Bailey 2005 Phronesis

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Logic and Music in Plato's "Phaedo"

Author(s): D. T. J. BaileySource: Phronesis, Vol. 50, No. 2 (2005), pp. 95-115Published by: BRILLStable URL: http://www.jstor.org/stable/4182771 .Accessed: 22/04/2011 19:05

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Logic and Music in Plato's PhaedoD. T. J. BAILEY

ABSTRACTThis paper aims to achieve a better understandingof what Socrates means by"OVug(pcoveiv"n the sections of the Phaedo in which he uses the word, and howits use contributes both to the articulationof the hypotheticalmethod and theproof of the soul's immortality.Section I sets out the well-known problems forthe most obvious readings of the relation,while Sections II and III argueagainsttwo remedies for these problems, the first an interpretationof what the oug-(pwvrivrelation consists in, the second an interpretationof what sorts of thingthe relationis meantto relate. My positive account in Section IV argues thatweshould take the musical connotations of the term seriously, and that Plato wasthinkingof a robust analogy between the way pitches form unities when relatedby certainintervals,and the way theoretical claims form unities when related byexplanatoryco-dependence. Section V surveys the work of IV from the point ofview of the initial difficulties and suggests further consequences for the hypo-thetical method, including the logical relation between the a qDO)vEivnd&tpavEiv relations,and the need for care in orderingthe results of a hypothesis."But anyhow I proceeded in this way: on each occasion hypothesising the X6yoqwhich I judged to be strongest,I put down as true the things that seem to me toC1t9)DVEiVwith it - both abouta causal account and any of the otherthings thatare - but those things that did not I put down as false." (Phaedo 100a3-7)."But if someone clung to the hypothesis itself, you would bid him goodbye andwouldn't answer him until you had examined its results, whether according toyou they aCupovri or Stiwpvri with one another." (Phaedo 101d3-5).

I. TheDifficultyThese two passagesfromSocrates'SecondVoyage in the Phaedo createa famousproblemabouthow to understandwhat Socratesmeansby theverb augppovriv. One would expect the word to be univocal across itsinstances n thepassages,partlybecause theyappear o close together anabruptchangein the use of the same wordin such a short space of timeseems unlikely),and partlybecauseSocrates s here attemptingo give asense to what looks like the technicalvocabularyof his new method.Itis in giving a sense to technicalvocabulary hat one should be as self-conscious as possible about one's language: t would thereforebe mostunfortunatef we had to attributeambiguoususe to Socrateswhen he is? KoninklijkeBrill NV, Leiden, 2005 Phronesis L12Also available online - www.brill.nl


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doingjust that. But,on the face of it, it seemsthat we must either mputean ambiguity to his use of autgyqvciv, or give him a lunatic methodologythatno one could possibly follow.Let me startwith the secondpassage.Socrates seems to be telling usnot to go aboutgiving a justification X6yo;)'for a hypothesisuntil wehave examined its results(opglqlOCvtcx)nd seen that they are consistentwith one another (&XXkijot; ujqpxwi). Then and only then should we tryto give an explanationof the hypothesis romwhich we got thoseresults.Thatpiece of advicemakes good methodological ense, for the followingreason. If you can show that inconsistentpropositionsfollow from ahypothesis, hat'senoughfor you to know straightaway thatthe hypoth-esis isfalse - in fact self-contradictory even thoughestablishing hatnoinconsistencyresults is not sufficientto establishthat the hypothesisistrue. In other words, checkingthe results of your hypothesis or consis-tency is the best way to respond o the impatientnterlocutorwho wantsa justification or the hypothesisas soon as you have made it. For theneedto defendthehypothesiswith a X6yo; s surely abnegatedf you findout that it is self-contradictory, hich it will be if it yields inconsistentresults.So checkingyourbpgsOC'vtaor consistencyofferstheprospectoffinding,as soon as possible, somethingsufficient or a final decisiononthe truth-value f your hypothesis:why cast about for a justification ora claim beforeyou have satisfiedyourselfthat the claim is not necessar-ily false?

But there are two problems with understanding asprivriv as "to beconsistentwith".Firstly,that is plainlynot how Plato uses the word inothercontexts,particularlyCratylus436dlff. HereSocrates and Cratylusagreethatthe etymologiesdiscussed n the dialoguethus far indicate hatnamesgenerallyaugpovsi one anotherby virtueof theirall saying thattheir referentsarein flux. Socratesgoes on to arguethatthiskindof con-sistencyis no guarantee f truth: or all the agreementamongnames dis-covered so far, the namegiver mighthave been wrongfromthe startinsupposing hatthe worldis as Heraclitus aidit is. Butthis relationamongnames, even though it cannot guarantee he correctnessof any one of

Accepted January 2005I translate X6yo; as "justification" dvisedly, but not without unease. Translatingit as "proposition"here would surely be too weak: it cannot mean just any proposi-tion, but a special kind of proposition,a justificatoryone. Butpreviously, n the famous"looking in X6yot"passage at 99d3-e6, it ought not to have meant "justification", inceit is hardly illuminating to be told to look in justifications when one is searching forthe truth about things.


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LOGICAND MUSIC IN PLATO'S PHAEDO 97

them, is surely stronger than the logical relation of consistency. A set ofpropositions is consistent if and only if there is no disagreement amongits members: propositions having nothing to do with one another are con-sistent provided they could be true together. But when Socrates andCratylus agree that all the names they have discussed so far a(TqV@v6ioneanother, there is more to be said for them than that they lack disagree-ment. They have more in common than that by virtue of all telling thesame story about the world.

But that the word crops up in other contexts meaning a relation strongerthan mere consistency is a trifling difficulty compared with the second.The consequences are much graver if we suppose that Socrates means "tobe consistent with" by his use of cuVywveiv in the first passage above.There Socrates tells us that he puts down as true whatever bears this rela-tion to the hypothesis. But for any hypothesis, there will be an infinitenumber of propositions which are consistent with it, but which we haveno independent reason to assert. Surely we are not to set these down astrue simply because they bear a relation as weak as consistency to thehypothesis. It sounds ludicrous - and of course it results in disaster. Foramong this infinite number of propositions there will be pairs of propo-sitions inconsistent with one another. So if Socrates means nothingmore than "to be consistent with" by augpov?iv, then he is effectivelytelling us in the first passage to assert contradictions, and his method isutterly crazy.

