Logic, Mathematics, and Computer Science: Modern Foundations with Practical Applications

399 Pages·2015·0.73 MB·English

by  Yves Nievergelt (auth.)

Yves Nievergelt
Logic, Mathematics,
and Computer
Science
Modern Foundations
with Practical Applications
Second Edition
Logic, Mathematics, and Computer Science

Yves Nievergelt
Logic, Mathematics,
and Computer Science
Modern Foundations with Practical
Applications
Second Edition
123
YvesNievergelt
DepartmentofMathematics
EasternWashingtonUniversity
Cheney,WA,USA
CoverartexcerptedfromtriadictruthtablesfromCharlesSandersPeirce’sLogicNotebook.
ISBN978-1-4939-3222-1 ISBN978-1-4939-3223-8 (eBook)
DOI10.1007/978-1-4939-3223-8
LibraryofCongressControlNumber:2015949237
MathematicsSubjectClassification(2010):Primary03-01;Secondary:68-01,91-01
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Preface
This second edition, entitled Logic, Mathematics, and Computer Science: Modern
Foundations with Practical Applications, has been adapted from Foundations of
Logic and Mathematics: Applications to Computer Science and Cryptography, ©
2002 by Birkhäuser, from which Chapters 1–5 have been retained but extensively
revised.Chapters6and7havebeenadded.
This text discusses the foundations where logic, mathematics, and computer
sciencebegin.Theintendedreadershipconsistsofundergraduatestudentsmajoring
inmathematicsorcomputersciencewhomustlearnsuchfoundationseitherfortheir
own interest or for further studies. For a motivated reader, there are no technical
prerequisites:youneednotknowanytechnicalsubjecttostartreadingthistext.
Although the text does not focus on the history and philosophy of the founda-
tions, the material cites copious references to the literature, where the reader may
find additional historical context. Consulting such references is neither suggested
nornecessarytostudythetheoryortoworkontheexercises,butindividualcitations
document the material by original sources, and all the citations together provide a
guidetothevariationsandchronologicaldevelopmentsoflogic,mathematics,and
computerscience.Forexample,Chapter1tracestheoriginofTruthtablestoCharles
Sanders Peirce’s unpublished 1909 Logic Notebook on philosophy and points out
theirapplicationsoveronehalfofacenturylatertothedesignofcomputersforuse
onEarthandonboardtheApollolunarspacecraft.
Along informal arguments, this text also shows the corresponding purely sym-
bolic manipulations of formulae, because they clarify the reasoning [11] and can
reveal hitherto hidden logical properties, such as the mutual independence of
different patterns of reasoning, or the impossibility of some proofs within some
logics:
Asforalgebra[oflogic],theveryideaoftheartisthatitpresentsformulaewhichcanbe
manipulated,andthatbyobservingtheeffectsofsuchmanipulationwefindpropertiesnot
tobeotherwisediscerned(CharlesSandersPeirce[104,p.182]).
If professionals are unable to learn some topics by any means other than the pure
manipulationofsymbols,thenitwouldseemunfairtoclaimthatalllearningmustbe
intuitiveandhidefromstudentssuchpurelymanipulativebutsuccessfulmethods.
v
vi Preface
The selection of topics also reflects major accomplishments from the twentieth
century: the foundation of all of mathematics, and later computer science, as well
ascomputer-assistedproofsofmathematicaltheorems,onaformallogicbasedon
only a few axioms, transformation rules, and postulates for set theory [47,50,54,
105,139].Also,whilenotwritteninformallogic,Nobel-Prizewinningapplications
tothesocialsciencesrelyonthesamefoundations,asshowninChapter7.
To introduce the foundations of logic, the provability theorem in Chapter 1
provides an algorithm to design proofs in propositional logic. Chapter 1 also
explains the concept of undecidability with multi-valued (“fuzzy”) logic and
presents a proof of unprovability. Chapter 2 introduces logical quantifiers.
Aworkingknowledgeoflogicalquantifiersiscrucialforthestudyofbasicconcepts
in modern mathematical analysis and topology, such as the uniform convergence
of a sequence of equicontinuous functions. Continuing with the foundations of
mathematics, Chapter 3 presents a version of the Zermelo–Fraenkel set theory. At
thejunctureofmathematicsandcomputerscience,Chapter4developstheconcepts
ofdefinitionandproofbyinduction.Chapter4thenusesinductionwithsettheory
todefinetheintegersandrationalnumbersandderivetheassociative,commutative,
and distributive laws, as well as algorithms, for their arithmetics. To give readers
someideaoftopicsatanintermediatelevel,Chapter5showsthatinawell-formed
theorysomeparadoxesdonotoccur,whileChapter6completesthefoundationsof
settheorywiththeaxiomofchoice.
