the game of logic by lewis carroll
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7/31/2019 The Game of Logic by Lewis Carroll
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SeetheSunisoverhead,|--y-------y'-|Shiningonus,FULLand|||RED!|x'||7|8|NowtheSunisgoneaway,-------------AndtheEMPTYskyisGREY!___
THEGAMEOFLOGIC
ByLewisCarrol
TomyChild-friend.
Icharminvain;forneveragain,AllkeenlyasmyglanceIbend,WillMemory,goddesscoy,EmbodyformyjoyDeparteddays,norletmegaze
Onthee,myfairyfriend!
Yetcouldthyface,inmysticgrace,Amomentsmileonme,'twouldsendFar-dartingraysoflightFromHeavenathwartthenight,BywhichtoreadinverydeedThyspirit,sweetestfriend!
SomaythestreamofLife'slongdreamFlowgentlyonwardtoitsend,Withmanyafloweretgay,Adownitswillowyway:
Maynosighvex,nocareperplex,Mylovinglittlefriend!
NOTABENE.
WitheachcopyofthisBookisgivenanEnvelope,containingaDiagram(similartothefrontispiece)oncard,andnineCounters,fourredandfivegrey.
TheEnvelope,&c.canbehadseparately,at3d.each.
TheAuthorwillbeverygratefulforsuggestions,especiallyfrombeginnersinLogic,ofanyalterations,orfurtherexplanations,thatmayseemdesirable.Lettersshouldbeaddressedtohimat"29,BedfordStreet,CoventGarden,London."
PREFACE
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"Therefoam'drebelliousLogic,gagg'dandbound."
ThisGamerequiresnineCounters--fourofonecolourandfiveofanother:sayfourredandfivegrey.
BesidesthenineCounters,italsorequiresonePlayer,ATLEAST.IamnotawareofanyGamethatcanbeplayedwithLESSthanthisnumber:whilethereareseveralthatrequireMORE:takeCricket,forinstance,whichrequirestwenty-two.Howmucheasieritis,whenyouwanttoplayaGame,tofindONEPlayerthantwenty-two.Atthesametime,thoughonePlayerisenough,agooddealmoreamusementmaybegotbytwoworkingatittogether,andcorrectingeachother'smistakes.
Asecondadvantage,possessedbythisGame,isthat,besidesbeinganendlesssourceofamusement(thenumberofarguments,thatmaybeworkedbyit,beinginfinite),itwillgivethePlayersalittleinstructionaswell.ButisthereanygreatharminTHAT,solongasyougetplentyofamusement?
CONTENTS.
CHAPTERPAGE
I.NEWLAMPSFOROLD.1.Propositions.......12.Syllogisms........203.Fallacies........32
II.CROSSQUESTIONS.1.Elementary........372.HalfofSmallerDiagram.Propositions
toberepresented.....403.Do.Symbolstobeinterpreted..424.SmallerDiagram.Propositionstoberepresented.......445.Do.Symbolstobeinterpreted..466.LargerDiagram.Propositionstoberepresented.......487.BothDiagramstobeemployed..51
III.CROOKEDANSWERS.1.Elementary........552.HalfofSmallerDiagram.Propositionsrepresented.......59
3.Do.Symbolsinterpreted...614.SmallerDiagram.Propositionsrepresented.625.Do.Symbolsinterpreted...656.LargerDiagram.Propositionsrepresented.677.BothDiagramsemployed....72
IV.HITORMISS.........85
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CHAPTERI.
NEWLAMPSFOROLD.
"Lightcome,lightgo."_________
1.Propositions.
"SomenewCakesarenice.""NonewCakesarenice.""Allnewcakesarenice."
Therearethree'PROPOSITIONS'foryou--theonlythreekindswearegoingtouseinthisGame:andthefirstthingtobedoneistolearnhowtoexpressthemontheBoard.
Letusbeginwith
"SomenewCakesarenice."
Butbeforedoingso,aremarkhastobemade--onethatisratherimportant,andbynomeanseasytounderstandallinamoment:sopleasetoreadthisVERYcarefully.
TheworldcontainsmanyTHINGS(suchas"Buns","Babies","Beetles"."Battledores".&c.);andtheseThingspossessmanyATTRIBUTES(suchas"baked","beautiful","black","broken",&c.:infact,whatevercanbe"attributedto",thatis"saidtobelongto",anyThing,isanAttribute).WheneverwewishtomentionaThing,weuseaSUBSTANTIVE:whenwewishtomentionanAttribute,weuseanADJECTIVE.Peoplehaveaskedthequestion"CanaThingexist
withoutanyAttributesbelongingtoit?"Itisaverypuzzlingquestion,andI'mnotgoingtotrytoanswerit:letusturnupournoses,andtreatitwithcontemptuoussilence,asifitreallywasn'tworthnoticing.But,iftheyputittheotherway,andask"CananAttributeexistwithoutanyThingforittobelongto?",wemaysayatonce"No:nomorethanaBabycouldgoarailway-journeywithnoonetotakecareofit!"Youneversaw"beautiful"floatingaboutintheair,orlitteredaboutonthefloor,withoutanyThingtoBEbeautiful,nowdidyou?
AndnowwhatamIdrivingat,inallthislongrigmarole?Itisthis.Youmayput"is"or"are"betweennamesoftwoTHINGS(forexample,"somePigsarefatAnimals"),orbetweenthenamesoftwo
ATTRIBUTES(forexample,"pinkislight-red"),andineachcaseitwillmakegoodsense.But,ifyouput"is"or"are"betweenthenameofaTHINGandthenameofanATTRIBUTE(forexample,"somePigsarepink"),youdoNOTmakegoodsense(forhowcanaThingBEanAttribute?)unlessyouhaveanunderstandingwiththepersontowhomyouarespeaking.Andthesimplestunderstandingwould,Ithink,bethis--thattheSubstantiveshallbesupposedtoberepeatedattheendofthesentence,sothatthesentence,ifwrittenoutinfull,wouldbe"somePigsarepink(Pigs)".Andnowtheword"are"makesquitegoodsense.
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Thus,inordertomakegoodsenseoftheProposition"somenewCakesarenice",wemustsupposeittobewrittenoutinfull,intheform"somenewCakesarenice(Cakes)".Nowthiscontainstwo'TERMS'--"newCakes"beingoneofthem,and"nice(Cakes)"theother."NewCakes,"beingtheonewearetalkingabout,iscalledthe'SUBJECT'oftheProposition,and"nice(Cakes)"the'PREDICATE'.AlsothisPropositionissaidtobea'PARTICULAR'one,sinceitdoesnotspeakoftheWHOLEofitsSubject,butonlyofaPARTofit.Theothertwokindsaresaidtobe'UNIVERSAL',becausetheyspeakoftheWHOLEoftheirSubjects--theonedenyingniceness,andtheotherassertingit,oftheWHOLEclassof"newCakes".Lastly,ifyouwouldliketohaveadefinitionoftheword'PROPOSITION'itself,youmaytakethis:--"asentencestatingthatsome,ornone,orall,oftheThingsbelongingtoacertainclass,calledits'Subject',arealsoThingsbelongingtoacertainotherclass,calledits'Predicate'".
Youwillfindthesesevenwords--PROPOSITION,ATTRIBUTE,TERM,SUBJECT,PREDICATE,PARTICULAR,UNIVERSAL--charminglyuseful,ifanyfriendshouldhappentoaskifyouhaveeverstudiedLogic.Mindyoubringallsevenwordsintoyouranswer,andyoufriendwillgoawaydeeplyimpressed--'asadderandawiserman'.
NowpleasetolookatthesmallerDiagramontheBoard,andsupposeittobeacupboard,intendedforalltheCakesintheworld(itwouldhavetobeagoodlargeone,ofcourse).Andletussupposeallthenewonestobeputintotheupperhalf(marked'x'),andalltherest(thatis,theNOT-newones)intothelowerhalf(marked'x'').ThusthelowerhalfwouldcontainELDERLYCakes,AGEDCakes,ANTE-DILUVIANCakes--ifthereareany:Ihaven'tseenmany,myself--andsoon.LetusalsosupposealltheniceCakestobeputintotheleft-handhalf(marked'y'),andalltherest(thatis,thenot-niceones)intotheright-handhalf(marked'y'').Atpresent,then,wemustunderstandxtomean"new",x'"not-new",y"nice",andy'"not-nice."
AndnowwhatkindofCakeswouldyouexpecttofindincompartmentNo.5?
Itispartoftheupperhalf,yousee;sothat,ifithasanyCakesinit,theymustbeNEW:anditispartoftheleft-handhalf;sothattheymustbeNICE.HenceifthereareanyCakesinthiscompartment,theymusthavethedouble'ATTRIBUTE'"newandnice":or,ifweuseletters,themustbe"xy."
Observethatthelettersx,yarewrittenontwooftheedgesofthiscompartment.ThisyouwillfindaveryconvenientruleforknowingwhatAttributesbelongtotheThingsinanycompartment.TakeNo.7,forinstance.IfthereareanyCakesthere,theymust
be"x'y",thatis,theymustbe"not-newandnice."
Nowletusmakeanotheragreement--thataredcounterinacompartmentshallmeanthatitis'OCCUPIED',thatis,thatthereareSOMECakesinit.(Theword'some,'inLogic,means'oneormore'sothatasingleCakeinacompartmentwouldbequiteenoughreasonforsaying"thereareSOMECakeshere").Alsoletusagreethatagreycounterinacompartmentshallmeanthatitis'EMPTY',thatisthatthereareNOCakesinit.InthefollowingDiagrams,Ishallput'1'(meaning'oneormore')whereyouaretoputaRED
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counter,and'0'(meaning'none')whereyouaretoputaGREYone.
AstheSubjectofourPropositionistobe"newCakes",weareonlyconcerned,atpresent,withtheUPPERhalfofthecupboard,wherealltheCakeshavetheattributex,thatis,"new."
Now,fixingourattentiononthisupperhalf,supposewefounditmarkedlikethis,
-----------||||1|||||-----------
thatis,witharedcounterinNo.5.Whatwouldthistellus,withregardtotheclassof"newCakes"?
WoulditnottellusthatthereareSOMEoftheminthexy-compartment?Thatis,thatsomeofthem(besideshavingtheAttributex,whichbelongstobothcompartments)havetheAttributey(thatis,"nice").Thiswemightexpressbysaying"somex-Cakesarey-(Cakes)",or,
puttingwordsinsteadofletters,
"SomenewCakesarenice(Cakes)",
or,inashorterform,
"SomenewCakesarenice".
AtlastwehavefoundouthowtorepresentthefirstProposition
ofthisSection.IfyouhavenotCLEARLYunderstoodallIhavesaid,gonofurther,butreaditoverandoveragain,tillyouDOunderstandit.Afterthatisoncemastered,youwillfindalltherestquiteeasy.
Itwillsavealittletrouble,indoingtheotherPropositions,ifweagreetoleaveouttheword"Cakes"altogether.IfinditconvenienttocallthewholeclassofThings,forwhichthecupboardisintended,the'UNIVERSE.'Thuswemighthavebegunthisbusinessbysaying"LetustakeaUniverseofCakes."(Soundsnice,doesn'tit?)
OfcourseanyotherThingswouldhavedonejustaswellasCakes.
WemightmakePropositionsabout"aUniverseofLizards",oreven"aUniverseofHornets".(Wouldn'tTHATbeacharmingUniversetolivein?)
Sofar,then,wehavelearnedthat
-----------||||1||
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|||-----------
means"somexandy,"i.e."somenewarenice."
Ithinkyouwillseewithoutfurtherexplanation,that
-----------|||||1||||-----------
means"somexarey',"i.e."somenewarenot-nice."
NowletusputaGREYcounterintoNo.5,andaskourselvesthemeaningof
-----------|||
|0|||||-----------
Thistellsusthatthexy-compartmentisEMPTY,whichwemayexpressby"noxarey",or,"nonewCakesarenice".ThisisthesecondofthethreePropositionsattheheadofthisSection.
Inthesameway,
-----------
|||||0||||-----------
wouldmean"noxarey',"or,"nonewCakesarenot-nice."
