University of Virginia Library

A Quantitative Solution of the Ambiguity of Three Texts
by
Antonín Hrubý

The traditional critical methods founded more or less on the principles of the Lachmanian school are unable to deal with the difficulties encountered by a textual critic. Awareness of this fact has had a profound influence on editorial practices of our times, especially in the field of medieval literature. Present practice favors the printing of a basic manuscript; and, if a contemporary editor tries at all to reconstruct a critical text, or if he tries to base certain theories and explanations on the results of textual criticism, he does so with the greatest apologies for his method and cannot refrain from stressing the subjective character of such an attempt.[1] Studies in textual criticism are not lacking, but a closer examination shows that with few exceptions scholars are today mainly concerned with the practical difficulties of applying the old principles of textual criticism and show a rather limited interest in purely theoretical problems.[2]

This attitude seems to be the result of a radical skepticism over an objective method in textual criticism, independently expressed in about 1913 in two related disciplines of philology. At that time, a series of students of classical philology based their criticism of the traditional

Lachmanian system on their experiences with papyruses,[3] and in the field of modern philology, Joseph Bédier explained in the introduction to his second edition of Jean Renart's Le Lai de l'Ombre the reasons which led him to repudiate the editorial practice of his time, his own previous attempts to establish the stemma of the lai and his decision to base the new edition mainly on a single manuscript.[4]

Though only a classical philologist can really judge the intricate questions raised by the papyrologists, even an outsider can recognize that classical philology reacted to the skepticism over method in a quite different manner from the medievalists. It is the general conclusion of the former that textual criticism is based on sure methods, the validity of which has not been weakened by denials of its authority; that the criterion of "common error" is only one of the indices which permit an objective criticism, and that its admitted inadequacy must be overcome by the strictest restriction of its use and definition[5] and by a systematic study of all the other indices which are available for retracing the history of the text.[6] Many studies, following the examples of A. C. Clark and Louis Havet,[7] proceed therefore to a systematic investigation of indices permitting one to retrace the history of the text and to arrive at a critical canon.[8] The most interesting result of this evolution of classical studies is the almost unanimous agreement of the critics that objective criticism can be achieved and that the notion of

"basic manuscript" and the practice of editing the so-called "best" manuscript in the most conservative manner has to be rejected.[9]

The medievalists are quite unanimous in a contrary view. Their practice favors following the "best" manuscript as closely as possible, and even such exhaustive studies of manuscript tradition as those by A. Micha, A. Henry or B. Edwards, result finally in a conservative reproduction of the basic manuscript.[10] The editing of a basic manuscript is based on the opinion that even the corrections of a scribe have their historical value and, at the same time, on the disbelief that the modern editor could achieve a better text through methodical recension than his medieval colleague could through sheer inspiration.[11] The opinion that any attempt to evolve a methodical reconstruction of a text with the present knowledge can only do damage to its authenticity, has been stated most explicity by Mario Roques,[12] but almost the same argument against the critical reconstruction of the text and the same methodical skepticism had been expressed already fifty years ago by Joseph Bédier: "Bref, nous renonçons à proposer un classement de nos manuscrits; non pas qu'il soit difficile d'en proposer un, . . . mais au contraire parce qu'il est trop facile d'en proposer plusiers" (Op. cit., p. xli.)

The main reason for the pronounced skepticism of the medievalists lies, to be sure, in the fact that in modern philology the traditional criterion of common error is even less applicable then in the classical branch. The impossibility of using the notion of error in medieval

texts as a discriminatory criterion for establishing the manuscript families has been stated again and again. A. Micha indicates, for example, that in all the numerous manuscripts of the no less numerous romances of Chrétien de Troyes he has found not one single clearly erroneous passage which has the necessary discriminatory value;[13] and, B. Edwards has made a similar statement concerning the eight manuscripts of Gui de Cambrai's Vengement Alixandre (A classification, p. 241). In A. Henry's introduction to the works of Adenet Le Roi, one can read of the difficulties of searching for a "discriminatory passage" in words which attest the professional honesty and care which the editor lavished upon the work of his choice (Adenet le Roi, I, p. 89).

