ILOSOPHY
.RE
\ncient ancl Medieval Universiry of Lor-rvain
Leuven (Iìclgium). Leuven
universiry Press 1L)97
TF{E, PERENNIAL TRAD ITI ON OF NEOPLATONISM
edited bv
JoHxJ. CTEARY
rx$hAtTsrmiT? {i rflo3ofrtA
rriv'r{ rAr*r rr
+dff-3 2 /6 g
- \_/ON PROCLUS' COMPARISON OF ARISTOTELIAN AND PARMENIDEAN LOGIC
Drno Buzznrrt
I. Interpreting Proclus' clairn
In his book The anatonry of l,leoplatonísm, a masterly account, as it is, ofthe conceptual structures underlying the fundamental themes of Neo- platonic philosophy, Professor Lloyd indulged in a particularly harsh judgement on "Proclus' comparison of Aristotelian and Parmenidean logic," by saying that Proclus' "criticism of Aristotelian formal logic is mostly too confused to amount to anything."l The following consider- ations are mainly concerned with countering this judgement and they simply airn to suggest a tentative interpretation of Proclus' clainr that the
"Eleatic rnethod" (In Parm. 1000.35; MD 351) is "much more com- plete" (1007.16;357) than the Aristotelian. Accordingly, I shall confine rnyseif to arguing that Proclus' contention is plausible, on both logical and historical grounds, and I shall by no means try to offer a thorough reconstruction of his views, an attempt that would go far beyond the linits of my present work.2
In his book, Professor Lloyd also n-raintains that Proclus' "mention" of Platonic diaiectic "does not make it any easier to make much sense of his claim that categorical syllogistic is heuristically the inferior, less 'neat'
for dispensing with hypotheses in the discovery of the 'connections
and differences of things'." But really is such a reference to Platonic dialectic altogether irrelevant from a logical point of view? Professor Lloyd hinrself points out that Proclus stresses "the superiority of Parnreniclean dialectic" by saying "that its premisses are reached by division."t So let us first consider how division stands to the other dialectic methods and to definition in particular.
II. The fourfold division of dialectic and Parrnenides' rnethod of hypotheses
The "division of 'dialectic' "+ into the "four subordinated procedures of definition, division, demonstration and analysis"(1 003.7-9; 353) was a
1 . A n t o n y C . L l o y d ( 1 9 9 0 ) , p p . 1 1 , 1 6 .
2 . C f . , i r r t h i s r e s p e c t , J o h n D i l l o n ( 1 9 8 7 ) , p p . 1 ( r 5 - 1 7 5 . 3 . L l o y d ( 1 e e o ) , p p . 1 2 , 1 6 .
1 . L l o y d ( l e e o ) , p . t ì .
3 3 2 l ) r N c r ÍJ v z z r : t t
"comrìlonplace"5 in later Neoplatonisnr. In his Pdrmenídts colnnrentary, Proclus says tl-rat "definition is a nrore allgllst and sovereigrr art than denronstration, and division in turn than definition," for "cJivision gives t o d e f i n i t i o n i t s f i r s t p r i n c i p l e s , b u t n o t v i c e v e r s a , " ( 9 8 2 . 1 1 - 1 5 ; 3 3 6 ) , a n d a g a i n " d e f i n i t i o n i s t h e b e g i n n i n g o f c l e n r o n s t r a t i o n " ( 9 8 0 . 3 3 - 3 , 1 ; 3 3 5 ) . Analysis proceeds frorn the sr-rbjects under investigatiorl "to tl-reir first principles," sonletinres "to the causr's, sonrctinres to the accessory causes (ouuaírLa), at other times to both"(1003.21-25 353), and "it serves as conrplement (c)urírcetrat) to denronstration," as well as to defìnition and division.r'
Ilut how are the four traditional parts of the dialectical nrethod formally related to one another? Proclus hinrself ofTers us an appropriatc answer by propounding a quite original reappraisal of Parnrenides' nrethod and of its connections with l)lato's clialectic. In his attenrpt to illustrate Parnrenides' dialectic "Proclus sets out to atone for a nrillenniurn of neglect,"T as he argues at length to convince his readers.
He wants ther-n "accept the view" that Parnrenides' "rnethod of hypotheses" and the "functions of dialectic" are "the sanle as each o t h e r " ( 6 5 1 . 1 0 - 1 6 ; 4 2 ) . T l - r e f u n c t i o n s o f c l i a l e c t i c h e h a s i n n r i n d a r c i t s
"diaeretic part" and its "definitional aspect," such as he finds thenr expoundecl in the Sophíst (253d) and nrentioned in the Phat'drus (2(r6b).8 And that is to sa1l, in other words, that Pannenides' method can be totally
"identified" with Platonic dialectic." A proper cxanrination of l)roclus'
"derol5atory conlparison"lt) o6 Aristotelian syllogistic with Parnrenides' method requires a closer examination of the articulations of dialectic.
