Springer Undergraduate Texts in Philosophy Harrie de Swart Philosophical and Mathematical Logic Springer Undergraduate Texts in Philosophy The Springer Undergraduate Texts in Philosophy offers a series of self-contained textbooksaimedtowardstheundergraduatelevelthatcoversallareasofphilosophy rangingfromclassicalphilosophytocontemporarytopicsinthefield.Thetextswill includeteachingaids(suchasexercisesandsummaries)andwillbeaimedmainly towardsmoreadvancedundergraduatestudentsofphilosophy. Theseriespublishes: • Allofthephilosophicaltraditions • Introduction books with a focus on including introduction books for specific topicssuchaslogic,epistemology,Germanphilosophyetc. • Interdisciplinaryintroductions–wherephilosophyoverlapswithotherscientific orpracticalareas Thisseriescoverstextbooksforallundergraduatelevelsinphilosophyparticu- larlythoseinterestedinintroductionstospecificphilosophytopics. We aim to make a first decision within 1 month of submission. 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For inquiries and submissions of proposals, authors can contact Ties.Nijssen@ Springer.com More information about this series at http://www.springer.com/series/13798 Harrie de Swart Philosophical and Mathematical Logic Harrie de Swart Faculty of Philosophy Erasmus University Rotterdam Rotterdam, The Netherlands Department of Philosophy Tilburg University Tilburg, The Netherlands ISSN 2569-8737 ISSN 2569-8753 (electronic) Springer Undergraduate Texts in Philosophy ISBN 978-3-030-03253-1 ISBN 978-3-030-03255-5 (eBook) https://doi.org/10.1007/978-3-030-03255-5 Library of Congress Control Number: 2018960381 © Springer Nature Switzerland AG 2018 This work is subject to copyright. 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[AndrzejGrzegorczyk] However,thissame[mathematical]formofthinking,thissamekindofconceptanal- ysis,isalsoapplicabletomanyotherareasthataredirectlyrelatedtotheimmediate realityofourdailylives.Andsuchabroaderapplicationofthemathematicalform ofthoughtseemstometobeofthehighestimportance.Afterall,theunparalleled development of the technique in a narrow sense, of the technical technique, one couldsay,isfollowedbyahardlylessimportantdevelopmentofthepsychological technique, of the advertising technique. propaganda technique, in short, of means toinfluencepeople.However,wehavefailedtostrengthenourdefenseequipment againstbeliefandsuggestionattemptsbyothersbyimprovingourthinkingtechnol- ogy.[...]Inthistangleofquestionsandshamquestionswecanfindaguideinthe conceptualanalysis,demonstratedinthemathematicalwayofthinking.Againstall these known and unknown psychic influences we can forge a weapon by improv- ingourthinkingtechnique.Andthatsuchareinforcementofourspiritisrequired, urgently needed, is my deepest conviction. [David van Dantzig, 1938, inaugural lecture,Delft,theNetherlands;translatedfromDutch] Thisbookisdedicated toJohanJ. deIongh (1915-1999) Myfriendandteacher viii It is the main task of a philosopher to show people that things do not have to be thewaytheyare,thattheymightbedifferentandthatinsomecasestheyshouldbe different.[JohandeIongh] Johan de Iongh (1915 - 1999) was a student of L.E.J. Brouwer (1881 - 1966), the foundingfatherofintuitionism.Hewasconvincedofthesoundnessoftheintuition- istic view of mathematics. He also had a great affinity with the signific position, representedbyGerritMannoury(1867-1956). He became professor in Nijmegen in 1961, where he was teaching the course on analysis for first-year students. Later de Iongh devoted most of his teaching to coursesonlogic,thefoundationsandthephilosophyofmathematics,andinparticu- larintuitionisticmathematics.Hewasverycarefulingivinganaccuratepresentation of Brouwer’s views. He took a great interest in the well-being of his students and founditimportanttoknowthempersonally. JohandeIonghwasasmuchaphilosopherasamathematician.HesharedPlato’s viewthatthestudyofmathematicsisthecorrectintroductiontophilosophy.Hehas published very little. His Platonic distrust towards the written word was great; his tendency to share his thoughts and ideas with friends, rather than