These two problems urge the thought that the at)(pwvriv relation mustbe stronger than that of consistency. So perhaps it is as strong as entail-ment. Socrates would then be telling us in the first passage to put downas true all and only those propositions that are entailed by the originalhypothesis. In that case he would not be inviting us to assert propositionswe have no reason to hold true: for indeed you do have good reason toaccept2 a proposition if it follows from a proposition you have alreadyasserted. Nor will he be inviting us to assert contradictions in the first pas-sage - or at least, if we are invited to assert a contradiction on the basisthat one followed from the hypothesis, that would be sufficient for us toknow that the hypothesis is necessarily false.

2 I need to write "accept"ratherthan "assert"here. After all, the billion-fold nega-tion of "It's teatime" follows from "It's teatime".But just because I assert that it'steatime, I have no reason to assert its billion-fold negation, although I better hadaccept the latteras true whenever I do assert that it is teatime.


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But sadly this interpretation ill not do either.For it fails to make senseof augpovsiv in the second passage. There, it seems likely that Socratesis talking of a relationship uch that, if it holds betweenthe results of ahypothesis, hen that is reason to continuesupposing he hypothesis rue,but if it fails to hold, that is reason to suppose the hypothesis false.Whatever his relation s, it betterhad not be entailment, or it need notbe the case that the resultsof truehypothesesentail one another. t is truethat water is H20; fromthis, it follows that wateris a compound;and italso followsthatwatercontainsoxygen.But neitherof these resultsentailsthe other. A stuff need not contain oxygen in order to be a compound,while there is at least one stuff which contains oxygen without being acompound,namely oxygen itself, or at any rate a pure sample of it. IfSocrates means "entail one another"by &XXijXoti;c coxvei,hen theresultsof the hypothesisthat water is H20 do not XkkiXoi;ai,(pov6i.But that is no reason to deem false the hypothesis hatwateris H20.And of course thereis a more serious reasonfor rejecting he entail-ment reading,for it makes out that the first passagethis time advocatesthe same outrageouspromiscuitywe foundon the "consistency"nterpre-tation.It was ludicrous o put downas trueall the propositions onsistentwith the hypothesis because this involved us in, among other things,assertingcontradictions.But it is also ludicrous o put down asfalse allthe propositionswhich are not entailedby thehypothesis.Foronce againtherewill be an infinitenumberof them, includingpropositionswe haveindependenteasonto supposetrue,and also contradictory airsof propo-sitions. Of course there is nothing wrong with puttingdown as false aconjunctionof contradictory airs: hatis preciselywhat one oughtto do.But it follows from what Socratessays in the firstpassage,on the entail-ment reading,that we will have to putdown as false each memberof acontradictorypair if neither is entailedby the hypothesis,and that isindeed disastrous.From the hypothesis hatwater is H20 it does not fol-low that I am in my office;nordoes it follow, from the samehypothesis,thatI am outside my office.Butwe had betternot putdownas false boththese non-entailedpropositions,or it cannot be thatI am neither n myoffice noroutsideit.3So once againwe have an interpretationhatgivesSocratesan unusablemethod.

I It cannot be, of course, provided that we accept the Law of Excluded Middle, amore controversialprinciple than the Law of Non-Contradiction.But even so, sup-posing that Socrates might not subscribeto the Law of Excluded Middle is obviouslymuch too high a price to pay in order to save the entailmentreading, and would not


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LOGIC AND MUSIC IN PLATO'S PHAEDO 99II. The Current View

The best current olution o thisproblem s dueto Jyl Gentzler.4Reflectingontheseproblemsorboth he"consistency"nd"entailment"nterpretationsof the cFuio)v6iv relation, she argues that Socrates must be talking aboutan interimrelation,stronger hanconsistencybutweakerthan entailment:thatis, a relation orwhich consistency s necessarybutnot sufficient,andfor whichentailment s sufficientbut not necessary.A propositionq mustbe consistent with p in order o avupzovri p, but thatby itself is notenoughfor q to axjipovrvi p; if q is entailed by p, then it will otipqovci p, althoughit need not do the former n orderto do the latter.In what follows I shall be arguing that Gentzleris basically right inthinking hat the relation ies somewherebetween consistencyand entail-ment, but the devil is in the detailsthatneed spelling out in showing whatthat involves, and in relating those details to the hypotheticalmethodingeneralandtheargumentativeontext nparticular.Gentzler'sdetails,sadly,do not add up to a genuinerelation: or some propositions an perfectlywell entail others with which they are inconsistent, namely their ownnegations.In fact Gentzler'srelationwill both hold and not hold betweena contradiction nd its negation,as I shall show. To putthe pointin con-text, if Socrates means Gentzler'srelation when he uses the word axoj-poweiv, then it follows that he will be unable to use that peculiarly helpfulstyle of reasoning,so belovedof Greekmathematicians, nownas argu-ing by reductioad absurdum,which exploits the properties f such propo-sitions. Whenwe argue by reductio ad absurdum,we give groundsforsupposinga proposition alse because it entails a contradiction,n whichcase it is self-contradictory. ow Gentzler'srelation s such thatq has tobe consistentwith p if p entails it, given that entailment s sufficientbutconsistencynecessary or p to axvu(po)evith q. But whatabout the situ-ationin whichq is none otherthan-np? f Gentzler'sSocrateswereto findout that his hypothesisentailedits own negation- that is, if he were torefute it by reductio ad absurdum - then far from doing the sensible thingandrejectinghis hypothesisas false because self-contradictory, e wouldhave to conclude thathe had made an invalid inferencesomewherealongtheway. Forthefact thatp and-'pareinconsistentwithone anotherwould

by itself be sufficient to save that reading anyhow, for Socrates would still be tellingus to deny many things we have no reason to deny.4 J. Gentzler,"oiugpwveivin Plato's Phaedo" in Phronesis 1991 pp. 265-276. Thispaper owes a good deal to Gentzler's presentationof the issues.


