Mann and E. P. Rosenbaum, 'Elementary Particles', Scientific American (July, 1957), p. 5 - my source for these comments. 17 Landscaped Tables Landscapes of the Mind Stones of Philosophy (New York: Pierre Matisse Gallery, 1952), p. -
Confrontation of Reism with Type-Theoretical Approach and Everyday Experience
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 27 (40) 2012 Witold Marciszewski CONFRONTATION OF REISM WITH TYPE-THEORETICAL APPROACH AND EVERYDAY EXPERIENCE 1. Grzegorczyk’s project for merging reism with type theory Quine’s famous essay “On What There Is” starts from the following remark. A curious thing about the ontological problem is its simplicity. It can be put in three Anglo-Saxon monosyllables” ‘What is there?’ It can be answered, moreover, in a word – ‘Everything’ – and everyone will accept this answer as true. Obviously, such accord is only apparent, since each philosophical doctrine offers its individual answer on what this everything does comprise. How- ever, the strategies of justifying their claims have something in common. Each philosopher tries to distinguish a category of entities whose existence would be most certain, beyond any doubt, and then – on this basis – to demonstrate the existence of other kinds of entities. Let the elements of such a basic category be called prime existents1 The history of philosophy can by summarized through mentioning kinds of entities which were acknowl- edged as prime existents by particular thinkers: by Plato – universal ideas; by Aristotle – substantial individuals; by Democritus – atoms; by Descartes – his own mind; by Hume – sense data; by Brentano (in one phase of his development) – mental phenomena, etc. What in the Aristotelian idiom is called individual substances can be translated into vernacular English as individual things or, even simpler, things – Latin res.2 1 In the use of the term “individual substances” I follow Moran’s [2000, p. 35] account on Brentano’s ontology. 2 Dictionaries contain a long list of everyday meanings of “thing”. -
Jerzy Andrzej Chmurzyński SEARCHING EUROPE's
DIALOGUE AND UNIVERSALISM No. 6–7/2002 Jerzy Andrzej ń SEARCHING ’DESTINATION ABSTRACT According to Andrzej ’vision, Europe has “reached” Christianity as its conjectural ideological termination. This view, though tempting, arouses reservations. First of all, Christianity is not a cultural monolith. There is a huge chasm between the “”with its ideal of perfection exhibited by rational effectiveness, and the Catholic and Orthodox ideal of perfecting oneself in ’sinfulness. Nor can one ig- nore ’deep cultural and religious split into a Latin West and a Byzantine East. Secondly, we have been witnessing diverse cultural and demographic transforma- tions in European civilization, some of which appear to be almost universal. They mani- fest themselves in a “”of individual Christianity, either in the form of “neolibertin- ism”, or in a separation of normative ethics from religious ideology. In this context, counter-cultural movements find ideological support in the American social conscious- ness, as in the ’phenomena of flower children and hippies. These tendencies may have evolved into the developing New Age ideology, although they do not exercise the same degree of influence on the population. Some of them stand in opposition to Judeo- Christian moral principles—especially in their advocacy of permissiveness. Others propagate the traditional values found in Oriental doctrines. All of them are apparently anti–Christian; they may lead Europe in a new, so far unknown direction. The decay of traditional morality was predictable, given the new permissiveness, es- pecially in the area of sexual morality. Some of the effects may be considered alarming in that they appear side by side with an increase in wealth. -
Kazimierz Kuratowski (Warsaw)
Kazimierz Kuratowski (Warsaw) THE PAST AND THE PRESENT OF THE POLISH SCHOOL OF MATHEMATICS I am concentrating in this article on two main subjects. Firstly: I am trying to answer the question what brought about such an “explosion” of mathematics in a country in whose scientific tradition there was hardly any mathematics and which happened at the time when after an over-one-century-long foreign rule the nation was trying hard to reconstruct its now independent country, ravaged by the First World War. Secondly: was this explosion a short-lived enthusiasm or, on the contrary, the Polish school of .mathematics struck roots so deeply that it was sub sequently able to survive the cataclysm of the Second World War and rebuild in the new circumastances — in People’s Poland — the internationally re cognized edifice of Polish mathematics? There will be in this article no mathematical theorems, no definitions or geometrical constructions. I shall be trying to use the language which can be understood without mathematical qualifications. It is therefore my hope that this text will be intelligible not only to mathematicians.1 1. PRECURSORS OF THE POLISH SCHOOL OF MATHEMATICS It was the years 1918—1920 when the Polish School of Mathematics was emerging. Before describing this period and the subsequent years one should, I think, review, be it only summarily, the contemporary state of Polish mathematics. I am going to mention those of its representatives the majority of whom had in fact been active in the 19th century but who also worked in the 20th century and so could influence the formation of the School of Mathematics being thus its precursors as it were. -
Contribution of Warsaw Logicians to Computational Logic
axioms Article Contribution of Warsaw Logicians to Computational Logic Damian Niwi ´nski Institute of Informatics, University of Warsaw, 02-097 Warsaw, Poland; [email protected]; Tel.: +48-22-554-4460 Academic Editor: Urszula Wybraniec-Skardowska Received: 22 April 2016; Accepted: 31 May 2016; Published: 3 June 2016 Abstract: The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal m-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the m-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations.