EnhancingtheDiagrammingMethodinInformalLogic
DaleJACQUETTE Bern
ABSTRACT
TheargumentdiagrammingmethoddevelopedbyMonroeC.Beardsleyinhis(1950)book
Practical Logic, which has since become the gold standard for diagramming arguments in informallogic,makesitpossibletomaptherelationbetweenpremisesandconclusionsof achainofreasoninginrelativelycomplexways.Themethodhassincebeenadaptedand developedinanumberofdirectionsbycontemporaryinformallogiciansandargumenta-tiontheorists.Ithasprovedusefulinpracticalapplicationsandespeciallypedagogicallyin teaching basic logic and critical reasoning skills at all levels of scientiic education. I pro-poseinthisessaytobuildonBeardsleydiagrammingtechniquestoreineandsupplement their structural tools for visualizing logical relationships in a number of categories not originally accommodated by the method, including dilemma and other disjunctive and conditional inferences,reductio ad absurdum arguments, efforts to contradict arguments, andlogicallycircularreasoning,withsugestionsforimproveddiagrammingoflogicalstruc-tures.
1.DIAGRAMMINGARGUMENTS
As a tool in understanding and evaluating arguments, diagramming techniquesofferausefulandelegantrepresentationofinferentialstruc-ture. Diagramming the informal interrelations between an argument’s assumptions and conclusions helps us to appreciate the logic of its im-plicationalconnections,andtoidentifyitsstrengthsandweaknesses1.
Forsimplicity,andbecausetheneededexamplestendtobemoreuni- vocalininformallogicalstructure,weconineattentionexclusivelytode-1 ThestandarddiagrammingmethodwasoriginallyproposedbyBeardsley(1950),and
furtherreinedbyThomas(1973)andScriven(1976).
ductive inferences. Modiications for inductive and other kinds of argu- mentdiagrammingareintuitiveandstraightforwardlymodeledstructural-lyoncorrespondingdeductiveparadigms,andlogicallyvalidonesatthat. Thereisaccordinglyscantmotivation,especiallyintheirstinstancewhen thebasicenhancementsofstandardargumentdiagrammingarebeingpre-sentedandproposed,forgoingbeyonddeductiveinferenceasthesimplest andthemostimportanttomakesureattheoutsetofgettingright.What kindofargumentdiagrammingcouldwereasonablybesaidtohave,ifwe cannotevenexplainhowtodiagramtheinferentialstructuresofdeductive-lyvalidarguments?Wehavetostartsomewhere,andwechooseforgood reasonstostartwithdeductivevalidityamongthekindsofargumentsthat informallogicismostoftenexpectedtoanalyze2.
2.STANDARDDIAGRAMMINGINTHESIMPLESTCASE
Theirststepinthestandardmethodofdiagrammingininformallogic istonumberanargument’sassumptionsandconclusions,typicallydis-tinguishedintheirordinarylanguageexpressionsbyinferenceindicator terms, like ‘thus’, ‘therefore’, ‘hence’. In the simplest case, where a sin-gle conclusion is supposed to follow from a sinsin-gle assumption, the as-sumptionandconclusionnumbersarewrittenouthorizontallywiththe conclusionbelowtheassumption,connectedbyaverticalarrowrunning fromassumptiontoconclusion.Hereisanargumentwiththisstraight-forwardstructure.
1.TodayisTuesday. —————————
2.TomorrowwillbeWednesday.
The argument can be represented by the following most basic dia-gram:
(1) ↓ (2)
More interesting arguments typically have more than one assump-tion, and sometimes more than one conclusion, and the assumptions and conclusions can be related together in any of several ways. The standard diagramming method is equipped with conventions to repre-sent arguments in which multiple assumptions contribute to a single conclusion,andsingleassumptionsimplymultipleconclusions.
3.ADDITIVEANDNONADDITIVEASSUMPTIONS
To begin with assumptions, the standard diagramming method depicts theseasrelatedintwoways,additivelyandnonadditively.
Severalassumptionstakentogetheraresometimesrequiredtosupport aconclusionormultipleconclusions,whichwouldnotfollowiftheas- sumptionswerenotcombinedorsupposedjointlytoholdtrue.Theseas-sumptions are said to be additive.They are diagrammed by connecting their numbers in the standard argument diagram with a ‘+’ sign, and drawing a horizontal line under all of the additive assumptions, as thoughtheywerebeingaddedtogetherinanadditioncolumninarith-metic.Finally,aninferencearrowisdrawn,runningfrombelowtheline totheconclusionorconclusionsthataresupposedtofollow.Hereisan example for the following argument, in which the conclusion follows fromthecombinedlogicalinputoftwodistinctassumptions.
1.Allrattlesnakesarepoisonous. 2.Thissnakeisarattlesnake. —————————
3.Thissnakeispoisonous.
The two assumptions in steps (1) and (2) are both required in order tosupporttheconclusionin(3).Neitherassumptionbyitselfissuficient toimplytheconclusion.Theadditiverelationshipbetweentheassump-tionsinupholdingtheconclusionisdiagrammedinthisway:
(1)+(2)
↓ (3)
have three, four, or, indeed, any number. Here is another example, this timeinvolvingthreeadditiveassumptions:
1.EithertheRepublicansortheDemocratswillwintheSenate.
2.IftheRepublicanswintheSenate,universalhealthcarewillbeindeinitelydelayed by their desire to appease conservative members of the AMA who oppose universal healthcare.
3. If the Democrats win the Senate, universal health care will be indeinitely delayed becauseofinternalpartydisagreementabouthowtoinancethebesthealthcarepak-kage.
—————————
4.Universalhealthcarewillbeindeinitelydelayed.
Thisargumentiscorrectlydiagrammedasinvolvingthreeadditiveas-sumptions in (1), (2), and (3), all of which are required to uphold the conclusionin(4):
(1)+(2)+(3) ——————
↓ (4)
Alternatively,assumptionscanbenonadditiveorindependentinthat theydonotallneedtobecombinedorsupposedjointlytoholdtruein ordertosupporttheconclusion.Thisoccurs,forexample,whenseveral different reasons each give suficient grounds to uphold a conclusion. Nonadditive or independent assumptions in argument are diagrammed by writing the assumptions’ numbers above the conclusion or conclu-sions, dispensing with the short horizontal line required in the case of additiveassumptions,anddrawingseparatearrowsfromeach,departing atanangleandconvergingonnumbersrepresentingtheconclusionor conclusions.