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be enough to indicate to anyone supposing here was such a relationasGentzler's hat -'p doesn't reallyoaNipoveiin the sense of "follow from")p! By such a method,Socratescould never find out if his hypothesiswasself-contradictory: his method would therefore be utterly useless.:An example might help here. Take the hypothesis hatheavier bodiesfall fasterthan ighterones. That claim soundedsafe andstrong o no lessa thinker hanAristotle.Now imaginea cannonball and a ping-pongballfalling. It follows from the hypothesis hat if we tie the two together, heproductwill fall more slowly than the cannon ball alone, since the ping-pong ball, by itself fallingmoreslowly, will act as a drag on the cannonball. But since the conjunction f the cannonball andthe ping-pongballwill be heavierthan the cannonball alone, even if only by a little, it alsofollows that it will fall fasterthan the cannonball alone.So a cannonballtiedto a ping-pongballwill fall bothfasterand slower thana cannonballalone:we have a contradiction.Now the lesson Galileofamouslydrewatthis point was that the initial Aristotelianhypothesis is false, simplybecause it entailsthis contradiction, ndhence all bodiesmust fall at thesamevelocity. But hadhe been workingwith Gentzler's upo,vmivrela-tion, he could not have drawnthis eminentlysensible conclusion.Foronrealisingthat the conclusion s inconsistentwiththe hypothesis for con-tradictions are inconsistent with every proposition- he would haveinferred hat it does not aiqv(pivei he hypothesis.In thatcase, it is notentailedby it,forentailments sufficientortheauppwvcivelation.Therefore

I Incorrespondence,AndrewBarkerhas suggesteda riposte o this argument:Socratesmight still work with a relation such as Gentzler'sbecause in calling his hypothesis"the strongest" n the first place, presumablyhe indicates that he is confidentthat itis not self-contradictory.I reply simply that such confidence would of course be noguaranteethat in fact his hypothesis is not self-contradictory,any more than Frege'sconfidence thatAxiom V was "purely logical" (Preface to Grungesetzeder ArithmetikVol. 1 (1893)) was enough to save it from Russell's paradox.The comparisonwithFrege aside, Barker's defence of Gentzlerseems to be prima facie implausibleon thefollowing grounds:the whole point of calling something a hypothesis in the first placeis to indicate its provisionality,the fact that it is not yet known for all that it is sup-posed, and is to that extent revisable. But if a proposition is treated as revisable inprinciple, pending the results of its theoretical development,why should we supposethat it is nonetheless exempt from the particularlyviolent form of revision involvedin rejectinga previously accepted propositionon the groundsthat it is self-contradic-tory?If a proposition s hypothesisedon the groundsthat it might turn out to be false,surely one should allow that it might turn out to be false in any way propositionscanbe false, including being false by virtue of being self-contradictory.


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he would have racked his brains for where he went wrong in pursuingthe bpgin0Ovtaof Aristotle's hypothesis, and mechanics would be all thepoorer. Certainly he would not have been justified in inferring that theAristotelian hypothesis is false.

Of course it is one thing to complain that there is something method-ologically wrong with the relation Gentzler reads into Socrates' use ofa@pwvriv, quite another thing to say that this is not the relation he meant.The considerations above about consistency and entailment are somewhatrefined, and may well not have occurred to Plato, for all that his probablefamiliarity with mathematics would have put him in the way of countlessreductio proofs (not to mention the variety of such arguments in the dia-logues). But for the moment I think that charity requires that we look foran alternative account of the relation if one is available: and, even pre-scinding from this problem, there is another motive for seeking a newinterpretationin any event. In order to work out what is going on here toour satisfaction, our focus must not be restricted to the aO9pwvrv rela-tion, whatever it is, alone. Socrates is here giving us a criterion for decid-ing on the truth and falsehood of propositions, and it is much too muchto hope that the relation he invokes will allow us to make truth-evalua-tions of any proposition whatsoever relative to the hypothesis. So in orderto get to his meaning, then at the same time as we are considering theaOup(ov6v relation we shall have to find some interpretation of whatthings - what class of propositions - we are supposed to classify as trueor false according to whether they bear the aiug(powev/8taopvriv relationsto our original hypothesis. It is just this that is missing both from the obvi-ously problematic interpretations of the ai4towvEiv relation, but also fromGentzler's interim solution.

III. Consistency on a Subject MatterOne possibility for answering the second problem above is that the rightcandidates for satisfying those relations will be propositions on the samesubject matter as the hypothesis.6 A proposition will auopwyvei he hypoth-esis if it is consistent with it and on the same subject matter; it will6ia(pwvr the hypothesis if it is inconsistent with it and on the same sub-ject matter; and if it is not on the same subject matter, it is not even acandidate for satisfying either the oV4powveiv or 8tapowev relations with

6 Nicholas Denyer suggested this to me.


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the hypothesis.Such a readingmighthave a ratherattractiveeature:pro-videdyou takeintoconsiderationherequirementboutbeingon the samesubjectmatter,the au,(pcoveivand 8ta(pwvEivelationsamount unprob-lematically o consistencyand inconsistency. n effect, the constraint hatthe candidatepropositionsbe on the same subjectmatteras the hypothe-sis synthesises herivalinterpretationsbove whichmadeSocrates' heoryout to be ridiculous: or any proposition onsistentwith a hypothesisandon thesamesubjectmatter houldalso be entailedby thathypothesis,pro-vided we construe"subjectmatter" n a sufficiently ine-grained ashion.7But this will not quitedo as it stands, for the opacity of the relationsbetween differentpropositions n the same subjectwill makeproceedingwith such a methodpractically mpossible.Forexample,Socratesexplic-itly says that the subjectmatterof his hypothesis s "a causal account"(100a6 neppiatia;): he is hypothesisingabout causation.But thenwhatattitude s he supposedto take, for instance,to another amous ancientclaim aboutcausation,Aristotle'sclaimthatthereare fourcauses?Is thatclaimconsistent rinconsistentwithSocrates'hypothesis?tcertainly eemsto be a candidate or such a question,given that it is apparently boutthe

samesubjectmatter,causation.Butherewe are in dangerof reasoningasfollows: "Aristotle'sclaim is on the same subjectmatteras Socrates';thereis no detectable nconsistencybetween the two, at this stage;8 o onthese groundswe shouldsupposeAristotle'sclaim true."Obviously,thiswould be a disastrousmethodof proceedingwhich would involve us inassertingmany falsehoods,simply because these means for decidingonthe truth-values f propositions rejust too slender.But if insteadwe rea-son that Aristotle's claim is not yet a candidatefor bearingthe a4up-

I There would still be a problem, I think, with the results of a hypothesis entailingone another. "Water s a compound"and "Water contains oxygen" are both proposi-tions consistent with one another and on the subject"chemical composition of water":but they do not appearto entail one another,as I argued above. It might be that thisproblem could be overcome by some suitabletweakingof the notion of a subjectmat-ter: one might argue that the two propositionsare not really on the same subject mat-ter, but that the former is about the chemical structureof water while the latter isabout the quite different subject of its constituents.But even if such a move could bemade successfully, it would be highly artificial and ad hoc; and I think the furtherobjections above are quite decisive against the subject matter interpretation.8 A sign of this, of course, is the vast scholarlyliteraturedebatingwhether the the-ory of causation at the end of the Phaedo countenancesfinal, efficient and materialcauses as well as formal ones. This debate could not have occurredif we could tellon the face of it that Aristotle's claim is inconsistentwith Socrates' hypothesis.


