Hempel and Oppenheim on Ex Plant Ion

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Philosophy of Science Association Hempel and Oppenheim on Explanation Author(s): Rolf Eberle, David Kaplan, Richard Montague Source: Philosophy of Science, Vol. 28, No. 4 (Oct., 1961), pp. 418-428 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/185484 Accessed: 20/10/2010 06:34 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/p
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  Philosophy of Science Association Hempel and Oppenheim on ExplanationAuthor(s): Rolf Eberle, David Kaplan, Richard MontagueSource: Philosophy of Science, Vol. 28, No. 4 (Oct., 1961), pp. 418-428Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/185484 Accessed: 20/10/2010 06:34 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=ucpress.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org. Philosophy of Science Association and The University of Chicago Press are collaborating with JSTOR todigitize, preserve and extend access to Philosophy of Science. http://www.jstor.org  HEMPELAND OPPENHEIMON EXPLANATION*ROLF EBERLE,DAVID KAPLAN,AND RICHARD MONTAGUE' University ofCalifornia, Los AngelesHempeland Oppenheim, intheirpaper'TheLogicofExplanation',have offeredan analysisof the notion of scientific explanation.The present paperadvances considera-tionsin thelightof which their analysis seems inadequate.Inparticular,several theo- rems are provedwith roughly the followingcontent: between almostany theory and al-most any singularsentence, certain relationsof explainabilityhold. Hempeland OppenheimhaveofferedinPartIIIof[3]ananalysis of thenotion of scientific explanation. Itis thepurposeof the present paper to ad-vance considerationsin thelightof which theiranalysisseems to usinadequate.LikeHempeland Oppenheim, we consideraformal languageLwhichhasthesyntacticalstructure of the lowerpredicate calculuswithout identity.The logical constantsofL are , , v D, , ( ), X, whichare the respectivesymbols of negation,conjunction,disjunction, implication,equivalence,universal quantification,and existentialquantification.Thevocabulary ofLcontainsin addition parentheses, individualvariables, individual constants,andpredicatesofanydesired finite degree.FormulasofL(which maycontainfreevariables) and sentences ofL(whichmay notcontainfreevariables)arecharacterized in the usualway.Throughoutthefollowingwe shall havethefixed language L in mind andmean by 'formula'and'sentence','formula ofL'and 'sentence of L' respectively. Weshall also suppose thataninterpretationforLhas beenfixed, sothat we may meaningfully speakof trueandfalsesentences.2Logical provability, logical derivability(ofasentencefrom a setofsentencesorfrom another sentence),andlogicalequivalencearetobe understoodinthe mannerusual withinthelowerpredicatecalculus withoutidentity.Weuse thenotations' - S' and'KI-S'forlogical provabilityandlogicalderiva-bility respectively.For the convenienceof thereader,the relevantnotions oftheHempeland Oppenheim paperaredefinedhere.A number ofancillarynotions havebeenomitted, and,consequently,definitional chainscompressed;but the reader will find thefollowingdefinitions immediateconsequencesof thosegivenby HempelandOppenheim.There isoneexception,notedin footnote 3, which corresponds toa minor correctionintroducedinorderto remain faithful towhatweregardas the authors'intention. * Received, September1960. 1 Thispaperwasprepared for publicationwhile thethird author held NationalScienceFoundation Grant NSFG-13226,andthesecondauthor wasagraduateFellowoftheNationalSciencefoundation. 2 Theinterpretationfor L issubjected byHempelandOppenheimto certainspecial require-ments that are hereirrelevant.418  HEMPELAND OPPENHEIMONEXPLANATION 419Asingularsentenceis oneinwhichno variablesoccur,an atomicsentenceone whichconsists ofa predicatefollowedbyindividualconstants,andabasic sentenceeitheran atomic sentenceor the negationofan atomicsentence.A fundamentallawis a truesentenceconsistingof oneormore universalquantifiersfollowedby anexpressionwithoutquantifiersor individualcon-stants.A sentenceS iscalledaderivative lawif (1)S consistsof oneor moreuniversal quantifiersfollowedbyanexpressionwithout quantifiers,(2)Sisnot logicallyequivalenttoanysingularsentence,(3)at leastone individualconstantoccursin S, and(4) thereis aclass K offundamentallawssuchthatSis logicallyderivablefromK.A lawis a sentencewhichiseitherafundamentallawor a derivativelaw.Afundamentaltheory is atruesentenceconsistingofone ormore quantifiersfollowedby an expressionwithoutquantifiersorindividual constants.A sen-tenceS