THÉMATISATION DU CUBISME PICTURAL DANS LA POÉSIE D’AVANT-GARDE

5.1 A recapitulation of ABPC

As was shown in the previous chapter, ABPC, for many centuries the one and only predicate logic in existence, is an exact parallel to standard propositional logic created by the Stoics. Every student of logic will remember the tradi-tional Square of Opposition, presented in Figure5.1a and expressed in terms of the Boethian symbolsA, I, E, and O (see also Section 2.2).

The entailments fromA to I and from E to O are known by the traditional names of positive and negative subalterns, respectively.A and E, moreover, are known as the universal positive and the universal negative, respectively, andI and O as the particular positive and the particular negative, respectively.

The relations expressed in the square representation must be extended by stipulating that I-type L-propositions ‘convert’, in the original Aristotelian sense of ‘conversion’ (antistrophe), defined in the Prior Analytics and discussed briefly in Section 3.4.1. ‘Antistrophe’ or conversion is the process whereby the F- and G-predicates are interchanged. In ABPC, it can be applied toI-sentences salva veritate:^SOMEF is G is equivalent to^SOME G is F (I
I!). Since O-sentences are I-sentences with a negated G-predicate,

SOME F is ^NOT-G is equivalent with ^{SOME NOT}-G is F. Moreover, as E-sentences are, in fact, negated I-sentences, the converse of^NOT[SOME F is G] is^NOT[^SOMEG is F]. Then, F and G are interchangeable inA-sentences provided both F and G are made negative (contraposition): ALL F IS

G is equivalent with^{ALL NOT}-G^{IS NOT}-F. Moreover,^ALLF^ISG entails, but is not equivalent with, ^SOME G ^IS F and hence ^SOME F ^IS G (the subaltern entailment).

As has been mentioned several times in the preceding chapters, we consider the Boethian notation in the form of the classic square unsatisfactory, even though it was in use for many centuries, mainly because it fails to express

the roles of external and internal negation and because it fails to express the Modulo*-Principle. For these reasons, we decided, in Section 2.2, to replace the symbols E and O with A* and I*, respectively, so as better to be able to express the fact thatE-sentences are in fact A-sentences (that is, universals) and thatO-sentences are in fact I-sentences (particulars), in both cases with an internal negation on the G-predicate, represented by the asterisk. We also adopted the sign ¬ for standard (external) negation so as to express the fact that contradictoriness is systematically caused by negation.

FIGURE5.1 ABPC represented as the Boethian Square of Opposition and as the strict natural square consisting of two isomorphic triangles

a. b.

F^IGURE5.2 Propositional calculus and ABPC as isomorphic complete octagonal graphs

It has been shown, in the preceding chapters, that, apart from the notation used for the vertices, ABPC is better represented not as a square with two crossing diagonals but as a combination of two triangles connected at two of their vertices by equivalence relations, as in Figure 5.1b, just as was done for propositional calculus in Figure 4.1. In Chapter 4, however, it was shown that the optimal representation for both propositional and predicate calculus is maximalist: it takes the form of an octagonal graph, as in Figure5.2, repeated from Figures 4.3 and 4.5.

Section4.1 has made it sufficiently clear that, under a strictly extensional ontology, ABPC suffers fromUNDUE EXISTENTIAL IMPORT(UEI). What interests us here is the historical development of ABPC and of its unknown variant Aristotelian-Abelardian predicate calculus or AAPC.

5.2 The not quite Aristotelian roots of ABPC 5.2.1 Aristotle’s own predicate logic

What did Aristotle’s own predicate logic, found mainly in his On Interpreta-tion (Int), look like?1 Specialists say, no doubt correctly, that Aristotle did not complete his system of predicate logic and elaborated only some aspects. And, as everyone does who is working at a new formal system, he also dwelled on aspects of the system that were not, in the end, incorporated into it.

Thus he spends some time on a sentence type called ‘adio´ristos’ in the first chapter of On Interpretation. This term means literally ‘indefinite’, but it is perhaps best rendered as ‘generic’. It is described formally by saying that it lacks any specific quantifier (ALLorSOME) and semantically, somewhat cryp-tically, as ‘about a universal class but not universal in character’ (Int14b7). His example is Man is white, where ‘man’ denotes a universal class but the sentence does not imply that all men are white. In fact, Aristotle allows for both Man is white and Man is not white to be true simultaneously despite the fact that, grammatically speaking, the latter is the negation of the former and should therefore be its logical contradictory (Int 17b31–32). He fails, however, to specify the quantificational truth conditions that would make this possible. It is easy to see that this sentence type, as long as it remains as ill-defined as it is, can hardly play a role in any sound predicate