Here is a simple case of two assumptions nonadditively or inde-pendentlysupportingthesameconclusion(therelationshipsdiagrammed belowcouldholdforanynumberoftwoormoreassumptions,andany numberofoneormoreconclusions).
1.5isanoddnumber. 2.7isanoddnumber. —————————
3.Thereisatleastoneoddnumber.
(1) (2)
(3)
Theintuitivetestforwhetherassumptionsareadditiveornonadditive iswhetherornottheconclusionwouldholdifoneoftheassumptionswere eliminated.Ifitappearsthattheconclusionwouldstillbeadequatelysup-ported even if an assumption did not hold, then most probably that as- sumptionisindependentoftheothers.If,ontheotherhand,itseemslike-lythattheconclusionwouldfailifanyassumptionwereeliminated,then theassumptionsareadditive,andmustbesupposedjointlytoholdinor-dertoimplytheconclusion.
Intheaboveexampleinvolvingtheoccurrenceofoddnumbers,ei- therassumption(1)byitselfor(2)byitselfwouldbeenoughtoguaran-tee the truth of the conclusion in (3) that there are at least some odd numbers.Thereasonisthatasingleexampleissuficienttoprovethat thereareatleastsomeinstancesofthekind.Ifweeliminateassumption (1),theconclusionstillfollowsonthestrengthof(2);ifweeliminateas-sumption (2), the conclusion still follows on the strength of (1). In the previoustwoexamples,bycontrast,theconclusiononrelectionappears tobeinadequatelysupportedifanyoftheassumptionsareeliminated.
4.DIVERGINGCONCLUSIONS
Theconclusionsthataresupposedtofollowfromeitheradditiveornon-additiveassumptionscanbealsobediverging.Thisoccurswhenasingle setofassumptionsinadditiveornonadditiveconigurationsupportsev-eral different conclusions. Consider, for example, the following argu-ment:
1.Tomisarationalanimal. —————————
2.Tomisrational. 3.Tomisananimal.
Thedivergingconclusionsin(2)and(3)arediagrammedtoshowthat theyareequallyimpliedbyassumption(1),inthisway:
(1)
Multipleconclusionscanalsodivergefromadditiveaswellassingle assumptionsinanargument.Hereisaspecimenargumentofthistype:
1.Picassowasagreatpainter.
2.Allgreatpaintersaretrueartistsandvisualpoets. —————————
3.Picassowasatrueartist. 4.Picassowasavisualpoet.
The assumptions in (1) and (2) are clearly additive, since neither of thedivergingconclusionsin(3)or(4)followsfrom(1)aloneor(2)alone. Thediagramforthisargumentwithdivergingconclusionsfromadditive assumptionshasthisform:
(1)+(2) —————
(3) (4)
Multiple conclusions diverging from nonadditive assumptions are represented simply as parallel basic inferences. This is exhibited in the followingargumentandaccompanyingstandarddiagram:
1.Tomisarationalanimal. 2.Oaksaredeciduoustrees. —————————
3.Tomisananimal. 4.Oaksaretrees.
(1) (2) ↓ ↓ (3) (4)
5.IMPLICITARGUMENTCOMPONENTSFORENTHYMEMES
re-ach a deductively valid inference, and, perhaps more importantly, to appreciate the argument on its merits more in the way that we may imagine it to have been intended. We may want to learn more from the argument by interpreting it as involving suppressed argument components by which a variety of background considerations can be said to have been justiiably taken for granted by the argument au-thor.
Hereisanexampleofanargumentthatbeneitsfromtheinterposi-tionofimplicitassumptions,alongwithadiagramrepresentingwhatwe mightplausiblyregardasitsrealmoredeeplyunderlying,ratherthanex-plicitlystated,structure.Theargumentisamodiicationoftheprevious example.First,weseetheargumentindeductivelyincompleteformasit might be most directly reconstructed from its enthymematic ordinary languageexpression.
1.Tomisacat. ————————— 2.Tomisananimal.
Theargumentevidentlyleavesoutanimportantassumptionneeded make the inference valid. This is obviously something like the assump-tion that ‘All cats are animals’, or ‘Whatever is a cat is also an animal’. Themissingorimplicitassumptionissuppliedinthisreconstruction,ac-cording to the principle of charity, using the standard bracketing con-ventionforimplicitargumentcomponentsdescribedabove:
1.Tomisacat.
[a.Allcatsareanimals.] ————————— 2.Tomisananimal.
Now the conclusion in (2) explicitly follows deductively from (1) and [a]. Diagramming this expanded version of the argument by the convention described for arguments with implicit assumptions, we have:
(1)+[a] ——————
↓ (2)
and implicit additive and nonadditive assumptions and stated and im-plicit converging and diverging conclusions. The standard method can beusedtodiagramanyinferentialrelationshipsthatbelongtothesecat-egories,bothsimpleandthemostinterestingandcomplex,builtupout ofsimpleinferenceunitsincomplicatedconigurations.Themethodcan also be extended to represent disputes involving the interrelation and contradictionofargumentsandcounterarguments.
6.SUMMARYOFSTANDARDDIAGRAMMINGTECHNIQUES
Thestandarddiagrammingpatternsforargumentswithsixtypesofrela-tionships between assumptions and conclusions are summarized in the followingchart.
STANDARDARGUMENTDIAGRAMMINGCONFIGURATIONS
(1) (1)+(2) (1) (2) (1)+[a] (1) (1)+(2)
————— ————— —————
↓ ↓
↓
(2) (3) (3) (2) (2) (3) (3) (4)
Basic Additive Assumptions
Nonadditive Independent Assumptions
Implicit
(Additive) Components
DivergingConclusions
Thebasicpatternsinthesestandardrulesfordiagrammingargumentsare foundinmostintroductoryinformallogictexts.Theyaresimilarinob- viouswaystothediagrammingtechniquesusedforanalyzingthenoun- verb-modiier(etc.)structureofsentencesinstandardtreatmentsofnat-urallanguagegrammars3.
An application of the standard diagramming method to a relatively complexargumentisseeninthefollowingexampleconcerningthemet-aphysicsofsubstance.
1.Substanceiseternal. 2.Timeisininite.
3.Iftimeisininite,thenwhateveriseternalisuncreatedandenduresthroughoutin-initetime.
