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Logic is the study of correct reasoning or good arguments. It is often defined in a more narrow sense as the science of deductively valid inferences or of ... Logic FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Studyofcorrectreasoning Thisarticleisaboutthestudyofcorrectreasoning.Forotheruses,seeLogic(disambiguation). "Logician"redirectshere.FortheschoolofChinesephilosophy,seeSchoolofNames.Forthethoroughbred,seeLogician(horse). PartofaseriesonPhilosophyPlatoKantNietzscheBuddhaConfuciusAverroes Branches Aesthetics Axiology Cosmology Epistemology Ethics Legal Linguistic Logic Mental Metaphilosophy Metaphysics Political Religious Scientific Social Periods Ancient Medieval Modern Contemporary Traditions Analytic Neopositivism Ordinarylanguage Aristotelian Buddhist Abhidharma Madhyamaka Pramāṇavāda Yogacara Cārvāka Christian Augustinian Humanist Scotist Thomist Occamist Confucian Neo-Confucianism NewConfucianism Continental Existentialism Phenomenology Hegelian Hindu Mīmāṃsā Nyāya-Vaiśeṣika Sāṃkhya Yoga Vedanta KashmirShaivism Navya-Nyāya Neo-Vedanta Integralyoga Islamic Ash'arism EarlyIslamic Averroist Avicennist Illuminationist Ismāʿīlism Sufi Jain Jewish Judeo-Islamic Kantian Legalism Platonist Neoplatonist Pragmatism Skepticism Taoistphilosophy Traditionsbyregion African Eastern Chinese Indian Middle-Eastern Egyptian Iranian Western Literature Aesthetics Epistemology Ethics Logic Metaphysics Politicalphilosophy Philosophers Aestheticians Epistemologists Ethicists Logicians Metaphysicians Socialandpoliticalphilosophers Womeninphilosophy Lists Index Outline Years Problems Publications Theories Glossary Philosophers Philosophyportalvte Logicisthestudyofcorrectreasoningorgoodarguments.Itisoftendefinedinamorenarrowsenseasthescienceofdeductivelyvalidinferencesoroflogicaltruths.Inthissense,itisequivalenttoformallogicandconstitutesaformalscienceinvestigatinghowconclusionsfollowfrompremisesinatopic-neutralwayorwhichpropositionsaretrueonlyinvirtueofthelogicalvocabularytheycontain.Whenusedasacountablenoun,theterm"alogic"referstoalogicalformalsystem.Formallogiccontrastswithinformallogic,whichisalsopartoflogicwhenunderstoodinthewidestsense.Thereisnogeneralagreementonhowthetwoaretobedistinguished.Oneprominentapproachassociatestheirdifferencewiththestudyofargumentsexpressedinformalorinformallanguages.Anothercharacterizesinformallogicasthestudyofampliativeinferences,incontrasttothedeductiveinferencesstudiedbyformallogic.Butitisalsocommontolinktheirdifferencetothedistinctionbetweenformalandinformalfallacies. Logicisbasedonvariousfundamentalconcepts.Itstudiesarguments,whicharemadeupofasetofpremisestogetherwithaconclusion.Premisesandconclusionsareusuallyunderstoodeitherassentencesoraspropositionsandarecharacterizedbytheirinternalstructure.Complexpropositionsaremadeupofotherpropositionslinkedtoeachotherbypropositionalconnectives.Simplepropositionshavesubpropositionalparts,likesingulartermsandpredicates.Ineithercase,thetruthofapropositionusuallydependsonthedenotationsofitsconstituents.Logicallytruepropositionsconstituteaspecialcasesincetheirtruthdependsonlyonthelogicalvocabularyusedinthem. Theargumentsorinferencesmadeupofthesepropositionscanbeeithercorrectorincorrect.Anargumentiscorrectifitspremisessupportitsconclusion.Thestrongestformofsupportisfoundindeductivearguments:itisimpossiblefortheirpremisestobetrueandtheirconclusiontobefalse.Thisisthecaseiftheyfollowaruleofinference,whichensuresthetruthoftheconclusionifthepremisesaretrue.Aconsequenceofthisisthatdeductiveargumentscannotarriveatanysubstantivenewinformationnotalreadyfoundintheirpremises.Theycontrastinthisrespectwithampliativearguments,whichmayprovidegenuinelynewinformation.Thiscomeswithanimportantdrawback:itispossibleforalltheirpremisestobetruewhiletheirconclusionisstillfalse.Manyargumentsfoundineverydaydiscourseandthesciencesareampliativearguments.Theyaresometimesdividedintoinductiveandabductivearguments.Inductiveargumentsusuallytaketheformofstatisticalgeneralizationswhileabductiveargumentsareinferencestothebestexplanation.Argumentsthatfallshortofthestandardsofcorrectreasoningarecalledfallacies.Forformalfallacies,thesourceoftheerrorisfoundintheformoftheargumentwhileinformalfallaciesusuallycontainerrorsonthelevelofthecontentorthecontext.Besidesthedefinitoryrulesoflogic,whichdeterminewhetheranargumentiscorrectornot,therearealsostrategicrules,whichdescribehowachainofcorrectargumentscanbeusedtoarriveatone'sintendedconclusion.Informallogic,formalsystemsareoftenusedtogiveaprecisedefinitionofcorrectreasoningusingaformallanguage. Systemsoflogicaretheoreticalframeworksforassessingthecorrectnessofreasoningandarguments.Aristotelianlogicfocusesonreasoningintheformofsyllogisms.Itstraditionaldominancewasreplacedbyclassicallogicinthemodernera.Classicallogicis"classical"inthesensethatitisbasedonvariousfundamentallogicalintuitionssharedbymostlogicians.Itconsistsofpropositionallogicandfirst-orderlogic.Propositionallogicignorestheinternalstructureofsimplepropositionsandonlyconsidersthelogicalrelationsonthelevelofpropositions.First-orderlogic,ontheotherhand,articulatesthisinternalstructureusingvariouslinguisticdevices,suchaspredicatesandquantifiers.Extendedlogicsacceptthebasicintuitionsbehindclassicallogicandextendittootherfields,suchasmetaphysics,ethics,andepistemology.Thishappensusuallybyintroducingnewlogicalsymbols,suchasmodaloperators.Deviantlogics,ontheotherhand,rejectcertainclassicalintuitionsandprovidealternativeaccountsofthefundamentallawsoflogic.Whilemostsystemsoflogicbelongtoformallogic,somesystemsofinformallogichavealsobeenproposed.Oneprominentapproachunderstandsreasoningasadialogicalgameofpersuasionwhileanotherfocusesontheepistemicroleofarguments.Logicisstudiedinandappliedtovariousfields,suchasphilosophy,mathematics,computerscience,andlinguistics.LogichasbeenstudiedsinceAntiquity,earlyapproachesincludingAristotelianlogic,Stoiclogic,Anviksiki,andthemohists.Modernformallogichasitsrootsintheworkoflate19th-centurymathematicianssuchasGottlobFrege. Contents 1Definition 2Formalandinformallogic 3Fundamentalconcepts 3.1Premises,conclusions,andtruth 3.1.1Premisesandconclusions 3.1.2Internalstructure 3.1.3Logicaltruth 3.2Argumentsandinferences 3.2.1Deductive 3.2.2Ampliative 3.3Fallacies 3.4Definitoryandstrategicrules 3.5Formalsystems 3.5.1Formallanguage 3.5.2Proofsystem 3.5.3Semantics 3.5.4Soundnessandcompleteness 4Systemsoflogic 4.1Aristotelian 4.2Classical 4.2.1Propositionallogic 4.2.2First-orderlogic 4.3Extended 4.3.1Modallogic 4.3.2Higherorderlogic 4.4Deviant 4.5Informal 5Areasofresearch 5.1Philosophyoflogicandphilosophicallogic 5.2Mathematicallogic 5.3Computationallogic 5.4Formalsemanticsofnaturallanguage 6Controversies 6.1"IsLogicEmpirical?" 6.2Toleratingtheimpossible 6.3Conceptionsoflogic 6.4Rejectionoflogicaltruth 7History 8Seealso 9Notes 10References 10.1Bibliography 11Externallinks Definition[edit] Theword"logic"originatesfromtheGreekword"logos",whichhasavarietyoftranslations,suchasreason,discourse,orlanguage.[1][2][3]Logicistraditionallydefinedasthestudyofthelawsofthoughtorcorrectreasoning.[4]Thisisusuallyunderstoodintermsofinferencesorarguments:reasoningmaybeseenastheactivityofdrawinginferences,whoseoutwardexpressionisgiveninarguments.[4][5]Aninferenceoranargumentisasetofpremisestogetherwithaconclusion.Logicisinterestedinwhetherargumentsaregoodorinferencesarevalid,i.e.whetherthepremisessupporttheirconclusions.[6][7][8] Thesegeneralcharacterizationsapplytologicinthewidestsensesincetheyaretruebothforformalandinformallogic.[9]Butmanydefinitionsoflogicfocusonformallogicbecauseitistheparadigmaticformoflogic.Inthisnarrowersense,logicisaformalsciencethatstudieshowconclusionsfollowfrompremisesinatopic-neutralway.[10][11]Asaformalscience,itcontrastswithempiricalsciences,likephysicsorbiology,becauseittriestocharacterizetheinferentialrelationsbetweenpremisesandconclusionsonlybasedonhowtheyarestructured.[12]Thismeansthattheactualcontentofthesepropositions,i.e.theirspecifictopic,isnotimportantforwhethertheinferenceisvalidornot.[10][11]Thiscanbeexpressedbydistinguishingbetweenlogicalandnon-logicalvocabulary:inferencesarevalidbecauseofthelogicaltermsusedinthem,independentofthemeaningsofthenon-logicalterms.[13][4]Validinferencesarecharacterizedbythefactthatthetruthoftheirpremisesensuresthetruthoftheirconclusion.Thismeansthatitisimpossibleforthepremisestobetrueandtheconclusiontobefalse.[14][8]Thegenerallogicalstructurescharacterizingvalidinferencesarecalledrulesofinference.[6]Inthissense,logicisoftendefinedasthestudyofvalidinference.[5]Thiscontrastswithanotherprominentcharacterizationoflogicasthescienceoflogicaltruths.[6]Apropositionislogicallytrueifitstruthdependsonlyonthelogicalvocabularyusedinit.Thismeansthatitistrueinallpossibleworldsandunderallinterpretationsofitsnon-logicalterms.[15]Thesetwocharacterizationsoflogicarecloselyrelatedtoeachother:aninferenceisvalidifthematerialconditionalfromitspremisestoitsconclusionislogicallytrue.[6] Theterm"logic"canalsobeusedinaslightlydifferentsenseasacountablenoun.Inthissense,alogicisalogicalformalsystem.Differentlogicsdifferfromeachotherconcerningtheformallanguagesusedtoexpressthemand,mostimportantly,concerningtherulesofinferencetheyacceptasvalid.[16][6][5]Startinginthe20thcentury,manynewformalsystemshavebeenproposed.Thereisanongoingdebateaboutwhichofthesesystemsshouldbeconsideredlogicsinthestrictsenseinsteadofnon-logicalformalsystems.[16][5]Suggestedcriteriaforthisdistinctionincludelogicalcompletenessandproximitytotheintuitionsgoverningclassicallogic.Accordingtothesecriteria,ithasbeenargued,forexample,thathigher-orderlogicsandfuzzylogicshouldnotbeconsideredlogicswhenunderstoodinastrictsense.[5][6] Formalandinformallogic[edit] Whenunderstoodinthewidestsense,logicencompassesbothformalandinformallogic.[5]Formallogicisthetraditionallydominantfield.[14]Variousproblemsinapplyingitsinsightstoactualeverydayargumentshavepromptedmoderndevelopmentsofinformallogic.[17][18][19]Theyoftenstressitssignificanceforvariouspracticalpurposeswhichformallogiconitsownisunabletoaddress.[14][19]Bothhaveincommonthattheyaimtoprovidecriteriaforassessingthecorrectnessofargumentsanddistinguishingthemfromfallacies.