THE“LAWOFSYLLOGISM”PROJECT
Overviewoftheproject:
Thisproject,whichwillbeamajorportionoftheGeometrypartofRIMASatExhibitionsin
December,containssixmainparts,describedinmoredetailonthenextpage.Itwillbe
presentedintheformofanaccordionbook.(Ifyou’venevermadeone,youcancheckoutthis
template:http://www.booklyn.org/education/isaccordion.pdf).
Themainpartsintheprojectareto:
1. Writeanintroduction.
2. Writeaseriesofpostulates(if-thenstatements)likein“GeraldtheGnu.”
3. Illustrateyourpostulateswithalegibleandwell-structuredEulerdiagram.
4. Createtheoremsbasedonyourpostulates.TheEulerdiagramshouldhelptovisualize
thevalidityofthetheorems.
5. Writealogicalproofusingdeductivereasoning,startingfromapremise(known
informationsuchas“Geraldisagnu”)andusingyourpostulatesandtheoremstodraw
conclusionsusingatleast5differentpostulates.
6. Writeapersonalreflectionhighlightingthekeylearningpointsoftheassignment.
BackgroundinformationontheLawofSyllogism:
Inmathematics,thetransitivepropertyisusedtomakedeductionsofthestyle:
ifa=bandb=c,thena=c.
Forexample,inthefigure,wearetoldthat∠𝐵is
congruentwith∠𝐴andalsowith∠𝐶.
Usingthetransitiveproperty,wecanthenconclude
that
∠𝐴and∠𝐶arealsocongruent.
Insimplifiedmathematicallanguage,wewould
write:
∠𝐴 ≅ ∠𝐵and∠𝐵 ≅ ∠𝐶!∠𝐴 ≅ ∠𝐶
Thelawofsyllogism,alsocalledreasoningbytransitivity,isalineofdeductivereasoning
similartothetransitiveproperty.Itfollowsthispattern:
Postulate1: Ifathenb.
Postulate2: Ifbthenc.
Byusingthelawofsyllogism,wecanthenprove:
Theorem:
Ifathenc.
Example:
Postulate1: IfyouliveinGnuJersey,thenyouareaquadruped.
Postulate2: Ifyouareaquadruped,thenyouhavefourlegs.
Theorem1-2:
IfyouliveinGnuJersey,thenyouhavefourlegs.
Proof:Let’ssayourpremisestatesthatGeraldlivesinGnuJersey.WeknowfromTheorem1-2
thatifyouliveinGnuJersey,thenyouhavefourlegs.ThereforeGeraldhasfourlegs.
Detailedrequirements:
Part1:Writeanintroduction
Inlessthanonepage,brieflydescribethegoalsoftheprojectandgivesomebackground
informationforthereader.Someonewhohasn’ttakentheclassshouldbeabletofollowwhat
youaredoing(youmaywanttoincludedefinitionsofthevocabularyyouwillbeusing,for
example).
Part2:Writeaseriesofconditionalstatementsor“postulates”.
Writeatleast9differentconditionalstatements(thatyouwillbeorganizinginanEuler
diagraminPart3)ofwhich:
o atleast1shouldbewrittenwiththeconclusionlocatedbeforethehypothesis
o atleast1shouldbewrittenwithoutusingtheword“if”
o atleast1shouldbewrittenincontrapositiveform
o atleast1shouldhavea2-parthypothesis(“if____and____”or“if____or____”)
o atleast1inverseorconverseofanotherpostulateshouldbepresent
o atleast1shouldbeunusableinyoursyllogism(trickystatement!)becauseusingit
wouldmakeyouguiltyofthe“converse”orthe“inverse”error
o atleast1shouldhaveanegatedhypothesisand/orconclusion
o Thecontentofseveralofthestatementsshouldhaveaconnectionwithyourschool
life(ideallysomethingyoulearnedinschool,forexampleinscience).
o Thestatementsshouldbefunorinterestingtoread.Grasptheattentionofyour
reader,andbecreative!
Part3:Organizeallyourconditionalstatementsinawell-structuredEulerdiagram
Eachhypothesisandeachconclusionfromyourconditionalstatementsshouldberepresented
withacircleorloopinyourEulerdiagram.Thesamephrase,ifitappearsinseveralconditional
statements,shouldonlyberepresentedonce.
MakesureyourEulerdiagram:
o islogicallycorrectandillustratestheconditionsandpossibilitiesexpressedinyour
postulates(intersectionsbetweenloopswhenapplicable,etc.)
o isneatandlegible,anditsstructureiseasytofollow
o containsakeyclearlyshowingthephraserepresentedineachloop
o wheneverpossible(anditwon’talwaysbe),theloopsshouldrepresentanaffirmative
set(what“is”,asopposedtowhat“isnot”).
Part4:Createtheoremsusingthelawofsyllogism
Usingatleast5ofyourpostulatesandthelawofsyllogism,derivesometheoremsthatarebased
onyourchainedpostulates.Onetheoremthatuses5postulatesismorecomplexandinteresting
than5theoremsthatuse2postulateseach.Organizeyourdeductionsinawaythatiseasyto
followbyanonwell-informedreader.
