The Logic Manual

193 Pages·2009·0.842 MB·English

by  Volker Halbach

THE LOGIC MANUAL
for Introduction to Logic
2008/2009
Volker Halbach
Oxford
8thAugust2008
Thistextistobeusedbycandidates
intheirfirstyearin2008/2009. The
settextforLiteraeHumanioresstu-
dentssittingModerationsin2009is
Hodges’sLogic.
Content
1 Sets,Relations,andArguments 5
1.1 Sets 5
1.2 BinaryRelations 7
1.3 Functions 13
1.4 Non-BinaryRelations 15
1.5 Arguments,ValidityandContradiction 16
1.6 Syntax,SemanticsandPragmatics 24
2 SyntaxandSemanticsofPropositionalLogic 26
2.1 Quotation 26
2.2 TheSyntaxoftheLanguageofPropositionalLogic 28
2.3 RulesforDroppingBrackets 30
2.4 TheSemanticsofPropositionalLogic 33
3 FormalisationinPropositionalLogic 50
3.1 Truth-Functionality 51
3.2 LogicalForm 55
3.3 FromLogicalFormtoFormalLanguage 60
3.4 Ambiguity 62
3.5 TheStandardConnectives 64
3.6 NaturalLanguageandPropositionalLogic 66
4 TheSyntaxofPredicateLogic 73
4.1 PredicatesandQuantification 73
4.2 TheSentencesofL 81
2
4.3 FreeandBoundOccurrencesofVariables 84
©VolkerHalbach 2008/2009
4.4 NotationalConventions 86
4.5 Formalisation 87
5 TheSemanticsofPredicateLogic 93
5.1 Structures 94
5.2 Truth 99
5.3 Validity,LogicalTruths,andContradictions 107
5.4 Counterexamples 108
6 NaturalDeduction 114
6.1 PropositionalLogic 116
6.2 PredicateLogic 128
7 FormalisationinPredicateLogic 141
7.1 Adequacy 141
7.2 Ambiguity 145
7.3 Extensionality 150
7.4 PredicateLogicandArgumentsinEnglish 156
8 IdentityandDefiniteDescriptions 165
8.1 QualitativeandNumericalIdentity 165
8.2 TheSyntaxofL 166
=
8.3 Semantics 167
8.4 ProofRulesforIdentity 170
8.5 Usesofidentity 173
8.6 IdentityasaLogicalConstant 182
NaturalDeductionRules 185
preface
TheLogicManualisarelativelybriefintroductiontologic. Ihavetriedto
focusonthecoretopicsandhaveneglectedsomeissuesthatarecovered
inmorecomprehensivebookssuchasForbes(1994),Guttenplan(1997),
Hodges(2001),Smith(2003),andTennant(1990). Inparticular,Ihave
triednottoincludematerialthatisinessentialtoPreliminaryExamina-
tionsandModerationsinOxford. Forvarioustopics,Icouldnotresist
addingfootnotesofferingextrainformationtothecuriousreader.
Logicisusuallytaughtinoneterm. Consequently,Ihavedividedthe
textintoeightchapters:
1. Sets,Relations,andArguments
2. SyntaxandSemanticsofPropositionalLogic
3. FormalisationinPropositionalLogic
4. TheSyntaxofPredicateLogic
5. TheSemanticsofPredicateLogic
6. NaturalDeduction
7. FormalisationinPredicateLogic
8. IdentityandDefiniteDescriptions
Ifthereaderwishestoreadselectively,chapters1–3constituteaself-con-
tainedpart,towhichSection6.1(NaturalDeductionforpropositional
logic)canbeadded;andchapters1–7yieldanintroductiontopredicate
logicwithoutidentity.
Ihavesetthecoredefinitions,explanations,andresultsinitalicslike
this. Thismightbeusefulforrevisionandforfindingimportantpassages
morequickly.
Insomecases,theendofanexampleoraproofismarkedbyasquare
◻. Theword‘iff’isshortfor‘ifandonlyif’.
IhavewrittenanExercisesBookletthatcanbeusedinconjunction
withthisLogicManual. ItisavailablefromWebLearn. Therealsosome
additionalteachingmaterialsmaybefoundsuchasfurtherexamplesof
proofsinthesystemofNaturalDeduction.
Iamindebtedtocolleaguesfordiscussionsandcommentsonprevious
versionsofthetext. Inparticular,IwouldliketothankStephenBlamey,
PaoloCrivelli, GeoffreyFerrari, LindsayJudson, OfraMagidor, David
McCarty,PeterMillican,AlexanderPaseau,AnnamariaSchiaparelli,Se-
bastianSequoiah-Grayson,MarkThakkar,GabrielUzquiano,andDavid
Wiggins. IamespeciallygratefultoJaneFriedmanandChristophervon
BülowfortheirhelpinpreparingthefinalversionoftheManual.
