QUANTUM LOGIC AND NONCLASSICAL LOGICS

QUANTUMLOGICANDNONCLASSICALLOGICSGIANPIEROCATTANEO,MARIALUISADALLACHIARA,ROBERTOGIUNTINI,ANDFRANCESCOPAOLI1.IntroductionClassicallogicissometimesdescribedas\thelogicofanomniscientmindinadeterministicuniverse.Fromanintuitivepointofviewthebasicfeaturesofclassicalsemanticscanbesummarizedasfollows:1)anyproblemissemanticallydecided:foranysentence®,either®oritsnegation:®istrue(excludedmiddleprinciple);atthesametime,asentence®anditsnegation:®cannotbebothtrue(non-contradictionprinciple).2)Meaningsbehaveinacompositionalway:themeaningofacompoundexpressionisdeterminedbythemeaningsofitsparts.3)Meaningsaresharpandunambiguous.Some(possiblyall)oftheseprincipleshavebeenputinquestionbydif-ferentformsofnonclassicallogic.Insomesigni¯cantcases,theobjectiveanddescriptionalnotionoftruth(whichischaracteristicofclassicallogic)hasbeenreplacedbyanepistemicconception.Accordingly,truthhasbeenidenti¯edwithwhatisknownbynon-omniscientminds,actinginauniversethatmaybeeitherdeterministicorindeterministic.The¯rstchoiceiscom-patiblewiththeintuitionisticapproachestologicandtomathematics,whilethesecondchoicerepresentsthebasicassumptionofthequantumlogicalin-vestigations.Inbothcases,theclassicalnotionoftruthhasbeenreplacedbythefollowingrelation:aninformationiforcesustoassertthetruthofasentence®:Onealsobrie°ysaysthattheinformationiforces(orveri¯es)thesentence®(andoneusuallywrites:ij=®).Shouldirepresentanoncontradictoryandcompleteinformation-system,ourforcingrelationwouldnaturallycollapseintotheclassicalnotionoftruth.However,humaninformationisgenerallyincompleteandnotnecessarilyconsistent.Asexpected,inthecaseofphysicaltheories,signi¯cantpiecesofinfor-mationcorrespondtowhatisknownbyanobserveraboutthephysicalsys-temsunderinvestigation.Inthisconnection,oneusuallyspeaksofphysicalstates(brie°y,states).Inthe\happiestsituations,astatemayrepresentamaximalknowledgeoftheobserver:apieceofinformationthatcannotKeywordsandphrases.quantumlogic,nonclassicallogics.WewarmlythankGiuseppeSergioliforhisprecioussuggestionsconcerningSection9.12G.CATTANEO,M.L.DALLACHIARA,R.GIUNTINI,ANDF.PAOLIbeconsistentlyextendedtoaricherinformation,intheframeworkofthetheory.Evenahypotheticalomniscientmindcouldnotknowmoreaboutthesysteminquestion(ifthetheoryiscorrect).Statesofthiskindareusuallycalledpurestates,bothinclassicalandinquantumphysics.Piecesofinformationthatarenotmaximalaregenerallyrepresentedbymixturesofpurestates(alsocalledmixedstates).Thereisanimportantdi®erencethatconcernsthelogicalbehaviorofclassicalandofquantumpurestates.Inclassicalmechanics,maximalityimplieslogicalcompleteness:anypurestatesemanticallydecidesanyphysicalproperty(orevent)thatmayholdforthesystemunderinvestigation(inotherwords,thestateattributestothesystemeitherthepropertyoritsnegation).Thisisinaccordancenotonlywithclassicallogic,butalsowithanumberofimportantnonclassicallogics(likeintuitionisticlogic),whereanynoncontradictoryandmaximalformalsystemislogicallycomplete.Quantumpurestates,instead,giverisetoasomewhat\mysteriousdivergencebetweenmaximalityandlogicalcompleteness,whichrepresentstheoriginofmostlogicalanomaliesofthequantumworld.Althoughrepresentingamaximalinformation,aquantumpurestateisneverlogicallycomplete.ThisisaconsequenceofHeisenberg'suncertaintyprinciple,accordingtowhichtherearepairsofcomplementaryeventsthatcannotbesimultaneouslydecidedbyanypurestate.Bothinclassicalandinquantummechanics,physicalstatesarerepre-sentedbyspecialkindsofmathematicalobjects.Inclassicalmechanics(CM),apurestateofasingleparticlecanberepresentedbyasequenceofsixrealnumbers(r1;:::;r6),wherethe¯rstthreenumberscorrespondtotheposition-coordinates,whilethelastonesarethemomentum-components.ThesetIR6ofallsextuplesofrealnumbersrepresentsthephase-spacefortheparticleinquestion.Similarlyforthecaseofcompoundsystems,con-sistingofa¯nitenumbernofparticles.Hence,anypurestateofaclassicalparticle-systemisrepresentedbyapointofanappropriatephasespace§.Howtorepresentthephysicaleventsthatmayoccurtoagivenparticle?Followingthestandardideasofclassical(extensional)semantics,itisquitenaturaltoassumethatthesucheventsaremathematicallyrepresentedbysuitablesubsetsof§.Whataboutthestructureofallevents?Asiswellknown,thepowersetofanysetgivesrisetoaBooleanalgebra.AndalsothesetF(§)ofallmeasurablesubsetsof§(whichismoretractablethanthefullpowersetof§,fromameasure-theoreticpointofview)turnsouttohaveaBooleanstructure.Hence,wemayrefertothefollowingBoolean¯eldofsets:EVC=hF(§);\;[;c;;;§)i;wheretheset-theoreticoperations\;[;crepresentrespectivelytheconjunc-tion,thedisjunctionandthenegationofclassicalevents.Asaconsequence,thelogicofCMturnsouttobeinperfectagreementwithclassicallogic.Furthermore,purestatesarelogicallycomplete:foranyQUANTUMLOGIC3pointpofthephase-space§andforanyeventEinF(§),eitherp2Eorp2Ec.Whathappensinthecaseofquantumtheory(QT)?Asopposedtoclassi-calmechanics,QTisessentiallyprobabilistic.Apurestategenerallyassignstoaquantumeventaprobability-value(arealnumberintheinterval[0;1]).Asaconsequence,aquantumeventmaybesemanticallyindeterminateforagivenpurestate,andthee