Logic and structure

266 Pages·2013·1.41 MB·English

by  van Dalen, Dirk

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Dirk van Dalen
Logic and Structure
Fifth Edition
DirkvanDalen
DepartmentofPhilosophy
UtrechtUniversity
Utrecht,TheNetherlands
BasedonapreviouseditionoftheWork:
LogicandStructure,4theditionbyDirkvanDalen
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ISBN978-1-4471-4557-8 ISBN978-1-4471-4558-5(eBook)
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Preface
Logicappearsina“sacred”andina“profane”form;thesacredformisdominantin
prooftheory,theprofaneforminmodeltheory.Thephenomenonisnotunfamiliar,
one also observes this dichotomy in other areas, e.g. set theory and recursion the-
ory.Someearlycatastrophes,suchasthediscoveryofthesettheoreticalparadoxes
(Cantor, Russell), or the definability paradoxes (Richard, Berry), make us treat a
subject for some time with the utmost awe and diffidence. Sooner or later, how-
ever, people start to treat the matter in a more free and easy way. Being raised in
the“sacred”tradition,myfirstencounterwiththeprofanetraditionwassomething
likeacultureshock.HartleyRogersintroducedmetoamorerelaxedworldoflogic
byhisexampleofteachingrecursiontheorytomathematiciansasifitwerejustan
ordinary course in, say, linear algebra or algebraic topology. In the course of time
I have come to accept this viewpoint as the didactically sound one: before going
intoesotericnicetiesoneshoulddevelopacertainfeelingforthesubjectandobtain
areasonableamountofplainworkingknowledge.Forthisreasonthisintroductory
textsetsoutintheprofaneveinandtendstowardsthesacredonlyattheend.
ThepresentbookhasdevelopedfromcoursesgivenattheMathematicsDepart-
ment of Utrecht University. The experience drawn from these courses and the re-
actionoftheparticipantssuggestedstronglythatoneshouldnotpracticeandteach
logicinisolation.Assoonaspossibleexamplesfromeverydaymathematicsshould
beintroduced;indeed,first-orderlogicfindsarichfieldofapplicationsinthestudy
ofgroups,rings,partiallyorderedsets,etc.
The role of logic in mathematics and computer science is twofold—a tool for
applications in both areas, and a technique for laying the foundations. The latter
rolewillbeneglectedhere,wewillconcentrateonthedailymattersofformalized
(orformalizable)science.Indeed,Ihaveoptedforapracticalapproach—Iwillcover
thebasicsofprooftechniquesandsemantics,andthengoontotopicsthatareless
abstract.ExperiencehastaughtusthatthenaturaldeductiontechniqueofGentzen
lends itself best to an introduction;it is close enough to actual informal reasoning
to enable students to devise proofs by themselves. Hardly any artificial tricks are
involved,andattheendthereisthepleasingdiscoverythatthesystemhasstriking
structuralproperties;inparticularitperfectlysuitstheconstructiveinterpretationof
v
vi Preface
logicanditallowsnormalforms.Thelattertopichasbeenaddedtothiseditionin
viewofitsimportanceintheoreticalcomputerscience.InChap.4wealreadyhave
enoughtechnicalpowertoobtainsomeofthetraditionaland(eventoday)surprising
modeltheoreticresults.
Thebookiswrittenforbeginnerswithoutknowledgeofmoreadvancedtopics;
noesotericsettheoryorrecursiontheoryisrequired.Thebasicingredientsarenatu-
raldeductionandsemantics,andthelatterispresentedinconstructiveandclassical
form.
InChap.6intuitionisticlogicistreatedonthebasisofnaturaldeductionwithout
theruleofreductioadabsurdum,andofKripkesemantics.Intuitionisticlogichas
graduallyfreeditselffromtheimageofeccentricityandnowitisrecognizedforits
usefulnessine.g.topostheoryandtypetheory;henceitsinclusioninanintroduc-
torytextisfullyjustified.Thechapteronnormalizationhasbeenaddedforthesame
reasons;normalizationplaysanimportantroleincertainpartsofcomputerscience.
Traditionallynormalization(andcutelimination)belongtoprooftheory,butgrad-
uallyapplicationsinotherareashavebeenintroduced.InChap.7weconsideronly
weaknormalization,andanumberofeasyapplicationsaregiven.
