Abstract
At the beginning of the book (§ 1, p. 1) we noted briefly that traditional syllogistic divides the valid syllogisms into four ‘figures’, and that these figures are formally distinguished from one another by the position of the middle term (the term which occurs in both premisses but not in the conclusion). In the first and fourth figures the middle term stands chiastically; in the second it stands at the end of both premisses, and in the third at the beginning of both. The first and fourth figures are distinguished by the fact that in the first figure the middle term stands at the beginning of the first and at the end of the second premiss, in the fourth at the end of the first and at the beginning of the second premiss.
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Notes
Aristotle has overlooked this possibility“ (AS, p. 23); ”His only mistake is the omission of these moods in the systematic division of the syllogisms“ (AS, p. 27).
I. M. Bochenski, La Logique de Théophraste,1947, p. 59: “It seems very likely, on the other hand, that APr. A 7 and particularly APr. B 1 were composed at some time later than the composition of APr. A 4–6. Aristotle would no longer have had the time to work out systematically the new discoveries which he had briefly indicated.” Why is Aristotle supposed not to have had the time? It is true that, although the Analytics are cited by the Metaphysics,the Eudemian and the Nicomachean Ethics and finally the Rhetoric whereas they refer to no other writing of Aristotle’s with the exception of the Topics,we cannot confidently use this as evidence of an early dating (pace Ross, APPA, p. 23) — for it is in the nature of the case that a scientific discussion should import its logical foundations from abroad, whereas a treatise on logic should have small opportunity to refer to the individual sciences. Nevertheless, it is reasonable to suppose that the last years of Aristotle’s life were filled by his wide-ranging scientific, historical and literary researches.
This use of the expression ’definite’ is due to Lorenzen (Einführung in die operative Logik undMathematik,1955,pp. 5 sq.); it seems to be of importance for philosophy too. Every ’scientific’ statement must be ’definite’, that is either ’definite qua provable’ or ’definite qua refutable’. An interpretation is ’refuted’ if we can prove a contradiction between statements in the interpretation itself, or between the interpretation and the text, or between consequences of the interpretation and of the text.
a) Stay 0ßv Spot Tp£ïç oStwç i:xwaty npòç àÂ,Â,riA.ouç YT£ Tbv iaxasov tv dap Elvat T@ µtaw Kai 1òv µtaov tv Ó%.({) TO npeotko fi EIvat fi µti Elvat àvàyKrl TOY 6Kpwv Elvat auAloylattbv Té£tov. (b) Ka? St teaov Jlty Ô Kai ant, tV S,Â.Â,w Kai âÂ.ÂA Év TOÚTq) tatty, S)(at ‘OA OtoEt yiv£Tat µécyov. (c) (aKpa St Tò aùíò TE tv Mg) Sy Kaì tv t) WÂo tasty (A 4, 25b32–37).
It is taken in this sense by Kirchmann, Erläuterungen zu Aristoteles’ Organon,Leipzig, 1877, p. 11 et passim.
Tò St tv 64) Elvat íTEpov taépw xai Tb xa’rà naysdç xanlyop£iaOat OaTtpou OâTEpov Taùsóv tasty (APr. A 1, 24b26–28).
Spa St rub; fatly tin iB1òT1isa ToO np(ttou ax7141atoç tafigavev Cintùv “Tdv iaxasov tv 6),w Eivat sß µtaw” àvsi sob tdv plcrov Toß taxdrou navrdç KaT7lyopeia0at (in APr. 72, 17–20).
Taùsdv yàp sò xasà naysbç Kai Tb tv act) Jlóvrl Tii axtaEt Stacptpovsa, câç Eivat, Slav Jlty tx ‘COO xasà Ttaysdç nottnjEOa Toùç aukkoytaµoùç, actCova Spov sdv Ttpttov, óµoìœç Kai rtpòsaanv LlEiCova Ttjv npeot iv, Stay St tic Tot) tv 64), sdv TEXEusaiov Kai Tilg SEuTtpav npósaaiv (ib. 78, 4–8 ).
This proposition, which has the form “If not-A then A” (Not p-÷p),is of course not a contradiction: in propositional logic it is equivalent to A,that is here to the proposition “there is a syllogism”. However, Aristotle treats a proposition of precisely this form as a contradiction (like “A and not-A”) at APr. B 4, 57b1214. On this cf. Patzig, ’Aristotle and Syllogisms from False Premisses’, Mind 68 (1959), 186–192 (See Appendix below.).
npòç µtv TilV TOO npOK£tittVO%) M ix ciaty àcyt)XXôyuYtot, IiXXo µévsot Tt t aùtCúv isst auXkoytcaoOat xaì SEt at (in APr. 109, 10–12).
Cf. pp. 44 sq. above.
otov Ei Tb µtv A trawl. TfSi, B Ttvi Tò St B Jlr1SEvì TO T. àvTlripE(poJltVwV yàp Tchv npOTàYEwv àvàyKrl TÒ r Tlvi TO A µr1 èiràpxcty. bµoiwç St Kànì Tüúv t’tpwv Yxr1tàtwv Ei yàp yivEtat Sul tf ç àvitaspocpr-lç cruaoytaµóç (A 7, 29a23–27).