Although the attitude of all the students of medieval philology is not always equally negative,[14] it appears nevertheless, that the skepticism of the medievalists is due not only to the difficulties of the problem, but equally to the fact that they think and work under the spell of the powerful personality of Joseph Bédier and leave unexploited the truly revolutionary innovations of the method, introduced into the matter namely by dom Henri Quentin, Sir Walter William Greg, and Jean Fourquet.

Joseph Bédier has treated the problems implied by textual criticism and the establishment of the genealogical stemma twice; first, in his introduction to the second edition of Jean Renart's Lai de l'Ombre and, fifteen years later, in his well known article in Romania.[15] In both studies, he explains the reasons which induced him to repudiate the traditional critical method and to reproduce as closely as possible one single manuscript. One of his reasons is revealed by his unusual observation that the application of Lachmanian principles almost inevitably leads to the construction of a dichotomous stemma. Since it is extremely unlikely that all of our extant manuscripts should invariably

belong to only two genetical groups, Bédier concluded that there is a hidden fallacy (un vice caché, as he says) in the method itself, which allows the editor to construct a dichotomous stemma, whenever he wishes. Bédier's second argument against the Lachmanian system is even more revealing. When Gaston Paris proposed a three-branch stemma in his review of Bédier's first edition of the Lai de l'Ombre (Rom, XIX, 611), Bédier, his confidence shaken in his own dichotomous solution, discovered to his surprise that he was able to construct as many as eleven different stemmata, all of which explain the facts equally well. As a reason for this profusion of different hypotheses, he reveals correctly the basic impossibility of establishing the genealogical relationship of three witnesses without using a qualitative criterion:

It is extremely interesting to note that Bédier describes here a basic fact which he discovered more or less intuitively and which, only a year before, W. W. Greg described and analyzed in his Calculus of Variants.[16] It is beyond doubt that both scholars discovered this principle, called by Greg quite appropriately the ambiguity of three texts, independently of each other. W. W. Greg formulated the principle of the ambiguity of three texts as a result of a thorough analysis of the problem of interpreting the variational groupings of manuscripts and has shown that it is impossible to determine the lineage of three extant manuscripts on logical grounds alone: "where three manuscripts only are concerned, no merely formal process can throw light on the relationship between them. Either the readings will be all divergent or else

the variant will be of type 1 [i.e.: A : BC, B : AC and C : AB], and since, in the latter case, the reading of the single divergent manuscript may always (theoretically at least) be unoriginal, it will never be possible to establish a common source for any pair of manuscripts to the exclusion of the third. Given three manuscripts, therefore, it is impossible either to prove or to disprove independent derivation. This fact, which I call the ambiguity of three texts, we shall find, meets us at every turn of the discussion, and it largely determines the nature of the calculus" (Calculus, p. 21).

We have quoted both passages in some length because they both reveal very clearly the essential problem implied by any attempt to apply a rigorous method in establishing the genealogical relationship of extant manuscripts. Since the problem remains basically the same whether we deal with three, four, or more manuscripts, it becomes apparent that an objectively valid methodology can be developed for this branch of philology only to the extent to which it will be possible to overcome the ambiguity of three texts. Unfortunately, however, neither Bédier nor Greg attacked directly the problem which they recognized with such a clarity.

Bédier concluded that, due to the inadequacy of the textual evidence, the problem is in practice insoluble; he therefore elevated the critics own taste, his knowledge and education, the goût as the only valid criterion of textual criticism, and had the fortune to form a school. Greg's Calculus on the contrary, has never been in favor with the textual critics.[17] The truly revolutionary importance of Greg's achievement, however, lies in the fact that Greg presented for the first time in the history of textual criticism a complete theory of interpreting the variational groupings and thereby created conditions for further investigation. Yet, he did not push his analysis far enough to recognize the intricate complexity between the frequencies of the variational groupings and their genealogical causes. Greg therefore overestimated the indicative value of certain variational groupings which he called simple or also, quite symptomatically, constant significant groupings. Believing that there is no formal solution to the problem of the ambiguity

of three texts, Greg accepted as a practical expedient the notion of originality, although he knew from experience that this criterion has only exceptionally the required distinctive value (Calculus, p. 53f).