Part5:Writeatleastonelogicalproofusingdeductivereasoning
Startingfromatleast1premise(thestatementtakenastrue,suchas“GeraldlivesinGnu
Jersey”),usedeductivereasoningandthetheoremsyoucreatedtoprovesomeconclusions.The
goalisnottofindmanyconclusions,butideallytofindatleastoneconclusionthatisbasedona
complexlogicalproof.
Part6:Writeareflection
Thisreflectionshouldsummarizeyourthoughtprocessandhighlightrelevantlearning
pointsandtrickypartsinthedeductivereasoning.Thisiswhereyoutellthereaderhowhard
youworkedandwhereyoushowoffalloftheknowledgethatyouacquired.Bethoughtfuland
philosophical,becomeanAncientGreek!(Youmaywanttowearatogaforexhibitionstoo!)
Organizeyourreflectionintoseveralparagraphs,eachcommentingonadifferentaspectofthe
project.Someexamplesinclude:
o CommentonthemakingoftheEulerdiagram:whataresomeoftheobstaclesyou
encounteredandovercame?Howarehypothesesandconclusionsillustrated?Whatisthe
meaningofintersections?Whyaresomesetsseparatedfromeachother?
o Commentonthekeypointsofthelogicalproof:mentiontheconverse/inverseerrorsyou
avoided,discussanyotherdifficultiesyouencountered.
o Commentonyourcontent:whatdidyouthinkwasinteresting/funaboutyour
statements?Howdidyougoaboutwritingthepostulates?Didyouhavetochangethem
much?Howdidthisassignmenthelpyoulearnmore?
o Overalllearning:whatdidyoulearnthroughoutthisproject?Howisisapplicabletoareas
ofknowledgeotherthanmath?
Structureandformatoftheaccordionbooklet
Theprojectshouldbepresentedinanaccordionbooklet,whichmakesitprettyandeasily
displayableduringexhibitions.(Seehttp://www.booklyn.org/education/isaccordion.pdf.)
Structure:
o Titlepage,withallpertinentinformation(name,date,course,titleofproject,…).
o Indexpage.
o Thesixmainpartsoftheproject
o Youmayuseextrapagesasneededforextrainformation,suchasaglossary,etc.
Presentationandcommunication
Youraccordionbookletshouldbeneatandorganized,visuallypleasing,andcreative.The
aestheticsshouldillustratethecontentwell;usecolorsandimagery.Allthewrittentextmustbe
typedinafontlargeenoughforexhibitions(atleast14pt).
Thecommunicationshouldbeeffective,sothatsomeonewhohasnottakenthisclassshouldbe
abletofollowwell.
Categories
Feedback&Performancelevel
1.Introduction
o Theaimoftheprojectisclearlydescribed.
o Anon-informedreaderisgiventhebackgroundneededto
followthecontentoftheproject.
o Technicalterminologyisexplainedandusedadequately.
Exceeds
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Developing Beginning
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Developing Beginning
2.Postulates
o Thevarietyintypesofconditionalstatementsmeetsthe
requirements(seeprojectdescription).
o Thestatementsareclearlywrittenandunambiguous.
o Thepostulatesarecreative/interestingand/orshowa
connectiontoatopicyouhavestudiedinschool.
o Thegrammarisappropriate:useofpronouns,useofconnecting
wordssuchas“if”,“when”,“all”,and“then”.
3.Eulerdiagram
o Eachhypothesis/conclusioninthepostulatesisrepresented.
o Theconcentricitiesandintersectionsbetweenloopscorrectly
reflectthecontentofallpostulatestogether(boththeirexplicit
andimplicitcontent).
o Thediagramiseasytoread,unambiguous,andakeyis
included.
4.Deductionoftheoremsusinglawofsyllogism
o Correctdeductivereasoningisusedtoderivetheorems.
o Thestructureofthelawofsyllogismiswellshown(seefirst
pageofprojectdescription).
o Thelevelofcomplexityisappropriate,withatleast5postulates
usedtoderivetheorems.
5.Logicalproofusingdeductivereasoning
o Theconclusionsreachedarerelevantandlogicallycorrect.
o Theformatofthelogicalproofisclearlystructured:
1. Startingpoint(givenpremise)
2. Theorem(relatingpremiseandconclusion)
3. Conclusion(endpointreachedbyconnectingpremisewith
theorem)
6.Reflection
o Summarizesthekeypointsinthethoughtprocess.
o Highlightsthecriticalthinking,questionsthatarose,anderrors
thatweredetected/avoided.
o Isthoughtfulandreasonable,andcommentsonthelearning
pointsandapplicabilityofthistypeofreasoning.
Presentation&Communication
o
o
o
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Acleartitlepageandindexarepresent.
Theaccordionbookletiswellmadeandvisuallypleasing.
Theaestheticsandvisualsusedenhancethecontent.
Thewritingiseffective,withfewornospellingand
grammaticalerrors.