3
4. If the world appears to change because substance is constantly changing, then the appearanceandsubstanceoftheworldareidentical.
5.Substancecannotremainthesamefrominstanttoinstant,becauseitsexistenceex-plainstheworld’scontinuouslychangingappearance.
6.Whateverenduresthroughtimeisconstantlychanging. —————————
7.Substanceenduresthroughoutininitetime. 8.Substanceisuncreated.
9.Substanceenduresthroughtime. 10.Substanceisconstantlychanging.
11.Theworldappearstochangebecausesubstanceisconstantlychanging. 12.Theappearanceandsubstanceoftheworldareidentical.
Thecomplexrelationsbetweentheassumptionsinpropositions(1)–(6), andthesubconclusionsandmainconclusionsinpropositions(7)–(12)can bestandardlydiagrammedinthisway:
7.LIMITATIONSOFSTANDARDDIAGRAMMING
Thediagrammingmethodprovidesawayofrepresentingsomeofthein- ferencerelationshipsthatholdbetweenanyargument’sstatedandimpli-cit assumptions and conclusions. There are nevertheless some important elementsofthelogicalstructureofanargumentthatthediagrammingme-thoddoesnotpicture.Thestandarddiagrammingmethodiscompletein asense,asfarasitgoes,butitisalsolimited.Itisnotasinformativeasit might be about the inferential relationship between certain kinds of as-sumptionsandconclusions.
Wenoticeatoncethatthediagrammingmethoddoesnotdistinguish betweenmanytypesofwidelydivergingargumentsthatallsharetheex-actsameinferencediagram.Thisisclearinthecaseofanyargumentin which two assumptions are additively required to deduce a conclusion. Allsuchargumentsmustbediagrammedinpreciselythesameway,re-gardless of the content of the assumptions and conclusion, and regard-lessofthelogicalconnectionsandrelationsthatmayobtainbetweenthe assumptions and conclusion by virtue of which the conclusion is sup-posed to follow from the assumptions. Here are two examples of quite distinct arguments which we are required by standard diagramming methodstopictureashavingtheverysameinferencestructure.
1.Ifitisraining,thentherooftopsarewet. 1.AliceistallerthanBob. 2.Itisraining. 2.SomeoneistallerthanAlice.
————————— —————————
3.Therooftopsarewet. 3.SomeoneistallerthanBob.
The two arguments are logically fundamentally very different from oneanother.Theargumentontheleftisafamiliarformofconditional detachment ormodus ponendo ponens. The argument on the right is an inferenceinvolvingthetransitivityoftherelationorrelationalproperty of being taller than. Both arguments are, nevertheless, standardly dia-grammedinexactlythesameway,bythefamiliaradditiveassumptions diagram, with a horizontal line beneath the two assumption numbers joinedbyanadditionsign,fromwhichaverticalarrowbelowpointsto the conclusion number. The two arguments share precisely the same standarddiagramform:
(1)+(2) ————
Although the standard diagramming method tells us something aboutanargument’sinferentialstructureintherelationbetweenitsas- sumptionsandconclusions,italsoleavesoutimportantfeaturesthatide-allywemightliketohaverepresented.Noticeinparticularthatinusing the standard diagramming method we have no good way to represent thebranchingstructureofdilemmaarguments,inwhichtwo(ormore) choicesleadtothesameconclusions.Thisisseeninacomparisonofthe followingtwoarguments:
1.Eitheritwillrainorsnow. 1.Rosesarered. 2.Ifitrains,thentherooftopswillbewet. 2.Violetsareblue. 3.Ifitsnows,thentherooftopswillbewet. 3.Sugarissweet.
————————— —————————
4.Therooftopswillbewet. 4.Rosesarered,violetsare
blue,andsugarissweet.
Theargumentontheleftisadisjunctivedilemma.Theargumenton therightisaconjunctiveinference,inwhichtheconclusionmerelycol-lectstogetherthethreeassertionsindividuallystatedbytheassumptions. Despitethesedifferences,bothargumentsonceagainmustbestandard- lydiagrammedashavingpreciselythesameinferentialstructure.Thear-gumentsareadditiveasabove,thoughthistimeeachincorporatesthree insteadoftwoadditiveassumptionsinsupportoftheconclusion.They sharethiscommonform:
(1)+(2)+(3) ——————
↓ (4)
Anotherlimitationofthestandarddiagrammingmethodisitsinabil-ity to depict the inferential relation between the conclusions and as- sumptionsofadeductivelycircularorquestion-beggingargumentorpe-titio principii. The standard method depicts logical inference in apetitio as extending from assumptions to conclusions, but fails to depict the backwardinferencefromconclusionstoassumptionsbyvirtueofwhich anargumentiscaughtincircularity.Thestandarddiagramofacircular argumentisindistinguishablefromthestandarddiagramofmanynon-circulararguments.Considertheseinferences:
1.Godexists. 1.Godexists.
————————— —————————
Bothargumentsarenaturallydiagrammedinthesameway,withthe same numbering of assumptions and conclusions, and an inference ar-row extending from the one and only assumption to the one and only conclusion,(1)→ (2).Yettheargumentontheleftisunmistakablycir-cular,whiletheargumentontherightisnot.Wecanwritethevertical equivalentof(1)→(1),(1)|(1)or(1)→(2),(1)|(2),where[(2)=(1)], to show that the argument’s conclusion merely repeats one of the as-sumptions. There are unfortunately objections to this practice in some cases,whichmakestheproposalunsuitablefordiagrammingallcircular reasoning. The problem is that in many arguments circularity does not appear simply between a proposition and itself, but, as far as the argu-ment’s inferential structure is concerned, by means of syntactically dis-tinctpropositions,insomeinstances,quitedistantfromoneanotherin immediatelexicalcontent.Moreimportantlystill,theproblemincircu-larargumentisnotmerelythat(1)→(2),where(2)insomewayrestates (1),butratherthat(2)→(1).Thisistrueoftheargumentaboveonthe left,butnotoftheargumentontheright.Toshowthisstructuralfeature of circular reasoning we need an arrow that literally circles back from aconclusionoftheargumentinwhichitoccursandsinglesouttheas-sumption by virtue of which its conclusion is trivialized. We need in compactdiagrammaticformthefactthatwhere(1)→(2),itisalsothe casethat(2)→(1),asprovidedintheenhancement.Ifwearesensitive totheparticularitiesofcircularargumentsmoregenerally,incomplexas well as in the simplest applications, then we will already be aware that circularitysometimesonlyaffectspartofanargument,andthatitisof-ten useful to know at a glance exactly which sub-inferences are caught upincircularity,andwhicharefreeofthatcomplaint.
thanparenthesesinthediagram.Weshallgenerallyfollowaversionof this convention, although other equally and potentially more informa-tivealternativegraphicdevicesmayalsobeavailable.
