Definition
Mathematicallogicisaformallysymbolizedandmathematicallogicinformallogic,anditstillbelongstothecategoryofintellectuallogicinessence.Mathematicallogicisalsocalledsymboliclogicandtheoreticallogic.Itisbothabranchofmathematicsandabranchoflogic.Itisadisciplinethatusesmathematicalmethodstostudylogicorformallogic.Theresearchobjectistheformalsystemaftersymbolizingthetwointuitiveconceptsofproofandcalculation.Mathematicallogicisanindispensablepartofbasicmathematics.Althoughtherearetwocharactersoflogicinthename,itdoesnotbelongtothecategoryofpurelogic.
Interpretation
Theso-calledmathematicalmethodisthegeneralmethodusedinexponentialscience,includingtheuseofsymbolsandformulas,existingmathematicalachievementsandmethods,especiallytheuseofformalaxioms.ThesystematicthinkingofstudyinglogicwithmathematicalmethodsgenerallytracedbacktoLeibniz.Hebelievedthatclassicaltraditionallogicmustbereformedanddevelopedtomakeitmorepreciseandeasytocalculate.LatergenerationsbasicallyworkedalongLeibniz'sideas.
Inshort,mathematicallogicispreciseandmathematicalformallogic.Itisthefoundationofmoderncomputertechnology.Thenewerawillbeaneraofgreatdevelopmentinmathematics,andmathematicallogicwillplayakeyroleinit.Logicisadisciplinethatexplores,elaborates,andestablishestheprinciplesofeffectivereasoning.ItwasfirstcreatedbytheancientGreekscholarAristotle.Thesubjectthatusesmathematicalmethodstostudyquestionsaboutreasoningandproofiscalledmathematicallogic.Alsocalledsymboliclogic.
Generation
Theideaofusingcomputationalmethodstoreplacethelogicalreasoningprocessinpeople'sthinkingwasproposedasearlyastheseventeenthcentury.Leibnizonceenvisionedwhetherhecouldcreatea"universalscientificlanguage"thatcoulduseformulastocalculatethereasoningprocesslikemathematics,soastodrawcorrectconclusions.Duetothesocialconditionsatthetime,hisideawasnotrealized.However,histhoughtsarethegermofpartofmodernmathematicallogic.Inthissense,Leibnizcanbesaidtobeapioneerofmathematicallogic.
In1847,BritishmathematicianBooleanpublished"MathematicalAnalysisofLogic",established"BooleanAlgebra",andcreatedasetofsymbolsystem,usingsymbolstorepresentvariousconceptsinlogic.Booleanestablishedaseriesofalgorithms,usedalgebraicmethodstostudylogicalproblems,andinitiallylaidthefoundationofmathematicallogic.
Attheendofthenineteenthcenturyandthebeginningofthetwentiethcentury,mathematicallogichasdevelopedconsiderably.In1884,theGermanmathematicianFregepublishedthebook"BasicsofArithmetic",whichintroducedthesymbolsofquantifiers.Makethesymbolsystemofmathematicallogicmorecomplete.ContributingtotheestablishmentofthisdisciplineisalsotheAmericanPierce,whoalsointroducedlogicalsymbolsinhiswritings.Asaresult,themostbasictheoreticalfoundationofmodernmathematicallogichasgraduallyformedandbecomeanindependentsubject.
Content
Whatisincludedinmathematicallogic?Broadlyspeaking,mathematicallogicincludessettheory,modeltheory,prooftheory,andrecursiontheory.Herewefirstintroduceitstwomostbasicandimportantcomponents,namely"propositionalcalculus"and"predicatecalculus".Propositionalcalculusisthestudyofhowpropositionsusesomelogicalconnectivestoformmorecomplexpropositionsandmethodsoflogicalreasoning.Apropositionisasentencethathasspecificmeaningandcanjudgewhetheritistrueorfalse.