Both quotations also show that given three extant collaterals, the critic will have to decide almost invariably between four hypotheses and it is generally recognized that with the help of the present techniques and criteria such a decision is extremely difficult, if not impossible. Greg's analysis of the ambiguity of three texts, on the other hand, reveals that the problem is completely insoluble on logical grounds alone. There is a tacit agreement among the critics that with a growing number of witnesses the decision might become eventually less difficult and for this reason Greg himself uses a group of six manuscripts to demonstrate his method of interpreting the variational groupings. This belief, however, is based on an incomplete analysis of the problem and is in reality unjustified. If, for instance, four collaterals are given and if they offer the variational evidence of two branches, i.e., they show a persistent variation AB : CD, then the number of hypothetical solutions grows to twenty, seven of them being significant with regard to the genealogy of the witnesses:

In the absence of some clearly distinctive criteria, any of the seven above stemmata accounts equally well for the variational evidence, obtained by the comparison of four collaterals. If there is insufficient evidence for the two branches, the number of possible significant stemmata grows to twenty-five. In the case of six collaterals, with variational evidence of the branches AB : CD : EF, the number of significant solutions is twenty-one and it grows to over three hundred in case the branches cannot be distinctly determined. Since our distinctive criteria remain equally unreliable in either of these cases, the difficulty of the solution obviously does not decrease, but increases with the growing number of witnesses. There is evidently little hope that the present criteria of common, peculiar, indicative, conjunctive and separative errors, the notions of originality and of directional variation, or even the different indices obtained from external evidence, will help the critic to eliminate several hundred hypothetical possibilities, especially if usually he does not even suspect their existence.

We see thus that in the absence of a formal way of overcoming the ambiguity of three texts, any system designed for the purpose of establishing the genealogical relationship of a given group of extant manuscripts is condemned to remain a mere technique, the validity of which will be determined by the validity of the assumptions introduced by the critic at the outset. It is therefore of great interest to investigate whether the unreliable and often subjective qualitative criteria may be excluded and whether a different approach to the problem permits us to overcome the ambiguity of three texts by a formal distinction. Since the only objective information with which the critic is left after excluding the qualitative criteria consists of statistical data concerning different types of agreements and disagreements between the compared witnesses; since this kind of information is of a distinctive quantitative character; and, finally, since the absolute majority of the changes introduced by the different scribes is random, an interpretation of the statistics of variational groupings by means of probability calculus appears at the outset a rather appropriate method for approaching the problem.

The idea of using statistics and probability calculus in textual criticism is not new. Statistics were used, to the best of my knowledge, for the first time by Dom Henri Quentin in 1926, whereas the probability calculus was used as far back as 1876 by Hermann Paul. In 1945, Jean Fourquet reintroduced the calculus of probability into the matter; atfer Fourquet, the probability calculus has been applied by Whitehead and Pickford and more recently by Castellani.[18] In all these cases, the

quantitative approach has served as a means of demonstrating an argument of a highly speculative character, and their results are therefore of no help for us.[19] However, although no attempt has been made as yet to attack the basic problem of the ambiguity of three texts, the following demonstration would have been impossible without the methodological innovations of Greg, Quentin, and Fourquet.

Reduced to its basic form, our problem is thus the following: is it possible, given three extant manuscripts, to obtain evidence about their way of descent from their common ancestor?