Whatwouldyoumakeofthis,Iwonder?
-----------|||
|1|1||||-----------
IhopeyouwillnothavemuchtroubleinmakingoutthatthisrepresentsaDOUBLEProposition:namely,"somexarey,ANDsomearey',"i.e."somenewarenice,andsomearenot-nice."
Thefollowingisalittleharder,perhaps:
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-----------||||0|0||||-----------
Thismeans"noxarey,ANDnonearey',"i.e."nonewarenice,ANDnonearenot-nice":whichleadstotherathercuriousresultthat"nonewexist,"i.e."noCakesarenew."Thisisbecause"nice"and"not-nice"makewhatwecallan'EXHAUSTIVE'divisionoftheclass"newCakes":i.e.betweenthem,theyEXHAUSTthewholeclass,sothatallthenewCakes,thatexist,mustbefoundinoneortheotherofthem.
Andnowsupposeyouhadtorepresent,withcountersthecontradictoryto"noCakesarenew",whichwouldbe"someCakesarenew",or,puttinglettersforwords,"someCakesarex",howwouldyoudoit?
Thiswillpuzzleyoualittle,Iexpect.EvidentlyyoumustputaredcounterSOMEWHEREinthex-halfofthecupboard,sinceyou
knowthereareSOMEnewCakes.ButyoumustnotputitintotheLEFT-HANDcompartment,sinceyoudonotknowthemtobeNICE:normayyouputitintotheRIGHT-HANDone,sinceyoudonotknowthemtobeNOT-NICE.
What,then,areyoutodo?IthinkthebestwayoutofthedifficultyistoplacetheredcounterONTHEDIVISION-LINEbetweenthexy-compartmentandthexy'-compartment.ThisIshallrepresent(asIalwaysput'1'whereyouaretoputaredcounter)bythediagram
-----------
||||-1-||||-----------
OuringeniousAmericancousinshaveinventedaphrasetoexpressthepositionofamanwhowantstojoinoneortheotheroftwoparties--suchastheirtwoparties'Democrats'and'Republicans'--butcan'tmakeuphismindWHICH.Suchamanissaidtobe"sittingonthefence."Nowthatisexactlythepositionoftheredcounteryouhavejustplacedonthedivision-line.HelikesthelookofNo.5,andhelikesthelookofNo.6,andhedoesn'tknowWHICHto
jumpdowninto.Sotherehesitsastride,sillyfellow,danglinghislegs,oneoneachsideofthefence!
NowIamgoingtogiveyouamuchharderonetomakeout.Whatdoesthismean?
-----------||||1|0|
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|||-----------
ThisisclearlyaDOUBLEProposition.Ittellsusnotonlythat"somexarey,"butalsothe"noxareNOTy."Hencetheresultis"ALLxarey,"i.e."allnewCakesarenice",whichisthelastofthethreePropositionsattheheadofthisSection.
Wesee,then,thattheUniversalProposition
"AllnewCakesarenice"
consistsofTWOPropositionstakentogether,namely,
"SomenewCakesarenice,"and"NonewCakesarenot-nice."
Inthesameway
-----------||||0|1||||-----------
wouldmean"allxarey'",thatis,
"AllnewCakesarenot-nice."
NowwhatwouldyoumakeofsuchaPropositionas"TheCakeyouhavegivenmeisnice"?IsitParticularorUniversal?
"Particular,ofcourse,"youreadilyreply."OnesingleCakeishardlyworthcalling'some,'even."
No,mydearimpulsiveReader,itis'Universal'.Rememberthat,fewastheyare(andIgrantyoutheycouldn'twellbefewer),theyare(orrather'itis')ALLthatyouhavegivenme!Thus,if(leaving'red'outofthequestion)IdividemyUniverseofCakes
intotwoclasses--theCakesyouhavegivenme(towhichIassigntheupperhalfofthecupboard),andthoseyouHAVEN'Tgivenme(whicharetogobelow)--Ifindthelowerhalffairlyfull,andtheupperoneasnearlyaspossibleempty.Andthen,whenIamtoldtoputanuprightdivisionintoeachhalf,keepingtheNICECakestotheleft,andtheNOT-NICEonestotheright,IbeginbycarefullycollectingALLtheCakesyouhavegivenme(sayingtomyself,fromtimetotime,"Generouscreature!HowshallIeverrepaysuchkindness?"),andpilingthemupintheleft-handcompartment.ANDITDOESN'TTAKELONGTODOIT!
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HereisanotherUniversalPropositionforyou."BarzillaiBeckaleggisanhonestman."Thatmeans"ALLtheBarzillaiBeckaleggs,thatIamnowconsidering,arehonestmen."(YouthinkIinventedthatname,nowdon'tyou?ButIdidn't.It'sonacarrier'scart,somewheredowninCornwall.)
ThiskindofUniversalProposition(wheretheSubjectisasingleThing)iscalledan'INDIVIDUAL'Proposition.
Nowletustake"NICECakes"astheSubjectofProposition:thatis,letusfixourthoughtsontheLEFT-HANDhalfofthecupboard,wherealltheCakeshaveattributey,thatis,"nice."
-----Supposewefinditmarkedlikethis:--|||1|Whatwouldthattellus?||-----||||||-----
Ihopethatitisnotnecessary,afterexplainingtheHORIZONTALoblongsofully,tospendmuchtimeovertheUPRIGHTone.Ihopeyouwillsee,foryourself,thatthismeans"someyarex",thatis,
"SomeniceCakesarenew."
"But,"youwillsay,"wehavehadthiscasebefore.YouputaredcounterintoNo.5,andyoutoldusitmeant'somenewCakesare
nice';andNOWyoutellusthatitmeans'someNICECakesareNEW'!CanitmeanBOTH?"
Thequestionisaverythoughtfulone,anddoesyouGREATcredit,dearReader!ItDOESmeanboth.Ifyouchoosetotakex(thatis,"newCakes")asyourSubject,andtoregardNo.5aspartofaHORIZONTALoblong,youmayreadit"somexarey",thatis,"somenewCakesarenice":but,ifyouchoosetotakey(thatis,"niceCake")asyourSubject,andtoregardNo.5aspartofanUPRIGHToblong,THENyoumayreadit"someyarex",thatis,"someniceCakesarenew".Theyaremerelytwodifferentwaysofexpressingtheverysametruth.
Withoutmorewords,Iwillsimplysetdowntheotherwaysinwhichthisuprightoblongmightbemarked,addingthemeaningineachcase.Bycomparingthemwiththevariouscasesofthehorizontaloblong,youwill,Ihope,beabletounderstandthemclearly.
Youwillfinditagoodplantoexamineyourselfonthistable,bycoveringupfirstonecolumnandthentheother,and'dodgingabout',asthechildrensay.
Alsoyouwilldowelltowriteoutforyourselftwoothertables--one
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fortheLOWERhalfofthecupboard,andtheotherforitsRIGHT-HANDhalf.
AndnowIthinkwehavesaidallweneedtosayaboutthesmallerDiagram,andmaygoontothelargerone.
_________________________________________________ |Symbols.|Meanings._______________|_________________________________ -----|||||||Someyarex';|||i.e.Somenicearenot-new.-----|||||1|||||-----||-----||||Noyarex;|0||i.e.Nonicearenew.|||
-----|[Observethatthisismerelyanotherwayof|||expressing"Nonewarenice."]||||||-----||-----|||||||Noyarex';|||i.e.Nonicearenot-new.-----|||||0||
|||-----||-----|||||1||Someyarex,andsomearex';|||i.e.Somenicearenew,andsomeare-----|not-new.||||1|||||-----||
-----|||||0||Noyarex,andnonearex';i.e.Noy|||exist;-----|i.e.NoCakesarenice.||||0|||||-----||
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-----|||||1||Allyarex;|||i.e.Allnicearenew.-----|||||0|||||-----||-----|||||0||Allyarex';|||i.e.Allnicearenot-new.-----|||||1|||||-----|_______________|_________________________________
Thismaybetakentobeacupboarddividedinthesamewayasthelast,butALSOdividedintotwoportions,fortheAttributem.Let
usgivetomthemeaning"wholesome":andletussupposethatallWHOLESOMECakesareplacedINSIDEthecentralSquare,andalltheUNWHOLESOMEonesOUTSIDEit,thatis,inoneorotherofthefourqueer-shapedOUTERcompartments.
Weseethat,justas,inthesmallerDiagram,theCakesineachcompartmenthadTWOAttributes,so,here,theCakesineachcompartmenthaveTHREEAttributes:and,justastheletters,representingtheTWOAttributes,werewrittenontheEDGESofthecompartment,so,here,theyarewrittenattheCORNERS.(Observethatm'issupposedtobewrittenateachofthefouroutercorners.)Sothatwecantellinamoment,bylookingatacompartment,whatthreeAttributesbelongtotheThingsinit.Forinstance,takeNo.12.Herewe
findx,y',m,atthecorners:soweknowthattheCakesinit,ifthereareany,havethetripleAttribute,'xy'm',thatis,"new,not-nice,andwholesome."Again,takeNo.16.Herewefind,atthecorners,x',y',m':sotheCakesinitare"not-new,not-nice,andunwholesome."(RemarkablyuntemptingCakes!)
ItwouldtakefartoolongtogothroughallthePropositions,containingxandy,xandm,andyandmwhichcanberepresentedonthisdiagram(thereareninety-sixaltogether,soIamsureyouwillexcuseme!)andImustcontentmyselfwithdoingtwoorthree,asspecimens.Youwilldowelltoworkoutalotmoreforyourself.
Takingtheupperhalfbyitself,sothatourSubjectis"newCakes",
howarewetorepresent"nonewCakesarewholesome"?
Thisis,writinglettersforwords,"noxarem."NowthistellsusthatnoneoftheCakes,belongingtotheupperhalfofthecupboard,aretobefoundINSIDEthecentralSquare:thatis,thetwocompartments,No.11andNo.12,areEMPTY.Andthis,ofcourse,isrepresentedby
-------------------
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||||_____|_____||||||||0|0|||||||-------------------
AndnowhowarewetorepresentthecontradictoryProposition"SOMExarem"?ThisisadifficultyIhavealreadyconsidered.IthinkthebestwayistoplacearedcounterONTHEDIVISION-LINEbetweenNo.11andNo.12,andtounderstandthistomeanthatONEofthetwocompartmentsis'occupied,'butthatwedonotatpresentknowWHICH.ThisIshallrepresentthus:--
-------------------||||_____|_____||||||||-1-|||||||-------------------
Nowletusexpress"allxarem."
Thisconsists,weknow,ofTWOPropositions,
"Somexarem,"and"Noxarem'."
Letusexpressthenegativepartfirst.ThistellsusthatnoneoftheCakes,belongingtotheupperhalfofthecupboard,aretobefoundOUTSIDEthecentralSquare:thatis,thetwocompartments,No.9andNo.10,areEMPTY.This,ofcourse,isrepresentedby
-------------------|0|0||_____|_____||||||||||||||||-------------------
Butwehaveyettorepresent"Somexarem."ThistellsusthatthereareSOMECakesintheoblongconsistingofNo.11andNo.12:soweplaceourredcounter,asinthepreviousexample,onthedivision-linebetweenNo.11andNo.12,andtheresultis
-------------------|0|0||_____|_____||||||||-1-|||||||-------------------
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Nowletustryoneortwointerpretations.
Whatarewetomakeofthis,withregardtoxandy?
-------------------||0||_____|_____||||||||1|0|||||||-------------------
Thistellsus,withregardtothexy'-Square,thatitiswholly'empty',sinceBOTHcompartmentsaresomarked.Withregardtothexy-Square,ittellsusthatitis'occupied'.True,itisonlyONEcompartmentofitthatissomarked;butthatisquiteenough,whethertheotherbe'occupied'or'empty',tosettlethefactthatthereisSOMETHINGintheSquare.
If,then,wetransferourmarkstothesmallerDiagram,soastogetridofthem-subdivisions,wehavearighttomarkit
-----------||||1|0||||-----------
whichmeans,youknow,"allxarey."