[14][9]Varioussuggestionshavebeenmadeconcerninghowtodrawthedistinctionbetweenthetwobutthereisnouniversallyacceptedanswer.Thesedifficultiesoftencoincidewiththewidedisagreementsabouthowinformallogicistobedefined.[20][19] Themostliteralapproachseestheterms"formal"and"informal"asapplyingtothelanguageusedtoexpressarguments.[21][17][18][19]Onthisview,formallogicstudiesargumentsexpressedinformallanguageswhileinformallogicstudiesargumentsexpressedininformalornaturallanguages.[14]Thismeansthattheinferencefromtheformulas" P {\displaystyleP} "and" Q {\displaystyleQ} "totheconclusion" P ∧ Q {\displaystyleP\landQ} "isstudiedbyformallogicwhiletheinferencefromtheEnglishsentences"Allitacigarette"and"Billstormedoutoftheroom"tothesentence"AllitacigaretteandBillstormedoutoftheroom"belongstoinformallogic.Formallanguagesarecharacterizedbytheirprecisionandsimplicity.[21]Theynormallycontainaverylimitedvocabularyandexactrulesonhowtheirsymbolscanbeusedtoconstructsentences,usuallyreferredtoaswell-formedformulas.[22]Thissimplicityandexactnessinturnmakeitpossibleforformallogictoformulatepreciserulesofinferencethatdeterminewhetheragivenargumentisvalid.[22]Thisapproachbringswithittheneedtotranslatenaturallanguageargumentsintotheformallanguagebeforetheirvaliditycanbeassessed,aprocedurethatcomeswithvariousproblemsofitsown.[6][12][19]Informallogicavoidssomeoftheseproblemsbyanalyzingnaturallanguageargumentsintheiroriginalformwithouttheneedoftranslation.[17][9]Butitfacesrelatedproblemsofitsown,associatedwiththeambiguity,vagueness,andcontext-dependenceofnaturallanguageexpressions.[23][24][14][19]Acloselyrelatedapproachappliestheterms"formal"and"informal"notjusttothelanguageused,butmoregenerallytothestandards,criteria,andproceduresofargumentation.[25] Anotherapproachdrawsthedistinctionaccordingtothedifferenttypesofinferencesanalyzed.[26][17]Thisperspectiveunderstandsformallogicasthestudyofdeductiveinferencesincontrasttoinformallogicasthestudyofnon-deductiveinferences,likeinductiveorabductiveinferences.[17]Thecharacteristicofdeductiveinferencesisthatthetruthoftheirpremisesensuresthetruthoftheirconclusion.Thismeansthatifallthepremisesaretrue,itisimpossiblefortheconclusiontobefalse.[14][8]Forthisreason,deductiveinferencesareinasensetrivialoruninterestingsincetheydonotprovidethethinkerwithanynewinformationnotalreadyfoundinthepremises.[6][27]Non-deductiveinferences,ontheotherhand,areampliative:theyhelpthethinkerlearnsomethingaboveandbeyondwhatisalreadystatedinthepremises.Theyachievethisatthecostofcertainty:evenifallpremisesaretrue,theconclusionofanampliativeargumentmaystillbefalse.[6][28][29] Onemoreapproachtriestolinkthedifferencebetweenformalandinformallogictothedistinctionbetweenformalandinformalfallacies.[25][19][17][30]Thisdistinctionisoftendrawninrelationtotheform,content,andcontextofarguments.Inthecaseofformalfallacies,theerrorisfoundontheleveloftheargument'sform,whereasforinformalfallacies,thecontentandcontextoftheargumentareresponsible.[31][32][33][34]Thisisconnectedtotheideathatformallogicabstractsawayfromtheargument'scontentandisonlyinterestedinitsform,specificallywhetheritfollowsavalidruleofinference.[10][11]Italsoconcernstheideathatit'snotimportantforthevalidityofaformalargumentwhetheritspremisesaretrueorfalse.Informallogic,ontheotherhand,alsotakesthecontentandcontextofanargumentintoconsideration.[14][19][21]Afalsedilemma,forexample,involvesanerrorofcontentbyexcludingviableoptions,asin"youareeitherwithusoragainstus;youarenotwithus;therefore,youareagainstus".[32][35]Forthestrawmanfallacy,ontheotherhand,theerrorisfoundonthelevelofcontext:aweakpositionisfirstdescribedandthendefeated,eventhoughtheopponentdoesnotholdthisposition.Butinanothercontext,againstanopponentthatactuallydefendsthestrawmanposition,theargumentiscorrect.[32][23] Otheraccountsdrawthedistinctionbasedoninvestigatinggeneralformsofargumentsincontrasttoparticularinstances,onthestudyoflogicalconstantsinsteadofsubstantiveconcepts,onthediscussionoflogicaltopicswithorwithoutformaldevices,orontheroleofepistemologyfortheassessmentofarguments.[14][19] Fundamentalconcepts[edit] Premises,conclusions,andtruth[edit] Premisesandconclusions[edit] Furtherinformation:premise Premisesandconclusionsarethebasicpartsofinferencesorargumentsandthereforeplayacentralroleinlogic.Inthecaseofavalidinferenceoracorrectargument,theconclusionfollowsfromthepremisesorthepremisessupporttheconclusion.[36][7]Forinstance,thepremises"Marsisred"and"Marsisaplanet"supporttheconclusion"Marsisaredplanet".Itisgenerallyacceptedthatpremisesandconclusionshavetobetruth-bearers.[36][7][i]Thismeansthattheyhaveatruthvalue,thattheyareeithertrueorfalse.Thuscontemporaryphilosophygenerallyseesthemeitheraspropositionsorassentences.[7]Propositionsarethedenotationsofsentencesandareusuallyunderstoodasabstractobjects.[37] Propositionaltheoriesofpremisesandconclusionsareoftencriticizedbecauseofthedifficultiesinvolvedinspecifyingtheidentitycriteriaofabstractobjectsorbecauseofnaturalistconsiderations.[7]Theseobjectionsareavoidedbyseeingpremisesandconclusionsnotaspropositionsbutassentences,i.e.asconcretelinguisticobjectslikethesymbolsdisplayedonthereader'scomputerscreen.Butthisapproachcomeswithnewproblemsofitsown:sentencesareoftencontext-dependentandambiguous,meaningthatwhetheranargumentisvalidwouldnotonlydependonitspartsbutalsoonitscontextandonhowitisinterpreted.[7][38] Inearlierwork,premisesandconclusionswereunderstoodinpsychologicaltermsasthoughtsorjudgments,anapproachknownas"psychologism".Thispositionwasheavilycriticizedaroundtheturnofthe20thcentury.[7][39][40] Internalstructure[edit] Acentralaspectofpremisesandconclusionsforlogic,independentofhowtheirnatureisconceived,concernstheirinternalstructure.Aspropositionsorsentences,theycanbeeithersimpleorcomplex.[41][42]Acomplexpropositionhasotherpropositionsasitsconstituents,whicharelinkedtoeachotherthroughpropositionalconnectiveslike"and"or"if-then".Simplepropositions,ontheotherhand,donothavepropositionalparts.Buttheycanalsobeconceivedashavinganinternalstructure:theyaremadeupofsubpropositionalparts,likesingulartermsandpredicates.[36][41][42]Forexample,thesimpleproposition"Marsisred"canbeformedbyapplyingthepredicate"red"tothesingularterm"Mars".[36]Incontrast,thecomplexproposition"MarsisredandVenusiswhite"ismadeupoftwosimplepropositionsconnectedbythepropositionalconnective"and".[36] Whetherapropositionistruedepends,atleastinpart,onitsconstituents.[42]Forcomplexpropositionsformedusingtruth-functionalpropositionalconnectives,theirtruthonlydependsonthetruth-valuesoftheirparts.[36]Butthisrelationismorecomplicatedinthecaseofsimplepropositionsandtheirsubpropositionalparts.Thesesubpropositionalpartshavemeaningsoftheirown,likereferringtoobjectsorclassesofobjects.[36][43][44][42]Whetherthesimplepropositiontheyformistruedependsontheirrelationtoreality,i.e.whattheobjectstheyrefertoarelike.Thistopicisstudiedbytheoriesofreference.[44] Logicaltruth[edit] Furtherinformation:logicaltruth Insomecases,asimpleoracomplexpropositionistrueindependentlyofthesubstantivemeaningsofitsparts.[13][4]Forexample,thecomplexproposition"ifMarsisred,thenMarsisred"istrueindependentofwhetheritsparts,i.e.thesimpleproposition"Marsisred",aretrueorfalse.Insuchcases,thetruthiscalledalogicaltruth:apropositionislogicallytrueifitstruthdependsonlyonthelogicalvocabularyusedinit.[13][15]Thismeansthatitistrueunderallinterpretationsofitsnon-logicalterms.Insomemodallogics,thisnotioncanbeunderstoodequivalentlyastruthatallpossibleworlds.[15]Logicaltruthplaysanimportantroleinlogicandsometheoristsevendefinelogicasthestudyoflogicaltruths.[6] Argumentsandinferences[edit] Furtherinformation:argumentandinference Logiciscommonlydefinedintermsofargumentsorinferencesasthestudyoftheircorrectness.[4][7]Anargumentisasetofpremisestogetherwithaconclusion.[45]Aninferenceistheprocessofreasoningfromthesepremisestotheconclusion.[7]Butthesetermsareoftenusedinterchangeablyinlogic.Sometimesadistinctionismadebetweensimpleandcomplexarguments.[7]Acomplexargumentismadeupofachainofsimplearguments.Thesesimpleargumentsconstituteachainbecausetheconclusionsoftheearlierargumentsareusedaspremisesinthelaterarguments.Foracomplexargumenttobesuccessful,eachlinkofthechainhastobesuccessful.[7] Argumentterminologyusedinlogic Acentralaspectofargumentsandinferencesisthattheyarecorrectorincorrect.Iftheyarecorrectthentheirpremisessupporttheirconclusion.Intheincorrectcase,thissupportismissing.Itcantakedifferentformscorrespondingtothedifferenttypesofreasoning.[6][28][46]Thestrongestformofsupportcorrespondstodeductivereasoning.Butevenargumentsthatarenotdeductivelyvalidmaystillconstitutegoodargumentsbecausetheirpremisesoffernon-deductivesupporttotheirconclusions.Forsuchcases,thetermampliativeorinductivereasoningisused.[6][28][46]Deductiveargumentsareassociatedwithformallogicincontrasttotherelationbetweenampliativeargumentsandinformallogic.[26][17] Deductive[edit] Adeductivelyvalidargumentisonewhosepremisesguaranteethetruthofitsconclusion.[14][8]Forinstance,theargument"Victoriaistall;Victoriahasbrownhair;thereforeVictoriaistallandhasbrownhair"isdeductivelyvalid.AlfredTarskiholdsthatdeductiveargumentshavethreeessentialfeatures:(1)theyareformal,i.e.theydependonlyontheformofthepremisesandtheconclusion;(2)theyareapriori,i.e.nosenseexperienceisneededtodeterminewhethertheyobtain;(3)theyaremodal,i.e.thattheyholdbylogicalnecessityforthegivenpropositions,independentofanyothercircumstances.[8] Becauseofthefirstfeature,thefocusonformality,deductiveinferenceisusuallyidentifiedwithrulesofinference.[6]Rulesofinferencespecifyhowthepremisesandtheconclusionhavetobestructuredfortheinferencetobevalid.Argumentsthatdonotfollowanyruleofinferencearedeductivelyinvalid.[6][47]Themodusponensisaprominentruleofinference.Ithastheform"ifA,thenB;A;thereforeB".[47] Thethirdfeaturecanbeexpressedbystatingthatdeductivelyvalidinferencesaretruth-preserving:itisimpossibleforthepremisestobetrueandtheconclusiontobefalse.[48][6][28][46]Becauseofthisfeature,itisoftenassertedthatdeductiveinferencesareuninformativesincetheconclusioncannotarriveatnewinformationnotalreadypresentinthepremises.[6][27]Butthispointisnotalwaysacceptedsinceitwouldmean,forexample,thatmostofmathematicsisuninformative.Adifferentcharacterizationdistinguishesbetweensurfaceanddepthinformation.