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E.W. Beth Als Logicus E.W. Beth als logicus ILLC Dissertation Series 2000-04 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ E.W. Beth als logicus Verbeterde electronische versie (2001) van: Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof.dr. J.J.M. Franse ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 26 september 2000, te 10.00 uur door Paul van Ulsen geboren te Diemen Promotores: prof.dr. A.S. Troelstra prof.dr. J.F.A.K. van Benthem Faculteit Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Copyright c 2000 by P. van Ulsen Printed and bound by Print Partners Ipskamp. ISBN: 90{5776{052{5 Ter nagedachtenis aan mijn vader v Inhoudsopgave Dankwoord xi 1 Inleiding 1 2 Levensloop 11 2.1 Beginperiode . 12 2.1.1 Leerjaren . 12 2.1.2 Rijpingsproces . 14 2.2 Universitaire carriere . 21 2.2.1 Benoeming . 21 2.2.2 Geleerde genootschappen . 23 2.2.3 Redacteurschappen . 31 2.2.4 Beth naar Berkeley . 32 2.3 Beth op het hoogtepunt van zijn werk . 33 2.3.1 Instituut voor Grondslagenonderzoek . 33 2.3.2 Oprichting van de Centrale Interfaculteit . 37 2.3.3 Logici en historici aan Beths leiband . 39 2.3.4 Beth naar Johns Hopkins . 41 2.3.5 Huiselijke aangelegenheden . 43 2.4 Lange noten . 45 3 Methodologie en ¯loso¯e 49 3.1 Filoso¯e en wetenschap . 49 3.1.1 Complementaire kengebieden . 50 3.1.2 Afwijkingen van het rechte pad . 55 3.1.3 Signi¯ca en taal¯loso¯e . 58 3.2 Logica en methodologie . 61 3.2.1 Logica en wiskunde . 61 3.2.2 Logica en andere wetenschappen . 68 3.3 Lange noten . 78 vii 4 Semantiek 81 4.1 Semantiek en algebra . 81 4.1.1 Achtergronden . 81 4.1.2 Ups en downs . 84 4.2 Afwijkende valuaties en hun afgeleiden . 93 4.2.1 Gereduceerde logica . 94 4.2.2 Pseudovaluaties . 102 4.3 Lange noten . 111 5 De¯nitietheorie 115 5.1 Beths de¯nitiestelling . 115 5.1.1 Beths globale omschrijving . 115 5.1.2 Begrippen . 117 5.1.3 Geschiedenis van de de¯nitietheorie . 122 5.2 Beths bijdragen . 126 5.2.1 Schets van het bewijs . 126 5.3 Directe reacties . 136 5.3.1 De¯nitiestelling: syntax of semantiek? . 136 5.3.2 Interpolatie . 139 5.3.3 Consistentiestelling . 147 5.4 Beths latere werk . 152 5.4.1 Modellen en de¯nieerbaaarheid . 152 5.4.2 De¯nitietheorie vervolgd . 158 6 Semantische tableaus 163 6.1 De¯nitie van semantische tableaus . 163 6.1.1 Inleiding . 164 6.1.2 Het begin bij Beth . 166 6.1.3 Oorsprong van Beths tableaus . 171 6.2 Achtergronden . 175 6.2.1 Tableausequenten . 175 6.2.2 Beths eisen . 176 6.2.3 Resultaten . 177 6.3 Prioriteitskwesties . 183 6.3.1 Verwante systemen . 183 6.3.2 Beth versus Hintikka . 183 6.3.3 Hintikka's modelverzamelingen . 185 6.3.4 Intuijtionisme . 187 7 De logische machine 191 7.1 Tableaus en bewijsmachines . 191 7.1.1 Beths kennismaking met mechanisch bewijzen . 191 7.1.2 Tableaus en Gentzens methoden . 194 7.2 Euratom-project . 201 7.2.1 Bestuurlijke achtergronden . 201 7.2.2 Loop van het onderzoek . 205 viii 7.3 Lange noten . 213 8 Deductieve tableaus 215 8.1 De¯nities . 215 8.1.1 Overwegingen vooraf . 215 8.1.2 Van reductie naar deductie . 220 8.2 Deductieve tableaus en logische systemen . 226 8.2.1 Klassieke deductieve tableaus . 226 8.2.2 Intuijtionistische deductieve tableaus . 227 8.2.3 Achtergronden . 230 8.3 Dialoogtableaus . 234 8.3.1 Overeenkomsten en verschillen . 234 8.3.2 Bedoeling van de dialoogtableaus . 235 9 Implicatieve systemen 241 9.1 Implicatieve systemen en aanverwanten . 241 9.1.1 Zuiver implicatief: intuijtionistisch en klassiek fragment . 242 9.1.2 Toevoeging van andere operatoren . 244 9.2 Kripke's hulpvaluaties . 250 9.2.1 Beths hulptableaus . 252 9.2.2 Derivatieve implicatieve logica . 254 9.2.3 Modale systemen . 262 10 Beth-modellen 267 10.1 De basis van de Beth-modellen . 267 10.1.1 Spreiding en tegenmodel . 267 10.1.2 Syntax en semantiek . 281 10.2 Constructie van Beth-modellen . 284 10.2.1 Boomconstructies . 284 10.2.2 De¯nities voor modelconstructies . 286 10.3 Volledigheid . 298 10.3.1 Aanloop tot volledigheid . 298 10.3.2 Volledigheidsstellingen . 302 10.3.3 Beth-modellen en topologie . 308 11 Supplementen 311 11.1 Sequenten en tableaus. 311 11.1.1 Beths sequenten . 311 11.1.2 Overzicht tableaus . 312 11.2 Varia Beth . 314 11.2.1 Leven E.W. Beth . 314 11.2.2 Bronnen . 