Validity (logic) - Wikipedia

Arguments Validity(logic) FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Forotheruses,seeValidity. Argumentwhoseconclusionmustbetrueifitspremisesare Inlogic,specificallyindeductivereasoning,anargumentisvalidifandonlyifittakesaformthatmakesitimpossibleforthepremisestobetrueandtheconclusionneverthelesstobefalse.[1]Itisnotrequiredforavalidargumenttohavepremisesthatareactuallytrue,[2]buttohavepremisesthat,iftheyweretrue,wouldguaranteethetruthoftheargument'sconclusion.Validargumentsmustbeclearlyexpressedbymeansofsentencescalledwell-formedformulas(alsocalledwffsorsimplyformulas). Thevalidityofanargumentcanbetested,provedordisproved,anddependsonitslogicalform.[3] Contents 1Arguments 2Validformula 3Statements 4Soundness 5Satisfiability 6Preservation 7Seealso 8References 9Furtherreading Arguments[edit] Argumentterminologyusedinlogic Inlogic,anargumentisasetofstatementsexpressingthepremises(whateverconsistsofempiricalevidencesandaxiomatictruths)andanevidence-basedconclusion. Anargumentisvalidifandonlyifitwouldbecontradictoryfortheconclusiontobefalseifallofthepremisesaretrue.[3]Validitydoesn'trequirethetruthofthepremises,insteaditmerelynecessitatesthatconclusionfollowsfromtheformerswithoutviolatingthecorrectnessofthelogicalform.Ifalsothepremisesofavalidargumentareproventrue,thisissaidtobesound.[3] Thecorrespondingconditionalofavalidargumentisalogicaltruthandthenegationofitscorrespondingconditionalisacontradiction.Theconclusionisalogicalconsequenceofitspremises. Anargumentthatisnotvalidissaidtobe"invalid". Anexampleofavalidargumentisgivenbythefollowingwell-knownsyllogism: Allmenaremortal. Socratesisaman. Therefore,Socratesismortal. Whatmakesthisavalidargumentisnotthatithastruepremisesandatrueconclusion,butthelogicalnecessityoftheconclusion,giventhetwopremises.Theargumentwouldbejustasvalidwerethepremisesandconclusionfalse.Thefollowingargumentisofthesamelogicalformbutwithfalsepremisesandafalseconclusion,anditisequallyvalid: Allcupsaregreen. Socratesisacup. Therefore,Socratesisgreen. Nomatterhowtheuniversemightbeconstructed,itcouldneverbethecasethattheseargumentsshouldturnouttohavesimultaneouslytruepremisesbutafalseconclusion.Theaboveargumentsmaybecontrastedwiththefollowinginvalidone: Allmenareimmortal. Socratesisaman. Therefore,Socratesismortal. Inthiscase,theconclusioncontradictsthedeductivelogicoftheprecedingpremises,ratherthanderivingfromit.Therefore,theargumentislogically'invalid',eventhoughtheconclusioncouldbeconsidered'true'ingeneralterms.Thepremise'Allmenareimmortal'wouldlikewisebedeemedfalseoutsideoftheframeworkofclassicallogic.However,withinthatsystem'true'and'false'essentiallyfunctionmorelikemathematicalstatessuchasbinary1sand0sthanthephilosophicalconceptsnormallyassociatedwiththoseterms. Astandardviewisthatwhetheranargumentisvalidisamatteroftheargument'slogicalform.Manytechniquesareemployedbylogicianstorepresentanargument'slogicalform.Asimpleexample,appliedtotwooftheaboveillustrations,isthefollowing:Lettheletters'P','Q',and'S'stand,respectively,forthesetofmen,thesetofmortals,andSocrates.Usingthesesymbols,thefirstargumentmaybeabbreviatedas: AllPareQ. SisaP. Therefore,SisaQ. Similarly,thesecondargumentbecomes: AllParenotQ. SisaP. Therefore,SisaQ. Anargumentistermedformallyvalidifithasstructuralself-consistency,i.e.ifwhentheoperandsbetweenpremisesarealltrue,thederivedconclusionisalwaysalsotrue.Inthethirdexample,theinitialpremisescannotlogicallyresultintheconclusionandisthereforecategorizedasaninvalidargument. Validformula[edit] Aformulaofaformallanguageisavalidformulaifandonlyifitistrueundereverypossibleinterpretationofthelanguage.Inpropositionallogic,theyaretautologies. Statements[edit] Astatementcanbecalledvalid,i.e.logicaltruth,ifitistrueinallinterpretations. Soundness[edit] Mainarticle:Soundness Validityofdeductionisnotaffectedbythetruthofthepremiseorthetruthoftheconclusion.Thefollowingdeductionisperfectlyvalid: AllanimalsliveonMars. Allhumansareanimals. Therefore,allhumansliveonMars. Theproblemwiththeargumentisthatitisnotsound.Inorderforadeductiveargumenttobesound,theargumentmustbevalidandallthepremisesmustbetrue.[3] Satisfiability[edit] Mainarticle:Satisfiability Modeltheoryanalyzesformulaewithrespecttoparticularclassesofinterpretationinsuitablemathematicalstructures.Onthisreading,formulaisvalidifallsuchinterpretationsmakeittrue.Aninferenceisvalidifallinterpretationsthatvalidatethepremisesvalidatetheconclusion.Thisisknownassemanticvalidity.[4] Preservation[edit] Intruth-preservingvalidity,theinterpretationunderwhichallvariablesareassignedatruthvalueof'true'producesatruthvalueof'true'. Inafalse-preservingvalidity,theinterpretationunderwhichallvariablesareassignedatruthvalueof'false'producesatruthvalueof'false'.[5] Preservationproperties Logicalconnectivesentences Trueandfalsepreserving: Proposition •Logicalconjunction(AND, ∧ {\displaystyle\land} ) •Logicaldisjunction(OR, ∨ {\displaystyle\lor} ) Truepreservingonly: Tautology( ⊤ {\displaystyle\top} ) •Biconditional(XNOR, ↔ {\displaystyle\leftrightarrow} ) •Implication( → {\displaystyle\rightarrow} ) •Converseimplication( ← {\displaystyle\leftarrow} ) Falsepreservingonly: Contradiction( ⊥ {\displaystyle\bot} ) •Exclusivedisjunction(XOR, ⊕ {\displaystyle\oplus} ) •Nonimplication( ↛ {\displaystyle\nrightarrow} ) •Conversenonimplication( ↚ {\displaystyle\nleftarrow} ) Non-preserving: Negation( ¬ {\displaystyle\neg} ) •Alternativedenial(NAND, ↑ {\displaystyle\uparrow} ) •Jointdenial(NOR, ↓ {\displaystyle\downarrow} ) Seealso[edit] Philosophyportal Logicalconsequence Reductioadabsurdum Mathematicalfallacy Soundness Ω-validity References[edit] ^ValidityandSoundness–InternetEncyclopediaofPhilosophy ^JcBeallandGregRestall,"LogicalConsequence",TheStanfordEncyclopediaofPhilosophy(Fall2014Edition). ^abcdGensler,HarryJ.,1945-(January6,2017).Introductiontologic(Third ed.).NewYork.ISBN 978-1-138-91058-4.OCLC 957680480.{{citebook}}:CS1maint:multiplenames:authorslist(link) ^L.T.F.Gamut,Logic,Language,andMeaning:IntroductiontoLogic,UniversityofChicagoPress,1991,p.115. ^RobertCogan,CriticalThinking:StepbyStep,UniversityPressofAmerica,1998,p.48. Furtherreading[edit] Barwise,Jon;Etchemendy,John.Language,ProofandLogic(1999):42. Beer,FrancisA."Validities:APoliticalSciencePerspective",SocialEpistemology7,1(1993):85-105. WiktionaryhasdefinitionsrelatedtoValidity vteLogic Outline History Fields Computerscience Formalsemantics(naturallanguage) Inference Philosophyoflogic Proof Semanticsoflogic Syntax Logics Classical Informal Criticalthinking Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analyticandsyntheticpropositions