Table Of ContentSpringer Undergraduate Texts in Philosophy
Andrea Iacona
LOGIC:
Lecture Notes
for Philosophy,
Mathematics, and
Computer Science
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Andrea Iacona
LOGIC: Lecture Notes for
Philosophy, Mathematics,
and Computer Science
AndreaIacona
CenterforLogic,LanguageandCognition
UniversityofTurin,Departmentof
PhilosophyandEducation
Torino,Italy
ISSN2569-8737 ISSN2569-8753 (electronic)
SpringerUndergraduateTextsinPhilosophy
ISBN978-3-030-64810-7 ISBN978-3-030-64811-4 (eBook)
https://doi.org/10.1007/978-3-030-64811-4
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Preface
Thisbookisanintroductiontologic.Itstartsfromzeroandcoversthemaintopics
that are usually taught in elementary and advanced logic courses. Its measured
pace, its controlled level of technicality, and its constant search for clarity and
precisionmakeitparticularlysuitedforphilosophystudents.Butalsomathematics
or computer science students may appreciate its style of exposition, especially if
theyhavesomeinclinationforpuretheoreticalissues.
The book is structured as follows. Chapters 1–3 provide some preliminary
clarifications and define the basic vocabulary of logic. Chapters 4–6 present a
propositionallanguageandexplainitssemantics.Chapters7and8setoutanatural
deductionsysteminthatlanguage.Chapters9and10outlineadeductivelyequiv-
alent axiomatic system and prove its consistency, soundness, and completeness.
Chapters 11–13 present a predicate language and explain its semantics. Chap-
ters14–15outlineanaxiomaticsysteminthatlanguageandproveitsconsistency,
soundness,andcompleteness.Chapters16–18dealwithsomegeneralresultsabout
first-order theories. Chapter 19 is devoted to Gödel’s incompleteness theorems.
Finally,Chapter20introducesthefundamentalsofmodallogic.
Since it is practically impossible to study this entire material within a single
course,someselectionofchaptersmustbemade,dependingonthekindofcourse
andthetimeavailable.Onewaytodividethebook,perhapsthesimplestway,isto
use Chapters 1–10 for an elementary course and Chapters 11–20 for an advanced
course. If instead the time available is not enough to cover ten chapters, which is
quite likely, one can use Chapters 1–8 for an elementary course and Chapters 9–
18 for an advanced course, leaving out Chapters 19 and 20. A third option, which
might be considered if one has only one course instead of two, and one wants to
includebothpropositionalandpredicatelogic,istouseChapters1–6and11–13.In
thiscase,onewillfocusonthesemanticsofpropositionalandpredicatelanguages,
leavingouttheprooftheoryandthemetatheory.
Each chapter includes ten exercises. Some of them are relatively simple and
mechanical,othersrequireafirmgraspofthecontentspresentedinthechapterand
abitofcreativity.Thereaderisnotexpectedtosolvealltheexercisesatfirstglance.
v
vi Preface
Butatleastitisworthtrying.Thesolutionsarelistedattheendofthebook,soone
canrelyonthemtomeasureone’sunderstandingandlearnfromone’smistakes.
I drew from the following sources, which would be good for supplemental
reading and are listed in the final bibliography: G. S. Boolos, J. P. Burgess, and
R. Jeffrey, Computability and Logic (2010); H. B. Enderton, A Mathematical
IntroductiontoLogic(1972);G.E.HughesandM.J.Cresswell,ANewIntroduction
to Modal Logic (1996); G. Hunter, Metalogic (1971); E. J. Lemmon, Beginning
Logic (1993); M. Sainsbury, Logical Forms (2001); and P. Smith, Introduction to
Gödel’s Theorems (2013). Further references on specific issues are mentioned in
thefootnotes.
ThisbookisbasedonasetoflecturenotesthatIhaveconsistentlyusedinmy
logic courses at the University of Turin, and that I have revised year after year.
Duetoinnumerablecorrections,adjustments,andmodifications,I’mnolongerina
positiontotellhowmanyversionsofthetextcirculatedamongstudents,andIdon’t
havealistofallthosewhofounderrorsorsuggestedimprovements.ButIknowfor
surethatIreceivedagreatdealofhelpfulcommentsfromGabrieleAbate,Martina
Calderisi, Daniel Crowley, Gabriele Chiriotti, Paolo Grugnetti, Martino Paschetto,
StefanoRomeo,GiulianoRosella,CristinaSagafrena,FrancescoScarpiello,Davide
Sutto,andMartinaZirattu.
Among the many intellectual debts that I have acquired in the course of
writing this book, some deserve special mention. Pasquale Frascolla has been an
inspiring presence ever since we started discussing about the logic exams of our
doctoralstudents.MatteoPlebaniandEricRaidlgavemeparticularlyextensiveand
extremelyhelpfulcomments,whichpromptednumeroussubstantialimprovements.
Finally,DiegoMarconirevisedthefinalversionofthemanuscriptandspottedsome
still remaining historical inaccuracies. I am very grateful to them all for the time
theyspentonmywork.