is calleda derivativetheoryif (1) Sconsistsofone or more quantifiersfollowed byan expressionwithoutquantifiers,(2)S is not logicallyequivalenttoanysingularsentence,(3)atleastoneindividualconstantoccursinS,and(4)thereis a classK offundamentaltheoriessuchthat S islogicallyderivablefromK.A theoryis a sentencewhichis eitherafundamentaltheoryoraderivativetheory.3Anorderedcouple (T,C)ofsentencesisanexplanansforasingularsentenceEif and onlyif the followingconditionsare satisfied:(1)T is atheory,(2)T is notlogicallyequivalenttoanysingularsentence,(3)C issingularandtrue,(4)Eislogicallyderivablefromtheset{T, C},and(5)thereis a classK ofbasicsentencessuch thatCislogicallyderivablefromK,andneitherEnor ' T islogicallyderivablefromK.AsingularsentenceE isexplainablebyatheoryTjustincasethereis asingular sentenceCsuch thatthe ordered couple(T,C)is anexplanansfor E.Thefollowingsimpleexampleissufficientinouropinionto indicateadivergencebetweenthenotiondefinedaboveand thecustomarynotionofexplainability.Let usassumethattherearenomermaidsand thattheEiffelToweris agoodconductorofheat.Thenthesingularsentence'the EiffelTowerisagoodconductorof heat'(orrather,itstranslationin thelanguage L)will beexplainableby'allmermaidsaregoodconductorsofheat'.To seethis,choose forC in thedefinitionof'explainable'thesingularsentence'if the 3The 'minor correction'mentionedabovearisesinconnectionwith thedefinitionof a deriva-tivetheory.Theauthors callasentenceessentiallyuniversalifit consistsof universalquantifiersfollowedbyanexpressionwithoutquantifiersandisnotequivalenttoasingularsentence.Onthe otherhand,theycalla sentenceessentially generalizedif it issimplynotequivalenttoasingularsentence.Theintention,however,seemsto betorequirein additionthat anessentiallygeneralizedsentenceconsistsofquantifiersfollowedbyanexpressionwithoutquantifiers.Thechoice betweenthe twopossibledefinitionsof theauxiliarynotion'essentiallygeneralized'affectsthe ensuingdefinitionof aderivativetheory,but does notessentiallyaffecttheresultsofthepresentpaper.  420 ROLFEBERLE,DAVIDKAPLAN,ANDRICHARD MONTAGUE Eiffel Tower is not a mermaid, then the Eiffel Tower is a good conductor ofheat', andforK in the definition of 'explanans' the class whose only memberis the basic sentence 'the Eiffel Tower is a mermaid'.There are innumerable obvious ways in which to circumvent this example.The definitionof'explainable' is, we think, more conclusively trivialized byTheorems 1-5 below, which show, roughly speaking, that according to thedefinition ofHempelandOppenheima relation ofexplainability holds betweenalmostany theoryand almostany true singular sentence.Butfirst severallemmas concerning the predicate calculus must be stated. Lemma 2 is provedin[1], Lemma 3iswell known,andLemma 4 is obvious.Lemma1.IfSis asingularsentence andTa sentenceconsisting of universalquantifiersfollowedbyanexpressionwithoutquantifiers or individual con-stants,and1- SDO T,then either1Sor1-T.Proof. Assumetothe contrarythatneitherF1 -~- S nor1-T. Then, bythe CompletenessTheoremofG6del([2]),thereare models M and N (cor-respondingto thelanguage L)such that S is true in M and Tis trueinN;wemayin fact chooseM andNsothat theiruniverses will bedisjoint.Formthe modelPas follows: theuniverseofP isto be the union ofthe universesofM andN;toeachpredicateofL,P istoassignanextension which istheunionofthetwoextensionsassigned byM andN;andthe individual con-stantsofLaretodesignateinPwhatthey designateinM.Itis theneasytoseethat bothS and -- T,inviewof their structure andthe factthattheyaretruein M andNrespectively,aretruein P.But thiscontradictstheassumptionthat I- SDT.Lemma2.IfS,T aresentencescontainingnocommonpredicates,andISvT,then1-Sor1-T.Lemma3.IfthesentenceS(bl,...,b.)isobtainedfrom the formulaS(xl,...,xx) by replacingallfree occurrencesofthevariablesxl,...,x.by theindividual constantsbl, bn, wherebibjifandonlyifxizx,,andIS(bl, ...,bn), thenalsoI (xi) ... (xn) S(xl, ..., XJ) Lemma4.IfS,T aresingularsentences with nocommonatomic sub-sentences,and1-SvT,then1-S or 1- T.Theorem1.Let Tbeanyfundamentallawand Eany singulartruesentencesuchthatneitherTnor E islogically provableandT,Ehave nopredicatesin common.Assumein addition thatthere are at leastasmanyindividualconstantsin LbeyondthoseoccurringinEastherearevariables inT,andat leastasmany one-place predicatesin Lbeyondthoseoccurringin TandFas there areindividual constantsin E. Thenthereis afundamentallawT'whichislogicallyderivablefrom T and such thatEisexplainable byT'.Proof.LetT and E have therespectiveforms (Y)...(Yn) To(Y..-Yn) EO(all..a,),
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