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LOGIC AND MUSIC IN PLATO'S PHAEDO 103opv6v/atx(po)v6^velations to the hypothesis, then on the present inter-

pretation there is no explanation for this - why is it not yet a candidate,given that it is at least about the same subject?

There is an even more serious problem with speaking of subject mat-ters in this context. Intuitively, it seems that a subject matter is determinedby the fact that such-and-such propositions are all the right ones. If youwere to ask me "What is the subject matter of Chemistry? What is it allabout?" then I suppose I should best answer you by showing you an ele-mentary Chemistry textbook - that is, by showing you a well-organisedcollection of true propositions. On the other hand, this thought cannot bequite correct: for when a boy makes a mistake in his Chemistry exam byfailing to balance his equation properly, it is not as if he has gone off thesubject altogether and started writing about a different topic such as his-tory or botany. Our intuitions conflict here. Take the once-believed propo-sition "There is a planet Vulcan between Mercury and the Sun". Is that aproposition about astronomy? Is it on the same subject matter as the truth"There are nine planets in the solar system"? Well... yes and no. In away it is about astronomy because it tries to talk about the things astron-omy deals with, planets and so forth. But in a way it is not about astron-omy because it is false: there is no such planet as Vulcan, so what itpurportsto speak of is not one of the objects of astronomy.

I infer from all this, even with the qualification that our intuitions some-how conflict, that the notion of a subject matter is just a little too closeto the concept of truth for it to be entirely helpful in explicating Socrates'method. The more we require candidates for the a4u(p(ovrv relation to beon the same subject matter as a hypothesis, the more we require them tobe true - that is, the more we deny ourselves the possibility of explain-ing how they can be false. And we should not want to deny ourselves thatpossibility. For one thing, as I have just said, Socrates' method has gotto be one with which we can proceed. It is a constraint on any usableheuristic procedure, which Socrates' method is surely meant to be, thatwe must be able to use it in advance of knowing the right answers, ofknowing what is true and what is false. Socrates must not be made outto be saying "Start from a true hypothesis, and then set down as true anypropositions that are materially equivalent with it": for that advice is nomore helpful to a philosopher than the recommendation "Bet on blackwhen the ball is going to land on black, and bet on red when it will landon red" is to a roulette player.9 We must be able to move forward even

I Additionally, it would abnegate the need to specify what constraints an initial


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from a false hypothesis,which meanswe must be able to supposethatfalsehoodsOax(PiOvEihe hypothesiswithout thereby ailing to follow themethod.How can we do that on the presentaccount f falsehoodsdo not,in some important ense, have a subjectmatter?IV. The Musical Connection

My own tack for responding to the problem of the aC(p(ovEiv relationinvolves examiningthe logically promising eatures of the musicalsys-tems suggestedby the word, and seeing if appropriate nalogues or themcan be found in good scientifictheories.That we should look for inspira-tion in musicto help us here, and supposethatPlatomight have done soas well, is not at all perverse.ThePhaedois a dialoguethoroughlyoakedin musical allusions above andbeyondthe connotations f theword01vg-p(oviv. See, for instance, the reference to the lover's lyre at 73d6, therequest for a "soothing song" at 77e8 (inaiietv, echoed at 114d7),Simmias'hypothesis hatthe soul is an attunement&piovia)at 85e5ff.,'?and so on. It shouldcome as no surprise o find Socratesdrawingdetailedphilosophicalnspirationrommusic and musicaltheoryin a dialoguesodominatedby the figureof Apollo."Musicalsystemsshould interestus in this contextbecause,at least atfirstglance, they exhibitsome of the propertieswe findin good theories.Thatis, theyarecomplex,well integratedystemswhose elements propo-sitions in the one case, pitches in the other) agree with one another n

hypothesis must satisfy: for one might just as well say "Proceedby putting down astrue all and only propositionsthat are materially equivalentwith '2 + 2 = 4'.10 The theory that the soul is an attunement may have originated from thePythagoreanphilosopher and harmonic theoristPhilolaus, to whom there is an allu-sion at Phaedo 61d6-7. For furtherdetails see A. D. Barker, Greek Musical WritingsVolume I: Harmonic and Acoustic Theory (Cambridge 1989) pp. 38-9. Everything Ihave to say about Greek music owes a great deal to this work, as well as its author'sgenerous and invaluable correspondence." If more justification is needed, I refer the reader to M. F. Burnyeat, "Plato onWhy Mathematics s Good for the Soul" in T. Smiley ed. Mathematicsand Necessity(London 2000) pp. 1-81. Burnyeat argues very plausibly that the key to understand-ing the connection between the content of mathematics and the moral value that amathematical rainingis supposed to impartto the rulersof the ideal city, accordingto the central books of the Republic, is to be found in the study of harmony (e.g.p. 47ff.). In other words, the real philosophical nature of the hypothetical method parexcellence, mathematics, is to be explicated best by an investigationof musical the-ory. That is exactly what I am suggesting for the hypotheticalmethod in the Phaedo.