[6][49][50]Onthisview,deductiveinferencesareuninformativeonthedepthlevelbutcanbehighlyinformativeonthesurfacelevel,asmaybethecaseforvariousmathematicalproofs.[6][49][50] Ampliative[edit] Ampliativeinferences,ontheotherhand,areinformativeevenonthedepthlevel.Theyaremoreinterestinginthissensesincethethinkermayacquiresubstantiveinformationfromthemandtherebylearnsomethinggenuinelynew.[6][28][29]Butthisfeaturecomeswithacertaincost:thepremisessupporttheconclusioninthesensethattheymakeitstruthmorelikelybuttheydonotensureitstruth.[6][28][29]Thismeansthattheconclusionofanampliativeargumentmaybefalseeventhoughallitspremisesaretrue.Thischaracteristiciscloselyrelatedtonon-monotonicityanddefeasibility:itmaybenecessarytoretractanearlierconclusionuponreceivingnewinformationorinthelightofnewinferencesdrawn.[4][48][6][28][46]Ampliativereasoningisofcentralimportancesincealotoftheargumentsfoundineverydaydiscourseandthesciencesareampliative.Ampliativeargumentsarenotautomaticallyincorrect.Instead,theyjustfollowdifferentstandardsofcorrectness.Animportantaspectofmostampliativeargumentsisthatthesupporttheyprovidefortheirconclusioncomesindegrees.[46][51][48]Inthissense,thelinebetweencorrectandincorrectargumentsisblurryinsomecases,aswhenthepremisesofferweakbutnon-negligiblesupport.Thiscontrastswithdeductivearguments,whichareeithervalidorinvalidwithnothingin-between.[46][51][48] Theterminologyusedtocategorizeampliativeargumentsisinconsistent.Someauthorsusetheterm"induction"tocoverallformsofnon-deductivearguments.[46][52][51][53]Butinamorenarrowsense,inductionisonlyonetypeofampliativeargumentbesidesabductivearguments.[48]Someauthorsalsoallowconductiveargumentsasonemoretype.[17][54]Inthisnarrowsense,inductionisoftendefinedasaformofstatisticalgeneralization.[55][56]Inthiscase,thepremisesofaninductiveargumentaremanyindividualobservationsthatallshowacertainpattern.Theconclusionthenisagenerallawthatthispatternalwaysobtains.[53]Inthissense,onemayinferthat"allelephantsaregray"basedonone'spastobservationsofthecolorofelephants.[48]Acloselyrelatedformofinductiveinferencehasasitsconclusionnotagenerallawbutonemorespecificinstance,aswhenitisinferredthatanelephantonehasnotseenyetisalsogray.[53]Sometheoristsstipulatethatinductiveinferencesrestonlyonstatisticalconsiderationsinordertodistinguishthemfromabductiveinference.[48] Abductiveinferencemayormaynottakestatisticalobservationsintoconsideration.Ineithercase,thepremisesoffersupportfortheconclusionbecausetheconclusionisthebestexplanationofwhythepremisesobtain.[48][57][ii]Inthissense,abductionisalsocalledtheinferencetothebestexplanation.[58]Forexample,giventhepremisethatthereisaplatewithbreadcrumbsinthekitchenintheearlymorning,onemayinfertheconclusionthatone'shouse-matehadamidnightsnackandwastootiredtocleanthetable.Thisconclusionisjustifiedbecauseitisthebestexplanationofthecurrentstateofthekitchen.[48]Forabduction,itisnotsufficientthattheconclusionexplainsthepremises.Forexample,theconclusionthataburglarbrokeintothehouselastnight,gothungryonthejob,andhadamidnightsnack,wouldalsoexplainthestateofthekitchen.Butthisconclusionisnotjustifiedbecauseitisnotthebestormostlikelyexplanation.[48][57][58] Fallacies[edit] Notallargumentsliveuptothestandardsofcorrectreasoning.Whentheydonot,theyareusuallyreferredtoasfallacies.Theircentralaspectisnotthattheirconclusionisfalsebutthatthereissomeflawwiththereasoningleadingtothisconclusion.[59]Sotheargument"itissunnytoday;thereforespidershaveeightlegs"isfallaciouseventhoughtheconclusionistrue.Sometheoristsgiveamorerestrictivedefinitionoffallaciesbyadditionallyrequiringthattheyappeartobecorrect.[30][23]Thisway,genuinefallaciescanbedistinguishedfrommeremistakesofreasoningduetocarelessness.Thisexplainswhypeopletendtocommitfallacies:becausetheyhaveanalluringelementthatseducespeopleintocommittingandacceptingthem.[30]However,thisreferencetoappearancesiscontroversialbecauseitbelongstothefieldofpsychology,notlogic,andbecauseappearancesmaybedifferentfordifferentpeople.[60][30] Fallaciesareusuallydividedintoformalandinformalfallacies.[31][32][33]Forformalfallacies,thesourceoftheerrorisfoundintheformoftheargument.Forexample,denyingtheantecedentisonetypeofformalfallacy,asin"ifOthelloisabachelor,thenheismale;Othelloisnotabachelor;thereforeOthelloisnotmale".[59][61]Butmostfallaciesfallintothecategoryofinformalfallacies,ofwhichagreatvarietyisdiscussedintheacademicliterature.Thesourceoftheirerrorisusuallyfoundinthecontentorthecontextoftheargument.[32][30][23]Informalfallaciesaresometimescategorizedasfallaciesofambiguity,fallaciesofpresumption,orfallaciesofrelevance.Forfallaciesofambiguity,theambiguityandvaguenessofnaturallanguageareresponsiblefortheirflaw,asin"feathersarelight;whatislightcannotbedark;thereforefeatherscannotbedark".[24][35][33][23]Fallaciesofpresumptionhaveawrongorunjustifiedpremisebutmaybevalidotherwise.[62][33]Inthecaseoffallaciesofrelevance,thepremisesdonotsupporttheconclusionbecausetheyarenotrelevanttoit.[35][33] Definitoryandstrategicrules[edit] Themainfocusofmostlogiciansistoinvestigatethecriteriaaccordingtowhichanargumentiscorrectorincorrect.Afallacyiscommittedifthesecriteriaareviolated.Inthecaseofformallogic,theyareknownasrulesofinference.[6]Theyconstitutedefinitoryrules,whichdeterminewhetheracertainlogicalmoveiscorrectorwhichmovesareallowed.Definitoryrulescontrastwithstrategicrules.[6][63][64]Strategicrulesspecifywhichinferentialmovesarenecessaryinordertoreachagivenconclusionbasedonacertainsetofpremises.[6][63][64]Thisdistinctiondoesnotjustapplytologicbutalsotovariousgamesaswell.Inchess,forexample,thedefinitoryrulesdictatethatbishopsmayonlymovediagonallywhilethestrategicrulesdescribehowtheallowedmovesmaybeusedtowinagame,forexample,bycontrollingthecenterandbydefendingone'sking.[6][63][64]Athirdtypeofrulesconcernsempiricaldescriptiverules.Theybelongtothefieldofpsychologyandgeneralizehowpeopleactuallydrawinferences.[6]Ithasbeenarguedthatlogiciansshouldgivemoreemphasistostrategicrulessincetheyarehighlyrelevantforeffectivereasoning.[6] Formalsystems[edit] Mainarticle:Formalsystem Aformalsystemoflogicconsistsofalanguage,aproofsystem,andasemantics.[65]Asystem'slanguageandproofsystemaresometimesgroupedtogetherasthesystem'ssyntax,sincetheybothconcerntheformratherthanthecontentofthesystem'sexpressions. Theterm"alogic"isoftenusedacountablenountorefertoaparticularformalsystemoflogic.Differentlogicscandifferfromeachotherintheirlanguage,proofsystem,ortheirsemantics.[16][6][5]Startinginthe20thcentury,manynewformalsystemshavebeenproposed.[iii][16][5][6] Formallanguage[edit] Mainarticle:Formallanguage Alanguageisasetofwellformedformulas.Forinstance,inpropositionallogic, P & Q {\displaystyleP\&Q} isaformulabut P Q & & & {\displaystylePQ\&\&\&} isnot.Languagesaretypicallydefinedbyprovidinganalphabetofbasicexpressionsandrecursivesyntacticruleswhichbuildthemintoformulas.[66][67][68][65] Proofsystem[edit] Mainarticle:Proofsystem Aproofsystemisacollectionofformalruleswhichdefinewhenaconclusionfollowsfromgivenpremises.Forinstance,theclassicalruleofconjunctionintroductionstatesthat P & Q {\displaystyleP\&Q} followsfromthepremises P {\displaystyleP} and Q {\displaystyleQ} .Rulesinaproofsystemsarealwaysdefinedintermsofformulas'syntacticform,neverintermsoftheirmeanings.Suchrulescanbeappliedsequentially,givingamechanicalprocedureforgeneratingconclusionsfrompremises.Thereareanumberofdifferenttypesofproofsystemsincludingnaturaldeductionandsequentcalculi.[69][70][65]Proofsystemsarecloselylinkedtophilosophicalworkwhichcharacterizeslogicasthestudyofvalidinference.[5] Semantics[edit] Mainarticle:Semanticsoflogic Asemanticsisasystemformappingexpressionsofaformallanguagetotheirdenotations.Inmanysystemsoflogic,denotationsaretruthvalues.Forinstance,thesemanticsforclassicalpropositionallogicassignstheformula P & Q {\displaystyleP\&Q} thedenotation"true"whenever P {\displaystyleP} istrueand Q {\displaystyleQ} istoo.Entailmentisasemanticrelationwhichholdsbetweenformulaswhenthefirstcannotbetruewithoutthesecondbeingtrueaswell.[71][72]Semanticsiscloselytiedtothephilosophicalcharacterizationoflogicasthestudyoflogicaltruth.[6] Soundnessandcompleteness[edit] Mainarticles:Soundness,completeness(logic),andmetalogic Asystemoflogicissoundwhenitsproofsystemcannotderiveaconclusionfromasetofpremisesunlessitissemanticallyentailedbythem.Inotherwords,itsproofsystemcannotleadtofalseconclusions,asdefinedbythesemantics.Asystemiscompletewhenitsproofsystemcanderiveeveryconclusionthatissemanticallyentailedbyitspremises.Inotherwords,itsproofsystemcanleadtoanytrueconclusion,asdefinedbythesemantics.Thus,soundnessandcompletenesstogetherdescribeasystemwhosenotionsofvalidityandentailmentlineupperfectly.[73][74][75] Thestudyofpropertiesofformalsystemsiscalledmetalogic.Otherimportantmetalogicalpropertiesincludeconsistency,decidability,andexpressivepower. Systemsoflogic[edit] Systemsoflogicaretheoreticalframeworksforassessingthecorrectnessofreasoningandarguments.Forovertwothousandyears,Aristotelianlogicwastreatedasthecannonoflogic.[16][76][77]Butmoderndevelopmentsinthisfieldhaveledtoavastproliferationoflogicalsystems.[78]Oneprominentcategorizationdividesmodernformallogicalsystemsintoclassicallogic,extendedlogics,anddeviantlogics.[5][78][79]ClassicallogicistobedistinguishedfromtraditionalorAristotelianlogic.Itencompassespropositionallogicandfirst-orderlogic.[6][4]Itis"classical"inthesensethatitisbasedonvariousfundamentallogicalintuitionssharedbymostlogicians.[6][4]Theseintuitionsincludethelawofexcludedmiddle,thedoublenegationelimination,theprincipleofexplosion,andthebivalenceoftruth.[80]Itwasoriginallydevelopedtoanalyzemathematicalargumentsandwasonlylaterappliedtootherfieldsaswell.Becauseofthisfocusonmathematics,itdoesnotincludelogicalvocabularyrelevanttomanyothertopicsofphilosophicalimportance,likethedistinctionbetweennecessityandpossibility,theproblemofethicalobligationandpermission,ortherelationsbetweenpast,present,andfuture.[81]Suchissuesareaddressedbyextendedlogics.Theybuildonthefundamentalintuitionsofclassicallogicandexpanditbyintroducingnewlogicalvocabulary.Thisway,theexactlogicalapproachisappliedtofieldslikeethicsorepistemologythatliebeyondthescopeofmathematics.[82][16][83] Deviantlogics,ontheotherhand,rejectsomeofthefundamentalintuitionsofclassicallogic.