III. A case for 'ascent':
defìnition and division
It is worth dwelling a whilc Llpoll Proclus' denronstration of the substantial identity of the Eleatic and Platonic rnethods in order to favour a better understanding of the relation between division and definition in Plato's dialectic. Division, one of "the two aspects of c-lialectic nrentioned i n t h e P h d e d r u s , " i s " t o d i v i d e t h e o n c i n t o t n a n y " ( 6 5 0 . 1 6 - 1 8 ; 4 1 ) a n d
5 . L l o y d ( r e e o ) , p . t ì .
6 . I n P t r u t . 9 8 2 . 2 4 - 2 8 ( M l ) 3 3 6 ) .
7 . Johrr I)illon, introduction to 1)ror/rr.s'Oomtnctttar),ott P l t t î o ' s ' P t n r r u r i d r ' - s ' , ( l ) r i r r c e t o n : l ) r i n c e t o n l J n i v . l ) r e s s , 1 9 t ì 7 ) , p . 1 1 .
fl. In Pann. 649-651 (Ml) .10-42).
9 . John I)illon, introduction to bk. 5, in 1)rtrr'1u-i' O o m m t l r f t t r y , p . 3 2 6 .
1 0 . L l o y d ( r e e o ) , p . 1 1 .
P l t o c l u s o N A t t r s - r t ) ' r ' E L I A N A N n P n t t v E N I l ) E A N L ( ) ( ; l ( . -) -)-)
1 a td e f i n i t i o n , t h e o t h e r a s p e c t , i s t o " c o l l e c t m a n y i n t o o n e ; " ( 6 4 9 . 2 5 - 2 6 ; 41); in exactly the satne way, "the Eleatic sage" in the Sophist "uses diaeresis to make nlany out of one and definition for getting one fronr n r a n y " ( 6 5 6 . 2 - 4 ; 4 5 ) . D i v i s i o n a n d d e f i r - r i t i o n a r e t h e n t o b e s e e n a s c o n v e r s e p r o c e d u r e s , t h a t a r e " c o n s o n a n t " w i t h t h e " f o u r p a r t s " ( 6 5 0 . 1 5 -
16; 41) of the "procedure of dialectic," as expounded by "the Eleatic wise nran" (649.36-650.1; 41) in the Sttphist
he who is capable of this perceives distinctly a single idea perwad- ing rrany, each of them positcd as distinct, and nrany ideas different from one another included ur-rder one enrbracing idea, and again a single idea pervading many others but united into one, and many ideas altogether distinct in every way (253d).
According to Proclus, what the wisc nran is actually saying is that "the dialectician's task is to make his way through such hypotheses as Parnre- n i d e s ' r n e t h o d g o e s t h r o u g h . " ( 6 5 0 . ' 1 0 - 1 2 ; 4 1 ) H e r e , t h e " t w o a s p e c t s o f d i a l e c t i c n r e n t i o n e d i n t h e P h a c d r u s " ( 6 5 0 . 1 6 - 1 8 ; 4 1 ) , i t s " d i a e r e t i c " a n d its "definitional" parts (650.35-37; 41), are further subdivided into two.
It nray be r-rseful to quote in full Proclus' explanatory renrarks, before conlnlenting upon thenr in ottr turn. Procltts' tttaintains, that
tl-re f:our parts in the above statellìent are consonant with the two aspects of dialectic nrentioned in the Plnt'drtr-s. One of thenr was to divide the one into tnany; this is the property of diaeresis, to sepa- rate a llenus into species. [i] The genus is " tlrc sin,qk idca" sprcad through many scparatc things anci cxi-ilinc in cadr o-f thcm; for the ,qcnus rs not an assenrblage of species, like a whole of parts, but is prcscttt itr each of thc spccies as existing before thern a,nd parÍicipated in both by tach o,f thc ,scparatc spccics ancl by the genr-rs itself. lril Tlrc -rpccic-s are tlre nrany ideas diflerent ftronr one another but ctrmprtlrcnded by onr sin,qle embracín,q idea, which is tlrc,qcnus; though it is outsicle thetn, as transcending tlrc sptcit's, yet it cotttait'ts thc caust:s of species; for to all tlrose who posit ldeas, rcdl ,qenera are thought to be both older dnd more esscntíal than the spccit's ranged under then-r; the realities exist- ing prior to species are not identical with the clnracters that exist itt t h c s p t c í e s b y p a r t i d p a Í í o n . ( 6 5 0 . 1 5 - 3 4 ; 4 1 )
H e n c e d i v i s i o n c o n s i s t s i n ( 1 ) s e e i n g o n c i c l e a ( " t h e ' s i n g l e
i d e a " ' ) r n m a r t y things ("spread through nlany separate thinus"), "existing in each of thenl," "posited as distir-rct" (e.g. the "genlls" present in each of the
" s p e c i e s , " o r " p a r t i c i p a t e d i n " b y " e a c h o f t h e s e p a r a t e s p e c i e s " ) ; o r e l s e ,
334 l)lNcr lSuzzÍ,.Trt
