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 Material Information Title: forall x: Introductory Textbook in Formal Logic Physical Description: Book Language: en-US Creator: Magnus, P. D. Publication Date: 1/9/2009
 Subjects Subjects / Keywords: arguments, sentences, formal logic, deductive validity, sequential logic, truth tables, quantified logic, formal semantics, proofs, modelsAbstract Reasoning, Logical Thinking, Philosophy, Thinking SkillsPhilosophy / Logic
 Record Information Source Institution: University of Florida Holding Location: University Press of Florida Rights Management: Copyright 2005-2008 by P.D. Magnus. Some rights reserved. This work is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. To view a copy, see: http://creativecommons.org/licenses/by-sa/3.0/ You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (a) â€¦ Resource Identifier: isbn - 9781616100353 System ID: AA00011647:00001

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Contents 1Whatislogic?5 1.1Arguments..............................6 1.2Sentences...............................6 1.3Twowaysthatargumentscangowrong..............7 1.4Deductivevalidity..........................8 1.5Otherlogicalnotions.........................10 1.6Formallanguages...........................12 PracticeExercises.............................15 2Sententiallogic17 2.1Sentenceletters............................17 2.2Connectives..............................19 2.3Othersymbolization.........................28 2.4SentencesofSL............................29 PracticeExercises.............................33 3Truthtables37 3.1Truth-functionalconnnectives....................37 3.2Completetruthtables........................38 3.3Usingtruthtables..........................41 3.4Partialtruthtables..........................42 PracticeExercises.............................44 4Quantiedlogic48 4.1Fromsentencestopredicates....................48 4.2BuildingblocksofQL........................50 4.3Quantiers..............................54 4.4TranslatingtoQL..........................57 4.5SentencesofQL............................68 4.6Identity................................71 PracticeExercises.............................76 5Formalsemantics83 5.1SemanticsforSL...........................84 3

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4 CONTENTS 5.2InterpretationsandmodelsinQL..................88 5.3Semanticsforidentity........................92 5.4Workingwithmodels.........................94 5.5TruthinQL..............................98 PracticeExercises.............................103 6Proofs107 6.1BasicrulesforSL...........................108 6.2Derivedrules.............................117 6.3Rulesofreplacement.........................119 6.4Rulesforquantiers.........................121 6.5Rulesforidentity...........................126 6.6Proofstrategy.............................128 6.7Proof-theoreticconcepts.......................129 6.8Proofsandmodels..........................131 6.9Soundnessandcompleteness.....................132 PracticeExercises.............................134 AOthersymbolicnotation140 BSolutionstoselectedexercises143 CQuickReference156

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Chapter1 Whatislogic? Logicisthebusinessofevaluatingarguments,sortinggoodonesfrombadones. Ineverydaylanguage,wesometimesusetheword`argument'torefertobelligerentshoutingmatches.Ifyouandafriendhaveanargumentinthissense, thingsarenotgoingwellbetweenthetwoofyou. Inlogic,wearenotinterestedintheteeth-gnashing,hair-pullingkindofargument.Alogicalargumentisstructuredtogivesomeoneareasontobelieve someconclusion.Hereisonesuchargument: Itisrainingheavily. Ifyoudonottakeanumbrella,youwillgetsoaked. : : Youshouldtakeanumbrella. Thethreedotsonthethirdlineoftheargumentmean`Therefore'andthey indicatethatthenalsentenceisthe conclusion oftheargument.Theother sentencesare premises oftheargument.Ifyoubelievethepremises,thenthe argumentprovidesyouwithareasontobelievetheconclusion. Thischapterdiscussessomebasiclogicalnotionsthatapplytoargumentsina naturallanguagelikeEnglish.Itisimportanttobeginwithaclearunderstandingofwhatargumentsareandofwhatitmeansforanargumenttobevalid. LaterwewilltranslateargumentsfromEnglishintoaformallanguage.We wantformalvalidity,asdenedintheformallanguage,tohaveatleastsomeof theimportantfeaturesofnatural-languagevalidity. 5

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6 forall x 1.1Arguments Whenpeoplemeantogivearguments,theytypicallyoftenusewordslike`therefore'and`because.'Whenanalyzinganargument,therstthingtodoisto separatethepremisesfromtheconclusion.Wordsliketheseareacluetowhat theargumentissupposedtobe,especiallyif|intheargumentasgiven|the conclusioncomesatthebeginningorinthemiddleoftheargument. premiseindicators: since,because,giventhat conclusionindicators: therefore,hence,thus,then,so Tobeperfectlygeneral,wecandenean argument asaseriesofsentences. Thesentencesatthebeginningoftheseriesarepremises.Thenalsentencein theseriesistheconclusion.Ifthepremisesaretrueandtheargumentisagood one,thenyouhaveareasontoaccepttheconclusion. Noticethatthisdenitionisquitegeneral.Considerthisexample: Thereiscoeeinthecoeepot. Thereisadragonplayingbassoononthearmoire. : : SalvadorDaliwasapokerplayer. Itmayseemoddtocallthisanargument,butthatisbecauseitwouldbe aterribleargument.Thetwopremiseshavenothingatalltodowiththe conclusion.Nevertheless,givenourdenition,itstillcountsasanargument| albeitabadone. 1.2Sentences Inlogic,weareonlyinterestedinsentencesthatcangureasapremiseor conclusionofanargument.Sowewillsaythata sentence issomethingthat canbetrueorfalse. Youshouldnotconfusetheideaofasentencethatcanbetrueorfalsewith thedierencebetweenfactandopinion.Often,sentencesinlogicwillexpress thingsthatwouldcountasfacts|suchas`Kierkegaardwasahunchback'or `Kierkegaardlikedalmonds.'Theycanalsoexpressthingsthatyoumightthink ofasmattersofopinion|suchas,`Almondsareyummy.' Also,therearethingsthatwouldcountas`sentences'inalinguisticsorgrammar coursethatwewillnotcountassentencesinlogic.

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ch.1whatislogic? 7 Questions Inagrammarclass,`Areyousleepyyet?'wouldcountasan interrogativesentence.Althoughyoumightbesleepyoryoumightbealert,the questionitselfisneithertruenorfalse.Forthisreason,questionswillnotcount assentencesinlogic.Supposeyouanswerthequestion:`Iamnotsleepy.'This iseithertrueorfalse,andsoitisasentenceinthelogicalsense.Generally, questions willnotcountassentences,but answers will. `Whatisthiscourseabout?'isnotasentence.`Nooneknowswhatthiscourse isabout'isasentence. Imperatives Commandsareoftenphrasedasimperativeslike`Wakeup!',`Sit upstraight',andsoon.Inagrammarclass,thesewouldcountasimperative sentences.Althoughitmightbegoodforyoutositupstraightoritmightnot, thecommandisneithertruenorfalse.Note,however,thatcommandsarenot alwaysphrasedasimperatives.`Youwillrespectmyauthority' is eithertrue orfalse|eitheryouwilloryouwillnot|andsoitcountsasasentenceinthe logicalsense. Exclamations `Ouch!'issometimescalledanexclamatorysentence,butit isneithertruenorfalse.Wewilltreat`Ouch,Ihurtmytoe!'asmeaningthe samethingas`Ihurtmytoe.'The`ouch'doesnotaddanythingthatcouldbe trueorfalse. 1.3Twowaysthatargumentscangowrong Considertheargumentthatyoushouldtakeanumbrellaonp.5,above.If premiseisfalse|ifitissunnyoutside|thentheargumentgivesyouno reasontocarryanumbrella.Evenifitisrainingoutside,youmightnotneedan umbrella.Youmightweararainpanchoorkeeptocoveredwalkways.Inthese cases,premisewouldbefalse,sinceyoucouldgooutwithoutanumbrella andstillavoidgettingsoaked. Supposeforamomentthatboththepremisesaretrue.Youdonotownarain pancho.Youneedtogoplaceswheretherearenocoveredwalkways.Nowdoes theargumentshowyouthatyoushouldtakeanumbrella?Notnecessarily. Perhapsyouenjoywalkingintherain,andyouwouldliketogetsoaked.In thatcase,eventhoughthepremisesweretrue,theconclusionwouldbefalse. Foranyargument,therearetwowaysthatitcouldbeweak.First,oneormore ofthepremisesmightbefalse.Anargumentgivesyouareasontobelieveits conclusiononlyifyoubelieveitspremises.Second,thepremisesmightfailto

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8 forall x supporttheconclusion.Evenifthepremisesweretrue,theformoftheargument mightbeweak.Theexamplewejustconsideredisweakinbothways. Whenanargumentisweakinthesecondway,thereissomethingwrongwith the logicalform oftheargument:Premisesofthekindgivendonotnecessarily leadtoaconclusionofthekindgiven.Wewillbeinterestedprimarilyinthe logicalformofarguments. Consideranotherexample: Youarereadingthisbook. Thisisalogicbook. : : Youarealogicstudent. Thisisnotaterribleargument.Mostpeoplewhoreadthisbookarelogic students.Yet,itispossibleforsomeonebesidesalogicstudenttoreadthis book.Ifyourroommatepickedupthebookandthumbedthroughit,theywould notimmediatelybecomealogicstudent.Sothepremisesofthisargument,even thoughtheyaretrue,donotguaranteethetruthoftheconclusion.Itslogical formislessthanperfect. Anargumentthathadnoweaknessofthesecondkindwouldhaveperfectlogical form.Ifitspremisesweretrue,thenitsconclusionwould necessarily betrue. Wecallsuchanargument`deductivelyvalid'orjust`valid.' Eventhoughwemightcounttheargumentaboveasagoodargumentinsome sense,itisnotvalid;thatis,itis`invalid.'Oneimportanttaskoflogicisto sortvalidargumentsfrominvalidarguments. 1.4Deductivevalidity Anargumentisdeductively valid ifandonlyifitisimpossibleforthepremises tobetrueandtheconclusionfalse. Thecrucialthingaboutavalidargumentisthatitisimpossibleforthepremises tobetrue atthesametime thattheconclusionisfalse.Considerthisexample: Orangesareeitherfruitsormusicalinstruments. Orangesarenotfruits. : : Orangesaremusicalinstruments. Theconclusionofthisargumentisridiculous.Nevertheless,itfollowsvalidly fromthepremises.Thisisavalidargument. If bothpremisesweretrue, then theconclusionwouldnecessarilybetrue.

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10 forall x Inductivearguments,evengoodinductivearguments,arenotdeductivelyvalid. Wewillnotbeinterestedininductiveargumentsinthisbook. 1.5Otherlogicalnotions Inadditiontodeductivevalidity,wewillbeinterestedinsomeotherlogical concepts. Truth-values Trueorfalseissaidtobethe truth-value ofsentence.Wedenedsentences asthingsthatcouldbetrueorfalse;wecouldhavesaidinsteadthatsentences arethingsthatcanhavetruth-values. Logicaltruth Inconsideringargumentsformally,wecareaboutwhatwouldbetrue if the premisesweretrue.Generally,wearenotconcernedwiththeactualtruthvalue ofanyparticularsentences|whethertheyare actually trueorfalse.Yetthere aresomesentencesthatmustbetrue,justasamatteroflogic. Considerthesesentences: 1.Itisraining. 2.Eitheritisraining,oritisnot. 3.Itisbothrainingandnotraining. Inordertoknowifsentence1istrue,youwouldneedtolookoutsideorcheckthe weatherchannel.Logicallyspeaking,itmightbeeithertrueorfalse.Sentences likethisarecalled contingent sentences. Sentence2isdierent.Youdonotneedtolookoutsidetoknowthatitistrue. Regardlessofwhattheweatherislike,itiseitherrainingornot.Thissentence is logicallytrue ;itistruemerelyasamatteroflogic,regardlessofwhatthe worldisactuallylike.Alogicallytruesentenceiscalleda tautology Youdonotneedtochecktheweathertoknowaboutsentence3,either.Itmust befalse,simplyasamatteroflogic.Itmightberaininghereandnotraining acrosstown,itmightberainingnowbutstoprainingevenasyoureadthis,but itisimpossibleforittobebothrainingandnotraininghereatthismoment.

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ch.1whatislogic? 11 Thethirdsentenceis logicallyfalse ;itisfalseregardlessofwhattheworldis like.Alogicallyfalsesentenceiscalleda contradiction Tobeprecise,wecandenea contingentsentence asasentencethatis neitheratautologynoracontradiction. Asentencemight always betrueandstillbecontingent.Forinstance,ifthere neverwereatimewhentheuniversecontainedfewerthanseventhings,then thesentence`Atleastseventhingsexist'wouldalwaysbetrue.Yetthesentence iscontingent;itstruthisnotamatteroflogic.Thereisnocontradictionin consideringapossibleworldinwhichtherearefewerthanseventhings.The importantquestioniswhetherthesentence must betrue,justonaccountof logic. Logicalequivalence Wecanalsoaskaboutthelogicalrelations between twosentences.Forexample: Johnwenttothestoreafterhewashedthedishes. Johnwashedthedishesbeforehewenttothestore. Thesetwosentencesarebothcontingent,sinceJohnmightnothavegoneto thestoreorwasheddishesatall.Yettheymusthavethesametruth-value.If eitherofthesentencesistrue,thentheybothare;ifeitherofthesentencesis false,thentheybothare.Whentwosentencesnecessarilyhavethesametruth value,wesaythattheyare logicallyequivalent Consistency Considerthesetwosentences: B1 MyonlybrotheristallerthanIam. B2 MyonlybrotherisshorterthanIam. Logicalonecannottelluswhich,ifeither,ofthesesentencesistrue.Yetwecan saythat if therstsentenceB1istrue, then thesecondsentenceB2must befalse.AndifB2istrue,thenB1mustbefalse.Itcannotbethecasethat bothofthesesentencesaretrue. Ifasetofsentencescouldnotallbetrueatthesametime,likeB1{B2,theyare saidtobe inconsistent .Otherwise,theyare consistent .

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ch.1whatislogic? 13 S is M All M sare C s. : :S is C Inbotharguments S standsforSocratesand M standsforman.Intherst argument, C standsformortal;inthesecond, C standsforcarrot.Bothargumentshavethisform,andeveryargumentofthisformisvalid.Soboth argumentsarevalid. Whatwedidherewasreplacewordslike`man'or`carrot'withsymbolslike `M'or`C'soastomakethelogicalformexplicit.Thisisthecentralidea behindformallogic.Wewanttoremoveirrelevantordistractingfeaturesofthe argumenttomakethelogicalformmoreperspicuous. Startingwithanargumentina naturallanguage likeEnglish,wetranslatethe argumentintoa formallanguage .PartsoftheEnglishsentencesarereplaced withlettersandsymbols.Thegoalistorevealtheformalstructureofthe argument,aswedidwiththesetwo. Thereareformallanguagesthatworklikethesymbolizationwegaveforthese twoarguments.AlogiclikethiswasdevelopedbyAristotle,aphilosopherwho livedinGreeceduringthe4thcenturyBC.AristotlewasastudentofPlatoand thetutorofAlexandertheGreat.Aristotle'slogic,withsomerevisions,wasthe dominantlogicinthewesternworldformorethantwomillennia. InAristoteleanlogic,categoriesarereplacedwithcapitalletters.Everysentence ofanargumentisthenrepresentedashavingoneoffourforms,whichmedieval logicianslabeledinthisway:AAll A sare B s.ENo A sare B s.ISome A is B .OSome A isnot B Itisthenpossibletodescribevalid syllogisms ,three-lineargumentslikethe twoweconsideredabove.Medievallogiciansgavemnemonicnamestoallof thevalidargumentforms.Theformofourtwoarguments,forinstance,was called Barbara .Thevowelsinthename,allAs,representthefactthatthetwo premisesandtheconclusionareallAformsentences. TherearemanylimitationstoAristoteleanlogic.Oneisthatitmakesno distinctionbetweenkindsandindividuals.Sotherstpremisemightjustas wellbewritten`All S sare M s':AllSocratesesaremen.Despiteitshistorical importance,Aristoteleanlogichasbeensuperceded.Theremainderofthisbook willdeveloptwoformallanguages. TherstisSL,whichstandsfor sententiallogic .InSL,thesmallestunitsare sentencesthemselves.Simplesentencesarerepresentedaslettersandconnected withlogicalconnectiveslike`and'and`not'tomakemorecomplexsentences.

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14 forall x ThesecondisQL,whichstandsfor quantiedlogic .InQL,thebasicunitsare objects,propertiesofobjects,andrelationsbetweenobjects. Whenwetranslateanargumentintoaformallanguage,wehopetomakeits logicalstructureclearer.Wewanttoincludeenoughofthestructureofthe Englishlanguageargumentsothatwecanjudgewhethertheargumentisvalid orinvalid.IfweincludedeveryfeatureoftheEnglishlanguage,allofthe subtletyandnuance,thentherewouldbenoadvantageintranslatingtoa formallanguage.WemightaswellthinkabouttheargumentinEnglish. Atthesametime,wewouldlikeaformallanguagethatallowsustorepresent manykindsofEnglishlanguagearguments.ThisisonereasontopreferQLto Aristoteleanlogic;QLcanrepresenteveryvalidargumentofAristoteleanlogic andmore. Sowhendecidingonaformallanguage,thereisinevitablyatensionbetween wantingtocaptureasmuchstructureaspossibleandwantingasimpleformal language|simplerformallanguagesleaveoutmore.Thismeansthatthereis noperfectformallanguage.Somewilldoabetterjobthanothersintranslating particularEnglish-languagearguments. Inthisbook,wemaketheassumptionthat true and false aretheonlypossible truth-values.Logicallanguagesthatmakethisassumptionarecalled bivalent whichmeans two-valued .Aristoteleanlogic,SL,andQLareallbivalent,but therearelimitstothepowerofbivalentlogic.Forinstance,somephilosophers haveclaimedthatthefutureisnotyetdetermined.Iftheyareright,then sentencesabout whatwillbethecase arenotyettrueorfalse.Someformal languagesaccommodatethisbyallowingforsentencesthatareneithertruenor false,butsomethinginbetween.Otherformallanguages,so-calledparaconsistentlogics,allowforsentencesthatarebothtrue and false. Thelanguagespresentedinthisbookarenottheonlypossibleformallanguages. However,mostnonstandardlogicsextendonthebasicformalstructureofthe bivalentlogicsdiscussedinthisbook.Sothisisagoodplacetostart. Summaryoflogicalnotions Anargumentisdeductively valid ifitisimpossibleforthepremisesto betrueandtheconclusionfalse;itis invalid otherwise. A tautology isasentencethatmustbetrue,asamatteroflogic. A contradiction isasentencethatmustbefalse,asamatteroflogic. A contingentsentence isneitheratautologynoracontradiction.

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ch.1whatislogic? 15 Twosentencesare logicallyequivalent iftheynecessarilyhavethe sametruthvalue. Asetofsentencesis consistent ifitislogicallypossibleforallthemembersofthesettobetrueatthesametime;itis inconsistent otherwise. PracticeExercises Attheendofeachchapter,youwillndaseriesofpracticeproblemsthat reviewandexplorethematerialcoveredinthechapter.Thereisnosubstitute foractuallyworkingthroughsomeproblems,becauselogicismoreaboutaway ofthinkingthanitisaboutmemorizingfacts.Theanswerstosomeofthe problemsareprovidedattheendofthebookinappendixB;theproblemsthat aresolvedintheappendixaremarkedwitha ? PartA Whichofthefollowingare`sentences'inthelogicalsense? 1.EnglandissmallerthanChina. 2.GreenlandissouthofJerusalem. 3.IsNewJerseyeastofWisconsin? 4.Theatomicnumberofheliumis2. 5.Theatomicnumberofheliumis 6.Ihateovercookednoodles. 7.Blech!Overcookednoodles! 8.Overcookednoodlesaredisgusting. 9.Takeyourtime. 10.Thisisthelastquestion. PartB Foreachofthefollowing:Isitatautology,acontradiction,oracontingentsentence? 1.CaesarcrossedtheRubicon. 2.SomeoneoncecrossedtheRubicon. 3.NoonehasevercrossedtheRubicon. 4.IfCaesarcrossedtheRubicon,thensomeonehas. 5.EventhoughCaesarcrossedtheRubicon,noonehasevercrossedthe Rubicon. 6.IfanyonehasevercrossedtheRubicon,itwasCaesar. ? PartC LookbackatthesentencesG1{G4onp.12,andconsidereachofthe followingsetsofsentences.Whichareconsistent?Whichareinconsistent?

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16 forall x 1.G2,G3,andG4 2.G1,G3,andG4 3.G1,G2,andG4 4.G1,G2,andG3 ? PartD Whichofthefollowingispossible?Ifitispossible,giveanexample. Ifitisnotpossible,explainwhy. 1.Avalidargumentthathasonefalsepremiseandonetruepremise 2.Avalidargumentthathasafalseconclusion 3.Avalidargument,theconclusionofwhichisacontradiction 4.Aninvalidargument,theconclusionofwhichisatautology 5.Atautologythatiscontingent 6.Twologicallyequivalentsentences,bothofwhicharetautologies 7.Twologicallyequivalentsentences,oneofwhichisatautologyandoneof whichiscontingent 8.Twologicallyequivalentsentencesthattogetherareaninconsistentset 9.Aconsistentsetofsentencesthatcontainsacontradiction 10.Aninconsistentsetofsentencesthatcontainsatautology

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Chapter2 Sententiallogic ThischapterintroducesalogicallanguagecalledSL.Itisaversionof sentential logic ,becausethebasicunitsofthelanguagewillrepresententiresentences. 2.1Sentenceletters InSL,capitallettersareusedtorepresentbasicsentences.Consideredonlyasa symbolofSL,theletter A couldmeananysentence.Sowhentranslatingfrom EnglishintoSL,itisimportanttoprovidea symbolizationkey .Thekeyprovides anEnglishlanguagesentenceforeachsentenceletterusedinthesymbolization. Forexample,considerthisargument: Thereisanappleonthedesk. Ifthereisanappleonthedesk,thenJennymadeittoclass. : : Jennymadeittoclass. ThisisobviouslyavalidargumentinEnglish.Insymbolizingit,wewantto preservethestructureoftheargumentthatmakesitvalid.Whathappensif wereplaceeachsentencewithaletter?Oursymbolizationkeywouldlooklike this: A: Thereisanappleonthedesk. B: Ifthereisanappleonthedesk,thenJennymadeittoclass. C: Jennymadeittoclass. Wewouldthensymbolizetheargumentinthisway: 17

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18 forall x A B : :C Thereisnonecessaryconnectionbetweensomesentence A ,whichcouldbeany sentence,andsomeothersentences B and C ,whichcouldbeanysentences. Thestructureoftheargumenthasbeencompletelylostinthistranslation. Theimportantthingabouttheargumentisthatthesecondpremiseisnot merely any sentence,logicallydivorcedfromtheothersentencesintheargument.Thesecondpremisecontainstherstpremiseandtheconclusion asparts Oursymbolizationkeyfortheargumentonlyneedstoincludemeaningsfor A and C ,andwecanbuildthesecondpremisefromthosepieces.Sowesymbolize theargumentthisway: A If A ,then C : :C Thispreservesthestructureoftheargumentthatmakesitvalid,butitstill makesuseoftheEnglishexpression`If ::: then ::: .'Althoughweultimately wanttoreplacealloftheEnglishexpressionswithlogicalnotation,thisisa goodstart. Thesentencesthatcanbesymbolizedwithsentencelettersarecalled atomic sentences ,becausetheyarethebasicbuildingblocksoutofwhichmorecomplex sentencescanbebuilt.Whateverlogicalstructureasentencemighthaveislost whenitistranslatedasanatomicsentence.FromthepointofviewofSL,the sentenceisjustaletter.Itcanbeusedtobuildmorecomplexsentences,butit cannotbetakenapart. Thereareonlytwenty-sixlettersofthealphabet,butthereisnologicallimit tothenumberofatomicsentences.Wecanusethesamelettertosymbolize dierentatomicsentencesbyaddingasubscript,asmallnumberwrittenafter theletter.Wecouldhaveasymbolizationkeythatlookslikethis: A 1 : Theappleisunderthearmoire. A 2 : ArgumentsinSLalwayscontainatomicsentences. A 3 : AdamAntistakinganairplanefromAnchoragetoAlbany. . A 294 : Alliterationangersotherwiseaableastronauts. Keepinmindthateachoftheseisadierentsentenceletter.Whenthereare subscriptsinthesymbolizationkey,itisimportanttokeeptrackofthem.

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ch.2sententiallogic 19 2.2Connectives Logicalconnectivesareusedtobuildcomplexsentencesfromatomiccomponents.TherearevelogicalconnectivesinSL.Thistablesummarizesthem, andtheyareexplainedbelow. symbol whatitiscalled whatitmeans : negation `Itisnotthecasethat ::: & conjunction `Both ::: and ::: disjunction `Either ::: or ::: conditional `If ::: then ::: \$ biconditional ` ::: ifandonlyif ::: Negation Considerhowwemightsymbolizethesesentences: 1.MaryisinBarcelona. 2.MaryisnotinBarcelona. 3.MaryissomewherebesidesBarcelona. Inordertosymbolizesentence1,wewillneedonesentenceletter.Wecan provideasymbolizationkey: B: MaryisinBarcelona. Notethatherewearegiving B adierentinterpretationthanwedidinthe previoussection.Thesymbolizationkeyonlyspecieswhat B means ina speciccontext .Itisvitalthatwecontinuetousethismeaningof B solong aswearetalkingaboutMaryandBarcelona.Later,whenwearesymbolizing dierentsentences,wecanwriteanewsymbolizationkeyanduse B tomean somethingelse. Now,sentence1issimply B Sincesentence2isobviouslyrelatedtothesentence1,wedonotwantto introduceadierentsentenceletter.ToputitpartlyinEnglish,thesentence means`Not B .'Inordertosymbolizethis,weneedasymbolforlogicalnegation. Wewilluse` : .'Nowwecantranslate`Not B 'to : B Sentence3isaboutwhetherornotMaryisinBarcelona,butitdoesnotcontain theword`not.'Nevertheless,itisobviouslylogicallyequivalenttosentence2.

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20 forall x Theybothmean:ItisnotthecasethatMaryisinBarcelona.Assuch,wecan translatebothsentence2andsentence3as : B Asentencecanbesymbolizedas : A ifitcanbeparaphrasedin Englishas`Itisnotthecasethat A .' Considerthesefurtherexamples: 4.Thewidgetcanbereplacedifitbreaks. 5.Thewidgetisirreplaceable. 6.Thewidgetisnotirreplaceable. Ifwelet R mean`Thewidgetisreplaceable',thensentence4canbetranslated as R Whataboutsentence5?Sayingthewidgetisirreplaceablemeansthatitis notthecasethatthewidgetisreplaceable.Soeventhoughsentence5isnot negativeinEnglish,wesymoblizeitusingnegationas : R Sentence6canbeparaphrasedas`Itisnotthecasethatthewidgetisirreplaceable.'Usingnegationtwice,wetranslatethisas :: R .Thetwonegationsina roweachworkasnegations,sothesentencemeans`Itisnotthecasethat ::: itisnotthecasethat :::R .'IfyouthinkaboutthesentenceinEnglish,itis logicallyequivalenttosentence4.SowhenwedenelogicalequivalenceisSL, wewillmakesurethat R and :: R arelogicallyequivalent. Moreexamples: 7.Elliottishappy. 8.Elliottisunhappy. Ifwelet H mean`Elliotishappy',thenwecansymbolizesentence7as H However,itwouldbeamistaketosymbolizesentence8as : H .IfElliottis unhappy,thenheisnothappy|butsentence8doesnotmeanthesamething as`ItisnotthecasethatElliottishappy.'Itcouldbethatheisnothappybut thatheisnotunhappyeither.Perhapsheissomewherebetweenthetwo.In ordertosymbolizesentence8,wewouldneedanewsentenceletter. Foranysentence A :If A istrue,then : A isfalse.If : A istrue,then A isfalse. Using`T'fortrueand`F'forfalse,wecansummarizethisina characteristic truthtable fornegation:

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ch.2sententiallogic 21 A : A T F F T Wewilldiscusstruthtablesatgreaterlengthinthenextchapter. Conjunction Considerthesesentences: 9.Adamisathletic. 10.Barbaraisathletic. 11.Adamisathletic,andBarbaraisalsoathletic. Wewillneedseparatesentencelettersfor9and10,sowedenethissymbolizationkey: A: Adamisathletic. B: Barbaraisathletic. Sentence9canbesymbolizedas A Sentence10canbesymbolizedas B Sentence11canbeparaphrasedas` A and B .'Inordertofullysymbolizethis sentence,weneedanothersymbol.Wewilluse`&.'Wetranslate` A and B as A & B .Thelogicalconnective`&'iscalled conjunction ,and A and B are eachcalled conjuncts Noticethatwemakenoattempttosymbolize`also'insentence11.Wordslike `both'and`also'functiontodrawourattentiontothefactthattwothingsare beingconjoined.Theyarenotdoinganyfurtherlogicalwork,sowedonotneed torepresenttheminSL. Somemoreexamples: 12.Barbaraisathleticandenergetic. 13.BarbaraandAdamarebothathletic. 14.AlthoughBarbaraisenergetic,sheisnotathletic. 15.Barbaraisathletic,butAdamismoreathleticthansheis. Sentence12isobviouslyaconjunction.Thesentencesaystwothingsabout Barbara,soinEnglishitispermissibletorefertoBarbaraonlyonce.Itmight

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ch.2sententiallogic 23 trueorfalse.WehaveusedthemtosymbolizedierentEnglishlanguagesentencesthatareallaboutpeoplebeingathletic,butthissimilarityiscompletely lostwhenwetranslatetoSL.Noformallanguagecancaptureallthestructure oftheEnglishlanguage,butaslongasthisstructureisnotimportanttothe argumentthereisnothinglostbyleavingitout. Foranysentences A and B A & B istrueifandonlyifboth A and B aretrue. Wecansummarizethisinthecharacteristictruthtableforconjunction: A B A & B T T T T F F F T F F F F Conjunctionis symmetrical becausewecanswaptheconjunctswithoutchangingthetruth-valueofthesentence.Regardlessofwhat A and B are, A & B is logicallyequivalentto B & A Disjunction Considerthesesentences: 16.EitherDenisonwillplaygolfwithme,orhewillwatchmovies. 17.EitherDenisonorEllerywillplaygolfwithme. Forthesesentenceswecanusethissymbolizationkey: D: Denisonwillplaygolfwithme. E: Ellerywillplaygolfwithme. M: Denisonwillwatchmovies. Sentence16is`Either D or M .'Tofullysymbolizethis,weintroduceanewsymbol.Thesentencebecomes D M .The` 'connectiveiscalled disjunction and D and M arecalled disjuncts Sentence17isonlyslightlymorecomplicated.Therearetwosubjects,butthe Englishsentenceonlygivestheverbonce.Intranslating,wecanparaphrase itas.`EitherDenisonwillplaygolfwithme,orEllerywillplaygolfwithme.' Nowitobviouslytranslatesas D E .

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24 forall x Asentencecanbesymbolizedas A B ifitcanbeparaphrased inEnglishas`Either A ,or B .'Eachofthedisjunctsmustbea sentence. SometimesinEnglish,theword`or'excludesthepossibilitythatbothdisjuncts aretrue.Thisiscalledan exclusiveor .An exclusiveor isclearlyintended whenitsays,onarestaurantmenu,`Entreescomewitheithersouporsalad.' Youmayhavesoup;youmayhavesalad;but,ifyouwant both soup and salad, thenyouhavetopayextra. Atothertimes,theword`or'allowsforthepossibilitythatbothdisjunctsmight betrue.Thisisprobablythecasewithsentence17,above.Imightplaywith Denison,withEllery,orwithbothDenisonandEllery.Sentence17merelysays thatIwillplaywith atleast oneofthem.Thisiscalledan inclusiveor Thesymbol` 'representsan inclusiveor .So D E istrueif D istrue,if E istrue,orifboth D and E aretrue.Itisfalseonlyifboth D and E arefalse. Wecansummarizethiswiththecharacteristictruthtablefordisjunction: A B A B T T T T F T F T T F F F Likeconjunction,disjunctionissymmetrical. A B islogicallyequivalentto B A Thesesentencesaresomewhatmorecomplicated: 18.Eitheryouwillnothavesoup,oryouwillnothavesalad. 19.Youwillhaveneithersoupnorsalad. 20.Yougeteithersouporsalad,butnotboth. Welet S 1 meanthatyougetsoupand S 2 meanthatyougetsalad. Sentence18canbeparaphrasedinthisway:`Either itisnotthecasethat you getsoup,or itisnotthecasethat yougetsalad.'Translatingthisrequiresboth disjunctionandnegation.Itbecomes : S 1 _: S 2 Sentence19alsorequiresnegation.Itcanbeparaphrasedas,` Itisnotthecase that eitherthatyougetsouporthatyougetsalad.'Weneedsomewayof indicatingthatthenegationdoesnotjustnegatetherightorleftdisjunct,but rathernegatestheentiredisjunction.Inordertodothis,weputparentheses

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ch.2sententiallogic 25 aroundthedisjunction:`Itisnotthecasethat S 1 S 2 .'Thisbecomessimply : S 1 S 2 Noticethattheparenthesesaredoingimportantworkhere.Thesentence : S 1 S 2 wouldmean`Eitheryouwillnothavesoup,oryouwillhavesalad.' Sentence20isan exclusiveor .Wecanbreakthesentenceintotwoparts.The rstpartsaysthatyougetoneortheother.Wetranslatethisas S 1 S 2 Thesecondpartsaysthatyoudonotgetboth.Wecanparaphrasethisas, `Itisnotthecaseboththatyougetsoupandthatyougetsalad.'Usingboth negationandconjunction,wetranslatethisas : S 1 & S 2 .Nowwejustneedto putthetwopartstogether.Aswesawabove,`but'canusuallybetranslatedas aconjunction.Sentence20canthusbetranslatedas S 1 S 2 & : S 1 & S 2 Although` 'isan inclusiveor ,wecansymbolizean exclusiveor inSL.Wejust needmorethanoneconnectivetodoit. Conditional Forthefollowingsentences,let R mean`Youwillcuttheredwire'and B mean `Thebombwillexplode.' 21.Iftheyoucuttheredwire,thenthebombwillexplode. 22.Thebombwillexplodeonlyifyoucuttheredwire. Sentence21canbetranslatedpartiallyas`If R ,then B .'Wewillusethe symbol` 'torepresentlogicalentailment.Thesentencebecomes R B .The connectiveiscalleda conditional .Thesentenceontheleft-handsideofthe conditional R inthisexampleiscalledthe antecedent .Thesentenceonthe right-handside B iscalledthe consequent Sentence22isalsoaconditional.Sincetheword`if'appearsinthesecond halfofthesentence,itmightbetemptingtosymbolizethisinthesamewayas sentence21.Thatwouldbeamistake. Theconditional R B saysthat if R weretrue, then B wouldalsobetrue.It doesnotsaythatyourcuttingtheredwireisthe only waythatthebombcould explode.Someoneelsemightcutthewire,orthebombmightbeonatimer. Thesentence R B doesnotsayanythingaboutwhattoexpectif R isfalse. Sentence22isdierent.Itsaysthattheonlyconditionsunderwhichthebomb willexplodeinvolveyourhavingcuttheredwire;i.e.,ifthebombexplodes, thenyoumusthavecutthewire.Assuch,sentence22shouldbesymbolizedas B R .