1 The most detailed and authoritative study on Aristotle’s On Interpretation (Perı` Herme¯neı´as) is Weidemann (1994), where one finds a translation that is better than most, along with a discussion of the chronology and authorship of the work, a complete survey of the manuscript tradition, the tradition of interpretation in Antiquity, the Arab world and the Middle Ages, and the translation tradition. What it does not have is an analysis of the actual logic involved.

logic. Such lapses are understandable, given the high degree of difficulty of predicate calculus and given the fact that Aristotle had to create logic out of nothing, without any existing terminology and without any formalization techniques to build on. Many things that were still opaque to him are clearer to us now.

In the matter of existential import, however, Aristotle appears to have seen the problem more sharply than most later logicians and many modern historians of logic, who tend to gloss over it too lightly or even fail to see it.

When one reads Aristotle’s text literally, one sees that he hedges precisely on the point where existential import becomes relevant. He rejects the classic Conversions and thus saves his logic from the blemish of undue existential import, as will be clear in a moment. That Aristotle may well have seen the danger of undue existential import looming at the horizon is seldom taken into consideration, presumably, one is inclined to think, be-cause his commentators, followers, and critics did not, on the whole, discern it as clearly as they could have. It was not until the late nineteenth century that the logical problem of existential import began to receive full attention.

Until that time, awareness of this problem seems to have been desultory and incomplete. Since we have no inclination to underestimate Aristotle, who is undoubtedly one of the greatest intellectual giants in Western history, we will have a closer look at the issue.

What we see, when we read the text of On Interpretation closely, is that Aristotle stops short of stating the Conversions—that is, the equivalence of^NO F is G and^ALLF is^NOT-G and of^SOMEF is G and^{NOT ALL}F is^NOT-G. The Kneales noticed this (Kneale and Kneale1962: 57):

Aristotle [. . . ] allowed that Every man is not white could be said to entail No man is white, but rejected the converse entailment.

Yet these authors failed to see the relevance of this rejection. Further down in their book, when discussing Abelard’s rejection of the Conversions and his appeal to Aristotle who, like Abelard, considered^{NOT EVERY}human is white and not^SOMEhuman is^NOT-white, to be the contradictory of^EVERYhuman is white, their comment is (Kneale and Kneale1962: 210):

It is true, of course, that Aristotle wrote Greek words corresponding to Non omnis homo est albus [Not every human is white], but it seems clear that he did not intend to convey by these words anything different from the doctrine later attributed to him by Boethius.

This not only contradicts what they wrote on p.57, but it is also, one must fear, just wrong. Why should Aristotle not be taken at his actual words?

If that is done, what results is a sound, though incomplete, system of predicate logic.

The actual passages in Aristotle are Int17b16–26 (translations mine):

I use the term ‘contradictorily opposed’ for positive and negative statements, the one attributing a property universally and the other not universally to the same objects, such as Every human is white versus Not every human is white, or No human is white versus Some human is white. The term ‘contrarily opposed’ is used for the pair universal positive and universal negative, such as Every human is white versus No human is white, or Every human is just versus No human is just. It is impossible for these latter two to be simultaneously true, but their contradictories can sometimes be simultaneously true of the same class, as in Not every human is white versus Some human is white.

and Int20a16–23:

Since the contrary of All animals are just is the statement meaning No animal is just, it is clear that these two will never be true at the same time nor be predicated of the same thing. But their contradictories, that is Not all animals are just and Some animal is just, will sometimes be simultaneously true. Then we have the following entailments:

All men are not-just entails No man is just, and Some man is just entails the former’s contradictory Not all men are not-just. For in that case there must be at least one just man.2

These passages are crucial. For if, as Aristotle actually does here, the Conversions are given up for one-way entailments (A* ‘ ¬I, and therefore alsoA ‘ ¬I*, because if A* ‘ ¬I then also A** ‘ ¬I*, and A**
A) but not vice versa, the logic is sound: it then no longer suffers from its central logical defect, undue existential import. This fact is too important for it to pass unnoticed and it seems less than fair to Aristotle to ascribe this crucial avoidance of a basic logical error to mere good luck on his part.

