Trends in Logic 37 Rafal Urbaniak Leśniewski's Systems of Logic and Foundations of Mathematics Trends in Logic Volume 37 Studia Logica Library For furthervolumes: http://www.springer.com/series/6645 TRENDS IN LOGIC Studia Logica Library Volume 37 Managing Editor Heinrich Wansing, Dresden University of Technology, Germany Editors Wieslaw Dziobiak, University of Puerto Rico at Mayagez, USA Melvin Fitting, City University of New York, USA Vincent F. Hendricks, Roskilde University, Denmark Daniele Mundici, University of Florence, Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Uppsala University, Sweden Ryszard Wójcicki, Polish Academy of Sciences, Warsaw, Poland SCOPE OF THE SERIES TrendsinLogicisabookseriescoveringessentiallythesameareaasthejournal Studia Logica—that is, contemporary formal logic and its applications and rela- tions to other disciplines. These include artificial intelligence, informatics, cog- nitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time. Volume Editor Heinrich Wansing Rafal Urbaniak Les´niewski’s Systems of Logic and Foundations of Mathematics 123 Rafal Urbaniak CentreforLogicandPhilosophyofScience Ghent University Ghent Belgium and Department of Philosophy, Sociologyand Journalism Gdan´skUniversity Gdan´sk Poland ISSN 1572-6126 ISSN 2212-7313 (electronic) ISBN 978-3-319-00481-5 ISBN 978-3-319-00482-2 (eBook) DOI 10.1007/978-3-319-00482-2 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013940092 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To my teachers Preface TheLvov-WarsawSchool ofLogic andAnalyticPhilosophy wasone ofthe most importantschoolsofphilosophicalthoughtintwentiethcentury.Inearly1910sits members already discussed the validity of the principles of excluded middle and contradiction. Among ideas developed in this school, one might count Łukasiewicz’sviewthatonecanbelieveacontradictionandthatcertainsentences canbeneithertruenorfalse.Thisledtotheconstructionofhisthree-valuedlogic. AnotherexampleisAjdukiewicz’sconventionalismaboutmeaningandhisformal work on definitions (it seems that it was Ajdukiewicz and Łukasiewicz who first focused on the consistency, translatability, and non-creativity conditions on defi- nitions, at least on the Polish ground). Other examples include Jas´kowski’s approach to natural deduction and his work on discussive logics, Lindenbaum’s lemmaonmaximallyconsistentsetsofformulas,Presburger’sworkonarithmetic, Kotarbin´ski’s semantical reism, andTarski’s work onformal semantics andtruth. One of the representatives of this school was Stanisław Les´niewski (1886–1939)(AlfredTarski,whoseimportanceintwentiethcenturylogicitishard to overestimate, was his only Ph.D. student). Les´niewskideveloped his system of foundations of mathematics as an alternative to the system of Principia Math- ematica. He constructed three systems: Protothetic, which is his version of a generalized propositional calculus, his own (higher-order) logic of predication called Ontology, and a theory of parthood called Mereology. Les´niewski’s work is interesting for a few reasons. • Ifoneisinterestedinhistoryoflogicingeneral,itishardtodenythatLes´niewski wasoneofthekeyfiguresinoneofthemostimportantschoolsoflogicintwentieth century. He devoted his research to developing an alternative to the system of PrincipiaMathematicaandthisattemptisworthstudyinginhisownright. • IfoneisinterestedinthedevelopmentofTarski’sthoughtitmightbeusefulto learn what his Ph.D. supervisor’s views were and how Les´niewski’s work and Tarski’s ideas are (or are not) related. • Philosophical discussions in which Les´niewski participated pertained to issues which are discussed quite lively even today. His approach to semantical and set-theoreticparadoxesandhisviewsonthevalidityoftheprincipleofexcluded middle and of the principle of contradiction are philosophically interesting. vii viii Preface • Les´niewski was a nominalist and his systems were a nominalistic attempt to provideasystemoffoundationsofmathematics.Itisamajorattemptofthissort and as such it is worth an examination. • His metalogic is quite specific. Nominalist as he was, he wanted to develop a purelyinscriptionalsyntacticdescriptionofhissystemsinawaythatdidnotmake anyreferencetoexpressiontypes.Itisinterestingtoseehowheproceeded. • Hissystemshavesomeinterestingproperties.Forinstance,inallofthemdefini- tionscanbecreative(andthisisnotconsideredtobeaproblem).Thegeneralityof