Theresultwouldhavebeenexactlythesame,ifthegivenoblonghadbeenmarkedthus:--
-------------------|1|0||_____|_____|||||||||0|||||||-------------------
Oncemore:howshallweinterpretthis,withregardtoxandy?
-------------------|0|1||_____|_____||||||||||||||||-------------------
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Thistellsus,astothexy-Square,thatONEofitscompartmentsis'empty'.Butthisinformationisquiteuseless,asthereisnomarkintheOTHERcompartment.Iftheothercompartmenthappenedtobe'empty'too,theSquarewouldbe'empty':and,ifithappenedtobe'occupied',theSquarewouldbe'occupied'.So,aswedonotknowWHICHisthecase,wecansaynothingaboutTHISSquare.
TheotherSquare,thexy'-Square,weknow(asinthepreviousexample)tobe'occupied'.
If,then,wetransferourmarkstothesmallerDiagram,wegetmerelythis:--
-----------|||||1||||-----------
whichmeans,youknow,"somexarey'."
Theseprinciplesmaybeappliedtoalltheother
oblongs.Forinstance,torepresent"ally'arem'"weshouldmarkthe-------RIGHT-HANDUPRIGHTOBLONG(theone||thathastheattributey')thus:--|---||0|||---|-1-||0|||---|||-------
and,ifweweretoldtointerpretthelowerhalfofthecupboard,markedasfollows,withregardtoxandy,
-------------------||||||||0||||||||-----|-----||1|0|-------------------
weshouldtransferittothesmallerDiagramthus,
-----------||||1|0||||-----------
andreadit"allx'arey."
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TwomoreremarksaboutPropositionsneedtobemade.
Oneisthat,ineveryPropositionbeginningwith"some"or"all",theACTUALEXISTENCEofthe'Subject'isasserted.If,forinstance,Isay"allmisersareselfish,"ImeanthatmisersACTUALLYEXIST.IfIwishedtoavoidmakingthisassertion,andmerelytostatetheLAWthatmiserlinessnecessarilyinvolvesselfishness,Ishouldsay"nomisersareunselfish"whichdoesnotassertthatanymisersexistatall,butmerelythat,ifanyDIDexist,theyWOULDbeselfish.
Theotheristhat,whenaPropositionbeginswith"some"or"no",andcontainsmorethattwoAttributes,theseAttributesmaybere-arranged,andshiftedfromoneTermtotheother,"adlibitum."Forexample,"someabcaredef"maybere-arrangedas"somebfareacde,"eachbeingequivalentto"someThingsareabcdef".Again"Nowiseoldmenarerashandrecklessgamblers"maybere-arrangedas"Norasholdgamblersarewiseandreckless,"eachbeingequivalentto"Nomenarewiseoldrashrecklessgamblers."
2.Syllogisms
NowsupposewedivideourUniverseofThingsinthreeways,withregardtothreedifferentAttributes.OutofthesethreeAttributes,wemaymakeupthreedifferentcouples(forinstance,iftheywerea,b,c,wemightmakeupthethreecouplesab,ac,bc).AlsosupposewehavetwoPropositionsgivenus,containingtwoofthesethreecouples,andthatfromthemwecanproveathirdPropositioncontainingthethirdcouple.(Forexample,ifwedivideourUniverseform,x,andy;andifwehavethetwoPropositionsgivenus,"nomarex'"and"allm'arey",containingthetwocouplesmxandmy,itmightbepossibletoprovefromthemathirdProposition,containingxandy.)
InsuchacasewecallthegivenPropositions'THEPREMISSES',thethirdone'THECONCLUSION'andthewholeset'ASYLLOGISM'.
Evidently,ONEoftheAttributesmustoccurinbothPremisses;orelseonemustoccurinONEPremiss,anditsCONTRADICTORYintheother.
Inthefirstcase(when,forexample,thePremissesare"somemarex"and"nomarey'")theTerm,whichoccurstwice,iscalled'THEMIDDLETERM',becauseitservesasasortoflinkbetweentheothertwoTerms.
Inthesecondcase(when,forexample,thePremissesare"no
marex'"and"allm'arey")thetwoTerms,whichcontainthesecontradictoryAttributes,maybecalled'THEMIDDLETERMS'.
Thus,inthefirstcase,theclassof"m-Things"istheMiddleTerm;and,inthesecondcase,thetwoclassesof"m-Things"and"m'-Things"aretheMiddleTerms.
TheAttribute,whichoccursintheMiddleTermorTerms,disappearsintheConclusion,andissaidtobe"eliminated",whichliterallymeans"turnedoutofdoors".
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NowletustrytodrawaConclusionfromthetwoPremisses--
"SomenewCakesareunwholesome;NoniceCakesareunwholesome."
Inordertoexpressthemwithcounters,weneedtodivideCakesinTHREEdifferentways,withregardtonewness,toniceness,andtowholesomeness.ForthiswemustusethelargerDiagram,makingxmean"new",y"nice",andm"wholesome".(EverythingINSIDEthecentralSquareissupposedtohavetheattributem,andeverythingOUTSIDEittheattributem',i.e."not-m".)
YouhadbetteradopttheruletomakemmeantheAttributewhichoccursintheMIDDLETermorTerms.(Ihavechosenmasthesymbol,because'middle'beginswith'm'.)
Now,inrepresentingthetwoPremisses,IprefertobeginwiththeNEGATIVEone(theonebeginningwith"no"),becauseGREYcounterscanalwaysbeplacedwithCERTAINTY,andwillthenhelptofixthepositionoftheredcounters,whicharesometimesalittleuncertainwheretheywillbemostwelcome.
Letusexpress,the"noniceCakesareunwholesome(Cakes)",i.e.
"noy-Cakesarem'-(Cakes)".ThistellsusthatnoneoftheCakesbelongingtothey-halfofthecupboardareinitsm'-compartments(i.e.theonesoutsidethecentralSquare).Hencethetwocompartments,No.9andNo.15,areboth'EMPTY';andwemustplaceagreycounterinEACHofthem,thus:--
-----------|0|||--|--|||||||--|-----|--||||||
|--|--||0||-----------
WehavenowtoexpresstheotherPremiss,namely,"somenewCakesareunwholesome(Cakes)",i.e."somex-Cakesarem'-(Cakes)".ThistellsusthatsomeoftheCakesinthex-halfofthecupboardareinitsm'-compartments.HenceONEofthetwocompartments,No.9andNo.10,is'occupied':and,aswearenottoldinWHICHofthesetwocompartmentstoplacetheredcounter,theusualrulewouldbetolayitonthedivision-linebetweenthem:but,inthiscase,theotherPremisshassettledthematterforus,bydeclaring
No.9tobeEMPTY.Hencetheredcounterhasnochoice,andMUSTgointoNo.10,thus:--
-----------|0|1||--|--|||||||--|-----|--||||||
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|--|--||0||-----------
AndnowwhatcounterswillthisinformationenableustoplaceintheSMALLERDiagram,soastogetsomePropositioninvolvingxandyonly,leavingoutm?Letustakeitsfourcompartments,onebyone.
First,No.5.AllweknowaboutTHISisthatitsOUTERportionisempty:butweknownothingaboutitsinnerportion.ThustheSquareMAYbeempty,oritMAYhavesomethinginit.Whocantell?SowedarenotplaceANYcounterinthisSquare.
Secondly,whatofNo.6?Herewearealittlebetteroff.WeknowthatthereisSOMETHINGinit,forthereisaredcounterinitsouterportion.Itistruewedonotknowwhetheritsinnerportionisemptyoroccupied:butwhatdoesTHATmatter?OnesolitaryCake,inonecorneroftheSquare,isquitesufficientexcuseforsaying"THISSQUAREISOCCUPIED",andformarkingitwitharedcounter.
AstoNo.7,weareinthesameconditionaswithNo.5--wefind
itPARTLY'empty',butwedonotknowwhethertheotherpartisemptyoroccupied:sowedarenotmarkthisSquare.
AndastoNo.8,wehavesimplynoinformationatall.
Theresultis
-------||1||---|---||||-------
Our'Conclusion',then,mustbegotoutoftherathermeagerpieceofinformationthatthereisaredcounterinthexy'-Square.HenceourConclusionis"somexarey'",i.e."somenewCakesarenot-nice(Cakes)":or,ifyouprefertotakey'asyourSubject,"somenot-niceCakesarenew(Cakes)";buttheotherlooksneatest.
WewillnowwriteoutthewholeSyllogism,puttingthesymbol&there4[*]for"therefore",andomitting"Cakes",forthesakeofbrevity,attheendofeachProposition.
[*][NOTEfromBrett:Theuseof"&there4"isaratherarbitraryselection.Thereisnofontavailableingeneralpractice
whichrendersthe"therefore"symbolcorrection(threedotsinatriangularformation).Thiscanbedone,however,inHTML,soifthisdocumentisreadinabrowser,thenthesymbolwillbeproperlyrecognized.Thisisapoorman'sexcuse.]
"SomenewCakesareunwholesome;NoniceCakesareunwholesome&there4SomenewCakesarenot-nice."
Andyouhavenowworkedout,successfully,yourfirst'SYLLOGISM'.
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Permitmetocongratulateyou,andtoexpressthehopethatitisbutthebeginningofalongandgloriousseriesofsimilarvictories!
WewillworkoutoneotherSyllogism--aratherharderonethanthelast--andthen,Ithink,youmaybesafelylefttoplaytheGamebyyourself,or(better)withanyfriendwhomyoucanfind,thatisableandwillingtotakeashareinthesport.
LetusseewhatwecanmakeofthetwoPremisses--
"AllDragonsareuncanny;AllScotchmenarecanny."
Remember,Idon'tguaranteethePremissestobeFACTS.Inthefirstplace,IneverevensawaDragon:and,inthesecondplace,itisn'toftheslightestconsequencetous,asLOGICIANS,whetherourPremissesaretrueorfalse:allWEhavetodoistomakeoutwhethertheyLEADLOGICALLYTOTHECONCLUSION,sothat,ifTHEYweretrue,ITwouldbetruealso.
Yousee,wemustgiveupthe"Cakes"now,orourcupboardwillbeofnousetous.Wemusttake,asour'Universe',someclass
ofthingswhichwillincludeDragonsandScotchmen:shallwesay'Animals'?And,as"canny"isevidentlytheAttributebelongingtothe'MiddleTerms',wewillletmstandfor"canny",xfor"Dragons",andyfor"Scotchmen".SothatourtwoPremissesare,infull,
"AllDragon-Animalsareuncanny(Animals);AllScotchman-Animalsarecanny(Animals)."
Andthesemaybeexpressed,usinglettersforwords,thus:--
"Allxarem';Allyarem."
ThefirstPremissconsists,asyoualreadyknow,oftwoparts:--
"Somexarem',"and"Noxarem."
Andthesecondalsoconsistsoftwoparts:--
"Someyarem,"and"Noyarem'."
Letustakethenegativeportionsfirst.
Wehave,then,tomark,onthelargerDiagram,first,"noxarem",andsecondly,"noyarem'".Ithinkyouwillsee,without
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furtherexplanation,thatthetworesults,separately,are
----------------------||||0|||--|--||--|--|||0|0||||||||--|--|--|--||--|--|--|--||||||||||||--|--||--|--|||||0||----------------------
andthatthesetwo,whencombined,giveus
-----------|0|||--|--|||0|0|||--|--|--|--||||||
|--|--||0||-----------
Wehavenowtomarkthetwopositiveportions,"somexarem'"and"someyarem".
Theonlytwocompartments,availableforThingswhicharexm',areNo.9andNo.10.Ofthese,No.9isalreadymarkedas'empty';soourredcountermustgointoNo.10.
Similarly,theonlytwo,availableforym,areNo.11andNo.13.
Ofthese,No.11isalreadymarkedas'empty';soourredcounterMUSTgointoNo.13.
Thefinalresultis
-----------|0|1||--|--|||0|0|||--|--|--|--|||1||||--|--|
|0||-----------
AndnowhowmuchofthisinformationcanusefullybetransferredtothesmallerDiagram?