To save time, we exclude all ancestral and mixed genealogies from our consideration. Excluded are thus the following types of descent and their permutations:

The types 1-3 can be easily identified by the fact that the extreme members of these groups cannot possibly agree (in theory at least) against their intermediary. Type number 4 can be determined only by a judgment of value concerning the originality and the directional value of the variations A : BC.[20]

After these exclusions, there are then for us only two possible types of derivation of three extant manuscripts:[21]

We see at the first glance that none of the six possible permutations of the first scheme (namely: ABC, ACB, BAC, BCA, CAB, CBA) is genealogically significant. In successive derivation, three out of the six possible permutations are significant, since in the second scheme ABC, BAC, CAB are genealogically identical to ACB, BCA, CBA, respectively. The character of the problem, however, remains the same in all three significant permutations of the successive derivation. It follows that the problem of the ambiguity of three texts will be solved if there is a formal way of distinguishing between independent and successive derivation.

By comparing three extant manuscripts, the critic gathers complete statistics of five different types of agreements and disagreements between the readings of the witnesses. Let us call them variations (not variants) and assign to them the symbols v1, v2, etc.:

• v1 = ABC
• v2 = AB : C
• v3 = AC : B
• v4 = CB : A
• v5 = A : B : C

These five variations remain the same when we deal with three collaterals whether they are derived independently or successively. The observable phenomena contain therefore, as we already pointed out, no indication of the genealogy of the manuscripts, if the critic abstains

from using a judgment of value.[22] We know, however, that the observable variations must be produced in quite a different way in either type of derivation.

In order to be able to proceed to a phenomenological analysis of our five variations, we have to introduce at this point a certain number of postulates and assumptions which have the function to eliminate, at the outset, extreme logical and technical complications:

• 1. The postulate that all extant manuscripts of a given work are derived by transcription from a single original.
• 2. The assumption of universal, persistent, and spontaneous variation.
• 3. The assumption of an archetype with no recognizable errors.
• 4. The postulate of impossibility of convergent variation and of uncontaminated tradition.

The extent of this report forbids me to comment on the implications of the preceding assumptions and postulates, but even without any additional explanation their character as mere working hypotheses is sufficiently clear. None of them is necessarily true, and every critic knows that especially the last one has very little chance to correspond to the reality; their only function thus is to permit us to create a scale of hypothetical norms which gives us the possibility of stating and defining, at a later stage, the "irregularities" of real cases. Our assumptions and postulates should therefore by no means be accepted as principles and the ultimate elimination of their implications becomes a necessity before attempting to establish the stemma of any actual group of extant manuscripts.

We shall now analyze, under the aforementioned assumptions and postulates, the generation of the observable variations in the hypothetical case of independent derivation. In such a case of derivation, all three witnesses descend directly from the archetype "A". They are not necessarily direct copies of it and we know that an undetermined number of lost hypothetical manuscripts may be placed on the lines

connecting "A" with A, B, and C. However, since all the changes introduced by the hypothetical manuscripts appear to us as changes of their terminal descendants, we can disregard the existence of the hypothetical manuscripts without affecting the correctness of our analysis. We can then say that, in case of independent derivation, all three scribes copy a certain given number of words contained in their common exemplar "A", the archetype. Since, from the point of view of the descendants, all words of "A" are "original", let us assign to them the symbol o. According to assumption two, the scribes of A, B, C, preserve a certain number of o and they change another number of them to, say, a, b, and c, respectively. The variation v1 is then generated whenever all three scribes preserve accidentally—according to 2 and 3—one and the same word of the archetype without change. The variations v2, v3, v4, are produced whenever two scribes preserve o and the third introduces a change of his own in any particular word of the archetype. The variation v5 arises whenever at least two, or else all three scribes, introduce a change in any particular word reading o.

