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1/14HOME TABLEOFCONTENTS2. 2 Squares of Opposi ti
onAsquareofoppositionshowsthelogicalrelationsamongcategoricalstatements.Therearetwosquaresofopposition:1.
theModernSquareofOpposition2.
theTraditionalSquareofOpposition2.2.1TheModernSquareofOppositionTheonlylogicalrelationintheModernSquareofOppositionisthecontradictoryrelation.I
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2/14TheContradictoryRelationTosaythattwostatementsarecontradictorytoeachothermeansthattheynecessarilyhaveoppositetruthvalue.Thatis,ifoneofthemistrue,theothermustbefalse,andifoneofthemisfalse,theothermustbetrue.Forexample,theAstatementAllbatsaremammalsandtheOstatementSomebatsarenotmammalscontradicteachother.ThecontradictoryrelationalsoexistsbetweentheEstatementNoswansareblackbirdsandtheIstatementsSomeswansareblackbirds.WecanuseVennDiagramstoillustratewhythecontradictoryrelationholdsbetweentheAandOstatements.WhentheAstatementistrue,theareaisempty.Butiftheareaisempty,thennomemberofScanbeinthearea.ThiscontradictstheOstatement,whichsaysthatthereisatleastonememberofSinthearea,i.e.,theareaisnotempty.7/1/2015
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3/14ThecontradictoryrelationexistsbetweentheAandOstatements,andbetweentheEandIstatements.Therelationcanalsobeexplicatedintermsoftheeliminationofcertaincasesinthetruthtable.Atruthtablelistsallpossibledistributionsoftruthvalues.Asinglestatementphastwopossibletruthvalues:truth(T)andfalsehood(F).Giventwostatementspandq,therearefourpossiblecombinationsoftruthvalues,rangingfrombothpandqbeingtrue(TT)tobothofthembeingfalse(FF).Accordingly,therearefourrows(cases)inthetruthtable.Ingeneral,givennstatements,thereare2nrowsinthetruthtable.GivenAandOstatements,therearefourpossibletruthvaluecombinations.WecanviewthecontradictoryrelationasrulingoutthelogicalpossibilitythatAandOarebothtrueandthelogicalpossibilitythattheyarebothfalse.ThesameholdsforEandI.Thereisoneadvantageofusingthetruthtabletounderstandthelogicalrelationsamongthecategoricalstatements.Thetablescanhelpusfigureoutwhetherthetruthvalueofastatementcanbedeterminedwhenthetruthvalueofanotherstatementisknown.7/1/2015
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4/142.2.2TheTraditionalSquareofOppositionIfweassumethatthesetdenotedbythesubjecttermcannotbeanemptyset,thentherearefourlogicalrelationsamongtheA,E,I,Ostatements.TheyareshownintheTraditionalSquareofOpposition.Thefourrelationsare:ContradictoryContrarySubcontraryImplicationInthediagrambelow,wecanactuallyseethecompletesquarethatshowsthelogicalrelationsamongtheA,E,I,Ostatements.7/1/2015
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5/14TheContraryRelationThecontradictoryrelationhasbeenexplicatedabove.Wenowlookatthecontraryrelation.Twostatementsarecontrarytoeachotheriftheycannotbothbetrue.ThecontraryrelationexistsbetweentheAandEstatements.WecanuseVennDiagramstoillustratewhytheAandEstatementscannotbothbetrue.SupposethatboththeAandEstatementsweretrue.IntermsoftheirVennDiagrams,thiswouldmeanthattheareaandtheareawereempty.Butifboththeareaandtheareawereempty,thenthesetSwouldbeempty.ThiswouldcontradicttheassumptionthatthesetScannotbeanemptyset.Thecontraryrelationcanalsobemadeclearintermsofthetruthtable.Noticethatthecontraryrelationrulesoutthetopcaseinthetable.7/1/2015
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6/14TheSubcontraryRelationTwostatementsaresubcontrarytoeachotheriftheycannotbothbefalse.ThesubcontraryrelationexistsbetweentheIandOstatements.Again,wecanshowwhytheIandOstatementscannotbothbefalsebyusingtheVennDiagrams.SupposethatboththeIandOstatementswerefalse.Accordingtothecontradictoryrelation,thiswouldamounttoboththeEandAstatementsbeingtrue.Butthenboththeareaandtheareawouldbeempty.Butifboththeareaandtheareawereempty,thenthesetSwouldbeempty.ThisagainwouldcontradicttheassumptionthatthesetScannotbeanemptyset.ThesubcontraryrelationdoesnotallowthelogicalpossibilityofbothIandObeingfalseinthetruthtable.7/1/2015
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7/14TheImplicationRelationImplicationisanimportantlogicconcept.Ifastatementpimpliesanotherstatementq,thenitcannotbethecasethatpistrue,butqisfalse.Therefore,theimplicationrelationrulesoutthesecondcaseinthetruthtable.IntheTraditionalSquaretheAstatementimpliestheIstatement,andtheEstatementimpliestheOstatement.Ifpimpliesq,italsomeansthatifqisfalse,thenpmustbefalse.7/1/2015
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8/14Exercise2.2I.