Yet it is true that Aristotle fails to be explicit on several points where one would have liked him to be a little less sparing of his words. For example, he gives no evidence of an explicit awareness of the subaltern entailments. Yet they follow directly from what he does present explicitly. He does say clearly and repeatedly that the truth of^ALLF is G requires the falsity of its contraryNOF is G. He also says explicitly and repeatedly that the contradictory of^NO F is G is its nonnegative counterpartSOMEF is G. Given this, it is hard to imagine

2 This last sentence is problematic owing to the extreme density of Aristotle’s style at this point. The literal translation of the Greek Ana´ngke¯ ga`r eı˜naı´ tina is ‘For there must be some’. Most existing translations leave the opacity of this sentence unclarified. I have followed Weidemann’s translation (Weidemann1994: 20): ‘denn notwendigerweise ist denn ja irgendeiner gerecht’, which makes perfect sense: if there is one just man, then it cannot be the case that all men are unjust.

that he failed to see that, therefore, the truth ofALL F is G requires the truth of

SOMEF is G—that is, the positive subaltern entailment. And analogously for

NO F is G, which, by contraposition, entails NOT ALL F is G—that is, the negative subaltern entailment for the negation of ^ALL F is G (though not necessarily for its supposed equivalent^SOMEF is^NOT-G). Moreover, he implies the validity of the subaltern entailments at Prior Analytics (25a8–14):

The positive sentence does convert [in the Aristotelian sense; PAMS], though not as a universal but as a particular, for example, if every pleasure is a good, then some good must be a pleasure. Of particular sentences the positive does convert (since if some pleasure is good, then some good will also be a pleasure), but the negative particular does not convert, because if some animal is not human, it does not follow that some human is not an animal.

This says that an A-sentence entails the converse (antistrophe) of the correspondingI-sentence and that an I-sentence and its converse are equiva-lent. It follows immediately, of course, that an A-sentence entails the corresponding I-sentence. Any impartial reader will agree that, therefore, Aristotle does have the subaltern entailments. He should, in any case, be given the benefit of the doubt in this respect.3

In similar manner we notice that Aristotle fails to posit the relation of subcontrariety explicitly. He certainly has no term for it. While his term for contraries was enantı´ai and for contradictories he used the terms antı´phasis and antiphatiko^s antikeı´menos (‘contradictorily opposed’), as at Int 17b16–17, or sometimes simply antikeı´menos (‘opposed’), as at Int17b24 (see also De Rijk2002: 103), there is no term for subcontraries. The first known occurrence of the concept of subcontrariety is in the logical treatise Perı´ Herme¯neı´as, written in the second century CE and reliably attributed to the Latin author Apuleius (best known for his Metamorphoses or The Golden Ass). In this treatise, the Latin term subpares (‘nearly equal’) is used for subcontraries, incongruae for contraries and alterutrae for contradictories (Sullivan1967: 65;

Londey and Johanson 1987: 56, 88–89, 111). (Apuleius has no term for the subalterns (Londey and Johanson 1987: 109) and is a little reticent on the subject of the Conversions (Sullivan 1967: 71), though he does use the term æquipollens for them.)

The first known occurrence of the Greek term hypenantı´ai, which underlies Latin subcontrariae, is in the fifth-century Greek commentary on Aristotle’s On Interpretation by Ammonius, who writes: ‘The particulars are called sub-contraries (hypenantı´ai), because they are placed below the sub-contraries [in the

3 See also Kneale and Kneale (1962: 58), where these authors express essentially the same conclusion.

Square; PAMS] and follow from them.’ (Busse1897: 92). (One may surmise that diagrams were commonly used in classroom teaching, or else it is hard to understand how Ammonius could use the expression ‘below the contraries’;

see also Section5.2.3.)

At Int17b25 and 20a17–18 (quoted above) Aristotle says that the contradicto-ries of contracontradicto-ries may be simultaneously true. But he does not, or not explicitly, draw the further consequence that the falsity of the one excludes the falsity of the other: they cannot be simultaneously false. Had he done so, he would have established the relation of subcontrariety between I-type and ¬A-type sentences. We shall be generous and assume that Aristotle did know about subcontrariety, even though he did not give it a name. Aristotle may not have seen all the details, but to deny him the insight that if two sentences cannot be simultaneously true, their contradictories cannot be simultaneously false, would seem to do him an injustice. In this we are supported by the Kneales, who take it that Aristotle was, in fact, aware of the relation of subcontrariety even though he had no term for it (Kneale and Kneale1962: 56):

The two particular statements [i.e.I and I*; PAMS] have been said by later logicians to be subaltern to the universal statements under which they occur in the figure and sub-contrary to each other. Although he does not use these expressions, Aristotle is interested in the relations so described, and assumes that sub-contraries cannot both be false though they may both be true. This is shown by his description of them as contradictories of contraries.