In successive derivation, the manuscripts A and X, descending directly from the archetype, will preserve a number of o and change another number of them to a and x, respectively. The scribes of B and C then find in their common exemplar not only o, preserved by X, but also a certain number of x. They will then preserve a number of o and change another number of them to, say, bo , co , respectively; also, they will preserve a number of x and change another number to, say, bx , cx , respectively. It must be noted that the generation of bx and cx is genealogically determined by the existence of x; as a consequence, one may very well find them, when comparing three witnesses, in variation with o or a of A, but obviously they never can be found in variation with o, bo , co , of C and B. We then can observe that in successive derivation, the variations v1, v2, v3, are generated in exactly the same way as the identical variations of independent derivation. The variation v4, however, will be generated in two additional cases, namely, whenever x, preserved both by B and C, meets through collation either with a or o of A. Compared to independent derivation, the variation v5 has in successive derivation six additional chances of arising, namely, whenever bxcx , bxx, xcx , of BC meet through collation either with a or with o of A. Graphically we can express these facts in the following way:

The table shows clearly that the five observable variations represent in either case of derivation quite different combinations of original and nonoriginal readings. Since in practice it is difficult or impossible to distinguish the original readings of the archetype and the changes of the different scribes, these combinations cannot be observed by the eye; but, they can be distinguished conceptually, and we know with certitude that they exist. We called the observable combinations variations; let us call those which can be distinguished only conceptually variants. We then can say that v1, v2, v3, are in both types of derivation identical; we shall call them simple variations, because they each represent only one possible variant. In independent derivation, v4 also is a simple variation, whereas in successive derivation v4 is a compound variation and represents three possible variants. The variation v5 represents four possible variants in independent derivation and ten variants in successive derivation.

The preceding statements already show the importance of the quantitative factor in our problem. We can expect that the difference between a simple and compound variation, although not observable by the eye, must be reflected by a difference in its frequency. We also notice that, with regard to independent derivation, in successive derivation the presence of X changes the character and therefore the frequencies of the variations. Finally, we see that the observable data obtained by a comparison of the witnesses, i.e. the variations, are causally determined, on the one hand, by the changes introduced by the scribes, and, on the other hand, by the genealogical relationship of the different extant and lost manuscripts of the group. If we then can express the expectancies of the different variations in both our types of derivation in mathematical terms, we should be able to "observe" the difference in the frequency of the variations produced by the presence of X. Such a possibility appears to be secured by the fact that all the extant manuscripts are descendants of one archetype, no matter how they are derived, and the total number of variations therefore may be conceived of as equal to the total number of potential changes contained in the archetype; the sum of the rates of variations is then a constant in any case of derivation. Thus, we can expect that the problem of the genealogy of three extant manuscripts will be soluble mathematically under the condition that the number of unknown factors does not exceed the number of independent bits of information offered by the statistics.[23]

To illustrate the preceding reasoning by a less abstract example, let us assume that we compared three states of one text and that we found the variation v1 six hundred times, and the variations v2, v3, v4, v5, one hundred times each. We then know that the total number of potential changes contained in the archetype of our three witnesses was: 600 + 100 + 100 + 100 + 100 = 1,000. Since it is more convenient to use rates in computations rather than integers, we express the preceding statement by saying that the rate of v1 is 600/1,000, and that the rates of the remaining four variations are 100/1,000 each. The same can be expressed even more conveniently in decimal numbers, and we then can write that, in our case, the sum of the rates of the five possible variations is: 0.6 + 0.1 + 0.1 + 0.1 + 0.1 = 1.0. In order to express this statement in general terms, we set the rate of v1 = f 1, v2 = f 2, etc. We then can state that in any case of derivation of three manuscripts the following must be true: f 1 + f 2 + f 3 + f 4 + f 5= 1.