Ifweregardpropositionsastheobjectsofoperations,likenumbers,lettersoralgebraicexpressionsinalgebra,andconsiderlogicalconnectivesasoperationalsymbols,justlikethe"add,subtract,multiply,anddivide"inalgebra."Inthatway,theprocessofcomposingcompoundpropositionsfromsimplepropositionscanberegardedastheprocessoflogicaloperations,thatis,thecalculationofpropositions.Suchlogicaloperationsalsohavecertainpropertieslikealgebraicoperationsandsatisfycertainrulesofoperation.Forexample,itsatisfiestheexchangelaw,associativelaw,anddistributionlaw,butalsosatisfiesthelogicalidentitylaw,absorptionlaw,doublenegationlaw,DiMorgan'slaw,syllogismlaw,andsoon.Usingtheselaws,wecancarryoutlogicalinferences,simplifycomplexpropositions,andinferwhethertwocomplexpropositionsareequivalent,thatis,whethertheirtruthtablesareexactlythesame,andsoon.
Aspecificmodelofpropositionalcalculusislogicalalgebra.Logicalalgebraisalsocalledswitchalgebra.Itsbasicoperationsarelogicaladdition,logicalmultiplication,andlogicalnegation,whichare"or","and",and"not"inpropositionalcalculus.Theoperandsareonlytwonumbers,0and1,whichareequivalent"True"and"False"inpropositionalcalculus.Theoperationalcharacteristicsoflogicalgebraareexactlythesameasthephenomenaofonandoff,highpotentialandlowpotential,conductionandcut-offincircuitanalysis.Thereareonlytwodifferentstates.Therefore,itiswidelyusedincircuitanalysis.
Theuseofelectroniccomponentscanformgatecircuitsequivalenttologicaladdition,logicalmultiplicationandlogicalnegation,whicharelogicalelements.Simplelogicelementscanalsobeformedintovariouslogicnetworks,sothatanycomplexlogicalrelationshipcanberealizedbylogicalelementsthroughappropriatecombination,sothatelectronicelementshavethefunctionoflogicaljudgment.Therefore,thereareimportantapplicationsinautomaticcontrol.Predicatecalculusisalsocalledpropositionalcalculus.Inpredicatecalculus,theinternalstructureofapropositionisanalyzedintoalogicalformwithsubjectandpredicate.Thepropositioniscomposedofpropositionalitems,logicalconnectivesandquantifiers,andthenthelogicalreasoningrelationshipbetweensuchpropositionsisstudied.
Thepropositionaltermreferstoalogicalformulathatcontainsvariabletermsinadditiontoconstantterms.Constantreferstocertainobjectsorcertainattributesandrelationships;variablereferstoanyonewithinacertainrange,andthisrangeiscalledthevariabledomainofthevariable.Propositionaltermsaredifferentfrompropositionalcalculus.Itdoesnotmatterwhetheritistrueorfalse.Ifthevariabletermisreplacedbyacertainobjectconcept,thenthepropositionalconnotationbecomesatrueorfalseproposition.Adduniversalquantifierorexistentialquantifiertopropositionalconnotation,thenitbecomesuniversalpropositionorspecialproposition.
Development
Aftertheestablishmentofthedisciplineofmathematicallogic,ithasdevelopedrapidly,andtherearemanyfactorsthatpromoteitsdevelopment.Forexample,theestablishmentofnon-Euclideangeometryhaspromptedpeopletostudythenon-contradictionbetweennon-EuclideangeometryandEuclideangeometry.
Theemergenceofsettheoryisamajoreventinthedevelopmentofmodernmathematics,butintheprocessofresearchonsettheory,therehasbeenaso-calledthirdcrisisinthehistoryofmathematics.Thiscrisiswascausedbythediscoveryoftheparadoxofsettheory.Whatistheparadox?Paradoxesarelogicalcontradictions.Settheoryisoriginallyabranchofverystrictargumentation,andisrecognizedasthefoundationofmathematics.
In1903,theBritishphilosopher,logician,andmathematicianRussellproposedthe"RussellParadox"namedafterhimforsettheory.Theproposalofthisparadoxalmostshooktheentiremathematicalfoundation.