division consists in (2) seeing many ideas (the "species") "c:ornprehended by one single embracing idea, which is the genus" (e.g. the "caLlses" of the s p e c i e s , c o n t a i n e d i n t h e " r e a l " g e n u s , " t r a n s c e n d i n g t h e s p e c i e s , " " o l d e r and more essential than the species" ranged under it). Here tl-re important fact is for us to notice that there is a logical difference between these "two kinds of the diaeretic part of dialectic": the first one gives rise to first- order statenrents about participated genera, the second one to second- order statenlents about real genera transcending the "characters that exist i n t h e s p e c i e s b y p a r t i c i p a t i o n " a n d c o n t a i n i n g th e i r c a u s c s . l l B r . r r l e t u s proceed with Proclus again:
Thus we see th:rt to distinguish between these tuto kinds is the task of the diaeretic part of dialectic; the distinction betr,veen the remain- ing tuto belongs to the delinitional aspect. This art [i] perceives a -sir1glc unifed ídea pewading many wholes - collectil{g the many ideas, each of which is a whole, into a single definition, r,veaving thenr together and frorr-r all these apprehendcd u,httles bringing about a sin- gle idea by grasping Íhe many as onc; and further [ii] it looks r-rpon //rc many ideas tt has collected as disÍinctboth-from one anttther antl-fronr the u,hole which arises-frttm them. This is what we should expect, for how could it make one out of nrany if it had not previously seen the rrrany as separate from one another? (650.3 4-651.9; 41-2)
So we see that, conversely, definition consists in (3) "collecting" Lnúny ideas, each of which is "apprehended" as "a whole," into oruc "single unified idea;" or else in (a) seeing "the many ideas" as distinct both "fronl one another and fron-r the" tvtc "whole which arises fronr thettt." Again, there is a logical difTerence between the first kind of the "definitional aspect" of dialectic, which gives rise to second-order statenrents, and its second kind, which brings about first-orcler statelnents.l2
Accordingly, we can distinguish (2) higher-order division (seeing the many in the one) and (3) higher order definition ("grasping the nrany :ìs one") frorn (1) first-order division (seeing the one in the nrany) and (a) first-order definition (seeing the n'rany as distinct fiorn the one). Higher- order statenlents posit the one, first-order statenlents posit the rnany. We nray iilustrate the connections between the two aspe cts of dialectic anc-l their subordinate kinds by nreans of the following diagranr, which shows how the four resulting procedures, cLlrrently known as the traclitional
1 1 . I n P a r m . 6 5 O . 2 - 3 6 ( M D 1 1 ) .
1 2 . r b i d . , 6 5 0 . 3 4 - 6 5 1 . 9 ( M l ) 1 l - 1 2 )
l ) t t O c ; l u s r t N A t t t s l r ) ' t L , L t A N A N I t I ) n t r r u 1 i r N I l ) l : A N L ( ) ( ; l ( 33s
rncthods of division, definition, clenronstratiot-t and analysis, can be related to each other:
Diacretic pdrt
Dc.lìrtitíorral Itlrt 2rtd-ordcr
statcmcLtts
l)ivision
()\
[)efinition (3) 'l st-ttrdcr
statcilt(nts
Analysis ( 1 )
I )errronstr:rtion (4)
This diagranl is sonrehow renriniscent of a 'sqttare
of opposition' and what it sholvs is (i) the conrplenrentarity of the corìtrary proceclures, sucl't as definition and division, or clernonstration ar-rcl analysis; and (ii) thc cornplenrentarity of the diftèrent logical levels of assertion, afTorcled by the corresponding lower ancl higher orders of predication, which rel:rtes definition to clernonstration and ciivision to analysis. l\efcrring tcl it, we can easily trnderstancl how "ciivision (2) gives to definition (3) its first p r i n c i p l e s " (9 8 2 . 1 3 - 1 1 ; 3 3 6 ) a n d w h y " d e f i n i t i o n ( 3 ) i s t h e b e g i n n i n g o f denronstration ('+)."(9tì0.33-34; 335) Ancl since "the prenrises fi-onr which clentonstrations proceecl" are "c--allses of what is cletnonstrated,"(1ll E u c l . 1 4 . 1 8 - 2 0 ; M 1 2 ) w e c a r ì a l s o s e e th a t a n a l y s i s ( 1 ) i s t h e c o n t r a r y (ciurírce trat) of denronstration (4), "inasr-nuch as it leads us to analyse eflects into their causes."(In Pann. 982.24-25: MI) 33(r) Moreover,
" i n a s n r u c h a s it p r o c e e d s f r o n i t h e c o n r p o s i t c t o t h e t r t o r e s i t n p l e " and
"inasr-rrLrch as it proceecls fronr the partictrlar to thc trnivers:ì1," (982.26- 28;336) analysis can be seen as opposcd, at a clifTcretrt level, to clefinition, which posits the sinrple containir-rs the cornpler, :.rnd to division, r,r..hicl-r p o s i t s : r s d i s t i n c t th e p a r t s c o n t a i n e d i n t h e w h o l e .