Let us now consider the case of a single scribe and let us again assume that his exemplar contained 1,000 potential changes; let us also assume that this particular scribe actually realized 100 out of the 1,000 possible changes and therefore necessarily preserved the version of his exemplar in the 900 remaining cases. The rate of the realized changes is then in our particular case 0.1, and the rate of unrealized changes is 0.9. One of the two possible variables is thus determined by the fact that the sum of the realized and unrealized changes of any particular transcript must be equal to the total number of potential changes contained in the exemplar (in our hypothetical case: 100 + 900 = 1,000), and that the sum of these rates must therefore be equal 1 (in our case: 0.1 + 0.9 = 1). To express this fact in general terms, we set the numerical value of potential changes contained in any archetype as n, the value of changes introduced by any transcript as p, and the value of its unrealized possible changes as q. The following relation then must be true: p + q = n. The immediate consequence of this definition is that Pp + Pq = 1, where Pp represents the rate of realized, and Pq the rate of unrealized changes of a particular transcript. The preceding statement implies that the following must be equally true: Pq = 1—Pp. If we then set the Pp rates of our three witnesses as Pa, Pb, Pc, respectively, we know that the corresponding rates of preservation must be (1 — Pa), (1 — Pb), (1 — Pc), respectively.

To make the quantitative analysis of our problem possible, we have now to introduce the notion of probability. Mathematical probability may be defined most conveniently for us as the ratio of the number of actual occurrences of an event to the number of possible

occurrences. The laws of probability tell us that if two events occur independently of each other, and if each event has a certain probability of happening, the chances of the events combined occurrence are equal to the product of their respective probabilities. Applied to textual transmission this means that if the scribe A preserves his copy readings at the rate of 0.9, and the scribe B preserves his copy readings at the rate of 0.8, the chance that A and B will preserve the copy reading at any given point is calculated by multiplying the two probabilities: 0.9 X 0.8 = 0.72. In other words, any given point of the copied text has the probability 0.72 of being preserved by both our scribes. That is to say that, in such a case, the rate of variation AB (that is: oo, say v1) is f 1 = 0.72. This implies that one may expect the variation v1 to appear in the statistics 72 times, in case the archetype contained 100 potential changes, 720 times, if the archetype contained 1,000 potential changes.

Since the chances of the combined occurrence of any number of independent events are equal to the product of their respective probabilities, we can easily spell out the expectancies of the different variations in independent as well as in successive derivation. The variation v1, for instance, can be expected to appear in the statistics produced by the comparison of three witnesses at the rate f 1 = (1—Pa) (1—Pb) (1—Pc), when dealing with independent derivation, whereas in successive derivation the expectancy of f 1 is only (1 — Pa) (1 — Px) (1 — Pb) (1 — Pc), because the preservation rate of B and C is, as shown by our preceding analysis, in such a case determined by the preservation rate of X. Analogically we then know that the expectancy of f 2 is (1—Pa) (1—Pb) Pc, in independent derivation, and only (1—Pa) (1—Px) (1—Pb) Pc in successive derivation.

In order to illustrate the different frequencies of the variations by a less abstract example, let us consider the hypothetical case of three collaterals and assume that we know all the unknown facts and factors. Let us thus assume that the archetype "A" consisted of one thousand words and that each of these words constitutes one potential change.[24]

Let us further assume that every scribe changes a constant number of words, say one hundred out of thousand. The rates of change Pa, Pb, Pc, Px, are thus in our hypothetical case all equal and have the numerical value of 1/10; the respective rates of preservation (1 — Pa), (1 — Px), etc., must therefore in our case all be: 1—1/10 = 9/10. This is to say that in independent derivation the expected rate f 1 is: 0.9 X 0.9 X 0.9 = 0.729; whereas in successive derivation the expected rate f 1 is only: 0.9 X 0.9 X 0.9 X 0.9 = 0.6561. Similarly, we can expect v2 to appear in the statistics at the rate of 0.081 in independent, and only at the rate of 0.0729 in successive derivation. If then the archetype really had only 1,000 words and all scribes really realized the identical rate of change of 1/10, the critic comparing three witnesses derived independently would find the variation ABC 1,000 X 0.729 = 729 times, the variation AB : C 1,000 X 0.081 = 81 times. If, on the contrary, the critic was comparing three witnesses derived successively, he would find the variation ABC only 1,000 X 0.6561 = 656 times, and the variation AB : C only 1,000 X 0.0729 = 73 times. In our hypothetical case, he would thus be able to identify easily independent and successive derivation by the respectively higher and lower rates f 1 and f 2.