Torino,Italy AndreaIacona
Contents
1 BasicNotions................................................................ 1
1.1 WhatIsLogic? ....................................................... 1
1.2 ArgumentsandTheirFormulation .................................. 3
1.3 ComplexReasoning.................................................. 4
1.4 TruthandFalsity ..................................................... 6
1.5 Bivalence ............................................................. 8
2 Validity....................................................................... 11
2.1 SomeSet-TheoreticalNotions....................................... 11
2.2 TruePremises......................................................... 12
2.3 ValidityasNecessaryTruthPreservation ........................... 13
2.4 OtherLogicalPropertiesandRelations............................. 15
2.5 ImportantFactsAboutValidity...................................... 16
2.6 ValidityIsNotEverything........................................... 20
3 Formality .................................................................... 25
3.1 FormalValidity....................................................... 25
3.2 FormalInvalidity..................................................... 28
3.3 FormalLanguage..................................................... 29
3.4 FormalSystem........................................................ 30
3.5 ObjectLanguageandMetalanguage ................................ 31
3.6 FurtherSet-TheoreticalNotions..................................... 32
4 TheSymbolsofPropositionalLogic...................................... 35
4.1 SentenceLetters...................................................... 35
4.2 SententialConnectives............................................... 36
4.3 Brackets............................................................... 38
4.4 ExpressiveCompleteness............................................ 39
4.5 Truth-FunctionalityandSubstitutivity .............................. 41
4.6 FormalizationinaPropositionalLanguage......................... 41
5 TheLanguageL............................................................. 45
5.1 FormationRules...................................................... 45
vii
viii Contents
5.2 SyntacticTrees ....................................................... 46
5.3 Scope.................................................................. 47
5.4 Interpretation ......................................................... 48
5.5 TruthTables .......................................................... 49
6 LogicalConsequenceinL.................................................. 53
6.1 DefinitionofLogicalConsequence.................................. 53
6.2 OtherLogicalPropertiesandRelations............................. 54
6.3 ImportantFactsAboutLogicalConsequence....................... 55
6.4 LogicalConsequenceasaTestforValidity......................... 56
6.5 EffectiveComputability.............................................. 57
7 TheSystemG ............................................................... 61
7.1 Derivation............................................................. 61
7.2 Rulesfor∼ ........................................................... 62
7.3 Rulesfor⊃ ........................................................... 64
7.4 Rulesfor∧ ........................................................... 65
7.5 Rulesfor∨ ........................................................... 68
8 DerivabilityinG............................................................ 71
8.1 DerivabilityandRelatedNotions.................................... 71
8.2 ImportantFactsAboutDerivability ................................. 72
8.3 SomeTips............................................................. 73
8.4 DerivedRules......................................................... 75
8.5 OtherNaturalDeductionSystems................................... 76
9 TheSystemL................................................................ 79
9.1 AxiomsandInferenceRule.......................................... 79
9.2 DeductionTheorem.................................................. 81
9.3 Explosion,DoubleNegation,Contraposition....................... 83
9.4 SubstitutionofEquivalents .......................................... 85
9.5 ReductioAdAbsurdum.............................................. 87
−
9.6 DeductiveEquivalenceBetweenG andL......................... 88
9.7 SystemsandTheories................................................ 89
10 Consistency,Soundness,Completeness .................................. 91
10.1 ConsistencyofL ..................................................... 91
10.2 DefinitionsofSoundnessandCompleteness........................ 92
10.3 SoundnessofL....................................................... 93
10.4 CompletenessofL ................................................... 93
−
10.5 ExtensiontoG ...................................................... 96
11 Quantification............................................................... 99
11.1 QuantifiedSentences................................................. 99
11.2 ABriefHistoricalSurvey............................................ 101
11.3 ExistentialImport .................................................... 103
11.4 MultipleGenerality .................................................. 104
11.5 DefiniteDescriptions................................................. 106
Contents ix
12 TheSymbolsofPredicateLogic .......................................... 109
12.1 Non-logicalExpressions............................................. 109
12.2 LogicalConstantsandAuxiliarySymbols.......................... 110
12.3 OtherSymbols........................................................ 111
12.4 NumericalExpressions............................................... 113
12.5 MultipleGeneralityandScopeAmbiguity.......................... 114
12.6 Existence.............................................................. 115
13 TheLanguageL ........................................................... 119
q
13.1 Syntax................................................................. 119
13.2 BasicSemanticNotions.............................................. 121
13.3 Satisfaction ........................................................... 122
13.4 Truth .................................................................. 123
13.5 LogicalConsequence ................................................ 126
13.6 Undecidability........................................................ 127
14 TheSystemQ ............................................................... 131
14.1 AxiomsandInferenceRule.......................................... 131
14.2 DerivabilityinQ ..................................................... 132
14.3 GeneralizationTheorem ............................................. 133
14.4 ValidityandDerivability............................................. 133
14.5 DeductionTheoremandOtherSyntacticResults................... 134
14.6 AlphabeticVariants .................................................. 135
15 Consistency,Soundness,Completeness .................................. 139
15.1 ConsistencyofQ..................................................... 139
15.2 SoundnessofQ....................................................... 140
15.3 CompletenessofQ................................................... 141
15.4 CompactnessTheorem............................................... 143
15.5 FinalRemarks........................................................ 144
16 UndecidabilityandRelatedResults ...................................... 147
16.1 UndecidabilityofQ ................................................. 147
16.2 GödelNumbering .................................................... 148
16.3 EffectiveEnumerabilityoftheTheoremsofQ..................... 149
16.4 AFurtherCorollary .................................................. 149
16.5 RecursiveAxiomatizationandDecidability ........................ 150
17 First-OrderLogic........................................................... 153
17.1 First-OrderLanguagesandSystems................................. 153
17.2 First-OrderLogicwithIdentity...................................... 154
17.3 First-OrderTheory ................................................... 155
17.4 TheLanguageofBasicArithmetic.................................. 156
17.5 PeanoArithmetic..................................................... 158
18 TheoriesandModels ....................................................... 161
18.1 Cardinality............................................................ 161
18.2 Löwenheim-SkolemTheorems...................................... 162