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26 forall x Itisimportanttorememberthattheconnective` 'saysonlythat,ifthe antecedentistrue,thentheconsequentistrue.Itsaysnothingaboutthe causal connectionbetweenthetwoevents.Translatingsentence22as B R does notmeanthatthebombexplodingwouldsomehowhavecausedyourcutting thewire.Bothsentence21and22suggestthat,ifyoucuttheredwire,your cuttingtheredwirewouldbethecauseofthebombexploding.Theydieron the logical connection.Ifsentence22weretrue,thenanexplosionwouldtell us|thoseofussafelyawayfromthebomb|thatyouhadcuttheredwire. Withoutanexplosion,sentence22tellsusnothing. Theparaphrasedsentence` A onlyif B 'islogicallyequivalentto`If A ,then B .' `If A then B 'meansthatif A istruethensois B .Soweknowthatifthe antecedent A istruebuttheconsequent B isfalse,thentheconditional`If A then B 'isfalse.Whatisthetruthvalueof`If A then B 'underother circumstances?Suppose,forinstance,thattheantecedent A happenedtobe false.`If A then B 'wouldthennottellusanythingabouttheactualtruthvalue oftheconsequent B ,anditisunclearwhatthetruthvalueof`If A then B wouldbe. InEnglish,thetruthofconditionalsoftendependsonwhat would bethecase iftheantecedent weretrue |evenif,asamatteroffact,theantecedentis false.ThisposesaproblemfortranslatingconditionalsintoSL.Consideredas sentencesofSL, R and B intheaboveexampleshavenothingintrinsictodo witheachother.Inordertoconsiderwhattheworldwouldbelikeif R were true,wewouldneedtoanalyzewhat R saysabouttheworld.Since R isan atomicsymbolofSL,however,thereisnofurtherstructuretobeanalyzed. Whenwereplaceasentencewithasentenceletter,weconsideritmerelyas someatomicsentencethatmightbetrueorfalse. InordertotranslateconditionalsintoSL,wewillnottrytocaptureallthe subtletiesoftheEnglishlanguage`If ::: then ::: .'Instead,thesymbol` 'will bea materialconditional .Thismeansthatwhen A isfalse,theconditional A B isautomaticallytrue,regardlessofthetruthvalueof B .Ifboth A and B aretrue,thentheconditional A B istrue. Inshort, A B isfalseifandonlyif A istrueand B isfalse.Wecansummarize thiswithacharacteristictruthtablefortheconditional. A B A B T T T T F F F T T F F T

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ch.2sententiallogic 27 Theconditionalis asymmetrical .Youcannotswaptheantecedentandconsequentwithoutchangingthemeaningofthesentence,because A B and B A arenotlogicallyequivalent. Notallsentencesoftheform`If ::: then ::: 'areconditionals.Considerthis sentence: 23.Ifanyonewantstoseeme,thenIwillbeontheporch. IfIsaythis,itmeansthatIwillbeontheporch,regardlessofwhetheranyone wantstoseemeornot|butifsomeonedidwanttoseeme,thentheyshould lookformethere.Ifwelet P mean`Iwillbeontheporch,'thensentence23 canbetranslatedsimplyas P Biconditional Considerthesesentences: 24.Thegureontheboardisatriangleonlyifithasexactlythreesides. 25.Thegureontheboardisatriangleifithasexactlythreesides. 26.Thegureontheboardisatriangleifandonlyifithasexactlythree sides. Let T mean`Thegureisatriangle'and S mean`Thegurehasthreesides.' Sentence24,forreasonsdiscussedabove,canbetranslatedas T S Sentence25isimportantlydierent.Itcanbeparaphrasedas,`Ifthegurehas threesides,thenitisatriangle.'Soitcanbetranslatedas S T Sentence26saysthat T istrue ifandonlyif S istrue;wecaninfer S from T andwecaninfer T from S .Thisiscalleda biconditional ,becauseitentails thetwoconditionals S T and T S .Wewilluse` \$ 'torepresentthe biconditional;sentence26canbetranslatedas S \$ T Wecouldabidewithoutanewsymbolforthebiconditional.Sincesentence26 means` T S and S T ,'wecouldtranslateitas T S & S T .We wouldneedparenthesestoindicatethat T S and S T areseparate conjuncts;theexpression T S & S T wouldbeambiguous. Becausewecouldalwayswrite A B & B A insteadof A \$ B ,we donotstrictlyspeaking need tointroduceanewsymbolforthebiconditional. Nevertheless,logicallanguagesusuallyhavesuchasymbol.SLwillhaveone, whichmakesiteasiertotranslatephraseslike`ifandonlyif.'

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28 forall x A \$ B istrueifandonlyif A and B havethesametruthvalue.Thisisthe characteristictruthtableforthebiconditional: A B A \$ B T T T T F F F T F F F T 2.3Othersymbolization WehavenowintroducedalloftheconnectivesofSL.Wecanusethemtogether totranslatemanykindsofsentences.Considertheseexamplesofsentencesthat usetheEnglish-languageconnective`unless': 27.Unlessyouwearajacket,youwillcatchcold. 28.Youwillcatchcoldunlessyouwearajacket. Let J mean`Youwillwearajacket'andlet D mean`Youwillcatchacold.' Wecanparaphrasesentence27as`Unless J D .'Thismeansthatifyoudonot wearajacket,thenyouwillcatchcold;withthisinmind,wemighttranslateit as : J D .Italsomeansthatifyoudonotcatchacold,thenyoumusthave wornajacket;withthisinmind,wemighttranslateitas : D J Whichoftheseisthecorrecttranslationofsentence27?Bothtranslationsare correct,becausethetwotranslationsarelogicallyequivalentinSL. Sentence28,inEnglish,islogicallyequivalenttosentence27.Itcanbetranslatedaseither : J D or : D J Whensymbolizingsentenceslikesentence27andsentence28,itiseasytoget turnedaround.Sincetheconditionalisnotsymmetric,itwouldbewrongto translateeithersentenceas J !: D .Fortunately,thereareotherlogically equivalentexpressions.Bothsentencesmeanthatyouwillwearajacketor| ifyoudonotwearajacket|thenyouwillcatchacold.Sowecantranslate themas J D .Youmightworrythatthe`or'hereshouldbean exclusiveor However,thesentencesdonotexcludethepossibilitythatyoumight both wear ajacket and catchacold;jacketsdonotprotectyoufromallthepossibleways thatyoumightcatchacold. Ifasentencecanbeparaphrasedas`Unless A B ,'thenitcanbe symbolizedas A B

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ch.2sententiallogic 29 Symbolizationofstandardsentencetypesissummarizedonp.156. 2.4SentencesofSL Thesentence`Applesarered,orberriesareblue'isasentenceofEnglish,and thesentence` A B 'isasentenceofSL.Althoughwecanidentifysentencesof Englishwhenweencounterthem,wedonothaveaformaldenitionof`sentence ofEnglish'.InSL,itispossibletoformallydenewhatcountsasasentence. ThisisonerespectinwhichaformallanguagelikeSLismoreprecisethana naturallanguagelikeEnglish. ItisimportanttodistinguishbetweenthelogicallanguageSL,whichweare developing,andthelanguagethatweusetotalkaboutSL.Whenwetalk aboutalanguage,thelanguagethatwearetalkingaboutiscalledthe object language .Thelanguagethatweusetotalkabouttheobjectlanguageis calledthe metalanguage TheobjectlanguageinthischapterisSL.ThemetalanguageisEnglish|not conversationalEnglish,butEnglishsupplementedwithsomelogicalandmathematicalvocabulary.Thesentence` A B 'isasentenceintheobjectlanguage, becauseitusesonlysymbolsofSL.Theword`sentence'isnotitselfpartofSL, however,sothesentence`ThisexpressionisasentenceofSL'isnotasentence ofSL.Itisasentenceinthemetalanguage,asentencethatweusetotalk about SL. Inthissection,wewillgiveaformaldenitionfor`sentenceofSL.'Thedenition itselfwillbegiveninmathematicalEnglish,themetalanguage. Expressions TherearethreekindsofsymbolsinSL: sentenceletters A;B;C;:::;Z withsubscripts,asneeded A 1 ;B 1 ;Z 1 ;A 2 ;A 25 ;J 375 ;::: connectives : ,&, \$ parentheses Wedenean expressionofsl asanystringofsymbolsofSL.Takeanyofthe symbolsofSLandwritethemdown,inanyorder,andyouhaveanexpression.

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30 forall x Well-formedformulae Sinceanysequenceofsymbolsisanexpression,manyexpressionsofSLwillbe gobbledegook.Ameaningfulexpressioniscalleda well-formedformula .Itis commontousetheacronym w ;thepluralisws. Obviously,individualsentenceletterslike A and G 13 willbews.Wecan formfurtherwsoutofthesebyusingthevariousconnectives.Usingnegation, wecanget : A and : G 13 .Usingconjunction,wecanget A & G 13 G 13 & A A & A ,and G 13 & G 13 .Wecouldalsoapplynegationrepeatedlytogetwslike :: A orapplynegationalongwithconjunctiontogetwslike : A & G 13 and : G 13 & : G 13 .Thepossiblecombinationsareendless,evenstartingwithjust thesetwosentenceletters,andthereareinnitelymanysentenceletters.So thereisnopointintryingtolistallthews. Instead,wewilldescribetheprocessbywhichwscanbeconstructed.Consider negation:Givenanyw A ofSL, : A isawofSL.Itisimportantherethat A isnotthesentenceletter A .Rather,itisavariablethatstandsinforany watall.Noticethatthisvariable A isnotasymbolofSL,so : A isnotan expressionofSL.Instead,itisanexpressionofthemetalanguagethatallowsus totalkaboutinnitelymanyexpressionsofSL:alloftheexpressionsthatstart withthenegationsymbol.Because A ispartofthemetalanguage,itiscalleda metavariable Wecansaysimilarthingsforeachoftheotherconnectives.Forinstance,if A and B arewsofSL,then A & B isawofSL.Providingclauseslike thisforalloftheconnectives,wearriveatthefollowingformaldenitionfora well-formedformulaofSL: 1.Everyatomicsentenceisaw. 2.If A isaw,then : A isawofSL. 3.If A and B arews,then A & B isaw. 4.If A and B arews,then A B isaw. 5.If A and B arews,then A B isaw. 6.If A and B arews,then A \$ B isaw. 7.AllandonlywsofSLcanbegeneratedbyapplicationsoftheserules. Noticethatwecannotimmediatelyapplythisdenitiontoseewhetheranarbitraryexpressionisaw.Supposewewanttoknowwhetherornot ::: D isawofSL.Lookingatthesecondclauseofthedenition,weknowthat ::: D isaw if :: D isaw.Sonowweneedtoaskwhetherornot :: D

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ch.2sententiallogic 31 isaw.Againlookingatthesecondclauseofthedenition, :: D isaw if : D is.Again, : D isaw if D isaw.Now D isasentenceletter,anatomic sentenceofSL,soweknowthat D isawbytherstclauseofthedenition. Soforacompoundformulalike ::: D ,wemustapplythedenitionrepeatedly. Eventuallywearriveattheatomicsentencesfromwhichthewisbuiltup. Denitionslikethisarecalled recursive .Recursivedenitionsbeginwithsome speciablebaseelementsanddenewaystoindenitelycompoundthebase elements.Justastherecursivedenitionallowscomplexsentencestobebuilt upfromsimpleparts,youcanuseittodecomposesentencesintotheirsimpler parts.Todeterminewhetherornotsomethingmeetsthedenition,youmay havetoreferbacktothedenitionmanytimes. Theconnectivethatyoulooktorstindecomposingasentenceiscalledthe mainlogicaloperator ofthatsentence.Forexample:Themainlogical operatorof : E F G isnegation, : .Themainlogicaloperatorof : E F G isdisjunction, Sentences Recallthatasentenceisameaningfulexpressionthatcanbetrueorfalse.Since themeaningfulexpressionsofSLarethewsandsinceeverywofSLiseither trueorfalse,thedenitionforasentenceofSListhesameasthedenitionfor aw.Noteveryformallanguagewillhavethisnicefeature.Inthelanguage QL,whichisdevelopedlaterinthebook,therearewswhicharenotsentences. TherecursivestructureofsentencesinSLwillbeimportantwhenweconsider thecircumstancesunderwhichaparticularsentencewouldbetrueorfalse. Thesentence ::: D istrueifandonlyifthesentence :: D isfalse,andsoon throughthestructureofthesentenceuntilwearriveattheatomiccomponents: ::: D istrueifandonlyiftheatomicsentence D isfalse.Wewillreturnto thispointinthenextchapter. Notationalconventions Awlike Q & R mustbesurroundedbyparentheses,becausewemightapply thedenitionagaintousethisaspartofamorecomplicatedsentence.Ifwe negate Q & R ,weget : Q & R .Ifwejusthad Q & R withouttheparentheses andputanegationinfrontofit,wewouldhave : Q & R .Itismostnatural toreadthisasmeaningthesamethingas : Q & R ,somethingverydierent than : Q & R .Thesentence : Q & R meansthatitisnotthecasethatboth Q and R aretrue; Q mightbefalseor R mightbefalse,butthesentencedoes nottelluswhich.Thesentence : Q & R meansspecicallythat Q isfalseand

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32 forall x that R istrue.Assuch,parenthesesarecrucialtothemeaningofthesentence. So,strictlyspeaking, Q & R withoutparenthesesis not asentenceofSL.When usingSL,however,wewilloftenbeabletorelaxtheprecisedenitionsoasto makethingseasierforourselves.Wewilldothisinseveralways. First,weunderstandthat Q & R meansthesamethingas Q & R .Asamatter ofconvention,wecanleaveoparenthesesthatoccur aroundtheentiresentence Second,itcansometimesbeconfusingtolookatlongsentenceswithmany, nestedpairsofparentheses.Weadopttheconventionofusingsquarebrackets `['and`]'inplaceofparenthesis.Thereisnologicaldierencebetween P Q and[ P Q ],forexample.Theunwieldysentence H I I H & J K couldbewritteninthisway: H I I H & J K Third,wewillsometimeswanttotranslatetheconjunctionofthreeormore sentences.Forthesentence`Alice,Bob,andCandiceallwenttotheparty', supposewelet A mean`Alicewent', B mean`Bobwent',and C mean`Candice went.'Thedenitiononlyallowsustoformaconjunctionoutoftwosentences, sowecantranslateitas A & B & C oras A & B & C .Thereisnoreason todistinguishbetweenthese,sincethetwotranslationsarelogicallyequivalent. Thereisnologicaldierencebetweentherst,inwhich A & B isconjoined with C ,andthesecond,inwhich A isconjoinedwith B & C .Sowemight aswelljustwrite A & B & C .Asamatterofconvention,wecanleaveout parentheseswhenweconjointhreeormoresentences. Fourth,asimilarsituationariseswithmultipledisjunctions.`EitherAlice,Bob, orCandicewenttotheparty'canbetranslatedas A B C oras A B C Sincethesetwotranslationsarelogicallyequivalent,wemaywrite A B C Theselattertwoconventionsonlyapplytomultipleconjunctionsormultipledisjunctions.Ifaseriesofconnectivesincludesbothdisjunctionsandconjunctions, thentheparenthesesareessential;aswith A & B C and A & B C .The parenthesesarealsorequiredifthereisaseriesofconditionalsorbiconditionals; aswith A B C and A \$ B \$ C Wehaveadoptedthesefourrulesas notationalconventions ,notaschangesto thedenitionofasentence.Strictlyspeaking, A B C isstillnotasentence. Instead,itisakindofshorthand.Wewriteitforthesakeofconvenience,but wereallymeanthesentence A B C Ifwehadgivenadierentdenitionforaw,thenthesecouldcountasws. Wemighthavewrittenrule3inthisway:If A B ::: Z arews,then

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ch.2sententiallogic 33 A & B & ::: & Z ,isaw."ThiswouldmakeiteasiertotranslatesomeEnglishsentences,butwouldhavethecostofmakingourformallanguagemore complicated.Wewouldhavetokeepthecomplexdenitioninmindwhenwe developtruthtablesandaproofsystem.Wewantalogicallanguagethatis expressivelysimple andallowsustotranslateeasilyfromEnglish,butwealsowant a formallysimple language.Adoptingnotationalconventionsisacompromise betweenthesetwodesires. PracticeExercises ? PartA Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoSL. M: Thosecreaturesaremeninsuits. C: Thosecreaturesarechimpanzees. G: Thosecreaturesaregorillas. 1.Thosecreaturesarenotmeninsuits. 2.Thosecreaturesaremeninsuits,ortheyarenot. 3.Thosecreaturesareeithergorillasorchimpanzees. 4.Thosecreaturesareneithergorillasnorchimpanzees. 5.Ifthosecreaturesarechimpanzees,thentheyareneithergorillasnormen insuits. 6.Unlessthosecreaturesaremeninsuits,theyareeitherchimpanzeesor theyaregorillas. PartB Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoSL. A: MisterAcewasmurdered. B: Thebutlerdidit. C: Thecookdidit. D: TheDuchessislying. E: MisterEdgewasmurdered. F: Themurderweaponwasafryingpan. 1.EitherMisterAceorMisterEdgewasmurdered. 2.IfMisterAcewasmurdered,thenthecookdidit. 3.IfMisterEdgewasmurdered,thenthecookdidnotdoit. 4.Eitherthebutlerdidit,ortheDuchessislying. 5.ThecookdiditonlyiftheDuchessislying.

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34 forall x 6.Ifthemurderweaponwasafryingpan,thentheculpritmusthavebeen thecook. 7.Ifthemurderweaponwasnotafryingpan,thentheculpritwaseither thecookorthebutler. 8.MisterAcewasmurderedifandonlyifMisterEdgewasnotmurdered. 9.TheDuchessislying,unlessitwasMisterEdgewhowasmurdered. 10.IfMisterAcewasmurdered,hewasdoneinwithafryingpan. 11.Thecookdidit,sothebutlerdidnot. 12.OfcoursetheDuchessislying! ? PartC Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoSL. E 1 : Avaisanelectrician. E 2 : Harrisonisanelectrician. F 1 : Avaisareghter. F 2 : Harrisonisareghter. S 1 : Avaissatisedwithhercareer. S 2 : Harrisonissatisedwithhiscareer. 1.AvaandHarrisonarebothelectricians. 2.IfAvaisareghter,thensheissatisedwithhercareer. 3.Avaisareghter,unlesssheisanelectrician. 4.Harrisonisanunsatisedelectrician. 5.NeitherAvanorHarrisonisanelectrician. 6.BothAvaandHarrisonareelectricians,butneitherofthemnditsatisfying. 7.Harrisonissatisedonlyifheisareghter. 8.IfAvaisnotanelectrician,thenneitherisHarrison,butifsheis,thenhe istoo. 9.AvaissatisedwithhercareerifandonlyifHarrisonisnotsatisedwith his. 10.IfHarrisonisbothanelectricianandareghter,thenhemustbesatised withhiswork. 11.ItcannotbethatHarrisonisbothanelectricianandareghter. 12.HarrisonandAvaarebothreghtersifandonlyifneitherofthemisan electrician. ? PartD Giveasymbolizationkeyandsymbolizethefollowingsentencesin SL. 1.AliceandBobarebothspies. 2.IfeitherAliceorBobisaspy,thenthecodehasbeenbroken. 3.IfneitherAlicenorBobisaspy,thenthecoderemainsunbroken.

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ch.2sententiallogic 35 4.TheGermanembassywillbeinanuproar,unlesssomeonehasbrokenthe code. 5.Eitherthecodehasbeenbrokenorithasnot,buttheGermanembassy willbeinanuproarregardless. 6.EitherAliceorBobisaspy,butnotboth. PartE GiveasymbolizationkeyandsymbolizethefollowingsentencesinSL. 1.IfGregorplaysrstbase,thentheteamwilllose. 2.Theteamwillloseunlessthereisamiracle. 3.Theteamwilleitherloseoritwon't,butGregorwillplayrstbaseregardless. 4.Gregor'smomwillbakecookiesifandonlyifGregorplaysrstbase. 5.Ifthereisamiracle,thenGregor'smomwillnotbakecookies. PartF Foreachargument,writeasymbolizationkeyandtranslatetheargumentaswellaspossibleintoSL. 1.IfDorothyplaysthepianointhemorning,thenRogerwakesupcranky. Dorothyplayspianointhemorningunlesssheisdistracted.SoifRoger doesnotwakeupcranky,thenDorothymustbedistracted. 2.ItwilleitherrainorsnowonTuesday.Ifitrains,Nevillewillbesad.If itsnows,Nevillewillbecold.Therefore,Nevillewilleitherbesadorcold onTuesday. 3.IfZoogrememberedtodohischores,thenthingsarecleanbutnotneat. Ifheforgot,thenthingsareneatbutnotclean.Therefore,thingsare eitherneatorclean|butnotboth. ? PartG Foreachofthefollowing:aIsitawofSL?bIsitasentenceof SL,allowingfornotationalconventions? 1. A 2. J 374 _: J 374 3. :::: F 4. : & S 5. G & : G 6. A A 7. A A & : F D \$ E 8.[ Z \$ S W ]&[ J X ] 9. F \$: D J C & D PartH

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36 forall x 1.ArethereanywsofSLthatcontainnosentenceletters?Whyorwhy not? 2.Inthechapter,wesymbolizedan exclusiveor using ,&,and : .How couldyoutranslatean exclusiveor usingonlytwoconnectives?Isthere anywaytotranslatean exclusiveor usingonlyoneconnective?

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Chapter3 Truthtables ThischapterintroducesawayofevaluatingsentencesandargumentsofSL. Althoughitcanbelaborious,thetruthtablemethodisapurelymechanical procedurethatrequiresnointuitionorspecialinsight. 3.1Truth-functionalconnnectives Anynon-atomicsentenceofSLiscomposedofatomicsentenceswithsentential connectives.Thetruth-valueofthecompoundsentencedependsonlyonthe truth-valueoftheatomicsentencesthatcompriseit.Inordertoknowthe truth-valueof D \$ E ,forinstance,youonlyneedtoknowthetruth-value of D andthetruth-valueof E .Connectivesthatworkinthiswayarecalled truth-functional Inthischapter,wewillmakeuseofthefactthatallofthelogicaloperators inSLaretruth-functional|itmakesitpossibletoconstructtruthtablesto determinethelogicalfeaturesofsentences.Youshouldrealize,however,that thisisnotpossibleforalllanguages.InEnglish,itispossibletoformanew sentencefromanysimplersentence X bysaying`Itispossiblethat X .'The truth-valueofthisnewsentencedoesnotdependdirectlyonthetruth-valueof X .Evenif X isfalse,perhapsinsomesense X could havebeentrue|thenthe newsentencewouldbetrue.Someformallanguages,called modallogics ,have anoperatorforpossibility.Inamodallogic,wecouldtranslate`Itispossible that X 'as X .However,theabilitytotranslatesentenceslikethesecomeat acost:The operatorisnottruth-functional,andsomodallogicsarenot amenabletotruthtables. 37

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38 forall x A : A T F F T A B A & B A B A B A \$ B T T T T T T T F F T F F F T F T T F F F F F T T Table3.1:ThecharacteristictruthtablesfortheconnectivesofSL. 3.2Completetruthtables Thetruth-valueofsentencesthatcontainonlyoneconnectiveisgivenbythe characteristictruthtableforthatconnective.Toputthemallinoneplace,the truthtablesfortheconnectivesofSLarerepeatedintable3.1. Thecharacteristictruthtableforconjunction,forexample,givesthetruthconditionsforanysentenceoftheform A & B .Eveniftheconjuncts A and B are long,complicatedsentences,theconjunctionistrueifandonlyifboth A and B aretrue.Considerthesentence H & I H .Weconsiderallthepossible combinationsoftrueandfalsefor H and I ,whichgivesusfourrows.Wethen copythetruth-valuesforthesentencelettersandwritethemunderneaththe lettersinthesentence. H I H & I H T T TTT T F TFT F T FTF F F FFF Nowconsiderthesubsentence H & I .Thisisaconjunction A & B with H as A andwith I as B H and I arebothtrueontherstrow.Sinceaconjunction istruewhenbothconjunctsaretrue,wewriteaTunderneaththeconjunction symbol.Wecontinuefortheotherthreerowsandgetthis: H I H & I H A & B T T T T TT T F T F FT F T F F TF F F F F FF Theentiresentenceisaconditional A B with H & I as A andwith H as B .Onthesecondrow,forexample, H & I isfalseand H istrue.Sincea conditionalistruewhentheantecedentisfalse,wewriteaTinthesecondrow

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ch.3truthtables 39 underneaththeconditionalsymbol.Wecontinuefortheotherthreerowsand getthis: H I H & I H A B T T T T T T F F T T F T F T F F F F T F ThecolumnofTsunderneaththeconditionaltellsusthatthesentence H & I I istrueregardlessofthetruth-valuesof H and I .Theycanbetrueorfalsein anycombination,andthecompoundsentencestillcomesouttrue.Itiscrucial thatwehaveconsideredallofthepossiblecombinations.Ifweonlyhadatwolinetruthtable,wecouldnotbesurethatthesentencewasnotfalseforsome othercombinationoftruth-values. Inthisexample,wehavenotrepeatedalloftheentriesineverysuccessivetable. Whenactuallywritingtruthtablesonpaper,however,itisimpracticaltoerase wholecolumnsorrewritethewholetableforeverystep.Althoughitismore crowded,thetruthtablecanbewritteninthisway: H I H & I H T T TTTTT T F TFFTT F T FFTTF F F FFFTF Mostofthecolumnsunderneaththesentenceareonlythereforbookkeeping purposes.Whenyoubecomemoreadeptwithtruthtables,youwillprobably nolongerneedtocopyoverthecolumnsforeachofthesentenceletters.Inany case,thetruth-valueofthesentenceoneachrowisjustthecolumnunderneath themainlogicaloperatorofthesentence;inthiscase,thecolumnunderneath theconditional. A completetruthtable hasarowforallthepossiblecombinationsofTand Fforallofthesentenceletters.Thesizeofthecompletetruthtabledependson thenumberofdierentsentencelettersinthetable.Asentencethatcontains onlyonesentenceletterrequiresonlytworows,asinthecharacteristictruth tablefornegation.Thisistrueevenifthesameletterisrepeatedmanytimes, asinthesentence[ C \$ C C ]& : C C .Thecompletetruthtable requiresonlytwolinesbecausethereareonlytwopossibilities: C canbetrue oritcanbefalse.AsinglesentencelettercanneverbemarkedbothTandF onthesamerow.Thetruthtableforthissentencelookslikethis:

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40 forall x C [ C \$ C C ]& : C C T TTTTT F FTTT F FTFFF F FFTF Lookingatthecolumnunderneaththemainconnective,weseethatthesentence isfalseonbothrowsofthetable;i.e.,itisfalseregardlessofwhether C istrue orfalse. Asentencethatcontainstwosentencelettersrequiresfourlinesforacomplete truthtable,asinthecharacteristictruthtablesandthetablefor H & I I Asentencethatcontainsthreesentencelettersrequireseightlines.Forexample: M N P M & N P T T T T T TTT T T F T T TTF T F T T T FTT T F F T F FFF F T T F F TTT F T F F F TTF F F T F F FTT F F F F F FFF Fromthistable,weknowthatthesentence M & N P mightbetrueorfalse, dependingonthetruth-valuesof M N ,and P Acompletetruthtableforasentencethatcontainsfourdierentsentenceletters requires16lines.Fiveletters,32lines.Sixletters,64lines.Andsoon.Tobe perfectlygeneral:Ifacompletetruthtablehas n dierentsentenceletters,then itmusthave2 n rows. Inordertollinthecolumnsofacompletetruthtable,beginwiththerightmostsentenceletterandalternateTsandFs.Inthenextcolumntotheleft, writetwoTs,writetwoFs,andrepeat.Forthethirdsentenceletter,writefour TsfollowedbyfourFs.Thisyieldsaneightlinetruthtableliketheoneabove. Fora16linetruthtable,thenextcolumnofsentencelettersshouldhaveeight TsfollowedbyeightFs.Fora32linetable,thenextcolumnwouldhave16Ts followedby16Fs.Andsoon.

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ch.3truthtables 41 3.3Usingtruthtables Tautologies,contradictions,andcontingentsentences RecallthatanEnglishsentenceisatautologyifitmustbetrueasamatterof logic.Withacompletetruthtable,weconsiderallofthewaysthattheworld mightbe.Ifthesentenceistrueoneverylineofacompletetruthtable,thenit istrueasamatteroflogic,regardlessofwhattheworldislike. Soasentenceisa tautologyinsl ifthecolumnunderitsmainconnectiveis Toneveryrowofacompletetruthtable. Conversely,asentenceisa contradictioninsl ifthecolumnunderitsmain connectiveisFoneveryrowofacompletetruthtable. Asentenceis contingentinsl ifitisneitheratautologynoracontradiction; i.e.ifitisTonatleastonerowandFonatleastonerow. Fromthetruthtablesintheprevioussection,weknowthat H & I H is atautology,that[ C \$ C C ]& : C C isacontradiction,andthat M & N P iscontingent. Logicalequivalence TwosentencesarelogicallyequivalentinEnglishiftheyhavethesametruth valueasamatterlogic.Onceagain,truthtablesallowustodeneananalogous conceptforSL:Twosentencesare logicallyequivalentinsl iftheyhave thesametruth-valueoneveryrowofacompletetruthtable. Considerthesentences : A B and : A & : B .Aretheylogicallyequivalent? Tondout,weconstructatruthtable. A B : A B : A & : B T T F TTT FT F FT T F F TTF FT F TF F T F FTT TF F FT F F T FFF TF T TF Lookatthecolumnsforthemainconnectives;negationfortherstsentence, conjunctionforthesecond.Ontherstthreerows,bothareF.Onthenal row,bothareT.Sincetheymatchoneveryrow,thetwosentencesarelogically equivalent.

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42 forall x Consistency AsetofsentencesinEnglishisconsistentifitislogicallypossibleforthemall tobetrueatonce.Asetofsentencesis logicallyconsistentinsl ifthere isatleastonelineofacompletetruthtableonwhichallofthesentencesare true.Itis inconsistent otherwise. Validity AnargumentinEnglishisvalidifitislogicallyimpossibleforthepremisesto betrueandfortheconclusiontobefalseatthesametime.Anargumentis validinsl ifthereisnorowofacompletetruthtableonwhichthepremises areallTandtheconclusionisF;anargumentis invalidinsl ifthereissuch arow. Considerthisargument: : L J L : L : :J Isitvalid?Tondout,weconstructatruthtable. J L : L J L : L J T T FT T TTT F T T T F TF T TTF T F T F T FT T FTT F T F F F TF F FFF T F F Yes,theargumentisvalid.TheonlyrowonwhichboththepremisesareTis thesecondrow,andonthatrowtheconclusionisalsoT. 3.4Partialtruthtables Inordertoshowthatasentenceisatautology,weneedtoshowthatitisTon everyrow.Soweneedacompletetruthtable.Toshowthatasentenceis not atautology,however,weonlyneedoneline:alineonwhichthesentenceisF. Therefore,inordertoshowthatsomethingisnotatautology,itisenoughto provideaone-line partialtruthtable |regardlessofhowmanysentenceletters thesentencemighthaveinit.