The translators generally follow Philoponus: Rolfes translates âvslasp£tpoµévwv TOTV npo’àa£wv “wenn man dann die Sätze umkehrt” and Sul tilg àvnaspocpïlç by “durch Umkehrung” (“immer ergibt sich durch Umkehrung ein Schluss”, p. 17); the otherwise excellent translation by A.J. Jenkinson (Oxford, 1928), runs “if the premisses are converted” and “by means of conversion”. Lukasiewicz, who translates the whole of A 7, 29a19–26, takes Sul Tflç àvitaspocprlç, as I do, to hint at the proof,not the condition,of the validity of these moods. However, he bestows not one word on the paradox: “whenever a syllogism does not resultchrw(133) a syllogism always results” (AS, p. 25). Nor does Ross say anything about it in his commentary. The Oxford translation softens the paradox by adding a word which is not in the Greek: “Whenever a proper syllogism does not result chrw(133)”.
Karl Zell, whom Waitz so sharply criticized (“haud paucis locis veram Aristotelis mentem eum non perspexisse apparet”, I, pp. XII-XIII), is to my knowledge the only translator who is right on this point: “so muss notwendig, da die Vordersätze sich umkehren lassen, C einigen A nicht zukommen” (Aristoteles’ Werke, Organon,Stuttgart,1836, p. 157). We shall shortly meet Zell again (below, n. 20), in connexion with his correct reading of a difficult and generally misunderstood point of Aristotle’s doctrine. Zell’s translation is not, of course, entirely free from error: but it is at least preferable to Rolfes’.
This section of the Analytics has often been wrongly interpreted in the past. Cf. Patzig, Mind 68 (1959), 186–192 (See Appendix below.).
XÉyw St µeiCov µtv âipov èv 4a Tò µÉaov tatty Ekattov St Tò Únò Tò µÉaov 6v (A 4, 26a21–23).
Zell (o.c. p. 128, n. 14) notes on the first half of (c): “This refers to the subsequent lettering of the terms in the order M N X”. However, Zell falls back on relative extensions to explain (d): “The middle term is ”outside“: i.e. it is not the case (as in the first figure) that it is contained in the major and contains the minor. It is ”the first“ since the two other terms are subordinate to it” (p. 141). Kirchmann (o.c. p. 128, n. 5, p. 32) expressly rejects Zell’s correct explanation of (c). Cf. further p. 125.
a) tàv St tf al)Tai Tò µtv rravtì Tò St µi]SEvì 6itâpxq, fi âµtpw itauti fl aaSEvì, Tò µtv axr’µa TÒ TOLOOTOV Kea Tpirov. (b) µÉaov S’ èv aka, Ka0’ o6 àµcpw Tà Kasr]yopoúµeva, dKpa St Tà Katr]yopoúµeva, (c) µetÇov S’ àxpov Tò i[0ppóUTepOv TOO µ6600, EXaTTOV Sè Tò tyyÚrspoV. (d) TiOETat St TÒ]1ÉaOV tfw µtv Taïv àxpwv, Éaxatov St t OÉaet (A 6, 28a10–15).
a) (Aka rzpßtov %r]rtttov Tàç 66o rtpotàaetç, et°’ o6tw SLatpextov eiç ‘rob; Spollç, µÉaov Sè OcTÉ0V Taw ópwv TÓV èV àµcpoT patç Taiç lcporecaecrt k yótavov. (b) àvàyKr] yap td µÉaov èv à uportpatç úrtàpxety tv àrtaat Totç axfigaaty. (c) tàv µtv oóv Katr]yopfj Kai Katr]yopftat Tò µèaov, fl afrò µtv Katr]yopj, fi ko St txefvou àrtapvrrrat, Tó rtpcbTov éatat axr}µa. (d) tàv St Kai Katr]yopfj Kai àrtapvf tat àrtó Ttvoç, TÓ µèaov (e) tat/ S’ ä7,.a èKE1you Kacr]yoptltat, fl Tò s v àrtapvijtat Tò St Katfyopfrat, Tò Éaxarov. (f) oútw yàp EIxev tv ketcrrp axr]µaTt Tò µèaov (A 32, 47a36-b6).
g) daa S’ èv rtp.efoat rtcpaivexat, Tot) JÉaou °tact yvwptot,µev Tò axfiga (A 32, 47b13–14).
It might be supposed that these definitions also cover the syllogisms appended in A 7 and B 1, which traditional logic places in the fourth figure; this is true for the premisses of Bamalip (BaA CaB) of Calemes (BaA CeB) and of Dimatis (BiA CaB). However, since sentence (c) admits to the first figure only those moods in which the premiss containing the middle term as predicate is affirmative it would not catch Fesapo and Fresison. It seems likely that when Aristotle came to write A 32 he was only thinking of the four valid moods of A 4; I leave open the question whether or not that is evidence that A 7 and B 1 were written after our passage.