I l u t n t o r e t o o u r p o i r r t , w c c e n r e a l i z e t h a t t h e r e is a l o g i c a l ' a s c e t r t ' t h a t nratches ntetaphysical "'ascerìt' frottt the sensible to the intelligible f o n n s . " ' 3 L " t u s r e c a l l t h a t : r c c o r c l i n g t o l l r o c l u s " p r i r t c i p l e s c 1 o n o t p o s s e s s t h e i r e x i s t e n c e c o n c e p t l r a l l y , b u t i n r e a l i t y , " ( 1 0 5 ' + . 2 7 - 2 8 ; 4 1 2 ) for "principles are principles thror,rgh their own :tttthority and not in v i r t u e o f o u r c o n c e p t s " ( 1 0 5 4 . 3 1 - 3 2 ; 1 1 2 ) . S o , i f " o b j c c t s o f t h o u g h t nlust exist prior to and indepenclentlv of the thinkirig ol- thent," we rright put the so-callecl "principle of Logical l{ealisnr" of the later Neoplatonistsla to b.tt"r Llse by nrakine it work hctrristically in reverse.
1 3 . L l o y d (r e e o ) , p . 1 - 1 .
14. Ilicharcl T. Wallis (,972), p.'12'{
3 3 6 I)rNc'r lJuzzr:tt
After all, logical insight is what is sought after here, and instead of saying
"that logical distinctions inrply ontological ones,"'' *" should adrnit in reverse that it is ontological distinctions that suggest logical ones. Hence ontological structures (whatever they nray mean) can be conveniently assurned as a heuristic model for logical forrr. And that is precisely what Proclus always purports to do. So, considering both kinds of the diaerctic part of dialectic, we should not be surprised, when we find that "an ontological shift tacitly occurs frorn the paronynrous entiry (in Aristotle's s e n s e ) t o i t s e p o n y m o u s ' q u a l i r y ' . " 1 ( ' F o r w e s h o u l d b e n r o r e a l e r t to t h e fact that a lo,qícal shift as well as tn ttntttltt.gícal one is here at stake.
We rriay now proceed and try to unravel the far reacl-ring implications ofProclus' assumption that in demonstration'ascent' is involved, because
"there is a necessary correpondence between the developnrent of denronstration and the hierarchv of the realities that are accounted for. " l7
IV. A case for 'ascent': dernonstration and analysis
Considering the relation between denronstration and analysis, rve shall c l e a r l y s e e th a t l o g i c a l ' a s c e n t ' i s a n e s s e n t i a l r e q u i r e n r c n t o f d i a l e c t i c . A s already noticed, there is a certain parallelisn-r between definition :rnd division on the one hancl ancl denronstr:rtion and analysis on the other.
Just as definition and division were seen as converse procedures, in the same way analysis was called "the converse of denronstration,"'n o,
" s y n t h e s i s , " i l l b e i n g it s " r e v e r s e p r o c e d u r e . " ( I n E u c l . 2 5 5 . 2 0 - 2 1 ; M 1 9 1 ì ) And as denronstration is to definition ("in denronstr:rtions and definitions the particular nrust", respectively in first- and higher-order assertions,
" b e strbordinate t o t h e u n i v e r s a l a n d t h e c l e f i n i t i o r t " ) ( I n P a m t . 9 8 1 . 5 - 7 ; M D 3 3 5 ) , s o a n a l y s i s i s t o d i v i s i o n . T h e y b o t h l e a d t o a " s t a r t i n g - p o i n t for derrronstration,"(In Eucl. 211.25; 166) but division, dealing with higher-order assertions abotrt first principles (repi rae apyríe)(57 .21-22),
"divides into its natural parts the geÍìLrs proposecl for erattri- nation,"(211.24-25; 166) whereas analysis, dealing with first-orcler asser- tions about what is consequent to first principles (rrepi ra p€Ta rae cipycie)(s7.23), "traces the desired result (rò ('qroupeu.;r.) back to :ì11 acknowledged principle. " (211 .20-21 ; 1 65).
As Professor Lloyd reminds Lls, "the star-rdard explanation" of analysis,
"Peripatetic as well as Platonist," 'was th:rt "of goine up," which is not
1 5 . r b i d .
1 ( r . Walter (lavini (1995), p . 1 3 1 . 1 7 . ' I ' l r c o l .
P l . 1 . 1 0 , . + 5 . 3 - 5 .
1 u . E Ì i a s , l r r I s q q . 3 7 . 2 1 2 2 .
