arguing from a definition aristotle on t

8/18/2019 Arguing From a Definition Aristotle on T

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RGUING FROM DEFINITION

RISTOTLE ON TRUTH ND

THE EXCLUDED MIDDLE

W C VINI


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At the very beginning of Metaphysics F 7, in 7 Bekker lines (and 77 Greekwords), Aristotle is able to give us

(a) a statement of the Principle of Excluded Middle (PEM),(b) a celebrated ‘semantic’ denition of truth and falsity,andc) an argument for the principle from the denition, the rst of seven

arguments in support of PEM to be found in the chapter.

The price paid for this impressive tour de force is the elusiveness of (c), apiece of both textual intricacy and Aristotelian obscura brevitas. Ancient Greekcommentators, Alexander of Aphrodisias and Asclepius of Tralles in particu-lar, already suggested two different readings of the text and consequently twodivergent understanding of the text’s line of argument; and the same holdstrue, to the best of my knowledge, for all modern editors and commentators,

except the most recent ones, Barbara Cassin and Michel Narcy (1989), whooffer a third, radically different, approach.In this paper, I would like to reopen discussion of this elusive argument

by suggesting a new interpretation of it, saving the traditional reading andpunctuation of the text, but also developing an intuition to be found in themost recent approach. In particular, I try to explain the contextual meaning ofAristotle’s ‘semantic’ denition of truth and falsity, namely its being both a‘semantic’ denition and the main premiss of a ‘semantic’ argument aiming torefute the possibility of a middle for contradictory pairs of affirmations andnegations; a premiss whose immediate consequence is the principle accordingto which, of every contradictory pair of affirmation and negation, necessarilyone member is true and the other false.

1. In both Ross’ and Jeager’s edition, the Greek text of our passage runsthis way:

1011 b 23 14/U01 ,u17v 065:} psrafb oivrztpoiaswg évééxaraz ezi/at24 0136:-fv, écll’ 0’cv0iym7 1? gooivaz 1? émoqaoivaz 5v K016?’ évég étzoziv.25 517/iov 52 rrpcbrov ,us v épzaayévozg ti I0‘ 0’cl176ég Kai i//8550;.26 to‘ ,uév yelp /léyszv 10 5v 17 afvaz if to‘ ‘mi 6v 85/Oil gusti-


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6 W. Cavini

27 50g, to 5:-E 10' 5v efvaz Kai rd /11) 6v pr) efiraz 6r/1176ég, 0301228 Kai 6 /léycov sziaz 1? ,u 6:/127626021 1? 1//séasmr dz/ll ’29 0512 to 5v /léysraz p saz 2? sfilaz 051 rd mi 5v.

The opening sentence states two things, both in modal terms, one nega-tive, the other afrmative:

(aa) But, on the other hand, it is not even possible that there should beanything in the middle of a contradictory pair,

(ab) but it is necessary either to afrm or to negate any one thing of onething.

Is PEM given by (aa), (ab), or both? Aristotle’s standards formulation ofthe principle is actually a more concise variant of (ab), namely either ‘adv (pdvaz1? éucoqooivaz’ or ‘miv civayrcazbv rpdvaz 1? cimorpoivaz’, with ‘miv’ instead of ‘§vrc0c0’ évo'g 6rzo6v’.‘ Compare for example the following texts:

T l [] rd: lcozvoi, ozbv 0 n nav rpdval I7 airroqoévaz (An. Post. A ll, 77a30)

T 2 léyco dé Kozvdzg ozbv to miv (péval 17 airrozpoivaz (An.P0st. A 32, 88a37-bl)

T 3 (léya) dé éznodezrcrzrcricg rdzg lcozvoig 50'fag éf cbv oinavrsg dszlcvzovazv)ozbv o rz rrdv dvaylcafov 1? (poivaz écvroqpoivaz (Met. B 3, 996b 27-29)?

So in the standard formulation of the principle there is no explicit exclu-sion of a middle3, but (ab) is clearly logically equivalent to (aa)4, i.e.