Letustakeitsfourcompartments,onebyone.
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AstoNo.5?This,wesee,iswholly'empty'.(Somarkitwithagreycounter.)
AstoNo.6?This,wesee,is'occupied'.(Somarkitwitharedcounter.)
AstoNo.7?Ditto,ditto.
AstoNo.8?Noinformation.
ThesmallerDiagramisnowprettyliberallymarked:--
-------|0|1||---|---||1||-------
AndnowwhatConclusioncanwereadofffromthis?Well,itisimpossibletopacksuchabundantinformationintoONEProposition:weshallhavetoindulgeinTWO,thistime.
First,bytakingxasSubject,weget"allxarey'",thatis,
"AllDragonsarenot-Scotchmen":
secondly,bytakingyasSubject,weget"allyarex'",thatis,
"AllScotchmenarenot-Dragons".
Letusnowwriteout,alltogether,ourtwoPremissesandourbrace
ofConclusions.
"AllDragonsareuncanny;AllScotchmenarecanny.&there4AllDragonsarenot-Scotchmen;AllScotchmenarenot-Dragons."
Letmemention,inconclusion,thatyoumayperhapsmeetwithlogicaltreatisesinwhichitisnotassumedthatanyThingEXISTSatall,by"somexarey"isunderstoodtomean"theAttributesx,yareCOMPATIBLE,sothataThingcanhavebothatonce",and"no
xarey"tomean"theAttributesx,yareINCOMPATIBLE,sothatnothingcanhavebothatonce".
Insuchtreatises,Propositionshavequitedifferentmeaningsfromwhattheyhaveinour'GameofLogic',anditwillbewelltounderstandexactlywhatthedifferenceis.
Firsttake"somexarey".HereWEunderstand"are"tomean"are,asanactualFACT"--whichofcourseimpliesthatsomex-ThingsEXIST.ButTHEY(thewritersoftheseothertreatises)onlyunderstand
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"are"tomean"CANbe",whichdoesnotatallimplythatanyEXIST.SotheymeanLESSthanwedo:ourmeaningincludestheirs(forofcourse"somexAREy"includes"somexCANBEy"),buttheirsdoesNOTincludeours.Forexample,"someWelshhippopotamiareheavy"wouldbeTRUE,accordingtothesewriters(sincetheAttributes"Welsh"and"heavy"arequiteCOMPATIBLEinahippopotamus),butitwouldbeFALSEinourGame(sincetherearenoWelshhippopotamitoBEheavy).
Secondly,take"noxarey".HereWEonlyunderstand"are"tomean"are,asanactualFACT"--whichdoesnotatallimplythatnoxCANbey.ButTHEYunderstandthePropositiontomean,notonlythatnoneAREy,butthatnoneCANPOSSIBLYbey.Sotheymeanmorethanwedo:theirmeaningincludesours(forofcourse"noxCANbey"includes"noxAREy"),butoursdoesNOTincludetheirs.Forexample,"noPolicemenareeightfeethigh"wouldbeTRUEinourGame(since,asanactualfact,nosuchsplendidspecimensareeverfound),butitwouldbeFALSE,accordingtothesewriters(sincetheAttributes"belongingtothePoliceForce"and"eightfeethigh"arequiteCOMPATIBLE:thereisnothingtoPREVENTaPolicemanfromgrowingtothatheight,ifsufficientlyrubbedwithRowland'sMacassarOil--whichsaidtomakeHAIRgrow,whenrubbedonhair,andsoofcoursewillmakeaPOLICEMANgrow,whenrubbedonaPoliceman).
Thirdly,take"allxarey",whichconsistsofthetwopartialPropositions"somexarey"and"noxarey'".Here,ofcourse,thetreatisesmeanLESSthanwedointheFIRSTpart,andmorethanwedointheSECOND.Butthetwooperationsdon'tbalanceeachother--anymorethanyoucanconsoleaman,forhavingknockeddownoneofhischimneys,bygivinghimanextradoor-step.
IfyoumeetwithSyllogismsofthiskind,youmayworkthem,quiteeasily,bythesystemIhavegivenyou:youhaveonlytomake'are'mean'areCAPABLEofbeing',andallwillgosmoothly.For"somexarey"willbecome"somexarecapableofbeingy",thatis,"theAttributesx,yareCOMPATIBLE".And"noxarey"will
become"noxarecapableofbeingy",thatis,"theAttributesx,yareINCOMPATIBLE".And,ofcourse,"allxarey"willbecome"somexarecapableofbeingy,andnonearecapableofbeingy'",thatis,"theAttributesx,yareCOMPATIBLE,andtheAttributesx,y'areINCOMPATIBLE."InusingtheDiagramsforthissystem,youmustunderstandaredcountertomean"theremayPOSSIBLYbesomethinginthiscompartment,"andagreyonetomean"therecannotPOSSIBLYbeanythinginthiscompartment."
3.Fallacies.
Andsoyouthink,doyou,thatthechiefuseofLogic,inreallife,istodeduceConclusionsfromworkablePremisses,andtosatisfyyourselfthattheConclusions,deducedbyotherpeople,arecorrect?Ionlywishitwere!Societywouldbemuchlessliabletopanicsandotherdelusions,andPOLITICALlife,especially,wouldbeatotallydifferentthing,ifevenamajorityofthearguments,thatscatteredbroadcastovertheworld,werecorrect!Butitisalltheotherway,Ifear.ForONEworkablePairofPremisses(ImeanaPairthatleadtoalogicalConclusion)thatyoumeetwithin
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readingyournewspaperormagazine,youwillprobablyfindFIVEthatleadtonoConclusionatall:and,evenwhenthePremissesAREworkable,forONEinstance,wherethewriterdrawsacorrectConclusion,thereareprobablyTENwherehedrawsanincorrectone.
Inthefirstcase,youmaysay"thePREMISSESarefallacious":inthesecond,"theCONCLUSIONisfallacious."
Thechiefuseyouwillfind,insuchLogicalskillasthisGamemayteachyou,willbeindetecting'FALLACIES'ofthesetwokinds.
ThefirstkindofFallacy--'FallaciousPremisses'--youwilldetectwhen,aftermarkingthemonthelargerDiagram,youtrytotransferthemarkstothesmaller.Youwilltakeitsfourcompartments,onebyone,andask,foreachinturn,"WhatmarkcanIplaceHERE?";andinEVERYonetheanswerwillbe"Noinformation!",showingthatthereisNOCONCLUSIONATALL.Forinstance,
"Allsoldiersarebrave;SomeEnglishmenarebrave.&there4SomeEnglishmenaresoldiers."
looksuncommonlyLIKEaSyllogism,andmighteasilytakeinalessexperiencedLogician.ButYOUarenottobecaughtbysuchatrick!YouwouldsimplysetoutthePremisses,andwouldthencalmlyremark"FallaciousPREMISSES!":youwouldn'tcondescendtoaskwhatCONCLUSIONthewriterprofessedtodraw--knowingthat,WHATEVERitis,itMUSTbewrong.Youwouldbejustassafeasthatwisemotherwas,whosaid"Mary,justgouptothenursery,andseewhatBaby'sdoing,ANDTELLHIMNOTTODOIT!"
TheotherkindofFallacy--'FallaciousConclusion'--youwillnotdetecttillyouhavemarkedBOTHDiagrams,andhavereadoffthecorrectConclusion,andhavecompareditwiththeConclusionwhichthewriterhasdrawn.
Butmind,youmustn'tsay"FALLACIOUSConclusion,"simplybecauseitisnotIDENTICALwiththecorrectone:itmaybeaPARTofthecorrectConclusion,andsobequitecorrect,ASFARASITGOES.Inthiscaseyouwouldmerelyremark,withapityingsmile,"DEFECTIVEConclusion!"Suppose,ofexample,youweretomeetwiththisSyllogism:--
"Allunselfishpeoplearegenerous;Nomisersaregenerous.&there4Nomisersareunselfish."
thePremissesofwhichmightbethusexpressedinletters:--
"Allx'arem;Noyarem."
HerethecorrectConclusionwouldbe"Allx'arey'"(thatis,"Allunselfishpeoplearenotmisers"),whiletheConclusion,drawn
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bythewriter,is"Noyarex',"(whichisthesameas"Nox'arey,"andsoisPARTof"Allx'arey'.")Hereyouwouldsimplysay"DEFECTIVEConclusion!"Thesamethingwouldhappen,ifyouwereinaconfectioner'sshop,andifalittleboyweretocomein,putdowntwopence,andmarchofftriumphantlywithasinglepenny-bun.Youwouldshakeyourheadmournfully,andwouldremark"DefectiveConclusion!Poorlittlechap!"AndperhapsyouwouldasktheyoungladybehindthecounterwhethershewouldletYOUeatthebun,whichthelittleboyhadpaidforandleftbehindhim:andperhapsSHEwouldreply"Sha'n't!"
Butif,intheaboveexample,thewriterhaddrawntheConclusion"Allmisersareselfish"(thatis,"Allyarex"),thiswouldbegoingBEYONDhislegitimaterights(sinceitwouldasserttheEXISTENCEofy,whichisnotcontainedinthePremisses),andyouwouldveryproperlysay"FallaciousConclusion!"
Now,whenyoureadothertreatisesonLogic,youwillmeetwithvariouskindsof(so-called)'Fallacies'whicharebynomeansALWAYSso.Forexample,ifyouweretoputbeforeoneoftheseLogiciansthePairofPremisses
"Nohonestmencheat;
Nodishonestmenaretrustworthy."
andweretoaskhimwhatConclusionfollowed,hewouldprobablysay"Noneatall!YourPremissesoffendagainstTWOdistinctRules,andareasfallaciousastheycanwellbe!"Thensupposeyouwereboldenoughtosay"TheConclusionis'Nomenwhocheataretrustworthy',"IfearyourLogicalfriendwouldturnawayhastily--perhapsangry,perhapsonlyscornful:inanycase,theresultwouldbeunpleasant.IADVISEYOUNOTTOTRYTHEEXPERIMENT!
"Butwhyisthis?"youwillsay."DoyoumeantotellusthatalltheseLogiciansarewrong?"Farfromit,dearReader!FromTHEIR
pointofview,theyareperfectlyright.Buttheydonotinclude,intheirsystem,anythinglikeALLthepossibleformsofSyllogisms.
TheyhaveasortofnervousdreadofAttributesbeginningwithanegativeparticle.Forexample,suchPropositionsas"Allnot-xarey,""Noxarenot-y,"arequiteoutsidetheirsystem.Andthus,having(fromsheernervousness)excludedaquantityofveryusefulforms,theyhavemaderuleswhich,thoughquiteapplicabletothefewformswhichtheyallowof,arenouseatallwhenyouconsiderallpossibleforms.
Letusnotquarrelwiththem,dearReader!Thereisroomenoughintheworldforbothofus.Letusquietlytakeourbroadersystem:
and,iftheychoosetoshuttheireyestoalltheseusefulforms,andtosay"TheyarenotSyllogismsatall!"wecanbutstandaside,andletthemRushupontheirFate!Thereisscarcelyanythingofyours,uponwhichitissodangeroustoRush,asyourFate.YoumayRushuponyourPotato-beds,oryourStrawberry-beds,withoutdoingmuchharm:youmayevenRushuponyourBalcony(unlessitisanewhouse,builtbycontract,andwithnoclerkoftheworks)andmaysurvivethefoolhardyenterprise:butifyouonceRushuponyourFATE--why,youmusttaketheconsequences!
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CHAPTERII.
CROSSQUESTIONS.
"TheManintheWildernessaskedofme'Howmanystrawberriesgrowinthesea?'"__________
1.Elementary.
1.Whatisan'Attribute'?Giveexamples.
2.Whenisitgoodsensetoput"is"or"are"betweentwonames?Giveexamples.
3.WhenisitNOTgoodsense?Giveexamples.
4.WhenitisNOTgoodsense,whatisthesimplestagreementtomake,inordertomakegoodsense?
5.Explain'Proposition','Term','Subject',and'Predicate'.Giveexamples.