8.PROPOSALTOENHANCESTANDARDDIAGRAMMING
The moral of these illustrations is not that standard diagramming is hopelesslyfaulty.Thepointisratherthatweshouldbeawareofsomeof thelimitationsofthediagrammingmethod,andnotexpectmoreinfor- mationfromthemethodthanitiscapableofproviding.Logicallyinter-esting features by which some arguments are distinguished even infor-mallyfromoneanotherareinvisibletostandarddiagramming.
Perhapsthemainfailureofthestandarddiagrammingmethodisits failuretoexhibitanyoftheinternalstructuralfeaturesofassumptions andconclusionsthatarerelevanttotheirinferentialrelationships.The standard method has no way of showing that one proposition is the negationofanother,orthatapropositionisdisjunctive,conjunctive, conditional,orbiconditionalinform.Thesecharacteristicsofproposi-tions are vitally important to the logical connecconditional,orbiconditionalinform.Thesecharacteristicsofproposi-tions that govern in-ference relations between an argument’s assumptions and conclu-sions.
Itisworthwhileforthesereasonstoconsidersubstantialrevisionsof thestandarddiagrammingmethod.Forthepresentitwillsuficetoillus-tratethepossibilitiesofenhancingstandarddiagrammingbyproposing innovationsthatwillmakeitpossibletodiagramtheinternalstructures of arguments more sensitively with respect to the internal logical form and content of their assumptions and conclusions, in these categories: (i) circular or question-begging arguments (petitio principii); (ii) disjunc-tive dilemma (and disjuncdisjunc-tive syllogism); (iii) conditional inferences (modus ponens,modus tollens, hypothetical syllogism, and combined types);(iv)reductioadabsurdumarguments.
9.DIAGRAMMINGCIRCULARITY
fromtoptobottominthediagram,asweexpectinanoncircularargu-ment,butalsobottomtotop,asthemarkofcircularreasoning4.
Toshowthecircularitythatobtainsinargumentswheretheconclu- sionmustalreadybeacceptedinordertoaccepttheargument’sassump-tionsinthesimplestcase,suchasthereiterativeexampleaboveinwhich theconclusionthatGodexistsisdeducedfromtheassumptionthatGod exists,wemakeuseofa(semi-)circleorloopinginferencepattern:
Insomeapplications,wecanrepresentinferentialcircularityasprevi-ouslymentionedbywriting(1)→(1)or(1)|(1)5
.Suchamethodispro-posed already by Beardsley, but it is clearly suboptimal6. If our purpose
istodifferentlynumbereverysyntacticallydistinctproposition,howev-er,thenthisstrictlyreiterativediagrammingdevicewillnotbeequalto thetask.IfIargue:Anininitespiritreignssupremethroughouttheuni-verse,therefore,Godexists,Iwillhaveengagedinanespeciallyblatant manifest circularity, but the reasoning is not readily represented by (1)→(1)or(1)|(1).Ifwetrytoillinthesuppressedassumptionsinthis way: (1) An ininite spirit reigning supreme in the universe exists + [(a)God=anininitespiritreigningsupremeintheuniverse]→(2)God exists,thenthecircularityisjustaspresentasbefore,butitisnotgraph-icallydisplayedeitheras(1)+(2)→(3),(1)+(2)→(1)or(1)+(2)→(2). Theirsteffortdoesnotgraphicallyindicatecircularityevenbylabeling, andthesecondtwodonotaccuratelynumberdistinctpropositionsex-pressingdifferentmeanings.
Circularity,eveninasingleargument,isoftenmoredificultthanthis to represent. Question-begging inferences that are spread out over a se-ries of subarguments can occur in which one conclusion or subconclu-sion is linked to another main conclusubconclu-sion in a circular coniguration. Hereisanexample:
(A) 1.Ifitishardtoindthecircularityinaseriesofarguments,thencircularitycantake usbysurprise.
2.Itissometimeshardtoindthecircularityinaseriesofarguments. —————————
3.Circularityinsomeseriesofargumentscantakeusbysurprise.
4 Cf.Jacquette(1993);Jacquette(1994a).
5 Throught,‘→’innotthematerialconditional,butahorizontalsignforthevertical
inferentialarrowinstandardargumentdiagramming.
(B) 1.Tobetakenbysurpriseistoencountertheunexpected.
2.Thecircularityinaseriesofargumentisunexpectedonlywhenitishardtoind. [a.Thecircularityinsomeseriesofargumentscantakeusbysurprise.]
[from(A)]
—————————
3.Itissometimeshardtoindthecircularityinaseriesofarguments.
Thecircularityisseeninthefactthattheconclusionof(B)followsin partfromtheconclusionof(A),whileitisalsoatthesametimeoneof(A)’s assumptions.Thesamepropositionappearsas(B3)and(A2),whichincon-tent are precisely identical. This constitutes a circularity in the series (A)– (B).Themainconclusionof(B),whichderivesinpartfrom(A),isalready assumedin(A),inthiscircularseriesofarguments.Wemustalreadyaccept theconclusionof (B)in order to accept thesecond assumption of(A),in ordertoderivetheconclusionof(B).Thus,weassumewhatwearetrying toprove.Thecircularitycanbediagrammedinaself-explanatoryway:
Diagramming circularity by a (semi-) circular or looping graphic de- viceistheeasiestandmostobviousmethodofenhancingstandarddia-gramming in order to represent an important inferential structural fea-tureofmanyargumentsthatisotherwiseoverlookedorgraphicallyless informativelydepictedbythestandardmethod.
10.CONVENTIONSFORDIAGRAMMINGCONTRADICTION ANDDISJUNCTION
Tocontinue,werequirespecialconventionsforrepresentingcontradiction anddisjunction.Wecanborrowaniconusedindiagrammingdisputesbe- tweenmultiplearguments,bywhichconlictarrowsbisectedbyshortslant-barsindicatethatthepropositionssorelatedaremutuallycontradictory7.