1 Sets, Relations, and Arguments
1.1 sets
Settheoryisemployedinmanydisciplines. Assuch,someacquaintance
withthemostbasicnotionsofsettheorywillbeusefulnotonlyinlogic,
butalsoinotherareasthatrelyonformalmethods. Settheoryisavast
areaofmathematicalresearchandofsignificantphilosophicalinterest.
Forthepurposesofthisbook,thereaderonlyneedstoknowafragment
ofthefundamentalsofsettheory.1
Asetisacollectionofobjects. Theseobjectsmaybeconcreteobjects
suchaspersons,carsandplanetsormathematicalobjectssuchasnumbers
orothersets.
Setsareidenticalifandonlyiftheyhavethesamemembers. Therefore,
thesetofallanimalswithkidneysandthesetofallanimalswithaheart
areidentical,becauseexactlythoseanimalsthathavekidneysalsohave
a heart and vice versa.2 In contrast, the property of having a heart is
usuallydistinguishedfromthepropertyofhavingkidneys,althoughboth
propertiesapplytothesameobjects.
That a is an element of the set M can be expressed symbolically by
1There are various mathematical introductions to set theory such as Devlin (1993),
Moschovakis(1994)orthemoreelementaryHalmos(1960). Incontrasttorigorous
expositionsofsettheory,Iwillnotproceedaxiomaticallyhere.
2I have added this footnote because there are regularly protests with respect to this
example.Forthisexample,onlycompleteandhealthyanimalsarebeingconsidered.I
havebeentoldthatplanarians(atypeofflatworms)areanexceptiontotheheart–kidney
rule,so,forthesakeoftheexample,Ishouldexcludethemaswell.
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1 Sets,Relations,andArguments 6
writing ‘a ∈ M’. If a is an element of M, one also says that a is in M or
that M contains a.
There is exactly one set that contains no elements, namely, the empty
set∅. Obviously,thereisonlyoneemptyset,becauseallsetscontaining
noelementscontainthesameelements,namelynone.
Therearevariouswaystodenotesets.
One can write down names of the elements, or other designations of
theelements,andenclosethislistincurlybrackets.
Theset{London,Munich},forinstance,hasexactlytwocitiesasits
elements. Theset{Munich,London}hasthesameelements. Therefore,
thesetsareidentical,thatis:
{London,Munich}={Munich,London}.
Thus, if a set is specified by including names for the elements in curly
brackets,theorderofthenamesbetweenthebracketsdoesnotmatter.
The set {the capital of England, Munich} is again the same set be-
cause‘thecapitalofEngland’isjustanotherwayofdesignatingLondon.
{London,Munich,thecapitalofEngland}isstillthesameset: adding
anothernameforLondon,namely,‘thecapitalofEngland’,doesnotadd
afurtherelementto{London,Munich}.
Thismethodofdesignatingsetshasitslimitations: sometimesone
lacks names for the elements. The method will also fail for sets with
infinitelymanyorevenjustimpracticallymanyelements.
Above I have designated a set by the phrase ‘the set of all animals
withaheart’. Onecanalsousethefollowingsemi-formalexpressionto
designatethisset:
{x ∶ x isananimalwithaheart}
Thisisreadas‘thesetofallanimalswithaheart’. Similarly,{x ∶ x isa
naturalnumberbiggerthan3}isthesetofnaturalnumbersbiggerthan3,
and{x ∶ x isbluealloverorx isredallover}isthesetofallobjectsthat
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1 Sets,Relations,andArguments 7
arebluealloverandallobjectsthatareredallover.3
1.2 binary relations
Theexpression‘isatiger’appliestosomeobjects,butnottoothers. There
isasetofallobjectstowhichitapplies,namelytheset{x ∶ x isatiger}
containingalltigersandnootherobjects. Theexpression‘isabiggercity
than’, incontrast, doesnotapplyto singleobjects; ratherit relatestwo
objects. It applies to London and Munich (in this order), for instance,
becauseLondonisabiggercitythanMunich. Onecanalsosaythatthe
expression‘isabiggercitythan’appliestopairsofobjects. Thesetofall
pairstowhichtheexpression‘isabiggercitythan’appliesiscalled‘the
binaryrelationofbeingabiggercitythan’orsimply‘therelationofbeing
abiggercitythan’.4 Thisrelationcontainsallpairswithobjectsd and e
suchthatd isabiggercitythan e.5
However,thesepairscannotbeunderstoodsimplyasthesets{d,e},
suchthatdisabiggercitythane,becauseelementsofasetarenotordered
bytheset: aspointedoutabove,theset{London,Munich}isthesame
set as {Munich, London}. So a set with two elements does not have a
first or second element. Since London is bigger than Munich, but not
viceversa,onlythepairwithLondonasfirstcomponentandMunichas
3Theassumptionthatanydescriptionofthiskindactuallydescribesasetisproblematic.
Theso-calledRussellparadoximposessomelimitationsonwhatsetsonecanpostulate.