[78][79]Becauseofthis,theyareusuallyseennotasitssupplementsbutasitsrivals.Deviantlogicalsystemsdifferfromeachothereitherbecausetheyrejectdifferentclassicalintuitionsorbecausetheyproposedifferentalternativestothesameissue.[78][79] Informallogicisusuallydoneinalesssystematicway.Itoftenfocusesonmorespecificissues,likeinvestigatingaparticulartypeoffallacyorstudyingacertainaspectofargumentation.[17]Nonetheless,somesystemsofinformallogichavealsobeenpresentedthattrytoprovideasystematiccharacterizationofthecorrectnessofarguments.[34][30][84][23] Aristotelian[edit] Whenunderstoodinthewidestsense,Aristotelianlogicencompassesagreatvarietyoftopics,includingmetaphysicalthesesaboutontologicalcategoriesandproblemsofscientificexplanation.[76][77][85]Butinamorenarrowsense,itreferstotermlogicorsyllogistics.Asyllogismisacertainformofargumentinvolvingthreepropositions:twopremisesandaconclusion.Eachpropositionhasthreeessentialparts:asubject,apredicate,andacopulaconnectingthesubjecttothepredicate.[76][77][85][86]Forexample,theproposition"Socratesiswise"ismadeupofthesubject"Socrates",thepredicate"wise",andthecopula"is".[77]Thesubjectandthepredicatearethetermsoftheproposition.Inthissense,Aristotelianlogicdoesnotcontaincomplexpropositionsmadeupofvarioussimplepropositions.[76]Itdiffersinthisaspectfrompropositionallogic,inwhichanytwopropositionscanbelinkedusingalogicalconnectivelike"and"toformanewcomplexproposition.[87] Aristotelianlogicdiffersfrompredicatelogicinthatthesubjectiseitheruniversal,particular,indefinite,orsingular.[76][85][86]Forexample,theterm"allhumans"isauniversalsubjectintheproposition"allhumansaremortal".Asimilarpropositioncouldbeformedbyreplacingitwiththeparticularterm"somehumans",theindefiniteterm"ahuman",orthesingularterm"Socrates".[85][86][76]Inpredicatelogic,ontheotherhand,universalandparticularpropositionswouldbeexpressedbyusingaquantifierandtwopredicates.[76]AnotherimportantdifferenceisthatAristotelianlogiconlyincludespredicatesforsimplepropertiesofentities,butlackspredicatescorrespondingtorelationsbetweenentities.[88]Thepredicatecanbelinkedtothesubjectintwoways:eitherbyaffirmingitorbydenyingit.[76][77]Forexample,theproposition"Socratesisnotacat"involvesthedenialofthepredicate"cat"tothesubject"Socrates".Usingdifferentcombinationsofsubjectsandpredicates,agreatvarietyofpropositionsandsyllogismscanbeformed.Syllogismsarecharacterizedbythefactthatthepremisesarelinkedtoeachotherandtotheconclusionbysharingonepredicateineachcase.[76][89][90]Thus,thesethreepropositionscontainthreepredicates,referredtoasmajorterm,minorterm,andmiddleterm.[89][90][77][85]ThecentralaspectofAristotelianlogicinvolvesclassifyingallpossiblesyllogismsintovalidandinvalidargumentsaccordingtohowthepropositionsareformed.[76][77][89][90]Forexample,thesyllogism"allmenaremortal;Socratesisaman;thereforeSocratesismortal"isvalid.Thesyllogism"allcatsaremortal;Socratesismortal;thereforeSocratesisacat",ontheotherhand,isinvalid. Classical[edit] Propositionallogic[edit] Mainarticle:Propositionalcalculus Propositionallogiccomprisesformalsystemsinwhichformulaearebuiltfromatomicpropositionsusinglogicalconnectives.Forinstance,propositionallogicrepresentstheconjunctionoftwoatomicpropositions P {\displaystyleP} and Q {\displaystyleQ} asthecomplexformula P & Q {\displaystyleP\&Q} .Unlikepredicatelogicwheretermsandpredicatesarethesmallestunits,propositionallogictakesfullpropositionswithtruthvaluesasitsmostbasiccomponent.[91]Thus,propositionallogicscanonlyrepresentlogicalrelationshipsthatarisefromthewaycomplexpropositionsarebuiltfromsimplerones;itcannotrepresentinferencesthatresultsfromtheinnerstructureofaproposition. First-orderlogic[edit] GottlobFrege'sBegriffschriftintroducedthenotionofquantifierinagraphicalnotation,whichhererepresentsthejudgementthat ∀ x . F ( x ) {\displaystyle\forallx.F(x)} istrue. Mainarticle:First-orderlogic First-orderlogicprovidesanaccountofquantifiersgeneralenoughtoexpressawidesetofargumentsoccurringinnaturallanguage.Forexample,BertrandRussell'sfamousbarberparadox,"thereisamanwhoshavesallandonlymenwhodonotshavethemselves"canbeformalisedbythesentence ( ∃ x ) ( man ( x ) ∧ ( ∀ y ) ( man ( y ) → ( shaves ( x , y ) ↔ ¬ shaves ( y , y ) ) ) ) {\displaystyle(\existsx)({\text{man}}(x)\wedge(\forally)({\text{man}}(y)\rightarrow({\text{shaves}}(x,y)\leftrightarrow\neg{\text{shaves}}(y,y))))} ,usingthenon-logicalpredicate man ( x ) {\displaystyle{\text{man}}(x)} toindicatethatxisaman,andthenon-logicalrelation shaves ( x , y ) {\displaystyle{\text{shaves}}(x,y)} toindicatethatxshavesy;allothersymbolsoftheformulaearelogical,expressingtheuniversalandexistentialquantifiers,conjunction,implication,negationandbiconditional. Thedevelopmentoffirst-orderlogicisusuallyattributedtoGottlobFrege,whoisalsocreditedasoneofthefoundersofanalyticphilosophy,buttheformulationoffirst-orderlogicmostoftenusedtodayisfoundinPrinciplesofMathematicalLogicbyDavidHilbertandWilhelmAckermannin1928.Theanalyticalgeneralityoffirst-orderlogicallowedtheformalizationofmathematics,drovetheinvestigationofsettheory,andallowedthedevelopmentofAlfredTarski'sapproachtomodeltheory.Itprovidesthefoundationofmodernmathematicallogic. Extended[edit] Modallogic[edit] Furtherinformation:Modallogic Manyextendedlogicstaketheformofmodallogicbyintroducingmodaloperators.Modallogicwereoriginallydevelopedtorepresentstatementsaboutnecessityandpossibility.Forinstancethemodalformula ◊ P {\displaystyle\DiamondP} canbereadas"possibly P {\displaystyleP} "while ◻ P {\displaystyle\BoxP} canbereadas"necessarily P {\displaystyleP} ".Modallogicscanbeusedtorepresentdifferentphenomenadependingonwhatflavorofnecessityandpossibilityisunderconsideration.When ◻ {\displaystyle\Box} isusedtorepresentepistemicnecessity, ◻ P {\displaystyle\BoxP} statesthat P {\displaystyleP} isknown.When ◻ {\displaystyle\Box} isusedtorepresentdeonticnecessity, ◻ P {\displaystyle\BoxP} statesthat P {\displaystyleP} isamoralorlegalobligation.Withinphilosophy,modallogicsarewidelyusedinformalepistemology,formalethics,andmetaphysics.Withinlinguisticsemantics,systemsbasedonmodallogicareusedtoanalyzelinguisticmodalityinnaturallanguages.[92][93][94]Otherfieldssuchcomputerscienceandsettheoryhaveappliedtherelationalsemanticsformodallogicbeyonditsoriginalconceptualmotivation,usingittoprovideinsightintopatternsincludingtheset-theoreticmultiverseandtransitionsystemsincomputation.[94][95] Higherorderlogic[edit] Furtherinformation:Higher-orderlogic Higher-orderlogicsextendclassicallogicnotbyusingmodaloperatorsbutbyintroducingnewformsofquantification.[7][96][97][98]Quantifierscorrespondtotermslike"all"or"some".Inclassicalfirst-orderlogic,quantifiersareonlyappliedtoindividuals.Theformula" ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle\existsx(Apple(x)\landSweet(x))} "(someapplesaresweet)isanexampleoftheexistentialquantifier" ∃ {\displaystyle\exists} "appliedtotheindividualvariable" x {\displaystylex} ".Inhigher-orderlogics,quantificationisalsoallowedoverpredicates.Thisincreasesitsexpressivepower.Forexample,toexpresstheideathatMaryandJohnsharesomequalities,onecouldusetheformula" ∃ Q ( Q ( m a r y ) ∧ Q ( j o h n ) ) {\displaystyle\existsQ(Q(mary)\landQ(john))} ".Inthiscase,theexistentialquantifierisappliedtothepredicatevariable" Q {\displaystyleQ} ".[7][96][97][98]Theaddedexpressivepowerisespeciallyusefulformathematicssinceitallowsformoresuccinctformulationsofmathematicaltheories.[7]Butithasvariousdrawbacksinregardtoitsmeta-logicalpropertiesandontologicalimplications,whichiswhyfirst-orderlogicisstillmuchmorewidelyused.[7][97] Deviant[edit] Agreatvarietyofdeviantlogicshavebeenproposed.Onemajorparadigmisintuitionisticlogic,whichrejectsthelawoftheexcludedmiddle.IntuitionismwasdevelopedbytheDutchmathematiciansL.E.J.BrouwerandArendHeytingtounderpintheirconstructiveapproachtomathematics,inwhichtheexistenceofamathematicalobjectcanonlybeprovenbyconstructingit.IntuitionismwasfurtherpursuedbyGerhardGentzen,KurtGödel,MichaelDummett,amongothers.Intuitionisticlogicisofgreatinteresttocomputerscientists,asitisaconstructivelogicandseesmanyapplications,suchasextractingverifiedprogramsfromproofsandinfluencingthedesignofprogramminglanguagesthroughtheformulae-as-typescorrespondence.ItiscloselyrelatedtononclassicalsystemssuchasGödel–Dummettlogicandinquisitivelogic.[99][100][101][102] Multi-valuedlogicsdepartfromclassicalitybyrejectingtheprincipleofbivalencewhichrequiresallpropositionstobeeithertrueorfalse.Forinstance,JanŁukasiewiczandStephenColeKleenebothproposedternarylogicswhichhaveathirdtruthvaluerepresentingthatastatement'struthvalueisindeterminate.[103][104][105]Theselogicshaveseenapplicationsincludingtopresuppositioninlinguistics.Fuzzylogicsaremultivaluedlogicsthathaveaninfinitenumberof"degreesoftruth",representedbyarealnumberbetween0and1.[106] Informal[edit] Thepragmaticordialogicalapproachtoinformallogicseesargumentsasspeechactsandnotmerelyasasetofpremisestogetherwithaconclusion.[34][30][84][23]Asspeechacts,theyoccurinacertaincontext,likeadialogue,whichaffectsthestandardsofrightandwrongarguments.[23][84]AprominentversionbyDouglasN.Waltonunderstandsadialogueasagamebetweentwoplayers.[30]Theinitialpositionofeachplayerischaracterizedbythepropositionstowhichtheyarecommittedandtheconclusiontheyintendtoprove.Dialoguesaregamesofpersuasion:eachplayerhasthegoalofconvincingtheopponentoftheirownconclusion.[23]Thisisachievedbymakingarguments:argumentsarethemovesofthegame.[23][84]Theyaffecttowhichpropositionstheplayersarecommitted.Awinningmoveisasuccessfulargumentthattakestheopponent'scommitmentsaspremisesandshowshowone'sownconclusionfollowsfromthem.[23]Thisisusuallynotpossiblestraightaway.Forthisreason,itisnormallynecessarytoformulateasequenceofargumentsasintermediarysteps,eachofwhichbringstheopponentalittleclosertoone'sintendedconclusion.Besidesthesepositiveargumentsleadingoneclosertovictory,therearealsonegativeargumentspreventingtheopponent'svictorybydenyingtheirconclusion.[23]Whetheranargumentiscorrectdependsonwhetheritpromotestheprogressofthedialogue.Fallacies,ontheotherhand,areviolationsofthestandardsofproperargumentativerules.[60][30]Thesestandardsalsodependonthetypeofdialogue:inthecontextofscience,thedialoguerulesaredifferentthaninthecontextofnegotiation.[84] Theepistemicapproachtoinformallogic,ontheotherhand,focusesontheepistemicroleofarguments.