Baldwin Demonstrative logic axiomatics denition Demonstration vrs Deduction John T Baldwin December 2 2007 brPage 2br Demonstration vrs Deduction John T Baldwin Demonstrative logic axiomatics denition Acknowledgement These notes paraphrase ID: 75520
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DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition DeductiveLogic Deductionistheprocessofextractinginformationimpliedbygivenpremisesregardlessofwhetherthosepremisesareknowntobetrueorevenwhethertheyaretrue. DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition Deduction(Proof) Corcoran Deductionistheprocessofextractinginformationimpliedbygivenpremisesregardlessofwhetherthosepremisesareknowntobetrueorevenwhethertheyaretrue.Afterall,evenfalsepropositionsimplylogicalconsequences;wecandeterminethatapremiseisfalsebydeducingfromitaconsequencewealreadyknowtobefalse.Adeductionfromunknownpremisesalsoproducesknowledgeofthefactthatitsconclusionfollowslogicallyfrom(isaconsequenceof)itspremises-notjustfromknowledgeofthetruthofitsconclusion. DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition Whatisaproof? ForAristotle,ademonstrationbeginswithpremisesthatareknowntobetrueandshowsbymeansofchainingofevidentstepsthatitsconclusionisalogicalconsequenceofitspremises.Thus,a demonstration isastep-by-stepdeductionwhosepremisesareknowntobetrue. DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition Whatisgeometry? NewView Geometryisthedeductionofconclusionfromacertainsetofgeometrichypotheses. ThesehypothesesmightbeEuclidean,spherical,hyperbolic,geometryetc.Whetherthesegeometricalhypothesesare"true"is not amathematicalquestion. DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition Denitions Adenitionisanabbreviation.WehavesomeconceptXwhichwecandescribeintermsofearliernotionsandwesayXholdsi thiscollectionofconditionshold.Keypoints: 1 newnotion 2 propertiesofoldnotion DemonstrationvrsDeduction JohnT.Baldwin Demonstrativelogic axiomatics denition ComplexExampleofDenition Problem DenearelationB((a1;a2);(b1;b2);(c1;c2))tointerpretcisbetweenaandb. IntuitiveSolution WemustdenealgebraicallyarelationBontriples(a1;a2);(b1;b2);(c1;c2)thatissatisedifandonlyif(c1;c2)isonthelinesegmentbetween(a1;a2);and(b1;b2).