SeeExercise7.6.
4Bythequalification‘binary’onedistinguishesrelationsapplyingtopairsfromrelations
applyingtotriplesandstringsofmoreobjects.Iwillreturntonon-binaryrelationsin
Section1.4.
5Oftenphilosophersdonotidentifyrelationswithsetsofpairs.Ontheirterminology
relationsneedtobedistinguishedfromsetsoforderedpairsinthesamewayproperties
needtobedistinguishedfromsets(seefootnote2).Insettheory,however,itiscommon
torefertosetsoforderedpairsasbinaryrelationsandIshallfollowthisusagehere.
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1 Sets,Relations,andArguments 8
secondcomponentshouldbeintherelationofbeingabiggercitythan,
butnotthepairwithMunichasfirstcomponentandLondonassecond
component.
Therefore, so-called ordered-pairs are used in set theory. They are
differentfromsetswithtwoelements. Orderedpairs,incontrasttosets
withtwoelements,haveafirstandasecondcomponent(andnofurther
components). Theorderedpair⟨London,Munich⟩hasLondonasitsfirst
componentandMunichasitssecond. ⟨Munich,London⟩isadifferent
orderedpair,becausethetwoorderedpairsdifferinboththeirfirstand
secondcomponents.6 Moreformally,anorderedpair⟨d,e⟩isidentical
with⟨f,g⟩ifandonlyifd = f and e = g. Theorderedpair⟨thelargest
city in Bavaria, the largest city in the UK⟩ is the same ordered pair as
⟨Munich,London⟩,becausetheycoincideintheirfirstandintheirsecond
component. Anorderedpaircanhavethesameobjectasfirstandsecond
component: ⟨London,London⟩,forinstance,hasLondonasitsfirstand
secondcomponent. ⟨Munich,London⟩and⟨London,London⟩aretwo
differentorderedpairs,becausetheydifferintheirfirstcomponents.Since
Iwillnotbedealingwithotherpairs,Iwilloftendropthequalification
‘ordered’from‘orderedpair’.
definition 1.1. A set is a binary relation if and only if it contains only
orderedpairs.
According to the definition, a set is a binary relation if it does not
containanythingthatisnotanorderedpair. Sincetheemptyset∅does
notcontainanything,itdoesnotcontainanythingthatisnotanordered
pair. Therefore,theemptysetisabinaryrelation.
The binary relation of being a bigger city than, that is, the relation
thatissatisfiedbyobjectsd and e ifandonlyifd isabiggercitythan e is
thefollowingset:
6Usinganicetrick,onecandispensewithorderedpairsbydefiningtheorderedpair
⟨d,e⟩as{{d},{d,e}}.Thetrickwillnotbeusedhere.
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1 Sets,Relations,andArguments 9
{⟨London,Munich⟩,⟨London,Oxford⟩,⟨Munich,Oxford⟩,
⟨Paris,Munich⟩,...}
InthefollowingdefinitionIwillclassifybinaryrelations. Later,Ishall
illustratethedefinitionsbyexamples. Here,andinthefollowing,Ishall
use‘iff’asanabbreviationfor‘ifandonlyif’.
definition1.2. AbinaryrelationR is
(i) reflexiveonasetSiffforalld inSthepair⟨d,d⟩isanelementofR;
(ii) symmetriciffforalld,e: if⟨d,e⟩∈ R then⟨e,d⟩∈ R;
(iii) asymmetricifffornod,e: ⟨d,e⟩∈ R and⟨e,d⟩∈ R;
(iv) antisymmetricifffornotwodistinctd,e: ⟨d,e⟩∈ R and⟨e,d⟩∈ R;
(v) transitive iff for all d,e, f: if ⟨d,e⟩ ∈ R and ⟨e, f ⟩ ∈ R, then also
⟨d, f ⟩∈ R;
(vi) an equivalence relation on S iff R is reflexive on S, symmetric and
transitive.
InthefollowingIshalloccasionallydropthequalification‘binary’.
Aslongastheyarenottoocomplicated,relationsandtheirproperties
–suchasreflexivityandsymmetry–canbevisualisedbydiagrams. For
everycomponentofanorderedpairintherelation,onewritesexactly
onename(orotherdesignation)inthediagram. Theorderedpairsinthe
relationarethenrepresentedbyarrows. Forinstance,therelation
{⟨France,Italy⟩,⟨Italy,Austria⟩,⟨France,France⟩,
⟨Italy,Italy⟩,⟨Austria,Austria⟩}
hasthefollowingdiagram:
(cid:6)(cid:6) (cid:6)(cid:6)
France Austria
(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:36)(cid:36) (cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:58)(cid:58)
Italy
(cid:86)(cid:86)
The arrow from ‘France’ to ‘Italy’ corresponds to the pair ⟨France,
Italy⟩,andthearrowfrom‘Italy’to‘Austria’correspondstothepair⟨Italy,
©VolkerHalbach 2008/2009