U*t‘P‘7* x(Pxv—1Px)—1O3P3x(—| Px/\——|Px).

The same connection is to be found in Met. I 4, 1055b 8-10:

l. Ev Kat?’ év0'g:


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Arguing from a denition 7

T 4 oivrupdoecog ,uév mix 5011 perafu, arepascog 5.2‘ rzvog § arzv- rov pévydp mix lirov rrdv, lbov 5’ dvzaov 013 ndv,

where an example of Aristotle’s PEM is given to justify (yép) the exclu-sion of a middle for contradictory pairs, namely

r/x(Pxv—|Px)——>—»3x(—\Px/\—|—|Px).

What our text is actually designed to argue for in what follows is (aa), theimpossibility of a middle for contradictory pairs, namely of a joint denial ofboth members of contradictory pairs of afrmations and negations, and not(ab), the necessity of afrmining or negating the same predicate of the samesubject. But arguing for (aa) is, by the same token, tantamount to arguing for(ab); so that the traditional interpretation of F 7 (cf., Ross’ title ‘Law ofexclud-ed middle proved’) has to be retained.

2. The rst argument for (aa) is then a ‘semantic’ one, an argument from themeaning of ‘true’ and ‘false’, i.e. from the very denition of truth and falsity5:

b25 dlov dé rrpcbrov pév épzaoguévozg ci to zit/l17t9ég Kai 1//2660;.

It is clear rst of all ifwe dene what is the true and the false.‘Among the interpreters, ancient and modern, there is no consensus on

how to understand argument (c):(.1) rd ,us v ydzp ls’)/szv to 5v ,u1j szilaz to ,u1j 5v sziwzz 1//£650; to 55 to 5v

sfvaz Kai 10 /uj 5v /117 sziiaz 6z).176e g,(2) (Ems mi 6 léycov sfvaz mi 0’:/1176813081 1//eziaerar(3) dz/ll’ 0516 10 6v /léysraz ,u1j ezilaz 1? eziraz 0516 10 ,u1j 5v.

5. A semantic argument, i.e. an argument from the meaning of a term or the denitionof a concept, is what Aristotle repeatedly recommends in arguing for the syllogistic princi-ples of non-contradiction (PNC) and excluded middle (PEM) in Met. F. Cf. in particularMet. F 4, 1006a18-b34;7, 1012a 21-24; 8, 1012b 5-8; see also F 7,1012a 3 and 11. The moveof arguing from a denition is a dialectical one; cf. Top. B 4, 111b 12-16; Primavesi 1996:157-60.

6. 517/10v [...] 6pza0guév0zg. cf. Rhet. B 7, 1385a 17 (épzaapévozg 1r 7v Xdpzv dlov 501011);An.P0sl. A 1, 71a 2-3 (qoavspov Jé 10610 Bswpomv 51:1‘ naacbv): the true and false:Top. A 2, 123a 15-17 (511 sf évdéxeraz dnolznszv to azj0r7,uév0v yévog rj rrjv dzaqoopoiv, ofovtuvxrjv To zczvefadaz 1? déav 10 a’z).1]0ég Kain//sdog); Met. B 2, 997a 12-15 (1ca00 ,100 ydp ‘ad/lzowa1caz 1révrwv 0’tp;(a1 r0 z dfzcépara éarzv, ez 1’ 5011 ,u1j 106 (pl/l0a0 (p0v, rivog Ema: nspi ouircbvdt/100 to Hswpaaz 16 013.1706; Kai 1;/s66og); the false or the true: Met. F 8, 1012b 7-8 (55épzapo 61a/lsrctéov /laévtag 11 cmuaivsz rd I//6550;‘ 1? 16 éclnég).


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8 W. Cavini

1) For to say of what is that it is not, or of what is not that it is, isfalse, while of what is that it is, and of what is not that it isnot, is true;

2) so that also he who says that it is or that it is not will speak truly offalsely;

3) but neither of what is it is said that it is not or that it is, nor of whatis not.