6.Whatare'Particular'and'Universal'Propositions?Giveexamples.
7.Givearuleforknowing,whenwelookatthesmallerDiagram,whatAttributesbelongtothethingsineachcompartment.
8.Whatdoes"some"meaninLogic?[Seepp.55,6]
9.Inwhatsensedoweusetheword'Universe'inthisGame?
10.Whatisa'Double'Proposition?Giveexamples.
11.WhenisaclassofThingssaidtobe'exhaustively'divided?Giveexamples.
12.Explainthephrase"sittingonthefence."
13.WhattwopartialPropositionsmakeup,whentakentogether,"allxarey"?
14.Whatare'Individual'Propositions?Giveexamples.
15.WhatkindsofPropositionsimply,inthisGame,theEXISTENCEoftheirSubjects?
16.WhenaPropositioncontainsmorethantwoAttributes,theseAttributesmayinsomecasesbere-arranged,andshiftedfromoneTermtotheother.Inwhatcasesmaythisbedone?Giveexamples.
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__________
BreakupeachofthefollowingintotwopartialPropositions:
17.Alltigersarefierce.
18.Allhard-boiledeggsareunwholesome.
19.Iamhappy.
20.Johnisnotathome.
__________
[Seepp.56,7]
21.Givearuleforknowing,whenwelookatthelargerDiagram,whatAttributesbelongtotheThingscontainedineachcompartment.
22.Explain'Premisses','Conclusion',and'Syllogism'.Giveexamples.
23.Explainthephrases'MiddleTerm'and'MiddleTerms'.
24.InmarkingapairofPremissesonthelargerDiagram,whyisitbesttomarkNEGATIVEPropositionsbeforeAFFIRMATIVEones?
25.Whyisitofnoconsequencetous,asLogicians,whetherthePremissesaretrueorfalse?
26.HowcanweworkSyllogismsinwhichwearetoldthat"somexarey"istobeunderstoodtomean"theAttributex,yareCOMPATIBLE",and"noxarey"tomean"theAttributesx,yareINCOMPATIBLE"?
27.Whatarethetwokindsof'Fallacies'?
28.Howmaywedetect'FallaciousPremisses'?
29.Howmaywedetecta'FallaciousConclusion'?
30.SometimestheConclusion,offeredtous,isnotidenticalwiththecorrectConclusion,andyetcannotbefairlycalled'Fallacious'.Whendoesthishappen?AndwhatnamemaywegivetosuchaConclusion?
[Seepp.57-59]
2.HalfofSmallerDiagram.
Propositionstoberepresented.
-----------|||
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|x||||--y-----y'-
__________
1.Somexarenot-y.
2.Allxarenot-y.
3.Somexarey,andsomearenot-y.
4.Noxexist.
5.Somexexist.
6.Noxarenot-y.
7.Somexarenot-y,andsomexexist.
__________
Takingx="judges";y="just";
8.Nojudgesarejust.
9.Somejudgesareunjust.
10.Alljudgesarejust.
__________
Takingx="plums";y="wholesome";
11.Someplumsarewholesome.
12.Therearenowholesomeplums.
13.Plumsaresomeofthemwholesome,andsomenot.
14.Allplumsareunwholesome.
[Seepp.59,60]
-----|||x|||--y--|||
|x'||-----
__________
Takingy="diligentstudents";x="successful";
15.Nodiligentstudentsareunsuccessful.
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16.Alldiligentstudentsaresuccessful.
17.Nostudentsarediligent.
18.Therearesomediligent,butunsuccessful,students.
19.Somestudentsarediligent.
[Seepp.60,1]
3.HalfofSmallerDiagram.
Symbolstobeinterpreted.
__________
-----------|||
|x||||--y-----y'-
__________
--------------||||||1.||0|2.|0|0|||||||--------------
--------------||||||3.|-|4.|0|1|||||||--------------
__________
Takingx="goodriddles";y="hard";
--------------
||||||5.|1||6.|1|0|||||||--------------
--------------||||||7.|0|0|8.|0||||||||--------------
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__________
[Seepp.61,2]
Takingx="lobster";y="selfish";
--------------||||||9.||1|10.|0||||||||--------------
--------------||||||11.|0|1|12.|1|1|||||||--------------
__________
-----||
x||||--y'-|||x'|||-----
Takingy="healthypeople";x="happy";
------------|0||||1||0|13.|---|14.|-1-|15.|---|16.|---|
|1||||1|||------------
[Seep.62]
4.SmallerDiagram.
Propositionstoberepresented.
-----------||||x||--y--|--y'-||x'||||-----------
__________
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1.Allyarex.
2.Someyarenot-x.
3.Nonot-xarenot-y.
4.Somexarenot-y.
5.Somenot-yarex.
6.Nonot-xarey.
7.Somenot-xarenot-y.
8.Allnot-xarenot-y.
9.Somenot-yexist.
10.Nonot-xexist.
11.Someyarex,andsomearenot-x.
12.Allxarey,andallnot-yarenot-x.
[Seepp.62,3]
Taking"nations"asUniverse;x="civilised";y="warlike";
13.Nouncivilisednationiswarlike.
14.Allunwarlikenationsareuncivilised.
15.Somenationsareunwarlike.
16.Allwarlikenationsarecivilised,andallcivilisednationsarewarlike.
17.Nonationisuncivilised.
__________
Taking"crocodiles"asUniverse;x="hungry";andy="amiable";
18.Allhungrycrocodilesareunamiable.
19.Nocrocodilesareamiablewhenhungry.
20.Somecrocodiles,whennothungry,areamiable;butsomearenot.
21.Nocrocodilesareamiable,andsomearehungry.
22.Allcrocodiles,whennothungry,areamiable;andallunamiablecrocodilesarehungry.
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23.Somehungrycrocodilesareamiable,andsomethatarenothungryareunamiable.
[Seepp.63,4]
5.SmallerDiagram.
Symbolstobeinterpreted.
__________
-----------||||x||--y--|--y'-||x'||||-----------
__________
--------------||||||1.|---|---|2.|---|---||1||||0|--------------
--------------||1||||3.|---|---|4.|---|---|
||0||0|0|--------------
__________
Taking"houses"asUniverse;x="builtofbrick";andy="two-storied";interpret
--------------|0|||||5.|---|---|6.|---|---||0|||-|----------|---
--------------||0||||7.|---|---|8.|---|---|||||0|1|--------------
[Seep.65]
Taking"boys"asUniverse;x="fat";andy="active";
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interpret
--------------|1|1|||0|9.|---|---|10.|---|---||||||1|--------------
--------------|0|1||1||11.|---|---|12.|---|---|||0||0|1|--------------
__________
Taking"cats"asUniverse;x="green-eyed";andy="good-tempered";interpret
--------------|0|0|||1|13.|---|---|14.|---|---|||0||1||--------------
--------------|1|||0|1|15.|---|---|16.|---|---|||0||1|0|--------------
[Seepp.65,6]
6.LargerDiagram.
Propositionstoberepresented.
__________
-----------||||--x--|||||||--y--m--y'-|
||||||--x'-||||-----------
__________
1.Noxarem.
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2.Someyarem'.
3.Allmarex'.
4.Nom'arey'.
5.Nomarex;Allyarem.
6.Somexarem;Noyarem.
7.Allmarex';Nomarey.
8.Nox'arem;Noy'arem'.
[Seepp.67,8]
Taking"rabbits"asUniverse;m="greedy";x="old";andy="black";represent
9.Nooldrabbitsaregreedy.
10.Somenot-greedyrabbitsareblack.
11.Allwhiterabbitsarefreefromgreediness.
12.Allgreedyrabbitsareyoung.
13.Nooldrabbitsaregreedy;Allblackrabbitsaregreedy.
14.Allrabbits,thatarenotgreedy,areblack;Nooldrabbitsarefreefromgreediness.
__________
Taking"birds"asUniverse;m="thatsingloud";x="well-fed";andy="happy";represent
15.Allwell-fedbirdssingloud;Nobirds,thatsingloud,areunhappy.
16.Allbirds,thatdonotsingloud,areunhappy;Nowell-fedbirdsfailtosingloud.
__________
Taking"persons"asUniverse;m="inthehouse";x="John";andy="havingatooth-ache";represent
17.Johnisinthehouse;Everybodyinthehouseissufferingfromtooth-ache.
18.ThereisnooneinthehousebutJohn;Nobody,outofthehouse,hasatooth-ache.
__________
[Seepp.68-70]
Taking"persons"asUniverse;m="I";x="thathastakenawalk";y="thatfeelsbetter";represent
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19.Ihavebeenoutforawalk;Ifeelmuchbetter.
__________
Choosingyourown'Universe'&c.,represent
20.Isenthimtobringmeakitten;Hebroughtmeakettlebymistake.
[Seepp.70,1]
7.BothDiagramstobeemployed.
__________
-----------|||-----------|--x--||||
||||||x||--y--m--y'-||--y--|--y'-|||||||x'||--x'-|||||||----------------------
__________
N.B.IneachQuestion,asmallDiagramshouldbedrawn,forxandyonly,andmarkedinaccordancewiththegivenlargeDiagram:andthenasmanyPropositionsaspossible,forxandy,shouldberead
offfromthissmallDiagram.
----------------------|0||||||--|--||--|--|||0|0||||0|1||1.|--|--|--|--|2.|--|--|--|--|||1|||||0||||--|--||--|--||0|||||----------------------
[Seep.72]
----------------------|||||0||--|--||--|--|||0|0|||||||3.|--|--|--|--|4.|--|--|--|--|||1|0||||0||||--|--||--|--||||||0|
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----------------------
__________
Mark,inalargeDiagram,thefollowingpairsofPropositionsfromtheprecedingSection:thenmarkasmallDiagraminaccordancewithit,&c.
5.No.13.[seep.49]9.No.17.6.No.14.10.No.18.7.No.15.11.No.19.[seep.50]8.No.16.12.No.20.
__________
Mark,onalargeDiagram,thefollowingPairsofPropositions:thenmarkasmallDiagram,&c.Theseare,infact,PairsofPREMISSESforSyllogisms:andtheresults,readofffromthesmallDiagram,aretheCONCLUSIONS.
13.Noexcitingbookssuitfeverishpatients;Unexcitingbooksmakeonedrowsy.
14.Some,whodeservethefair,gettheirdeserts;Nonebutthebravedeservethefair.
15.Nochildrenarepatient;Noimpatientpersoncansitstill.
[Seepp.72-5]
16.Allpigsarefat;Noskeletonsarefat.
17.Nomonkeysaresoldiers;Allmonkeysaremischievous.
18.Noneofmycousinsarejust;Nojudgesareunjust.
19.Somedaysarerainy;Rainydaysaretiresome.
20.Allmedicineisnasty;Sennaisamedicine.
21.SomeJewsarerich;AllPatagoniansareGentiles.
22.Allteetotalerslikesugar;Nonightingaledrinkswine.
23.Nomuffinsarewholesome;Allbunsareunwholesome.
24.Nofatcreaturesrunwell;Somegreyhoundsrunwell.
25.Allsoldiersmarch;Someyouthsarenotsoldiers.
26.Sugarissweet;Saltisnotsweet.
27.Someeggsarehard-boiled;Noeggsareuncrackable.
28.TherearenoJewsinthehouse;TherearenoGentilesinthegarden.
[Seepp.75-82]
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29.Allbattlesarenoisy;Whatmakesnonoisemayescapenotice.
30.NoJewsaremad;AllRabbisareJews.
31.Therearenofishthatcannotswim;Someskatesarefish.
32.Allpassionatepeopleareunreasonable;Someoratorsarepassionate.
[Seepp.82-84]
CHAPTERIII.
CROOKEDANSWERS.
"Iansweredhim,asIthoughtgood,'Asmanyasred-herringsgrowinthewood'."
__________
1.Elementary.
1.Whatevercanbe"attributedto",thatis"saidtobelongto",aThing,iscalledan'Attribute'.Forexample,"baked",whichcan(frequently)beattributedto"Buns",and"beautiful",whichcan(seldom)beattributedto"Babies".