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ch.3truthtables 43 Consider,forexample,thesentence U & T S & W .Wewanttoshowthat itis not atautologybyprovidingapartialtruthtable.WellinFfortheentire sentence.Themainconnectiveofthesentenceisaconditional.Inorderforthe conditionaltobefalse,theantecedentmustbetrueTandtheconsequent mustbefalseF.Sowelltheseinonthetable: S T U W U & T S & W T F F Inorderforthe U & T tobetrue,both U and T mustbetrue. S T U W U & T S & W T T TTT F F Nowwejustneedtomake S & W false.Todothis,weneedtomakeatleast oneof S and W false.Wecanmakeboth S and W falseifwewant.All thatmattersisthatthewholesentenceturnsoutfalseonthisline.Makingan arbitrarydecision,wenishthetableinthisway: S T U W U & T S & W F T T F TTT F FFF Showingthatsomethingisacontradictionrequiresacompletetruthtable. Showingthatsomethingis not acontradictionrequiresonlyaone-linepartial truthtable,wherethesentenceistrueonthatoneline. Asentenceiscontingentifitisneitheratautologynoracontradiction.So showingthatasentenceiscontingentrequiresa two-line partialtruthtable: Thesentencemustbetrueononelineandfalseontheother.Forexample,we canshowthatthesentenceaboveiscontingentwiththistruthtable: S T U W U & T S & W F T T F TTT F FFF F T F F FFT T FFF Notethattherearemanycombinationsoftruthvaluesthatwouldhavemade thesentencetrue,sotherearemanywayswecouldhavewrittenthesecond line. Showingthatasentenceis not contingentrequiresprovidingacompletetruth table,becauseitrequiresshowingthatthesentenceisatautologyorthatitisa contradiction.Ifyoudonotknowwhetheraparticularsentenceiscontingent, thenyoudonotknowwhetheryouwillneedacompleteorpartialtruthtable.

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44 forall x YES NO tautology? completetruthtable one-linepartialtruthtable contradiction? completetruthtable one-linepartialtruthtable contingent? two-linepartialtruthtable completetruthtable equivalent? completetruthtable one-linepartialtruthtable consistent? one-linepartialtruthtable completetruthtable valid? completetruthtable one-linepartialtruthtable Table3.2:Doyouneedacompletetruthtableorapartialtruthtable?It dependsonwhatyouaretryingtoshow. Youcanalwaysstartworkingonacompletetruthtable.Ifyoucompleterows thatshowthesentenceiscontingent,thenyoucanstop.Ifnot,thencompletethetruthtable.Eventhoughtwocarefullyselectedrowswillshowthat acontingentsentenceiscontingent,thereisnothingwrongwithllinginmore rows. Showingthattwosentencesarelogicallyequivalentrequiresprovidingacompletetruthtable.Showingthattwosentencesare not logicallyequivalentrequiresonlyaone-linepartialtruthtable:Makethetablesothatonesentence istrueandtheotherfalse. Showingthatasetofsentencesisconsistentrequiresprovidingonerowofatruth tableonwhichallofthesentencesaretrue.Therestofthetableisirrelevant, soaone-linepartialtruthtablewilldo.Showingthatasetofsentencesis inconsistent,ontheotherhand,requiresacompletetruthtable:Youmust showthatoneveryrowofthetableatleastoneofthesentencesisfalse. Showingthatanargumentisvalidrequiresacompletetruthtable.Showing thatanargumentis invalid onlyrequiresprovidingaone-linetruthtable:If youcanproducealineonwhichthepremisesarealltrueandtheconclusionis false,thentheargumentisinvalid. Table3.2summarizeswhenacompletetruthtableisrequiredandwhenapartial truthtablewilldo. PracticeExercises Ifyouwantadditionalpractice,youcanconstructtruthtablesforanyofthe sentencesandargumentsintheexercisesforthepreviouschapter. ? PartA Determinewhethereachsentenceisatautology,acontradiction,ora contingentsentence.Justifyyouranswerwithacompleteorpartialtruthtable

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ch.3truthtables 45 whereappropriate. 1. A A 2. : B & B 3. C !: C 4. : D D 5. A \$ B \$: A \$: B 6. A & B B & A 7. A B B A 8. : [ A B A ] 9. A & B B A 10. A \$ [ A B & : B ] 11. : A B \$ : A & : B 12. : A & B \$ A 13. A & B & : A & B & C 14. A B C 15.[ A & B & C ] B 16. A & : A B C 17. : C A B 18. B & D \$ [ A \$ A C ] ? PartB Determinewhethereachpairofsentencesislogicallyequivalent. Justifyyouranswerwithacompleteorpartialtruthtablewhereappropriate. 1. A : A 2. A A A 3. A A A \$ A 4. A _: B A B 5. A & : A : B \$ B 6. : A & B : A _: B 7. : A B : A !: B 8. A B : B !: A 9.[ A B C ],[ A B C ] 10.[ A B & C ],[ A B & C ] ? PartC Determinewhethereachsetofsentencesisconsistentorinconsistent. Justifyyouranswerwithacompleteorpartialtruthtablewhereappropriate. 1. A A : A !: A A & A A A 2. A & B C !: B C 3. A B A C B C 4. A B B C A : C 5. B & C A A B : B C 6. A B B C C !: A

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ch.3truthtables 47 1. ?A B 2. ?A & B 3. ?A \$ B WecouldhavealanguagethatisequivalenttoSLwithonlynegationand disjunctionasconnectives.Showthis:Usingonlyparentheses,sentenceletters, negation : ,anddisjunction ,writesentencesthatarelogicallyequivalent toeachofthefollowing. 4. A & B 5. A B 6. A \$ B The Sheerstroke isalogicalconnectivewiththefollowingcharacteristictruthtable: A B A j B T T F T F T F T T F F T 7.WriteasentenceusingtheconnectivesofSLthatislogicallyequivalent to A j B EverysentencewrittenusingaconnectiveofSLcanberewrittenasalogically equivalentsentenceusingoneormoreSheerstrokes.UsingonlytheSheer stroke,writesentencesthatareequivalenttoeachofthefollowing. 8. : A 9. A & B 10. A B 11. A B 12. A \$ B

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Chapter4 Quantiedlogic ThischapterintroducesalogicallanguagecalledQL.Itisaversionof quantied logic ,becauseitallowsforquantierslike all and some .Quantiedlogicisalso sometimescalled predicatelogic ,becausethebasicunitsofthelanguageare predicatesandterms. 4.1Fromsentencestopredicates Considerthefollowingargument,whichisobviouslyvalidinEnglish: Ifeveryoneknowslogic,theneithernoonewillbeconfusedoreveryonewill.Everyonewillbeconfusedonlyifwetrytobelievea contradiction.Thisisalogicclass,soeveryoneknowslogic. : : Ifwedon'ttrytobelieveacontradiction,thennoonewillbe confused. InordertosymbolizethisinSL,wewillneedasymbolizationkey. L: Everyoneknowslogic. N: Noonewillbeconfused. E: Everyonewillbeconfused. B: Wetrytobelieveacontradiction. Noticethat N and E arebothaboutpeoplebeingconfused,buttheyaretwo separatesentenceletters.Wecouldnotreplace E with : N .Whynot? : N means`Itisnotthecasethatnoonewillbeconfused.'Thiswouldbethecase 48

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ch.4quantiedlogic 49 ifevenonepersonwereconfused,soitisalongwayfromsayingthat everyone willbeconfused. Oncewehaveseparatesentencelettersfor N and E ,however,weeraseany connectionbetweenthetwo.Theyarejusttwoatomicsentenceswhichmight betrueorfalseindependently.InEnglish,itcouldneverbethecasethat bothnooneandeveryonewasconfused.AssentencesofSL,however,thereisa truth-valueassignmentforwhich N and E arebothtrue. Expressionslike`noone',`everyone',and`anyone'arecalled quantiers .By translating N and E asseparateatomicsentences,weleaveoutthe quantier structure ofthesentences.Fortunately,thequantierstructureisnotwhat makesthisargumentvalid.Assuch,wecansafelyignoreit.Toseethis,we translatetheargumenttoSL: L N E E B L : : : B N ThisisavalidargumentinSL.Youcandoatruthtabletocheckthis. Nowconsideranotherargument.ThisoneisalsovalidinEnglish. Willardisalogician.Alllogicianswearfunnyhats. : : Willardwearsafunnyhat. TosymbolizeitinSL,wedeneasymbolizationkey: L: Willardisalogician. A: Alllogicianswearfunnyhats. F: Willardwearsafunnyhat. Nowwesymbolizetheargument: L A : :F Thisis invalid inSL.Again,youcanconrmthiswithatruthtable.Thereis somethingverywronghere,becausethisisclearlyavalidargumentinEnglish. ThesymbolizationinSLleavesoutalltheimportantstructure.Onceagain,

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50 forall x thetranslationtoSLoverlooksquantierstructure:Thesentence`Alllogicians wearfunnyhats'isaboutbothlogiciansandhat-wearing.Bynottranslating thisstructure,welosetheconnectionbetweenWillard'sbeingalogicianand Willard'swearingahat. SomeargumentswithquantierstructurecanbecapturedinSL,liketherst example,eventhoughSLignoresthequantierstructure.Otherargumentsare completelybotchedinSL,likethesecondexample.Noticethattheproblem isnotthatwehavemadeamistakewhilesymbolizingthesecondargument. Thesearethebestsymbolizationswecangiveforthesearguments inSL Generally,ifanargumentcontainingquantierscomesout validinSL ,then theEnglishlanguageargumentisvalid.Ifitcomesout invalidinSL ,thenwe cannotsaytheEnglishlanguageargumentisinvalid.Theargumentmightbe validbecauseofquantierstructurewhichthenaturallanguageargumenthas andwhichtheargumentinSLlacks. Similarly,ifasentencewithquantierscomesoutasa tautologyinSL ,thenthe Englishsentenceislogicallytrue.Ifcomesoutas contingentinSL ,thenthis mightbebecauseofthestructureofthequantiersthatgetsremovedwhenwe translateintotheformallanguage. Inordertosymbolizeargumentsthatrelyonquantierstructure,weneedto developadierentlogicallanguage.Wewillcallthislanguagequantiedlogic, QL. 4.2BuildingblocksofQL Justassentenceswerethebasicunitofsententiallogic,predicateswillbethe basicunitofquantiedlogic.Apredicateisanexpressionlike`isadog.'This isnotasentenceonitsown.Itisneithertruenorfalse.Inordertobetrueor false,weneedtospecifysomething:Whoorwhatisitthatisadog? Thedetailsofthiswillbeexplainedintherestofthechapter,buthereis thebasicidea:InQL,wewillrepresentpredicateswithcapitalletters.For instance,wemightlet D standfor` isadog.'Wewilluselower-case lettersasthenamesofspecicthings.Forinstance,wemightlet b standfor Bertie.Theexpression Db willbeasentenceinQL.Itisatranslationofthe sentence`Bertieisadog.' Inordertorepresentquantierstructure,wewillalsohavesymbolsthatrepresentquantiers.Forinstance,` 9 'willmean`Thereissome .'Sotosay thatthereisadog,wecanwrite 9 xDx ;thatis:Thereissome x suchthat x is adog.

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ch.4quantiedlogic 51 Thatwillcomelater.Westartbydeningsingulartermsandpredicates. SingularTerms InEnglish,a singularterm isawordorphrasethatreferstoa specic person, place,orthing.Theword`dog'isnotasingularterm,becausethereareagreat manydogs.Thephrase`Philip'sdogBertie'isasingularterm,becauseitrefers toaspeciclittleterrier. A propername isasingulartermthatpicksoutanindividualwithoutdescribingit.Thename`Emerson'isapropername,andthenamealonedoesnottell youanythingaboutEmerson.Ofcourse,somenamesaretraditionallygivento boysandotheraretraditionallygiventogirls.If`JackHathaway'isusedasa singularterm,youmightguessthatitreferstoaman.However,thenamedoes notnecessarilymeanthatthepersonreferredtoisaman|oreventhatthe creaturereferredtoisaperson.Jackmightbeagiraeforallyoucouldtelljust fromthename.Thereisagreatdealofphilosophicalactionsurroundingthis issue,buttheimportantpointhereisthatanameisasingulartermbecauseit picksoutasingle,specicindividual. Othersingulartermsmoreobviouslyconveyinformationaboutthethingto whichtheyrefer.Forinstance,youcantellwithoutbeingtoldanythingfurther that`Philip'sdogBertie'isasingulartermthatreferstoadog.A definite description picksoutanindividualbymeansofauniquedescription.In English,denitedescriptionsareoftenphrasesoftheform`thesuch-and-so.' Theyreferto the specicthingthatmatchesthegivendescription.Forexample, `thetallestmemberofMontyPython'and`therstemperorofChina'are denitedescriptions.Adescriptionthatdoesnotpickoutaspecicindividual isnotadenitedescription.`AmemberofMontyPython'and`anemperorof China'arenotdenitedescriptions. Wecanusepropernamesanddenitedescriptionstopickoutthesamething. Thepropername`MountRainier'namesthelocationpickedoutbythedenite description`thehighestpeakinWashingtonstate.'Theexpressionsrefertothe sameplaceindierentways.YoulearnnothingfrommysayingthatIamgoing toMountRainier,unlessyoualreadyknowsomegeography.Youcouldguess thatitisamountain,perhaps,buteventhisisnotasurething;forallyou knowitmightbeacollege,likeMountHolyoke.YetifIweretosaythatIwas goingtothehighestpeakinWashingtonstate,youwouldknowimmediately thatIwasgoingtoamountaininWashingtonstate. InEnglish,thespecicationofasingulartermmaydependoncontext;`Willard' meansaspecicpersonandnotjustsomeonenamedWillard;`P.D.Magnus' asalogicalsingulartermmeans me andnottheotherP.D.Magnus.Welive withthiskindofambiguityinEnglish,butitisimportanttokeepinmindthat

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52 forall x singulartermsinQLmustrefertojustonespecicthing. InQL,wewillsymbolizesingulartermswithlower-caseletters a through w Wecanaddsubscriptsifwewanttousesomelettermorethanonce.So a;b;c;:::w;a 1 ;f 32 ;j 390 ,and m 12 arealltermsinQL. Singulartermsarecalled constants becausetheypickoutspecicindividuals. Notethat x;y ,and z arenotconstantsinQL.Theywillbe variables ,letters whichdonotstandforanyspecicthing.Wewillneedthemwhenweintroduce quantiers. Predicates Thesimplestpredicatesarepropertiesofindividuals.Theyarethingsyoucan sayaboutanobject.` isadog'and` isamemberofMontyPython' arebothpredicates.IntranslatingEnglishsentences,thetermwillnotalways comeatthebeginningofthesentence:`Apianofellon 'isalsoapredicate. Predicateslikethesearecalled one-place or monadic ,becausethereisonly oneblanktollin.Aone-placepredicateandasingulartermcombinetomake asentence. Otherpredicatesareaboutthe relation betweentwothings.Forinstance,` isbiggerthan ',` istotheleftof ',and` owesmoneyto .'Theseare two-place or dyadic predicates,becausetheyneedtobe lledinwithtwotermsinordertomakeasentence. Ingeneral,youcanthinkaboutpredicatesasschematicsentencesthatneedtobe lledoutwithsomenumberofterms.Conversely,youcanstartwithsentences andmakepredicatesoutofthembyremovingterms.Considerthesentence, `VinnieborrowedthefamilycarfromNunzio.'Byremovingasingularterm,we canrecognizethissentenceasusinganyofthreedierentmonadicpredicates: borrowedthefamilycarfromNunzio. Vinnieborrowed fromNunzio. Vinnieborrowedthefamilycarfrom Byremovingtwosingularterms,wecanrecognizethreedierentdyadicpredicates: Vinnieborrowed from borrowedthefamilycarfrom borrowed fromNunzio. Byremovingallthreesingularterms,wecanrecognizeone three-place or

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ch.4quantiedlogic 53 triadic predicate: borrowed from IfwearetranslatingthissentenceintoQL,shouldwetranslateitwithaone-, two-,orthree-placepredicate?Itdependsonwhatwewanttobeabletosay. Iftheonlythingthatwewilldiscussbeingborrowedisthefamilycar,then thegeneralityofthethree-placepredicateisunnecessary.Iftheonlyborrowing weneedtosymbolizeisdierentpeopleborrowingthefamilycarfromNunzio, thenaone-placepredicatewillbeenough. Ingeneral,wecanhavepredicateswithasmanyplacesasweneed.Predicates withmorethanoneplacearecalled polyadic .Predicateswith n places,for somenumber n ,arecalled n-place or n-adic InQL,wesymbolizepredicateswithcapitalletters A through Z ,withorwithout subscripts.Whenwegiveasymbolizationkeyforpredicates,wewillnotuse blanks;instead,wewillusevariables.Byconvention,constantsarelistedatthe endofthekey.Sowemightwriteakeythatlookslikethis: Ax: x isangry. Hx: x ishappy. T 1 xy: x isastallortallerthan y T 2 xy: x isastoughortougherthan y Bxyz: y isbetween x and z d: Donald g: Gregor m: Marybeth Wecansymbolizesentencesthatuseanycombinationofthesepredicatesand terms.Forexample: 1.Donaldisangry. 2.IfDonaldisangry,thensoareGregorandMarybeth. 3.MarybethisatleastastallandastoughasGregor. 4.DonaldisshorterthanGregor. 5.GregorisbetweenDonaldandMarybeth. Sentence1isstraightforward: Ad .The` x 'inthekeyentry` Ax 'isjusta placeholder;wecanreplaceitwithothertermswhentranslating. Sentence2canbeparaphrasedas,`If Ad ,then Ag and Am .'QLhasallthe truth-functionalconnectivesofSL,sowetranslatethisas Ad Ag & Am Sentence3canbetranslatedas T 1 mg & T 2 mg .

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54 forall x Sentence4mightseemasifitrequiresanewpredicate.Ifweonlyneeded tosymbolizethissentence,wecoulddeneapredicatelike Sxy tomean` x isshorterthan y .'However,thiswouldignorethelogicalconnectionbetween `shorter'and`taller.'ConsideredonlyassymbolsofQL,thereisnoconnection between S and T 1 .Theymightmeananythingatall.Insteadofintroducinga newpredicate,weparaphrasesentence4usingpredicatesalreadyinourkey:`It isnotthecasethatDonaldisastallerortallerthanGregor.'Wecantranslate itas : T 1 dg Sentence5requiresthatwepaycarefulattentiontotheorderoftermsinthe key.Itbecomes Bdgm 4.3Quantiers Wearenowreadytointroducequantiers.Considerthesesentences: 6.Everyoneishappy. 7.EveryoneisatleastastoughasDonald. 8.Someoneisangry. Itmightbetemptingtotranslatesentence6as Hd & Hg & Hm .Yetthiswould onlysaythatDonald,Gregor,andMarybetharehappy.Wewanttosaythat everyone ishappy,evenifwehavenotdenedaconstanttonamethem.In ordertodothis,weintroducethe` 8 'symbol.Thisiscalledthe universal quantifier Aquantiermustalwaysbefollowedbyavariableandaformulathatincludes thatvariable.Wecantranslatesentence6as 8 xHx .ParaphrasedinEnglish, thismeans`Forall x x ishappy.'Wecall 8 x an x-quantier .Theformulathat followsthequantieriscalledthe scope ofthequantier.Wewillgiveaformal denitionofscopelater,butintuitivelyitisthepartofthesentencethatthe quantierquantiesover.In 8 xHx ,thescopeoftheuniversalquantieris Hx Sentence7canbeparaphrasedas,`Forall x x isatleastastoughasDonald.' Thistranslatesas 8 xT 2 xd Inthesequantiedsentences,thevariable x isservingasakindofplaceholder. Theexpression 8 x meansthatyoucanpickanyoneandputtheminas x .There isnospecialreasontouse x ratherthansomeothervariable.Thesentence 8 xHx meansexactlythesamethingas 8 yHy 8 zHz ,and 8 x 5 Hx 5 Totranslatesentence8,weintroduceanothernewsymbol:the existential quantifier 9 .Liketheuniversalquantier,theexistentialquantierrequires avariable.Sentence8canbetranslatedas 9 xAx .Thismeansthatthereis

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ch.4quantiedlogic 55 some x whichisangry.Moreprecisely,itmeansthatthereis atleastone angry person.Onceagain,thevariableisakindofplaceholder;wecouldjustaseasily havetranslatedsentence8as 9 zAz Considerthesefurthersentences: 9.Nooneisangry. 10.Thereissomeonewhoisnothappy. 11.Noteveryoneishappy. Sentence9canbeparaphrasedas,`Itisnotthecasethatsomeoneisangry.' Thiscanbetranslatedusingnegationandanexistentialquantier: :9 xAx Yetsentence9couldalsobeparaphrasedas,`Everyoneisnotangry.'Withthis inmind,itcanbetranslatedusingnegationandauniversalquantier: 8 x : Ax Bothoftheseareacceptabletranslations,becausetheyarelogicallyequivalent. Thecriticalthingiswhetherthenegationcomesbeforeorafterthequantier. Ingeneral, 8 x A islogicallyequivalentto :9 x : A .Thismeansthatanysentence whichcanbesymbolizedwithauniversalquantiercanbesymbolizedwithan existentialquantier,andviceversa.Onetranslationmightseemmorenatural thantheother,butthereisnologicaldierentintranslatingwithonequantier ratherthantheother.Forsomesentences,itwillsimplybeamatteroftaste. Sentence10ismostnaturallyparaphrasedas,`Thereissome x suchthat x is nothappy.'Thisbecomes 9 x : Hx .Equivalently,wecouldwrite :8 xHx Sentence11ismostnaturallytranslatedas :8 xHx .Thisislogicallyequivalent tosentence10andsocouldalsobetranslatedas 9 x : Hx AlthoughwehavetwoquantiersinQL,wecouldhaveanequivalentformal languagewithonlyonequantier.Wecouldproceedwithonlytheuniversal quantier,forinstance,andtreattheexistentialquantierasanotationalconvention.Weusesquarebrackets[]tomakesomesentencesmorereadable,but weknowthatthesearereallyjustparentheses.Inthesameway,wecould write` 9 x 'knowingthatthisisjustshorthandfor` :8 x : .'Thereisachoice betweenmakinglogicformallysimpleandmakingitexpressivelysimple.With QL,weoptforexpressivesimplicity.Both 8 and 9 willbesymbolsofQL. UniverseofDiscourse Giventhesymbolizationkeywehavebeenusing, 8 xHx means`Everyoneis happy.'Whoisincludedinthis everyone ?Whenweusesentenceslikethisin English,weusuallydonotmeaneveryonenowaliveontheEarth.Wecertainly donotmeaneveryonewhowaseveraliveorwhowilleverlive.Wemean

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56 forall x somethingmoremodest:everyoneinthebuilding,everyoneintheclass,or everyoneintheroom. Inordertoeliminatethisambiguity,wewillneedtospecifya universeof discourse |abbreviatedUD.TheUDisthesetofthingsthatwearetalking about.SoifwewanttotalkaboutpeopleinChicago,wedenetheUDtobe peopleinChicago.Wewritethisatthebeginningofthesymbolizationkey,like this: UD: peopleinChicago Thequantiers rangeover theuniverseofdiscourse.GiventhisUD, 8 x means `EveryoneinChicago'and 9 x means`SomeoneinChicago.'Eachconstant namessomememberoftheUD,sowecanonlyusethisUDwiththesymbolizationkeyaboveifDonald,Gregor,andMarybethareallinChicago.Ifwe wanttotalkaboutpeopleinplacesbesidesChicago,thenweneedtoinclude thosepeopleintheUD. InQL,theUDmustbe non-empty ;thatis,itmustincludeatleastonething. ItispossibletoconstructformallanguagesthatallowforemptyUDs,butthis introducescomplications. EvenallowingforaUDwithjustonemembercanproducesomestrangeresults. Supposewehavethisasasymbolizationkey: UD: theEielTower Px: x isinParis. Thesentence 8 xPx mightbeparaphrasedinEnglishas`EverythingisinParis.' Yetthatwouldbemisleading.Itmeansthateverything intheUD isinParis. ThisUDcontainsonlytheEielTower,sowiththissymbolizationkey 8 xPx justmeansthattheEielTowerisinParis. Non-referringterms InQL,eachconstantmustpickoutexactlyonememberoftheUD.Aconstant cannotrefertomorethanonething|itisa singular term.Eachconstantmust stillpickout something .Thisisconnectedtoaclassicphilosophicalproblem: theso-calledproblemofnon-referringterms. Medievalphilosopherstypicallyusedsentencesaboutthe chimera toexemplify thisproblem.Chimeraisamythologicalcreature;itdoesnotreallyexist. Considerthesetwosentences:

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ch.4quantiedlogic 57 12.Chimeraisangry. 13.Chimeraisnotangry. Itistemptingjusttodeneaconstanttomean`chimera.'Thesymbolization keywouldlooklikethis: UD: creaturesonEarth Ax: x isangry. c: chimera Wecouldthentranslatesentence12as Ac andsentence13as : Ac Problemswillarisewhenweaskwhetherthesesentencesaretrueorfalse. Oneoptionistosaythatsentence12isnottrue,becausethereisnochimera. Ifsentence12isfalsebecauseittalksaboutanon-existentthing,thensentence 13isfalseforthesamereason.Yetthiswouldmeanthat Ac and : Ac would bothbefalse.Giventhetruthconditionsfornegation,thiscannotbethecase. Sincewecannotsaythattheyarebothfalse,whatshouldwedo?Anotheroption istosaythatsentence12is meaningless becauseittalksaboutanon-existent thing.So Ac wouldbeameaningfulexpressioninQLforsomeinterpretations butnotforothers.Yetthiswouldmakeourformallanguagehostagetoparticularinterpretations.Sinceweareinterestedinlogicalform,wewanttoconsider thelogicalforceofasentencelike Ac apartfromanyparticularinterpretation. If Ac weresometimesmeaningfulandsometimesmeaningless,wecouldnotdo that. Thisisthe problemofnon-referringterms ,andwewillreturntoitlatersee p.74.TheimportantpointfornowisthateachconstantofQL must referto somethingintheUD,althoughtheUDcanbeanysetofthingsthatwelike. Ifwewanttosymbolizeargumentsaboutmythologicalcreatures,thenwemust deneaUDthatincludesthem.Thisoptionisimportantifwewanttoconsider thelogicofstories.Wecantranslateasentencelike`SherlockHolmeslivedat 221BBakerStreet'byincludingctionalcharacterslikeSherlockHolmesinour UD. 4.4TranslatingtoQL WenowhaveallofthepiecesofQL.Translatingmorecomplicatedsentences willonlybeamatterofknowingtherightwaytocombinepredicates,constants, quantiers,andconnectives.Considerthesesentences:

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58 forall x 14.Everycoininmypocketisaquarter. 15.Somecoinonthetableisadime. 16.Notallthecoinsonthetablearedimes. 17.Noneofthecoinsinmypocketaredimes. Inprovidingasymbolizationkey,weneedtospecifyaUD.Sincewearetalking aboutcoinsinmypocketandonthetable,theUDmustatleastcontainallof thosecoins.Sincewearenottalkingaboutanythingbesidescoins,weletthe UDbeallcoins.Sincewearenottalkingaboutanyspeciccoins,wedonot needtodeneanyconstants.Sowedenethiskey: UD: allcoins Px: x isinmypocket. Tx: x isonthetable. Qx: x isaquarter. Dx: x isadime. Sentence14ismostnaturallytranslatedwithauniversalquantier.TheuniversalquantiersayssomethingabouteverythingintheUD,notjustaboutthe coinsinmypocket.Sentence14meansthat,foranycoin, if thatcoinisinmy pocket then itisaquarter.Sowecantranslateitas 8 x Px Qx Sincesentence14isaboutcoinsthatarebothinmypocket and thatarequarters,itmightbetemptingtotranslateitusingaconjunction.However,the sentence 8 x Px & Qx wouldmeanthateverythingintheUDisbothinmy pocketandaquarter:Allthecoinsthatexistarequartersinmypocket.This iswouldbeacrazythingtosay,anditmeanssomethingverydierentthan sentence14. Sentence15ismostnaturallytranslatedwithanexistentialquantier.Itsays thatthereissomecoinwhichisbothonthetableandwhichisadime.Sowe cantranslateitas 9 x Tx & Dx Noticethatweneededtouseaconditionalwiththeuniversalquantier,but weusedaconjunctionwiththeexistentialquantier.Whatwoulditmeanto write 9 x Tx Dx ?Probablynotwhatyouthink.Itmeansthatthereissome memberoftheUDwhichwouldsatisfythesubformula;roughlyspeaking,there issome a suchthat Ta Da istrue.InSL, A B islogicallyequivalentto : A B ,andthiswillalsoholdinQL.So 9 x Tx Dx istrueifthereissome a suchthat : Ta Da ;i.e.,itistrueifsomecoinis either notonthetable or isadime.Ofcoursethereisacointhatisnotthetable|therearecoinslots ofotherplaces.So 9 x Tx Dx istriviallytrue.Aconditionalwillusually bethenaturalconnectivetousewithauniversalquantier,butaconditional withinthescopeofanexistentialquantiercandoverystrangethings.Asa generalrule,donotputconditionalsinthescopeofexistentialquantiersunless youaresurethatyouneedone.

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ch.4quantiedlogic 59 Sentence16canbeparaphrasedas,`Itisnotthecasethateverycoinonthe tableisadime.'Sowecantranslateitas :8 x Tx Dx .Youmightlook atsentence16andparaphraseitinsteadas,`Somecoinonthetableisnota dime.'Youwouldthentranslateitas 9 x Tx & : Dx .Althoughitisprobably notobvious,thesetwotranslationsarelogicallyequivalent.Thisisdueto thelogicalequivalencebetween :8 x A and 9 x : A ,alongwiththeequivalence between : A B and A & : B Sentence17canbeparaphrasedas,`Itisnotthecasethatthereissomedime inmypocket.'Thiscanbetranslatedas :9 x Px & Dx .Itmightalsobe paraphrasedas,`Everythinginmypocketisanon-dime,'andthencouldbe translatedas 8 x Px !: Dx .Againthetwotranslationsarelogicallyequivalent.Botharecorrecttranslationsofsentence17. Wecannowtranslatetheargumentfromp.49,theonethatmotivatedtheneed forquantiers: Willardisalogician.Alllogicianswearfunnyhats. : : Willardwearsafunnyhat. UD: people Lx: x isalogician. Fx: x wearsafunnyhat. w: Willard Translating,weget: Lw 8 x Lx Fx : :Fw ThiscapturesthestructurethatwasleftoutoftheSLtranslationofthisargument,andthisisavalidargumentinQL. Emptypredicates ApredicateneednotapplytoanythingintheUD.Apredicatethatappliesto nothingintheUDiscalledan empty predicate. Supposewewanttosymbolizethesetwosentences: 18.Everymonkeyknowssignlanguage.

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60 forall x 19.Somemonkeyknowssignlanguage. Itispossibletowritethesymbolizationkeyforthesesentencesinthisway: UD: animals Mx: x isamonkey. Sx: x knowssignlanguage. Sentence18cannowbetranslatedas 8 x Mx Sx Sentence19becomes 9 x Mx & Sx Itistemptingtosaythatsentence18entailssentence19;thatis:ifevery monkeyknowssignlanguage,thenitmustbethatsomemonkeyknowssign language.ThisisavalidinferenceinAristoteleanlogic:All M sare S : : some M is S .However,theentailmentdoesnotholdinQL.Itispossibleforthe sentence 8 x Mx Sx tobetrueeventhoughthesentence 9 x Mx & Sx is false. Howcanthisbe?Theanswercomesfromconsideringwhetherthesesentences wouldbetrueorfalse iftherewerenomonkeys Wehavedened 8 and 9 insuchawaythat 8 A isequivalentto :9: A .As such,theuniversalquantierdoesn'tinvolvetheexistenceofanything|only non-existence.Ifsentence18istrue,thenthereare no monkeyswhodon'tknow signlanguage.Iftherewerenomonkeys,then 8 x Mx Sx wouldbetrue and 9 x Mx & Sx wouldbefalse. Weallowemptypredicatesbecausewewanttobeabletosaythingslike,`Ido notknowifthereareanymonkeys,butanymonkeysthatthereareknowsign language.'Thatis,wewanttobeabletohavepredicatesthatdonotormight notrefertoanything. Whathappensifweaddanemptypredicate R totheinterpretationabove?For example,wemightdene Rx tomean` x isarefrigerator.'Nowthesentence 8 x Rx Mx willbetrue.Thisiscounterintuitive,sincewedonotwantto saythatthereareawholebunchofrefrigeratormonkeys.Itisimportantto remember,though,that 8 x Rx Mx meansthatanymemberoftheUDthat isarefrigeratorisamonkey.SincetheUDisanimals,therearenorefrigerators intheUDandsothesetenceistriviallytrue. Ifyouwereactuallytranslatingthesentence`Allrefrigeratorsaremonkeys', thenyouwouldwanttoincludeappliancesintheUD.Thenthepredicate R wouldnotbeemptyandthesentence 8 x Rx Mx wouldbefalse.