There are clearly three steps:

1) an articulated denition of falsity and truth in terms of n0n-con-tradictory pairs of affirmations and negations;

2) an immediate consequence of the denition;and 3) a nal assertion introduced by M/10?.

As for the modern editors, Bekker, Schwegler and Tredennick put a com-ma between 1) and 2), and a full stop between 2) and 3); Bonitz, Christ,Ross and Jaeger also put a comma between 1) and 2), but an ano stigme asemi-colon) between 2) and 3):

ca) 1), come 2). Mid 3) 1), ohms 2); rial/10 : 3)

Besides, they all read 3) as 3a), i.e. with léyaraz following Ab,’ not /léyezfollowing EJ. Cassin and Narcy, on the contrary, put an ano stigme a semi-colon) between 1) and 2), and a comma between 2) and 3):

cb) l); 05012 2), é/lid 3)

and read 3) as 3b), i.e. with /léysz having the same subject, 0 /Léywv as 2))following, as they normally do, EJ.“ »

The main difference between ca) and cb) is that according to the former the interpretatio vulgata) a conclusion 4) has to be supplied”:

ca) 1), chars 2); dike‘: 3a);

while according to Cassin and Narcy the Parisienses for short) the argu-ment is a self-contained one, with a protasis, 1), and an apodosis, 2)+ 3),

7. Cf. also the Medieval Translatio Anonyma Media: ‘dicitur’.

8. Cf. also AIP AscP and Moerbekes’ Latin translation.

9. As Schwegler 1847: III 182) already observed, the argument is ‘sehr liickenhafte’.


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introduced by a conclusive (bare governing both (2) and (3), and articulated bya non-adversative but conditional 0’:/l/lot:

(cb) (1), a )0'r.


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10 W. Cavini

without any reference to the middle and with a rt implied: ‘if(1), then also(?) ifone says that it is or is not he will speak truly or falsely’;while step (3) must be understood in the same sense as Alexander’s reading,namely: ‘but it is not said of what is or of what is not thatit is or is not, ’; and then we get the sameimplicit conclusion:

(cab) (1), 0301:: (2b); 0’:/Md: (3a);

where (2b) is simply an aside with no deductive import, and the oppo-nent, also in this case, is not speaking either truly or falsely, but simply sayingnothing at all. That is one of the main drawbacks of both (caa) and (cab)‘2, foraccording to Aristotle (cf. (aa)) the denial of PEM is not only false, but impos-sible, i.e. necessarily false.“ The other main misunderstanding to be found inthe interpretatio vulgata is its interpretation of the middle (M) as a subject ofpredication (cf. (2a) and (3a)) and not as the joint denial of a contradictorypair, i.e. as a (false) assertion, e.g.

M opts on/a9ov oirce oinc imam» to uiyuot (F 7, 10120: 27-28).“

The Parisienses’ interpretation (cb) is radically different from both (caa)and (cab). Step (2) has to be read as (2c), namely as (2b) so far as concerns thetext, but with no tz implied so far as concerns its interpretation:

(2c) et c’est pourquoi celui qui dit ‘est’ou ‘n’est pas’ dira vrai ou dirafaux.‘5 -

Step (2) is not a parenthetical consequence of (l), for abate governs both(2) and (3) (the two clauses being connected by a conditional, not an adversa-

12. Infecting also Syrianus’ (in Met. 78.22-32) and Thomas Aquinas’ (in Met. IV xvi721) interpretations, which sound quite different from both (caa) and (cab) (but in Spiazzi’sedition Thomas’ text on this point makes no sense: ‘Et ita nec afrmans nec negans, deneccessitate dicit (‘? ) verum vel falsum’).