7
Thus,inthediagrambelow,propositions(1)and(2)areshowntobelogi-callyincompatibleorcontradictories:
Exploiting the fact that in classical inference semantics a contradic-tionimpliesanyandeveryproposition(sometimesknownasoneform of the paradox of material implication and one form of the paradox of strictimplication),wecanfurtheruseconlictarrowstogetherwiththe horizontal line and inference arrow to diagram the inference of any proposition(3)fromanymutuallylogicallyincompatibleorcontradicto-rypropositions(1)and(2),inthefollowingT-patterndiagram:
Thisdevicewillproveusefullaterindiagrammingconditionalinfer-ences and reductio ad absurdum arguments. Alternative conventions might be developed for nonclassical valid inference systems, such as thoseavailableinrelevanceandparaconsistentlogics.
Anothernewdiagrammingmethodisrequiredtorepresentthecom- positionofadisjunctionbyitsdisjuncts.Ifproposition(1)isthedisjunc-tionaorb,thenwecanpictureitsinternaldisjunctivelogicalstructure inthefollowingintuitivewayasadeltapattern.
Proposition(1)isshowntoconsistofthetwodisjuncts,aandb.The twodisjunctsarelabeledbylettersofthealphabetratherthannumbers, enclosedwithinboxes,attheendsofdiverginglines.Thisconventionis appropriate,becausebyhypothesisthedisjunctsarenotexplicitlygiven in the argument statement as distinct propositions, but are contained withinthedisjunction.Thereisnothingsigniicantaboutthechoiceof
boxes to enclose the terms representing the disjuncts, but something comparableisneededtodistinguishthemgraphicallyfromtheparenthe- sesusedtorepresentstatedargumentcomponentsandthesquarebrack-ets used to represent the implicit argument components in the recon-structionofenthymemes.Theboxedtermsappearattheendofdiverging linesinthedeltaconigurationtodepictthefactthatthedisjunctional- lowsthetruthofeitheroneortheotherorbothpossibilitiesrepresent-edbyitsdisjuncts.Theideaistoshowsomethinglikealternativepaths orchannelsthatmightbetaken.Thelinesaretippedwithinferencear-rowheads. Divergent arrows terminating in boxed letters as opposed to parantheticalnumbersdonotindicatedivergentconclusions,butdiver-gentpossibleconclusionsadivergentdisjunctivepossibilities.
11.DIAGRAMMINGDISJUNCTIVEINFERENCES
The conventions proposed in the previous section make it relatively easy to diagram disjunctive arguments. We notice that in the absence of such methodsdisjunctiveargumentsarenotdistinguishedashavinganyspecial inferentialstructurebystandarddiagramming.Thestandardmethoddoes notexhibittheinternallogicalconnectivesbywhichsimplerpropositions aretruth-functionallycombinedintomorecomplexcomposites.
Consider an elementary instantiation of argument by disjunctive (sometimescalledconstructive)dilemma:
1.EithertodayisMondayortodayisTuesday. 2.IftodayisMonday,thentodayisaweekday. 3.IftodayisTuesday,thentodayisaweekday. —————————
4.Todayisaweekday.
Weshallreturntothisexampleinthenextsectionimmediatelyfol-lowing. There we will further exhibit the logic of conditional inference from the boxed disjunctsa andb in the disjunctive composition of (1) andtheconditionalpropositionsin(2)and(3)(unanalyzedhere)ascon-vergingontheconclusionin(4).
Thefruitfulnessofthisenhancementofstandarddiagrammingtech-niques is seen in the following application. Disjunctive syllogism is a popularargumentforminwhichadisjunctionisadvanced,andallbut one of the disjuncts are rejected, from which the remaining disjunct is validly inferred. This is also the underlying logic of reasoning by ‘ex-hausting the alternatives’, leaving the one unrejected possibility as the only conclusion. By the enlargement of the diagramming method pro-posedhere,inferencesbydisjunctivesyllogismarereadilyidentiiableas specialcaseinstancesofdisjunctivedilemma.
Takethefollowingdisjunctivesyllogismasanexample.
1.EitherthePro-LifeadvocatesorthePro-ChoiceadvocateswilltriumphintheirSu-premeCourtbattle.
2.ButthePro-Lifeadvocateswillnottriumph. —————————
3.ThePro-ChoiceadvocateswilltriumphintheirSupremeCourtbattle.
Thenotationbywhichapropositioninthediagramisidentiiedun- derdifferenttermsprovidesausefulreminderoftheproposition’scon-tent, or what it expresses, but is unnecessary in applying the enhanced diagrammingmethodtorepresentmoresensitivelytheinternalproposi-tionalrelationswithinaninferentialstructure.
12.DIAGRAMMINGCONDITIONALINFERENCES
Thefactthatconditionalpropositionsaretruth-functionallyreducibleto disjunctions, in which the negation of the conditional’s antecedent is disjoinedwiththeconditional’sconsequent,canbeinvokedtoenhance thestandarddiagrammingmethodinrepresentingconditionalinferenc- essuchasmodusponens,modustollens,hypotheticalsyllogism,andrelat- edforms.Anyconditionalpropositioncanbediagrammedasadisjunc-tionoftwodisjuncts,toadoptaunivocalconvention,withthenegation oftheantecedentontheleftfork,andtheconsequentontheright.
This choice represents an interpretation of the conditional in infor- mallogicasamaterialconditional,andassuchrequiresjustiication.Al-though the diagramming method developed here is intended as an ad-juncttoinformaldeductivereasoningintheparadigm,weconsiderthe materialconditionalalsoasitisdeinedinformalsymboliclogic.Theex-planationisthat:
(1)Itissimpler,moreunivocalandbetterunderstoodthanothercon-ditionals.
(2)Thereisnothinginherentlyformalistaboutunderstandingcondi-tionalstatementsinrelationtodisjunctionandnegation,which,indeed, canbedoneformallyorinformally.
(3)Byinterpretingconditionalseveninformallyasthematerialcon-ditional, also interpreted informally, we establish and illustrate the fact thattheresourcesfordeiningthematerialconditionalarealreadyavail-abletoinformallogic,ancillarytotheconceptofamathematicallogical truthfunction.