In reality, however, the rates of change are variables, and the identification of the two types of derivation is therefore somewhat more difficult to carry out, although its principle remains basically the same. We noticed already that the most characteristic difference between the two types of derivation lies in the fact that the variation v4 is simple in independent and compound in successive derivation. If we then estimate the frequency of v4 in successive derivation, we must not assume—as we did in the preceding simplified hypothetical case—that the scribes B and C realize the same rate of change in copying the errors and changes introduced by their exemplar X, as in copying the readings of the archetype, preserved by X. Such an assumption is, to be sure, most unlikely to correspond to reality; first, because even the most inattentive scribe will introduce a certain number of his own emendations, and second, because there might be another number of changes and emendations caused by the scribe's recollection of other versions

of the text. Whatever the scribe's psychological reasons might be, however, they are irrelevant with regard to the frequency of v4 in successive derivation. The only relevant factor in our context is the fact that the presence of X in successive derivation may influence the rates of B and C, whereas in independent derivation such a possibility is not given at all. We thus express this new fact by adopting, in successive derivation, the rates Pb1, Pc1, for changing o, and the rates Pb2, Pc2, for changing x; the corresponding rates of preserving o and x are then: (1—Pb1), (1—Pc1), and (1—Pb2), (1—Pc2), respectively:[25]

The preceding table shows very clearly that the case of independent derivation must be soluble mathematically, because the statistics offer four bits of independent information (namely: the frequencies f 1, f 2, f 3, f 4), and the case has only three possible variables. In successive derivation, on the other hand, it is impossible to compute all variables, because there are six possible variables and still only four independent bits of information. The case of successive derivation is then mathematically insoluble, and this fact, secured by our analysis of the observable phenomena, permits us to make the desired distinction between the two types of derivation.

We see, for instance, that the four equations of independent derivation can be solved for the values of the three variables by straight computation:

It is more convenient for us, however, to solve the equations for the corresponding rates of preservation:

In the same way, we compute:

The case of successive derivation is different. Given the four equations of table III and six variables, we can compute the two following variables only:

The third operation, however, which in independent derivation allowed us to compute the variable (1-Pa), will in successive derivation yield the following result:

The preceding facts, secured by our analysis, offer us the possibility of making the desired distinction between independent and successive derivation.

For instance, as far as independent derivation is concerned, we know that:

Therefore, we know that in independent derivation the following relation must be true:

The same relation is expressed by writing:

On the other hand we know that in successive derivation, the same operation cannot possibly yield the result equal 1, because we are unable to compute the correct value (1—Pa). The operation will therefore give the following result in successive derivation (compare Table III):

We see thus that the above operation can be considered as a test operation and that the result equal 1 identifies the independent, whereas the result not equal 1 identifies the successive derivation.

We shall now demonstrate in theory the functioning of the identification test by comparing one hypothetical case of independent and one of successive derivation. Let us again assume that the archetype contained 1,000 potential changes and that we know all the unknown facts and factors.

The examples thus show that, in agreement with our theoretical expectation, the test result allows us to identify the case of independent derivation as well as the case of successive derivation. This simple test therefore permits us to prove or disprove independent derivation and thereby overcome the ambiguity of three texts. Since the ambiguity remains the same whether we deal with three or more texts, we may conclude that the quantitative solution, demonstrated here on three manuscripts only, may be applicable to any number of extant manuscripts. From Joseph Bédier's demonstration and W. W. Greg's analysis of the problem, we retained that neither the methods based on qualitative criteria nor the application of logic permit us to make a formal distinction between the two basic types of derivation. The application of probability calculus, on the contrary, promises to place the methodology of textual criticism on a different ground because it offers, in theory at least, the possibility of overcoming the ambiguity of three texts by a formal distinction.