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ch.4quantiedlogic 61 AUDmusthave atleast onemember. Apredicatemayapplytosome,all,ornomembersoftheUD. Aconstantmustpickout exactly onememberoftheUD. AmemberoftheUDmaybepickedoutbyoneconstant,many constants,ornoneatall. PickingaUniverseofDiscourse TheappropriatesymbolizationofanEnglishlanguagesentenceinQLwilldependonthesymbolizationkey.Insomeways,thisisobvious:Itmatterswhether Dx means` x isdainty'or` x isdangerous.'ThemeaningofsentencesinQL alsodependsontheUD. Let Rx mean` x isarose,'let Tx mean` x hasathorn,'andconsiderthis sentence: 20.Everyrosehasathorn. Itistemptingtosaythatsentence20shouldbetranslatedas 8 x Rx Tx .If theUDcontainsallroses,thatwouldbecorrect.YetiftheUDismerely things onmykitchentable ,then 8 x Rx Tx wouldonlymeanthateveryroseon mykitchentablehasathorn.Iftherearenorosesonmykitchentable,the sentencewouldbetriviallytrue. TheuniversalquantieronlyrangesovermembersoftheUD,soweneedto includeallrosesintheUDinordertotranslatesentence20.Wehavetwo options.First,wecanrestricttheUDtoincludeallrosesbut only roses.Then sentence20becomes 8 xTx .ThismeansthateverythingintheUDhasathorn; sincetheUDjustisthesetofroses,thismeansthateveryrosehasathorn. Thisoptioncansaveustroubleifeverysentencethatwewanttotranslateusing thesymbolizationkeyisaboutroses. Second,wecanlettheUDcontainthingsbesidesroses:rhododendrons,rats, ries,andwhatallelse.Thensentence20mustbe 8 x Rx Tx Ifwewantedtheuniversalquantiertomean every thing,withoutrestriction, thenwemighttrytospecifyaUDthatcontainseverything.Thiswouldlead toproblems.Does`everything'includethingsthathaveonlybeenimagined, likectionalcharacters?Ontheonehand,wewanttobeabletosymbolize argumentsaboutHamletorSherlockHolmes.Soweneedtohavetheoptionof includingctionalcharactersintheUD.Ontheotherhand,weneverneedto talkabouteverythingthatdoesnotexist.Thatmightnotevenmakesense.

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62 forall x Therearephilosophicalissuesherethatwewillnottrytoaddress.Wecan avoidthesedicultiesbyalwaysspecifyingtheUD.Forexample,ifwemeanto talkaboutplants,people,andcities,thentheUDmightbe`livingthingsand places.' Supposethatwewanttotranslatesentence20and,withthesamesymbolization key,translatethesesentences: 21.Esmereldahasaroseinherhair. 22.EveryoneiscrosswithEsmerelda. WeneedaUDthatincludesrosessothatwecansymbolizesentence20anda UDthatincludespeoplesowecantranslatesentence21{22.Hereisasuitable key: UD: peopleandplants Px: x isaperson. Rx: x isarose. Tx: x hasathorn. Cxy: x iscrosswith y Hxy: x has y intheirhair. e: Esmerelda Sincewedonothaveapredicatethatmeans` ::: hasaroseinherhair',translatingsentence21willrequireparaphrasing.Thesentencesaysthatthereisa roseinEsmerelda'shair;thatis,thereissomethingwhichisbotharoseandis inEsmerelda'shair.Soweget: 9 x Rx & Hex Itistemptingtotranslatesentence22as 8 xCxe .Unfortunately,thiswould meanthateverymemberoftheUDiscrosswithEsmerelda|bothpeopleand plants.Itwouldmean,forinstance,thattheroseinEsmerelda'shairiscross withher.Ofcourse,sentence22doesnotmeanthat. `Everyone'meanseveryperson,noteverymemberoftheUD.Sowecanparaphrasesentence22as,`EverypersoniscrosswithEsmerelda.'Weknowhowto translatesentenceslikethis: 8 x Px Cxe Ingeneral,theuniversalquantiercanbeusedtomean`everyone'iftheUD containsonlypeople.IftherearepeopleandotherthingsintheUD,then `everyone'mustbetreatedas`everyperson.' Translatingpronouns WhentranslatingtoQL,itisimportanttounderstandthestructureofthe sentencesyouwanttotranslate.WhatmattersisthenaltranslationinQL,

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ch.4quantiedlogic 63 andsometimesyouwillbeabletomovefromanEnglishlanguagesentence directlytoasentenceofQL.Othertimes,ithelpstoparaphrasethesentence oneormoretimes.Eachsuccessiveparaphraseshouldmovefromtheoriginal sentenceclosertosomethingthatyoucantranslatedirectlyintoQL. Forthenextseveralexamples,wewillusethissymbolizationkey: UD: people Gx: x canplayguitar. Rx: x isarockstar. l: Lemmy Nowconsiderthesesentences: 23.IfLemmycanplayguitar,thenheisarockstar. 24.Ifapersoncanplayguitar,thenheisarockstar. Sentence23andsentence24havethesameconsequent` ::: heisarockstar', buttheycannotbetranslatedinthesameway.Ithelpstoparaphrasethe originalsentences,replacingpronounswithexplicitreferences. Sentence23canbeparaphrasedas,`IfLemmycanplayguitar,then Lemmy is arockstar.'Thiscanobviouslybetranslatedas Gl Rl Sentence24mustbeparaphraseddierently:`Ifapersoncanplayguitar,then thatperson isarockstar.'Thissentenceisnotaboutanyparticularperson, soweneedavariable.Translatinghalfway,wecanparaphrasethesentenceas, `Foranyperson x ,if x canplayguitar,then x isarockstar.'Nowthiscanbe translatedas 8 x Gx Rx .Thisisthesameas,`Everyonewhocanplayguitar isarockstar.' Considerthesefurthersentences: 25.Ifanyonecanplayguitar,thenLemmycan. 26.Ifanyonecanplayguitar,thenheorsheisarockstar. Thesetwosentenceshavethesameantecedent`Ifanyonecanplayguitar ::: ', buttheyhavedierentlogicalstructures. Sentence25canbeparaphrased,`Ifsomeonecanplayguitar,thenLemmycan playguitar.'Theantecedentandconsequentareseparatesentences,soitcan besymbolizedwithaconditionalasthemainlogicaloperator: 9 xGx Gl .

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64 forall x Sentence26canbeparaphrased,`Foranyone,ifthatonecanplayguitar,then thatoneisarockstar.'Itwouldbeamistaketosymbolizethiswithan existentialquantier,becauseitistalkingabouteverybody.Thesentenceis equivalentto`Allguitarplayersarerockstars.'Itisbesttranslatedas 8 x Gx Rx TheEnglishwords`any'and`anyone'shouldtypicallybetranslatedusingquantiers.Asthesetwoexamplesshow,theysometimescallforanexistentialquantierasinsentence25andsometimesforauniversalquantierasinsentence 26.Ifyouhaveahardtimedeterminingwhichisrequired,paraphrasethe sentencewithanEnglishlanguagesentencethatuseswordsbesides`any'or `anyone.' Quantiersandscope Inthesentence 9 xGx Gl ,thescopeoftheexistentialquantieristheexpression Gx .Woulditmatterifthescopeofthequantierwerethewholesentence? Thatis,doesthesentence 9 x Gx Gl meansomethingdierent? Withthekeygivenabove, 9 xGx Gl meansthatifthereissomeguitarist, thenLemmyisaguitarist. 9 x Gx Gl wouldmeanthatthereissomeperson suchthatifthatpersonwereaguitarist,thenLemmywouldbeaguitarist. Recallthattheconditionalhereisamaterialconditional;theconditionalistrue iftheantecedentisfalse.Lettheconstant p denotetheauthorofthisbook, someonewhoiscertainlynotaguitarist.Thesentence Gp Gl istruebecause Gp isfalse.Sincesomeonenamely p satisesthesentence,then 9 x Gx Gl istrue.Thesentenceistruebecausethereisanon-guitarist,regardlessof Lemmy'sskillwiththeguitar. Somethingstrangehappenedwhenwechangedthescopeofthequantier,becausetheconditionalinQLisamaterialconditional.Inordertokeepthe meaningthesame,wewouldhavetochangethequantier: 9 xGx Gl means thesamethingas 8 x Gx Gl ,and 9 x Gx Gl meansthesamethingas 8 xGx Gl Thisodditydoesnotarisewithotherconnectivesorifthevariableisinthe consequentoftheconditional.Forexample, 9 xGx & Gl meansthesamething as 9 x Gx & Gl ,and Gl !9 xGx meansthesamethingsas 9 x Gl Gx Ambiguouspredicates Supposewejustwanttotranslatethissentence:

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68 forall x Theantecedentoftheconditionalisstructurallyjustlikesentence33,with y and x inplaceofImreandKarl.Sosentence34canbecompletelytranslated inthisway 9 x 8 y 8 z Lxz Lyz Lxy Whensymbolizingsentenceswithmultiplequantiers,itisbesttoproceedby smallsteps.ParaphrasetheEnglishsentencesothatthelogicalstructureis readilysymbolizedinQL.Thentranslatepiecemeal,replacingthedaunting taskoftranslatingalongsentencewiththesimplertaskoftranslatingshorter formulae. 4.5SentencesofQL Inthissection,weprovideaformaldenitionfora well-formedformula w and sentence ofQL. Expressions TherearesixkindsofsymbolsinQL: predicates A;B;C;:::;Z withsubscripts,asneeded A 1 ;B 1 ;Z 1 ;A 2 ;A 25 ;J 375 ;::: constants a;b;c;:::;w withsubscripts,asneeded a 1 ;w 4 ;h 7 ;m 32 ;::: variables x;y;z withsubscripts,asneeded x 1 ;y 1 ;z 1 ;x 2 ;::: connectives : ,&, \$ parentheses quantiers 8 ; 9 Wedenean expressionofql asanystringofsymbolsofSL.Takeanyofthe symbolsofQLandwritethemdown,inanyorder,andyouhaveanexpression. Well-formedformulae Bydenition,a termofql iseitheraconstantoravariable. An atomicformulaofql isann-placepredicatefollowedby n terms.

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ch.4quantiedlogic 69 JustaswedidforSL,wewillgivea recursive denitionforawofQL.In fact,mostofthedenitionwilllooklikethedenitionofforawofSL:Every atomicformulaisaw,andyoucanbuildnewwsbyapplyingthesentential connectives. Wecouldjustaddaruleforeachofthequantiersandbedonewithit.For instance:If A isaw,then 8 x A and 9 x A arews.However,thiswould allowforbizarresentenceslike 8 x 9 xDx and 8 xDw .Whatcouldthesepossibly mean?Wecouldadoptsomeinterpretationofsuchsentences,butinsteadwe willwritethedenitionofawsothatsuchabominationsdonotevencount aswell-formed. Inorderfor 8 x A tobeaw, A mustcontainthevariable x andmustnot alreadycontainanx-quantier. 8 xDw willnotcountasawbecause` x 'does notoccurin Dw ,and 8 x 9 xDx willnotcountasawbecause 9 xDx contains anx-quantier 1.Everyatomicformulaisaw. 2.If A isaw,then : A isaw. 3.If A and B arews,then A & B ,isaw. 4.If A and B arews, A B isaw. 5.If A and B arews,then A B isaw. 6.If A and B arews,then A \$ B isaw. 7.If A isaw, x isavariable, A containsatleastoneoccurrenceof x ,and A containsno x -quantiers,then 8 xA isaw. 8.If A isaw, x isavariable, A containsatleastoneoccurrenceof x ,and A containsno x -quantiers,then 9 xA isaw. 9.AllandonlywsofQLcanbegeneratedbyapplicationsoftheserules. Noticethatthe` x 'thatappearsinthedenitionaboveisnotthevariable x .It isa meta-variable thatstandsinforanyvariableofQL.So 8 xAx isaw,but soare 8 yAy 8 zAz 8 x 4 Ax 4 ,and 8 z 9 Az 9 Wecannowgiveaformaldenitionforscope:The scope ofaquantieristhe subformulaforwhichthequantieristhemainlogicaloperator.

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70 forall x Sentences Asentenceissomethingthatcanbeeithertrueorfalse.InSL,everywwasa sentence.ThiswillnotbethecaseinQL.Considerthefollowingsymbolization key: UD: people Lxy: x loves y b: Boris Considertheexpression Lzz .Itisanatomicforumula:atwo-placepredicate followedbytwoterms.Allatomicformulaarews,so Lzz isaw.Doesit meananything?Youmightthinkthatitmeansthat z loveshimself,inthe samewaythat Lbb meansthatBorisloveshimself.Yet z isavariable;itdoes notnamesomepersonthewayaconstantwould.Thew Lzz doesnottell ushowtointerpret z .Doesitmeaneveryone?anyone?someone?Ifwehad az-quantier,itwouldtellushowtointerpret z .Forinstance, 9 zLzz would meanthatsomeonelovesthemself. Someformallanguagestreatawlike Lzz asimplicitlyhavingauniversal quantierinfront.WewillnotdothisforQL.Ifyoumeantosaythateveryone lovesthemself,thenyouneedtowritethequantier: 8 zLzz Inordertomakesenseofavariable,weneedaquantiertotellushowto interpretthatvariable.Thescopeofanx-quantier,forinstance,isthethe partoftheformulawherequantiertellshowtointerpret x Inordertobepreciseaboutthis,wedenea boundvariable tobeanoccurrenceofavariable x thatiswithinthescopeofan x -quantier.A free variable isanoccuranceofavariablethatisnotbound. Forexample,considerthew 8 x Ex Dy !9 z Ex Lzx .Thescopeof theuniversalquantier 8 x is Ex Dy ,sotherst x isboundbytheuniversal quantierbutthesecondandthird x sarefree.Thereisnoty-quantier,so the y isfree.Thescopeoftheexistentialquantier 9 z is Ex Lzx ,soboth occurrencesof z areboundbyit. Wedenea sentence ofQLasawofQLthatcontainsnofreevariables. Notationalconventions WewilladoptthesamenotationalconventionsthatwedidforSLp.31.First, wemayleaveotheoutermostparenthesesofaformula.Second,wewilluse squarebrackets`['and`]'inplaceofparenthesestoincreasethereadabilityof

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ch.4quantiedlogic 71 formulae.Third,wewillleaveoutparenthesesbetweeneachpairofconjuncts whenwritinglongseriesofconjunctions.Fourth,wewillleaveoutparentheses betweeneachpairofdisjunctswhenwritinglongseriesofdisjunctions. Substitutioninstance If A isaw, c aconstant,and x avariable,then A [ c j x ]isthewmadeby replacingeachoccuranceof x in A with c .Thisiscalleda substitution instance of 8 x A and 9 x A ; c iscalledthe instantiatingconstant Forexample: Aa Ba Af Bf ,and Ak Bk areallsubstitutioninstances of 8 x Ax Bx ;theinstantiatingconstantsare a f ,and k ,respectively. Raj Rdj ,and Rjj aresubstitutioninstancesof 9 zRzj ;theinstantiatingconstants are a d ,and j ,respectively. Thisdenitionwillbeusefullater,whenwedenetruthandderivabilityinQL. If 8 xPx istrue,theneverysubstitutioninstance Pa Pb Pc ...istrue.Toput thepointinformally,ifeverythingisa P ,then a isa P b isa P c isa P ,andso on.Conversely,ifsomesubstitutioninstanceof 9 xPx suchas Pa istrue,then 9 xPx mustbetrue.Informally,ifsomespecic a isa P ,thenthereissome P 4.6Identity Considerthissentence: 35.Pavelowesmoneytoeveryoneelse. LettheUDbepeople;thiswillallowustotranslate`everyone'asauniversal quantier.Let Oxy mean` x owesmoneyto y ',andlet p meanPavel.Nowwe cansymbolizesentence35as 8 xOpx .Unfortunately,thistranslationhassome oddconsequences.ItsaysthatPavelowesmoneytoeverymemberoftheUD, includingPavel;itentailsthatPavelowesmoneytohimself.However,sentence 35doesnotsaythatPavelowesmoneytohimself;heowesmoneytoeveryone else .Thisisaproblem,because 8 xOpx isthebesttranslationwecangiveof thissentenceintoQL. ThesolutionistoaddanothersymboltoQL.Thesymbol`='isatwo-place predicate.Sinceithasaspeciallogicalmeaning,wewriteitabitdierently: Fortwoterms t 1 and t 2 t 1 = t 2 isanatomicformula. Thepredicate x = y means` x isidenticalto y .'Thisdoesnotmeanmerely that x and y areindistinguishableorthatallofthesamepredicatesaretrueof

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72 forall x them.Rather,itmeansthat x and y aretheverysamething. Whenwewrite x 6 = y ,wemeanthat x and y arenotidentical.Thereisnoreason tointroducethisasanadditionalpredicate.Instead, x 6 = y isanabbreviation of : x = y Nowsupposewewanttosymbolizethissentence: 36.PavelisMisterCheckov. Lettheconstant c meanMisterCheckov.Sentence36canbesymbolizedas p = c .Thismeansthattheconstants p and c bothrefertothesameguy. Thisisallwellandgood,buthowdoesithelpwithsentence35?Thatsentence canbeparaphrasedas,`EveryonewhoisnotPavelisowedmoneybyPavel.' Thisisasentencestructurewealreadyknowhowtosymbolize:`Forall x ,if x isnotPavel,then x isowedmoneybyPavel.'InQLwithidentity,thisbecomes 8 x x 6 = p Opx Inadditiontosentencesthatusetheword`else',identitywillbehelpfulwhen symbolizingsomesentencesthatcontainthewords`besides'and`only.'Consider theseexamples: 37.NoonebesidesPavelowesmoneytoHikaru. 38.OnlyPavelowesHikarumoney. Weaddtheconstant h ,whichmeansHikaru. Sentence37canbeparaphrasedas,`NoonewhoisnotPavelowesmoneyto Hikaru.'Thiscanbetranslatedas :9 x x 6 = p & Oxh Sentence38canbeparaphrasedas,`PavelowesHikaru and noonebesidesPavel owesHikarumoney.'Wehavealreadytranslatedoneoftheconjuncts,andthe otherisstraightforward.Sentence38becomes Oph & :9 x x 6 = p & Oxh Expressionsofquantity Wecanalsouseidentitytosayhowmanythingsthereareofaparticularkind. Forexample,considerthesesentences: 39.Thereisatleastoneappleonthetable. 40.Thereareatleasttwoapplesonthetable. 41.Thereareatleastthreeapplesonthetable.

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ch.4quantiedlogic 73 LettheUDbe thingsonthetable ,andlet Ax mean` x isanapple.' Sentence39doesnotrequireidentity.Itcanbetranslatedadequatelyas 9 xAx : Thereissomeappleonthetable|perhapsmany,butatleastone. Itmightbetemptingtoalsotranslatesentence40withoutidentity.Yetconsider thesentence 9 x 9 y Ax & Ay .Itmeansthatthereissomeapple x intheUD andsomeapple y intheUD.Sincenothingprecludes x and y frompickingout thesamememberoftheUD,thiswouldbetrueeveniftherewereonlyone apple.Inordertomakesurethattherearetwo dierent apples,weneedan identitypredicate.Sentence40needstosaythatthetwoapplesthatexistare notidentical,soitcanbetranslatedas 9 x 9 y Ax & Ay & x 6 = y Sentence41requirestalkingaboutthreedierentapples.Itcanbetranslated as 9 x 9 y 9 z Ax & Ay & Az & x 6 = y & y 6 = z & x 6 = z Continuinginthisway,wecouldtranslate`Thereareatleast n applesonthe table.'Thereisasummaryofhowtosymbolizesentencesliketheseonp.157. Nowconsiderthesesentences: 42.Thereisatmostoneappleonthetable. 43.Thereareatmosttwoapplesonthetable. Sentence42canbeparaphrasedas,`Itisnotthecasethatthereareatleast two applesonthetable.'Thisisjustthenegationofsentence40: :9 x 9 y Ax & Ay & x 6 = y Sentence42canalsobeapproachedinanotherway.Itmeansthatanyapples thatthereareonthetablemustbetheselfsameapple,soitcanbetranslated as 8 x 8 y Ax & Ay x = y .Thetwotranslationsarelogicallyequivalent,so botharecorrect. Inasimilarway,sentence43canbetranslatedintwoequivalentways.Itcan beparaphrasedas,`Itisnotthecasethatthereare three ormoredistinct apples',soitcanbetranslatedasthenegationofsentence41.Usinguniversal quantiers,itcanalsobetranslatedas 8 x 8 y 8 z Ax & Ay & Az x = y x = z y = z : Seep.157forthegeneralcase. Theexamplesabovearesentencesaboutapples,butthelogicalstructureofthe sentencestranslatesmathematicalinequalitieslike a 3, a 2,andsoon.We alsowanttobeabletotranslatestatementsofequalitywhichsayexactlyhow manythingsthereare.Forexample:

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74 forall x 44.Thereisexactlyoneappleonthetable. 45.Thereareexactlytwoapplesonthetable. Sentence44canbeparaphrasedas,`Thereis atleast oneappleonthetable, andthereis atmost oneappleonthetable.'Thisisjusttheconjunction ofsentence39andsentence42: 9 xAx & 8 x 8 y Ax & Ay x = y .Thisis asomewhatcomplicatedwayofgoingaboutit.Itisperhapsmorestraightforwardtoparaphrasesentence44as,`Thereisathingwhichistheonly appleonthetable.'Thoughtofinthisway,thesentencecanbetranslated 9 x Ax & :9 y Ay & x 6 = y Similarly,sentence45maybeparaphrasedas,`Therearetwodierentapples onthetable,andthesearetheonlyapplesonthetable.'Thiscanbetranslated as 9 x 9 y Ax & Ay & x 6 = y & :9 z Az & x 6 = z & y 6 = z Finally,considerthissentence: 46.Thereareatmosttwothingsonthetable. Itmightbetemptingtoaddapredicatesothat Tx wouldmean` x isathing onthetable.'However,thisisunnecessary.SincetheUDisthesetofthings onthetable,allmembersoftheUDareonthetable.Ifwewanttotalk abouta thingonthetable ,weneedonlyuseaquantier.Sentence46canbe symbolizedlikesentence43whichsaidthattherewereatmosttwoapples, butleavingoutthepredicateentirely.Thatis,sentence46canbetranslatedas 8 x 8 y 8 z x = y x = z y = z Techniquesforsymbolizingexpressionsofquantity`atmost',`atleast',and `exactly'aresummarizedonp.157. Denitedescriptions RecallthataconstantofQLmustrefertosomememberoftheUD.Thisconstraintallowsustoavoidtheproblemofnon-referringterms.GivenaUDthat includedonlyactuallyexistingcreaturesbutaconstant c thatmeant`chimera' amythicalcreature,sentencescontaining c wouldbecomeimpossibletoevaluate. ThemostwidelyinuentialsolutiontothisproblemwasintroducedbyBertrand Russellin1905.Russellaskedhowweshouldunderstandthissentence: 47.ThepresentkingofFranceisbald.

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ch.4quantiedlogic 75 Thephrase`thepresentkingofFrance'issupposedtopickoutanindividualby meansofadenitedescription.However,therewasnokingofFrancein1905 andthereisnonenow.Sincethedescriptionisanon-referringterm,wecannot justdeneaconstanttomean`thepresentkingofFrance'andtranslatethe sentenceas Kf Russell'sideawasthatsentencesthatcontaindenitedescriptionshaveadifferentlogicalstructurethansentencesthatcontainpropernames,eventhough theysharethesamegrammaticalform.Whatdowemeanwhenweuseanunproblematic,referringdescription,like`thehighestpeakinWashingtonstate'? Wemeanthatthereissuchapeak,becausewecouldnottalkaboutitotherwise.Wealsomeanthatitistheonlysuchpeak.Iftherewasanotherpeak inWashingtonstateofexactlythesameheightasMountRainier,thenMount Rainierwouldnotbe the highestpeak. Accordingtothisanalysis,sentence47issayingthreethings.First,itmakes an existence claim:ThereissomepresentkingofFrance.Second,itmakesa uniqueness claim:ThisguyistheonlypresentkingofFrance.Third,itmakes aclaimof predication :Thisguyisbald. Inordertosymbolizedenitedescriptionsinthisway,weneedtheidentitypredicate.Withoutit,wecouldnottranslatetheuniquenessclaimwhichaccording toRussellisimplicitinthedenitedescription. LettheUDbe peopleactuallyliving ,let Fx mean` x isthepresentkingof France',andlet Bx mean` x isbald.'Sentence47canthenbetranslatedas 9 x Fx & :9 y Fy & x 6 = y & Bx .Thissaysthatthereissomeguywhoisthe presentkingofFrance,heistheonlypresentkingofFrance,andheisbald. Understoodinthisway,sentence47ismeaningfulbutfalse.Itsaysthatthis guyexists,buthedoesnot. Theproblemofnon-referringtermsismostvexingwhenwetrytotranslate negations.Soconsiderthissentence: 48.ThepresentkingofFranceisnotbald. AccordingtoRussell,thissentenceisambiguousinEnglish.Itcouldmean eitheroftwothings: 48a.ItisnotthecasethatthepresentkingofFranceisbald. 48b.ThepresentkingofFranceisnon-bald. Bothpossiblemeaningsnegatesentence47,buttheyputthenegationindierentplaces.

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76 forall x Sentence48aiscalleda wide-scopenegation ,becauseitnegatestheentire sentence.Itcanbetranslatedas :9 x Fx & :9 y Fy & x 6 = y & Bx .Thisdoes notsayanythingaboutthepresentkingofFrance,butrathersaysthatsome sentenceaboutthepresentkingofFranceisfalse.Sincesentence47iffalse, sentence48aistrue. Sentence48bsayssomethingaboutthepresentkingofFrance.Itsaysthathe lacksthepropertyofbaldness.Likesentence47,itmakesanexistenceclaim andauniquenessclaim;itjustdeniestheclaimofpredication.Thisiscalled narrow-scopenegation .Itcanbetranslatedas 9 x Fx & :9 y Fy & x 6 = y & : Bx .SincethereisnopresentkingofFrance,thissentenceisfalse. Russell'stheoryofdenitedescriptionsresolvestheproblemofnon-referring termsandalsoexplainswhyitseemedsoparadoxical.Beforewedistinguished betweenthewide-scopeandnarrow-scopenegations,itseemedthatsentences like48shouldbebothtrueandfalse.Byshowingthatsuchsentencesare ambiguous,Russellshowedthattheyaretrueunderstoodonewaybutfalse understoodanotherway. ForamoredetaileddiscussionofRussell'stheoryofdenitedescriptions,includingobjectionstoit,seethePeterLudlow'sentry`descriptions'in TheStanford EncyclopediaofPhilosophy :Summer2005edition,editedbyEdwardN.Zalta, http://plato.stanford.edu/archives/sum2005/entries/descriptions/ PracticeExercises PartA Identifywhichvariablesareboundandwhicharefree. 1. 9 xLxy & 8 yLyx 2. 8 xAx & Bx 3. 8 x Ax & Bx & 8 y Cx & Dy 4. 8 x 9 y [ Rxy Jz & Kx ] Ryx 5. 8 x 1 Mx 2 \$ Lx 2 x 1 & 9 x 2 Lx 3 x 2 ? PartB 1.Identifywhichofthefollowingaresubstitutioninstancesof 8 xRcx : Rac Rca Raa Rcb Rbc Rcc Rcd Rcx 2.Identifywhichofthefollowingaresubstitutioninstancesof 9 x 8 yLxy : 8 yLby 8 xLbx Lab 9 xLxa ? PartC Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQL.

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ch.4quantiedlogic 77 UD: allanimals Ax: x isanalligator. Mx: x isamonkey. Rx: x isareptile. Zx: x livesatthezoo. Lxy: x loves y a: Amos b: Bouncer c: Cleo 1.Amos,Bouncer,andCleoallliveatthezoo. 2.Bouncerisareptile,butnotanalligator. 3.IfCleolovesBouncer,thenBouncerisamonkey. 4.IfbothBouncerandCleoarealligators,thenAmoslovesthemboth. 5.Somereptilelivesatthezoo. 6.Everyalligatorisareptile. 7.Anyanimalthatlivesatthezooiseitheramonkeyoranalligator. 8.Therearereptileswhicharenotalligators. 9.Cleolovesareptile. 10.Bouncerlovesallthemonkeysthatliveatthezoo. 11.AllthemonkeysthatAmosloveslovehimback. 12.Ifanyanimalisanreptile,thenAmosis. 13.Ifanyanimalisanalligator,thenitisareptile. 14.EverymonkeythatCleolovesisalsolovedbyAmos. 15.ThereisamonkeythatlovesBouncer,butsadlyBouncerdoesnotreciprocatethislove. PartD ThesearesyllogisticguresidentiedbyAristotleandhissuccessors, alongwiththeirmedievalnames.TranslateeachargumentintoQL. Barbara All B sare C s.All A sare B s. : : All A sare C s. Baroco All C sare B s.Some A isnot B : : Some A isnot C Bocardo Some B isnot C .All A sare B s. : : Some A isnot C Celantes No B sare C s.All A sare B s. : : No C sare A s. Celarent No B sare C s.All A sare B s. : : No A sare C s. Cemestres No C sare B s.No A sare B s. : : No A sare C s. Cesare No C sare B s.All A sare B s. : : No A sare C s. Dabitis All B sare C s.Some A is B : : Some C is A Darii All B sare C s.Some A is B : : Some A is C .

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78 forall x Datisi All B sare C s.Some A is B : : Some A is C Disamis Some B is C .All A sare B s. : : Some A is C Ferison No B sare C s.Some A is B : : Some A isnot C Ferio No B sare C s.Some A is B : : Some A isnot C Festino No C sare B s.Some A is B : : Some A isnot C Baralipton All B sare C s.All A sare B s. : : Some C is A Frisesomorum Some B is C .No A sare B s. : : Some C isnot A PartE Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQL. UD: allanimals Dx: x isadog. Sx: x likessamuraimovies. Lxy: x islargerthan y b: Bertie e: Emerson f: Fergis 1.Bertieisadogwholikessamuraimovies. 2.Bertie,Emerson,andFergisarealldogs. 3.EmersonislargerthanBertie,andFergisislargerthanEmerson. 4.Alldogslikesamuraimovies. 5.Onlydogslikesamuraimovies. 6.ThereisadogthatislargerthanEmerson. 7.IfthereisadoglargerthanFergis,thenthereisadoglargerthanEmerson. 8.NoanimalthatlikessamuraimoviesislargerthanEmerson. 9.NodogislargerthanFergis. 10.AnyanimalthatdislikessamuraimoviesislargerthanBertie. 11.ThereisananimalthatisbetweenBertieandEmersoninsize. 12.ThereisnodogthatisbetweenBertieandEmersoninsize. 13.Nodogislargerthanitself. 14.Everydogislargerthansomedog. 15.Thereisananimalthatissmallerthaneverydog. 16.Ifthereisananimalthatislargerthananydog,thenthatanimaldoes notlikesamuraimovies. PartF Foreachargument,writeasymbolizationkeyandtranslatetheargumentintoQL.

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ch.4quantiedlogic 79 1.Nothingonmydeskescapesmyattention.Thereisacomputeronmy desk.Assuch,thereisacomputerthatdoesnotescapemyattention. 2.Allmydreamsareblackandwhite.OldTVshowsareinblackandwhite. Therefore,someofmydreamsareoldTVshows. 3.NeitherHolmesnorWatsonhasbeentoAustralia.Apersoncouldseea kangarooonlyiftheyhadbeentoAustraliaortoazoo.AlthoughWatson hasnotseenakangaroo,Holmeshas.Therefore,Holmeshasbeentoa zoo. 4.NooneexpectstheSpanishInquisition.NooneknowsthetroublesI've seen.Therefore,anyonewhoexpectstheSpanishInquisitionknowsthe troublesI'veseen. 5.Anantelopeisbiggerthanabreadbox.Iamthinkingofsomethingthat isnobiggerthanabreadbox,anditiseitheranantelopeoracantaloupe. Assuch,Iamthinkingofacantaloupe. 6.Allbabiesareillogical.Nobodywhoisillogicalcanmanageacrocodile. Bertholdisababy.Therefore,Bertholdisunabletomanageacrocodile. ? PartG Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQL. UD: candies Cx: x haschocolateinit. Mx: x hasmarzipaninit. Sx: x hassugarinit. Tx: Borishastried x Bxy: x isbetterthan y 1.Borishasnevertriedanycandy. 2.Marzipanisalwaysmadewithsugar. 3.Somecandyissugar-free. 4.Theverybestcandyischocolate. 5.Nocandyisbetterthanitself. 6.Borishasnevertriedsugar-freechocolate. 7.Borishastriedmarzipanandchocolate,butnevertogether. 8.Anycandywithchocolateisbetterthananycandywithoutit. 9.Anycandywithchocolateandmarzipanisbetterthananycandythat lacksboth. PartH Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQL. UD: peopleanddishesatapotluck Rx: x hasrunout. Tx: x isonthetable.

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80 forall x Fx: x isfood. Px: x isaperson. Lxy: x likes y e: Eli f: Francesca g: theguacamole 1.Allthefoodisonthetable. 2.Iftheguacamolehasnotrunout,thenitisonthetable. 3.Everyonelikestheguacamole. 4.Ifanyonelikestheguacamole,thenElidoes. 5.Francescaonlylikesthedishesthathaverunout. 6.Francescalikesnoone,andnoonelikesFrancesca. 7.Elilikesanyonewholikestheguacamole. 8.Elilikesanyonewholikesthepeoplethathelikes. 9.Ifthereisapersononthetablealready,thenallofthefoodmusthave runout. ? PartI Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQL. UD: people Dx: x dancesballet. Fx: x isfemale. Mx: x ismale. Cxy: x isachildof y Sxy: x isasiblingof y e: Elmer j: Jane p: Patrick 1.AllofPatrick'schildrenareballetdancers. 2.JaneisPatrick'sdaughter. 3.Patrickhasadaughter. 4.Janeisanonlychild. 5.AllofPatrick'sdaughtersdanceballet. 6.Patrickhasnosons. 7.JaneisElmer'sniece. 8.PatrickisElmer'sbrother. 9.Patrick'sbrothershavenochildren. 10.Janeisanaunt. 11.Everyonewhodancesballethasasisterwhoalsodancesballet. 12.Everymanwhodancesballetisthechildofsomeonewhodancesballet.