P l { o c L U S c t N A t t t s . t c ) f E I I A N a N t l P R t t H , t t r N I l ) E A N L c l c ; t < ;
3 3 7
o n l y , a s w e s h a l l s e e , g o i r r g " b a c k , " i . e . " f r o t t r a n e n d t o a b e g i n n i n g " , b u t a l s o i n r p l i e s a g a i n t h e i d e a o f a c e r t a i n ' a s c e n t ' . T h e " l t t r p e r i a l l o g i c i a n s w e r e r e a d y t o a s s u l n e , " P r o f e s s o r L l o y d n o t i c e s , " t h a t w h e n ' p a n d q , t h e r e f o r e r ' w a s a s s e r t e d ' r ,
i f 1t and q'could be asserte6.,l() This shows analysis being the converse of demonstration, for dentonstration goes frorl certain pretnises, p and q, to the conclusion, r, whereas analysis proves that a conclusion, r, holds only granted the prenrises, p and 4, froni which it can be derived. But nlore considerations are requircd in order to e x p l a i n h o w a n a l y s i s i r n p l i e s l o g i c a l ' a s c e n t ' .
In purest metaphysical jargon, Proclus lloes at length to show, in his Parnrcttidc.s conlnlentary, that "logical proofs depend on" such entities "as fonns-in-soul," which are also "the causes of the principles of deuton- s t r a t i o n . " I ) e t r t o n s t r a t i o n , t h e t r , r e q u i r e s a r t l c t a p h y s i c a l ' a S c e l l t ' t o entities that "are prior itr t.ìature and not relative to uS, and 1ìlore honourable than what is den-ronstrated by nreans of thetn;" but they are
"universal, not particular;" thLls, "even those who do not believe in the Forllls," i.e. Aristotle, "pay honour in their writings on l)ettrottstratiort20 to tl-re universal," for "it is this that tnakes logical proofs such as they d e c l a r e t h e n t t o b e . " 2 l l J u t h o w a r e w e e x a c t l y t o d e s c r i b e a c o r r e - s p o n d i r r g , i f e v e r , l o , g i c a l ' a s c e t t t ' ?
T h e p r i r i c i p l e , t h a t w h e n ' p l r t d q , therefore r' are asserted 'r,
rf p ancl q' cl.n be asserted, inrplies a logical 'ascent'
because it is a logical nretatheorent equivalent to a deduction theorenr, as I shall try to sho*.22 However, before delving deeper irlto logical nìatters an historical diversion is here in orcler.
V. Material vs. forrnal validity
In this respect, it is worth recalling the kind of argutnents that have been thoroughly discussecl byJonathan lJarnes in a recent series of lecrures.23 These arslrnrents are the so called Stoic rlpe0ri6oe nepaít'ouree )'óyoL tnd their counterparts discussed by late aritiquity logicians. According to B a r n e s , " w e s h o u l d n o t s u p p o s e t h a t t h e y a l l h a v e s o n r e t h i n g p o s i t i v c i r l conlnlon;" \Me should rather content ourselves with saying tl-rey are
"characterised by t lack of sonrething." More precisely, wl-rat they lack for is "sonre formdl rrrlc wl-rich valiclates" theur. Hence "Llntttethodically
1 9 . L l o y c l (1 e e o ) , p . l { ) . 2 0 . C f . A n . P o - s t . I . 1 1 , 77 a5ff . 2 1 . h t P a r m . 9 t l 0 . 3 - 2 9 ( M l ) 3 3 4 ) . 2 2 . S e e s e c t i o t - r 7 b e l o w .
2 3 . J o n a t h a n L l a r n c s ( 1 9 9 0 ) , p p . 7 - 1 1 9
3 3 8 l)rNcr |lu'zz.ettt
concludirjg argunìents" are not to be thought of as "forrlally valid argunìents," but rather as "nraterially valid argunìents."24 To explain the difference, and quite significantly to our pLrrposes, Ilarnes refers to the
"ntedievai distinction," as proposed by_ tsr-rridan, between ".otlscEtentiae formales arrd ctmsequenfiae tnateridles."" (Thcre is llrore point in this
remark, than we nìay perceive at first; but we shall conìe back to it further on.)2n Occasionally, nraterially valid argunìcnts are conceivecl as
"trlrncate d"27 or inconrplete arÉlunrents, that can be transfonrrecl into complete argunrents by adding a ceneral prenrise or principle, which would render thern fornrally valid. We nray cite, by way of illr-rstration, the two exanrples given by Barnes, which he calls the "Euclidean Argument":
(1) Things equal to the sanre thing are equal to one another, (2) CA is eqtral to All,
(3) CB is equal to AlJ,
therefore, (,+) CA is equal to CIì;
and the "Truncated Argunìent"28:
( 1 ) C A i s e q u a l t o A l J , (3) CIf is equal to AIJ,
therefore, (4) CA is equal to CI-ì.
But what is the relation between a conrplete arllunrent ancJ the corre- sponding truncated :ìrgunlent? The "'non-logical' rule of infcrence by which the conclusion is correctly but trnnrc'thodically derived" in the unnrethodical argunìent "is just one of the prenrisses.frttm u,hich the sarne conclusion is botl-r correctly and nrethodically derived" in the nrethodical arÉlunrent. IJut why "is not" the unnrethoclical argullrcnt to be said "a truncated version" of the corresponding nrethodical argunlent?2" Tl',e answer turns on the logical status of the proposition that acts in trlrn as a nrissing prenrise and as arì asserted one : as we shall see, its logical statlls is not quite the sarne in the two cases.