[34][30]Itisbasedontheideathatargumentsaimtoincreaseourknowledge.Theyachievethisbylinkingjustifiedbeliefstobeliefsthatarenotyetjustified.[107]Correctargumentssucceedatexpandingknowledgewhilefallaciesareepistemicfailures:theydonotjustifythebeliefintheirconclusion.[34][30]Inthissense,logicalnormativityconsistsinepistemicsuccessorrationality.[107]Forexample,thefallacyofbeggingthequestionisafallacybecauseitfailstoprovideindependentjustificationforitsconclusion,eventhoughitisdeductivelyvalid.[107][35]TheBayesianapproachisoneexampleofanepistemicapproach.[30]CentraltoBayesianismisnotjustwhethertheagentbelievessomethingbutthedegreetowhichtheybelieveit,theso-calledcredence.Degreesofbeliefareunderstoodassubjectiveprobabilitiesinthebelievedproposition,i.e.ashowcertaintheagentisthatthepropositionistrue.[108][109][110]Onthisview,reasoningcanbeinterpretedasaprocessofchangingone'scredences,ofteninreactiontonewincominginformation.[30]Correctreasoning,andtheargumentsitisbasedon,followsthelawsofprobability,forexample,theprincipleofconditionalization.Badorirrationalreasoning,ontheotherhand,violatestheselaws.[109][111][34] Areasofresearch[edit] Logicisstudiedinvariousfields.Inmanycases,thisisdonebyapplyingitsformalmethodtospecifictopicsoutsideitsscope,liketoethicsorcomputerscience.[4][5]Inothercases,logicitselfismadethesubjectofresearchinanotherdiscipline.Thiscanhappenindiverseways,likebyinvestigatingthephilosophicalpresuppositionsoffundamentallogicalconcepts,byinterpretingandanalyzinglogicthroughmathematicalstructures,orbystudyingandcomparingabstractpropertiesofformallogicalsystems.[4][112][113][5] Philosophyoflogicandphilosophicallogic[edit] Mainarticles:PhilosophyoflogicandPhilosophicallogic Philosophyoflogicisthephilosophicaldisciplinestudyingthescopeandnatureoflogic.[7][4]Itinvestigatesmanypresuppositionsimplicitinlogic,likehowtodefineitsfundamentalconceptsorthemetaphysicalassumptionsassociatedwiththem.[16]Itisalsoconcernedwithhowtoclassifythedifferentlogicalsystemsandconsiderstheontologicalcommitmentstheyincur.[4]Philosophicallogicisoneimportantareawithinthephilosophyoflogic.Itstudiestheapplicationoflogicalmethodstophilosophicalproblemsinfieldslikemetaphysics,ethics,andepistemology.[81][16]Thisapplicationusuallyhappensintheformofextendedordeviantlogicalsystems.[83][6] Mathematicallogic[edit] Mainarticle:Mathematicallogic Mathematicallogicisthestudyoflogicwithinmathematics.Majorsubareasincludemodeltheory,prooftheory,settheory,andcomputabilitytheory.[114][115] Researchinmathematicallogiccommonlyaddressesthemathematicalpropertiesofformalsystemsoflogic.However,itcanalsoincludeattemptstouselogictoanalyzemathematicalreasoningortoestablishlogic-basedfoundationsofmathematics.[116]Thelatterwasamajorconcerninearly20thcenturymathematicallogic,whichpursuedtheprogramoflogicismpioneeredbyphilosopher-logicianssuchasGottlobFregeandBertrandRussell.Mathematicaltheoriesweresupposedtobelogicaltautologies,andtheprogrammewastoshowthisbymeansofareductionofmathematicstologic.[117]Thevariousattemptstocarrythisoutmetwithfailure,fromthecripplingofFrege'sprojectinhisGrundgesetzebyRussell'sparadox,tothedefeatofHilbert'sprogrambyGödel'sincompletenesstheorems. SettheoryoriginatedinthestudyoftheinfinitebyGeorgCantor,andithasbeenthesourceofmanyofthemostchallengingandimportantissuesinmathematicallogic,fromCantor'stheorem,throughthestatusoftheAxiomofChoiceandthequestionoftheindependenceofthecontinuumhypothesis,tothemoderndebateonlargecardinalaxioms. Recursiontheorycapturestheideaofcomputationinlogicalandarithmeticterms;itsmostclassicalachievementsaretheundecidabilityoftheEntscheidungsproblembyAlanTuring,andhispresentationoftheChurch–Turingthesis.[118]Todayrecursiontheoryismostlyconcernedwiththemorerefinedproblemofcomplexityclasses—whenisaproblemefficientlysolvable?—andtheclassificationofdegreesofunsolvability.[119] Computationallogic[edit] Mainarticles:ComputationallogicandLogicincomputerscience Asimpletogglingcircuitisexpressedusingalogicgateandasynchronousregister. Incomputerscience,logicisstudiedaspartofthetheoryofcomputation.Keyareasoflogicthatarerelevanttocomputingincludecomputabilitytheory,modallogic,andcategorytheory.Earlycomputermachinerywasbasedonideasfromlogicsuchasthelambdacalculus.[120][121][122][123][124][125]Computerscientistsalsoapplyconceptsfromlogictoproblemsincomputingandviceversa.Forinstance,modernartificialintelligencebuildsonlogicians'workinargumentationtheory,whileautomatedtheoremprovingcanassistlogiciansinfindingandcheckingproofs.InlogicprogramminglanguagessuchasProlog,aprogramcomputestheconsequencesoflogicalaxiomsandrulestoansweraquery. Formalsemanticsofnaturallanguage[edit] Mainarticle:Formalsemantics(naturallanguage) Formalsemanticsisasubfieldofbothlinguisticsandphilosophywhichuseslogictoanalyzemeaninginnaturallanguage.Itisanempiricalfieldwhichseekstocharacterizethedenotationsoflinguisticexpressionsandexplainhowthosedenotationsarecomposedfromthemeaningsoftheirparts.ThefieldwasdevelopedbyRichardMontagueandBarbaraParteeinthe1970s,andremainsanactiveareaofresearch.Centralquestionsincludescope,binding,andlinguisticmodality.[126][127][128][129] Controversies[edit] "IsLogicEmpirical?"[edit] Furtherinformation:IsLogicEmpirical? Whatistheepistemologicalstatusofthelawsoflogic?Whatsortofargumentisappropriateforcriticizingpurportedprinciplesoflogic?Inaninfluentialpaperentitled"IsLogicEmpirical?"[130]HilaryPutnam,buildingonasuggestionofW.V.Quine,arguedthatingeneralthefactsofpropositionallogichaveasimilarepistemologicalstatusasfactsaboutthephysicaluniverse,forexampleasthelawsofmechanicsorofgeneralrelativity,andinparticularthatwhatphysicistshavelearnedaboutquantummechanicsprovidesacompellingcaseforabandoningcertainfamiliarprinciplesofclassicallogic:ifwewanttoberealistsaboutthephysicalphenomenadescribedbyquantumtheory,thenweshouldabandontheprincipleofdistributivity,substitutingforclassicallogicthequantumlogicproposedbyGarrettBirkhoffandJohnvonNeumann.[131] AnotherpaperofthesamenamebyMichaelDummettarguesthatPutnam'sdesireforrealismmandatesthelawofdistributivity.[132]Distributivityoflogicisessentialfortherealist'sunderstandingofhowpropositionsaretrueoftheworldinjustthesamewayashehasarguedtheprincipleofbivalenceis.Inthisway,thequestion,"IsLogicEmpirical?"canbeseentoleadnaturallyintothefundamentalcontroversyinmetaphysicsonrealismversusanti-realism. Toleratingtheimpossible[edit] Mainarticle:Paraconsistentlogic GeorgWilhelmFriedrichHegelwasdeeplycriticalofanysimplifiednotionofthelawofnon-contradiction.ItwasbasedonGottfriedWilhelmLeibniz'sideathatthislawoflogicalsorequiresasufficientgroundtospecifyfromwhatpointofview(ortime)onesaysthatsomethingcannotcontradictitself.Abuilding,forexample,bothmovesanddoesnotmove;thegroundforthefirstisoursolarsystemandforthesecondtheearth.InHegeliandialectic,thelawofnon-contradiction,ofidentity,itselfreliesupondifferenceandsoisnotindependentlyassertable. Closelyrelatedtoquestionsarisingfromtheparadoxesofimplicationcomesthesuggestionthatlogicoughttotolerateinconsistency.Relevancelogicandparaconsistentlogicarethemostimportantapproacheshere,thoughtheconcernsaredifferent:akeyconsequenceofclassicallogicandsomeofitsrivals,suchasintuitionisticlogic,isthattheyrespecttheprincipleofexplosion,whichmeansthatthelogiccollapsesifitiscapableofderivingacontradiction.GrahamPriest,themainproponentofdialetheism,hasarguedforparaconsistencyonthegroundsthatthereareinfact,truecontradictions.[133][clarificationneeded] Conceptionsoflogic[edit] Mainarticle:Conceptionsoflogic Logicarosefromaconcernwithcorrectnessofargumentation.Modernlogiciansusuallywishtoensurethatlogicstudiesjustthoseargumentsthatarisefromappropriatelygeneralformsofinference.Forexample,ThomasHofweberwritesintheStanfordEncyclopediaofPhilosophythatlogic"doesnot,however,covergoodreasoningasawhole.Thatisthejobofthetheoryofrationality.Ratheritdealswithinferenceswhosevaliditycanbetracedbacktotheformalfeaturesoftherepresentationsthatareinvolvedinthatinference,betheylinguistic,mental,orotherrepresentations."[134] Theideathatlogictreatsspecialformsofargument,deductiveargument,ratherthanargumentingeneral,hasahistoryinlogicthatdatesbackatleasttologicisminmathematics(19thand20thcenturies)andtheadventoftheinfluenceofmathematicallogiconphilosophy.Aconsequenceoftakinglogictotreatspecialkindsofargumentisthatitleadstoidentificationofspecialkindsoftruth,thelogicaltruths(withlogicequivalentlybeingthestudyoflogicaltruth),andexcludesmanyoftheoriginalobjectsofstudyoflogicthataretreatedasinformallogic.RobertBrandomhasarguedagainsttheideathatlogicisthestudyofaspecialkindoflogicaltruth,arguingthatinsteadonecantalkofthelogicofmaterialinference(intheterminologyofWilfredSellars),withlogicmakingexplicitthecommitmentsthatwereoriginallyimplicitininformalinference.[135][page needed] Rejectionoflogicaltruth[edit] Thephilosophicalveinofvariouskindsofskepticismcontainsmanykindsofdoubtandrejectionofthevariousbasesonwhichlogicrests,suchastheideaoflogicalform,correctinference,ormeaning,sometimesleadingtotheconclusionthattherearenologicaltruths.Thisisincontrastwiththeusualviewsinphilosophicalskepticism,wherelogicdirectsskepticalenquirytodoubtreceivedwisdoms,asintheworkofSextusEmpiricus. FriedrichNietzscheprovidesastrongexampleoftherejectionoftheusualbasisoflogic:hisradicalrejectionofidealizationledhimtorejecttruthasa"... mobilearmyofmetaphors,metonyms,andanthropomorphisms—inshort ...metaphorswhicharewornoutandwithoutsensuouspower;coinswhichhavelosttheirpicturesandnowmatteronlyasmetal,nolongerascoins".[136]Hisrejectionoftruthdidnotleadhimtorejecttheideaofeitherinferenceorlogiccompletelybutrathersuggestedthat"logic[came]intoexistenceinman'shead[out]ofillogic,whoserealmoriginallymusthavebeenimmense.Innumerablebeingswhomadeinferencesinawaydifferentfromoursperished".[137]Thusthereistheideathatlogicalinferencehasauseasatoolforhumansurvival,butthatitsexistencedoesnotsupporttheexistenceoftruth,nordoesithavearealitybeyondtheinstrumental:"Logic,too,alsorestsonassumptionsthatdonotcorrespondtoanythingintherealworld".[138] ThispositionheldbyNietzschehowever,hascomeunderextremescrutinyforseveralreasons.Somephilosophers,suchasJürgenHabermas,claimhispositionisself-refuting—andaccuseNietzscheofnotevenhavingacoherentperspective,letaloneatheoryofknowledge.