2.WhentheyaretheNamesoftwoThings(forexample,"thesePigsarefatAnimals"),oroftwoAttributes(forexample,"pinkislightred").
3.WhenoneistheNameofaThing,andtheothertheNameofanAttribute(forexample,"thesePigsarepink"),sinceaThingcannot
actuallyBEanAttribute.
4.ThattheSubstantiveshallbesupposedtoberepeatedattheendofthesentence(forexample,"thesePigsarepink(Pigs)").
5.A'Proposition'isasentencestatingthatsome,ornone,orall,oftheThingsbelongingtoacertainclass,calledthe'Subject',arealsoThingsbelongingtoacertainotherclass,calledthe'Predicate'.Forexample,"somenewCakesarenotnice",thatis(writteninfull)"somenewCakesarenotniceCakes";wherethe
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class"newCakes"istheSubject,andtheclass"not-niceCakes"isthePredicate.
6.AProposition,statingthatSOMEoftheThingsbelongingtoitsSubjectareso-and-so,iscalled'Particular'.Forexample,"somenewCakesarenice","somenewCakesarenotnice."
AProposition,statingthatNONEoftheThingsbelongingtoitsSubject,orthatALLofthem,areso-and-so,iscalled'Universal'.Forexample,"nonewCakesarenice","allnewCakesarenotnice".
7.TheThingsineachcompartmentpossessTWOAttributes,whosesymbolswillbefoundwrittenontwooftheEDGESofthatcompartment.
8."Oneormore."
9.AsanameoftheclassofThingstowhichthewholeDiagramisassigned.
10.APropositioncontainingtwostatements.Forexample,"somenewCakesareniceandsomearenot-nice."
11.Whenthewholeclass,thusdivided,is"exhausted"amongthesetsintowhichitisdivided,therebeingnomemberofitwhich
doesnotbelongtosomeoneofthem.Forexample,theclass"newCakes"is"exhaustively"dividedinto"nice"and"not-nice"sinceEVERYnewCakemustbeoneortheother.
12.Whenamancannotmakeuphismindwhichoftwopartieshewilljoin,heissaidtobe"sittingonthefence"--notbeingabletodecideonwhichsidehewilljumpdown.
13."Somexarey"and"noxarey'".
14.AProposition,whoseSubjectisasingleThing,iscalled'Individual'.Forexample,"Iamhappy","Johnisnotathome".TheseareUniversalPropositions,beingthesameas"alltheI's
thatexistarehappy","ALLtheJohns,thatIamnowconsidering,arenotathome".
15.Propositionsbeginningwith"some"or"all".
16.Whentheybeginwith"some"or"no".Forexample,"someabcaredef"maybere-arrangedas"somebfareacde",eachbeingequivalentto"someabcdefexist".
17.Sometigersarefierce,Notigersarenot-fierce.
18.Somehard-boiledeggsareunwholesome,Nohard-boiledeggsarewholesome.
19.SomeI'sarehappy,NoI'sareunhappy.
20.SomeJohnsarenotathome,NoJohnsareathome.
21.TheThings,ineachcompartmentofthelargerDiagram,possessTHREEAttributes,whosesymbolswillbefoundwrittenatthreeoftheCORNERSofthecompartment(exceptinthecaseofm',whichisnotactuallyinsertedintheDiagram,butisSUPPOSEDtostandateachofitsfouroutercorners).
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22.IftheUniverseofThingsbedividedwithregardtothreedifferentAttributes;andiftwoPropositionsbegiven,containingtwodifferentcouplesoftheseAttributes;andiffromthesewecanproveathirdProposition,containingthetwoAttributesthathavenotyetoccurredtogether;thegivenPropositionsarecalled'thePremisses',thethirdone'theConclusion',andthewholeset'aSyllogism'.Forexample,thePremissesmightbe"nomarex'"and"allm'arey";anditmightbepossibletoprovefromthemaConclusioncontainingxandy.
23.IfanAttributeoccursinbothPremisses,theTermcontainingitiscalled'theMiddleTerm'.Forexample,ifthePremissesare"somemarex"and"nomarey'",theclassof"m-Things"is'theMiddleTerm.'
IfanAttributeoccursinonePremiss,anditscontradictoryintheother,theTermscontainingthemmaybecalled'theMiddleTerms'.Forexample,ifthePremissesare"nomarex'"and"allm'arey",thetwoclassesof"m-Things"and"m'-Things"maybecalled'theMiddleTerms'.
24.BecausetheycanbemarkedwithCERTAINTY:whereasAFFIRMATIVEPropositions(thatis,thosethatbeginwith"some"or"all")
sometimesrequireustoplacearedcounter'sittingonafence'.
25.BecausetheonlyquestionweareconcernedwithiswhethertheConclusionFOLLOWSLOGICALLYfromthePremisses,sothat,ifTHEYweretrue,ITalsowouldbetrue.
26.Byunderstandingaredcountertomean"thiscompartmentCANbeoccupied",andagreyonetomean"thiscompartmentCANNOTbeoccupied"or"thiscompartmentMUSTbeempty".
27.'FallaciousPremisses'and'FallaciousConclusion'.
28.Byfinding,whenwetrytotransfermarksfromthelarger
Diagramtothesmaller,thatthereis'noinformation'foranyofitsfourcompartments.
29.ByfindingthecorrectConclusion,andthenobservingthattheConclusion,offeredtous,isneitheridenticalwithitnorapartofit.
30.WhentheofferedConclusionisPARTofthecorrectConclusion.Inthiscase,wemaycallita'DefectiveConclusion'.
2.HalfofSmallerDiagram.
Propositionsrepresented.
__________
--------------
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||||||1.||1|2.|0|1|||||||--------------
--------------||||||3.|1|1|4.|0|0|||||||--------------
--------------||||||5.|1|6.||0|||||||--------------
-------|||7.|1|1|Itmightbethoughtthattheproper|||--------------|||
Diagramwouldbe|11|,inordertoexpress"some|||-------xexist":butthisisreallycontainedin"somexarey'."Toputaredcounteronthedivision-linewouldonlytellus"ONEOFTHEcompartmentsisoccupied",whichweknowalready,inknowingthatONEisoccupied.-------|||8.Noxarey.i.e.|0|||||-------
-------|||9.Somexarey'.i.e.||1||||-------
-------|||10.Allxarey.i.e.|1|0||||-------
-------
|||11.Somexarey.i.e.|1|||||-------
-------|||12.Noxarey.i.e.|0|||||-------
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-------|||13.Somexarey,andsomearey'.i.e.|1|1||||-------
-------|||14.Allxarey'.i.e.|0|1||||----------||15.Noyarex'.i.e.|---||0|---
---|1|16.Allyarex.i.e.|---||0|---
---|0|17.Noyexist.i.e.|---||0|---
---||18.Someyarex'.i.e.|---||1|---
---
||15.Someyexist.i.e.|-1-|||---
3.HalfofSmallerDiagram.
Symbolsinterpreted.
__________
1.Noxarey'.
2.Noxexist.
3.Somexexist.
4.Allxarey'.
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5.Somexarey.i.e.Somegoodriddlesarehard.
6.Allxarey.i.e.Allgoodriddlesarehard.
7.Noxexist.i.e.Noriddlesaregood.
8.Noxarey.i.e.Nogoodriddlesarehard.
9.Somexarey'.i.e.Somelobstersareunselfish.
10.Noxarey.i.e.Nolobstersareselfish.
11.Allxarey'.i.e.Alllobstersareunselfish.
12.Somexarey,andsomearey'.i.e.Somelobstersareselfish,andsomeareunselfish.
13.Ally'arex'.i.e.Allinvalidsareunhappy.
14.Somey'exist.i.e.Somepeopleareunhealthy.
15.Somey'arex,andsomearex'.i.e.Someinvalidsarehappy,andsomeareunhappy.
16.Noy'exist.i.e.Nobodyisunhealthy.
4.SmallerDiagram.
Propositionsrepresented.
__________
--------------|1|||||1.|---|---|2.|---|---||0|||1||--------------
--------------|||||1|3.|---|---|4.|---|---|||0||||--------------
--------------||1||||5.|---|---|6.|---|---|||||0||--------------
--------------||||||7.|---|---|8.|---|---|
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||1||0|1|--------------
--------------||||||9.|---|-1-|10.|---|---|||||0|0|--------------
--------------|1|||1|0|11.|---|---|12.|---|---||1||||1|--------------
-------|||13.Nox'arey.i.e.|---|---||0||-------
-------||0|
14.Ally'arex'.i.e.|---|---|||1|-------
-------|||15.Somey'exist.i.e.|---|-1-||||-------
-------|1|0|
16.Allyarex,andallxarey.i.e.|---|---||0||--------------|||17.Nox'exist.i.e.|---|---||0|0|-------
-------|0|1|18.Allxarey'.i.e.|---|---||||
-------
-------|0||19.Noxarey.i.e.|---|---||||-------
-------
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|||20.Somex'arey,andsomearey'.i.e.|---|---||1|1|-------
-------|0|1|21.Noyexist,andsomexexist.i.e.|---|---||0||-------
-------||1|22.Allx'arey,andally'arex.i.e.|---|---||1|0|-------
-------|1||17.Somexarey,andsomex'arey'.i.e.|---|---|||1|-------
5.SmallerDiagram.
Symbolsinterpreted.
__________
1.Someyarenot-x,or,Somenot-xarey.
2.Nonot-xarenot-y,or,Nonot-yarenot-x.
3.Nonot-yarex.
4.Nonot-xexist.i.e.NoThingsarenot-x.
5.Noyexist.i.e.Nohousesaretwo-storied.
6.Somex'exist.i.e.Somehousesarenotbuiltofbrick.
7.Noxarey'.Or,noy'arex.i.e.Nohouses,builtofbrick,areotherthantwo-storied.Or,nohouses,that
arenottwo-storied,arebuiltofbrick.
8.Allx'arey'.i.e.Allhouses,thatarenotbuiltofbrick,arenottwo-storied.
9.Somexarey,andsomearey'.i.e.Somefatboysareactive,andsomearenot.
10.Ally'arex'.i.e.Alllazyboysarethin.
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11.Allxarey',andally'arex.i.e.Allfatboysarelazy,andalllazyonesarefat.
12.Allyarex,andallx'arey.i.e.Allactiveboysarefat,andallthinonesarelazy.
13.Noxexist,andnoy'exist.i.e.Nocatshavegreeneyes,andnonehavebadtempers.
14.Somexarey',andsomex'arey.Orsomeyarex',andsomey'arex.i.e.Somegreen-eyedcatsarebad-tempered,andsome,thathavenotgreeneyes,aregood-tempered.Or,somegood-temperedcatshavenotgreeneyes,andsomebad-temperedoneshavegreeneyes.
15.Somexarey,andnox'arey'.Or,someyarex,andnoy'arex'.i.e.Somegreen-eyedcatsaregood-tempered,andnone,thatarenotgreen-eyed,arebad-tempered.Or,somegood-temperedcatshavegreeneyes,andnone,thatarebad-tempered,havenotgreeneyes.
16.Allxarey',andallx'arey.Or,allyarex',andally'arex.i.e.Allgreen-eyedcatsarebad-temperedandall,thathavenotgreeneyes,aregood-tempered.Or,all
good-temperedoneshaveeyesthatarenotgreen,andallbad-temperedoneshavegreeneyes.
6.LargerDiagram.
Propositionsrepresented.