Variouspeoplehavecontributedtotheshapingofthetextatonetimeoranother;
DanaScott,JaneBridge,HenkBarendregtandJeffZuckerhavebeenmosthelpful
for the preparation of the first edition. Since then many colleagues and students
havespottedmistakesandsuggestedimprovements;thiseditionbenefitedfromthe
remarks of Eleanor McDonnell, A. Scedrov and Karst Koymans. To all of these
criticsandadvisersIamgrateful.Progresshasdictatedthatthetraditionaltypewriter
shouldbereplacedbymoremoderndevices;thisbookhasbeenredoneinLATEXby
Addie Dekker and my wife, Doke. Addie led the way with the first three sections
of Chap. 2 and Doke finished the rest of the manuscript; I am indebted to both of
them,especiallyto Dokewhofoundtimeand courageto masterthe secretsofthe
LATEX trade. Thanks go to Leen Kievit for putting together the derivations and for
adding the finer touches required for a LATEX manuscript. Paul Taylor’s macro for
prooftreeshasbeenusedforthenaturaldeductionderivations.
June1994 DirkvanDalen
The conversion to TEX has introduced a number of typos that are corrected in
thepresentnewprinting.Manyreadershavebeenkindenoughtosendmetheircol-
lection of misprints, and I am grateful to them for their help. In particular I want
to thank Jan Smith, Vincenzo Scianna, A. Ursini, Mohammad Ardeshir and Nori-
hiroKamide.HereinUtrechtmylogicclasseshavebeenveryhelpful;inparticular
MarkoHollenberg,whotaughtpartofacourse,hasprovidedmewithusefulcom-
ments.Thanksgotothemtoo.
I have used the occasion to incorporate a few improvements. The definition of
“subformula”hasbeenstreamlined—togetherwiththenotionofpositiveandneg-
ative occurrence. There is also a small addendum on “induction on the rank of a
formula”.
January1997 DirkvanDalen
Preface vii
AttherequestofusersIhaveaddedachapterontheincompletenessofarithmetic.It
makesthebookmoreself-contained,andaddsusefulinformationonbasicrecursion
theoryandarithmetic.Thecodingofformalarithmeticmakesuseoftheexponen-
tial;thisisnotthemostefficientcoding,butfortheheartoftheargumentthatisnot
oftheutmostimportance.Inordertoavoidextraworktheformalsystemofarith-
meticcontainstheexponential.Astheprooftechniqueofthebookisthatofnatural
deduction,thecodingofthenotionofderivabilityisalsobasedonit.Thereareof
coursemanyotherapproaches.Thereaderisencouragedtoconsulttheliterature.
Thematerialof thischapteris byandlargethatof a coursegivenin Utrechtin
1993. Students have been most helpful in commenting on the presentation, and in
preparingTEXversions.W.Deanhaskindlypointedoutsomemorecorrectionsin
theoldtext.
The final text has benefited from the comments and criticism of a number of
colleagues and students. I am grateful for the advice of Lev Beklemishev, John
Kuiper, Craig Smoryn´ski and Albert Visser. Thanks are due to Xander Schrijen,
whosevaluableassistancehelpedtoovercometheTEXproblems.
May2003 DirkvanDalen
A number of corrections have been provided by Tony Hurkens; furthermore, I am
indebtedtohimandHaroldHodesforpointingoutthatthedefinitionof“freefor”
wasinneedofimprovement.SjoerdZwartfoundanastytypothathadescapedme
andall(ormost)readers.
April2008 DirkvanDalen
Tothefiftheditionanewsectiononultraproductshasbeenadded.Thetopichasa
longhistoryanditpresentsanelegantandinstructiveapproachtotheroleofmodels
inlogic.
Again I have received comments and suggestions from readers. It is a pleasure
tothankDiegoBarreiro,VictorKrivtsov,EinamLivnat,ThomasOpfer,Masahiko
Rokuyama,KatsuhikoSano,PatrickSkevikandIskenderTasdelen.