(4)Therearemanycandidateconditionalsbeyondthematerialcon-ditionalwhosedeductiveinferentialstructure,asin modusponensandtol-lens, is properly explicated in relation to disjunction and negation (or equivalently,butgraphicallymorecomplexly,toconjunctionandnega-tion), and it would be prejudicial even to their informal critique to choosefromamongthem.
typesofconditionalsoverotherswithnoavailablegraphicguidelinesas to how each might be applied to ostensibly different colloquial argu-ments, where there is in fact no better way to represent the inferential logicalstructureinvolvinganynonmaterialconditionalpropositionex-ceptbythepreviouslydiscountedequivocal(X)→(Y).
(6) Finally, informal logic need not be seen as isolated from formal symboliclogic,butratherasanally,partofaspectrumofmethodscon-tributingindifferentwaystoacompletelogicalanalysisofanargument. Wemaybegininformallyatsomelevelorpointofhistoricalorigin,and end with the most sophisticated notations, axiomatizations and algo-rithmsofcontemporarymathematicallogic,settheory,andtheirformal semantics. If informal logic is understood as at least potentially part of suchaspectrumofmethods,ifwedonothavegoodreasonsforexclud-ing it as such, then enhanced diagrammsuchaspectrumofmethods,ifwedonothavegoodreasonsforexclud-ing in informal logic, without shaming its informal ethos, can nevertheless welcome the graphic dia- gramming,itselfaninherentlyformalactivityandend-result,ofinferen-tial relations in ways that dovetail smoothly with their counterparts in formal symbolic logic8. The example of note in the present application
is that of reading the logical structure of conditionals in enhanced dia-grammingasthatofmaterialconditionals.
There still remains the greatest difference in the world between for-malandinformallogicaltreatmentofthematerialconditional,sinceour preferred choices for enhanced argument diagramming do not, signii-cantly, have anything to say about truth values, the conditional as atruthfunction,abouttruthtabledeinitionsoftheconditionalbycas-es,oraboutdecisionmethodsfortheconditionaloranyoftherestofan algebraic propositional logic to which the material conditional might formallybelong,andintermsofwhoseinterrelationswithwhichitcan onlybefullyunderstood.Enhancedargumentdiagramming,thoughstill squarelypartofinformallogic,therebymakesapointofpositivecontact with symbolic logic and facilitates one transition of analytic methods fromestablishedinformaltoestablishedformalconcepts,notations,and techniques. For these reasons, and with this informal justiication, we considerthedeductivelogical-inferentialstructureofconditionalreason-ingtobethatofthematerialconditional,forpurposesofadvancingan enhanceddiagrammingmethodininformallogic.Thealternative,again, seemsonlytobetheratherlogicallyopaqueandindistinctone-size-its-all diagramming of conditionals generally as (X)→ (Y). We show a de-ductivelyinferentiallyrelevantinternalstructurebelongingto(X)→(Y)
byrelatingthe→symboltopriorgraphicallyinterpretedandintuitively transparent informally understood devices for diagramming negation anddisjunction.Ifthereisbothapreferrednonmaterialconditionalthat contributes to deductively valid reasoning and can be distinctively dia- grammedastoitslogicalstructureandcontributiontodeductiveinfer-ence as something more logically informative than (X)→ (Y), then it would make a splendid addition to enhanced diagramming in formal logic to set beside the proposed graphic analysis of the material condi- tional.Suficeittosaythatsuchaproposalhasyettoappearontheho-rizon.
Continuingnowwiththemostbasicformofmodusponens,werepre- senttheinferencebytheidenticaldiagramusedabovetodepictthelog-icalstructureofonebasicformofdisjunctivesyllogism,indicatingtheir logicalequivalenceandinterreducibility.Hereisaconditionalmoduspo-nensrephrasingoftheargumentusedtoillustratedisjunctivesyllogism:
1.IfthePro-Lifeadvocatesdonottriumph,thenthePro-Choiceadvocateswilltriumph intheirSupremeCourtbattle.
2.ButthePro-Lifeadvocateswillnottriumph. —————————
3.ThePro-ChoiceadvocateswilltriumphintheirSupremeCourtbattle.
Thediagramfortherevisedargumentbytheproposedenhancement hastheverysamepictorialstructure,avariationofthediamondpattern, asthatpresentedfordisjunctivesyllogism:
1.IfthePro-Lifeadvocatesdonottriumph,thenthePro-Choiceadvocateswilltriumph intheirSupremeCourtbattle.
2.ButthePro-Choiceadvocateswillnottriumph. —————————
3.ThePro-ChoiceadvocateswilltriumphintheirSupremeCourtbattle.
The diagram formodus tollens has this corresponding form. Conver-genceontheconclusionthistimeiseffectedbycontradictionwiththe argument’sminorassumptionontheright,ratherthan,asinthecaseof modusponens,ontheleft.
By extensions of the same diagramming methods, the logical infer-enceinhypotheticalsyllogism,involvingtwoconditionalassumptions, and a conditional conclusion, in which the antecedent of the irst as- sumptionandtheconsequentofthesecondassumptionarecondition-allyrelatedasantecedentandconsequentoftheconclusion,canalsobe depicted.Hereisadiagraminwhatweshallcallthebutterlypatternfor asimpleexampleofhypotheticalsyllogism:
1.Ifwewinthematch,wewinthegame. 2.Ifwewinthegame,wewinthetournament. —————————
3.Ifwewinthematch,wewinthetournament.
Structureslikethisarenowobtainedforrelatedinferencescombining thefeaturesofhypotheticalsyllogismwithmodusponensormodustollens. The irst is an argument form in which conditional assumptions of the form,IfP,thenQ;IfQ,thenR;andanassumptionoftheirstcondition-al’santecedentP;supportingtheconclusion,therefore,R.Thepatternof interpretation by reduction of conditionals to equivalent disjunctive formsispredictableenoughatthispointtoseeataglancehowtheen- hanceddiagrammingmethodworkswhenappliedtoreductionsofsuc-cessiveconditionalsasdisjunctions.
1.Ifyougiveamouseacracker,shemightwantacookie. 2.Ifamousemightwantacookie,shemightalsowantamufin. 3.Yougiveamouseacracker.
—————————
4.Amousemightalsowantamufin.