__________
------------------------------|||||||---|---||---|---|||0|0|||||||1.|---|---|---|---|2.|-1-|---|---|---||||||||||||---|---||---|---|||||||------------------------------
------------------------------
|||||0||---|---||---|---|||0|0|||||||3.|---|---|---|---|4.|---|---|---|---|||-||||||||---|---||---|---||||||0|------------------------------
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------------------------------|0||||||---|---||---|---|||0|0||||0|1||5.|---|---|---|---|6.|---|---|---|---|||1|||||0||||---|---||---|---||0|||||------------------------------
------------------------------|||||0||---|---||---|---|||0|0|||||||7.|---|---|---|---|8.|---|---|---|---|||0|1||||0|0|||---|---||---|---||||||0|------------------------------
---------------|||
|---|---|||0|0||9.Noxarem.i.e.|---|---|---|---|||0||||---|---||||---------------
---------------||||---|---||||||
10.Somem'arey.i.e.|-1-|---|---|---|||||||---|---||||---------------
---------------||||---|---||||0||11.Ally'arem'.i.e.|---|---|---|-1-||||0||
|---|---||||---------------
---------------||||---|---|||0|0||12.Allmarex'.i.e.|---|---|---|---|||1||
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|---|---||||---------------
---------------|0|||---|---|||0|0||13.Noxarem;i.e.|---|---|---|---|Allyarem.||1||||---|---||0||---------------
---------------|0|0||---|---||||||14.Allm'arey;i.e.|---|---|---|---|Noxarem'.||||||---|---||1|0|---------------
---------------|0|0||---|---|||1|0||15.Allxarem;i.e.|---|---|---|---|Nomarey'.|||0|||---|---||||---------------
---------------|0|0|
|---|---||||||16.Allm'arey';i.e.|---|---|---|---|Noxarem'.||||||---|---||0|1|---------------
---------------|0|0||---|---|||1|0||17.Allxarem;i.e.|---|---|---|---|
Allmarey.|||0|||---|---|[SeeremarksonNo.7,p.60.]|||---------------
---------------|0|||---|---||||||18.Nox'arem;i.e.|---|---|---|---|
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Nom'arey.||0|0|||---|---||0||---------------
---------------||||---|---|||1|0||19.Allmarex;i.e.|---|---|---|---|Allmarey.||0|0|||---|---||||---------------
20.Wehadbettertake"persons"asUniverse.Wemaychoose"myself"as'MiddleTerm',inwhichcasethePremisseswilltaketheform
Iama-person-who-sent-him-to-bring-a-kitten;Iama-person-to-whom-he-brought-a-kettle-by-mistake.
Orwemaychoose"he"as'MiddleTerm',inwhichcasethePremisseswilltaketheform
Heisa-person-whom-I-sent-to-bring-me-a-kitten;Heisa-person-who-brought-me-a-kettle-by-mistake.
Thelatterformseemsbest,astheinterestoftheanecdoteclearlydependsonHISstupidity--notonwhathappenedtoME.Letusthenmakem="he";x="personswhomIsent,&c.";andy="personswhobrought,&c."
Hence,Allmarex;Allmarey.andtherequiredDiagramis
---------------||||---|---|||1|0|||---|---|---|---|||0|0|||---|---||||---------------
7.BothDiagramsemployed.
-------|0||1.|---|---|i.e.Allyarex'.|1||-------
-------||1|
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2.|---|---|i.e.Somexarey';or,Somey'arex.|||-------
-------|||3.|---|---|i.e.Someyarex';or,Somex'arey.|1||-------
-------|||4.|---|---|i.e.Nox'arey';or,Noy'arex'.||0|-------
-------|0||5.|---|---|i.e.Allyarex'.i.e.Allblackrabbits|1||areyoung.-------
-------|||
6.|---|---|i.e.Someyarex'.i.e.Someblack|1||rabbitsareyoung.-------
-------|1|0|7.|---|---|i.e.Allxarey.i.e.Allwell-fedbirds|||arehappy.-------
-------|||i.e.Somex'arey'.i.e.Somebirds,
8.|---|---|thatarenotwell-fed,areunhappy;||1|or,Someunhappybirdsarenot-------well-fed.
-------|1|0|9.|---|---|i.e.Allxarey.i.e.Johnhasgota|||tooth-ache.-------
-------
|||10.|---|---|i.e.Nox'arey.i.e.Noone,butJohn,|0||hasgotatooth-ache.-------
-------|1||11.|---|---|i.e.Somexarey.i.e.Someone,who|||hastakenawalk,feelsbetter.
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-------
-------|1||i.e.Somexarey.i.e.Someone,12.|---|---|whomIsenttobringmeakitten,|||broughtmeakettlebymistake.-------
---------------||0||---|---|||0|0||13.|-1-|---|---|---|-------|||||||0||---|---||---|---|||0||||----------------------
Let"books"beUniverse;m="exciting",x="thatsuitfeverishpatients";y="thatmakeonedrowsy".
Nomarex;&there4Noy'arex.Allm'arey.
i.e.Nobookssuitfeverishpatients,exceptsuchasmakeonedrowsy.
---------------||||---|---|||1|0||14.|---|---|---|---|-------|||0|||1|||---|---||---|---|
||||||----------------------
Let"persons"beUniverse;m="thatdeservethefair";x="thatgettheirdeserts";y="brave".
Somemarex;&there4Someyarex.Noy'arem.
i.e.Somebravepersonsgettheirdeserts.
---------------
|0|||---|---|||0|0||15.|---|---|---|---|-------||||||0|||---|---||---|---||0|||||----------------------
Let"persons"beUniverse;m="patient";
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x="children";y="thatcansitstill".
Noxarem;&there4Noxarey.Nom'arey.
i.e.Nochildrencansitstill.
---------------|0|0||---|---|||0|1||16.|---|---|---|---|-------||0||||0|1||---|---||---|---|||||||----------------------
Let"things"beUniverse;m="fat";x="pigs";y="skeletons".
Allxarem;&there4Allxarey'.Noyarem.
i.e.Allpigsarenot-skeletons.
---------------||||---|---|||0|0||17.|---|---|---|---|-------||1|0||||||---|---||---|---|||||1||----------------------
Let"creatures"beUniverse;m="monkeys";x="soldiers";y="mischievous".
Nomarex;&there4Someyarex'.Allmarey.
i.e.Somemischievouscreaturesarenotsoldiers.
---------------|0|||---|---|||0|0||
18.|---|---|---|---|-------||||||0|||---|---||---|---||0|||||----------------------
Let"persons"beUniverse;m="just";x="mycousins";y="judges".
Noxarem;&there4Noxarey.
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Noyarem'.
i.e.Noneofmycousinsarejudges.
---------------||||---|---|||1|0||19.|---|---|---|---|-------||||||1|||---|---||---|---|||||||----------------------
Let"periods"beUniverse;m="days";x="rainy";y="tiresome".
Somemarex;&there4Somexarey.Allxmarey.
i.e.Somerainyperiodsaretiresome.
N.B.ThesearenotlegitimatePremisses,sincetheConclusionisreallypartofthesecondPremiss,sothatthe
firstPremississuperfluous.Thismaybeshown,inletters,thus:--
"Allxmarey"contains"Somexmarey",whichcontains"Somexarey".Or,inwords,"Allrainydaysaretiresome"contains"Somerainydaysaretiresome",whichcontains"Somerainyperiodsaretiresome".
Moreover,thefirstPremiss,besidesbeingsuperfluous,isactuallycontainedinthesecond;sinceitisequivalentto"Somerainydaysexist",which,asweknow,isimpliedintheProposition"Allrainydaysaretiresome".
Altogether,amostunsatisfactoryPairofPremisses!
---------------|0|||---|---|||1|||20.|---|---|---|---|-------||0|0|||1|||---|---||---|---||0|||0||----------------------
Let"things"beUniverse;m="medicine";
x="nasty";y="senna".
Allmarex;&there4Allyarex.Allyarem.
i.e.Sennaisnasty.
[SeeremarksonNo.7,p60.]
---------------
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||||---|---|||0|1||21.|-1-|---|---|---|-------||0|||||1||---|---||---|---|||||||----------------------
Let"persons"beUniverse;m="Jews";x="rich";y="Patagonians".
Somemarex;&there4Somexarey'.Allyarem'.
i.e.SomerichpersonsarenotPatagonians.
---------------|0|||---|---|||-||22.|---|---|---|---|-------||0|0||||||---|---||---|---|
|0|||0||----------------------
Let"creatures"beUniverse;m="teetotalers";x="thatlikesugar";y="nightingales".
Allmarex;&there4Noyarex'.Noyarem'.
i.e.Nonightingalesdislikesugar.
---------------|||
|---|---|||0|0||23.|-1-|---|---|---|-------||0|||||||---|---||---|---|||||||----------------------
Let"food"beUniverse;m="wholesome";x="muffins";y="buns".
Noxarem;Allyarem.
Thereis'noinformation'forthesmallerDiagram;sonoConclusioncanbedrawn.
---------------||||---|---|||0|0||24.|---|---|---|---|-------||1||||||
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|---|---||---|---|||||1||----------------------
Let"creatures"beUniverse;m="thatrunwell";x="fat";y="greyhounds".
Noxarem;&there4Someyarex'.Someyarem.
i.e.Somegreyhoundsarenotfat.
---------------||||---|---|||-||25.|-1-|---|---|---|-------||0|0||||||---|---||---|---|||||||----------------------
Let"persons"beUniverse;m="soldiers";
x="thatmarch";y="youths".
Allmarex;Someyarem'.
Thereis'noinformation'forthesmallerDiagram;sonoConclusioncanbedrawn.
---------------|0|0||---|---|||0|1||26.|---|---|---|---|-------
||0||||0|1||---|---||---|---||1|||1||----------------------
Let"food"beUniverse;m="sweet";x="sugar";y="salt".
Allxarem;&there4Allxarey'.Allyarem'.Allyarex'.
i.e.Sugarisnotsalt.Saltisnotsugar.
---------------||||---|---|||1|0||27.|---|---|---|---|-------|||0|||1|||---|---||---|---|
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||||||----------------------
Let"Things"beUniverse;m="eggs";x="hard-boiled";y="crackable".
Somemarex;&there4Somexarey.Nomarey'.
i.e.Somehard-boiledthingscanbecracked.
---------------|0|||---|---|||0|0||28.|---|---|---|---|-------||||||0|||---|---||---|---||0|||||----------------------
Let"persons"beUniverse;m="Jews";x="thatareinthehouse";y="thatareinthegarden".
Nomarex;&there4Noxarey.Nom'arey.
i.e.Nopersons,thatareinthehouse,arealsointhegarden.
---------------|0|0||---|---|||-||29.|---|---|---|---|-------||||||||
|---|---||---|---||1|0||1||----------------------
Let"Things"beUniverse;m="noisy";x="battles";y="thatmayescapenotice".
Allxarem;&there4Somex'arey.Allm'arey.
i.e.Somethings,thatarenotbattles,mayescapenotice.
---------------|0|||---|---|||0|0||30.|---|---|---|---|-------||1||||0|||---|---||---|---||0|||1||----------------------
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Let"persons"beUniverse;m="Jews";x="mad";y="Rabbis".
Nomarex;&there4Allyarex'.Allyarem.
i.e.AllRabbisaresane.
---------------||||---|---|||1|||31.|---|---|---|---|-------||0|0|||1|||---|---||---|---|||||||----------------------
Let"Things"beUniverse;m="fish";x="thatcanswim";y="skates".
Nomarex';&there4Someyarex.Someyarem.
i.e.Someskatescanswim.
---------------||||---|---|||0|0||32.|---|---|---|---|-------||1|||||||---|---||---|---|||||1||----------------------
Let"people"beUniverse;m="passionate";
x="reasonable";y="orators".
Allmarex';&there4Someyarex'.Someyarem.
i.e.Someoratorsareunreasonable.
[SeeremarksonNo.7,p.60.]
CHAPTERIV.
HITORMISS.
"Thoucanstnothitit,hitit,hitit,Thoucanstnothitit,mygoodman."
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__________
1.Painiswearisome;Nopainiseagerlywishedfor.
2.Nobaldpersonneedsahair-brush;Nolizardshavehair.
3.Allthoughtlesspeopledomischief;Nothoughtfulpersonforgetsapromise.
4.IdonotlikeJohn;SomeofmyfriendslikeJohn.
5.Nopotatoesarepine-apples;Allpine-applesarenice.
6.Nopinsareambitious;Noneedlesarepins.
7.Allmyfriendshavecolds;Noonecansingwhohasacold.
8.Allthesedishesarewell-cooked;Somedishesareunwholesomeifnotwell-cooked.
9.Nomedicineisnice;Sennaisamedicine.