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ch.4quantiedlogic 81 PartJ Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQLwithidentity.Thelastsentenceisambiguousandcanbe translatedtwoways;youshouldprovidebothtranslations.Hint:Identityis onlyrequiredforthelastfoursentences. UD: people Kx: x knowsthecombinationtothesafe. Sx: x isaspy. Vx: x isavegetarian. Txy: x trusts y h: Hofthor i: Ingmar 1.Hofthorisaspy,butnovegetarianisaspy. 2.NooneknowsthecombinationtothesafeunlessIngmardoes. 3.Nospyknowsthecombinationtothesafe. 4.NeitherHofthornorIngmarisavegetarian. 5.Hofthortrustsavegetarian. 6.EveryonewhotrustsIngmartrustsavegetarian. 7.EveryonewhotrustsIngmartrustssomeonewhotrustsavegetarian. 8.OnlyIngmarknowsthecombinationtothesafe. 9.IngmartrustsHofthor,butnooneelse. 10.Thepersonwhoknowsthecombinationtothesafeisavegetarian. 11.Thepersonwhoknowsthecombinationtothesafeisnotaspy. ? PartK Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQLwithidentity.Thelasttwosentencesareambiguousandcan betranslatedtwoways;youshouldprovidebothtranslationsforeach. UD: cardsinastandarddeck Bx: x isblack. Cx: x isaclub. Dx: x isadeuce. Jx: x isajack. Mx: x isamanwithanaxe. Ox: x isone-eyed. Wx: x iswild. 1.Allclubsareblackcards. 2.Therearenowildcards. 3.Thereareatleasttwoclubs. 4.Thereismorethanoneone-eyedjack. 5.Thereareatmosttwoone-eyedjacks. 6.Therearetwoblackjacks.

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82 forall x 7.Therearefourdeuces. 8.Thedeuceofclubsisablackcard. 9.One-eyedjacksandthemanwiththeaxearewild. 10.Ifthedeuceofclubsiswild,thenthereisexactlyonewildcard. 11.Themanwiththeaxeisnotajack. 12.Thedeuceofclubsisnotthemanwiththeaxe. PartL Usingthesymbolizationkeygiven,translateeachEnglish-language sentenceintoQLwithidentity.Thelasttwosentencesareambiguousandcan betranslatedtwoways;youshouldprovidebothtranslationsforeach. UD: animalsintheworld Bx: x isinFarmerBrown'seld. Hx: x isahorse. Px: x isaPegasus. Wx: x haswings. 1.Thereareatleastthreehorsesintheworld. 2.Thereareatleastthreeanimalsintheworld. 3.ThereismorethanonehorseinFarmerBrown'seld. 4.TherearethreehorsesinFarmerBrown'seld. 5.ThereisasinglewingedcreatureinFarmerBrown'seld;anyothercreaturesintheeldmustbewingless. 6.ThePegasusisawingedhorse. 7.TheanimalinFarmerBrown'seldisnotahorse. 8.ThehorseinFarmerBrown'selddoesnothavewings.

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Chapter5 Formalsemantics Inthischapter,wedescribea formalsemantics forSLandforQL.Theword `semantics'comesfromthegreekwordfor`mark'andmeans`relatedtomeaning.'Soaformalsemanticswillbeamathematicalaccountofmeaninginthe formallanguage. Aformal,logicallanguageisbuiltfromtwokindsofelements:logicalsymbols andnon-logicalsymbols.Connectiveslike`&'andquantierslike` 8 'are logicalsymbols,becausetheirmeaningisspeciedwithintheformallanguage. Whenwritingasymbolizationkey,youarenotallowedtochangethemeaning ofthelogicalsymbols.Youcannotsay,forinstance,thatthe` : 'symbolwill mean`not'inoneargumentand`perhaps'inanother.The` : 'symbolalways meanslogicalnegation.ItisusedtotranslatetheEnglishlanguageword`not', butitisasymbolofaformallanguageandisdenedbyitstruthconditions. ThesentenceslettersinSLarenon-logicalsymbols,becausetheirmeaningis notdenedbythelogicalstructureofSL.Whenwetranslateanargumentfrom EnglishtoSL,forexample,thesentenceletter M doesnothaveitsmeaning xedinadvance;instead,weprovideasymbolizationkeythatsayshow M shouldbeinterpretedinthatargument.InQL,thepredicatesandconstants arenon-logicalsymbols. IntranslatingfromEnglishtoaformallanguage,weprovidedsymbolization keyswhichwereinterpretationsofallthenon-logicalsymbolsweusedinthe translation.An interpretation givesameaningtoallthenon-logicalelements ofthelanguage. Itispossibletoprovidedierentinterpretationsthatmakenoformaldierence. InSL,forexample,wemightsaythat D means`TodayisTuesday';wemightsay insteadthat D means`TodayisthedayafterMonday.'Thesearetwodierent 83

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84 forall x interpretations,becausetheyusedierentEnglishsentencesforthemeaningof D .Yet,formally,thereisnodierencebetweenthem.Allthatmattersoncewe havesymbolizedthesesentencesiswhethertheyaretrueorfalse.Inorderto characterizewhatmakesadierenceintheformallanguage,weneedtoknow whatmakessentencestrueorfalse.Forthis,weneedaformalcharacterization of truth WhenwegavedenitionsforasentenceofSLandforasentenceofQL,we distinguishedbetweenthe objectlanguage andthe metalanguage .The objectlanguageisthelanguagethatweare talkingabout :eitherSLorQL.The metalanguageisthelanguagethatweusetotalkabouttheobjectlanguage: English,supplementedwithsomemathematicaljargon.Itwillbeimportantto keepthisdistinctioninmind. 5.1SemanticsforSL Thissectionprovidesarigorous,formalcharacterizationof truthinSL which buildsonwhatwealreadyknowfromdoingtruthtables.Wewereabletouse truthtablestoreliablytestwhetherasentencewasatautologyinSL,whether twosentenceswereequivalent,whetheranargumentwasvalid,andsoon.For instance: A isatautologyinSLifitisToneverylineofacompletetruthtable. Thisworkedbecauseeachlineofatruthtablecorrespondstoawaytheworld mightbe.WeconsideredallthepossiblecombinationsofTandFforthe sentencelettersthatmadeadierencetothesentenceswecaredabout.The truthtableallowedustodeterminewhatwouldhappengiventhesedierent combinations. Onceweconstructatruthtable,thesymbols`T'and`F'aredivorcedfromtheir metalinguisticmeaningof`true'and`false'.We interpret `T'asmeaning`true', buttheformalpropertiesofTaredenedbythecharacteristictruthtables forthevariousconnectives.Thetableswouldbethesameifwehadusedthe symbols`1'and`0',andcomputerscanbeprogrammedtollouttruthtables withouthavinganysensethat1meanstrueand0meansfalse. Formally,whatwewantisafunctionthatassignsa1or0toeachofthesentences ofSL.WecaninterpretthisfunctionasadenitionoftruthforSLifitassigns 1toallofthetruesentencesofSLand0toallofthefalsesentencesofSL.Call thisfunction` v 'for`valuation'.Wewant v toabeafunctionsuchthatfor anysentence A v A =1if A istrueand v A =0if A isfalse. RecallthattherecursivedenitionofawforSLhadtwostages:Therststep saidthatatomicsentencessolitarysentencelettersarews.Thesecondstage allowedforwstobeconstructedoutofmorebasicws.Therewereclausesof

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ch.5formalsemantics 85 thedenitionforallofthesententialconnectives.Forexample,if A isaw, then : A isaw. Ourstrategyfordeningthetruthfunction, v ,willalsobeintwosteps.The rststepwillhandletruthforatomicsentences;thesecondstepwillhandle truthforcompoundsentences. TruthinSL HowcanwedenetruthforanatomicsentenceofSL?Consider,forexample, thesentence M .Withoutaninterpretation,wecannotsaywhether M istrue orfalse.Itmightmeananything.Ifweuse M tosymbolize`Themoonorbits theEarth',then M istrue.Ifuse M tosymbolize`Themoonisagiantturnip', then M isfalse. Moreover,thewayyouwoulddiscoverwhetherornot M istruedependson what M means.If M means`Itismonday,'thenyouwouldneedtochecka calendar.If M means`Jupiter'smoonIohassignicantvolcanicactivity,'then youwouldneedtocheckanastronomytext|andastronomersknowbecause theysentsatellitestoobserveIo. WhenwegiveasymbolizationkeyforSL,weprovideaninterpretationofthe sentencelettersthatweuse.ThekeygivesanEnglishlanguagesentenceforeach sentenceletterthatweuse.Inthisway,theinterpretationspecieswhateachof thesentenceletters means .However,thisnotenoughtodeterminewhetheror notthatsentenceistrue.Thesentencesaboutthemoon,forinstance,require thatyouknowsomerudimentaryastronomy.Imagineasmallchildwhobecame convincedthatthemoonisagiantturnip.Shecouldunderstandwhatthe sentence`Themoonisagiantturnip'means,butmistakenlythinkthatitwas true. Consideranotherexample:If M means`Itismorningnow',thenwhetheritis trueornotdependsonwhenyouarereadingthis.Iknowwhatthesentence means,but|sinceIdonotknowwhenyouwillbereadingthis|Idonotknow whetheritistrueorfalse. Soaninterpretationalonedoesnotdeterminewhetherasentenceistrueor false.Truthorfalsitydependsalsoonwhattheworldislike.If M meant`The moonisagiantturnip'andtherealmoonwereagiantturnip,then M would betrue.Toputthepointinageneralway,truthorfalsityisdeterminedbyan interpretation plus awaythattheworldis. INTERPRETATION+STATEOFTHEWORLD= TRUTH/FALSITY

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86 forall x Inprovidingalogicaldenitionoftruth,wewillnotbeabletogiveanaccount ofhowanatomicsentenceismadetrueorfalsebytheworld.Instead,we willintroducea truthvalueassignment .Formally,thiswillbeafunctionthat tellsusthetruthvalueofalltheatomicsentences.Callthisfunction` a 'for `assignment'.Wedene a forallsentenceletters P ,suchthat a P = 1if P istrue ; 0otherwise. Thismeansthat a takesanysentenceofSLandassignsiteitheraoneora zero;oneifthesentenceistrue,zeroifthesentenceisfalse.Thedetailsofthe function a aredeterminedbythemeaningofthesentenceletterstogetherwith thestateoftheworld.If D means`Itisdarkoutside',then a D =1atnight orduringaheavystorm,while a D =0onaclearday. Youcanthinkof a asbeinglikearowofatruthtable.Whereasatruthtable rowassignsatruthvaluetoafewatomicsentences,thetruthvalueassignment assignsavaluetoeveryatomicsentenceofSL.Thereareinnitelymanysentence letters,andthetruthvalueassignmentgivesavaluetoeachofthem.When constructingatruthtable,weonlycareaboutsentencelettersthataectthe truthvalueofsentencesthatinterestus.Assuch,weignoretherest.Strictly speaking,everyrowofatruthtablegivesa partial truthvalueassignment. Itisimportanttonotethatthetruthvalueassignment, a ,isnotpartofthe languageSL.Rather,itispartofthemathematicalmachinerythatweareusing todescribeSL.Itencodeswhichatomicsentencesaretrueandwhicharefalse. Wenowdenethetruthfunction, v ,usingthesamerecursivestructurethatwe usedtodeneawofSL. 1.If A isasentenceletter,then v A = a A 2.If A is : B forsomesentence B ,then v A = 1if v B =0 ; 0otherwise. 3.If A is B & C forsomesentences B ; C ,then v A = 1if v B =1and v C =1, 0otherwise. Itmightseemasifthisdenitioniscircular,becauseitusestheword`and' intryingtodene`and.'Notice,however,thatthisisnotadenitionofthe Englishword`and';itisadenitionoftruthforsentencesofSLcontainingthe logicalsymbol`&.'Wedenetruthforobjectlanguagesentencescontaining thesymbol`&'usingthemetalanguageword`and.'Thereisnothingcircular aboutthat.

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ch.5formalsemantics 87 4.If A is B C forsomesentences B ; C ,then v A = 0if v B =0and v C =0, 1otherwise. 5.If A is B C forsomesentences B ; C ,then v A = 0if v B =1and v C =0, 1otherwise. 6.If A is B \$ C forsomesentences B ; C ,then v A = 1if v B = v C ; 0otherwise. Sincethedenitionof v hasthesamestructureasthedenitionofaw,we knowthat v assignsavalueto every wofSL.SincethesentencesofSLand thewsofSLarethesame,thismeansthat v returnsthetruthvalueofevery sentenceofSL. TruthinSLisalwaystruth relativeto sometruthvalueassignment,because thedenitionoftruthforSLdoesnotsaywhetheragivensentenceistrueor false.Rather,itsayshowthetruthofthatsentencerelatestoatruthvalue assignment. OtherconceptsinSL WorkingwithSLsofar,wehavedonewithoutaprecisedenitionof`tautology', `contradiction',andsoon.Truthtablesprovidedawayto checkif asentence wasatautologyinSL,buttheydidnot dene whatitmeanstobeatautology inSL.WewillgivedenitionsoftheseconceptsforSLintermsofentailment. Therelationofsemanticentailment,` A entails B ',meansthatthereisnotruth valueassignmentforwhich A istrueand B isfalse.Putdierently,itmeans that B istrueforanyandalltruthvalueassignmentsforwhich A istrue. Weabbreviatethiswithasymbolcalledthe doubleturnstile : A j = B means` A semanticallyentails B .' Wecantalkaboutentailmentbetweenmorethantwosentences: f A 1 ; A 2 ; A 3 ; gj = B meansthatthereisnotruthvalueassignmentforwhichallofthesentencesin theset f A 1 ; A 2 ; A 3 ; g aretrueand B isfalse.

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88 forall x Wecanalsousethesymbolwithjustonesentence: j = C meansthat C istruefor alltruthvalueassignments.Thisisequivalenttosayingthatthatthesentence isentailedbyanything. Thedoubleturnstilesymbolallowsustogiveconcisedenitionsforvarious conceptsofSL: A tautologyinsl isasentence A suchthat j = A A contradictioninsl isasentence A suchthat j = : A Asentenceis contingentinsl ifandonlyifitisneitheratautologynoracontradiction. Anargument P 1 ; P 2 ; : : C "is validinsl ifandonlyif f P 1 ; P 2 ; gj = C Twosentences A and B are logicallyequivalentinsl ifand onlyifboth A j = B and B j = A Logicalconsistencyissomewhathardertodeneintermsofsemanticentailment.Instead,wewilldeneitinthisway: Theset f A 1 ; A 2 ; A 3 ; g is consistentinsl ifandonlyifthereis atleastonetruthvalueassignmentforwhichallofthesentencesare true.Thesetis inconsistentinsl ifandifonlythereisnosuch assignment. 5.2InterpretationsandmodelsinQL InSL,aninterpretationorsymbolizationkeyspecieswhateachofthesentence lettersmeans.Theinterpretationofasentenceletteralongwiththestateofthe worlddetermineswhetherthesentenceletteristrueorfalse.Sincethebasic unitsaresentenceletters,aninterpretationonlymattersinsofarasitmakes sentenceletterstrueorfalse.Formally,thesemanticsforSLisstrictlyinterms oftruthvalueassignments.Twointerpretationsarethesame,formally,ifthey makeforthesametruthvalueassignment. WhatisaninterpretationinQL?LikeasymbolizationkeyforQL,aninterpretationrequiresaUD,aschematicmeaningforeachofthepredicates,andan objectthatispickedoutbyeachconstant.Forexample: UD: comicbookcharacters Fx: x ghtscrime. b: theBatman

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ch.5formalsemantics 89 w: BruceWayne Considerthesentence Fb .Thesentenceistrueonthisinterpretation,but| justasinSL|thesentenceisnottrue justbecause oftheinterpretation.Most peopleinourcultureknowthatBatmanghtscrime,butthisrequiresamodicumofknowledgeaboutcomicbooks.Thesentence Fb istruebecauseofthe interpretation plus somefactsaboutcomicbooks.Thisisespeciallyobvious whenweconsider Fw .BruceWayneisthesecretidentityoftheBatmanin thecomicbooks|theidentityclaim b = w istrue|so Fw istrue.Sinceitis a secret identity,however,othercharactersdonotknowthat Fw istrueeven thoughtheyknowthat Fb istrue. Wecouldtrytocharacterizethisasatruthvalueassignment,aswedidforSL. Thetruthvalueassignmentwouldassign0or1toeachatomicw: Fb Fw andsoon.Ifweweretodothat,however,wemightjustaswelltranslatethe sentencesfromQLtoSLbyreplacing Fb and Fw withsentenceletters.We couldthenrelyonthedenitionoftruthforSL,butatthecostofignoringall thelogicalstructureofpredicatesandterms.Inwritingasymbolizationkey forQL,wedonotgiveseparatedenitionsfor Fb and Fw .Instead,wegive meaningsto F b ,and w .Thisisessentialbecausewewanttobeabletouse quantiers.Thereisnoadequatewaytotranslate 8 xFx intoSL. Sowewantaformalcounterparttoaninterpretationforpredicatesandconstants,notjustforsentences.Wecannotuseatruthvalueassignmentforthis, becauseapredicateisneithertruenorfalse.Intheinterpretationgivenabove, F istrue of theBatmani.e., Fb istrue,butitmakesnosenseatalltoask whether F onitsownistrue.ItwouldbelikeaskingwhethertheEnglish languagefragment` ::: ghtscrime'istrue. Whatdoesaninterpretationdoforapredicate,ifitdoesnotmakeittrueor false?Aninterpretationhelpstopickouttheobjectstowhichthepredicate applies.Interpreting Fx tomean` x ghtscrime'picksoutBatman,Superman, Spiderman,andotherheroesasthethingsthatare F s.Formally,thisisa setofmembersoftheUDtowhichthepredicateapplies;thissetiscalledthe extension ofthepredicate. Manypredicateshaveindenitelylargeextensions.Itwouldbeimpractical totryandwritedownallofthecomicbookcrimeghtersindividually,so insteadweuseanEnglishlanguageexpressiontointerpretthepredicate.This issomewhatimprecise,becausetheinterpretationalonedoesnottellyouwhich membersoftheUDareintheextensionofthepredicate.Inordertogure outwhetheraparticularmemberoftheUDisintheextensionofthepredicate togureoutwhetherBlackLightningghtscrime,forinstance,youneedto knowaboutcomicbooks.Ingeneral,theextensionofapredicateistheresult ofaninterpretation alongwith somefacts.

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90 forall x Sometimesitispossibletolistallofthethingsthatareintheextensionofa predicate.InsteadofwritingaschematicEnglishsentence,wecanwritedown theextensionasasetofthings.Supposewewantedtoaddaone-placepredicate M tothekeyabove.Wewant Mx tomean` x livesinWayneManor',sowe writetheextensionasasetofcharacters: extension M = f BruceWayne,Alfredthebutler,DickGrayson g Youdonotneedtoknowanythingaboutcomicbookstobeabletodetermine that,onthisinterpretation, Mw istrue:BruceWayneisjustspeciedtobeone ofthethingsthatis M .Similarly, 9 xMx isobviouslytrueonthisinterpretation: ThereisatleastonememberoftheUDthatisan M |infact,therearethree ofthem. Whataboutthesentence 8 xMx ?Thesentenceisfalse,becauseitisnottrue thatallmembersoftheUDare M .Itrequiresthebarestminimumofknowledge aboutcomicbookstoknowthatthereareothercharactersbesidesjustthese three.Althoughwespeciedtheextensionof M inaformallypreciseway,we stillspeciedtheUDwithanEnglishlanguagedescription.Formallyspeaking, aUDisjustasetofmembers. Theformalsignicanceofapredicateisdeterminedbyitsextension,butwhat shouldwesayaboutconstantslike b and w ?ThemeaningofaconstantdetermineswhichmemberoftheUDispickedoutbytheconstant.Theindividual thattheconstantpicksoutiscalledthe referent oftheconstant.Both b and w havethesamereferent,sincetheybothrefertothesamecomicbook character.Youcanthinkofaconstantletterasanameandthereferentasthe thingnamed.InEnglish,wecanusethedierentnames`Batman'and`Bruce Wayne'torefertothesamecomicbookcharacter.Inthisinterpretation,we canusethedierentconstants` b 'and` w 'torefertothesamememberofthe UD. Sets Weusecurlybrackets` f 'and` g 'todenotesets.Themembersofthesetcanbe listedinanyorder,separatedbycommas.Thefactthatsetscanbeinanyorder isimportant,becauseitmeansthat f foo,bar g and f bar,foo g arethesameset. Itispossibletohaveasetwithnomembersinit.Thisiscalledthe empty set .Theemptysetissometimeswrittenas fg ,butusuallyitiswrittenasthe singlesymbol ; .

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ch.5formalsemantics 91 Models Aswehaveseen,aninterpretationinQLisonlyformallysignicantinsofaras itdeterminesaUD,anextensionforeachpredicate,andareferentforeach constant.Wecallthisformalstructurea model forQL. Toseehowthisworks,considerthissymbolizationkey: UD: PeoplewhoplayedaspartoftheThreeStooges Hx: x hadheadhair. f: MisterFine IfyoudonotknowanythingabouttheThreeStooges,youwillnotbeableto saywhichsentencesofQLaretrueonthisinterpretation.Perhapsyoujust rememberLarry,Curly,andMoe.Isthesentence Hf trueorfalse?Itdepends onwhichofthestoogesisMisterFine. Whatisthemodelthatcorrespondstothisinterpretation?Thereweresix peoplewhoplayedaspartoftheThreeStoogesovertheyears,sotheUD willhavesixmembers:LarryFine,MoeHoward,andCurlyHoward,Shemp Howard,JoeBesser,andCurlyJoeDeRita.Curly,Joe,andCurlyJoewerethe onlycompletelybaldstooges.Theresultisthismodel: UD= f Larry,Curly,Moe,Shemp,Joe,CurlyJoe g extension H = f Larry,Moe,Shemp g referent f =Larry YoudonotneedtoknowanythingabouttheThreeStoogesinordertoevaluate whethersentencesaretrueorfalseinthis model Hf istrue,sincethereferent of f Larryisintheextensionof H .Both 9 xHx and 9 x : Hx aretrue,since thereisatleastonememberoftheUDthatisintheextensionof H andatleast onememberthatisnotintheextensionof H .Inthisway,themodelcaptures alloftheformalsignicanceoftheinterpretation. Nowconsiderthisinterpretation: UD:wholenumberslessthan10 Ex: x iseven. Nx: x isnegative. Lxy: x islessthan y Txyz: x times y equals z Whatisthemodelthatgoeswiththisinterpretation?TheUDistheset f 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 g .

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92 forall x Theextensionofaone-placepredicatelike E or N isjustthesubsetoftheUD ofwhichthepredicateistrue.Roughlyspeaking,theextensionofthepredicate E isthesetof E sintheUD.Theextensionof E isthesubset f 2 ; 4 ; 6 ; 8 g .There aremanyevennumbersbesidesthesefour,butthesearetheonlymembersof theUDthatareeven.TherearenonegativenumbersintheUD,so N hasan emptyextension;i.e.extension N = ; Theextensionofatwo-placepredicatelike L issomewhatvexing.Itseemsasif theextensionof L oughttocontain1,since1islessthanalltheothernumbers; itoughttocontain2,since2islessthanalloftheothernumbersbesides1;and soon.EverymemberoftheUDbesides9islessthansomememberoftheUD. Whatwouldhappenifwejustwroteextension L = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 g ? Theproblemisthatsetscanbewritteninanyorder,sothiswouldbethesame aswritingextension L = f 8 ; 7 ; 6 ; 5 ; 4 ; 3 ; 2 ; 1 g .Thisdoesnottelluswhichof themembersofthesetarelessthanwhichothermembers. Weneedsomewayofshowingthat1islessthan8butthat8isnotlessthan 1.Thesolutionistohavetheextensionof L consistofpairsofnumbers. An orderedpair islikeasetwithtwomembers,exceptthattheorder does matter.Wewriteorderedpairswithanglebrackets` < 'and` > '.Theordered pair < foo,bar > isdierentthantheorderedpair < bar,foo > .Theextension of L isacollectionoforderedpairs,allofthepairsofnumbersintheUDsuch thattherstnumberislessthanthesecond.Writingthisoutcompletely: extension L = f < 1,2 > < 1,3 > < 1,4 > < 1,5 > < 1,6 > < 1,7 > < 1,8 > < 1,9 > < 2,3 > < 2,4 > < 2,5 > < 2,6 > < 2,7 > < 2,8 > < 2,9 > < 3,4 > < 3,5 > < 3,6 > < 3,7 > < 3,8 > < 3,9 > < 4,5 > < 4,6 > < 4,7 > < 4,8 > < 4,9 > < 5,6 > < 5,7 > < 5,8 > < 5,9 > < 6,7 > < 6,8 > < 6,9 > < 7,8 > < 7,9 > < 8,9 > g Three-placepredicateswillworksimilarly;theextensionofathree-placepredicateisasetoforderedtripleswherethepredicateistrueofthosethreethings inthatorder .Sotheextensionof T inthismodelwillcontainorderedtriples like < 2,4,8 > ,because2 4=8. Generally,theextensionofann-placepredicateisasetofallorderedn-tuples suchthat a 1 { a n aremembersoftheUDandthepredicateis trueof a 1 { a n inthatorder. 5.3Semanticsforidentity IdentityisaspecialpredicateofQL.Wewriteitabitdierentlythanother two-placepredicates: x = y insteadof Ixy .Wealsodonotneedtoincludeit

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ch.5formalsemantics 93 inasymbolizationkey.Thesentence x = y alwaysmeans` x isidenticalto y ,' anditcannotbeinterpretedtomeananythingelse.Inthesameway,whenyou constructamodel,youdonotgettopickandchoosewhichorderedpairsgo intotheextensionoftheidentitypredicate.Italwayscontainsjusttheordered pairofeachobjectintheUDwithitself. Thesentence 8 xIxx ,whichcontainsanordinarytwo-placepredicate,iscontingent.Whetheritistrueforaninterpretationdependsonhowyouinterpret I andwhetheritistrueinamodeldependsontheextensionof I Thesentence 8 xx = x isatautology.Theextensionofidentitywillalways makeittrue. Noticethatalthoughidentityalwayshasthesameinterpretation,itdoesnot alwayshavethesameextension.TheextensionofidentitydependsontheUD. IftheUDinamodelistheset f Doug g ,thenextension=inthatmodelis f < Doug,Doug > g .IftheUDistheset f Doug,Omar g ,thenextension=in thatmodelis f < Doug,Doug > < Omar,Omar > g .Andsoon. Ifthereferentoftwoconstantsisthesame,thenanythingwhichistrueofone istrueoftheother.Forexample,ifreferent a =referent b ,then Aa \$ Ab Ba \$ Bb Ca \$ Cb Rca \$ Rcb 8 xRxa \$8 xRxb ,andsoonforanytwo sentencescontaining a and b .However,thereverseisnottrue. Itispossiblethatanythingwhichistrueof a isalsotrueof b ,yetfor a and b still tohavedierentreferents.Thismayseempuzzling,butitiseasytoconstruct amodelthatshowsthis.Considerthismodel: UD= f Rosencrantz,Guildenstern g referent a =Rosencrantz referent b =Guildenstern forallpredicates P ,extension P = ; extension== f < Rosencrantz,Rosencrantz > < Guildenstern,Guildenstern > g ThisspeciesanextensionforeverypredicateofQL:Alltheinnitely-many predicatesareempty.Thismeansthatboth Aa and Ab arefalse,andtheyare equivalent;both Ba and Bb arefalse;andsoonforanytwosentencesthatcontain a and b .Yet a and b refertodierentthings.Wehavewrittenouttheextensionofidentitytomakethisclear:Theorderedpair < referent a ; referent b > isnotinit.Inthismodel, a = b isfalseand a 6 = b istrue.

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94 forall x 5.4Workingwithmodels WewillusethedoubleturnstilesymbolforQLmuchaswedidforSL.` A j = B meansthat` A entails B ':When A and B aretwosentencesofQL, A j = B meansthatthereisnomodelinwhich A istrueand B isfalse. j = A means that A istrueineverymodel. ThisallowsustogivedenitionsforvariousconceptsinQL.Becauseweare usingthesamesymbol,thesedenitionswilllooksimilartothedenitionsin SL.Remember,however,thatthedenitionsinQLareintermsof models rather thanintermsoftruthvalueassignments. A tautologyinql isasentence A thatistrueineverymodel; i.e., j = A A contradictioninql isasentence A thatisfalseineverymodel; i.e., j = : A Asentenceis contingentinql ifandonlyifitisneitheratautologynoracontradiction. Anargument P 1 ; P 2 ; : : C "is validinql ifandonlyifthere isnomodelinwhichallofthepremisesaretrueandtheconclusion isfalse;i.e., f P 1 ; P 2 ; gj = C .Itis invalidinql otherwise. Twosentences A and B are logicallyequivalentinsl ifand onlyifboth A j = B and B j = A Theset f A 1 ; A 2 ; A 3 ; g is consistentinql ifandonlyifthereis atleastonemodelinwhichallofthesentencesaretrue.Thesetis inconsistentinql ifandifonlythereisnosuchmodel. Constructingmodels Supposewewanttoshowthat 8 xAxx Bd is not atautology.Thisrequires showingthatthesentenceisnottrueineverymodel;i.e.,thatitisfalseinsome model.Ifwecanprovidejustonemodelinwhichthesentencefalse,thenwe willhaveshownthatthesentenceisnotatautology. Whatwouldsuchamodellooklike?Inorderfor 8 xAxx Bd tobefalse,the antecedent 8 xAxx mustbetrue,andtheconsequent Bd mustbefalse. Toconstructsuchamodel,westartwithaUD.Itwillbeeasiertospecify extensionsforpredicatesifwehaveasmallUD,sostartwithaUDthathas justonemember.Formally,thissinglemembermightbeanything.Let'ssayit isthecityofParis.

