# Mathematical logic

## Generation

Theideaof​​usingcomputationalmethodstoreplacethelogicalreasoningprocessinpeople'sthinkingwasproposedasearlyastheseventeenthcentury.Leibnizonceenvisionedwhetherhecouldcreatea"universalscientificlanguage"thatcoulduseformulastocalculatethereasoningprocesslikemathematics,soastodrawcorrectconclusions.Duetothesocialconditionsatthetime,hisideawasnotrealized.However,histhoughtsarethegermofpartofmodernmathematicallogic.Inthissense,Leibnizcanbesaidtobeapioneerofmathematicallogic.

In1847,BritishmathematicianBooleanpublished"MathematicalAnalysisofLogic",established"BooleanAlgebra",andcreatedasetofsymbolsystem,usingsymbolstorepresentvariousconceptsinlogic.Booleanestablishedaseriesofalgorithms,usedalgebraicmethodstostudylogicalproblems,andinitiallylaidthefoundationofmathematicallogic.

## Development

Becauseitisanewlyemergingandfast-developingsubject,italsohasmanyproblemsthatneedtobestudiedindepth.Manymathematiciansarenowstudyingtheproblemsofmathematicallogicitself.

## 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.