10.Someoystersaresilent;Nosilentcreaturesareamusing.
11.Allwisemenwalkontheirfeet;Allunwisemenwalkontheirhands.
12."Mindyourownbusiness;Thisquarrelisnobusinessofyours."
13.Nobridgesaremadeofsugar;Somebridgesarepicturesque.
14.Noriddlesinterestmethatcanbesolved;Alltheseriddlesareinsoluble.
15.Johnisindustrious;Allindustriouspeoplearehappy.
16.Nofrogswritebooks;Somepeopleuseinkinwritingbooks.
17.Nopokersaresoft;Allpillowsaresoft.
18.Noantelopeisungraceful;Gracefulanimalsdelighttheeye.
19.Someunclesareungenerous;Allmerchantsaregenerous.
20.Nounhappypeoplechuckle;Nohappypeoplegroan.
21.Audiblemusiccausesvibrationintheair;Inaudiblemusicisnotworthpayingfor.
22.Hegavemefivepounds;Iwasdelighted.
23.NooldJewsarefatmillers;Allmyfriendsareoldmillers.
24.Flourisgoodforfood;Oatmealisakindofflour.
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25.Somedreamsareterrible;Nolambsareterrible.
26.Norichmanbegsinthestreet;Allwhoarenotrichshouldkeepaccounts.
27.Nothievesarehonest;Somedishonestpeoplearefoundout.
28.Allwaspsareunfriendly;Allpuppiesarefriendly.
29.Allimprobablestoriesaredoubted;Noneofthesestoriesareprobable.
30."Hetoldmeyouhadgoneaway.""Heneversaysonewordoftruth."
31.Hissongsneverlastanhour;Asong,thatlastsanhour,istedious.
32.Nobride-cakesarewholesome;Unwholesomefoodshouldbeavoided.
33.Nooldmisersarecheerful;Someoldmisersarethin.
34.Allduckswaddle;Nothingthatwaddlesisgraceful.
35.NoProfessorsareignorant;Someignorantpeopleareconceited.
36.Toothacheisneverpleasant;Warmthisneverunpleasant.
37.Boresareterrible;Youareabore.
38.Somemountainsareinsurmountable;Allstilescanbesurmounted.
39.NoFrenchmenlikeplumpudding;AllEnglishmenlike
plumpudding.
40.Noidlerswinfame;Somepaintersarenotidle.
41.Nolobstersareunreasonable;Noreasonablecreaturesexpectimpossibilities.
42.Nokinddeedisunlawful;Whatislawfulmaybedonewithoutfear.
43.Nofossilscanbecrossedinlove;Anyoystermaybecrossedinlove.
44."Thisisbeyondendurance!""Well,nothingbeyondendurancehaseverhappenedtome."
45.Alluneducatedmenareshallow;Allthesestudentsareeducated.
46.Allmycousinsareunjust;Nojudgesareunjust.
47.Nocountry,thathasbeenexplored,isinfestedbydragons;Unexploredcountriesarefascinating.
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48.Nomisersaregenerous;Someoldmenarenotgenerous.
49.Aprudentmanshunshyaenas;Nobankerisimprudent.
50.Somepoetryisoriginal;Nooriginalworkisproducibleatwill.
51.Nomisersareunselfish;Nonebutmiserssaveegg-shells.
52.Allpalepeoplearephlegmatic;Noone,whoisnotpale,lookspoetical.
53.Allspidersspinwebs;Somecreatures,thatdonotspinwebs,aresavage.
54.Noneofmycousinsarejust;Alljudgesarejust.
55.Johnisindustrious;Noindustriouspeopleareunhappy.
56.Umbrellasareusefulonajourney;Whatisuselessonajourneyshouldbeleftbehind.
57.Somepillowsaresoft;Nopokersaresoft.
58.Iamoldandlame;Nooldmerchantisalamegambler.
59.Noeventfuljourneyiseverforgotten;Uneventfuljourneysarenotworthwritingabookabout.
60.Sugarissweet;Somesweetthingsarelikedbychildren.
61.Richardisoutoftemper;NoonebutRichardcanridethathorse.
62.Alljokesaremeanttoamuse;NoActofParliamentisajoke.
63."Isawitinanewspaper.""Allnewspaperstelllies."
64.Nonightmareispleasant;Unpleasantexperiencesarenotanxiouslydesired.
65.Prudenttravellerscarryplentyofsmallchange;Imprudenttravellerslosetheirluggage.
66.Allwaspsareunfriendly;Nopuppiesareunfriendly.
67.Hecalledhereyesterday;Heisnofriendofmine.
68.Noquadrupedscanwhistle;Somecatsarequadrupeds.
69.Nocookedmeatissoldbybutchers;Nouncookedmeatisservedatdinner.
70.Goldisheavy;Nothingbutgoldwillsilencehim.
71.Somepigsarewild;Therearenopigsthatarenotfat.
72.Noemperorsaredentists;Alldentistsaredreadedby
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children.
73.All,whoarenotold,likewalking;NeitheryounorIareold.
74.Allbladesaresharp;Somegrassesareblades.
75.Nodictatorialpersonispopular;Sheisdictatorial.
76.Somesweetthingsareunwholesome;Nomuffinsaresweet.
77.Nomilitarymenwritepoetry;Nogeneralsarecivilians.
78.Boresaredreaded;Aboreisneverbeggedtoprolonghisvisit.
79.Allowlsaresatisfactory;Someexcusesareunsatisfactory.
80.Allmycousinsareunjust;Alljudgesarejust.
81.Somebunsarerich;Allbunsarenice.
82.Nomedicineisnice;Nopillsareunmedicinal.
83.Somelessonsaredifficult;Whatisdifficultneedsattention.
84.Nounexpectedpleasureannoysme;Yourvisitisanunexpectedpleasure.
85.Caterpillarsarenoteloquent;Jonesiseloquent.
86.Somebaldpeoplewearwigs;Allyourchildrenhavehair.
87.Allwaspsareunfriendly;Unfriendlycreaturesarealwaysunwelcome.
88.Nobankruptsarerich;Somemerchantsarenotbankrupts.
89.Weaselssometimessleep;Allanimalssometimessleep.
90.Ill-managedconcernsareunprofitable;Railwaysareneverill-managed.
91.Everybodyhasseenapig;Nobodyadmiresapig.
______________
ExtractaPairofPremissesoutofeachofthefollowing:anddeducetheConclusion,ifthereisone:--
92."TheLion,asanyonecantellyouwhohasbeenchasedbythemasoftenasIhave,isaverysavageanimal:andtherearecertainindividualsamongthem,thoughIwillnotguaranteeitasagenerallaw,whodonotdrinkcoffee."
93."Itwasmostabsurdofyoutoofferit!Youmighthaveknown,ifyouhadhadanysense,thatnooldsailorseverlikegruel!"
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"ButIthought,ashewasanuncleofyours--"
"Anuncleofmine,indeed!Stuff!"
"Youmaycallitstuff,ifyoulike.AllIknowis,MYunclesarealloldmen:andtheylikegruellikeanything!"
"Well,thenYOURunclesare--"
94."Docomeaway!Ican'tstandthissqueezinganymore.Nocrowdedshopsarecomfortable,youknowverywell."
"Well,whoexpectstobecomfortable,outshopping?"
"Why,Ido,ofcourse!AndI'msuretherearesomeshops,furtherdownthestreet,thatarenotcrowded.So--"
95."Theysaynodoctorsaremetaphysicalorganists:andthatletsmeintoalittlefactaboutYOU,youknow."
"Why,howdoyoumakeTHATout?Youneverheardmeplaytheorgan."
"No,doctor,butI'veheardyoutalkaboutBrowning'spoetry:and
thatshowedmethatyou'reMETAPHYSICAL,atanyrate.So--"
___________________
ExtractaSyllogismoutofeachofthefollowing:andtestitscorrectness:--
96."Don'ttalktome!I'veknownmorerichmerchantsthanyouhave:andIcantellyounotONEofthemwaseveranoldmisersincetheworldbegan!"
"AndwhathasthatgottodowitholdMr.Brown?"
"Why,isn'theveryrich?"
"Yes,ofcourseheis.Andwhatthen?"
"Why,don'tyouseethatit'sabsurdtocallhimamiserlymerchant?Eitherhe'snotamerchant,orhe'snotamiser!"
97."ItISsokindofyoutoenquire!I'mreallyfeelingagreatdealbetterto-day."
"AndisitNature,orArt,thatistohavethecreditofthishappychange?"
"Art,Ithink.TheDoctorhasgivenmesomeofthatpatentmedicineofhis."
"Well,I'llnevercallhimahumbugagain.There'sSOMEBODY,atanyrate,thatfeelsbetteraftertakinghismedicine!"
98."No,Idon'tlikeyouonebit.AndI'llgoandplaywithmydoll.DOLLSareneverunkind."
"Soyoulikeadollbetterthanacousin?Ohyoulittlesilly!"
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"OfcourseIdo!COUSINSareneverkind--atleastnocousinsI'veeverseen."
"Well,andwhatdoesTHATprove,I'dliketoknow!Ifyoumeanthatcousinsaren'tdolls,whoeversaidtheywere?"
99."Whatareyoutalkingaboutgeraniumsfor?Youcan'ttelloneflowerfromanother,atthisdistance!Igrantyouthey'reallREDflowers:itdoesn'tneedatelescopetoknowTHAT."
"Well,somegeraniumsarered,aren'tthey?"
"Idon'tdenyit.Andwhatthen?Isupposeyou'llbetellingmesomeofthoseflowersaregeraniums!"
"Ofcoursethat'swhatIshouldtellyou,ifyou'dthesensetofollowanargument!Butwhat'sthegoodofprovinganythingtoYOU,Ishouldliketoknow?"
100."Boys,you'vepassedafairlygoodexamination,allthingsconsidered.NowletmegiveyouawordofadvicebeforeIgo.Rememberthatall,whoarereallyanxioustolearn,workHARD."
"Ithankyou,Sir,inthenameofmyscholars!AndproudamItothinkthereareSOMEofthem,atleast,thatarereallyANXIOUStolearn."
"Verygladtohearit:andhowdoyoumakeitouttobeso?"
"Why,Sir,Iknowhowhardtheywork--someofthem,thatis.Whoshouldknowbetter?"
___________________
ExtractfromthefollowingspeechaseriesofSyllogisms,orargumentshavingtheformofSyllogisms:andtesttheircorrectness.
Itissupposedtobespokenbyafondmother,inanswertoafriend'scautioussuggestionthatsheisperhapsaLITTLEoverdoingit,inthewayoflessons,withherchildren.
101."Well,they'vegottheirownwaytomakeintheworld.WEcan'tleavethemafortuneapiece.Andmoney'snottobehad,asYOUknow,withoutmoney'sworth:theymustWORKiftheywanttolive.Andhowaretheytowork,iftheydon'tknowanything?Takemywordforit,there'snoplaceforignoranceinTHESEtimes!Andallauthoritiesagreethatthetimetolearniswhenyou'reyoung.One'sgotnomemoryafterwards,worthspeakingof.Achildwilllearnmoreinanhourthanagrownmaninfive.Sothose,that
havetolearn,mustlearnwhenthey'reyoung,ifeverthey'retolearnatall.Ofcoursethatdoesn'tdounlesschildrenareHEALTHY:IquiteallowTHAT.Well,thedoctortellsmenochildrenarehealthyunlessthey'vegotagoodcolourintheircheeks.Andonlyjustlookatmydarlings!Why,theircheeksbloomlikepeonies!Well,now,theytellmethat,tokeepchildreninhealth,youshouldnevergivethemmorethansixhoursaltogetheratlessonsintheday,andatleasttwohalf-holidaysintheweek.Andthat'sEXACTLYourplanIcanassureyou!Wenevergobeyondsixhours,andeveryWednesdayandSaturday,aseveris,notonesyllableof
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lessonsdotheydoaftertheironeo'clockdinner!SohowyoucanimagineI'mrunninganyriskintheeducationofmypreciouspetsismorethanIcanunderstand,Ipromiseyou!"
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