Usetheinteractiveprogramtodeterminethetruthvalues.WriteTforTrue,FforFalseand?forUndetermined.1.
GiventhattheAstatementisfalse,usetheModernSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.FI:SomeSareP. O:SomeSarenotP.Check Answers2.
GiventhattheIstatementistrue,usetheModernSquaretodeterminethetruthvalueoftheotherthreestatements.7/1/2015
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9/14A:AllSareP. E:NoSareP.TI:SomeSareP. O:SomeSarenotP.Check
Answers3.
GiventhattheIstatementisfalse,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.FI:SomeSareP. O:SomeSarenotP.Check Answers4.
GiventhattheEstatementisfalse,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP. F7/1/2015
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10/14I:SomeSareP. O:SomeSarenotP.Check Answers5.
GiventhattheAstatementistrue,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.TI:SomeSareP. O:SomeSarenotP.Check Answers6.
GiventhattheOstatementistrue,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.7/1/2015
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11/14 TI:SomeSareP. O:SomeSarenotP.Check Answers7.
GiventhattheAstatementisfalse,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.FI:SomeSareP. O:SomeSarenotP.Check Answers8.
GiventhattheEstatementistrue,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP. T7/1/2015
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12/14I:SomeSareP. O:SomeSarenotP.Check Answers9.
GiventhattheIstatementistrue,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.TI:SomeSareP. O:SomeSarenotP.Check Answers10.
GiventhattheOstatementisfalse,usetheTraditionalSquaretodeterminethetruthvalueoftheotherthreestatements.A:AllSareP.
E:NoSareP.7/1/2015
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13/14 FI:SomeSareP. O:SomeSarenotP.Check AnswersII.
Youcannotseewhatisinsideabox,butaretoldthatitcontainscoloredshapes.Ineachofthefollowingquestions,supposethegivensentenceistheonlyinformationyouhaveaboutthecoloredshapesinthebox.Trytoseeifyoucandeterminewhethertheothersentencesaretrue(T)orfalse(F).Ifyouthinkthetruthvaluecannotbedetermined,identifyitasundetermined(?).1.
Somesquaresareblue.Allsquaresareblue.Nosquaresareblue.Somesquaresarenotblue.2.
Somecirclesarenotgreen.Allcirclesaregreen.Nocirclesaregreen.Somecirclesaregreen.3.
Notrianglesarered.Alltrianglesarered.Sometrianglesarered.Sometrianglesarenotred.4.
Itisfalsethatalloctagonsaregray.Nooctagonsaregray.Someoctagonsaregray.Someoctagonsarenotgray.7/1/2015
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14/145.
Itisfalsethatsomegreenshapesarehexagons.Allgreenshapesarehexagons.Nogreenshapesarehexagons.Somegreenshapesarenothexagons.Check
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