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LOGIC AND MUSIC IN PLATO'S PHAEDO 105some important ense,'2while also standing n significantand determinaterelations o one another.A musical&pRovias notjust a bunchof differentpitches, ust as a correct heory s not just a mass of truepropositions inbothcases theordering f, andrelationsbetween, he elements s important.'3The idea that musical systems are significantly tructureds not some-thingI have liftedanachronisticallyromcontemporary iatonicmusic.Itlay at the heartof Greekmusic as well. Its origins appearto lie in thethought hatpitchesrelatedby certain ntervals'4 the octave,the perfectfifth and the perfect fourth- blend togetherto form a unified whole,a a- t(pwvia,while notes that are otherwise related do not form suchintegrations.'5The various familiar species of Greek &ploviat, Dorian,Mixolydianand so forth, are thenconstructedby permuting therpitchesin variousways on to a basic structure f av (poviat.axpgoviaLre there-fore ordered structuresof pitches constitutedby relations of differentstrengths:weak relations which must hold between all the notes in aapgovha in orderfor it to count as a complete musical structure,andstrongerrelations which hold between certain pitches in the &pIoviawhich are sufficient or those pitches to form mq(pxovicl.t is the latterthatespeciallyconcernus in explicatingmy view of the Phaedo's conceptOfai,UgpovEiv.Plato gives us a characterization of these special intervals in theTimaeus,when he has Timaeus argue that differentpitches succeed in

12 See Symposium187b4-5, where Erixymachussays that musicalharmony s a kindof agreement - aulgpwvia 68 ogokoyxa Tn;.'" It is also worth bearing in mind the fact that musical systems and collections ofk6yot are similar at a quite vertiginous level by virtue of the fact that they are bothsomehow imitative. The imitative nature of X6yot is enough, in the Phaedo, to rec-ommend them as the objects of investigationfor the hypotheticalmethod we are dis-cussing. The imitative nature of the &p,tovitt is enough to recommend the thought,adumbrated n Republic 397a-401b, that they must be restrictedand regimentedin theideal city lest in aping undesirableemotions they do a disservice to the education ofthe young.

14 More precisely: pitches related by intervals expressible in terms of ratios thatareeither multiple (n:1) or epimoric (n+l:n). This seems to be the closest one can get toa general characterisationof the a sqxpvixt, although it expresses neithernecessarynor sufficientconditionsfor being a lgoc(pvia. Not all such ratiosexpressacsg(pov'tfr(e.g. 3:1); and at least one interval forming a auV(pwovax,he octave plus fourth, can-not be expressed either as multiple or epimoric (its ratio is 8:3).'5 One of these intervals is actually mentioned by name when Socrates is illustrat-ing the "subtle" hings thatwill not receive opposites althoughthey are not themselvesopposites. Among this class, we read at 105bl, is rO+uto6Xtov, he ratio of the fifth(3:2). This is yet furtherevidence that Plato has music in mind in our passage.


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forming au4(poviat when "by attaching a similarity, they blend togethera single experience out of high-pitched and low-pitched movement"(o6,olo6 a iLpoovtaave;,piavc o6 ia;, alapiiai; Y1UVEKEpaaaVto,arCv80b4-5). Aristotle was also interested n the Gu4lgpovial,iving them avery similardescription t De Sensu 448a9-11. Pitches standing n appro-priateratiosto one another orma mixturewherebyeach note is perceivedsimultaneouslywhen theaozqxoviaas a whole is heardas one thing: 00&t&agEyjuEva aija (X6yot yaip Eicntv a&VtK lpEVoV, otov tb 8ua' tasov icat8tb 7iVtre), av 1ii 6) 'ev XiGO6VlyrXl.'6

Now the precisemusicaldetails neednot concernus here.'7All we needis the thought hat, within a musicalstructure, &pgovia,certainpitchesare so related to one another hat they form some kind of unity,whichnotes that are not so relateddo not form. When notes are in the ratios2:1, 3:2 or 4:3, they blendtogether o form u(povikx. Notes withintheappovia that are not so relateddo not form this strong unity together,although hey are still partof the same structure.s therea relationhold-ing between the o6yot f a good theoryaboutcausation hat is analogousto this special unifying relation holding between the pitches in a axji-povia? That is, can we find two propositions o related thatthey form,as it were, some kind of unityover and above theirbeing merelyassertedtogether, ust as two pitchesin the ratio2:1 or 3:2 blend to forma oup-povia (an octave or a fifth) over and above theirbeingeach sounded?I think there is enough information in the Phaedo to suggest thatSocrates'hypothesisand its resultsmightstand in such a relation o oneanother a relation ometimesdiscussed n recentphilosophyof science.'8This relationholds betweenpropositionsf they standas explanationsof

16 Both these passagesare discussed n A. D. Barker,Op. Cit. pp. 76 & 62 respectively.17 This is convenient: for if they did then we would need an argumentabout theextent of Plato's detailed knowledge of harmonictheory, which would be extremelydifficult to formulate.M. F. Burnyeat, Op. Cit. p. 15ff., argues that he was at leastfamiliar enough with the writings of the Pythagoreanharmonic theoristArchytasofTarentum o quote him (consciously or unconsciously) at Republic 530d8.18 I have in mind: P. Kitcher, "Two Approaches to Explanation"in Journal ofPhilosophy LXXXII 1985 pp. 632-639; W. Salmon,"Scientific Explanation:Causation

and Unification"in Causality and Explanation(Oxford 1998) pp. 68-78. The exam-ple I use is taken from Salmon. A more general statement of the view that explana-tions can both explain something and be themselves explained by it is to be found inW. V. 0. Quine and J. Ullian, The Webof Belief (New York 1970) p. 79: "... therecan be mutual reinforcementbetween an explanationand what it explains. Not onlydoes a supposed truthgain credibilityif we can thinkof something thatwould explain


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LOGIC AND MUSIC IN PLATO'S PHAEDO 107

one another,albeit in differentways. Take the two propositions"Copperis a conductor"and "Pennies conductelectricity".These claims form akind of unitybecausethey provideone anotherwith mutualexplanatorysupport.The claim thatcopper s a conductor tandsas an explanation orthe claim that penniesconductby being moregeneral than the latter: tsays what it is aboutpenniesthat makes themconduct,and thatwhatpen-nies are made of conductswhether t is in pennyform or not. One mightsimilarly explain in turn the claim that copperis a conductorwith theclaim that metals are conductors.Again, we say what it is aboutcopperthat makes it a conductor.And one might again explainthis propositionwith the claim that electronsin such-and-sucha relation conduct elec-tricity,and so on. I shall call this generalisingexplanation:'9 e explainparticular acts by subsuming hem undermore and more general laws -"higher"principles,as Plato would call them. But there is also what Ishall call particularisingexplanation,which seeks to explain away themost general explanatory ormulaeby characterizing uch truthsas lawsabout causal relationsbetween particular hings. The idea is that the lawsare shown to be trueby virtue of the explanatory upport hey get fromtheirinstantiations.n this way, we can give a kind of explanation or theclaim thatcopperis a conductorby pointingout that thereare particularcopper things, namely pennies, and that they all conductelectricity.20I think that both these kinds of explanation are on display in thePhaedo. Socrates' startingpoint is the generalone aboutthe causal powerof Forms,which we might representschematicallyas "There s such a