13. I owe this point to Mario Mignucci. '

14. Cf. also F 4, 1008a 1-7: ravré rs 05v avttaivsz mtg /léyovm r0'v /léyov tovtov, Kat‘ 6'11013K dzvoiylcry 17 (pdvaz 0’z1r0(/roivaz. efyoip (il170ég 6'11 0 zv49pa)n'og rcafoézc 5zvt9pam0g, 517/lov 51:1Kai 051 évpwnog 051 mix aivpwnog £o ra|- tozjioip 500111 6:50 éutotpdastg, was ,uz'a éf égugooiérceivn, Kat‘ am ,uz'a (iv £177 olvrzxstuévn, where the joint denial of a contradictory pair is(wrongly) taken as the contradictory (6zv1ucsz,uévr7: cf. De Int. 6, 17a 32 and 34) of the corre-sponding joint assertion; see Kirwan 1971: 103.

15. Which is, to my mind, the most important intution to be developed in this newapproach.


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Arguing from a denition l 1

tive, oil/102 , and 6 /léywv at 2c is also the understood subject of léysz at 3b :

cb 1 ; chore 2c , provided that 3b .

But, as we have seen, according to the Parisienses, the resulting argumentis a direct proof not a proof by refutation or a proof followed by a refutation ,contrary to what Aristotle explicitly says d propos of the Principle of Non-Contradiction PNC at F 4, 1006a 5 ff., and repeats £1 propos of PEM andmore generally at F 8, 1012b 5-8. Besides, argument c has to be taken in this

interpretation as a direct proof not only of PEM but also of PNC, for 2c , provided that 3b

actually means

[ 2c ] [o]n peut bien dire vrai ou faux soit quand on dit ‘est’ soit quand on_ dit ‘n’est pas’; mais [ 3b ] cela ne veut pas dire que l’alternative “‘est” ou

“n’est pas”’ puisse étre dite entiérement c’est-a-dire dans ses deux mem-bres simultanément vraie ou entiérement fausse ce qui serait l’intérmediairede la contradiction p. 261

namely, what is excluded is not only the possibility of a joint denial ofboth members of a contradictory pair ‘que l’alternative [...] puisse étre diteentierement [...] fausse ce qui serait l’intermédiaire de la contradiction ’ , but alsoof a joint assertion of them ‘que l’alternative [...] puisse étre dite entierement[...] vraie’ . So, both principles, PEM and PNC, would be directly proved. Andnally, the interpretation of o’r t/lo‘: as a conditional particle ‘a condition que’sounds quite strained, and the choice of /léyaz at 3b seems very doubtful, for/léyaz is more easily understood as a trivialfalsa lectio of a compendium léye +oblique stroke for léyaraz .

3. Therefore all the interpretations of argument c propounded up tonow, namely

caa 1 , bars 2a ; obi/1.0‘: 3a ; cab 1 , 03012 2b ; d/Md 3a ; cd 1 ; chars 2c , provided that 3b ,

have serious drawbacks. As we can see, they all converge as regards step 1 , andall diverge as regards step 2 , while step 3 divides the interpretatio vulgata fromthe Parisienses, together with the presence of absence of an implicit conclusion proof by refutation or direct proof . I would like now to suggest a new interpreta-tion, in particular of step 2 , saving the traditional reading and punctuation of thetext, but also developing an intuition of the Parisienses.


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12 W. Cavini

Beginning from the beginning, step (1) is a denition (cf. b 25) consistingof two asyndetic clauses“, antithetically balanced and strikingly symmetrical,but with exact symmetry avoided through a variation of structure (17/rcai) :

(1) (a) to pév yap léyszv to 5v mi ctitozt I7 10' mi 5v ciitaz 1//8660;‘,(b) to 6 5 10' 5v cfvott lcat'r0' mi dv. mi eziiaz on/ir76ég,V

(a) 1//£650; to mfv léyctv 10' 5v mi sfvaz to an 5v etitaz(b) 0’:/lnég to at /iéyszv 10' 5v sfvaz 10' an o'v an stitat

As we can see, clause (a) denes falsity through a non-contradictory pairof (false) negation and (false), afrmation: and, symmetrically, clause (b) de-nes truth through a non-contradictory pair of (true) afrmation and (true)negation. But if horizontally we have two non-contradictory pairs of affirma-tions and negations (ro' 5v mi sfvott / 10' mi 5v ciiiott and 10' 5v slilaz / 10' mi 5v misiiraz), being respectively both false and both true, vertically, on the contrary,we get two contradictory pairs of affirmations and negations (ro' o”v an aziraz / to6v aziiaz and tomi 5v eii/at / tomi 5v mi stiiaz), namely two pairs of afrmationsand negations whose members are neither both false (as in (a)) nor both true(as in (b)), but one false and the other true.