ItisimportanttonoticethatalthoughpropositionQissharedbythe irstandsecondconditionalassumptions,asconsequentoftheirstand antecedentofthesecond,thesametermsdonotappearassharedbythe conditionalswhentheyarereducedtodisjunctiveform.Thisisbecause the consequentQ in the irst conditional is retained upon reduction to the equivalent disjunction, while the antecedent of the second condi-tional is reduced in the equivalent disjunction as not-Q. These are ac-cordinglydiagrammedwithdifferentboxedalphabetletters,bforQ,and cfornot-Q,andtheirmutuallogicalincompatibilityorcontradictionis representedbytheconlictarrowconvention.
Thesecondexampleisthemirror-imageoftheirst,inthesameway and for the same reason that the modiied diamond diagram formodus tollensis the mirror-image of the modiied diamond diagram formodus ponens. Consider the problem of diagramming conditional arguments likethefollowing:
1.IfIcouldaffordit,I’dbuyyouanewcar. 2.I’dbuyyouanewcar,ifmoneygrowsontrees. 3.Moneydoesn’tgrowontrees.
—————————
Asainalexampleinthiscategory,letusreturntothedisjunctivedi-lemma considered earlier. The irst assumption is a disjunction of the form,PorQ,andthesecondandthirdassumptionsareconditionalsof the general form, IfP, thenR, and IfQ, thenR. The conclusion thatR follows by dilemma. Now we can represent the logical structure of the argumentinmoredetailbyfurtherreducingthesecondandthirdcon-ditional assumptions to disjunctive form. When we do this, the en- hanceddiagrammingmethodyieldsthefollowingvariationofabutter-lydiagramofitsinferentialrelations:
13.DIAGRAMMINGINDIRECTPROOFORREDUCTIOADABSURDUM
In order to diagramreductio ad absurdum arguments, it is necessary to haveaconventionforrepresentingassumptionsintroducedonlyashy- pothesesforpurposesofindirectproof,toberejectedwhenacontradic-tionisderivedinpartorinwholefromthem.
Whereasinnon-reductioargumentstheassumptionswithwhichan argument begins remain the argument’s assumptions throughout, in reductio arguments the negations of false assumptions are asserted as the argument’s conclusion. For this reason, it is appropriate in dia-grammingreductioargumentstodistinguishreductioassumptionsfrom the other (sincere) assumptions in an argument by placing the num- bers(oralphabetlettersinthecaseofimplicitreductioassumptionsre-quired in the reconstruction of enthymemes) that represent them withinanglebrackets:<>.
analogous to the procedure involving square brackets in reconstructing enthymemeswithimplicitargumentcomponents.Considerthisreductio argument:
<1.SusanwillnotrunforPresident.>
2.IfSusandoesnotrunforPresident,thenMarkwillnotrunforVice-President. 3.MarkwillrunforVice-President.
—————————
4.MarkwillnotrunforVice-President. 5.SusanwillrunforPresident.
Theinferenceisnotdificulttorepresent.Thebasicreductiostrategy can nevertheless be built into much more complicated argument struc-tures. The logic of the argument is pictured in this diagram. The essen-tialdiagrammingelementistheT-pattern:
Theargumentisdiagrammedbyindicatingtheessentialfeaturesof reductio assumptions and the absurd consequences that are supposed to follow from them. Angle brackets mark thereductio assumption to be rejected upon deduction of a logically contradictory consequence. To indicate that a logically contradictory consequence has been de-duced,weuseconlictarrows,aspreviously,torepresentthefactthat the propositions represented by corresponding numbers are logically incompatible.
Onefurtherconventionthatwedeliberatelydonotadoptistheuse of such a device as ‘<’ and ‘>’ or ‘<<’ and ‘>>’ to mark theconclusions inareductioinferencethatfollowonlyfromthereductiohypothesisor hypotheses.Themainreasonforthisdecisionisthatmanyreductioin- ferencesaredeductivelyvaliddespiteundercertaintyastowhichcon-clusionshouldberegardedasdependentonthereductiohypothesisor hypotheses.Thisisoftenamatterofdisputebetweenanargumentau- thorandtheargument’scritics,andweshouldnotprejudicetheprop-er interpretation merely in diagramming the argument’s inferential structure.
trackofthedirectinferencesfromreductiohypothesesintheargument. Suchinformationmaybeusefulwhenwehaveit,butoftenindiagram-ming arguments we do not know areductio argument author’s inten- tions.However,themostimportantobjectiontomakingthedirectcon-clusionsofreductioinferencesstandoutinthediagrammingconvention isthatitrequiresdiagrammertohavealreadyfullyinterpretedtheargu-mentbeforediagrammingitsstructure,whereasthepurposeofargument diagrammingispreciselytofacilitateanunderstandingoftheargument’s inferentialstructure.Makingthatkindoflabelingarequirementfordia-gramming puts the cart before the horse, and undermines one of the principalreasonsfordiagrammingthelogicalinferentialstructuresofde-ductive and other arguments. If we must know what conclusions are meantinearnest,andwhicharemerelytheimplicationsofhypotheses weknowtobefalse,inordertodiagramanargument,thenthereseems littlepointinactuallymakingthediagram.Atleastinthosegeneralin- stanceswhenweareusingdiagramminginordertounderstandinferen-tial structure, we may prefer not to mark the diagrams more than we need to in order to relect the argument’s internal inferential relations. Wedonotwanttoimposerequirementsondiagrammingmethodsthat mightnotbeobviouslyorunivocallyfulilledinallrelevantapplications, that do not in any case contribute to our understanding of deductive connectionswithintheargument,andespeciallynotiftheirrestrictions precludeitfrombeingusedinallofthesortsofwaysexpectedofargu-mentdiagramming.Thediagramsareneverthelesstouseasweseeit,so we certainly can mark them in any way we choose, including annotat-ing the conclusions ofreductio inferences, if doing so serves a practical purpose.
Here is another example, a simpliied variation of Euclid’s famous proof that there is no greatest prime number. Suppose I want to prove thatthereisnogreatestevennumber.Iassumetheoppositeofthecon-clusionIhopetoestablish,bybeginningwiththepropositionthatthere is a greatest even number, which I call ‘N’. IfN is an even number, no matterhowgreat,IcanalwaysobtainanevennumbergreaterthanNby adding N+2.Thus,Ihavereducedtheassumptionthatthereisagreat-est even number to an absurdity, an outright logical contradiction — thatNisthegreatestevennumber,andNisnotthegreatestevennumber (sinceN+2isanevennumbergreaterthanN)9.