[139]GeorgLukács,inhisbookTheDestructionofReason,assertsthat,"WerewetostudyNietzsche'sstatementsinthisareafromalogico-philosophicalangle,wewouldbeconfrontedbyadizzychaosofthemostluridassertions,arbitraryandviolentlyincompatible."[140]BertrandRusselldescribedNietzsche'sirrationalclaimswith"Heisfondofexpressinghimselfparadoxicallyandwithaviewtoshockingconventionalreaders"inhisbookAHistoryofWesternPhilosophy.[141] History[edit] Mainarticle:Historyoflogic Aristotle,384–322 BCE. Logicwasdevelopedindependentlyinseveralculturesduringantiquity.OnemajorearlycontributorwasAristotle,whodevelopedtermlogicinhisOrganonandPriorAnalytics.[142][143]Inthisapproach,judgementsarebrokendownintopropositionsconsistingoftwotermsthatarerelatedbyoneofafixednumberofrelation.Inferencesareexpressedbymeansofsyllogismsthatconsistoftwopropositionssharingacommontermaspremise,andaconclusionthatisapropositioninvolvingthetwounrelatedtermsfromthepremises.Aristotle'smonumentalinsightwasthenotionthatargumentscanbecharacterizedintermsoftheirform.ThelaterlogicianŁukasiewiczdescribedthisinsightas"oneofAristotle'sgreatestinventions".[143]Aristotle'ssystemoflogicwasalsoresponsiblefortheintroductionofhypotheticalsyllogism,[144]temporalmodallogic,[145][146]andinductivelogic,[147]aswellasinfluentialvocabularysuchasterms,predicables,syllogismsandpropositions.Aristotelianlogicwashighlyregardedinclassicalandmedievaltimes,bothinEuropeandtheMiddleEast.ItremainedinwideuseintheWestuntiltheearly19thcentury.[148]Ithasnowbeensupersededbylaterwork,thoughmanyofitskeyinsightsliveoninmodernsystemsoflogic. Adepictionfromthe15thcenturyofthesquareofopposition,whichexpressesthefundamentaldualitiesofsyllogistic. IbnSina(Avicenna)(980–1037CE)wasthefounderofAvicennianlogic,whichreplacedAristotelianlogicasthedominantsystemoflogicintheIslamicworld,[149]andalsohadanimportantinfluenceonWesternmedievalwriterssuchasAlbertusMagnus[150]andWilliamofOckham.[151][152]Avicennawroteonthehypotheticalsyllogism[153]andonthepropositionalcalculus.[154]Hedevelopedanoriginal"temporallymodalized"syllogistictheory,involvingtemporallogicandmodallogic.[155]Healsomadeuseofinductivelogic,suchasthemethodsofagreement,difference,andconcomitantvariationwhicharecriticaltothescientificmethod.[153]Fakhral-Dinal-Razi(b.1149)criticisedAristotle's"firstfigure"andformulatedanearlysystemofinductivelogic,foreshadowingthesystemofinductivelogicdevelopedbyJohnStuartMill(1806–1873).[156] InEuropeduringthelatermedievalperiod,majoreffortsweremadetoshowthatAristotle'sideaswerecompatiblewithChristianfaith.DuringtheHighMiddleAges,logicbecameamainfocusofphilosophers,whowouldengageincriticallogicalanalysesofphilosophicalarguments,oftenusingvariationsofthemethodologyofscholasticism.Initially,medievalChristianscholarsdrewontheclassicsthathadbeenpreservedinLatinthroughcommentariesbysuchfiguressuchasBoethius,latertheworkofIslamicphilosopherssuchasAvicennaandAverroesweredrawnon,whichexpandedtherangeofancientworksavailabletomedievalChristianscholarssincemoreGreekworkwasavailabletoMuslimscholarsthathadbeenpreservedinLatincommentaries.In1323,WilliamofOckham'sinfluentialSummaLogicaewasreleased.Bythe18thcentury,thestructuredapproachtoargumentshaddegeneratedandfallenoutoffavour,asdepictedinHolberg'ssatiricalplayErasmusMontanus. TheChineselogicalphilosopherGongsunLong(c.325–250BCE)proposedtheparadox"Oneandonecannotbecometwo,sinceneitherbecomestwo."[115][iv]InChina,thetraditionofscholarlyinvestigationintologic,however,wasrepressedbytheQindynastyfollowingthelegalistphilosophyofHanFeizi. InIndia,theAnviksikischooloflogicwasfoundedbyMedhātithi(c.6thcenturyBCE).[157]Innovationsinthescholasticschool,calledNyaya,continuedfromancienttimesintotheearly18thcenturywiththeNavya-Nyāyaschool.Bythe16thcentury,itdevelopedtheoriesresemblingmodernlogic,suchasGottlobFrege's"distinctionbetweensenseandreferenceofpropernames"andhis"definitionofnumber",aswellasthetheoryof"restrictiveconditionsforuniversals"anticipatingsomeofthedevelopmentsinmodernsettheory.[v]Since1824,IndianlogicattractedtheattentionofmanyWesternscholars,andhashadaninfluenceonimportant19th-centurylogicianssuchasCharlesBabbage,AugustusDeMorgan,andGeorgeBoole.[158]Inthe20thcentury,WesternphilosopherslikeStanislawSchayerandKlausGlashoffhaveexploredIndianlogicmoreextensively. ThesyllogisticlogicdevelopedbyAristotlepredominatedintheWestuntilthemid-19thcentury,wheninterestinthefoundationsofmathematicsstimulatedthedevelopmentofsymboliclogic(nowcalledmathematicallogic).In1854,GeorgeBoolepublishedTheLawsofThought,[159]introducingsymboliclogicandtheprinciplesofwhatisnowknownasBooleanlogic.In1879,GottlobFregepublishedBegriffsschrift,whichinauguratedmodernlogicwiththeinventionofquantifiernotation,reconcilingtheAristotelianandStoiclogicsinabroadersystem,andsolvingsuchproblemsforwhichAristotelianlogicwasimpotent,suchastheproblemofmultiplegenerality.From1910to1913,AlfredNorthWhiteheadandBertrandRussellpublishedPrincipiaMathematica[117]onthefoundationsofmathematics,attemptingtoderivemathematicaltruthsfromaxiomsandinferencerulesinsymboliclogic.In1931,Gödelraisedseriousproblemswiththefoundationalistprogramandlogicceasedtofocusonsuchissues. ThedevelopmentoflogicsinceFrege,Russell,andWittgensteinhadaprofoundinfluenceonthepracticeofphilosophyandtheperceivednatureofphilosophicalproblems(seeanalyticphilosophy)andphilosophyofmathematics.Logic,especiallysententiallogic,isimplementedincomputerlogiccircuitsandisfundamentaltocomputerscience.Logiciscommonlytaughtbyuniversityphilosophy,sociology,advertisingandliteraturedepartments,oftenasacompulsorydiscipline. Seealso[edit] Philosophyportal Argument –Attempttopersuadeortodeterminethetruthofaconclusion Argumentationtheory –Studyofhowconclusionsarereachedthroughlogicalreasoning;oneoffourrhetoricalmodes Criticalthinking –Analysisoffactstoformajudgment Digitalelectronics –Electroniccircuitsthatutilizedigitalsignals(alsoknownasdigitallogicorlogicgates) Fallacies Listoffallacies –Typesofreasoningthatarelogicallyincorrect Listoflogicians –None Listoflogicjournals –Wikipedialistarticle Listoflogicsymbols –None Logicpuzzle –Puzzlederivingfromthemathematicsfieldofdeduction Mathematics –Areaofknowledge Listofmathematicsarticles Outlineofmathematics –Overviewofandtopicalguidetomathematics Metalogic –Studyofthepropertiesoflogicalsystems Outlineoflogic –Overviewofandtopicalguidetologic Philosophy –Studyofthetruthsandprinciplesofbeing,knowledge,orconduct Listofphilosophytopics Outlineofphilosophy –Overviewofandtopicalguidetophilosophy Logos –TerminWesternphilosophy,theology,psychology,andrhetoric Logicalreasoning –Useoflogictomakesenseofthings Reason –Capacityforconsciouslymakingsenseofthings Truth –Propertyofbeinginaccordwithfactorreality Vectorlogic Notes[edit] ^Thoughseeimperativelogic,dynamicsemantics,andinquisitivesemanticsforlogicalsystemswhichnarroworgeneralizethenotionofvalidinferencetootherkindsofobjects. ^Onabductivereasoning,see: Magnani,L.2001.Abduction,Reason,andScience:ProcessesofDiscoveryandExplanation.NewYork:KluwerAcademicPlenumPublishers.xvii.ISBN 0-306-46514-0. Josephson,JohnR.,andSusanG.Josephson.1994.AbductiveInference:Computation,Philosophy,Technology.NewYork:CambridgeUniversityPress.viii.ISBN 0-521-43461-0. Bunt,H.andW.Black.2000.Abduction,BeliefandContextinDialogue:StudiesinComputationalPragmatics,(NaturalLanguageProcessing1).Amsterdam:JohnBenjamins.vi.ISBN 90-272-4983-0,1-55619-794-2. ^Theterm"alogic"issometimesreservedforjustthesystem'ssyntax,i.e.itslanguageandprooftheory.Inthephilosophicalliterature,thetermissometimesfurtherrestrictedtoreferonlytoparticularlogic-basedformalsystemssuchasthosewhicharecompleteormotivatedbyintuitionsclosetothosewhichmotivatedclassicallogic. ^ThefourCatuṣkoṭilogicaldivisionsareformallyveryclosetothefouropposedpropositionsoftheGreektetralemma,whichinturnareanalogoustothefourtruthvaluesofmodernrelevancelogic. (cf.Belnap,Nuel.1977."Ausefulfour-valuedlogic."InModernUsesofMultiple-ValuedLogic,editedbyDunnandEppstein.Boston:Reidel; Jayatilleke,K.N..1967."TheLogicofFourAlternatives."InPhilosophyEastandWest.UniversityofHawaiiPress.) ^Chakrabarti,KisorKumar.1976."SomeComparisonsBetweenFrege'sLogicandNavya-NyayaLogic."PhilosophyandPhenomenologicalResearch36(4):554–63.doi:10.2307/2106873JSTOR 2106873. "Thispaperconsistsofthreeparts.ThefirstpartdealswithFrege'sdistinctionbetweensenseandreferenceofpropernamesandasimilardistinctioninNavya-Nyayalogic.InthesecondpartwehavecomparedFrege'sdefinitionofnumbertotheNavya-Nyayadefinitionofnumber.Inthethirdpartwehaveshownhowthestudyoftheso-called'restrictiveconditionsforuniversals'inNavya-Nyayalogicanticipatedsomeofthedevelopmentsofmodernsettheory." References[edit] ^fromGreek:λογική,logikḗ,'possessedofreason,intellectual,dialectical,argumentative',relatedtoλόγος(logos),"word,thought,idea,argument,account,reason,orprinciple."(LiddellandScott,1999). ^Pépin,Jean."Logos".EncyclopediaofReligion. ^"logic".www.etymonline.com.Retrieved29December2021. ^abcdefghijkl"Philosophyoflogic".www.britannica.com.Retrieved21November2021. ^abcdefghijklHaack,Susan(1978)."1.'Philosophyoflogics'".PhilosophyofLogics.LondonandNewYork:CambridgeUniversityPress. ^abcdefghijklmnopqrstuvwxyzaaabacadaeafagJaakko,Hintikka;Sandu,Gabriel(2006)."WhatisLogic?".PhilosophyofLogic.NorthHolland.pp. 13–39. ^abcdefghijklmnopAudi,Robert."Philosophyoflogic".TheCambridgeDictionaryofPhilosophy.CambridgeUniversityPress. ^abcdeMcKeon,Matthew."LogicalConsequence".InternetEncyclopediaofPhilosophy.Retrieved20November2021. ^abcBlair,J.Anthony;Johnson,RalphH.(2000)."InformalLogic:AnOverview".InformalLogic.20(2).doi:10.22329/il.v20i2.2262. ^abcMacFarlane,John(2017)."LogicalConstants:4.Topicneutrality".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved4December2021. ^abcCorkum,Philip(2015)."GeneralityandLogicalConstancy".RevistaPortuguesadeFilosofia.71(4):753–767.doi:10.17990/rpf/2015_71_4_0753.ISSN 0870-5283.JSTOR 43744657. ^abMagnus,P.D.(2005)."1.4Deductivevalidity".ForallX:AnIntroductiontoFormalLogic.Victoria,BC,Canada:StateUniversityofNewYorkOerServices. ^abcMacFarlane,John(2017)."LogicalConstants".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved21November2021. ^abcdefghijCraig,Edward(1996)."Formalandinformallogic".RoutledgeEncyclopediaofPhilosophy.Routledge. ^abcGómez-Torrente,Mario(2019)."LogicalTruth".