No extragalactic asteroid has yet been found with the universal laws of logic
engraved in it. Consequently, not just one logic but many different logics have
beeninvented.Differentlogicsleadtodifferentmathematicsanddifferentcomputer
sciences. However, the acid test for adopting a particular logic is its ability
to make predictions that are born by subsequent experiments. Formal logic is
thus a mathematical model of rational thought processes. In this aspect, logic,
mathematics, and computer science are experimental sciences. Only one logic has
passed all such tests, which is the one used throughout this text. Other logics
are outlined in Chapter 1 as a pedagogical device and to show some of their
shortcomings.
Acknowledgments
I thank Dr. Stephen P. Keeler at the Boeing Company for having corrected many
errorsinthefirstedition.Newandremainingerrorsaremine.IalsothankDr.Mary
Keeler for guiding me to Charles Sanders Peirce’s unpublished work on logic and
philosophy. I commend the editors at Birkhäuser and Springer, in particular, Ann
Kostant and Elizabeth Loew, for their hard work, patience, encouragements, and
positiveattitudethroughthedevelopmentandproductionofbothtexts.
Cheney,WA,USA YvesNievergelt
30June2015
Contents
1 PropositionalLogic:ProofsfromAxiomsandInferenceRules......... 1
1.1 Introduction ............................................................. 1
1.1.1 AnExampleDemonstratingtheUseofLogicin
RealLife....................................................... 2
1.2 ThePurePropositionalCalculus ....................................... 4
1.2.1 Formulae,Axioms,InferenceRules,andProofs............. 5
1.3 ThePurePositiveImplicationalPropositionalCalculus .............. 9
1.3.1 ExamplesofProofsintheImplicationalCalculus........... 9
1.3.2 DerivedRules:ImplicationsSubjecttoHypotheses......... 11
1.3.3 AGuideforProofs:anImplicationalDeductionTheorem.. 14
1.3.4 Example:LawofAssertionfromtheDeductionTheorem.. 18
1.3.5 More Examples to Design Proofs of
ImplicationalTheorems....................................... 21
1.3.6 AnotherGuideforProofs:SubstitutivityofEquivalences .. 23
1.3.7 MoreDerivedRulesofInference............................. 25
1.3.8 TheLawsofCommutationandofAssertion................. 27
1.3.9 ExercisesontheClassicalImplicationalCalculus........... 28
1.3.10 EquivalentImplicationalAxiomSystems.................... 29
1.3.11 ExercisesonKleene’sAxioms................................ 30
1.3.12 ExercisesonTarski’sAxioms................................. 31
1.4 ProofsbytheConverseLawofContraposition........................ 32
1.4.1 ExamplesofProofsintheFullPropositionalCalculus...... 32
1.4.2 GuidesforProofsinthePropositionalCalculus............. 34
1.4.3 ProofsbyReductioadAbsurdum ............................ 35
1.4.4 ProofsbyCases................................................ 36
1.4.5 ExercisesonFrege’sandChurch’sAxioms.................. 37
1.5 OtherConnectives....................................................... 38
1.5.1 DefinitionsofOtherConnectives............................. 38
1.5.2 ExamplesofProofsofTheoremswithConjunctions........ 38
1.5.3 ExamplesofProofsofTheoremswithEquivalences........ 41
1.5.4 ExamplesofProofsofTheoremswithDisjunctions......... 44
vii
viii Contents
1.5.5 ExamplesofProofswithConjunctionsandDisjunctions... 46
1.5.6 ExercisesonOtherConnectives.............................. 47
1.6 PatternsofDeductionwithOtherConnectives........................ 48
1.6.1 ConjunctionsofImplications................................. 48
1.6.2 ProofsbyCasesorbyContradiction ......................... 53
1.6.3 ExercisesonPatternsofDeduction........................... 54
1.6.4 EquivalentClassicalAxiomSystems......................... 55
1.6.5 ExercisesonKleene’s,Rosser’s,andTarski’sAxioms...... 56
1.7 Completeness,Decidability,Independence,Provability,
andSoundness........................................................... 56
1.7.1 Multi-ValuedFuzzyLogics................................... 56
1.7.2 SoundMulti-ValuedFuzzyLogics ........................... 57
1.7.3 IndependenceandUnprovability ............................. 59
1.7.4 CompleteMulti-ValuedFuzzyLogics........................ 61
1.7.5 Peirce’sLawasaDenialoftheAntecedent.................. 62
1.7.6 ExercisesonChurch’sandŁukasiewicz’s
TriadicSystems................................................ 62
1.8 BooleanLogic........................................................... 63
1.8.1 TheTruthTableoftheLogicalImplication.................. 63
1.8.2 BooleanLogiconEarthandinSpace........................ 65