it, but also conversely: an explanation gains credibility if it accounts for somethingwe suppose to be true.""9 Kitcher and Salmon call this style of explanation"bottom-up",referringto thedirectionwe move in going from the explanandum o the explanans. But many philoso-phers use "top-down"as an expression for explanation of the particularby the moregeneral. Hence I avoid the terminologyaltogether.20 I have occasionally encounteredresistance to the thoughtthat "Pennies conduct"is any kind of explanation of "Copperis a conductor".Perhaps the example is tootrivial to express the point. Certainly I grant that it is not a generalising explanationof the latter; and I grant too that in science we are often more interested in general-ising explanationsthanother kinds since it is the particularswith which we arealready

familiarthroughperception. But these concessions are neither individually norjointlysufficient for scepticismabout the possibility of particularising xplanations.It is merequestion begging to suppose either that generalising explanationsare the only expla-nations, or that whenever we want an explanation we already have a grasp on somequite particular act and are casting about for a more general one.


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thing as the (p,and if anythingelse happensto be tp, then it is (p by thep".From this we get a handfulof opiOnEvta:1. When something s (p,then (p-ness s somehow in thatthing.2. Anything hat is (pby the (p is different rom that by which it is (p.(It is by Simmias'largeness that he overtopsSocrates,not by his

nature.)3. If (pand r are opposites, then the p or v in things will not toler-ate the advance of the other:either it will retreator perishon the

approachof its opposite.It seems to me that in movingfrom the originalhypothesis o 1, 2 and 3,we get successive particularising xplanations:1, 2 and 3 explain howSocrates'hypothesiscan be trueby exploringwhat happens n particularapplications f thatvery generalclaim. 1for instanceexplicates he detailsof what is meant by "p by the (p":when something is "(pby the (p", thatis becausesomethingof the(p s in that thing.Likewise,2 gives some par-ticularising xplanation or 1: it says a little abouthow the agentsof cau-sation are in theirsubjects.Since they are different rom theirsubjects,theyare in themwithoutbeingthem.Andin turn3 provides urther xpla-nation for 2 by suggesting that the domain over which the variable(p ranges,containing he properties hat are in things withoutbeing them,is populatedby things that have opposites.The thingsthatcan have (p nthem- Simmias'size, Socrates'fingers,Phaedo's face - are not the sortof things that have opposites:this is why such thingsmust be differentfromthe causalagentsin them, as 2 asserts.2 gives particularisingxpla-nation for 1 while 3 explains2 in a similarfashion.Butequally,we have generalising xplanationshere.Supposesomeonewere to ask "How is it that certainpropertiesof a thingare capableofdoingsuch thingsas advancing,retreating r perishing?Thatis, how is itthat3 is true?"We can explainthis with the moregeneral2: certainprop-erties arecapableof doingsuchthingsbecausetheyare different rom thethings that have them, and hence can advanceinto or retreat rom thehosts from which they are differentwithoutimpairing heiridentity.Wecan explainthis in turnby ascending o 1. Thingshave propertieshataredifferent rom themselveswhile still characterisinghem when they havethem because, when they have them, those propertiesare somehow inthem. And if we ask for a more generalexplanationof this fact, we getit from Socrates' original hypothesis:propertiesare in things because,more generally, when something other than the (p itself is (p, it is so in


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LOGIC AND MUSIC IN PLATO'S PHAEDO 109virtue of, because of, something else, the (p. In fact Socrates is quiteexplicit that he intendsto reach his conclusion using this kind of gener-alising explanation,when he says at lOOb7-9hat aC'i Rot 8ibcOqTE KaGVY(JpEi; E?1Val WiVT, ?AXnt'iX (Ot1 ?K TOV'TOW'U'V al"Xla(V int&10E15V MXIavDUp4orivc); &Oivarov j Thehopeis thathe can explainthepar-ticularfact thatthe soul is immortalby showinghow it is related o truthsabout comingto be and passing away generally.I have been arguingthat the same sort of relationshipholds betweenSocrates' hypothesisand its variousopRT1OEvVTx,akenas pairsof propo-sitions, as holds between the propositions"Copper s a conductor"and"Penniesconductelectricity".As pairs of propositions, hey form a unityin virtue of supporting ne another n differentways, by providingdiffer-ent kinds of explanation,generalisingor particularising,or one another.Both kinds of explanation hold, in different directions, between thehypothesisand 1, 1 and 2, and 2 and 3. And this is stronglyanalogoustothe way in which pairs of pitches form a unity by blending with oneanother.Two notes an octave apartdo not bearthe same relationto oneanother.Forwhile blending s a symmetrical elation, he precise relationof lying in such and such a ratio to another pitch is asymmetrical.2'Likewise, the propositionsabout copper and pennies above do not bearthe samerelation o one another.Forwhile explanation,as I have argued,is a symmetrical relation, the precise relations of generalisingly explainingor particularisingly explaining are asymmetrical. And in both cases, themusicaland thetheoretical, omethingnew emergesfrom the combinationof the two thingsso related- a blended unity- which does not emerge,for instance,when we sounda pitch togetherwith anothersix semi-tonesabove it, or when we conjoin truths such as "Cats are mammals"and

21 This is not an end to their logical similarity. In addition to being asymmetrical,concordant relations are irreflexive- no pitch blends into a aict(pvia with itself -and non-transitive, n that in some cases where pitch a blends with b and b with c, awill also blend with c (as when b is a perfect fifth above a and c a perfect fourthabove b, where c will be an octave above a), but in other cases a will blend with band b with c without a also blending with c (I give an example of this later on in themain text). These logical propertiesalso hold of explanation.The kind of explanationsin which I am interested are irreflexive: the proposition p is surely neithera general-ising nor a particularisingexplanation of itself (although I grant that some proposi-tions can, in some sense, be self-explanatory). Such explanations are also non-transitive,in that sometimes where p explains q and q explains r, p will explain r, butsometimes not.


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"Sydneyis in Australia".The two pitches do not form a xugpwviandblendinto a musical unity,just as the two propositionsdo not blendintoan explanatory unity.