2012 DirkvanDalen
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 PropositionalLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 PropositionsandConnectives . . . . . . . . . . . . . . . . . . . . 5
2.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 SomePropertiesofPropositionalLogic . . . . . . . . . . . . . . . 20
2.4 NaturalDeduction . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 TheMissingConnectives . . . . . . . . . . . . . . . . . . . . . . 46
3 PredicateLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 TheLanguageofaSimilarityType . . . . . . . . . . . . . . . . . 56
3.4 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 SimplePropertiesofPredicateLogic . . . . . . . . . . . . . . . . 68
3.6 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 NaturalDeduction . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.9 AddingtheExistentialQuantifier . . . . . . . . . . . . . . . . . . 91
3.10 NaturalDeductionandIdentity . . . . . . . . . . . . . . . . . . . 93
4 CompletenessandApplications . . . . . . . . . . . . . . . . . . . . . 97
4.1 TheCompletenessTheorem . . . . . . . . . . . . . . . . . . . . . 97
4.2 CompactnessandSkolem–Löwenheim . . . . . . . . . . . . . . . 104
4.3 SomeModelTheory . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 SkolemFunctionsorHowtoEnrichYourLanguage . . . . . . . . 127
4.5 Ultraproducts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Second-OrderLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 IntuitionisticLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.1 ConstructiveReasoning . . . . . . . . . . . . . . . . . . . . . . . 155
ix
x Contents
6.2 IntuitionisticPropositionalandPredicateLogic . . . . . . . . . . . 157
6.3 KripkeSemantics . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4 SomeModelTheory . . . . . . . . . . . . . . . . . . . . . . . . . 174
7 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.1 Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 NormalizationforClassicalLogic . . . . . . . . . . . . . . . . . . 191
7.3 NormalizationforIntuitionisticLogic . . . . . . . . . . . . . . . . 197
7.4 AdditionalRemarks: StrongNormalizationand the Church–
RosserProperty . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8 Gödel’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1 PrimitiveRecursiveFunctions . . . . . . . . . . . . . . . . . . . . 209
8.2 PartialRecursiveFunctions . . . . . . . . . . . . . . . . . . . . . 218
8.3 RecursivelyEnumerableSets . . . . . . . . . . . . . . . . . . . . 229
8.4 SomeArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.5 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.6 Derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.7 Incompleteness. . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Chapter 1
Introduction
Withoutadoptingoneofthevariousviewsadvocatedinthefoundationsofmathe-
matics,wemayagreethatmathematiciansneedandusealanguage,ifonlyforthe
communicationoftheirresultsandtheirproblems.Whilemathematicianshavebeen
claimingthegreatestpossibleexactnessfortheirmethods,theyhavebeenlesssen-
sitive as to their means of communication.It is well knownthat Leibniz proposed
toputthepracticeofmathematicalcommunicationandmathematicalreasoningon
afirmbase;itwas,however,notbeforethenineteenthcenturythatthoseenterprises
were (more) successfully undertaken by G. Frege and G. Peano. No matter how
ingeniously and rigorously Frege, Russell, Hilbert, Bernays and others developed
mathematicallogic,itwasonlyinthesecondhalfofthiscenturythatlogicandits
languageshowedanyfeaturesofinteresttothegeneralmathematician.Thesophis-
ticatedresultsofGödelwereofcourseimmediatelyappreciated,butforalongtime
theyremainedtechnicalhighlightswithoutpracticaluse.EvenTarski’sresultonthe
decidability of elementary algebra and geometry had to bide its time before any
applicationsturnedup.
Nowadays the applications of logic to algebra, analysis, topology, etc. are nu-
merous and well recognized. It seems strange that quite a number of simple facts,
withinthegraspofanystudent,wereoverlookedforsuchalongtime.Itisnotpos-
sible to give proper credit to all those who opened up this new territory. Any list
would inevitably show the preferences of the author, and neglect some fields and
persons.
Letusnotethatmathematicshasafairlyregular,canonicalwayofformulating
its material, partly by its nature, partly under the influence of strong schools, like
the one of Bourbaki. Furthermore the crisis at the beginning of this century has
forcedmathematicianstopayattentiontothefinerdetailsoftheirlanguageandto
theirassumptionsconcerningthenatureandtheextentofthemathematicaluniverse.
Thisattentionstartedtopayoffwhenitwasdiscoveredthattherewasinsomecases
acloseconnectionbetweenclassesofmathematicalstructuresandtheirsyntactical
description.Hereisanexample:
ItiswellknownthatasubsetofagroupGwhichisclosedundermultiplication
andinverse,isagroup;however,asubsetofanalgebraicallyclosedfieldF which
D.vanDalen,LogicandStructure,Universitext,DOI10.1007/978-1-4471-4558-5_1, 1
©Springer-VerlagLondon2013