Yet while one appreciates the Kneales’s generosity vis-a`-vis Aristotle as regards subcontrariety, one must note their apparent confusion on the subject of the Conversions. For in Aristotle’s system of predicate logic (AAPC), the pair < I,I*> is not part of the relation of subcontrariety, whereas the pair <I,¬A> is, as are the pairs < I*,¬A*>, <¬A,¬I*>, and <¬I,¬A*>, as one can read from Figures 5.3 and 5.6a. Aristotle merely accepted the one-way entailments fromA to ¬I* and from A* to ¬I and therefore from I* to ¬A and fromI to ¬A*, which makes A and I*, and A* and I, contraries, but not

¬I and ¬I*, so that I and I* cannot be subcontraries.

What appears to be the authentic Aristotelian system as it can be culled from his texts is based on the two independent stipulations (i) and (ii), from which further theorems follow (‘><’ stands for contrariety; ‘’ for subcontrariety):

(i) A ><¬I (contraries) (ii)A* ‘ ¬I (entailment) From this follow the theorems: From this follow the theorems:

¬A  I subcontraries) A* >< I (contraries) A‘ I (positive subaltern) ¬A*  ¬I (subcontraries)

¬I ‘ ¬A (contraposition) I ‘ ¬A* (contraposition) A^{* ><}¬I* (Modulo*-Principle) A >< I* (Modulo*-Principle)

¬A*  I* (subcontraries) ¬A  ¬I* (subcontraries) A^*‘ I* (negative subaltern) A ‘ ¬I* (Modulo*-Principle)

¬I* ‘ ¬A* (contraposition) I* ‘ ¬A(contraposition)

This system, incomplete as it is, is presented in Figure5.3, which differs from Kneale and Kneale (1962: 55) and Parsons (2006: 4), but is in agreement with Thompson (1953: 259). The relation of contradictoriness of any pair < T,

¬T> (T stands for any of the sentence types defined) has been added because it is implicit in Aristotle’s definition of sentential negation. This system is, though not fully elaborated, logically sound, other than ABPC, which is not.

As has been shown, Aristotle rejected the Conversions, cutting them down to the one-way entailments specified above. That this is sufficient to avoid undue existential import is not apparent from the incomplete Figure5.3, but it is from Figure5.7a, where all the missing logical relations have been filled in.

There one sees that the types¬A, ¬I, ¬A*, and ¬I* do not entail any I-type or I*-type sentence, which absolves the system of the charge of undue existential import. In Figure 5.2b, however, which represents ABPC, one sees that all

I A

¬I A*

I*

¬I*

CD

CD

C C

¬A

¬A*

CD

CD C

C

CD: contradictories C: contraries SC: subcontraries

>: entailment SC

SC SC

SC

FIGURE5.3 Aristotle’s predicate logic as presented in his texts

sentence types as represented by the vertices entail anI-type or an I*-type sentence, which restricts the system to situations where [[F]]6¼
.

5.2.2 The ancient commentators

There is a remarkable contrast between, on the one hand, the very long history of predicate logic and, on the other, the amount of persistent misun-derstanding and misattribution.4 There is no doubt that the Master laid the foundations for predicate logic, and in better-informed circles (e.g. Kneale and Kneale 1962; De Rijk 2002) it is also known that his own system of predicate logic differs in certain important respects from traditional ABPC as developed by his commentators, especially the Latin author Apuleius (125–180 CE), the Greek Ammonius (440–520 CE), and his younger Roman contemporary Boethius (480–524 CE)—that is, five to eight cen-turies after Aristotle.

One should realize that formal predicate logic, as presented in Aristotle’s texts, was not at all a popular subject in ancient times. There was a rich tradition of writing commentaries on Aristotle’s works, in the context of higher-education teaching of Aristotelian philosophy, but the interest was almost entirely focused on questions of the compatibility or incompatibility of Platonic and Aristotelian doctrines and, after the third century CE, also the more mystically oriented philosophy introduced by Plotinus known as Neo-Platonism. There was a secondary focus on Aristotle’s theory of the syllogism, but clearly no focus at all on predicate calculus (for documentation, see Sorabji1990, 2004).

Until the advent of the Middle Ages, formal predicate logic was but a sideshow of a sideshow. Compared with the large number of known ancient commentators on Aristotelian philosophy in its each and every aspect (see Sorabji2004), the number of commentaries written on Aristotelian predicate

Until the advent of the Middle Ages, formal predicate logic was but a sideshow of a sideshow. Compared with the large number of known ancient commentators on Aristotelian philosophy in its each and every aspect (see Sorabji2004), the number of commentaries written on Aristotelian predicate