317 Afkortingen 319 Abstract 329 ix Bibliogra¯e 331 x Dankwoord Er zijn tal van personen en instellingen die aan het tot stand komen van deze dissertatie hebben bijgedragen. In de eerste instantie zijn dit de beide promotoren, de professoren A.S. Troel- stra en J.F.A.K. van Benthem. Niet alleen tijdens het leveren van commentaar op het aangedragen werk, maar ook in de dagelijkse omgang op het instituut heb ik het genoegen mogen smaken in hun nabijheid te hebben kunnen verkeren. Eveneens was dit het geval met de andere leden van de vakgroep. Naast de beide promotoren valt naar de moeite die de leden van promotie- commissie zich getroost hebben te refereren. Deze commissie bestond uit doc- tor D.H.J. de Jongh (Amsterdam), professor A.J. Kox (Amsterdam), professor A. Visser (Utrecht) en professor H. Visser (Maastricht). Van de mensen met wie ik in de loop der tijd de te behandelen stof heb kun- nen bespreken vallen met name W.J. Blok, H.C. Doets, A. Hendriks, D.H.J. de Jongh, E.C.W. Krabbe en M. van Lambalgen te noemen. Van de leden van het onderzoeksinstituut ILLC kunnen in het bijzonder enkelen van de afdeling theoretische informatica, P. van Emde Boas, Th. Janssen en L. Torenvliet, genoemd worden. Al in een vroeg stadium is door hun bemid- deling de mogelijkheid tot electronische tekstverwerking en werkruimte verkre- gen. De laatste was des te meer van belang, daar deze boven de bibliotheek van het Mathematisch Instituut gelegen was. Aangezien ik telkens weer historische teksten heb moeten raadplegen is dit voor mij van groot belang geweest. De medewerking van met name de bibliothecaris F. Kroon heeft het nodige daartoe bijgedragen. Het raadplegen van archieven was voor de tot standkoming van deze dis- sertatie essentieel. H. Visser en A.S. Troelstra zijn als beheerders van respec- tievelijk het Beth Archief en het Heyting Archief zo bereidwillig geweest deze archieven tot dit doel open te stellen. Tenslotte hebben uit andere hoofde E. Gaskill, F. van der Kolk, E.J. van der Linden, M. Pauly, M. Vervoort, G. Winkel en de behulpzame mensen van de Computer Netwerk Groep het nodige gedaan om het voltooien van deze dissertatie tot een goed einde te brengen. xi Amsterdam, augustus 2000 Paul van Ulsen Voor het verbeteren van de electronische versie in 2001 is mij de ruimte gegeven door de bibliotheek van de Faculteit Natuurwetenschappen, Wiskunde en Informatica; met name kan hier H. Harmsen genoemd worden. De verbeteringem betre®en vooral tekstuele. Schrappen van overbodige zin- nen of delen daarvan of stukjes tekst met herhaling van al eerder gegeven of overbodige informatie. Verder is er gekeken naar het beter laten verlopen van overgangen: hierdoor zijn enkele paragrafen omgewisseld. Onduidelijkheden zijn verbeterd, evenals drie fouten (schrappen van Beth bij een bespreking bij von Muralt (p. 29); een verticale kolomstreep links i.p.v. rechts t.o.v. de for- mules A(x); A(p) (p. 225, regel 7b); een 0 i.p.v. een 1 in een valuatiekolom op p. 259 9(v op ((A B) A) A). 0 ! ! ! Amsterdam, 2001 P. van Ulsen xii Hoofdstuk 1 Inleiding \There is a certain tendency to belittle the value of my work" 1 Evert Willem Beth, wiskundige, ¯losoof en organisator. Evert Willem Beth, 1908 { 1964, was hoogleraar in de logica en haar geschiedenis en de ¯loso¯e van de exacte wetenschappen aan de Universiteit van Amsterdam, vanaf 1946 tot aan zijn dood. Beth opereerde op een moeilijk grensvlak van disciplines. Hij werd door de ¯losofen als een wiskundige en logicus afgeschreven, door de wiskundigen en logici, evenmin positief bedoeld, voor een ¯losoof gehouden. Dit had inder- tijd, en ook nu nog, te maken met de onbekendheid van zijn vakgebied en de miskenning van het belang van zijn verdere bezigheden. Niettemin kwam de combinatie van logica, grondslagenonderzoek van de wiskunde, didactiek van de wiskunde, geschiedenis van de wetenschaps¯loso¯e (in de meest ruime zin), ¯loso¯e, wetenschaps¯loso¯e en organisatorisch vermogen in meer of mindere mate ook bij anderen voor. Die anderen waren, in Nederland in de tijd van Beth, niet de minsten. Men telde hieronder L.E.J. Brouwer, J. Clay, G. Man- noury, D. van Dantzig en A. Heyting.2 Volledig is de omschrijving op Beth van toepassing. Beth behoorde niet tot de wereldtop van de logici in die dagen. Hij speelde in Nederland wel een grote rol, maar men kan daar tegen inbrengen, dat dit nogal voor de hand lag in een land, arm aan dienaren van de logica en ¯loso¯e van de wiskunde.
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