TherearemanyexamplesinRussell’sParadox,oneofwhichispopularandfamousisthe"BarberParadox":Thereisabarberinacertaincountry,andonedayheannouncedthathewouldonlyshavehimself.Manshaves.Thenaquestionarises:Doesthebarbershavehimselfornot?Ifheshaveshimself,heistheonewhoshaveshimself.Accordingtohisprinciple,heshouldnotshavehimself;ifhedoesnotshavehimself,thenheistheonewhodoesnotshavehimself,accordingtohisInprinciple,heshouldshavehimselfagain.Thiscreatesacontradiction.
Theproposingoftheparadoxhaspromptedmanymathematicianstostudythenon-contradictoryproblemofsettheory,whichhasledtoanimportantbranchofmathematicallogic-axiomsettheory.
Theemergenceofnon-Euclideangeometryandthediscoveryoftheparadoxofsettheoryindicatethattherearestillmanyproblemsinmathematics.Asaresearchobject,studyingthelogicalstructureofmathematicalsystemsandthelawsofproofs,thishasproducedanotherbranchofmathematicallogic-prooftheory.
Mathematicallogichasrecentlydevelopedmanynewbranches,suchasrecursiontheoryandmodeltheory.Recursiontheorymainlystudiesthetheoryofcomputability,whichiscloselyrelatedtothedevelopmentandapplicationofcomputers.Modeltheorymainlystudiestherelationshipbetweenformalsystemsandmathematicalmodels.Mathematicallogichasdevelopedrapidlyinrecentyears.Themainreasonisthatthissubjecthashadasignificantimpactonthedevelopmentofotherbranchesofmathematicssuchassettheory,numbertheory,algebra,topology,etc.,especiallythenewlyformedcomputerscience..Inturn,thedevelopmentofotherdisciplineshasalsopromotedthedevelopmentofmathematicallogic.
Becauseitisanewlyemergingandfast-developingsubject,italsohasmanyproblemsthatneedtobestudiedindepth.Manymathematiciansarenowstudyingtheproblemsofmathematicallogicitself.
Inshort,theimportanceofthissubjectisalreadyveryobvious,andithasarousedtheconcernandattentionofmanypeople.
System
Themainbranchesofmathematicallogicinclude:logicalcalculus(includingpropositionalcalculusandpredicatecalculus),modeltheory,prooftheory,recursiontheoryandaxiomaticsettheory.Therearemanyoverlapsbetweenmathematicallogicandcomputerscience,andbothbelongtosciencethatsimulatesthemechanismofhumancognition.Manypioneersofcomputersciencearebothmathematiciansandlogicians,suchasAlanTuringandChurch.
Theresearchofprocedurallinguisticsandsemanticsisderivedfrommodeltheory,whiletheprogramverificationisderivedfrommodelcheckingofmodeltheory.
Curry-Howardisomorphismgivestheequivalenceof"proof"and"procedure".Thisresultisrelatedtoprooftheory,andintuitionisticlogicandlinearlogicplayabigrolehere.Calculussuchaslambdacalculusandcombinatorlogicarenowidealprogramminglanguages.
Computerscience'sachievementsinautomaticverificationandautomaticsearchforprooftechniqueshavecontributedtothestudyoflogic,suchasautomatictheoremprovingandlogicprogramming.
Computer
Whentherearemorethantwologicstatesinthelogicalgebra(suchas0,1,2ormorestates),therearetwobasiclogicsofthegeneralmodel.Oneisthelogicofchangingfromonestatetoanother,whichisaunarylogic;theotheristhelogicofchoosingoneofthetwostatesaccordingtoacertainrule(suchascomparingthesize).Thisisabinarylogic.
Accordingtothesetwologics,anylogicalrelationshipofanynumberofstatescanbeexpressed,thatis,thesmallestexpression.Thatis,thelogicofanymulti-stateiscomplete.Whenthenumberoflogicalstatesexpandsbyarationalorderofmagnitudeormore.Anymathematicaloperationcanbeexpressedjointlybytwooperationalrelations:additionandsubtractionandcomparisonofmagnitude.