V. The Solution proves its worthAt thispoint one might objectthat the musicalanalogy amounts o a mereinterestingobservationrather than a plausible theory about how Platomighthave understood he special kind of agreement hat occursbetweenpropositions when they aop9ovri one another. It might be that Plato hasSocratesdevelophis hypothesis n theway in whichhe does withouteverthinkingof musicalauppoviat.at all, and any workableanalogywith con-cordantrelations n musicaltheoryis pure coincidence.And I admit thatthis is perfectlypossible.But let me restatethe case for the analogynowthatI have explicated ts non-musicalaspect.It seems to me thatwe needan answer to the following question.Given the likelihoodthat Platowasasking himself "Whatmore is requiredof a propositionbesides consis-tencywiththehypothesisf it is to countamong hathypothesis' pgrnOCvrn?"',to whatkindof structurewouldhe turnhis mindfor an answer? t seemslikely that he would first think of a mathematical tructure,given thatmathematicss the most obvious exampleof a science that reasons fromhypotheses,as Socratestakes himself to be doing in our passage.But ifhe was thinkingof mathematics, e would almostcertainlyhave thoughtof harmonictheory, especially since music has already been such animportant heme in the Phaedo. And if he was thinkingof harmonic he-ory, he might well have been struckby the way certainprivileged nter-vals form strongerand more interestingrelationsbetween their pitchesthan other equally musical intervalswithin the axpjovicx,ust as somepropositions orm stronger ndmoreinterestingelationswithpropositionsthey explain,and are explainedby, than otherpropositions quallycon-sistent with them.Let us turn once again to the problems facing the method outlinedabove to see if the new interpretationvercomes hem. We saw thattherewas a problemaboutcashing out alu(pxvriv solely in terms of consis-tency and "beingon the same subjectmatter" s a hypothesis.We won-deredhow one could arbitrate etweenpropositions n the same subjectmatteras a falsehood,because it isn't clearhow a false proposition anhavea subjectmatter.Thingsare muchclearerwhenwe deal insteadwiththe notionof explanation.For it is perfectlypossiblefor a propositionobe consistent with anotherand explanatoryof it, and vice versa, even


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when both are false. My explanation akes into accountthe possibilityofsomeoneassertinga false hypothesisand then mistakenlyputtingdownastrue falsehoods that genuinely augipovCi it without thereby failing to fol-low the method.In otherwordsI can clearly explainhow we can proceedwith Socrates' methodwithoutalready having the right answers,whichthe "subjectmatter" nterpretationould not - which means I can con-tinue to hold for the moment that the hypotheticalmethod is genuinelyheuristic.Again, it was not clear what we should do with a propositionon thesame subjectmatter as a hypothesis,but neither clearly consistent norinconsistentwith it. Aristotle'sclaim thatthereare four causes bore thisrelation o Socrates'hypothesis.But if Socrates s looking for propositionsthat are consistentwithhis hypothesisbutalsoexplainit and are explainedby it, thenit seemswe areon saferground.Forwhile we may notbe ableto tell initiallywhethera propositionon the same subjectmatteras ourhypothesis is consistent or inconsistentwith it, we can tell with muchgreaterease whetherthe propositionbearsan explanatory elationto thehypothesisor one of its results.Explanation s much more transparentto our minds than consistency, as the career of Frege testifies. SinceAristotle'sclaim that thereare fourcausesdoes not seem to coherein theright way with either the hypothesisor the resultsmentionedso far, thatis a sufficient eason not to putthe claim down as true. But since there isnot any detectable nconsistencyeither, we ought not to put it down asfalse. So it mustbe one of the infinitenumberof propositionswhich isnot a candidatefor either of the aug(p veiv/8ta(pcov6v elations.Hereagain his nterpretationcoreswellagainst he"subjectmatter" nedescribedabove.There s nodangerof supposingAristotle's hesistruesimplybecauseit is on the samesubjectmatteras the hypothesisand not detectably ncon-sistentwith it: for it does not adequately xplainit. We also have a gen-eral answerto the question"whatmore is required rom a propositionabout a certainsubjectmatterfor us to be able to proceedwith it (i.e.makea decisionaboutits truth-value)ather han temporarilyay it asideandcast about for others?"The accountsays this:we can proceed f andonly if it is eitherdetectablynconsistentwith the hypothesis,or bothcon-sistentwith it and a good explanationof it in the sense elucidated.If aproposition atisfiesneitherof these conditions, hatis sufficient easonforsupposing hat t cannotyet satisfytheavu(pcvciv/8ta(pow*vvelations,andhence is not yet a candidate or truth-evaluation.Fromthis we can also derive the interesting esult that the 6la(P(vEIvrelationis the contrary,not the contradictory, f the avi(ppvCiv elation.


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For I am saying that a propositionwill augqoveiv another f it satisfies aconjunctionof conditions: t must be consistentwith it and explain it.Since the negationof a conjunctions the disjunction f the negatedcon-juncts, thenany propositionwill satisfy thatrelationrelativeto another fit is either inconsistentwith it or fails to explainit in the right way. Soif the iaypcoveivelationwere the contradictory f caipqoweiv, hen wewould once again be lost in a ruinouswilderness,puttingdown as falseall thosepropositionswhichfail to explainourhypothesis,which will onceagain includepropositionswe have independent easons to suppose trueand,worse,contradictory airs.Therefore he 8la(pOvEivelationhad bet-ter be its contrary, onsisting simplyin inconsistencywith the hypothesisor one of its results.Put anotherway, the tawpowsvvelationis not thenegationof the aup(ovEv relation,but its opposite.This shouldcome asno surpriseat all given that Socrates'SecondVoyage embodiesthe mostopposition-fixatedassage of an opposition-fixatedialogue.The musicalanalogyalso helps us to give a little colourto the "yet"sat the end of the last paragraph utone. As I have said, we have to sup-pose at least initiallythatAristotle'sclaim that there are four causes isnot yet a candidate or the a-opwvEv/6ltapxowvvelations.But if we areto have a fully developedtheoryof causation t surelymustbecomeoneat somepoint.After all, a completelysatisfactoryheoryof causationmustbe one thatsays, among otherthings,how many sorts of causation hereare. In fact we have to supposegenerallythat in the developmentof atheory a large numberof propositions tartout beyondthe pale of con-siderationbut become candidates ater on. Fortunately, xactly the samething can happenin music. To illustrate his, let us simplify things byputting he pointin modem terms.Take, for instance,a note of pitch C,whichis going to be the startingpitchandbasis for a scale or attunementjust as an initial hypothesisis the startingpoint and basis for a theoryaboutsomething.The C an octave higherwill forma augipwviawith it.So will the F a fourthabove middleC and the G a fifthabove it. Whataboutthe D thatlies a tone above the startingpoint?It does not formaauw(pwcviwith the initial C, when just those two notes are considered:tones are not ciwpwviat. But it does forma aci(povie with the G a fifthaboveC, since it lies a perfectfourthbelow it. Hence,as the complexofrelations is developed furtherand further,new notes form caujt)xOvialwithinthesystemthatwere incapableof formingsuch unitiesearlierwhenthe right pitcheswith which to blendwere not yet available.C does notblend with D by itself. But since C blendswith G, and G blendswith D,