Ei àvàyKrl µév n aaßcty rtpòç àµcpw xotvóv, Tobro S’ év8txtrat tptxtSç (rj yàp Tò A Tot) T Kai Tò T tob B Kat1yop1laavtaç, i Tò T xaT’ àµcpoiv, p àµrPw xara to T), Tabla S’ éatt tà EtptpIÉya axMµata, 1pavepòv fin rtàvta auaaoytaµóy àvàywu ytvecrOat Stet toútwv Ttvòç ‘OW oxullàtcov (APr. A 23, 41a13–18).
According to Apuleius (Opera, ed. Thomas, IV, 193, 16–20) the ’subaltern’ moods were introduced by Ariston of Alexandria (c. 50 BC), a Peripatetic. Apuleius calls the move “ineptum”. Incidentally, Bocheríski (FL, p. 161; HFL, p. 140) must be corrected on this point: he has let himself be confused by Apuleius’ statement that three of Ariston’s new moods belong to the first figure and two to the second. He has not taken into account the fact that Apuleius, like Theophrastus, counts the moods of the later fourth figure as moods of the first. In addition to Barbari and Celaront Ariston constructed Celantop (in the first figure, according to Apuleius). Therefore Bochelíski’s 24.273 should read, not “A to all B; B to some C; therefore C to some A”, but rather “A to no B; B to all C; therefore C not to some A” - the subaltern form of Celantes (cf. Bochenski, FL, p. 116; HFL, p. 101). With this improvement the “riddle of the text is solved” not “only partially”, but completely. - Prantl, who here for once follows Apuleius and calls Ariston’s move “colossally stupid” (I, p. 557), conjectures the third new mood of the first figure to be AaB BaC-’ CiA, that is the traditional Bamalip. But Apuleius himself has already admitted the mood in this, its Theophrastian, form as the second mood of the first figure (cf. I, p. 557 and 366); the moods introduced by Ariston must of course be ones which Apuleius did not allow into his system.
I suppose here, for the sake of argument only, that such extensional definitions are possible. Cf. pp. 97 sqq.
Ross writes: “Thus he (sc. Aristotle) recognizes the validity of all the inferences which later logicians treated as moods of a fourth figure, but he treats them, more sensibly,by way of two appendixes to his treatment of the first figure” (APPA, p. 35 - my italics). Aristotle never explicitly states that the five moods in question belong to the first figure; nor, of course, that they belong to an appendix of the first figure. It is not clear from Ross’ words whether the “more sensibly” only reproaches those logicians who introduced a fourth figure, or whether it also applies to those who, like Theophrastus, regarded the moods in question as legitimate members of the first figure (and not as bastard appendages to it). Whereas Uberweg (SdL5, p. 339) thought that Theophrastus was simply working out Aristotle’s own suggestions, Prantl (I, p. 367: “a superficial interpretation”) and Maier (SdA II, 1, p. 98: “disastrous for the future development”) violently attacked the introduction of the five indirect moods into the first figure.
Cf. p. 85, n. 17 and p. 127, n. 2.
E.g. Zeller, Die Philosophie der Griechen II, 22, p. 164: “Aristotle does not take explicit note of the fourth case, in which it (sc. the middle term) is subject of the lower and predicate of the higher term; however, we shall blame him the less for this, since such a case cannot in fact occur in a pure and rigorous procedure”. Cf. Prantl I, p. 272.
Kant, Die falsche Spitzfindigkeit der vier syllogistischen Figuren,1762, § 4: “The sort of inference in this figure is so unnatural chrw(133) ” etc. Ross, APPA, p. 35: “The fourth figure draws a completely unnatural conclusion where a completely natural conclusion is possible”.
Prantl (I p. 367) calls the sixth Theophrastian mood of the first figure (Celantes = Calemes (IV)) “utterly worthless”; the eight and ninth (Fapesmo = Fesapo (IV) and Frisesomorum = Fresison (IV)) “can have chrw(133) no importance whatsoever as special moods of argument” (ib.); finally “the whole fourth figure chrw(133) must be labelled a vast and pointless game” (I p. 574).
xakeo St µtaov µtv ô xaì aútò tv Mao Kai äß,7,,o tv tonto) lath/chrw(133) äxpa St tò aúíò tE Iv äa,X0 Öv Kai tv (I â%,%o tativ.
Cf. also Alexander, in APr. 72, 17 sqq.
Cf. also Democritus, frag. B 26, Diels-Kranz.
Thus Alexander says (72, 21–26) that the predicate of a proposition of the form AaB is necessarily more extended than its subject, but that this does not hold for propositions of the form AeB.
B. Einarson, ’On certain mathematical terms in Aristotle’s Logic’, AJP (1936), 33 sqq. and 151 sqq.
Iulius Pacius, In Porphyrii Isagogen et Aristotelis Organum commentarius analyticus,Genf, 1605, p. 122: “Merito autem has vocat ‘figuras’: nam per figuras mathematicas declarantur”.
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Patzig, G. (1968). The Figures. In: Aristotle’s Theory of the Syllogism. Synthese Library, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0787-9_4
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