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ch.5formalsemantics 95 Wewant 8 xAxx tobetrue,sowewantallmembersoftheUDtobepaired withthemselfintheextensionof A ;thismeansthattheextensionof A must be f < Paris,Paris > g Wewant Bd tobefalse,sothereferentof d mustnotbeintheextensionof B Wegive B anemptyextension. SinceParisistheonlymemberoftheUD,itmustbethereferentof d .The modelwehaveconstructedlookslikethis: UD= f Paris g extension A = f < Paris,Paris > g extension B = ; referent d =Paris Strictlyspeaking,amodelspeciesanextensionfor every predicateofQLand areferentfor every constant.Assuch,itisgenerallyimpossibletowritedown acompletemodel.Thatwouldrequirewritingdowninnitelymanyextensions andinnitelymanyreferents.However,wedonotneedtoconsidereverypredicateinordertoshowthattherearemodelsinwhich 8 xAxx Bd isfalse. Predicateslike H andconstantslike f 13 makenodierencetothetruthor falsityofthissentence.Itisenoughtospecifyextensionsfor A and B anda referentfor d ,aswehavedone.Thisprovidesa partialmodel inwhichthe sentenceisfalse. Perhapsyouarewondering:Whatdoesthepredicate A meaninEnglish?The partialmodelcouldcorrespondtoaninterpretationlikethisone: UD: Paris Axy : x isinthesamecountryas y Bx : x wasfoundedinthe20thcentury. d : theCityofLights However,allthatthepartialmodeltellsusisthat A isapredicatewhichistrue ofParisandParis.ThereareindenitelymanypredicatesinEnglishthathave thisextension. Axy mightinsteadtranslate` x isthesamesizeas y 'or` x and y arebothcities.'Similarly, Bx issomepredicatethatdoesnotapplytoParis;it mightinsteadtranslate` x isonanisland'or` x isasubcompactcar.'Whenwe specifytheextensionsof A and B ,wedonotspecifywhatEnglishpredicates A and B shouldbeusedtotranslate.Weareconcernedwithwhetherthe 8 xAxx Bd comesouttrueorfalse,andallthatmattersfortruthandfalsity inQListheinformationinthemodel:theUD,theextensionsofpredicates, andthereferentsofconstants. Wecanjustaseasilyshowthat 8 xAxx Bd isnotacontradiction.Weneed onlyspecifyamodelinwhich 8 xAxx Bd istrue;i.e.,amodelinwhicheither 8 xAxx isfalseor Bd istrue.Hereisonesuchpartialmodel:

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96 forall x UD= f Paris g extension A = f < Paris,Paris > g extension B = f Paris g referent d =Paris Wehavenowshownthat 8 xAxx Bd isneitheratautologynoracontradiction.Bythedenitionof`contingentinQL,'thismeansthat 8 xAxx Bd is contingent.Ingeneral,showingthatasentenceiscontingentwillrequiretwo models:oneinwhichthesentenceistrueandanotherinwhichthesentenceis false. Supposewewanttoshowthat 8 xSx and 9 xSx arenotlogicallyequivalent. Weneedtoconstructamodelinwhichthetwosentenceshavedierenttruth values;wewantoneofthemtobetrueandtheothertobefalse.Westartby specifyingaUD.Again,wemaketheUDsmallsothatwecanspecifyextensions easily.Wewillneedatleasttwomembers.LettheUDbe f Duke,Miles g .If wechoseaUDwithonlyonemember,thetwosentenceswouldendupwith thesametruthvalue.Inordertoseewhy,tryconstructingsomepartialmodels withone-memberUDs. Wecanmake 9 xSx truebyincludingsomethingintheextensionof S ,andwe canmake 8 xSx falsebyleavingsomethingoutoftheextensionof S .Itdoes notmatterwhichoneweincludeandwhichoneweleaveout.MakingDukethe only S ,wegetapartialmodelthatlookslikethis: UD= f Duke,Miles g extension S = f Duke g Thispartialmodelshowsthatthetwosentencesare not logicallyequivalent. Backonp.66,wesaidthatthisargumentwouldbeinvalidinQL: Rc & K 1 c & Tc : :Tc & K 2 c Inordertoshowthatitisinvalid,weneedtoshowthatthereissomemodelin whichthepremisesaretrueandtheconclusionisfalse.Wecanconstructsuch amodeldeliberately.Hereisonewaytodoit: UD= f Bjork g extension T = f Bjork g extension K 1 = f Bjork g extension K 2 = ; extension R = f Bjork g referent c =Bjork Similarly,wecanshowthatasetofsentencesisconsistentbyconstructinga

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ch.5formalsemantics 97 Table5.1:Itisrelativelyeasytoansweraquestionifyoucandoitbyconstructingamodelortwo.Itismuchharderifyouneedtoreasonaboutallpossible models.Thistableshowswhenconstructingmodelsisenough. YESNO Is A atautology?showthat A mustbe trueinanymodel constructamodel in which A isfalse Is A acontradiction?showthat A mustbe falseinanymodel constructamodel in which A istrue Is A contingent? constructtwomodels oneinwhich A istrue andanotherinwhich A isfalse eithershowthat A isa tautologyorshowthat A isacontradiction Are A and B equivalent? showthat A and B musthavethesame truthvalueinany model constructamodel in which A and B have dierenttruthvalues Istheset A consistent? constructamodel in whichallthesentences in A aretrue showthatthesentencescouldnotallbe trueinanymodel Istheargument ` P : : C 'valid? showthatanymodelin which P istruemust beamodelinwhich C istrue constructamodel in which P istrueand C isfalse modelinwhichallofthesentencesaretrue. Reasoningaboutallmodels Wecanshowthatasentenceis not atautologyjustbyprovidingonecarefully speciedmodel:amodelinwhichthesentenceisfalse.Toshowthatsomething isatautology,ontheotherhand,itwouldnotbeenoughtoconstructten,one hundred,orevenathousandmodelsinwhichthesentenceistrue.Itisonly atautologyifitistruein every model,andthereareinnitelymanymodels. Thiscannotbeavoidedjustbyconstructingpartialmodels,becausethereare innitelymanypartialmodels. Consider,forexample,thesentence Raa \$ Raa .Therearetwologicallydistinct partialmodelsofthissentencethathavea1-memberUD.Thereare32distinct partialmodelsthathavea2-memberUD.Thereare1526distinctpartialmodels thathavea3-memberUD.Thereare262,144distinctpartialmodelsthathave

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98 forall x a4-memberUD.Andsoontoinnity.Inordertoshowthatthissentenceis atautology,weneedtoshowsomethingaboutallofthesemodels.Thereisno hopeofdoingsobydealingwiththemoneatatime. Nevertheless, Raa \$ Raa isobviouslyatautology.Wecanproveitwitha simpleargument: Therearetwokindsofmodels:thoseinwhich < referent a ; referent a > isintheextensionof R andthoseinwhichitisnot.Intherst kindofmodel, Raa istrue;bythetruthtableforthebiconditional, Raa \$ Raa isalsotrue.Inthesecondkindofmodel, Raa isfalse; thismakes Raa \$ Raa true.Sincethesentenceistrueinbothkinds ofmodel,andsinceeverymodelisoneofthetwokinds, Raa \$ Raa istrueineverymodel.Therefore,itisatautology. Thisargumentisvalid,ofcourse,anditsconclusionistrue.However,itisnot anargumentinQL.Rather,itisanargumentinEnglish about QL;itisan argumentinthemetalanguage.Thereisnoformalprocedureforevaluating orconstructingnaturallanguageargumentslikethisone.Theimprecisionof naturallanguageistheveryreasonwebeganthinkingaboutformallanguages. Therearefurtherdicultieswiththisapproach. Considerthesentence 8 x Rxx Rxx ,anotherobvioustautology.Itmight betemptingtoreasoninthisway:` Rxx Rxx istrueineverymodel,so 8 x Rxx Rxx mustbetrue.'Theproblemisthat Rxx Rxx is not truein everymodel.Itisnotasentence,andsoitis neither true nor false.Wedonot yethavethevocabularytosaythatwewanttosayabout Rxx Rxx .Inthe nextsection,weintroducetheconceptof satisfaction ;afterdoingso,wewillbe betterabletoprovideanargumentthat 8 x Rxx Rxx isatautology. Itisnecessarytoreasonaboutaninnityofmodelstoshowthatasentence isatautology.Similarly,itisnecessarytoreasonaboutaninnityofmodels toshowthatasentenceisacontradition,thattwosentencesareequivalent, thatasetofsentencesisinconsistent,orthatanargumentisvalid.Thereare otherthingswecanshowbycarefullyconstructingamodelortwo.Table5.1 summarizeswhichthingsarewhich. 5.5TruthinQL ForSL,wesplitthedenitionoftruthintotwoparts:atruthvalueassignment a forsentencelettersandatruthfunction v forallsentences.Thetruth functioncoveredthewaythatcomplexsentencescouldbebuiltoutofsentence lettersandconnectives.

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ch.5formalsemantics 99 InthesamewaythattruthforSLisalways truthgivenatruthvalueassignment truthforQLis truthinamodel .ThesimplestatomicsentenceofQLconsists ofaone-placepredicatefollowedbyaconstant,like Pj .Itistrueinamodel M ifandonlyifthereferentof j isintheextensionof P in M Wecouldgooninthiswaytodenetruthforallatomicsentencesthatcontain onlypredicatesandconstants:Consideranysentenceoftheform Rc 1 ::: c n where R isann-placepredicateandthe c sareconstants.Itistruein M ifand onlyif < referent c 1 ;:::; referent c n > isinextension R in M Wecouldthendenetruthforsentencesbuiltupwithsententialconnectivesin thesamewaywedidforSL.Forexample,thesentence Pj Mda istruein M ifeither Pj isfalsein M or Mda istruein M Unfortunately,thisapproachwillfailwhenweconsidersentencescontaining quantiers.Consider 8 xPx .Whenisittrueinamodel M ?Theanswercannot dependonwhether Px istrueorfalsein M ,becausethe x in Px isafree variable. Px isnotasentence.Itisneithertruenorfalse. WewereabletogivearecursivedenitionoftruthforSLbecauseeverywellformedformulaofSLhasatruthvalue.ThisisnottrueinQL,sowecannot denetruthbystartingwiththetruthofatomicsentencesandbuildingup.We alsoneedtoconsidertheatomicformulaewhicharenotsentences.Inorder todothiswewilldene satisfaction ;everywell-formedformulaofQLwillbe satisedornotsatised,evenifitdoesnothaveatruthvalue.Wewillthenbe abletodene truth forsentencesofQLintermsofsatisfaction. Satisfaction Theformula Px says,roughly,that x isoneofthe P s.Thiscannotbequite right,however,because x isavariableandnotaconstant.Itdoesnotnameany particularmemberoftheUD.Instead,itsmeaninginasentenceisdetermined bythequantierthatbindsit.Thevariable x muststand-inforeverymember oftheUDinthesentence 8 xPx ,butitonlyneedstostand-inforonemember in 9 xPx .Sincewewantthedenitionofsatisfactiontocover Px withoutany quantierwhatsoever,wewillstartbysayinghowtointerpretafreevariable likethe x in Px Wedothisbyintroducinga variableassignment .Formally,thisisafunction thatmatchesupeachvariablewithamemberoftheUD.Callthisfunction `a.'The`a'isfor`assignment',butthisisnotthesameasthetruthvalue assignmentthatweusedindeningtruthforSL. Theformula Px issatisedinamodel M byavariableassignment a ifandonly if a x ,theobjectthat a assignsto x ,isinthetheextensionofPin M .

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100 forall x Whenis 8 xPx satised?Itisnotenoughif Px issatisedin M by a ,because thatjustmeansthat a x isinextension P 8 xPx requiresthateveryother memberoftheUDbeinextension P aswell. Soweneedanotherbitoftechnicalnotation:ForanymemberoftheUDand anyvariable x ,let a [ j x ]bethevariableassignmentthatassignsto x but agreeswith a inallotherrespects.Wehaveused,theGreekletterOmega, tounderscorethefactthatitissomememberoftheUDandnotsomesymbol ofQL.Suppose,forexample,thattheUDispresidentsoftheUnitedStates. Thefunction a [GroverCleveland j x ]assignsGroverClevelandtothevariable x regardlessofwhat a assignsto x ;foranyothervariable, a [GroverCleveland j x ] agreeswith a Wecannowsayconciselythat 8 xPx issatisedinamodel M byavariable assignment a ifandonlyif,foreveryobjectintheUDof M Px issatised in M by a [ j x ]. Youmayworrythatthisiscircular,becauseitgivesthesatisfactionconditions forthesentence 8 xPx usingthephrase`foreveryobject.'However,itisimportanttorememberthedierencebetweenalogicalsymbollike` 8 'andanEnglish languagewordlike`every.'Thewordispartofthemetalanguagethatweusein deningsatisfactionconditionsforobjectlanguagesentencesthatcontainthe symbol. Wecannowgiveageneraldenitionofsatisfaction,extendingfromthecaseswe havealreadydiscussed.Wedeneafunction s for`satisfaction'inamodel M suchthatforanyw A andvariableassignment a s A ;a =1if A issatised in M by a ;otherwise s A ;a =0. 1.If A isanatomicwoftheform Pt 1 ::: t n and i istheobjectpickedout by t i ,then s A ;a = 1if < 1 ::: n > isinextension P in M ; 0otherwise. Foreachterm t i :If t i isaconstant,then i =referent t i .If t i isa variable,then i = a t i 2.If A is : B forsomew B ,then s A ;a = 1if s B ;a =0 ; 0otherwise. 3.If A is B & C forsomews B ; C ,then s A ;a = 1if s B ;a =1and s C ;a =1, 0otherwise.

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ch.5formalsemantics 101 4.If A is B C forsomews B ; C ,then s A ;a = 0if s B ;a =0and s C ;a =0, 1otherwise. 5.If A is B C forsomews B ; C ,then s A ;a = 0if s B ;a =1and s C ;a =0, 1otherwise. 6.If A is B \$ C forsomesentences B ; C ,then s A ;a = 1if s B ;a = s C ;a ; 0otherwise. 7.If A is 8 xB forsomew B andsomevariable x ,then s A ;a = 1if s B ;a [ j x ]=1foreverymemberoftheUD ; 0otherwise. 8.If A is 9 xB forsomew B andsomevariable x ,then s A ;a = 1if s B ;a [ j x ]=1foratleastonememberoftheUD ; 0otherwise. ThisdenitionfollowsthesamestructureasthedenitionofawforQL,so weknowthateverywofQLwillbecoveredbythisdenition.Foramodel M andavariableassignment a ,anywwilleitherbesatisedornot.Nowsare leftoutorassignedconictingvalues. Truth Considerasimplesentencelike 8 xPx .Bypart7inthedenitionofsatisfaction, thissentenceissatisedif a [ j x ]satises Px in M foreveryintheUD.By part1ofthedenition,thiswillbethecaseifeveryisintheextension of P .Whether 8 xPx issatiseddoesnotdependontheparticularvariable assignment a .Ifthissentenceissatised,thenitistrue.Thisisaformalization ofwhatwehavesaidallalong: 8 xPx istrueifeverythingintheUDisinthe extensionof P ThesamethingholdsforanysentenceofQL.Becauseallofthevariablesare bound,asentenceissatisedornotregardlessofthedetailsofthevariable assignment.Sowecandenetruthinthisway:Asentence A is truein M ifandonlyifsomevariableassignmentsatises A in M ; A is falsein M otherwise.

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102 forall x TruthinQLis truthinamodel .SentencesofQLarenotat-footedlytrue orfalseasmeresymbols,butonlyrelativetoamodel.Amodelprovidesthe meaningofthesymbols,insofarasitmakesanydierencetotruthandfalsity. Reasoningaboutallmodelsreprise Attheendofsection5.4,wewerestymiedwhenwetriedtoshowthat 8 x Rxx Rxx isatautology.Havingdenedsatisfaction,wecannowreasoninthisway: Considersomearbitrarymodel M .NowconsideranarbitrarymemberoftheUD;forthesakeofconvenience,callit.Itmustbe thecaseeitherthat < ; > isintheextensionof R orthatitis not.If < ; > isintheextensionof R ,then Rxx issatisedbya variableassignmentthatassignsto x bypart1ofthedenition ofsatisfaction;sincetheconsequentof Rxx Rxx issatised, theconditionalissatisedbypart5.If < ; > isnotinthe extensionof R ,then Rxx isnotsatisedbyavariableassignment thatassignsto x bypart1;sinceantecedentof Rxx Rxx is notsatised,theconditionalissatisedbypart5.Ineithercase, Rxx Rxx issatised.ThisistrueforanymemberoftheUD,so 8 x Rxx Rxx issatisedbyanytruthvalueassignmentbypart 7.So 8 x Rxx Rxx istruein M bythedenitionoftruth. ThisargumentholdsregardlessoftheexactUDandregardlessof theexactextensionof R ,so 8 x Rxx Rxx istrueinanymodel. Therefore,itisatautology. Givingargumentsaboutallpossiblemodelstypicallyrequiresclevercombinationoftwostrategies: 1.Dividecasesbetweentwopossiblekinds,suchthateverycasemustbeone kindortheother.Intheargumentonp.98,forexample,wedistinguished twokindsofmodelsbasedonwhetherornotaspecicorderedpairwasin extension R .Intheargumentabove,wedistinguishedcasesinwhichanorderedpairwasinextension R andcasesinwhichitwasnot. 2.Consideranarbitraryobjectasawayofshowingsomethingmoregeneral.In theargumentabove,itwascrucialthatwasjustsomearbitrarymemberof theUD.Wedidnotassumeanythingspecialabout.Assuch,whateverwecould showtoholdofmustholdofeverymemberoftheUD|ifwecouldshow itfor,wecouldshowitforanything.Inthesameway,wedidnotassume anythingspecialabout M ,andsowhateverwecouldshowabout M musthold forallmodels.

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ch.5formalsemantics 103 Consideronemoreexample.Theargument 8 x Hx & Jx : : 8 xHx isobviously valid.Wecanonlyshowthattheargumentisvalidbyconsideringwhatmust betrueineverymodelinwhichthepremiseistrue. Consideranarbitrarymodel M inwhichthepremise 8 x Hx & Jx istrue.Theconjunction Hx & Jx issatisedregardlessofwhatis assignedto x ,so Hx mustbealsobypart3ofthedenitionof satisfaction.Assuch, 8 x Hx issatisedbyanyvariableassignmentbypart7ofthedenitionofsatisfactionandtruein M by thedenitionoftruth.Sincewedidnotassumeanythingabout M besides 8 x Hx & Jx beingtrue, 8 x Hx mustbetrueinanymodel inwhich 8 x Hx & Jx istrue.So 8 x Hx & Jx j = 8 xHx Evenforasimpleargumentlikethisone,thereasoningissomewhatcomplicated. Forlongerarguments,thereasoningcanbeinsuerable.Theproblemarises becausetalkingaboutaninnityofmodelsrequiresreasoningthingsoutin English.Whatarewetodo? Wemighttrytoformalizeourreasoningaboutmodels,codifyingthedivide-andconquerstrategiesthatweusedabove.Thisapproach,originallycalled semantic tableaux ,wasdevelopedinthe1950sbyEvertBethandJaakkoHintikka.Their tableauxarenowmorecommonlycalled truthtrees Amoretraditionalapproachistoconsiderdeductiveargumentsasproofs.A proofsystem consistsofrulesthatformallydistinguishbetweenlegitimateand illegitimatearguments|withoutconsideringmodelsorthemeaningsofthe symbols.Inthenextchapter,wedevelopproofsystemsforSLandQL. PracticeExercises ? PartA Determinewhethereachsentenceistrueorfalseinthemodelgiven. UD= f Corwin,Benedict g extension A = f Corwin,Benedict g extension B = f Benedict g extension N = ; referent c =Corwin 1. Bc 2. Ac \$: Nc 3. Nc Ac Bc 4. 8 xAx 5. 8 x : Bx

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104 forall x 6. 9 x Ax & Bx 7. 9 x Ax Nx 8. 8 x Nx _: Nx 9. 9 xBx !8 xAx ? PartB Determinewhethereachsentenceistrueorfalseinthemodelgiven. UD= f Waylan,Willy,Johnny g extension H = f Waylan,Willy,Johnny g extension W = f Waylan,Willy g extension R = f < Waylan,Willy > < Willy,Johnny > < Johnny,Waylan > g referent m =Johnny 1. 9 x Rxm & Rmx 2. 8 x Rxm Rmx 3. 8 x Hx \$ Wx 4. 8 x Rxm Wx 5. 8 x Wx Hx & Wx 6. 9 xRxx 7. 9 x 9 yRxy 8. 8 x 8 yRxy 9. 8 x 8 y Rxy Ryx 10. 8 x 8 y 8 z Rxy & Ryz Rxz PartC Determinewhethereachsentenceistrueorfalseinthemodelgiven. UD= f Lemmy,Courtney,Eddy g extension G = f Lemmy,Courtney,Eddy g extension H = f Courtney g extension M = f Lemmy,Eddy g referent c =Courtney referent e =Eddy 1. Hc 2. He 3. Mc Me 4. Gc _: Gc 5. Mc Gc 6. 9 xHx 7. 8 xHx 8. 9 x : Mx 9. 9 x Hx & Gx 10. 9 x Mx & Gx 11. 8 x Hx Mx

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ch.5formalsemantics 105 12. 9 xHx & 9 xMx 13. 8 x Hx \$: Mx 14. 9 xGx & 9 x : Gx 15. 8 x 9 y Gx & Hy PartD Writeoutthemodelthatcorrespondstotheinterpretationgiven. UD:naturalnumbersfrom10to13 Ox: x isodd. Sx: x islessthan7. Tx: x isatwo-digitnumber. Ux: x isthoughttobeunlucky. Nxy: x isthenextnumberafter y PartE Showthateachofthefollowingiscontingent. 1. ?Da & Db 2. ? 9 xTxh 3. ?Pm & :8 xPx 4. 8 zJz \$9 yJy 5. 8 x Wxmn _9 yLxy 6. 9 x Gx !8 yMy ? PartF Showthatthefollowingpairsofsentencesarenotlogicallyequivalent. 1. Ja Ka 2. 9 xJx Jm 3. 8 xRxx 9 xRxx 4. 9 xPx Qc 9 x Px Qc 5. 8 x Px !: Qx 9 x Px & : Qx 6. 9 x Px & Qx 9 x Px Qx 7. 8 x Px Qx 8 x Px & Qx 8. 8 x 9 yRxy 9 x 8 yRxy 9. 8 x 9 yRxy 8 x 9 yRyx PartG Showthatthefollowingsetsofsentencesareconsistent. 1. f Ma, : Na,Pa, : Qa g 2. f Lee Lef : Lfe : Lff g 3. f: Ma & 9 xAx Ma Fa 8 x Fx Ax g 4. f Ma Mb Ma !8 x : Mx g 5. f8 yGy 8 x Gx Hx 9 y : Iy g 6. f9 x Bx Ax 8 x : Cx 8 x Ax & Bx Cx g

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106 forall x 7. f9 xXx 9 xYx 8 x Xx \$: Yx g 8. f8 x Px Qx 9 x : Qx & Px g 9. f9 z Nz & Ozz 8 x 8 y Oxy Oyx g 10. f:9 x 8 yRxy 8 x 9 yRxy g PartH Constructmodelstoshowthatthefollowingargumentsareinvalid. 1. 8 x Ax Bx : : 9 xBx 2. 8 x Rx Dx 8 x Rx Fx : : 9 x Dx & Fx 3. 9 x Px Qx : : 9 xPx 4. Na & Nb & Nc : : 8 xNx 5. Rde 9 xRxd : :Red 6. 9 x Ex & Fx 9 xFx !9 xGx : : 9 x Ex & Gx 7. 8 xOxc 8 xOcx : : 8 xOxx 8. 9 x Jx & Kx 9 x : Kx 9 x : Jx : : 9 x : Jx & : Kx 9. Lab !8 xLxb 9 xLxb : :Lbb PartI 1. ? Showthat f: Raa; 8 x x = a Rxa g isconsistent. 2. ? Showthat f8 x 8 y 8 z x = y y = z x = z ; 9 x 9 yx 6 = y g isconsistent. 3. ? Showthat f8 x 8 yx = y; 9 xx 6 = a g isinconsistent. 4.Showthat 9 x x = h & x = i iscontingent. 5.Showthat f9 x 9 y Zx & Zy & x = y : Zd d = s g isconsistent. 6.Showthat` 8 x Dx !9 yTyx : : 9 y 9 zy 6 = z 'isinvalid. PartJ 1.Manylogicbooksdeneconsistencyandinconsistencyinthisway:Aset f A 1 ; A 2 ; A 3 ; g isinconsistentifandonlyif f A 1 ; A 2 ; A 3 ; gj = B & : B forsomesentence B .Asetisconsistentifitisnotinconsistent." Doesthisdenitionleadtoanydierentsetsbeingconsistentthanthe denitiononp.88?Explainyouranswer. 2. ? Ourdenitionoftruthsaysthatasentence A is truein M ifandonlyif somevariableassignmentsatises A in M .Woulditmakeanydierence ifwesaidinsteadthat A is truein M ifandonlyif every variable assignmentsatises A in M ?Explainyouranswer.

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Chapter6 Proofs ConsidertwoargumentsinSL: ArgumentA P Q : P : : Q ArgumentB P Q P : : Q Clearly,thesearevalidarguments.Youcanconrmthattheyarevalidby constructingfour-linetruthtables.ArgumentAmakesuseofaninferenceform thatisalwaysvalid:Givenadisjunctionandthenegationofoneofthedisjuncts, theotherdisjunctfollowsasavalidconsequence.Thisruleiscalled disjunctive syllogism ArgumentBmakesuseofadierentvalidform:Givenaconditionalandits antecedent,theconsequentfollowsasavalidconsequence.Thisiscalled modus ponens Whenweconstructtruthtables,wedonotneedtogivenamestodierentinferenceforms.Thereisnoreasontodistinguishmodusponensfromadisjunctive syllogism.Forthissamereason,however,themethodoftruthtablesdoesnot clearlyshow why anargumentisvalid.Ifyouweretodoa1028-linetruthtable foranargumentthatcontainstensentenceletters,thenyoucouldchecktosee iftherewereanylinesonwhichthepremiseswerealltrueandtheconclusion werefalse.Ifyoudidnotseesuchalineandprovidedyoumadenomistakes inconstructingthetable,thenyouwouldknowthattheargumentwasvalid. Yetyouwouldnotbeabletosayanythingfurtheraboutwhythisparticular argumentwasavalidargumentform. 107

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108 forall x Theaimofa proofsystem istoshowthatparticularargumentsarevalidina waythatallowsustounderstandthereasoninginvolvedintheargument.We beginwithbasicargumentforms,likedisjunctivesyllogismandmodusponens. Theseformscanthenbecombinedtomakemorecomplicatedarguments,like thisone: : L J L : L : :J Bymodusponens,andentail J L .Thisisan intermediateconclusion Itfollowslogicallyfromthepremises,butitisnottheconclusionwewant.Now J L andentail J ,bydisjunctivesyllogism.Wedonotneedanewrule forthisargument.Theproofoftheargumentshowsthatitisreallyjusta combinationofruleswehavealreadyintroduced. Formally,a proof isasequenceofsentences.Therstsentencesofthesequence areassumptions;thesearethepremisesoftheargument.Everysentencelater inthesequencefollowsfromearliersentencesbyoneoftherulesofproof.The nalsentenceofthesequenceistheconclusionoftheargument. ThischapterbeginswithaproofsystemforSL,whichisthenextendedtocover QLandQLplusidentity. 6.1BasicrulesforSL Indesigningaproofsystem,wecouldjuststartwithdisjunctivesyllogismand modusponens.Wheneverwediscoveredavalidargumentwhichcouldnotbe provenwithruleswealreadyhad,wecouldintroducenewrules.Proceedingin thisway,wewouldhaveanunsystematicgrabbagofrules.Wemightaccidently addsomestrangerules,andwewouldsurelyendupwithmorerulesthanwe need. Instead,wewilldevelopwhatiscalleda naturaldeduction system.Ina naturaldeductionsystem,therewillbetworulesforeachlogicaloperator:an introduction rulethatallowsustoproveasentencethathasitasthemain logicaloperatorandan elimination rulethatallowsustoprovesomething givenasentencethathasitasthemainlogicaloperator. Inadditiontotherulesforeachlogicaloperator,wewillalsohaveareiterationrule.Ifyoualreadyhaveshownsomethinginthecourseofaproof,the reiterationruleallowsyoutorepeatitonanewline.Forinstance:

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ch.6proofs 109 1 A 2 A R1 Whenweaddalinetoaproof,wewritetherulethatjustiesthatline.Wealso writethenumbersofthelinestowhichtherulewasapplied.Thereiteration ruleaboveisjustiedbyoneline,thelinethatyouarereiterating.Sothe`R1' online2oftheproofmeansthatthelineisjustiedbythereiterationruleR appliedtoline1. Obviously,thereiterationrulewillnotallowustoshowanything new .Forthat, wewillneedmorerules.Theremainderofthissectionwillgiveintroduction andeliminationrulesforallofthesententialconnectives.Thiswillgiveus acompleteproofsystemforSL.Laterinthechapter,weintroducerulesfor quantiersandidentity. Alloftherulesintroducedinthischapteraresummarizedstartingonp.158. Conjunction Thinkforamoment:Whatwouldyouneedtoshowinordertoprove E & F ? Ofcourse,youcouldshow E & F byproving E andseparatelyproving F .This holdsevenifthetwoconjunctsarenotatomicsentences.Ifyoucanprove [ A J V ]and[ V L \$ F N ],thenyouhaveeectivelyproven [ A J V ]&[ V L \$ F N ] : Sothiswillbeourconjunctionintroductionrule,whichweabbreviate&I: m A n B A & B &I m n Alineofproofmustbejustiedbysomerule,andherewehave`&Im,n.' Thismeans:Conjunctionintroductionappliedtoline m andline n .Theseare variables,notreallinenumbers; m issomelineand n issomeotherline.In anactualproof,thelinesarenumbered1 ; 2 ; 3 ;::: andrulesmustbeapplied tospeciclinenumbers.Whenwedenetherule,however,weusevariables tounderscorethepointthattherulemaybeappliedtoanytwolinesthatare alreadyintheproof.Ifyouhave K online8and L online15,youcanprove K & L atsomelaterpointintheproofwiththejustication`&I8,15.' Now,considertheeliminationruleforconjunction.Whatareyouentitledto concludefromasentencelike E & F ?Surely,youareentitledtoconclude E ;if

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110 forall x E & F weretrue,then E wouldbetrue.Similarly,youareentitledtoconclude F .Thiswillbeourconjunctioneliminationrule,whichweabbreviate&E: m A & B A &E m B &E m Whenyouhaveaconjunctiononsomelineofaproof,youcanuse&Etoderive eitheroftheconjuncts.The&Erulerequiresonlyonesentence,sowewrite onelinenumberasthejusticationforapplyingit. Evenwithjustthesetworules,wecanprovidesomeproofs.Considerthis argument. [ A B C D ]&[ E F G H ] : : [ E F G H ]&[ A B C D ] Themainlogicaloperatorinboththepremiseandconclusionisconjunction. Sinceconjunctionissymmetric,theargumentisobviouslyvalid.Inorderto provideaproof,webeginbywritingdownthepremise.Afterthepremises,we drawahorizontalline|everythingbelowthislinemustbejustiedbyarule ofproof.Sothebeginningoftheprooflookslikethis: 1 [ A B C D ]&[ E F G H ] Fromthepremise,wecangeteachoftheconjunctsby&E.Theproofnow lookslikethis: 1 [ A B C D ]&[ E F G H ] 2 [ A B C D ]&E1 3 [ E F G H ]&E1 Therule&Irequiresthatwehaveeachoftheconjunctsavailablesomewhere intheproof.Theycanbeseparatedfromoneanother,andtheycanappearin anyorder.Sobyapplyingthe&Iruletolines3and2,wearriveatthedesired conclusion.Thenishedprooflookslikethis:

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ch.6proofs 111 1 [ A B C D ]&[ E F G H ] 2 [ A B C D ]&E1 3 [ E F G H ]&E1 4 [ E F G H ]&[ A B C D ]&I3,2 Thisproofistrivial,butitshowshowwecanuserulesofprooftogetherto demonstratethevalidityofanargumentform.Also:Usingatruthtableto showthatthisargumentisvalidwouldhaverequiredastaggering256lines, sincethereareeightsentencelettersintheargument. Disjunction If M weretrue,then M N wouldalsobetrue.Sothedisjunctionintroduction rule Iallowsustoderiveadisjunctionifwehaveoneofthetwodisjuncts: m A A B I m B A I m Noticethat B canbe any sentencewhatsoever.Sothefollowingisalegitimate proof: 1 M 2 M [ A \$ B C & D ] \$ [ E & F ] I1 Itmayseemoddthatjustbyknowing M wecanderiveaconclusionthat includessentenceslike A B ,andtherest|sentencesthathavenothingtodo with M .Yettheconclusionfollowsimmediatelyby I.Thisisasitshouldbe: Thetruthconditionsforthedisjunctionmeanthat,if A istrue,then A B is trueregardlessofwhat B is.Sotheconclusioncouldnotbefalseifthepremise weretrue;theargumentisvalid. Nowconsiderthedisjunctioneliminationrule.Whatcanyouconcludefrom M N ?Youcannotconclude M .Itmightbe M 'struththatmakes M N true,asintheexampleabove,butitmightnot.From M N alone,youcannot concludeanythingabouteither M or N specically.Ifyoualsoknewthat N wasfalse,however,thenyouwouldbeabletoconclude M Thisisjustdisjunctivesyllogism,itwillbethedisjunctioneliminationrule E.