2 4 . I b i d . , p . 7 9 . 2 5 . I b i c 1 . , p . 1 ( r i f . 2 6 . S e e n o t e . 1 5 b c l o w . 2 7 . I b i c ì . , p . 8 l f L 2 8 . I b i c . l . , p . f ì . 2 9 . ' W a l t e r
( l a v i n i (1993), p. t)5.
P t t c t c : l . u s c l N A t t t s r c ) l E L I A N n N t l P n t t u E N I I ) E A N L c l c ; l <
VI. A rnodern (and rnedieval) counterpart
33e
In order to clarify the rnatter let us proceed further in our digression. The whole story here is strikingly reminiscent of "the theory of the Syllogisnr propounded" by John Stuart Mill "in the second Book of the Logic."
Mill's theory grew out of an idea "respecting the use of axiorns in ratioc- ination," which he carne upon "when reading a second or third tirne the chapters on lì-easoning in the second volunte of Dugald Stewart."3"
Dugald Stewart had'justly renrarked" that "axionls are not the four-rda- tions or first principles of p;eornetry,-from which all the other truths of the science are synthetically deduced," but "are nrerely necessary assllrllp- tions, self-evident indeed, and the denial of which would annihilate all demonstration, but -frorn which, as prenrises, nothing can be der-rlon- strated."3t S., .on.eived, general propositions :rrc nrerely "forntulae for making" inferences, and
the nrajor prenrise of a syllogism, consequently, is a forn-tula of this description: and the conclusion is not an inferencc drawn ,from the fonnula, but:rn inference drawn according ro the formula.s2
The idea of an infèrence drawn according to a formula brings back the flavour of unmethodicals; on the other hand, its close resetnblance to contenrportry infarcnce-ticket or inference-lict'nce theories have already been pointed out quite appropriately,3s and I have stressed elsewhere the intportance of the (lanrbridge tradition, up to Iì.anrsey and Wittgenstein, in conveying Mill's views to Ryle and Toulnrit't, who brought therrr forward ,r'r.*.34
According to Toultnin, in what he calls "working" as opposed to
"idealized logic," given sonle "facts" or " ddta (l))," we seek to establish a
" claint (C)" or "conclusion" by bringing forward "not further data," but
"rules, principles," o, "general, hypothetical statetÌìents," which act as
"Lt,arratlts" to license our inference.tt On his own aclnrission-"', Toultttin had drawn ot-ì lì.yle's distinction betweren lau'-statemcnts tnd staternents of ,fact. lnhis Concept ttf'Mind, Ryle says:
3 0 . J o h n S t u a r t M i l l ( 1 9 8 1 ) , p p . 1 f ì 9 - l 9 l . 3 ' l . John Stuart Mill (1e74), pp. 1 90- 1 91 . 3 2 . I b i d . , p . 1 e 3 .
3 3 . A l a n l l y a n ( 1 9 7 0 ) , p . 3 2 f T .
3 4 . l ) i n o l J u z z e t t i ( 1 9 8 7 ) , p p . 1 0 1 - l 1 9 .
3 5 . S t e p l r c n E . T o u l n r i n ( 1 9 5 8 ) , p p . 9 7 L ) t ) .
3 6 . I b i d . , p . 2 7 0 .
3,10 l ) r N o l J u z . z . r : t t
Law-statenlents are true or false but they c1o r-rot state truths or fàlsehoods of the sanre type as those asserted by the statetnents of fact to which they apply or are sr,rpposcd to apply,
and specifies that
a law is used as, so to speak, arr inlterence-ticket (a season ticket) which licenses its possessors to nlove fi-onr asserting factual st:rte- nlents to asserting other factual statenrctrts.lT
-We slrould notice that both Touhnin and l{yle contrast the ltyltttthttical nature of law-statenìents (they call thertr respectively "general, hypothetical statenlents"38 and "open hypothetical statelllents"se) to the catc,qorical llature of factual statelllents (respectively, again, they ntelition
"categorical statenlents of fact"f(' and "sitrgular categorical state- r n e n t s " 4 l ) , a f a c t w h i c h r e t n i t r d s u s o f P r o c l u s ' p r e f e r e n c e f o r
"hypothetical fonns of argtuÌrent"(ln Pdnu. 1007.27; Ml) 357).Ilut we sl-rall conle back later to this point. It has also been observed that
"Touhnin's distir-rction between prerlrises .forl which olle reasons ancl rules in accordance tt,íth which inferer-rces are drawn" is "a sound olle" and that it is "canonical in nrodern logical theory"+2 - but not ortly rnodcnr iogical theory, we should actually say. ln a brief note entitled "The Re- descovery of the Topics: Profèssor Touhnin's Inference-'Warrants", Otto lJird has shown that what the nrcdieval logicians called a "Topical Maxirn" or a " mtxima propttsititt" are the "traditional logic counterparts of Toulnrin's 'Warrants,"
for in the rrredieval cloctrir-re on the Topics "the Maxint perfonns the sanre ftrnction rìs a wàrrAnt"*'' and it is actually described as "a confìnn:ttory rule that proves a conseqtlcltce."+* If, at t nraxinr, we assume a logical law, which is true by necessity, we obtain a c()ns(quentia-fortnalis; on the other hand, ifwe :ìssunrc as r tnaxirtr a general proposition, contingently true in virtue of its tenns, we obtain a, c()ttsc- quentia matcridlis.