to write them down,muchgreater.Yetsometextsfromhimhavebeenpreserved,andmanyofhis ideashavebeenworkedoutinPh.D.thesesandpapersbyhisstudents. His broad scholarship was impressive. He read Greek and Latin authors in the original.Hisinterestinsciencereachedfarbeyondmathematicsandhewaswidely readinworldliterature. He was a convinced Catholic and his thinking on mathematics and philosophy has developed in continuing discussion with St Augustine, St. Thomas Aquinas, St.ThomasMoreandNicholasofCusa.Healwaysstartedhislectureswithashort prayerinLatin:Spiritussanctigratiailluminetsensusetcordanostra[Maythegrace of the Holy Spirit illuminate our senses and our hearts]. And he always finished his lectures with the following prayer: Gratias tibi agimus, Domine, pro omnibus beneficiistuis[Wethankyou,myLord,forallyourblessings]. Itwasaprivilegetobehisstudent,hisPhDstudent,hisassistantandhisfriend. Foreword The following quotation is from Lewis Carroll, Symbolic Logic and The Game of Logic;Introduction. Thelearner,whowishestotrythequestionfairly,whetherthislittlebookdoes,ordoes not,supplythematerialsforamostinterestingrecreation,isearnestlyadvisedtoadoptthe followingRules: (1)Beginatthebeginning,anddonotallowyourselftogratifyamereidlecuriosityby dippingintothebook,hereandthere.Thiswouldverylikelyleadtoyourthrowingitaside, withtheremark‘Thisismuchtoohardforme!’,andthuslosingthechanceofaddinga verylargeitemtoyourstockofmentaldelights....Youwillfindthelatterparthopelessly unintelligible,ifyoureaditbeforereachingitinregularcourse. (2) Don’t begin any fresh Chapter, or Section, until you are certain that you thoroughly understandthewholebookuptothatpoint,andthatyouhaveworked,correctly,mostif notalloftheexampleswhichhavebeenset.Solongasyouareconsciousthatalltheland you have passed through is absolutely conquered, and that you are leaving no unsolved difficultiesbehindyou,whichwillbesuretoturnupagainlateron,yourtriumphalprogress willbeeasyanddelightful.Otherwise,youwillfindyourstateofpuzzlementgetworseand worseasyouproceed,tillyougiveupthewholethinginutterdisgust. (3)Whenyoucometoanypassageyoudon’tunderstand,readitagain:ifyoustilldon’t understandit,readitagain:ifyoufail,evenafterthreereadings,verylikelyyourbrainis gettingalittletired.Inthatcase,putthebookaway,andtaketootheroccupations,andnext day,whenyoucometoitfresh,youwillverylikelyfindthatitisquiteeasy. (4)Ifpossible,findsomegenialfriend,whowillreadthebookalongwithyou,andwilltalk overthedifficultieswithyou.Talkingisawonderfulsmoother-overofdifficulties.WhenI comeuponanything-inLogicorinanyotherhardsubject-thatentirelypuzzlesme,Ifind itacapitalplantotalkitover,aloud,evenwhenIamallalone.Onecanexplainthingsso clearlytoone’sself!Andthen,youknow,oneissopatientwithone’sself:onenevergets irritatedatone’sownstupidity! If,dearReader,youwillfaithfullyobservetheseRules,andsogivemylittlebookareally fairtrial,Ipromiseyou,mostconfidently,thatyouwillfindSymbolicLogictobeoneof themost,ifnotthemost,fascinatingofmentalrecreations! ... Mentalrecreationisathingthatweallofusneedforourmentalhealth;andyoumayget muchhealthyenjoyment,nodoubt,fromGames,suchasBack-gammon,Chess,andthe newGame‘Halma’.Butafterall,whenyouhavemadeyourselfafirst-rateplayeratany ix x Foreword oneoftheseGames,youhavenothingrealtoshowforit,asaresult!YouenjoyedtheGame, andthevictory,nodoubt,atthetime;butyouhavenoresultthatyoucantreasureupand getrealgoodoutof.And,allthewhile,youhavebeenleavingunexploredaperfectmineof wealth.OncemasterthemachineryofSymbolicLogic,andyouhaveamentaloccupation alwaysathand,ofabsorbinginterest,andonethatwillbeofrealusetoyouinanysubject youmaytakeup.Itwillgiveyouclearnessofthought-theabilitytoseeyourwaythrough apuzzle-thehabitofarrangingyourideasinanorderlyandget-at-ableform-and,more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments,whichyouwillsocontinuallyencounterinbooks,innewspapers,inspeeches, andeveninsermons,andwhichsoeasilydeludethosewhohavenevertakenthetroubleto masterthisfascinatingArt.Tryit.ThatisallIaskofyou! [FromLewisCarroll,SymbolicLogicandTheGameofLogic.Introduction;Dover Publications,Mineola,NY,1958.]