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112 forall x m A B n : B A E m n m A B n : A B E m n Conditional Considerthisargument: R F : : : R F Theargumentiscertainlyavalidone.Whatshouldtheconditionalintroduction rulebe,suchthatwecandrawthisconclusion? Webegintheproofbywritingdownthepremiseoftheargumentanddrawing ahorizontalline,likethis: 1 R F Ifwehad : R asafurtherpremise,wecouldderive F bythe Erule.Wedonot have : R asapremiseofthisargument,norcanwederiveitdirectlyfromthe premisewedohave|sowecannotsimplyprove F .Whatwewilldoinsteadis starta subproof ,aproofwithinthemainproof.Whenwestartasubproof,we drawanotherverticallinetoindicatethatwearenolongerinthemainproof. Thenwewriteinanassumptionforthesubproof.Thiscanbeanythingwe want.Here,itwillbehelpfultoassume : R .Ourproofnowlookslikethis: 1 R F 2 : R Itisimportanttonoticethatwearenotclaimingtohaveproven : R .Wedo notneedtowriteinanyjusticationfortheassumptionlineofasubproof.You canthinkofthesubproofasposingthequestion:Whatcouldweshow if : R weretrue?Foronething,wecanderive F .Sowedo: 1 R F 2 : R 3 F E1,2

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ch.6proofs 113 Thishasshownthat if wehad : R asapremise, then wecouldprove F .In eect,wehaveproven : R F .Sotheconditionalintroductionrule Iwill allowustoclosethesubproofandderive : R F inthemainproof.Ournal prooflookslikethis: 1 R F 2 : R 3 F E1,2 4 : R F I2{3 Noticethatthejusticationforapplyingthe Iruleistheentiresubproof. Usuallythatwillbemorethanjusttwolines. Itmayseemasiftheabilitytoassumeanythingatallinasubproofwouldlead tochaos:Doesitallowyoutoproveanyconclusionfromanypremises?The answerisno,itdoesnot.Considerthisproof: 1 A 2 B 3 B R2 Itmayseemasifthisisaproofthatyoucanderiveanyconclusions B from anypremise A .Whentheverticallineforthesubproofends,thesubproofis closed .Inordertocompleteaproof,youmustcloseallofthesubproofs.And youcannotclosethesubproofandusetheRruleagainonline4toderive B in themainproof.Onceyoucloseasubproof,youcannotreferbacktoindividual linesinsideit. Closingasubproofiscalled discharging theassumptionsofthatsubproof.So wecanputthepointthisway:Youcannotcompleteaproofuntilyouhave dischargedalloftheassumptionsbesidestheoriginalpremisesoftheargument. Ofcourse,itislegitimatetodothis: 1 A 2 B 3 B R2 4 B B I2{3

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114 forall x Thisshouldnotseemsostrange,though.Since B B isatautology,noparticularpremisesshouldberequiredtovalidlyderiveit.Indeed,aswewillsee,a tautologyfollowsfromanypremises. Putinageneralform,the Irulelookslikethis: m A want B n B A B I m { n Whenweintroduceasubproof,wetypicallywritewhatwewanttoderivein thecolumn.Thisisjustsothatwedonotforgetwhywestartedthesubproofif itgoesonforveortenlines.Thereisno`want'rule.Itisanotetoourselves andnotformallypartoftheproof. Althoughitisalwayspermissibletoopenasubproofwithanyassumptionyou please,thereissomestrategyinvolvedinpickingausefulassumption.Starting asubproofwithanarbitrary,wackyassumptionwouldjustwastelinesofthe proof.Inordertoderiveaconditionalbythe I,forinstance,youmustassume theantecedentoftheconditionalinasubproof. The Irulealsorequiresthattheconsequentoftheconditionalbethelastline ofthesubproof.Itisalwayspermissibletocloseasubproofanddischargeits assumptions,butitwillnotbehelpfultodosountilyougetwhatyouwant. Nowconsidertheconditionaleliminationrule.Nothingfollowsfrom M N alone,butifwehaveboth M N and M ,thenwecanconclude N .Thisrule, modusponens,willbetheconditionaleliminationrule E. m A B n A B E m n Nowthatwehaverulesfortheconditional,considerthisargument: P Q Q R : :P R Webegintheproofbywritingthetwopremisesasassumptions.Sincethemain logicaloperatorintheconclusionisaconditional,wecanexpecttousethe Irule.Forthat,weneedasubproof|sowewriteintheantecedentofthe conditionalasassumptionofasubproof:

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ch.6proofs 115 1 P Q 2 Q R 3 P Wemade P availablebyassumingitinasubproof,allowingustouse Eon therstpremise.Thisgivesus Q ,whichallowsustouse Eonthesecond premise.Havingderived R ,weclosethesubproof.Byassuming P wewereable toprove R ,soweapplythe Iruleandnishtheproof. 1 P Q 2 Q R 3 P want R 4 Q E1,3 5 R E2,4 6 P R I3{5 Biconditional Therulesforthebiconditionalwillbelikedouble-barreledversionsoftherules fortheconditional. Inordertoderive W \$ X ,forinstance,youmustbeabletoprove X by assuming W and prove W byassuming X .Thebiconditionalintroduction rule \$ Irequirestwosubproofs.Thesubproofscancomeinanyorder,and thesecondsubproofdoesnotneedtocomeimmediatelyaftertherst|but schematically,theruleworkslikethis: m A want B n B p B want A q A A \$ B \$ I m { n p { q Thebiconditionaleliminationrule \$ Eletsyoudoabitmorethantheconditionalrule.Ifyouhavetheleft-handsubsentenceofthebiconditional,youcan derivetheright-handsubsentence.Ifyouhavetheright-handsubsentence,you

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116 forall x canderivetheleft-handsubsentence.Thisistherule: m A \$ B n A B \$ E m n m A \$ B n B A \$ E m n Negation HereisasimplemathematicalargumentinEnglish: Assumethereissomegreatestnaturalnumber.Callit A Thatnumberplusoneisalsoanaturalnumber. Obviously, A +1 >A Sothereisanaturalnumbergreaterthan A Thisisimpossible,since A isassumedtobethegreatestnaturalnumber. : : Thereisnogreatestnaturalnumber. Thisargumentformistraditionallycalleda reductio .ItsfullLatinnameis reductioadabsurdum ,whichmeans`reductiontoabsurdity.'Inareductio, weassumesomethingforthesakeofargument|forexample,thatthereis agreatestnaturalnumber.Thenweshowthattheassumptionleadstotwo contradictorysentences|forexample,that A isthegreatestnaturalnumber andthatitisnot.Inthisway,weshowthattheoriginalassumptionmusthave beenfalse. Thebasicrulesfornegationwillallowforargumentslikethis.Ifweassume somethingandshowthatitleadstocontradictorysentences,thenwehave proventhenegationoftheassumption.Thisisthenegationintroduction : I rule: m A forreductio n B n +1 : B n +2 : A : I m { n +1 Fortheruletoapply,thelasttwolinesofthesubproofmustbeanexplicit contradiction:somesentencefollowedonthenextlinebyitsnegation.We write`forreductio'asanotetoourselves,areminderofwhywestartedthe subproof.Itisnotformallypartoftheproof,andyoucanleaveitoutifyou nditdistracting.

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ch.6proofs 117 Toseehowtheruleworks,supposewewanttoprovethelawofnon-contradiction: : G & : G .Wecanprovethiswithoutanypremisesbyimmediatelystartinga subproof.Wewanttoapply : Itothesubproof,soweassume G & : G .We thengetanexplicitcontradictionby&E.Theprooflookslikethis: 1 G & : G forreductio 2 G &E1 3 : G &E1 4 : G & : G : I1{3 The : Erulewillworkinmuchthesameway.Ifweassume : A andshowthat itleadstoacontradiction,wehaveeectivelyproven A .Sotherulelookslike this: m : A forreductio n B n +1 : B n +2 A : E m { n +1 6.2Derivedrules Therulesofthenaturaldeductionsystemaremeanttobesystematic.Thereis anintroductionandaneliminationruleforeachlogicaloperator,butwhythese basicrulesratherthansomeothers?Manynaturaldeductionsystemshavea disjunctioneliminationrulethatworkslikethis: m A B n A C o B C C m n o Itmightseemasiftherewillbesomeproofsthatwecannotdowithourproof system,becausewedonothavethisrule.Yetthisisnotthecase.Ifyoucan doaproofwiththisrule,youcandoaproofwiththebasicrulesofthenatural deductionsystem.Considerthisproof:

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118 forall x 1 A B 2 A C 3 B C want C 4 : C forreductio 5 A forreductio 6 C I2,5 7 : C R4 8 : A : I5{7 9 B forreductio 10 C I3,9 11 : C R4 12 B E1,8 13 : B : I9{11 14 C : E4{13 A B ,and C aremeta-variables.TheyarenotsymbolsofSL,butstand-insfor arbitrarysentencesofSL.Sothisisnot,strictlyspeaking,aproofinSL.Itis morelikearecipe.Itprovidesapatternthatcanproveanythingthatthe rulecanprove,usingonlythebasicrulesofSL.Thismeansthatthe isnot reallynecessary.Addingittothelistofbasicruleswouldnotallowustoderive anythingthatwecouldnotderivewithoutit. Nevertheless,the rulewouldbeconvenient.Itwouldallowustodoinone linewhatrequireselevenlinesandseveralnestedsubproofswiththebasicrules. Sowewilladd totheproofsystemasaderivedrule. A derivedrule isaruleofproofthatdoesnotmakeanynewproofspossible. Anythingthatcanbeprovenwithaderivedrulecanbeprovenwithoutit.You canthinkofashortproofusingaderivedruleasshorthandforalongerproof thatusesonlythebasicrules.Anytimeyouusethe rule,youcouldalways taketenextralinesandprovethesamethingwithoutit. Forthesakeofconvenience,wewilladdseveralotherderivedrules.Oneis modustollens MT.

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ch.6proofs 119 m A B n : B : A MT m n Weleavetheproofofthisruleasanexercise.Notethatifwehadalreadyproven theMTrule,thentheproofofthe rulecouldhavebeendoneinonlyve lines. WealsoaddhypotheticalsyllogismHSasaderivedrule.Wehavealready givenaproofofitonp.115. m A B n B C A C HS m n 6.3Rulesofreplacement Considerhowyouwouldprovethisargument: F G & H : :F G Perhapsitistemptingtowritedownthepremiseandapplythe&Eruletothe conjunction G & H .Thisisimpermissible,however,becausethebasicrulesof proofcanonlybeappliedtowholesentences.Weneedtoget G & H onaline byitself.Wecanprovetheargumentinthisway: 1 F G & H 2 F want G 3 G & H E1,2 4 G &E3 5 F G I2{4 Wewillnowintroducesomederivedrulesthatmaybeappliedtopartofa sentence.Thesearecalled rulesofreplacement ,becausetheycanbeused toreplacepartofasentencewithalogicallyequivalentexpression.Onesimple ruleofreplacementiscommutivityabbreviatedComm,whichsaysthatwe canswaptheorderofconjunctsinaconjunctionortheorderofdisjunctsina disjunction.Wedenetherulethisway:

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120 forall x A & B B & A A B B A A \$ B B \$ A Comm Theboldarrowmeansthatyoucantakeasubformulaononesideofthearrow andreplaceitwiththesubformulaontheotherside.Thearrowisdouble-headed becauserulesofreplacementworkinbothdirections. Considerthisargument: M P P & M : : P M M & P Itispossibletogiveaproofofthisusingonlythebasicrules,butitwillbelong andinconvenient.WiththeCommrule,wecanprovideaproofeasily: 1 M P P & M 2 P M P & M Comm1 3 P M M & P Comm2 AnotherruleofreplacementisdoublenegationDN.WiththeDNrule,you canremoveorinsertapairofnegationsanywhereinasentence.Thisisthe rule: :: A A DN TwomorereplacementrulesarecalledDeMorgan'sLaws,namedforthe19thcenturyBritishlogicianAugustDeMorgan.AlthoughDeMorgandiddiscover theselaws,hewasnotthersttodoso.Therulescaptureusefulrelations betweennegation,conjunction,anddisjunction.Herearetherules,whichwe abbreviateDeM: : A B : A & : B : A & B : A _: B DeM Because A B isa materialconditional ,itisequivalentto : A B .Afurther replacementrulecapturesthisequivalence.WeabbreviatetheruleMC,for `materialconditional.'Ittakestwoforms: A B : A B A B : A B MC Nowconsiderthisargument: : P Q : :P & : Q

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ch.6proofs 121 Asalways,wecouldprovethisargumentusingonlythebasicrules.Withrules ofreplacement,though,theproofismuchsimpler: 1 : P Q 2 : : P Q MC1 3 :: P & : Q DeM2 4 P & : Q DN3 Analreplacementrulecapturestherelationbetweenconditionalsandbiconditionals.Wewillcallthisrulebiconditionalexchangeandabbreviateit \$ ex. [ A B & B A ] A \$ B \$ ex 6.4Rulesforquantiers ForproofsinQL,weuseallofthebasicrulesofSLplusfournewbasicrules: bothintroductionandeliminationrulesforeachofthequantiers. SinceallofthederivedrulesofSLarederivedfromthebasicrules,theywill alsoholdinQL.Wewilladdanotherderivedrule,areplacementrulecalled quantiernegation. Universalelimination Ifyouhave 8 xAx ,itislegitimatetoinferthatanythingisan A .Youcaninfer Aa Ab Az Ad 3 |inshort,youcaninfer A c foranyconstant c .Thisisthe generalformoftheuniversaleliminationrule 8 E: m 8 xA A [ c j x ] 8 E m A [ c j x ]isasubstitutioninstanceof 8 xA .Thesymbolsforasubstitutioninstance arenotsymbolsofQL,soyoucannotwritetheminaproof.Instead,you writethesubsitutedsentencewiththeconstant c replacingalloccurancesof thevariable x in A .Forexample:

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122 forall x 1 8 x Mx Rxd 2 Ma Rad 8 E1 3 Md Rdd 8 E1 Existentialintroduction Whenisitlegitimatetoinfer 9 xAx ?Ifyouknowthatsomethingisan A |for instance,ifyouhave Aa availableintheproof. Thisistheexistentialintroductionrule 9 I: m A 9 xA [ x jj c ] 9 I m Itisimportanttonoticethat A [ x jj c ]isnotthesameasasubstitutioninstance. Wewriteitwithtwobarstoshowthatthevariable x doesnotneedtoreplace alloccurrencesoftheconstant c .Youcandecidewhichoccurrencestoreplace andwhichtoleaveinplace.Forexample: 1 Ma Rad 2 9 x Ma Rax 9 I1 3 9 x Mx Rxd 9 I1 4 9 x Mx Rad 9 I1 5 9 y 9 x Mx Ryd 9 I4 6 9 z 9 y 9 x Mx Ryz 9 I5 Universalintroduction Auniversalclaimlike 8 xPx wouldbeprovenifeverysubstitutioninstanceofit hadbeenproven,ifeverysentence Pa Pb ::: wereavailableinaproof.Alas, thereisnohopeofproving every substitutioninstance.Thatwouldrequire proving Pa Pb ::: Pj 2 ::: Ps 7 ::: ,andsoontoinnity.Thereareinnitely manyconstantsinQL,andsothisprocesswouldnevercometoanend. Considerasimpleargument: 8 xMx : : 8 yMy Itmakesnodierencetothemeaningofthesentencewhetherweusethevariable

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ch.6proofs 123 x orthevariable y ,sothisargumentisobviouslyvalid.Supposewebeginin thisway: 1 8 xMx want 8 yMy 2 Ma 8 E1 Wehavederived Ma .Nothingstopsusfromusingthesamejusticationto derive Mb ::: Mj 2 ::: Ms 7 ::: ,andsoonuntilwerunoutofspaceor patience.Wehaveeectivelyshownthewaytoprove M c foranyconstant c Fromthis, 8 xMx follows. 1 8 xMx 2 Ma 8 E1 3 8 yMy 8 I2 Itisimportantherethat a wasjustsomearbitraryconstant.Wehadnotmade anyspecialassumptionsaboutit.If Ma wereapremiseoftheargument,then thiswouldnotshowanythingabout all y .Forexample: 1 8 xRxa 2 Raa 8 E1 3 8 yRyy notallowed! Thisistheschematicformoftheuniversalintroductionrule 8 I: m A 8 xA [ x j c ] 8 I m c mustnotoccurinanyundischargedassumptions. Notethatwecandothisforanyconstantthatdoesnotoccurinanundischarged assumptionandforanyvariable. Notealsothattheconstantmaynotoccurinany undischarged assumption,but itmayoccurastheassumptionofasubproofthatwehavealreadyclosed.For example,wecanprove 8 z Dz Dz withoutanypremises.

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124 forall x 1 Df want Df 2 Df R1 3 Df Df I1{2 4 8 z Dz Dz 8 I3 Existentialelimination Asentencewithanexistentialquantiertellsusthatthereis some member oftheUDthatsatisesaformula.Forexample, 9 xSx tellsusroughlythat thereisatleastone S .Itdoesnottellus which memberoftheUDsatises S however.Wecannotimmediatelyconclude Sa Sf 23 ,oranyothersubstitution instanceofthesentence.Whatcanwedo? Supposethatweknewboth 9 xSx and 8 x Sx Tx .Wecouldreasoninthis way: Since 9 xSx ,thereissomethingthatisan S .Wedonotknowwhich constantsrefertothisthing,ifanydo,socallthisthing.From 8 x Sx Tx ,itfollowsthatifisan S ,thenitisa T .Therefore isa T .Becauseisa T ,weknowthat 9 xTx Inthisparagraph,weintroducedanameforthethingthatisan S .Wecalled it,sothatwecouldreasonaboutitandderivesomeconsequencesfromthere beingan S .Sinceisjustabogusnameintroducedforthepurposeofthe proofandnotagenuineconstant,wecouldnotmentionitintheconclusion. Yetwecouldderiveasentencethatdoesnotmention;namely, 9 xTx .This sentencedoesfollowfromthetwopremises. Wewanttheexistentialeliminationruletoworkinasimilarway.Yetsince GreekletterslikearenotsymbolsofQL,wecannotusetheminformalproofs. Instead,wewilluseconstantsofQLwhichdonototherwiseappearintheproof. Aconstantthatisusedtostandinforwhateveritisthatsatisesanexistential claimiscalleda proxy .Reasoningwiththeproxymustalloccurinsidea subproof,andtheproxycannotbeaconstantthatisdoingworkelsewherein theproof. Thisistheschematicformoftheexistentialeliminationrule 9 E:

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ch.6proofs 125 m 9 xA n A [ c j x ] p B B 9 E m n { p Theconstant c mustnotappearin 9 xA ,in B ,orinanyundischargedassumption. Withthisrule,wecangiveaformalproofthat 9 xSx and 8 x Sx Tx together entail 9 xTx .ThestructureoftheproofiseectivelythesameastheEnglishlanguageargumentwithwhichwebegan,exceptthatthesubproofusesthe constant` a 'ratherthanthebogusname. 1 9 xSx 2 8 x Sx Tx want 9 xTx 3 Sa 4 Sa Ta 8 E2 5 Ta E3,4 6 9 xTx 9 I5 7 9 xTx 9 E1,3{6 Quantiernegation WhentranslatingfromEnglishtoQL,wenotedthat :9 x : A islogicallyequivalentto 8 x A .InQL,theyareprovablyequivalent.Wecanproveonehalfof theequivalencewitharathergruesomeproof:

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126 forall x 1 8 xAx want :9 x : Ax 2 9: Ax forreductio 3 : Ac for 9 E 4 8 xAx forreductio 5 Ac 8 E1 6 : Ac R3 7 :8 xAx : I4{6 8 8 xAx R1 9 :8 xAx 9 E3{7 10 :9: Ax : I2{9 Inordertoshowthatthetwosentencesaregenuinelyequivalent,weneeda secondproofthatassumes :9 x : A andderives 8 x A .Weleavethatproofasan exerciseforthereader. Itwilloftenbeusefultotranslatebetweenquantiersbyaddingorsubtracting negationsinthisway,soweaddtwoderivedrulesforthispurpose.Theserules arecalledquantiernegationQN: :8 xA 9 x : A :9 xA 8 x : A QN SinceQNisareplacementrule,itcanbeusedonwholesentencesoronsubformulae. 6.5Rulesforidentity TheidentitypredicateisnotpartofQL,butweadditwhenweneedtosymbolizecertainsentences.Forproofsinvolvingidentity,weaddtworulesofproof. Supposeyouknowthatmanythingsthataretrueof a arealsotrueof b .For example: Aa & Ab Ba & Bb : Ca & : Cb Da & Db : Ea & : Eb ,andsoon. Thiswouldnotbeenoughtojustifytheconclusion a = b .Seep.93.Ingeneral, therearenosentencesthatdonotalreadycontaintheidentitypredicatethat couldjustifytheconclusion a = b .Thismeansthattheidentityintroduction rulewillnotjustify a = b oranyotheridentityclaimcontainingtwodierent constants.

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ch.6proofs 127 However,itisalwaystruethat a = a .Ingeneral,nopremisesarerequired inordertoconcludethatsomethingisidenticaltoitself.Sothiswillbethe identityintroductionrule,abbreviated=I: c = c =I Noticethatthe=Iruledoesnotrequirereferringtoanypriorlinesoftheproof. Foranyconstant c ,youcanwrite c = c onanypointwithonlythe=Iruleas justication. Ifyouhaveshownthat a = b ,thenanythingthatistrueof a mustalsobetrueof b .Foranysentencewith a init,youcanreplacesomeoralloftheoccurrencesof a with b andproduceanequivalentsentence.Forexample,ifyoualreadyknow Raa ,thenyouarejustiedinconcluding Rab Rba Rbb .Recallthat A [ a jj b ]is thesentenceproducedbyreplacing a in A with b .Thisisnotthesameasa substitutioninstance,because b mayreplacesomeoralloccurrencesof a .The identityeliminationrule=Ejustiesreplacingtermswithothertermsthat areidenticaltoit: m a = b n A A [ a jj b ]=E m n A [ b jj a ]=E m n Toseetherulesinaction,considerthisproof:

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128 forall x 1 8 x 8 yx = y 2 9 xBx 3 8 x Bx !: Cx want :9 xCx 4 Be 5 8 ye = y 8 E1 6 e = f 8 E5 7 Bf =E6,4 8 Bf !: Cf 8 E3 9 : Cf E8,7 10 : Cf 9 E2,4{9 11 8 x : Cx 8 I10 12 :9 xCx QN11 6.6Proofstrategy Thereisnosimplerecipeforproofs,andthereisnosubstituteforpractice. Here,though,aresomerulesofthumbandstrategiestokeepinmind. Workbackwardsfromwhatyouwant. Theultimategoalistoderivethe conclusion.Lookattheconclusionandaskwhattheintroductionruleisforits mainlogicaloperator.Thisgivesyouanideaofwhatshouldhappen justbefore thelastlineoftheproof.Thenyoucantreatthislineasifitwereyourgoal. Askwhatyoucoulddotoderivethisnewgoal. Forexample:Ifyourconclusionisaconditional A B ,plantousethe I rule.Thisrequiresstartingasubproofinwhichyouassume A .Inthesubproof, youwanttoderive B Workforwardsfromwhatyouhave. Whenyouarestartingaproof,look atthepremises;later,lookatthesentencesthatyouhavederivedsofar.Think abouttheeliminationrulesforthemainoperatorsofthesesentences.These willtellyouwhatyouroptionsare. Forexample:Ifyouhave 8 x A ,thinkaboutinstantiatingitforanyconstant thatmightbehelpful.Ifyouhave 9 x A andintendtousethe 9 Erule,thenyou

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ch.6proofs 129 shouldassume A [ c j x ]forsome c thatisnotinuseandthenderiveaconclusion thatdoesnotcontain c Forashortproof,youmightbeabletoeliminatethepremisesandintroduce theconclusion.Alongproofisformallyjustanumberofshortproofslinked together,soyoucanllthegapbyalternatelyworkingbackfromtheconclusion andforwardfromthepremises. Changewhatyouarelookingat. Replacementrulescanoftenmakeyour lifeeasier.Ifaproofseemsimpossible,tryoutsomedierentsubstitutions. Forexample:Itisoftendiculttoproveadisjunctionusingthebasicrules.If youwanttoshow A B ,itisofteneasiertoshow : A B andusetheMC rule. Showing :9 x A canalsobehard,anditisofteneasiertoshow 8 x : A anduse theQNrule. Somereplacementrulesshouldbecomesecondnature.Ifyouseeanegated disjunction,forinstance,youshouldimmediatelythinkofDeMorgan'srule. Donotforgetindirectproof. Ifyoucannotndawaytoshowsomething directly,tryassumingitsnegation. Rememberthatmostproofscanbedoneeitherindirectlyordirectly.Oneway mightbeeasier|orperhapsonesparksyourimaginationmorethantheother| buteitheroneisformallylegitimate. Repeatasnecessary. Onceyouhavedecidedhowyoumightbeabletoget totheconclusion,askwhatyoumightbeabletodowiththepremises.Then considerthetargetsentencesagainandaskhowyoumightreachthem. Persist. Trydierentthings.Ifoneapproachfails,thentrysomethingelse. 6.7Proof-theoreticconcepts Wewillusethesymbol` ` 'toindicatethataproofispossible.Thissymbolis calledthe turnstile .Sometimesitiscalleda singleturnstile ,tounderscorethe factthatthisisnotthedoubleturnstilesymbol j =thatweusedtorepresent semanticentailmentinch.5.

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130 forall x Whenwewrite f A 1 ; A 2 ;::: g` B ,thismeansthatitispossibletogiveaproof of B with A 1 A 2 ::: aspremises.Withjustonepremise,weleaveoutthe curlybraces,so A ` B meansthatthereisaproofof B with A asapremise. Naturally, ` C meansthatthereisaproofof C thathasnopremises. Often,logicalproofsarecalled derivations .So A ` B canbereadas` B is derivablefrom A .' A theorem isasentencethatisderivablewithoutanypremises;i.e., T isa theoremifandonlyif ` T Itisnottoohardtoshowthatsomethingisatheorem|youjusthavetogive aproofofit.Howcouldyoushowthatsomethingis not atheorem?Ifits negationisatheorem,thenyoucouldprovideaproof.Forexample,itiseasy toprove : Pa & : Pa ,whichshowsthat Pa & : Pa cannotbeatheorem.For asentencethatisneitheratheoremnorthenegationofatheorem,however, thereisnoeasywaytoshowthis.Youwouldhavetodemonstratenotjustthat certainproofstrategiesfail,butthatnoproofispossible.Evenifyoufailin tryingtoproveasentenceinathousanddierentways,perhapstheproofis justtoolongandcomplexforyoutomakeout. Twosentences A and B are provablyequivalent ifandonlyifeachcanbe derivedfromtheother;i.e., A ` B and B ` A Itisrelativelyeasytoshowthattwosentencesareprovablyequivalent|itjust requiresapairofproofs.Showingthatsentencesare not provablyequivalent wouldbemuchharder.Itwouldbejustashardasshowingthatasentence isnotatheorem.Infact,theseproblemsareinterchangeable.Canyouthink ofasentencethatwouldbeatheoremifandonlyif A and B wereprovably equivalent? Thesetofsentences f A 1 ; A 2 ;::: g is provablyinconsistent ifandonlyifa contradictionisderivablefromit;i.e.,forsomesentence B f A 1 ; A 2 ;::: g` B and f A 1 ; A 2 ;::: g`: B Itiseasytoshowthatasetisprovablyinconsistent:Youjustneedtoassume thesentencesinthesetandproveacontradiction.Showingthatasetis not provablyinconsistentwillbemuchharder.Itwouldrequiremorethanjust providingaproofortwo;itwouldrequireshowingthatproofsofacertainkind are impossible .

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ch.6proofs 131 6.8Proofsandmodels Asyoumightalreadysuspect,thereisaconnectionbetween theorems and tautologies Thereisaformalwayofshowingthatasentenceisatheorem:Proveit.For eachline,wecanchecktoseeifthatlinefollowsbythecitedrule.Itmaybe hardtoproduceatwentylineproof,butitisnotsohardtocheckeachline oftheproofandconrmthatitislegitimate|andifeachlineoftheproof individuallyislegitimate,thenthewholeproofislegitimate.Showingthata sentenceisatautology,though,requiresreasoninginEnglishaboutallpossible models.Thereisnoformalwayofcheckingtoseeifthereasoningissound. Givenachoicebetweenshowingthatasentenceisatheoremandshowingthat itisatautology,itwouldbeeasiertoshowthatitisatheorem. Contrawise,thereisnoformalwayofshowingthatasentenceis not atheorem. WewouldneedtoreasoninEnglishaboutallpossibleproofs.Yetthereisa formalmethodforshowingthatasentenceisnotatautology.Weneedonly constructamodelinwhichthesentenceisfalse.Givenachoicebetweenshowing thatasentenceisnotatheoremandshowingthatitisnotatautology,itwould beeasiertoshowthatitisnotatautology. Fortunately,asentenceisatheoremifandonlyifitisatautology.Ifwe provideaproofof ` A andthusshowthatitisatheorem,itfollowsthat A isa tautology;i.e., j = A .Similarly,ifweconstructamodelinwhich A isfalseand thusshowthatitisnotatautology,iffollowsthat A isnotatheorem. Ingeneral, A ` B ifandonlyif A j = B .Assuch: Anargumentis valid ifandonlyif theconclusionisderivablefromthe premises . Twosentencesare logicallyequivalent ifandonlyiftheyare provably equivalent . Asetofsentencesis consistent ifandonlyifitis notprovablyinconsistent Youcanpickandchoosewhentothinkintermsofproofsandwhentothinkin termsofmodels,doingwhicheveriseasierforagiventask.Table6.1summarizes whenitisbesttogiveproofsandwhenitisbesttogivemodels. Inthisway,proofsandmodelsgiveusaversatiletoolkitforworkingwith arguments.IfwecantranslateanargumentintoQL,thenwecanmeasureits logicalweightinapurelyformalway.Ifitisdeductivelyvalid,wecangivea formalproof;ifitisinvalid,wecanprovideaformalcounterexample.

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132 forall x YES NO Is A atautology? prove ` A giveamodelinwhich A isfalse Is A acontradiction? prove `: A giveamodelinwhich A istrue Is A contingent? giveamodelinwhich A istrueandanother inwhich A isfalse prove ` A or `: A Are A and B equivalent? prove A ` B and B ` A giveamodelinwhich A and B havedierent truthvalues Istheset A consistent? giveamodelinwhich allthesentencesin A aretrue takingthesentencesin A ,prove B and : B Istheargument ` P : : C 'valid? prove P ` C giveamodelinwhich P istrueand C isfalse Table6.1:Sometimesitiseasiertoshowsomethingbyprovidingproofsthan itisbyprovidingmodels.Sometimesitistheotherwayround.Itdependson whatyouaretryingtoshow. 6.9Soundnessandcompleteness Thistoolkitisincrediblyconvenient.Itisalsointuitive,becauseitseemsnatural thatprovabilityandsemanticentailmentshouldagree.Yet,donotbefooled bythesimilarityofthesymbols` j ='and` ` .'Thefactthatthesetwoarereally interchangeableisnotasimplethingtoprove. Whyshouldwethinkthatanargumentthat canbeproven isnecessarilya valid argument?Thatis,whythinkthat A ` B implies A j = B ? Thisistheproblemof soundness .Aproofsystemis sound ifthereareno proofsofinvalidarguments.Demonstratingthattheproofsystemissound wouldrequireshowingthat any possibleproofistheproofofavalidargument. Itwouldnotbeenoughsimplytosucceedwhentryingtoprovemanyvalid argumentsandtofailwhentryingtoproveinvalidones. Fortunately,thereisawayofapproachingthisinastep-wisefashion.Ifusing the&Eruleonthelastlineofaproofcouldneverchangeavalidargument intoaninvalidone,thenusingtherulemanytimescouldnotmakeanargument invalid.Similarly,ifusingthe&Eand Erulesindividuallyonthelastline ofaproofcouldneverchangeavalidargumentintoaninvalidone,thenusing themincombinationcouldnoteither.

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ch.6proofs 133 Thestrategyistoshowforeveryruleofinferencethatitalonecouldnotmakea validargumentintoaninvalidone.Itfollowsthattherulesusedincombination wouldnotmakeavalidargumentinvalid.Sinceaproofisjustaseriesof lines,eachjustiedbyaruleofinference,thiswouldshowthateveryprovable argumentisvalid. Consider,forexample,the&Erule.Supposeweuseittoadd A & B toavalid argument.Inorderfortheruletoapply, A and B mustalreadybeavailablein theproof.Sincetheargumentsofarisvalid, A and B areeitherpremisesofthe argumentorvalidconsequencesofthepremises.Assuch,anymodelinwhich thepremisesaretruemustbeamodelinwhich A and B aretrue.According tothedenitionof truthinql ,thismeansthat A & B isalsotrueinsuch amodel.Therefore, A & B validlyfollowsfromthepremises.Thismeansthat usingthe&Eruletoextendavalidproofproducesanothervalidproof. Inordertoshowthattheproofsystemissound,wewouldneedtoshowthisfor theotherinferencerules.Sincethederivedrulesareconsequencesofthebasic rules,itwouldsucetoprovidesimilarargumentsforthe16otherbasicrules. Thistediousexercisefallsbeyondthescopeofthisbook. Givenaproofthattheproofsystemissound,itfollowsthateverytheoremisa tautology. Itisstillpossibletoask:Whythinkthat every validargumentisanargument thatcanbeproven?Thatis,whythinkthat A j = B implies A ` B ? Thisistheproblemof completeness .Aproofsystemis complete ifthereis aproofofeveryvalidargument.CompletenessforalanguagelikeQLwasrst provenbyKurtGodelin1929.Theproofisbeyondthescopeofthisbook. Theimportantpointisthat,happily,theproofsystemforQLisbothsoundand complete.Thisisnotthecaseforallproofsystemsandallformallanguages. BecauseitistrueofQL,wecanchoosetogiveproofsorconstructmodels| whicheveriseasierforthetaskathand. Summaryofdenitions Asentence A isa theorem ifandonlyif ` A . Twosentences A and B are provablyequivalent ifandonlyif A ` B and B ` A . f A 1 ; A 2 ;::: g is provablyinconsistent ifandonlyif,forsomesentence B f A 1 ; A 2 ;::: g` B & : B .

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134 forall x PracticeExercises ? PartA Provideajusticationruleandlinenumbersforeachlineofproof thatrequiresone. 1 W !: B 2 A & W 3 B J & K 4 W 5 : B 6 J & K 7 K 1 L \$: O 2 L _: O 3 : L 4 : O 5 L 6 : L 7 L 1 Z C & : N 2 : Z N & : C 3 : N C 4 : N & : C 5 Z 6 C & : N 7 C 8 : C 9 : Z 10 N & : C 11 N 12 : N 13 N C ? PartB GiveaproofforeachargumentinSL. 1. K & L : :K \$ L 2. A B C : : A & B C 3. P & Q R P !: R : :Q E 4. C & D E : :E D 5. : F G F H : :G H 6. X & Y X & Z : X & D D M: :M PartC GiveaproofforeachargumentinSL. 1. Q Q & : Q : : : Q 2. J !: J : : : J 3. E F F G : F : :E & G

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ch.6proofs 135 4. A \$ B B \$ C : :A \$ C 5. M N M : : : M !: N 6. S \$ T : :S \$ T S 7. M N & O P N P : P : :M & O 8. Z & K K & M K D : :D PartD ShowthateachofthefollowingsentencesisatheoreminSL. 1. O O 2. N _: N 3. : P & : P 4. : A !: C A C 5. J \$ [ J L & : L ] PartE ShowthateachofthefollowingpairsofsentencesareprovablyequivalentinSL. 1. :::: G G 2. T S : S !: T 3. R \$ E E \$ R 4. : G \$ H : G \$ H 5. U I : U & : I PartF Provideproofstoshoweachofthefollowing. 1. M & : N !: M ` N & M _: M 2. f C E & G : C G g` G 3. f Z & K \$ Y & M D & D M g` Y Z 4. f W X Y Z X Y : Z g` W Y PartG Forthefollowing,provideproofsusingonlythebasicrules.Theproofs willbelongerthanproofsofthesameclaimswouldbeusingthederivedrules. 1.ShowthatMTisalegitimatederivedrule.Usingonlythebasicrules, provethefollowing: A B : B : : : A 2.ShowthatCommisalegitimateruleforthebiconditional.Usingonlythe basicrules,provethat A \$ B and B \$ A areequivalent. 3.Usingonlythebasicrules,provethefollowinginstanceofDeMorgan's Laws: : A & : B : : : A B 4.WithoutusingtheQNrule,prove :9 x : A `8 x A 5.Showthat \$ exisalegitimatederivedrule.Usingonlythebasicrules, provethat D \$ E and D E & E D areequivalent.