My suggestion is that step (2) has to be read in this light, as statement of

the principle, immediately deriving from denition (1), according to which, ofevery contradictory pair, necessarily one member is true and the other false, aprinciple now cleverly detected by Weem Whitaker in Chapters 7-9 of De In-terpretatione, where three kinds of exceptions to it are discussed, and dubbedby him ‘Rule of Contradictory Pairs’ (RCP).‘8 So I suggest to read step (2) as

(2d) so that also ifone says that it is or that it is not he will speak trulyor falsely”

namely as a formulation of RCP, on a pair with other parallel formulations ofthe same principle to be found in De Interpretatione, e. g.

T 5 civoiymy rv ,ua v oi/117911 rnv dé 2//£0511 stiiocz zivrigoaozv (8, 12a 26-27)

l6. Independent, but juxtaposed, and paratactically related by miv 58' half asyn-deton (Denniston 1952: 99).

17. Leal Carretero 1983: 54 ff.

18. Whitaker 1996: 79-82.

19. Futurum necessitatis: if one asserts one member of a contradictory pair (6 léycovstiiai mi), then necessarily he speaks truly or falsely.


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T 6 o’zvoZy1c17 t1?v icatoitpaozv 1? t1?v o’t1ré(oozo'zv (211791? 1? 1//£051? siilaz (9, 18a28-29)

T 7 ndaa icaroigoaozg 1? 0’c1r0'(0aozg 0’:/i1101?g 1? 1,usv51?g (9, 18a 34)T 8 o’tvoiy1c17 ,uév Goirepov ,uo'pz0v 11?; oivtzlpoioewg o’c1i116’ég 211/011 1? 1,1/@550;

(9, 19a 36-37)

Now, the assumption of a middle M, i.e. of the joint denial of a contradic-tory pair, clearly conicts with RCP, and therefore also with denition (1),from which RCP immediately derives. So

ifone assumes the very

possibility ofa middle for contradictory pairs one must deny, by the same token, the stand-ard denition of truth and falsity, and that sounds evidently like a reductio adabsurdum of the opponent of PEM. That is also the way in which Aristotleresumes at F 8, 1012b 5-11, the argument from denition of F 7, lucidly statingthis time (a truncated variant of) the standard formulation of RCP”:

T 9 cilia 1rpo'gnoivtagrozigtozoérovgloyovgaezodaz5ezI1coz6oi1repé}.é;(t9nKai év toil, énoivco Zoyozg, 0611' efvai 11 1? ,u1? sfvaz, (3:11/lot o*17,uaz'vszv rz,03015 éf opzquozi 51a/isméov /laévrag ti o17,uaz'vez to 1,/@550; 1? too’c).11t9s'g. aide‘ ,u170e'v oi)./lo 10' l179ég gooivaz oinozpoivaz 2//6650;écmv, adovatov noivta 1//.9051? sz?>az- a’zvéy1c1] yap rg oivriqraiaewg0ci1apov aiiiai ,u6pi0v o’r}.1]0ég.