Prothetic admits interesting extensions (intuitionistic (see López-Escobar and Miraglia2002)ormodal(seetheworksofSuszkoandingeneral,seeSect.3.7for references).ThelanguageofOntology(which,inaway,canbeviewedasoneof thefirstfreeformallogics)is,arguably,moresuitableforcapturingcertainaspects ofpredicationandabstractnounphrasesastheyworkinnaturallanguage. This book is devoted to a presentation of Les´niewski’s achievements and their critical evaluation. I discuss his philosophical views, describe his systems, and evaluatetheroletheycanplayinthefoundationsofmathematics.Itwasmypurpose tofocusonprimarysourcesandpresentLes´niewski’sownviewsandresultsrather thanthosepresentinsecondaryliterature.Forthisreason,laterdevelopmentsarenot treated in detail but rather either mentioned in passing, or described in sections devotedtosecondaryliteratureincludedinsomechapters.Theintendedaudienceof this book includes philosophy majors, graduate students, and professional philos- ophersinterestedinlogic,mathematics,andtheirphilosophyandhistory. PartsofthisbookstartedasmyPh.D.dissertationwrittenunderthesupervision of Richard Zach and defended in 2008 at the University of Calgary. Other parts report on research which went beyond the dissertation (in particular, Chap. 6 was written together with Severi K. Hämäri). Ultimately, in 2011 my wife took me to India, where she pursued her research in Indian philosophy and forced me to use those few months to write the whole book anew. Varanasi, India, March 2012 Rafal Urbaniak Department of Philosophy Sociology and Journalism Gdan´sk University Gdan´sk, Poland Centre for Logic and Philosophy of Science Ghent University Ghent, Belgium References López-Escobar, E. & Miraglia, F. (2002). Definitions: The primitive concept of logics or the Les´niewski-Tarskilegacy.DissertationesMathematicae401,PolskaAkademiaNauk,Instytut Matematyczny,Warszawa. Acknowledgments This book started as a Ph.D. thesis written under supervision of Richard Zach at theUniversityofCalgaryandIowehimgratitudeforhistime,effort,andpatience. I am also grateful for all the comments on my dissertation which I received from Nicole Wyatt, Jack MacIntosh, and John Kearns. I would also like to thank my other teachers. Words cannot express my grati- tudetomyfirstphilosophyteachers,Martyna,andRobertKoszkało,fromwhomI stole (well, borrowed and never returned) my first book on Les´niewski’s systems and without whose motivation I would probably end up being a helicopter pilot (and we all know how unexciting and underpaid that job would be). I owe grat- itude tomy M.A. supervisor, Jarosław Mrozek, who allowed me toteachmy first seminar on Les´niewski before I graduated and to my M.A. referee, Andrzej Włodzimierz Mostowski. As for Chap. 6, many thanks for their comments to John MacFarlane, Nuel Belnap, Wilfrid Hodges, Paolo Mancosu, Øystein Linnebo, and Jan von Plato. WorkonthispartwassupportedbytheSpecialResearchFundofGhentUniversity through project [BOF07/GOA/019]. Later parts were discussed at various places where I was giving talks: Ghent, Riga, Edinburgh, Szklarska Pore˛ba, Hejnice, Geneva, Frankfurt, Dublin, St. Andrews, Auckland, and Oxford, among others. I am grateful to all the audience membersfortheirfeedback.Theapproachtohigher-orderquantificationdiscussed in the last chapter was also partially shaped in discussion with Øystein Linnebo, who was my host at Bristol during my British Academy Visiting Fellowship, Hannes Leitgeb and Leon Horsten—other Bristol Philosophy Department mem- bers at the time of my stay in Bristol—and Stewart Shapiro, whom I ran into and tended to disagree with on various occasions. I am grateful to my students, Paweł Pawłowski and Paweł Siniło, who proof- readmajorpartsofthistextandhelpedmetoavoidatleastsomeofthemistakes. IwasalsoabletodiscussalatedraftofthisbookwithPeterSimonsduringmy stayinDublinasaLongRoomHubVisitingFellow,soIwouldalsoliketothank the Long Room Hub and Peter Simons for their support. ix x Acknowledgments I am deeply thankful to myparents, Halina, and Andrzej Urbaniak—the list of things I am grateful for would probably be longer than the book itself. Finally, I am grateful to my wife, Agnieszka Rostalska, for her patience in general, for causing my voluntary exile in India and for turning it into a writing spree.