Contradiction Paradox Antinomy Deduction Deductiveclosure Definition Description Entailment Linguistic Form Induction Logicaltruth Name Necessityandsufficiency Premise Probability Reference Statement Substitution Truth Validity Liststopics Mathematicallogic Booleanalgebra Settheory other Logicians Rulesofinference Paradoxes Fallacies Logicsymbols  Philosophyportal Category WikiProject (talk) changes vteMathematicallogicGeneral Axiom First-orderlogic Formalproof Logicalconsequence Formalsemantics Foundationsofmathematics Model Set Theorem Theory Typetheory Theorems (list) Gödel'scompleteness and incompletenesstheorems Tarski'sundefinability Banach–Tarskiparadox Cantor's theorem, paradox and diagonalargument Lindström's Löwenheim–Skolem Compactness LogicsTraditional Classicallogic Proposition Inference Logicalequivalence Argument Validity Syllogism Squareofopposition Venndiagram Propositional Booleanalgebra Booleanfunctions Logicalconnectives Propositionalcalculus Propositionalformula Truthtables Many-valuedlogic Predicate First-order Quantifiers Predicate Second-order Monadicpredicatecalculus Settheory Set Element Orderedpair Ordinalnumber Russell'sparadox Powerset Subset Relation Function/Map Domain Codomain Image Extensionality Forcing Cardinality Emptyset Inhabitedset Enumeration Countableset Uncountableset Finiteset Infiniteset Class Schröder–Bernsteintheorem Settheories Zermelo–Fraenkel Axiomofchoice Continuumhypothesis General Kripke–Platek Morse–Kelley Naive Tarski–Grothendieck VonNeumann–Bernays–Gödel Syntax & Language Alphabet Arity Expression Ground Extension Relation Formal Grammar Language Proof System Theory Formationrule Formula Atomic Closed Ground Open Free/boundvariable Logicalconnective ¬ ∨ ∧ → ↔ = Predicate Functional Variable Propositionalvariable Quantifier ∃ ∀ Sentence Atomic Signature String Substitution Symbol Function Logical/Constant Non-logical Variable Term Prooftheory Formalproof Naturaldeduction Logicalconsequence Ruleofinference Sequentcalculus Theorem Systems Formal Axiomatic Deductive Hilbert Completetheory Independence (from ZFC) Modeltheory Interpretation Model Equivalence Finite Saturated Substructure Non-standardmodel ofarithmetic Diagram Elementary Categoricaltheory Modelcompletetheory Strength Semanticsoflogic Theoriesoftruth Semantic Tarski's Kripke's T-schema Transferprinciple Truthpredicate Truthvalue Type Ultraproduct Validity Computabilitytheory Church–Turingthesis Computablyenumerable Computablefunction Computableset Decisionproblem Haltingproblem Primitiverecursivefunction Recursion Recursiveset  Mathematicsportal Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Validity_(logic)&oldid=1091537429" Categories:ArgumentsConceptsinlogicDeductivereasoningLogicaltruthHiddencategories:CS1maint:multiplenames:authorslistUsemdydatesfromFebruary2019ArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataUseAmericanEnglishfromFebruary2019AllWikipediaarticleswritteninAmericanEnglishPagesthatuseadeprecatedformatofthemathtags Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةБългарскиCatalàDeutschEestiEspañolفارسیFrançais한국어ÍslenskaItalianoעבריתМакедонски日本語PortuguêsRomânăРусскийSimpleEnglishکوردیСрпски/srpskiSvenskaУкраїнська粵語中文 Editlinks