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136 forall x ? PartH Provideajusticationruleandlinenumbersforeachlineofproof thatrequiresone. 1 8 x 9 y Rxy Ryx 2 8 x : Rmx 3 9 y Rmy Rym 4 Rma Ram 5 : Rma 6 Ram 7 9 xRxm 8 9 xRxm 1 8 x 9 yLxy !8 zLzx 2 Lab 3 9 yLay !8 zLza 4 9 yLay 5 8 zLza 6 Lca 7 9 yLcy !8 zLzc 8 9 yLcy 9 8 zLzc 10 Lcc 11 8 xLxx 1 8 x Jx Kx 2 9 x 8 yLxy 3 8 xJx 4 Ja 5 Ja Ka 6 Ka 7 8 yLay 8 Laa 9 Ka & Laa 10 9 x Kx & Lxx 11 9 x Kx & Lxx 1 : 9 xMx _8 x : Mx 2 :9 xMx & :8 x : Mx 3 :9 xMx 4 8 x : Mx 5 :8 x : Mx 6 9 xMx _8 x : Mx ? PartI Provideaproofofeachclaim. 1. `8 xFx _:8 xFx 2. f8 x Mx \$ Nx ;Ma & 9 xRxa g`9 xNx 3. f8 x : Mx Ljx ; 8 x Bx Ljx ; 8 x Mx Bx g`8 xLjx 4. 8 x Cx & Dt `8 xCx & Dt 5. 9 x Cx Dt `9 xCx Dt PartJ ProvideaproofoftheargumentaboutBillyonp.65.

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ch.6proofs 137 PartK LookbackatPartDonp.77.Provideproofstoshowthateachofthe argumentformsisvalidinQL. PartL Aristotleandhissuccessorsidentiedothersyllogisticforms.Symbolize eachofthefollowingargumentformsinQLandaddtheadditionalassumptions `Thereisan A 'and`Thereisa B .'Thenprovethatthesupplementedarguments formsarevalidinQL. Darapti: All A sare B s.All A sare C s. : : Some B is C .1.IfeveryMisL andeveryMisS,thensomeSisLDarapti. Felapton: No B sare C s.All A sare B s. : : Some A isnot C Barbari: All B sare C s.All A sare B s. : : Some A is C Camestros: All C sare B s.No A sare B s. : : Some A isnot C Celaront: No B sare C s.All A sare B s. : : Some A isnot C Cesaro: No C sare B s.All A sare B s. : : Some A isnot C Fapesmo: All B sare C s.No A sare B s. : : Some C isnot A PartM Provideaproofofeachclaim. 1. 8 x 8 yGxy `9 xGxx 2. 8 x 8 y Gxy Gyx `8 x 8 y Gxy \$ Gyx 3. f8 x Ax Bx ; 9 xAx g`9 xBx 4. f Na !8 x Mx \$ Ma ;Ma; : Mb g`: Na 5. `8 z Pz _: Pz 6. `8 xRxx !9 x 9 yRxy 7. `8 y 9 x Qy Qx PartN Showthateachpairofsentencesisprovablyequivalent. 1. 8 x Ax !: Bx :9 x Ax & Bx 2. 8 x : Ax Bd 8 xAx Bd 3. 9 xPx Qc 8 x Px Qc 4. Rca \$8 xRxa 8 x Rca \$ Rxa PartO Showthateachofthefollowingisprovablyinconsistent. 1. f Sa Tm Tm Sa Tm & : Sa g 2. f9 xRxa 8 x 8 yRyx g 3. f:9 x 9 yLxy Laa g

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138 forall x 4. f8 x Px Qx 8 z Pz Rz 8 yPy : Qa & : Rb g ? PartP Writeasymbolizationkeyforthefollowingargument,translateit, andproveit: Thereissomeonewholikeseveryonewholikeseveryonethathelikes. Therefore,thereissomeonewholikeshimself. PartQ Provideaproofofeachclaim. 1. f Pa Qb;Qb b = c; : Pa g` Qc 2. f m = n n = o;An g` Am Ao 3. f8 xx = m;Rma g`9 xRxx 4. :9 xx 6 = m `8 x 8 y Px Py 5. 8 x 8 y Rxy x = y ` Rab Rba 6. f9 xJx; 9 x : Jx g`9 x 9 yx 6 = y 7. f8 x x = n \$ Mx ; 8 x Ox & Mx g` On 8. f9 xDx; 8 x x = p \$ Dx g` Dp 9. f9 x Kx & 8 y Ky x = y & Bx ;Kd g` Bd 10. ` Pa !8 x Px x 6 = a PartR LookbackatPartFonp.78.Foreachargument:IfitisvalidinQL, giveaproof.Ifitisinvalid,constructamodeltoshowthatitisinvalid. ? PartS Foreachofthefollowingpairsofsentences:Iftheyarelogically equivalentinQL,giveproofstoshowthis.Iftheyarenot,constructamodel toshowthis. 1. 8 xPx Qc 8 x Px Qc 2. 8 xPx & Qc 8 x Px & Qc 3. Qc _9 xQx 9 x Qc Qx 4. 8 x 8 y 8 zBxyz 8 xBxxx 5. 8 x 8 yDxy 8 y 8 xDxy 6. 9 x 8 yDxy 8 y 9 xDxy ? PartT Foreachofthefollowingarguments:IfitisvalidinQL,giveaproof. Ifitisinvalid,constructamodeltoshowthatitisinvalid. 1. 8 x 9 yRxy : : 9 y 8 xRxy 2. 9 y 8 xRxy : : 8 x 9 yRxy 3. 9 x Px & : Qx : : 8 x Px !: Qx 4. 8 x Sx Ta Sd : :Ta

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ch.6proofs 139 5. 8 x Ax Bx 8 x Bx Cx : : 8 x Ax Cx 6. 9 x Dx Ex 8 x Dx Fx : : 9 x Dx & Fx 7. 8 x 8 y Rxy Ryx : :Rjj 8. 9 x 9 y Rxy Ryx : :Rjj 9. 8 xPx !8 xQx 9 x : Px : : 9 x : Qx 10. 9 xMx !9 xNx :9 xNx : : 8 x : Mx PartU 1.Ifyouknowthat A ` B ,whatcanyousayabout A & C ` B ?Explain youranswer. 2.Ifyouknowthat A ` B ,whatcanyousayabout A C ` B ?Explain youranswer.

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AppendixA Symbolicnotation Inthehistoryofformallogic,dierentsymbolshavebeenusedatdierenttimes andbydierentauthors.Often,authorswereforcedtousenotationthattheir printerscouldtypeset. Inonesense,thesymbolsusedforvariouslogicalconstantsisarbitrary.There isnothingwritteninheaventhatsaysthat` : 'mustbethesymbolfortruthfunctionalnegation.Wemighthavespeciedadierentsymboltoplaythat part.Oncewehavegivendenitionsforwell-formedformulaewandfor truthinourlogiclanguages,however,using` : 'isnolongerarbitrary.Thatis thesymbolfornegationinthistextbook,andsoitisthesymbolfornegation whenwritingsentencesinourlanguagesSLorQL. Thisappendixpresentssomecommonsymbols,sothatyoucanrecognizethem ifyouencountertheminanarticleorinanotherbook. summaryofsymbols negation : conjunction&, ^ disjunction conditional biconditional \$ Negation Twocommonlyusedsymbolsarethe hoe ,` : ',andthe swungdash ` .'Insomemoreadvancedformalsystemsitisnecessarytodistinguishbetweentwokindsofnegation;thedistinctionissometimesrepresentedbyusing both` : 'and` .' Disjunction Thesymbol` 'istypicallyusedtosymbolizeinclusivedisjunction. Conjunction Conjunctionisoftensymbolizedwiththe ampersand ,`&.'The ampersandisactuallyadecorativeformoftheLatinword`et'whichmeans `and';itiscommonlyusedinEnglishwriting.Asasymbolinaformalsys140

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appendix:symbolicnotation 141 tem,theampersandisnottheword`and';itsmeaningisgivenbytheformal semanticsforthelanguage.Perhapstoavoidthisconfusion,somesystemsuse adierentsymbolforconjunction.Forexample,` ^ 'isacounterparttothe symbolusedfordisjunction.Sometimesasingledot,` ',isused.Insomeolder texts,thereisnosymbolforconjunctionatall;` A and B 'issimplywritten ` AB .' MaterialConditional Therearetwocommonsymbolsforthematerialconditional:the arrow ,` ',andthe hook ,` .' MaterialBiconditional The double-headedarrow ,` \$ ',isusedinsystems thatusethearrowtorepresentthematerialconditional.Systemsthatusethe hookfortheconditionaltypicallyusethe triplebar ,` ',forthebiconditional. Quantiers TheuniversalquantieristypicallysymbolizedasanupsidedownA,` 8 ',andtheexistentialquantierasabackwardsE,` 9 .'Insometexts, thereisnoseparatesymbolfortheuniversalquantier.Instead,thevariableis justwritteninparenthesesinfrontoftheformulathatitbinds.Forexample, `all x are P 'iswritten x Px Insomesystems,thequantiersaresymbolizedwithlargerversionsofthesymbolsusedforconjunctionanddisjunction.Althoughquantiedexpressionscannotbetranslatedintoexpressionswithoutquantiers,thereisaconceptual connectionbetweentheuniversalquantierandconjunctionandbetweenthe existentialquantieranddisjunction.Considerthesentence 9 xPx ,forexample. Itmeansthat either therstmemberoftheUDisa P or thesecondoneis, or thethirdoneis,....Suchasystemusesthesymbol` W 'insteadof` 9 .' Polishnotation ThissectionbrieydiscussessententiallogicinPolishnotation,asystemof notationintroducedinthelate1920sbythePolishlogicianJanLukasiewicz. Lowercaselettersareusedassentenceletters.Thecapitalletter N isused fornegation. A isusedfordisjunction, K forconjunction, C fortheconditional, E forthebiconditional.`A'isforalternation,anothernameforlogical disjunction.`E'isforequivalence. notationPolish ofSLnotation : N & K A C \$ E InPolishnotation,abinaryconnectiveiswritten before thetwosentencesthat itconnects.Forexample,thesentence A & B ofSLwouldbewritten Kab in Polishnotation.

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142 forall x Thesentences : A B and : A B areverydierent;themainlogical operatoroftherstistheconditional,butthemainconnectiveofthesecond isnegation.InSL,weshowthisbyputtingparenthesesaroundtheconditional inthesecondsentence.InPolishnotation,parenthesesareneverrequired.The left-mostconnectiveisalwaysthemainconnective.Therstsentencewould simplybewritten CNab andthesecond NCab ThisfeatureofPolishnotationmeansthatitispossibletoevaluatesentences simplybyworkingthroughthesymbolsfromrighttoleft.Ifyouwereconstructingatruthtablefor NKab ,forexample,youwouldrstconsiderthe truth-valuesassignedto b and a ,thenconsidertheirconjunction,andthen negatetheresult.ThegeneralruleforwhattoevaluatenextinSLisnotnearly sosimple.InSL,thetruthtablefor : A & B requireslookingat A and B thenlookinginthemiddleofthesentenceattheconjunction,andthenatthe beginningofthesentenceatthenegation.Becausetheorderofoperationscan bespeciedmoremechanicallyinPolishnotation,variantsofPolishnotation areusedastheinternalstructureformanycomputerprogramminglanguages.

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AppendixB Solutionstoselected exercises Manyoftheexercisesmaybeansweredcorrectlyindierentways.Wherethat isthecase,thesolutionhererepresentsonepossiblecorrectanswer. Chapter1PartC 1.consistent 2.inconsistent 3.consistent 4.consistent Chapter1PartD 1,2,3,6,8,and10arepossible. Chapter2PartA 1. : M 2. M _: M 3. G C 4. : C & : G 5. C : G & : M 6. M C G Chapter2PartC 1. E 1 & E 2 143

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144 forall x 2. F 1 S 1 3. F 1 E 1 4. E 2 & : S 2 5. : E 1 & : E 2 6. E 1 & E 2 & : S 1 S 2 7. S 2 F 2 8. : E 1 !: E 2 & E 1 E 2 9. S 1 \$: S 2 10. E 2 & F 2 S 2 11. : E 2 & F 2 12. F 1 & F 2 \$ : E 1 & : E 2 Chapter2PartD A: Aliceisaspy. B: Bobisaspy. C: Thecodehasbeenbroken. G: TheGermanembassywillbeinanuproar. 1. A & B 2. A B C 3. : A B !: C 4. G C 5. C _: C & G 6. A B & : A & B Chapter2PartG 1.anobno 2.anobyes 3.ayesbyes 4.anobno 5.ayesbyes 6.anobno 7.anobyes 8.anobyes 9.anobno Chapter3PartA 1.tautology 2.contradiction 3.contingent

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solutionsforch.3 145 4.tautology 5.tautology 6.contingent 7.tautology 8.contradiction 9.tautology 10.contradiction 11.tautology 12.contingent 13.contradiction 14.contingent 15.tautology 16.tautology 17.contingent 18.contingent Chapter3PartB 2,3,5,6,8,and9arelogicallyequivalent. Chapter3PartC 1,3,6,7,and8areconsistent. Chapter3PartD 3,5,8,and10arevalid. Chapter3PartE 1. A and B havethesametruthvalueoneverylineofacompletetruthtable, so A \$ B istrueoneveryline.Itisatautology. 2.Thesentenceisfalseonsomelineofacompletetruthtable.Onthatline, A and B aretrueand C isfalse.Sotheargumentisinvalid. 3.Sincethereisnolineofacompletetruthtableonwhichallthreesentences aretrue,theconjunctionisfalseoneveryline.Soitisacontradiction. 4.Since A isfalseoneverylineofacompletetruthtable,thereisnolineon which A and B aretrueand C isfalse.Sotheargumentisvalid. 5.Since C istrueoneverylineofacompletetruthtable,thereisnolineon which A and B aretrueand C isfalse.Sotheargumentisvalid. 6.Notmuch. A B isatautologyif A and B aretautologies;itisacontradictioniftheyarecontradictions;itiscontingentiftheyarecontingent. 7. A and B havedierenttruthvaluesonatleastonelineofacomplete truthtable,and A B willbetrueonthatline.Onotherlines,itmight betrueorfalse.So A B iseitheratautologyoritiscontingent;itis not acontradiction. Chapter3PartF 1. : A B 2. : A !: B

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146 forall x 3. : [ A B !: B A ] Chapter4PartB 1. Rca Rcb Rcc ,and Rcd aresubstitutioninstancesof 8 xRcx 2.Oftheexpressionslisted,only 8 yLby isasubstitutioninstanceof 9 x 8 yLxy Chapter4PartC 1. Za & Zb & Zc 2. Rb & : Ab 3. Lcb Mb 4. Ab & Ac Lab & Lac 5. 9 x Rx & Zx 6. 8 x Ax Rx 7. 8 x Zx Mx Ax 8. 9 x Rx & : Ax 9. 9 x Rx & Lcx 10. 8 x Mx & Zx Lbx 11. 8 x Mx & Lax Lxa 12. 9 xRx Ra 13. 8 x Ax Rx 14. 8 x Mx & Lcx Lax 15. 9 x Mx & Lxb & : Lbx Chapter4PartG 1. :9 xTx 2. 8 x Mx Sx 3. 9 x : Sx 4. 9 x [ Cx & :9 yByx ] 5. :9 xBxx 6. :9 x Cx & : Sx & Tx 7. 9 x Cx & Tx & 9 x Mx & Tx & :9 x Cx & Mx & Tx 8. 8 x [ Cx !8 y : Cy Bxy ] 9. 8 x )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( Cx & Mx !8 y [ : Cy & : My Bxy ] Chapter4PartI 1. 8 x Cxp Dx 2. Cjp & Fj 3. 9 x Cxp & Fx

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solutionsforch.4 147 4. :9 xSxj 5. 8 x Cxp & Fx Dx 6. :9 x Cxp & Mx 7. 9 x Cjx & Sxe & Fj 8. Spe & Mp 9. 8 x Sxp & Mx !:9 yCyx 10. 9 x Sxj & 9 yCyx & Fj 11. 8 x Dx !9 y Sxy & Fy & Dy 12. 8 x Mx & Dx !9 y Cxy & Dy Chapter4PartK 1. 8 x Cx Bx 2. :9 xWx 3. 9 x 9 y Cx & Cy & x 6 = y 4. 9 x 9 y Jx & Ox & Jy & Oy & x 6 = y 5. 8 x 8 y 8 z Jx & Ox & Jy & Oy & Jz & Oz x = y x = z y = z 6. 9 x 9 y )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Jx & Bx & Jy & By & 8 z [ Jz & Bz x = z y = z ] 7. 9 x 1 9 x 2 9 x 3 9 x 4 Dx 1 & Dx 2 & Dx 3 & Dx 4 & x 1 6 = x 2 & x 1 6 = x 3 & x 1 6 = x 4 & x 2 6 = x 3 & x 2 6 = x 4 & x 3 6 = x 4 & :9 y Dy & y 6 = x 1 & y 6 = x 2 & y 6 = x 3 & y 6 = x 4 8. 9 x )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Dx & Cx & 8 y [ Dy & Cy x = y ]& Bx 9. 8 x Ox & Jx Wx & 9 x Mx & 8 y My x = y & Wx 10. 9 x )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Dx & Cx & 8 y [ Dy & Cy x = y ]& Wx !9 x 8 y Wx \$ x = y 11.widescope: :9 x Mx & 8 y My x = y & Jx narrowscope: 9 x Mx & 8 y My x = y & : Jx 12.widescope: :9 x 9 z )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Dx & Cx & Mz & 8 y [ Dy & Cy x = y ]& 8 y [ My z = y & x = z ] narrowscope: 9 x 9 z )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Dx & Cx & Mz & 8 y [ Dy & Cy x = y ]& 8 y [ My z = y & x 6 = z ] Chapter5PartA 2,3,4,6,8,and9aretrueinthemodel. Chapter5PartB 2,4,5,and7aretrueinthemodel. Chapter5PartD UD= f 10,11,12,13 g extension O = f 11,13 g extension S = ; extension T = f 10,11,12,13 g extension U = f 13 g extension N = f < 11,10 > < 12,11 > < 13,12 > g referent m =Johnny Chapter5PartE

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148 forall x 1.Thesentenceistrueinthismodel: UD= f Stan g extension D = f Stan g referent a =Stan referent b =Stan Anditisfalseinthismodel: UD= f Stan g extension D = ; referent a =Stan referent b =Stan 2.Thesentenceistrueinthismodel: UD= f Stan g extension T = f < Stan,Stan > g referent h =Stan Anditisfalseinthismodel: UD= f Stan g extension T = ; referent h =Stan 3.Thesentenceistrueinthismodel: UD= f Stan,Ollie g extension P = f Stan g referent m =Stan Anditisfalseinthismodel: UD= f Stan g extension P = ; referent m =Stan Chapter5PartF Therearemanypossiblecorrectanswers.Herearesome: 1.Makingtherstsentencetrueandthesecondfalse: UD= f alpha g extension J = f alpha g extension K = ; referent a =alpha 2.Makingtherstsentencetrueandthesecondfalse: UD= f alpha,omega g extension J = f alpha g referent m =omega 3.Makingtherstsentencefalseandthesecondtrue: UD= f alpha,omega g extension R = f < alpha,alpha > g

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solutionsforch.5 149 4.Makingtherstsentencefalseandthesecondtrue: UD= f alpha,omega g extension P = f alpha g extension Q = ; referent c =alpha 5.Makingtherstsentencetrueandthesecondfalse: UD= f iota g extension P = ; extension Q = ; 6.Makingtherstsentencefalseandthesecondtrue: UD= f iota g extension P = ; extension Q = f iota g 7.Makingtherstsentencetrueandthesecondfalse: UD= f iota g extension P = ; extension Q = f iota g 8.Makingtherstsentencetrueandthesecondfalse: UD= f alpha,omega g extension R = f < alpha,omega > < omega,alpha > g 9.Makingtherstsentencefalseandthesecondtrue: UD= f alpha,omega g extension R = f < alpha,alpha > < alpha,omega > g Chapter5PartI 1.Therearemanypossibleanswers.Hereisone: UD= f Harry,Sally g extension R = f < Sally,Harry > g referent a =Harry 2.Therearenopredicatesorconstants,soweonlyneedtogiveaUD.Any UDwith2memberswilldo. 3.Weneedtoshowthatitisimpossibletoconstructamodelinwhichthese arebothtrue.Suppose 9 xx 6 = a istrueinamodel.Thereissomethingin theuniverseofdiscoursethatis not thereferentof a .Sothereareatleast twothingsintheuniverseofdiscourse:referent a andthisotherthing. Callthisotherthing |weknow a 6 = .Butif a 6 = ,then 8 x 8 yx = y isfalse.Sotherstsentencemustbefalseifthesecondsentenceistrue. Assuch,thereisnomodelinwhichtheyarebothtrue.Therefore,they areinconsistent. Chapter5PartJ

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150 forall x 2.No,itwouldnotmakeanydierence.Thesatisfactionofasentencedoes notdependonthevariableassignment.Soasentencethatissatisedby some variableassignmentissatisedby every othervariableassignment aswell. Chapter6PartA 1 W !: B 2 A & W 3 B J & K 4 W &E2 5 : B E1,4 6 J & K E3,5 7 K &E6 1 L \$: O 2 L _: O 3 : L 4 : O E2,3 5 L \$ E1,4 6 : L R3 7 L : E3{6 1 Z C & : N 2 : Z N & : C 3 : N C 4 : N & : C DeM3 5 Z 6 C & : N E1,5 7 C &E6 8 : C &E4 9 : Z : I5{8 10 N & : C E2,9 11 N &E10 12 : N &E4 13 N C : E3{12 Chapter6PartB 1. 1 K & L want K \$ L 2 K want L 3 L &E1 4 L want K 5 K &E1 6 K \$ L \$ I2{3,4{5

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solutionsforch.6 151 2. 1 A B C want A & B C 2 A & B want C 3 A &E2 4 B C E1,3 5 B &E2 6 C E4,5 7 A & B C I2{6 3. 1 P & Q R 2 P !: R want Q E 3 P &E1 4 : R E2,3 5 Q R &E1 6 Q E5,4 7 Q E I6 4. 1 C & D E want E D 2 : E want D 3 C & D E1,2 4 D &E3 5 : E D I2{4 6 E D MC5 5. 1 : F G 2 F H want G H 3 : G want H 4 :: F MT1,3 5 F DN4 6 H E2,5 7 : G H I3{6 8 G H MC7

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152 forall x 6. 1 X & Y X & Z 2 : X & D 3 D M want M 4 : X forreductio 5 : X _: Y I4 6 : X & Y DeM5 7 X & Z E1,6 8 X &E7 9 : X R4 10 X : E4{9 11 : M forreductio 12 D E3,11 13 X & D &I10,12 14 : X & D R2 15 M : E11{14 Chapter6PartH 1 8 x 9 y Rxy Ryx 2 8 x : Rmx 3 9 y Rmy Rym 8 E1 4 Rma Ram 5 : Rma 8 E2 6 Ram E4,5 7 9 xRxm 9 I6 8 9 xRxm 9 E3,4{7 1 8 x 9 yLxy !8 zLzx 2 Lab 3 9 yLay !8 zLza 8 E1 4 9 yLay 9 I2 5 8 zLza E3,4 6 Lca 8 E5 7 9 yLcy !8 zLzc 8 E1 8 9 yLcy 9 I5 9 8 zLzc E7,8 10 Lcc 8 E9 11 8 xLxx 8 I10

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solutionsforch.6 153 1 8 x Jx Kx 2 9 x 8 yLxy 3 8 xJx 4 Ja 8 E3 5 Ja Ka 8 E1 6 Ka E5,4 7 8 yLay 8 Laa 8 E7 9 Ka & Laa &I6,8 10 9 x Kx & Lxx 9 I9 11 9 x Kx & Lxx 9 E2,7{10 1 : 9 xMx _8 x : Mx 2 :9 xMx & :8 x : Mx DeM1 3 :9 xMx &E2 4 8 x : Mx QN3 5 :8 x : Mx &E2 6 9 xMx _8 x : Mx : E1{5 Chapter6PartI 1. 1 : 8 xFx _:8 xFx forreductio 2 :8 xFx & ::8 xFx DeM1 3 :8 xFx &E2 4 ::8 xFx &E2 5 8 xFx _:8 xFx : E1{4 2. 1 8 x Mx \$ Nx 2 Ma & 9 xRxa want 9 xNx 3 Ma \$ Na 8 E1 4 Ma &E2 5 Na \$ E3,4 6 9 xNx 9 I5

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154 forall x 3. 1 8 x : Mx Ljx 2 8 x Bx Ljx 3 8 x Mx Bx want 8 xLjx 4 : Ma Lja 8 E1 5 Ma Lja _! 4 6 Ba Lja 8 E2 7 Ma Ba 8 E3 8 Lja 7,5,6 9 8 xLjx 8 I8 4. 1 8 x Cx & Dt want 8 xCx & Dt 2 Ca & Dt 8 E1 3 Ca &E2 4 8 xCx 8 I3 5 Dt &E2 6 8 xCx & Dt &I4,5 5. 1 9 x Cx Dt want 9 xCx Dt 2 Ca Dt for 9 E 3 : 9 xCx Dt forreductio 4 :9 xCx & : Dt DeM3 5 : Dt &E4 6 Ca E2,5 7 9 xCx 9 I6 8 :9 xCx &E4 9 9 xCx Dt : E3{8 10 9 xCx Dt 9 E1,2{9 Chapter6PartP Regardingthetranslationofthisargument,seep.67.

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solutionsforch.6 155 1 9 x 8 y [ 8 z Lxz Lyz Lxy ] 2 8 y [ 8 z Laz Lyz Lay ] 3 8 z Laz Laz Laa 8 E2 4 :9 xLxx forreductio 5 8 x : Lxx QN4 6 : Laa 8 E5 7 :8 z Laz Laz MT5,6 8 Lab 9 Lab R8 10 Lab Lab I8{{9 11 8 z Laz Laz 8 I10 12 :8 z Laz Laz R7 13 9 xLxx : E4{{12 14 9 xLxx 9 E1,2{{13 Chapter6PartS 2,3,and5arelogicallyequivalent. Chapter6PartT 2,4,5,7,and10arevalid.Herearecompleteanswersfor someofthem: 1. UD= f mocha,freddo g extension R = f < mocha,freddo > < freddo,mocha > g 2. 1 9 y 8 xRxy want 8 x 9 yRxy 2 8 xRxa 3 Rba 8 E2 4 9 yRby 9 I3 5 8 x 9 yRxy 8 I4 6 8 x 9 yRxy 9 E1,2{5

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QuickReference CharacteristicTruthTables A : A T F F T A B A & B A B A B A \$ B T T T T T T T F F T F F F T F T T F F F F F T T Symbolization SententialConnectives chapter2 Itisnotthecasethat P : P Either P ,or Q P Q Neither P ,nor Q : P Q or : P & : Q Both P ,and Q P & Q If P ,then Q P Q P onlyif Q P Q P ifandonlyif Q P \$ Q Unless P Q P unless Q P Q Predicates chapter4 All F sare G s. 8 x Fx Gx Some F sare G s. 9 x Fx & Gx Notall F sare G s. :8 x Fx Gx or 9 x Fx & : Gx No F sare G s. 8 x Fx !: Gx or :9 x Fx & Gx Identity section4.6 Only j is G 8 x Gx \$ x = j Everythingbesides j is G 8 x x 6 = j Gx The F is G 9 x Fx & 8 y Fy x = y & Gx `TheFisnotG'canbetranslatedtwoways: ItisnotthecasethattheFisG.wide :9 x Fx & 8 y Fy x = y & Gx The F isnonG .narrow 9 x Fx & 8 y Fy x = y & : Gx 156

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Usingidentitytosymbolizequantities Thereareatleast F s. one 9 xFx two 9 x 1 9 x 2 Fx 1 & Fx 2 & x 1 6 = x 2 three 9 x 1 9 x 2 9 x 3 Fx 1 & Fx 2 & Fx 3 & x 1 6 = x 2 & x 1 6 = x 3 & x 2 6 = x 3 four 9 x 1 9 x 2 9 x 3 9 x 4 Fx 1 & Fx 2 & Fx 3 & Fx 4 & x 1 6 = x 2 & x 1 6 = x 3 & x 1 6 = x 4 & x 2 6 = x 3 & x 2 6 = x 4 & x 3 6 = x 4 n 9 x 1 9 x n Fx 1 & & Fx n & x 1 6 = x 2 & & x n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 6 = x n Thereareatmost F s. Onewaytosay`atmost n thingsare F 'istoputanegationsigninfrontofone ofthesymbolizationsaboveandsay : `atleast n +1thingsare F .'Equivalently: one 8 x 1 8 x 2 Fx 1 & Fx 2 x 1 = x 2 two 8 x 1 8 x 2 8 x 3 Fx 1 & Fx 2 & Fx 3 x 1 = x 2 x 1 = x 3 x 2 = x 3 three 8 x 1 8 x 2 8 x 3 8 x 4 Fx 1 & Fx 2 & Fx 3 & Fx 4 x 1 = x 2 x 1 = x 3 x 1 = x 4 x 2 = x 3 x 2 = x 4 x 3 = x 4 n 8 x 1 8 x n +1 Fx 1 & & Fx n +1 x 1 = x 2 __ x n = x n +1 Thereareexactly F s. Onewaytosay`exactly n thingsare F 'istoconjointwoofthesymbolizations aboveandsay`atleast n thingsare F '&`atmost n thingsare F .'Thefollowing equivalentformulaeareshorter: zero 8 x : Fx one 9 x Fx & :9 y Fy & x 6 = y two 9 x 1 9 x 2 Fx 1 & Fx 2 & x 1 6 = x 2 & :9 y )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(Fy & y 6 = x 1 & y 6 = x 2 three 9 x 1 9 x 2 9 x 3 Fx 1 & Fx 2 & Fx 3 & x 1 6 = x 2 & x 1 6 = x 3 & x 2 6 = x 3 & :9 y Fy & y 6 = x 1 & y 6 = x 2 & y 6 = x 3 n 9 x 1 9 x n Fx 1 & & Fx n & x 1 6 = x 2 & & x n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 6 = x n & :9 y Fy & y 6 = x 1 & & y 6 = x n SpecifyingthesizeoftheUD Removing F fromthesymbolizationsaboveproducessentencesthattalkabout thesizeoftheUD.Forinstance,`thereareatleast2thingsintheUD'may besymbolizedas 9 x 9 y x 6 = y 157

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BasicRulesofProof Reiteration m A A R m ConjunctionIntroduction m A n B A & B &I m n ConjunctionElimination m A & B A &E m B &E m DisjunctionIntroduction m A A B I m B A I m DisjunctionElimination m A B n : B A E m n m A B n : A B E m n ConditionalIntroduction m A want B n B A B I m { n ConditionalElimination m A B n A B E m n BiconditionalIntroduction m A want B n B p B want A q A A \$ B \$ I m { n p { q BiconditionalElimination m A \$ B n B A \$ E m n m A \$ B n A B \$ E m n NegationIntroduction m A forreductio n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 B n : B : A : I m { n NegationElimination m : A forreductio n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 B n : B A : E m { n

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QuantierRules ExistentialIntroduction m A 9 xA [ x jj c ] 9 I m x mayreplacesomeoralloccurrencesof c in A ExistentialElimination m 9 xA n A [ c j x ] p B B 9 E m n { p Theconstant c mustnotappearin 9 xA ,in B ,or inanyundischargedassumption. UniversalIntroduction m A 8 xA [ x j c ] 8 I m c mustnotoccurinanyundischargedassumptions. UniversalElimination m 8 xA A [ c j x ] 8 E m IdentityRules c = c =I m c = d n A A [ c jj d ]=E m n A [ d jj c ]=E m n Oneconstantmayreplacesomeoralloccurrences oftheother. DerivedRules Dilemma m A B n A C p B C C m n p ModusTollens m A B n : B : A MT m n HypotheticalSyllogism m A B n B C A C HS m n ReplacementRules Commutivity Comm A & B B & A A B B A A \$ B B \$ A DeMorgan DeM : A B : A & : B : A & B : A _: B DoubleNegation DN :: A A MaterialConditional MC A B : A B A B : A B BiconditionalExchange \$ ex [ A B & B A ] A \$ B QuantifierNegation QN :8 xA 9 x : A :9 xA 8 x : A

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TRACE ROUTE

Total Execution Time: 2 Milliseconds

MILLISECOND   CLASS.METHODMESSAGE
0sobekcm_page_globals.constructor
0sobekcm_page_globals.constructorApplication State validated or built
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.constructorNavigation Object created from URI query string
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.display_itemRetrieving item or group information
0sobekcm_page_globals.get_entire_collection_hierarchyRetrieving hierarchy information
0sobekcm_assistant.get_entire_collection_hierarchy
0cached_data_manager.retrieve_item_aggregation
0cached_data_manager.retrieve_item_aggregationFound item aggregation on local cache
0item_aggregation_builder.get_item_aggregationFound 'all' item aggregation in cache