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the structureCGD forms a unity held togetherby the middleterm,G.22And one can imagine the same sort of thing happeningto Aristotle'sclaim. It does not blend with Socrates' hypothesisor any of the resultsadumbratedn our section of the Phaedo becauseit does not explain anyof themin therightway. But it maywell be thatit will blend with propo-sitionsadded ater on in the developmentof the theory,which themselvesblend explanatorilywith earlier results, in which case it ought to becounted as true.This musical understanding f axul(poveiv lso helps us, I think, tounderstandbetterwhy Socratesis so keen to stress the value of orderwhen he concludesthe introduction f the methodthat relies upon thisrelation.Consideration f the explanatory elationsI have been using tocharacterize he meaningof iOg(pxvEiv sharpens he issue. For it seemsto be perfectlypossiblethattwo propositions ould be consistentwithyourhypothesis and explanatory of it, without bearing that relation to oneanother.23orinstance,starting rom the hypothesis"Metalsconductelec-tricity",we mightput down as true"Copper onductselectricity"on thegrounds hat it is consistentwith the hypothesisand affords t some par-ticularising explanation.But for the very same reasons, we might putdown as truethe claim "Mercury onductselectricity".So shouldwe putboth of themdown as true,on the basis of the indifference rgument hat,since thereis not yet more reasonto assert one rather hanthe other, andone needsto assertsomethingn order o get anywhere, herefore ne mustassertboth?No: thereare reasons or thinkingwe ought notto assertboth,on the present nterpretation. oreven thoughthe claim aboutcopperandthe claim aboutmercuryareperfectlyconsistentwith one another, t is farfrom clear that they afford each other explanatory upportof the sort Ihave been discussing, since neither is more generalnor more particularthan the other.They are, as it were, propositions"of the same level".There could only be collateral supportbetween them. But it is doubtfulthat there is any such thing. It just does not seem to matter to the fact

22 I do not of course mean that you will get a aug(powviczf you sound C, G and Dtogether.The tone intervalbetweenC and D will disruptthe purityof the fifthbetweenC and G. But if one goes from C to G, and thence to D, one always proceeds accord-ing to intervalsof the rightkind. This recalls the sense suggestedat the end of RepublicVI in which the hypothetical method involves somehow moving from one propositionto anotherin a sequence, a thing only dialectic does from a properfirst principle.

23 I owe this observation to Ben Morison.


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that copper is a conductor,by itself, whetheror not mercury s one aswell. But if we take ourselves o be followingPlato'shypotheticalmethodin this set-up, it ought to matter.For Socratessays thatyou have finisheddevelopingyour hypothesisonly when you know that its resultsbear theright relation,not just to it, but to one another. All the opgTrOnEvtaust&XXXoi;aiu,upovcvin order or thereasoner o proceed o giving a ko6yo;for the hypothesis tself.This is a nettleI am prepared o grasp: t seems to me in the natureofthe aug(pcoveivelationthat,at any stage, many candidates or satisfyingit - p, q, r and so on - will emerge, only one of which can be selectedat a time and put down as true, say p, on the grounds hatthe others, qand r, do not cohere with p in the right way, even thoughthey are con-sistentwith it and cohereno less in the rightway with thehypothesis.ForanyonefollowingPlato'smethod romthehypothesis hat metalsconductelectricity,he will have to decidewhich of the two particularisingxpla-nations o go for, assertthat,andstay silent at least for the momentaboutthe others,howeverstrange hat looks. Presumably,he ideawould be todevelop thehypothesisvia one proposition, ndthenstartagainby puttingdown as truetheother,and thencomparinghedegreeto whichtheresult-ing sets of propositions ormadequate heoriesof the kind the hypothet-ical method s after.It shouldnotbe a surpriseby now thatthesamethingcan happen n music. Fromthe startingpointC, one will get a aiigipovkviwith the F a fourthabove it, or alternativelyhe G a fifth above - bothnotes are suitablyrelated to the original.But one cannot,as it were, gofor bothof these pitches together.Forthe F andthe G do not blendintoa auwpwvia.They are not relatedto one anotheras each is to C. Onemust choose one of them at a time if one intends to develop a musicalcomplexof the rightsort fromC, andtheremay at firstbe no more rea-son to choose one ratherthan the other. This strain of indeterminacybrings out the element of unexplained alent that will be required or asuccessfulapplicationof the method.Socrateshas no storyto tell aboutwhy you shouldput downas trueone particularropositionromthemanycandidates hat will emerge.But he does at leastrealisethatthe pluralityof candidates s going to mean that one mustpay specialattention o theorderin whichone is proceeding order orceduponthereasonerby thenatureof the aigbpoveiv relation perhapsso that the hypothesiser aneffectivelyretracehis steps and startdevelopinghis hypothesisall overagainwith differentpropositionshat j.tqovri it withoutalso bearing hatrelationto the firstassertedbatchof bpgnT0EvTr.


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I hope that this has gone some way to establishingmy view that, forPlato, theory building is a kind of propositionalorchestration.There ismuchof which I am uncertain,and much thatthis paper acks the spaceto considerin detail, but I am convincedof the appropriatenessf thatgeneraldescription.Of course the analogy goes lamein this respect: hereare presumably etterand worse ways of developinga hypothesisvia theright coherencerelationsas describedabove, but it is presumablynot thecase that there are betterand worse ways of following a startingpitchwith the right concordant ntervals. But nonetheless in developing theanalogywe have found that the hypotheticalmethod is much richer andmore sophisticated han Socrates'brief sketch conveys, which is some-thingwe shouldprefer especially given the reasonsI have set out indi-catingthatPlato has music in mindanyhowin this context.24Corpus Christi CollegeOxford

24 I am very gratefulto Nicholas Denyer and David Sedley, both of whom read thispiece several times and saved me from many mistakes, as well as providing muchhelpful input and encouragement.But I am especially grateful to Andrew Barker,with-out whom it could not have been written at all.

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