3 7 . ( ì i f b e r t l \ v l c ( 1 e a e ) , p p . I 1 6 - 1 1 7 . 3 l J . T o u l n r i n ( 1 9 5 8 ) , p . t ) l ì .
3 9 . l ì . y l e ( l e a e ) , p . 1 1 t ì . , 1 0 . T o u h r r i r r (1 9 5 8 ) , p . ' l { ) 5 . 4 1 . l \ y l e ( l e a e ) , p . 1 1 8 . 4 2 . E r n e s r N a g e l ( 1 9 5 4 ) , p . - 1 { ) 5 . 1 3 . ( ) t t o i l i r c l ( 1 e 6 1 ) , p . 5 3 7 .
' 1 4 . Albert of Saxony, Pcrutilis / t l g i i i r ( V e r r i c e : 1 5 2 3 ) , f . 3 3 r a
P t t o c ; r - u s o N A l t l s l c ) T F . u A N a N t t I ) R t t u E N I l ) l , A N L ( ) ( ; l ( l
VII. A rationale for logical 'ascent'
Having thus come full circle, are we now in a bettcr position ro clari$z Proclus' "confused" criticisrrr of Aristotelian syllogistic? To our purposes, the important fact about the distinction between forn-ral and rlaterial consequence is pointed out by the Pscudo-Scot, who nraintains that a material consequence "can be reducecl to a fonnal conseqLlence through the assumption of a necessary proposition."45 W. can illustrate this point by way of a diagranr:
311
Conscqucntia matcrialis
Cttnscquuúia forrnalis
Inference:
p therefore, r7
p è q
t)
ttr.r.nrr.', q
R u l e :
p - ì q
( ( p - + q ) & p ) - + q
The irnplication is that prcnrises and rules can be interchanged. As Ernest Nagel observes, this nreans th:rt "a rule of inference cen in gencral be replaced by t prcrtrisc - provicled, of cottrse, that soltle rules are retained; and in the case of nraterial rules of inference this can apparently always be done." In additiorr Nagel infonns us that "the above lnanoeuvre" can also be "introducecl in reverse," and
one or nrore nraterial prenrises can be elit-ninated fror-n :ìn argu- nrent without distroying its validity, provided that this elinrinatiotr is conrpensatecl by the introduction of appropriate tnaterial leading principles which pernrit the derivation of the original conclusion b y t h e r e n r a i n i t r g p r e t t t i s c s . + t '
Mill was playing the very salrle !5anle rvith the nr.rjor prentisc of the syllogisnr, as we can see fronr the following rcconstrttctiotr:
. + 5 . "(ìonsecluentirr n r a t e r i a l i s . . . e s t i l l l , c l u a e p o t c s t r c c l t t c i a c l f b r t t t a l e r t ì , p e r a s s l l l l l p - t i c ' r n e r r r u u i u s p r o p o s i t i o n i s n c c e s s a r i a c . " P s e n c l o - S c o t , 1 l l i b n u n p r i n t r n r P r i o n u n A r r o l y , - titttrurn ArisÍottlis Qrrac-sfionr'-s, inJohorrtris /)ul-s Sctrli ()pt'rt (\uttitr, (L,vttn: l)urartd, 1639;
r e p r i n t , w i t h a f c , r c w o r c l b y T u l l i o ( ì r c g o r y , H i l c l e s h c i n r : (ì . C ) h n s , 1 9 ( r t ì ) , 1 . 2 8 7 1 ' > .
4 6 . N a g e l ( 1 9 5 4 ) , p p . 4 0 5 - , 1 { ) 6 .
342 I)rNcr LJvz.z.rtlt
Rt'al inlerutct:
Syllo,qism
l n f e r e n c e :
Ua
[ìule:
(,r)( U-r -+ M.r)
( ( r ) ( U r - - > M r ) & ( E , r ) ( U " r ) ) -+ ( E . r ) ( M r )
therefore. Ma
He was taking the rnajor prcnrise of the syllogisnr, :ì general proposition construed as an inrplicatioll, as a lllaterial rule for what he callecl a "real"
inferetrce. llut we should not foruet, as Rylc renrincls us, that such an
"'open' hypothetical statenlent" and all "1aw-statellÌents" of this kinci
"belong to a different and nrore sophisticated level ofdiscourse front that, or those, to which belong the stater-nents of the fàcts that satisfy tlìellì,"
just as "algebraic statenlents" are "on a diflerent level of cliscourse fron-r the arithnletical statenlcnts which s:rtisfy thenr."+7 What we have hcre is, in other words, proper logical 'ascent'.