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved22November2021. ^abcdefghJacquette,Dale(2006)."Introduction:Philosophyoflogictoday".PhilosophyofLogic.NorthHolland. ^abcdefghiGroarke,Leo(2021)."InformalLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved31December2021. ^abAudi,Robert."Informallogic".TheCambridgeDictionaryofPhilosophy.CambridgeUniversityPress. ^abcdefghiJohnson,RalphH.(1999)."TheRelationBetweenFormalandInformalLogic".Argumentation.13(3):265–274.doi:10.1023/A:1007789101256.S2CID 141283158. ^vanEemeren,FransH.;Garssen,Bart;Krabbe,ErikC.W.;SnoeckHenkemans,A.Francisca;Verheij,Bart;Wagemans,JeanH.M.(2021)."InformalLogic".HandbookofArgumentationTheory.SpringerNetherlands.pp. 1–45.doi:10.1007/978-94-007-6883-3_7-1.ISBN 978-94-007-6883-3. ^abcHonderich,Ted(2005)."logic,informal".TheOxfordCompaniontoPhilosophy.OxfordUniversityPress. ^abCraig,Edward(1996)."Formallanguagesandsystems".RoutledgeEncyclopediaofPhilosophy.Routledge. ^abcdefghijklWalton,DouglasN.(1987)."1.Anewmodelofargument".InformalFallacies:TowardsaTheoryofArgumentCriticisms.JohnBenjamins. ^abEngel,S.Morris(1982)."2.Themediumoflanguage".WithGoodReasonanIntroductiontoInformalFallacies. ^abBlair,J.Anthony;Johnson,RalphH.(1987)."TheCurrentStateofInformalLogic".InformalLogic.9(2).doi:10.22329/il.v9i2.2671. ^abWeddle,Perry(26July2011)."36.Informallogicandtheeductive-inductivedistinction".Argumentation3.DeGruyterMouton.doi:10.1515/9783110867718.383.ISBN 978-3-11-086771-8. ^abD'Agostino,Marcello;Floridi,Luciano(2009)."TheEnduringScandalofDeduction:IsPropositionalLogicReallyUninformative?".Synthese.167(2):271–315.doi:10.1007/s11229-008-9409-4.hdl:2299/2995.ISSN 0039-7857.JSTOR 40271192.S2CID 9602882. ^abcdefgBackmann,Marius(1June2019)."VarietiesofJustification—How(Not)toSolvetheProblemofInduction".ActaAnalytica.34(2):235–255.doi:10.1007/s12136-018-0371-6.ISSN 1874-6349.S2CID 125767384. ^abc"DeductiveandInductiveArguments".InternetEncyclopediaofPhilosophy.Retrieved4December2021. ^abcdefghijklmHansen,Hans(2020)."Fallacies".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved18March2021. ^abVleet,VanJacobE.(2010)."Introduction".InformalLogicalFallacies:ABriefGuide.Upa. ^abcdeDowden,Bradley."Fallacies".InternetEncyclopediaofPhilosophy.Retrieved19March2021. ^abcdeStump,DavidJ."Fallacy,Logical".www.encyclopedia.com.Retrieved20March2021. ^abcdefKorb,Kevin(2004)."BayesianInformalLogicandFallacy".InformalLogic.24(1):41–70.doi:10.22329/il.v24i1.2132. ^abcdMackie,J.L.(1967)."Fallacies".www.encyclopedia.com.Retrieved19March2021. ^abcdefgHonderich,Ted(2005)."philosophicallogic".TheOxfordCompaniontoPhilosophy.OxfordUniversityPress. ^Falguera,JoséL.;Martínez-Vidal,Concha;Rosen,Gideon(2021)."AbstractObjects".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved7January2022. ^Pietroski,Paul(2021)."LogicalForm:1.PatternsofReason".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved4December2021. ^Kusch,Martin(2020)."Psychologism".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved30November2021. ^Rush,Penelope(2014)."Introduction".TheMetaphysicsofLogic.CambridgeUniversityPress. ^abKing,JeffreyC.(2019)."StructuredPropositions".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved4December2021. ^abcdPickel,Bryan(1July2020)."Structuredpropositionsandtrivialcomposition".Synthese.197(7):2991–3006.doi:10.1007/s11229-018-1853-1.ISSN 1573-0964.S2CID 49729020. ^Craig,Edward(1996)."Philosophyoflogic".RoutledgeEncyclopediaofPhilosophy.Routledge. ^abMichaelson,Eliot;Reimer,Marga(2019)."Reference".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved4December2021. ^Blackburn,Simon(1January2008)."argument".TheOxfordDictionaryofPhilosophy.OxfordUniversityPress.ISBN 978-0-19-954143-0. ^abcdefgIEPStaff."DeductiveandInductiveArguments".Retrieved6January2022. ^abBlackburn,Simon(24March2016)."ruleofinference".ADictionaryofPhilosophy.OxfordUniversityPress.ISBN 978-0-19-873530-4. ^abcdefghijDouven,Igor(2021)."Abduction".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^abSagüillo,JoséM.(2014)."HintikkaonInformationandDeduction".Teorema:RevistaInternacionaldeFilosofía.33(2):75–88.ISSN 0210-1602.JSTOR 43047609. ^abHintikka,Jaakko(1970)."Information,Deduction,andtheAPriori".Noûs.4(2):135–152.doi:10.2307/2214318.ISSN 0029-4624.JSTOR 2214318. ^abcHawthorne,James(2021)."InductiveLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved6January2022. ^Wilbanks,JanJ.(1March2010)."DefiningDeduction,Induction,andValidity".Argumentation.24(1):107–124.doi:10.1007/s10503-009-9131-5.ISSN 1572-8374.S2CID 144481717. ^abcBorchert,Donald(2006).MacmillanEncyclopediaofPhilosophy,2ndEdition.Macmillan. ^Possin,Kevin(2016)."ConductiveArguments:WhyisThisStillaThing?".InformalLogic.36(4):563–593.doi:10.22329/il.v36i4.4527. ^Scott,John;Marshall,Gordon(2009)."analyticinduction".ADictionaryofSociology.OxfordUniversityPress.ISBN 978-0-19-953300-8. ^HOUDE,R.;CAMACHO,L."Induction".NewCatholicEncyclopedia. ^abKoslowski,Barbara(2017)."Abductivereasoningandexplanation".TheRoutledgeInternationalHandbookofThinkingandReasoning.Routledge.doi:10.4324/9781315725697.ISBN 978-1-315-72569-7. ^abCummings,Louise(5April2010)."Abduction".TheRoutledgePragmaticsEncyclopedia.Routledge.ISBN 978-1-135-21457-9. ^ab"Thought".EncyclopediaBritannica.Retrieved14October2021. ^abWalton,DouglasN.(1987)."3.Logicofpropositions".InformalFallacies:TowardsaTheoryofArgumentCriticisms.JohnBenjamins. ^Stone,MarkA.(2012)."DenyingtheAntecedent:ItsEffectiveUseinArgumentation".InformalLogic.32(3):327–356.doi:10.22329/il.v32i3.3681. ^Engel,S.Morris(1982)."4.Fallaciesofpresumption".WithGoodReasonanIntroductiontoInformalFallacies. ^abcPedemonte,Bettina(25June2018)."StrategicvsDefinitoryRules:TheirRoleinAbductiveArgumentationandtheirRelationshipwithDeductiveProof".EurasiaJournalofMathematics,ScienceandTechnologyEducation.14(9):em1589.doi:10.29333/ejmste/92562.ISSN 1305-8215.S2CID 126245285. ^abc"Logicalsystems".www.britannica.com.Retrieved4December2021. ^abcShapiro,Stewart;KouriKissel,Teresa(2021)."ClassicalLogic".InZalta,EdwardN.(ed.).TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^Gamut,L.T.F.(1991).Logic,LanguageandMeaningVol1:IntroductiontoLogic.UniversityofChicagoPress.pp. 26–27.ISBN 978-0226280851. ^Enderton,Herbert(2001).Amathematicalintroductiontologic.Elsevier.Chapter1.ISBN 978-0122384523. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.pp. 30–31.ISBN 978-0199575589. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.pp. 47–48.ISBN 978-0199575589. ^Gamut,L.T.F.(1991).Logic,LanguageandMeaningVol1:IntroductiontoLogic.UniversityofChicagoPress.pp. 114–155.ISBN 978-0226280851. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.pp. 34–42.ISBN 978-0199575589. ^Shapiro,Stewart;KouriKissel,Teresa(2021)."ClassicalLogic".InZalta,EdwardN.(ed.).TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^Gamut,L.T.F.(1991).Logic,LanguageandMeaningVol1:IntroductiontoLogic.UniversityofChicagoPress.pp. 148–150.ISBN 978-0226280851. ^Enderton,Herbert(2001).Amathematicalintroductiontologic.Elsevier.Chapter2.5.ISBN 978-0122384523. ^vanDalen,Dirk(1994).LogicandStructure.Springer.Chapter1.5.ISBN 0-387-57839-0. ^abcdefghijSmith,Robin(2020)."Aristotle'sLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved1January2022. ^abcdefgGroarke,LouisF."Aristotle:Logic".InternetEncyclopediaofPhilosophy.Retrieved1January2022. ^abcdHaack,Susan(1996)."1.'Alternative'in'AlternativeLogic'".DeviantLogic,FuzzyLogic:BeyondtheFormalism.ChicagoandLondon:UniversityofChicagoPress. ^abcWolf,RobertG.(1978)."AreRelevantLogicsDeviant?".Philosophia.7(2):327–340.doi:10.1007/BF02378819.S2CID 143697796. ^Shapiro,Stewart;KouriKissel,Teresa(2021)."ClassicalLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved4December2021. ^abBurgess,JohnP.(2009)."1.Classicallogic".PhilosophicalLogic.Princeton,NJ,USA:PrincetonUniversityPress. ^Borchert,Donald(2006)."Logic,Non-Classical".MacmillanEncyclopediaofPhilosophy,2ndEdition.Macmillan. ^abGoble,Lou(2001)."Introduction".TheBlackwellGuidetoPhilosophicalLogic.Wiley-Blackwell. ^abcdeRitola,Juho(1December2008)."Walton'sInformalLogic:APragmaticApproach".InformalLogic.28(4):335.doi:10.22329/il.v28i4.2856. ^abcdeBobzien,Susanne(2020)."AncientLogic:2.Aristotle".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved3January2022. ^abc"historyoflogic-Aristotle".www.britannica.com. ^Magnus,P.D.(2005)."2.2Connectives".ForallX:AnIntroductiontoFormalLogic.Victoria,BC,Canada:StateUniversityofNewYorkOerServices. ^Westerståhl,Dag(1989)."AristotelianSyllogismsandGeneralizedQuantifiers".StudiaLogica.48(4):577–585.doi:10.1007/BF00370209.S2CID 32089424. ^abcHurley,PatrickJ.(2015)."4.CategoricalSyllogisms".Logic:TheEssentials.Wadsworth.ISBN 978-1-305-59041-0. ^abcCopi,IrvingMarmer;Cohen,Carl;McMahon,Kenneth(1953)."6.CategoricalSyllogisms".IntroductiontoLogic.NewYork,NY,USA:Macmillan. ^Brody,BoruchA.(2006).Encyclopediaofphilosophy.Vol. 5.DonaldM.Borchert(2nd ed.).Detroit:ThomsonGale/MacmillanReferenceUSA.pp. 535–536.ISBN 0-02-865780-2.OCLC 61151356.Thetwomostimportanttypesoflogicalcalculiarepropositional(orsententialcalaculiandfunctional(orpredicate)calculi.Apropositionalcalculusisasystemcontainingpropositionalvariablesandconnectives(somealsocontainpropositionalconstants)butnotindividualorfunctionalvariablesorconstants.Intheextendedpropositionalcalculus,quantifierswhoseoperatorvariablesarepropositionalvariablesareadded. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.Chapter2.3.ISBN 978-0199575589. ^Gamut,L.T.F.(1991).Logic,LanguageandMeaningVol1:IntroductiontoLogic.UniversityofChicagoPress.2.3.ISBN 978-0226280851. ^abBlackburn,Patrick;deRijke,Maarten;Venema,Yde(2001).ModalLogic.CambridgeTractsinTheoreticalComputerScience.CambridgeUniversityPress. ^vanBenthem,Johan(2010).ModalLogicforOpenMinds(PDF).CSLI.S2CID 62162288.Archivedfromtheoriginal(PDF)on19February2020. ^abVäänänen,Jouko(2021)."Second-orderandHigher-orderLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved23November2021. ^abcKetland,Jeffrey(2005)."SecondOrderLogic".EncyclopediaofPhilosophy. ^ab"predicatecalculus".ADictionaryofComputing. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.Chapter3.5.ISBN 978-0199575589. ^Bezhanishvili,Nick;deJongh,Dick(2009).IntuitionisticLogic(PDF). ^Moschovakis,Joan(2018)."IntuitionisticLogic".InZalta,EdwardN.(ed.).TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^Ciardelli,I.(2009)."Inquisitivesemanticsandintermediatelogics"(PDF).MasterThesis,ILLCUniversityofAmsterdam. ^Sider,Theodore(2010).LogicforPhilosophy.OxfordUniversityPress.Chapter3.4.ISBN 978-0199575589. ^Gamut,L.T.F.(1991).Logic,LanguageandMeaningVol1:IntroductiontoLogic.UniversityofChicagoPress.5.5.ISBN 978-0226280851. ^Zegarelli,Mark(2010),LogicForDummies,JohnWiley&Sons,p. 30,ISBN 978-1-118-05307-2 ^Hájek,Petr(2006)."FuzzyLogic".InZalta,EdwardN.(ed.).StanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^abcSiegel,Harvey;Biro,John(1997)."EpistemicNormativity,Argumentation,andFallacies".Argumentation.11(3):277–292.doi:10.1023/A:1007799325361.S2CID 126269789. ^Olsson,ErikJ.(2018)."BayesianEpistemology".IntroductiontoFormalPhilosophy.Springer.pp. 431–442. ^abHájek,Alan;Lin,Hanti(2017)."ATaleofTwoEpistemologies?".ResPhilosophica.94(2):207–232.doi:10.5840/resphilosophica201794264(inactive28February2022).{{citejournal}}:CS1maint:DOIinactiveasofFebruary2022(link) ^Hartmann,Stephan;Sprenger,Jan(2010)."BayesianEpistemology".TheRoutledgeCompaniontoEpistemology.London:Routledge.pp. 609–620. ^Talbott,William(2016)."BayesianEpistemology".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved6March2021. ^Gödel,Kurt(1984)."Russell'smathematicallogic".PhilosophyofMathematics:SelectedReadings(2 ed.).CambridgeUniversityPress.pp. 447–469.ISBN 978-0-521-29648-9. ^Monk,J.Donald(1976)."Introduction".MathematicalLogic.Springer.pp. 1–9.doi:10.1007/978-1-4684-9452-5_1.ISBN 978-1-4684-9452-5. ^Hinman,PeterG.(2005).Fundamentalsofmathematicallogic.Wellesley,Mass.:AKPeters.ISBN 978-1-315-27553-6.OCLC 958798526. ^ab"Supplement#3:NotesonLogic|Logic|Argument|Free30-dayTrial".Scribd.Retrieved27May2020. ^Stolyar,AbramA.(1983).IntroductiontoElementaryMathematicalLogic.DoverPublications.p. 3.ISBN 978-0-486-64561-2. ^abWhitehead,AlfredNorth;Russell,Bertrand(1967).PrincipiaMathematicato*56.CambridgeUniversityPress.ISBN 978-0-521-62606-4. ^Brookshear,J.Glenn(1989)."Computability:FoundationsofRecursiveFunctionTheory".Theoryofcomputation:formallanguages,automata,andcomplexity.RedwoodCity,Calif.:Benjamin/CummingsPub.Co.ISBN 978-0-8053-0143-4. ^Brookshear,J.Glenn(1989)."Complexity".Theoryofcomputation:formallanguages,automata,andcomplexity.RedwoodCity,Calif.:Benjamin/CummingsPub.Co.ISBN 978-0-8053-0143-4. ^Barr,Michael;CharlesWells(1998).CategoryTheoryforComputingScience(PDF).CentredeRecherchesMathématiques. ^Lewis,HarryR.(1981).ElementsoftheTheoryofComputation.PrenticeHall. ^Davis,Martin(11May1995)."InfluencesofMathematicalLogiconComputerScience".InRolfHerken(ed.).TheUniversalTuringMachine.SpringerVerlag.ISBN 9783211826379.Retrieved26December2013. ^DeLoach,Scott;ThomasHartrum(June2000)."ATheoryBasedRepresentationforObject-OrientedDomainModels".IEEETransactionsonSoftwareEngineering.25(6):500–517.doi:10.1109/32.852740. ^Blackburn,Patrick;deRijke,Maarten;Venema,Yde(2001).ModalLogic.CambridgeTractsinTheoreticalComputerScience.CambridgeUniversityPress. ^Hofstadter,DouglasR.(5February1999).Gödel,Escher,Bach:AnEternalGoldenBraid.BasicBooks.ISBN 978-0465026562. ^Partee,Barbara(2016)."FormalSemantics".InAloni,Maria;Dekker,Paul(eds.).CambridgeHandbookofFormalSemantics.CambridgeUniversityPress.ISBN 978-1-107-02839-5. ^Coppock,Elizabeth;Champollion,Lucas(2019).Invitationtoformalsemantics(PDF). ^IreneHeim;AngelikaKratzer(1998).Semanticsingenerativegrammar.Wiley-Blackwell.ISBN 978-0-631-19713-3. ^Kroeger,Paul(2019).AnalyzingMeaning.LanguageSciencePress.ISBN 978-3-96110-136-8. ^Putnam,H.(1969)."IsLogicEmpirical?".BostonStudiesinthePhilosophyofScience.5:216–241.doi:10.1007/978-94-010-3381-7_5.ISBN 978-94-010-3383-1. ^Birkhoff,G.;vonNeumann,J.(1936)."TheLogicofQuantumMechanics".AnnalsofMathematics.37(4):823–843.doi:10.2307/1968621.JSTOR 1968621. ^Dummett,M.(1978)."IsLogicEmpirical?".TruthandOtherEnigmas.ISBN 978-0-674-91076-8. ^Priest,Graham(2008)."Dialetheism".InZalta,EdwardN.(ed.).StanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^Hofweber,T.(2004)."LogicandOntology".InZalta,EdwardN(ed.).StanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity. ^Brandom,Robert(2000).ArticulatingReasons.Cambridge,MA:HarvardUniversityPress.ISBN 978-0-674-00158-9. ^Nietzsche,1873,OnTruthandLiesinaNonmoralSense. ^Nietzsche,1882,TheGayScience. ^Nietzsche,1878,Human,AllTooHuman ^BabetteBabich,Habermas,Nietzsche,andCriticalTheory ^GeorgLukács."TheDestructionofReasonbyGeorgLukács1952".Marxists.org.Retrieved16June2013. ^Russell,Bertrand(1945),AHistoryofWesternPhilosophyAndItsConnectionwithPoliticalandSocialCircumstancesfromtheEarliestTimestothePresentDay(PDF),SimonandSchuster,p. 762,archivedfromtheoriginalon28May2014 ^E.g.,Kline(1972,p.53)wrote"AmajorachievementofAristotlewasthefoundingofthescienceoflogic". ^abŁukasiewicz,Jan(1957).Aristotle'ssyllogisticfromthestandpointofmodernformallogic(2nd ed.).OxfordUniversityPress.p. 7.ISBN 978-0-19-824144-7. ^JonathanLear(1986)."AristotleandLogicalTheory".CambridgeUniversityPress.p.34.ISBN 0-521-31178-0 ^SimoKnuuttila(1981)."Reforgingthegreatchainofbeing:studiesofthehistoryofmodaltheories".SpringerScience&Business.p.71.ISBN 90-277-1125-9 ^MichaelFisher,DovM.Gabbay,LluísVila(2005)."Handbookoftemporalreasoninginartificialintelligence".Elsevier.p.119.ISBN 0-444-51493-7 ^HaroldJosephBerman(1983)."Lawandrevolution:theformationoftheWesternlegaltradition".HarvardUniversityPress.p.133.ISBN 0-674-51776-8 ^"AristotleArchived7June2010attheWaybackMachine",MTUDepartmentofChemistry. ^DagNikolausHasse(19September2008)."InfluenceofArabicandIslamicPhilosophyontheLatinWest".StanfordEncyclopediaofPhilosophy.Retrieved13October2009. ^RichardF.Washell(1973),"Logic,Language,andAlberttheGreat",JournaloftheHistoryofIdeas34(3),pp.445–450[445]. ^Knealep.229 ^Kneale:p.266;Ockham:SummaLogicaei.14;Avicenna:AvicennaeOperaVenice1508f87rb ^abGoodman,LennEvan(2003),IslamicHumanism,p.155,OxfordUniversityPress,ISBN 0-19-513580-6. ^Goodman,LennEvan(1992);Avicenna,p.188,Routledge,ISBN 0-415-01929-X. ^"Historyoflogic:Arabiclogic".EncyclopædiaBritannica.Archivedfromtheoriginalon12October2007. ^MuhammadIqbal,TheReconstructionofReligiousThoughtinIslam,"TheSpiritofMuslimCulture"(cf.[1]and[2]) ^Vidyabhusana,S.C.1971.AHistoryofIndianLogic:Ancient,Mediaeval,andModernSchools.pp.17–21. ^JonardonGaneri(2001).Indianlogic:areader.Routledge.pp. vii,5,7.ISBN 978-0-7007-1306-6. ^Boole,George.1854.AnInvestigationoftheLawsofThoughtonWhichareFoundedtheMathematicalTheoriesofLogicandProbabilities. Bibliography[edit] Barwise,J.(1982).HandbookofMathematicalLogic.Elsevier.ISBN 978-0-08-093364-1. Belnap,N.(1977)."Ausefulfour-valuedlogic".InDunn&Eppstein,Modernusesofmultiple-valuedlogic.Reidel:Boston. Bocheński,J.M.(1959).Aprécisofmathematicallogic.TranslatedfromtheFrenchandGermaneditionsbyOttoBird.D.Reidel,Dordrecht,SouthHolland. Bocheński,J.M.(1970).Ahistoryofformallogic.2ndEdition.TranslatedandeditedfromtheGermaneditionbyIvoThomas.ChelseaPublishing,NewYork. Brookshear,J.Glenn(1989).Theoryofcomputation:formallanguages,automata,andcomplexity.RedwoodCity,Calif.:Benjamin/CummingsPub.Co.ISBN 978-0-8053-0143-4. Cohen,R.S,andWartofsky,M.W.(1974).LogicalandEpistemologicalStudiesinContemporaryPhysics.BostonStudiesinthePhilosophyofScience.D.ReidelPublishingCompany:Dordrecht,Netherlands.ISBN 90-277-0377-9. Finkelstein,D.(1969)."Matter,Space,andLogic".inR.S.CohenandM.W.Wartofsky(eds.1974). Gabbay,D.M.,andGuenthner,F.(eds.,2001–2005).HandbookofPhilosophicalLogic.13vols.,2ndedition.KluwerPublishers:Dordrecht. Haack,Susan(1996).DeviantLogic,FuzzyLogic:BeyondtheFormalism,UniversityofChicagoPress. Harper,Robert(2001)."Logic".OnlineEtymologyDictionary.Retrieved8May2009. Hilbert,D.,andAckermann,W,(1928).GrundzügedertheoretischenLogik(PrinciplesofMathematicalLogic).Springer-Verlag.OCLC 2085765 Hodges,W.(2001).Logic.AnintroductiontoElementaryLogic,PenguinBooks. Hofweber,T.(2004),LogicandOntology.StanfordEncyclopediaofPhilosophy.EdwardN.Zalta(ed.). Hughes,R.I.G.(1993,ed.).APhilosophicalCompaniontoFirst-OrderLogic.HackettPublishing. Kline,Morris(1972).MathematicalThoughtFromAncienttoModernTimes.OxfordUniversityPress.ISBN 978-0-19-506135-2. Kneale,William,andKneale,Martha,(1962).TheDevelopmentofLogic.OxfordUniversityPress,London,UK. Liddell,HenryGeorge;Scott,Robert."Logikos".AGreek-EnglishLexicon.PerseusProject.Retrieved8May2009. Mendelson,Elliott,(1964).IntroductiontoMathematicalLogic.Wadsworth&Brooks/ColeAdvancedBooks&Software:Monterey,Calif.OCLC 13580200 Smith,B.(1989)."LogicandtheSachverhalt".TheMonist72(1):52–69. Whitehead,AlfredNorthandBertrandRussell(1910).PrincipiaMathematica.CambridgeUniversityPress:Cambridge,England.OCLC 1041146 Externallinks[edit] LogicatWikipedia'ssisterprojectsDefinitionsfromWiktionaryMediafromCommonsNewsfromWikinewsQuotationsfromWikiquoteTextsfromWikisourceTextbooksfromWikibooksResourcesfromWikiversity LibraryresourcesaboutLogic Resourcesinyourlibrary Resourcesinotherlibraries LogicatPhilPapers LogicattheIndianaPhilosophyOntologyProject "Logic".InternetEncyclopediaofPhilosophy. "Logicalcalculus",EncyclopediaofMathematics,EMSPress,2001[1994] AnOutlineforVerbalLogic Introductionsandtutorials "AnIntroductiontoPhilosophicalLogic,byPaulNewall".Archivedfromtheoriginalon3April2008.aimedatbeginners. forallx:anintroductiontoformallogic,byP.D.Magnus,coverssententialandquantifiedlogic. LogicSelf-Taught:AWorkbook(originallypreparedforon-linelogicinstruction). NicholasRescher.(1964).IntroductiontoLogic,St.Martin'sPress. Essays "SymbolicLogic"and"TheGameofLogic",LewisCarroll,1896. Math&Logic:Thehistoryofformalmathematical,logical,linguisticandmethodologicalideas.InTheDictionaryoftheHistoryofIdeas. OnlineTools InteractiveSyllogisticMachineAweb-basedsyllogisticmachineforexploringfallacies,figures,terms,andmodesofsyllogisms. ALogicCalculatorAweb-basedapplicationforevaluatingsimplestatementsinsymboliclogic. Referencematerial TranslationTips,byPeterSuber,fortranslatingfromEnglishintologicalnotation. OntologyandHistoryofLogic.AnIntroductionwithanannotatedbibliography. 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