1.9 AutomatedTheoremProving........................................... 67
1.9.1 TheProvabilityTheorem...................................... 67
1.9.2 TheCompletenessTheorem .................................. 69
1.9.3 Example:Peirce’sLawfromtheCompletenessTheorem... 69
1.9.4 ExercisesontheDeductionTheorem ........................ 72
2 First-OrderLogic:ProofswithQuantifiers............................... 75
2.1 Introduction ............................................................. 75
2.2 ThePurePredicateCalculusofFirstOrder............................ 75
2.2.1 LogicalPredicates............................................. 75
2.2.2 Variables,Quantifiers,andFormulae......................... 77
2.2.3 ProperSubstitutionsofFreeorBoundVariables ............ 78
2.2.4 AxiomsandRulesforthePurePredicateCalculus.......... 80
2.2.5 ExercisesonQuantifiers ...................................... 82
2.2.6 ExampleswithImplicationalandPredicateCalculi......... 82
2.2.7 ExampleswithPurePropositionalandPredicateCalculi ... 86
2.2.8 OtherAxiomaticSystemsforthePurePredicateCalculus.. 87
2.2.9 ExercisesonKleene’s,Margaris’s,andRosser’sAxioms... 89
2.3 MethodsofProofforthePurePredicateCalculus .................... 90
2.3.1 SubstitutingEquivalentFormulae ............................ 90
2.3.2 DischargingHypotheses ...................................... 91
2.3.3 PrenexNormalForm.......................................... 95
2.3.4 ProofswithMorethanOneQuantifier ....................... 96
2.3.5 ExercisesontheSubstitutivityofEquivalence............... 97
Contents ix
2.4 PredicateCalculuswithOtherConnectives............................ 98
2.4.1 UniversalQuantifiersandConjunctionsorDisjunctions.... 98
2.4.2 ExistentialQuantifiersandConjunctionsorDisjunctions... 100
2.4.3 ExercisesonQuantifierswithOtherConnectives............ 101
2.5 Equality-Predicates ..................................................... 101
2.5.1 First-OrderPredicateCalculiwithanEquality-Predicate... 102
2.5.2 Simple Applied Predicate Calculi
withanEquality-Predicate.................................... 103
2.5.3 OtherAxiomSystemsfortheEquality-Predicate............ 106
2.5.4 DefinedRanking-Predicates .................................. 107
2.5.5 ExercisesonEquality-Predicates............................. 107
3 SetTheory:ProofsbyDetachment,Contraposition,and
Contradiction................................................................. 109
3.1 Introduction ............................................................. 109
3.2 SetsandSubsets......................................................... 110
3.2.1 EqualityandExtensionality................................... 110
3.2.2 TheEmptySet................................................. 114
3.2.3 SubsetsandSupersets......................................... 114
3.2.4 ExercisesonSetsandSubsets ................................ 118
3.3 Pairing,Power,andSeparation......................................... 119
3.3.1 Pairing ......................................................... 119
3.3.2 PowerSets..................................................... 122
3.3.3 SeparationofSets ............................................. 124
3.3.4 ExercisesonPairing,Power,andSeparationofSets ........ 126
3.4 UnionsandIntersectionsofSets ....................................... 127
3.4.1 UnionsofSets................................................. 127
3.4.2 IntersectionsofSets........................................... 132
3.4.3 UnionsandIntersectionsofSets.............................. 135
3.4.4 ExercisesonUnionsandIntersectionsofSets............... 139
3.5 CartesianProductsandRelations ...................................... 142
3.5.1 CartesianProductsofSets .................................... 142
3.5.2 CartesianProductsofUnionsandIntersections ............. 147
3.5.3 MathematicalRelationsandDirectedGraphs ............... 149
3.5.4 ExercisesonCartesianProductsofSets...................... 153
3.6 MathematicalFunctions ................................................ 154
3.6.1 MathematicalFunctions....................................... 154
3.6.2 ImagesandInverseImagesofSetsbyFunctions............ 159
3.6.3 ExercisesonMathematicalFunctions........................ 162
3.7 CompositeandInverseFunctions...................................... 164
3.7.1 CompositionsofFunctions ................................... 164
3.7.2 Injective,Surjective,Bijective,andInverseFunctions ...... 166
3.7.3 TheSetofallFunctionsfromaSettoaSet.................. 171
3.7.4 ExercisesonInjective,Surjective,andInverseFunctions... 173