But against all such arguments we must postulate, as we said above,not that something is or is not, but that people mean something, sothat we must argue from a denition, having got What falsity ortruth means. Ifthat which it is true to afrm is nothing other thanthat which it is false to negate, it is impossible that all assertionsshould be false, for necessarily, of a contradictory pair, one memberis true 2‘

that is to say, if (1), then -10 M, for RCP.The last step of our argument, namely the adversative assertion

(3) obi/1’ 051.9 10' 5v léyetaz ,u1? sfvaz 1? siivaz 0512 to ,u1? 5v,

has then to be taken as Aristotle ’s objection to the opponent of PEM22,which asserts neither one of the members of the two possible contradictorypairs (to 5v ,u1? siirotz 1? efvotz / to /11? 5v ,u1? afvaz 1? 2111011), but denies both:

20. Unnoticed by Whitaker, Cf. also F 7, 1012a 2-4.

21. Ross’ translation slightly modied.

22. And not, pace Kirwan (1971: 117), as the opponenfs assertion.


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(30) but neither of what is it is said that it is not or that it is, nor of whatis not,

thus contradicting RCP by saying that, of a contradictory pair, both membersare false.

Our semantic argument (c) is therefore a reductio ad absurdum of the op-ponent of PEM (more precisely of (aa)) having the following structure:

(1) DfTF by Assumption(2) RCP from (1)(3) M by Assumption -.RCP from (3) RCP /\ -1 RCP from (2) by Conjunction —» M from (3) by RAA

or more compactly:

(cc) (1), ébars (2d); 0’zMo'c (3c); .

Summing up: an immediate consequence of the standard ‘semantic’ de-nition of truth and falsity is the principle according to which asserting onemember of a contradictory pair is true and asserting the other false; but ifneither one of the members of a contradictory pair is asserted, but both aredenied, we are clearly contradicting this principle, and therefore also the de-nition from which it immediately derives. That is to say, we are self-contradict-ing ourselves.*

References

Cassin/Nancy 1989Barbara Cassin and Michel Nancy, La Décision du sens: Le livre Gam-ma de la Métaphysique d ’Arist0te, Introduction, texte, traduction etcommentaire, Paris, 1989. .,

* This paper is the rst written version, and the last (?) metamorphosis, of an oralpresentation on the same topic I made in 1996/97 before different audiences: the PaduaGraduate Seminar in Philosophy, the Cambridge B Club, and the Thessaloniki Symposi-um. I am deeply indebted to all the pariticpants, and very grateful for the perceptivenessand patience with which they all attended my confused attempts to grapple with the prob-lem. Special thanks are due to Michel Narcy for saving me from misunderstanding his inter-pretation, and to David Sedley (among many other things) for improving my English.


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Bonitz 1849Hermann Bonitz, Aristotelis Metaphysica: Commentarius (1849),Darmstadt 1960.

Kirwan 1971~ Christopher Kirwan, Aristotle’s Metaphysics Books, 1 , A, E, Trans-

lated with Notes, Oxford, 1971.Denniston 1960

J. D. Denniston, Greek Prose Style (1952), Oxford 19602.Leal Carretero 1983

Fernando Miguel Leal Carretero, Der aristotelische Wahrheitsbegriffand die Aufgabe der Semantik, Diss. Koln 1983.

Mignucci 1975Mario Mignucci, L argomentazione dimostrativa in Aristotele: Com-mento agli Analitici Secondi, vol. 1, Padova 1975.

Primavesi 1996Oliver Primavesi, Die Aristotelische Topik: Ein Interpretationsmodellund seine Erprobung am Beispiel von Topik B, Miinchen 1996.

Ross 1953W. D. Ross, Aristotle ’s Metaphysics (1924), A Revised Text with In-

troduction and Commentary, 2 vol., Oxford 19532.Schwegler 1847Albert Schwegler, Die Metaphysik des Aristoteles, vol. 3 (1847),Frankfurt am Main 1960.

Wittgenstein 1961Ludwig Wittgenstein, Notebooks 1914-1916, Edited by G. H. vonWright and G. E. M. Anscombe, with an English translation by G.E. M. Anscombe, Oxford 1961.

Whitaker 1996 6C. W. A. Whitaker, Aristotle ’s De Interpretatione: Contradiction andDialectic, Oxford 1996.


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arguing from a definition aristotle on t

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