Thereductioargumenttoprovethatthereisnogreatestevennumber canirstbereconstructedinthefollowingway:
<1.Thereisagreatestevennumber,N.>
2.IfNisanevennumber,thenN+2isanevennumbergreaterthanN. —————————
3.N+2isanevennumbergreaterthanN. 4.Nisthegreatestevennumber.
5.Nisnotthegreatestevennumber. 6.Thereisnogreatestevennumber.
The diagram for this basicreductioargument has this form, where againtheT-patterndepictsthemainpointoflogicalinterest:
14.ENHANCEDMETHODILLUSTRATED
Todemonstratetheadvantagesoftheproposedenhanceddiagramming method,weshallconsideracontrivedlogictextbookargumentofgratu-itousreductio reasoning, for purposes of illustrating enhanced diagram-mingofitsinferentialstructure,andcomparebothkindsofdiagrams.If webeginwiththereconstructedinference,thenwedonotidentifyany assumptionsashypothesesofthereductio.Theargumentstates:
1.Allhumansaremortal.
2.NoteveryonehasthegoodfortunetovisitCarthage.
3.Eitherit’snotthecasethatallhumansaremortal,orI’mamonkey’suncle. 4. If not everyone has the good fortune to visit Carthage, then I’m not a monkey’s uncle.
5.Ifit’snotthecasethatallhumansaremortal,thennoteveryonehasthegoodfor-tunetovisitCarthage.
————————— 6.I’mamonkey’suncle.
7.EveryonehasthegoodfortunetovisitCarthage. 8.It’snotthecasethatallhumansaremortal.
9.NoteveryonehasthegoodfortunetovisitCarthage.
Thissequenceiseasilyidentiiedasavalidinference.Howshouldit bediagrammedforpurposesofanalysisininformallogic?Thestandard diagrammingmethodhasthefollowingform:
(1)+(3) ————
↓ (6)+(4) ————
↓ (7)+(2) ————
↓
(8)+(5) (4)+(5)
———— ————
↓ ↓
(9) + (10) ——————————
(11)
15.EXTRACREDITPROBLEMONSHERLOCKHOLMES’LOGIC
Asainalexample,hereisachallengeforthereader.Theargumentappe-arsaspartofSherlockHolmes’reasoninginTheAdventureoftheCardboard Box.SirArthurConanDoyle’smasterdetectiveiscalledintoinvestigatea peculiaroccurrenceinwhichawomanhasbeensentasmallcardboardbox packedwithsaltandcontainingtwoseveredhumanears.
Holmesisevidentlyengagedinatrickybitofcogitationaboutthepos-sibilitiesposedbyMissCushing’sreceiptofthemysteriousandgrislybox. We can reconstruct this argument, as far as Holmes takes it in the above passage, with a bit of additional information from later in the story, and then diagram it according to the standard method as a preliminary step leadingtowardamorecompletecriticalevaluation.Theargumentisinter-esting,becauseitisenthymematic,andinvolvesmultipleimplicitinference components.Itcontainsanimplicitdoublereductioadabsurdumfromtwo distinctbutrelatedreductioassumptions,aninferencebydisjunctivesyllo-gism, and several types of conditional inferences. Thereductio subargu- mentsarenotcompleteduntillaterinthestory.Forconvenience,therel-evantassumption,unstatedinthepassagequotedabove,isintroducedin [e]asimplicit.Thefollowingreconstructionseemsappropriate:
1.Adoublemurderhasbeencommittedofthetwopersonswhoseearswerecontained inthebox.
2. Oneoftheearsissmall,inelyformed,andpiercedforanearring.
[a.Asmall,inelyformedearpiercedforanearringprobablybelongstoawoman.] 3.Theotherearbelongstoaman,andissunburned,discolored,andpiercedforanear-ring.
4. If the two persons to whom the ears belong were not dead (murdered), we would haveheardaboutwhathappenedtothembeforenow.
[b. We have not heard about what happened to the persons to whom the ears be-longbeforenow.]
5.TheearsweremailedtoMissCushingonThursdaymorning.
6.Ifthetwopeopleweremurdered,thenitwasthemurdererwhosenttheearstoMiss Cushing.
<7.IfthemurderersenttheearstoMissCushing,thenthemurdererhasastrongrea-sonforsendingtheearstoMissCushing.>
<8.IfthemurdererhadastrongreasonforsendingtheearstoMissCushing,theneit-her the murderer sent the ears to Miss Cushing to inform <8.IfthemurdererhadastrongreasonforsendingtheearstoMissCushing,theneit-her that the murder was done,ortocauseherpain.>
9. If the murderer sent the ears to Miss Cushing to inform her that the murder was done,thenMissCushingknowswhothemurdereris.
10. If Miss Cushing knows who the murderer is, then she would not have called the policeaboutreceivingtheboxofears.
[c.MissCushingcalledthepoliceaboutreceivingtheboxofears.]
11. If Miss Cushing knows who the murderer is and wishes to shield the murderer’s identity,thenshewouldhaveburiedtheears.
[d.MissCushingdidnotburytheears.]
[e.ThemurdererdidnotsendtheearstoMissCushingtocauseherpain.]
[f.IfitisnotthecasethatifthemurderersenttheearstoMissCushing,thenthe murdererhasastrongreasonforsendingtheearstoMissCushing,thenthemurderer mayhavesenttheearstoMissCushingbymistake.]
—————————
[h.Thetwopersonstowhomtheearsbelongaredead(murdered).]
12. The murder of the two persons to whom the ears belong occurred before Thurs-day.
[i.ThemurderersenttheearstoMissCushing.]
[j.ThemurdererhasastrongreasonforsendingtheearstoMissCushing.]
[k.EitherthemurderersenttheearstoMissCushingtoinformherthatthemurder wasdone,ortocauseherpain.]
13.MissCushingdoesnotknowwhothemurdereris.
[l.MissCushingdoesnotwishtoshieldthemurderer’sidentity.]
[m.ThemurdererdidnotsendtheearstoMissCushingtoinformherthatthemur-derwasdone.]
[n.ThemurderersenttheearstoMissCushingtocauseherpain.]
[o.ItisnotthecasethatthemurderersenttheearstoMissCushingeithertoinform herthatthemurderwasdone,ortocauseherpain.]
[p.ItisnotthecasethatifthemurderersenttheearstoMissCushing,thenthemur-dererhasastrongreasonforsendingtheearstoMissCushing.]
[q.ThemurderermayhavesenttheearstoMissCushingbymistake.]