All these facts rnay be expressed in :ì nlore concise and alrrrost forrnal way by rnodifi.ing, to conlply rvith both fornral and nraterial argunìents, a generally accepted "principle about the connection bctween validity and logical truth" in the following way:
A (fornral) argunrent is valid if and only if the conditittnrll st:ìtenlent, whose antecedent is the conjur-rction of the prenrises of the argu- n l e n t a n d w h o s c c o r l s c q u c n t i s t h e c o n c l u s i o n o f t h e a r g u n r e n t , i s (logically) true.as
The principle stated here is ag:rin a nrodern counterpart of a principle repeatedly invoked by Sextr-rs Enrpirictrs:
They say that an argullrcnt is conclusive (out,arcrtrri) rvhen the conditional rvhich begins fronr the conjunction clf its prcnrises ancl ends in its conclusion is tnle .4"
4 7 . l ì . y l e ( l e a e ) , p . 1 1 8 .
, 1 u . I quote fronr Karel Lanrbcrt ancl IJ.rs ( 1 . van Fraassen ( 1 9 7 2 ) , p . 1 2 . I h a v e r r c l d c c l , i n brackets, thc r,vord 'fornral'
ar-rci bracketed the
"vord
'krgically'.
4 9 . S e x t u s E n r p i r i c u s , A d r t . M a t h . V I I I - 1 2 6 ( H i i l s e r , 1 0 6 2 ) S e e a l s o V I I I 3 0 . + , :t 1 5 - 1 8 ( H ù l s e r , 1 0 5 9 , 1 0 6 0 , 1 0 ( r 5 ) a n d P ) , r r h . H y p o Í . l l 1 3 7 - 3 8 ( H i i l s e r , 1 0 5 u , 1 0 6 . 1 ) .
therefore, Ma
( x ) ( U x - + M x ) ( U a - + M a )
Uu
l ) l r c l c t t . u s r t N A t u s ' t c ) l - E L t A N n N l t P A I t M E N I I ) E A N L c l t ; l t : -)+-)
1 i )The fonnal relation between a valicl argunlent and a true cotrditional w a s t h e n c l e a r l y r e c o g n i z e d b y " l r n p e r i a l l o g i c i a n s " : t h e i r a s s u l l ì p t i o l l
"that when 'p
and q, therefore r' w:ìs asserted 'r,
rf p and 4' cor-rld be asserted" can be confidently ascribcd to th:rt recognition. For we crìrl now justify our contentior-r that rvhat they "were ready to assullle" is i n d e e d a l o g i c a l n r c t a t h e o r . r l r . 5 " T h e f o l l o w i n g d e d u c t i o n t h e o r e r n
Inference:
p q
,h.'*fbt.J iff
R u l e :
( p & q ) - + r
shows that the conditional 'r,
ifp and q' acts :is a rule for a tnaterial forn-r of inferer-rce, such as 'p
and 4, therefore r'. Accordingly, Professor Lloycl r,vas certainly right in pointing o.tj. that "this is why analysis was called 'the c o n v e r s e o f d e r n o n s t r a t i o n ' ; " ' r b u t t h i s i s t h e c a s e p r e c i s e l y b e c a u s e t h e cor"rdition al'r, tf p and q' was assurÌrecl as :r rnetalogical rule, a trtove that clearly requires a fontr of logical 'ascent'.
VIII. A plausible vindication
Are we now able, in thc encl, to find son're logical significance in Procltts' clainr?'We nray tentatively answer by observing two facts.
In the first place, by discharging a propositiort AS a frretrlise atld assunring it as a nrle we change radically its logical status. As a prenrissc, it :lcts as all asscrted object-laneuagc proposition' as e rttle, it is either (i) a nretalinguisti c statcnl cn t c:ontai n i n g nretali n guistic variables th :rt stancl for the actual prenrises and conclusion and their tenns, or (ii) a corre- sponding second-orcJer object-langllage st:ìtelììent abottt wl-rat sttcl't terlr'rs or actual propositions purport to signify. To say it with l\ylc, thcy
"belong to a clifferent and rÌìort: sopl-risticatecl level of discourse .""
S e c o n d l y , w e c 2 ì n e s s l u l l e a p r o p o s i t i o n a s a r u l e i f i t h a s a c o r t c l i t i o n a l fonl. Hence, thc "non-fbnnal restriction of thc conclitionals" Professor Lloycl is allucling to, rvhen he observcs that
the substittrtictn of'rvhat bclongs apparently to a propositi<-rnal lttgic o f t h e f o n n ' p -)/by w h a t a p p a r e n t l y b e l o n g s t o a l o g i c o f t e ' r t r t s
5 0 . S c c s c c t i o n . { l t r t i t t o t c I 9 r t b t t v c .
