7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 1/14HOME TABLEOFCONTENTS2. 2 Squares of Opposi ti onAsquareofoppositionshowsthelogicalrelationsamongcategoricalstatements.Therearetwosquaresofopposition:1. theModernSquareofOpposition2. theTraditionalSquareofOpposition2.2.1TheModernSquareofOppositionTheonlylogicalrelationintheModernSquareofOppositionisthecontradictoryrelation.I L O G I C7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 2/14TheContradictoryRelationTosaythattwostatementsarecontradictorytoeachothermeansthattheynecessarilyhaveoppositetruthvalue.Thatis,ifoneofthemistrue,theothermustbefalse,andifoneofthemisfalse,theothermustbetrue.Forexample,theAstatementAllbatsaremammalsandtheOstatementSomebatsarenotmammalscontradicteachother.ThecontradictoryrelationalsoexistsbetweentheEstatementNoswansareblackbirdsandtheIstatementsSomeswansareblackbirds.WecanuseVennDiagramstoillustratewhythecontradictoryrelationholdsbetweentheAandOstatements.WhentheAstatementistrue,theareaisempty.Butiftheareaisempty,thennomemberofScanbeinthearea.ThiscontradictstheOstatement,whichsaysthatthereisatleastonememberofSinthearea,i.e.,theareaisnotempty.7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 4/142.2.2TheTraditionalSquareofOppositionIfweassumethatthesetdenotedbythesubjecttermcannotbeanemptyset,thentherearefourlogicalrelationsamongtheA,E,I,Ostatements.TheyareshownintheTraditionalSquareofOpposition.Thefourrelationsare:ContradictoryContrarySubcontraryImplicationInthediagrambelow,wecanactuallyseethecompletesquarethatshowsthelogicalrelationsamongtheA,E,I,Ostatements.7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 5/14TheContraryRelationThecontradictoryrelationhasbeenexplicatedabove.Wenowlookatthecontraryrelation.Twostatementsarecontrarytoeachotheriftheycannotbothbetrue.ThecontraryrelationexistsbetweentheAandEstatements.WecanuseVennDiagramstoillustratewhytheAandEstatementscannotbothbetrue.SupposethatboththeAandEstatementsweretrue.IntermsoftheirVennDiagrams,thiswouldmeanthattheareaandtheareawereempty.Butifboththeareaandtheareawereempty,thenthesetSwouldbeempty.ThiswouldcontradicttheassumptionthatthesetScannotbeanemptyset.Thecontraryrelationcanalsobemadeclearintermsofthetruthtable.Noticethatthecontraryrelationrulesoutthetopcaseinthetable.7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 6/14TheSubcontraryRelationTwostatementsaresubcontrarytoeachotheriftheycannotbothbefalse.ThesubcontraryrelationexistsbetweentheIandOstatements.Again,wecanshowwhytheIandOstatementscannotbothbefalsebyusingtheVennDiagrams.SupposethatboththeIandOstatementswerefalse.Accordingtothecontradictoryrelation,thiswouldamounttoboththeEandAstatementsbeingtrue.Butthenboththeareaandtheareawouldbeempty.Butifboththeareaandtheareawereempty,thenthesetSwouldbeempty.ThisagainwouldcontradicttheassumptionthatthesetScannotbeanemptyset.ThesubcontraryrelationdoesnotallowthelogicalpossibilityofbothIandObeingfalseinthetruthtable.7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 7/14TheImplicationRelationImplicationisanimportantlogicconcept.Ifastatementpimpliesanotherstatementq,thenitcannotbethecasethatpistrue,butqisfalse.Therefore,theimplicationrelationrulesoutthesecondcaseinthetruthtable.IntheTraditionalSquaretheAstatementimpliestheIstatement,andtheEstatementimpliestheOstatement.Ifpimpliesq,italsomeansthatifqisfalse,thenpmustbefalse.7/1/2015 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 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 SquaresofOppositionfile:///C:/Users/VP%20for%20Administratio/Desktop/Squares%20of%20Opposition.html 14/145. Itisfalsethatsomegreenshapesarehexagons.Allgreenshapesarehexagons.Nogreenshapesarehexagons.Somegreenshapesarenothexagons.Check AnswersThisworkislicensedunderaCreativeCommonsAttributionNoncommercialNoDerivativeWorks3.0UnitedStatesLicense.PREVIOUSNEXT