In order to illustrate the preceding abstract reasoning by examples which speak more directly to our imagination and to demonstrate at the same time the practical applicability of the proposed method, we

shall now apply the identification test to four manuscripts belonging to the family of Jean Renart's, Le Lai de l'Ombre.[26] It is the same family which served Bédier as a basis for his repudiation of the traditional methods.

Although we are using an actual case to substantiate the following demonstration, our interest remains predominantly theoretical. It is not our concern here to analyze the problems and intricacies of the practical application of the proposed method, and we shall therefore concentrate on those aspects of the actual case only, which permit us to verify the validity of the preceding deductions. In order to save lengthy explanations and to allow us a better control of the demonstration, we shall create a hypothetical group which, according to my results, is analogous to the manuscripts A, B, D, F, of the actual Lai de l'Ombre family. A simple comparison of both cases will then provide a sufficient basis for judging the verification value of the different operations.

Let us thus assume that we are faced with the manuscripts A, B, D, F, and that we again know all the unknown facts and factors. We then assume that the genealogical relationship of our four witnesses is defined by the following stemma:

We therefore know that the four witnesses, considered by groups of three manuscripts, must form two groups independently and two groups successively derived:

We also know that in such a case, the comparison of the witnesses by groups of three manuscripts will produce the usual five variations which we define in the following way:

• 2. Group ABF The variations, variants and rates of group ABF are analogous to the ones of group ABD. We obtain them by using Table VI and writing: F, f o, f x, Pf1, Pf2, instead of D, do, dx , Pd1, Pd2, respectively.
• 3. Group ADF The definition of group ADF is obtained by following the model of successive derivation, Table III, and by writing D, F, d, f, Pd1, Pd2, Pf1, Pf2, instead of: B, C, b, c, Pb1, Pb2, Pc1, Pc2, respectively.
• 4. Group BDF The definition of group BDF is obtained by following Table III and by writing in addition to all the substitutions of group ADF: B, b, and Pb, instead of A, a, and Pa.

If we now assign specific values to all the variables, we can easily compute the expected rates of variations in all four groups by following

the above indications. Let us therefore assume that, in copying their respective exemplars, the scribes A, B, D, F, and X, realized the following rates of change:

Under these circumstances then, the statistics will read as follows:

As a next step, we apply the test operation

and obtain the following results:

--> The results of the tests allow us, as we see, to identify the four partial stemmata which we have drawn in diagram 6, and thereby to determine exactly the genealogical relationship of the manuscripts A, B, D, F, because we are able to draw the composite scheme which, in this hypothetical case, we chose as a definition of our example. Now, we want to compare the hypothetical case to the four actual manuscripts of the Lai de l'Ombre family. Before dealing with actual manuscript groups, however, we must deal, at least briefly, with the implications of our postulates and assumptions.[27] Postulate number one need not concern us here, because its specific implications do not affect the validity of our analysis nor the possibility of applying probability calculus for interpreting the statistics.[28] Assumption number two also creates no additional problem, since any transcript failing to satisfy it is identical with its exemplar and therefore mechanically assumes its function and position in the genealogical scheme.

Different are the implications of assumption number three, because the possibility of a heavily corrupted archetype certainly cannot be ruled out. If such were the case, the critic would encounter complications which would make it difficult or even impossible to identify the genealogy of the witnesses. Since we are not concerned here with the limits of the application, but rather with the correctness of the theory in general, I have chosen, in order to prevent lengthy explanations, an unproblematic example in which the corruption of the archetype is so small that it has no effect on the calculus.

Assumption number four has little chance to correspond to reality because uncontaminated families and families without convergent variation seem to be extremely rare in practice. The problem of contamination and convergent variation is therefore a major problem of any critical method. I must here refer to my report on "Statistical Methods in Textual Criticism," where I have analyzed the problems of the practical application of this theory and proposed a technique of dealing with contamination by mechanical dissolution of the conflicts.[29] In

order to give to the reader the possibility of judging the extent of the dissolution as well as its effect on the calculus, we shall list the statistics and the results of the test both before and after the dissolution of the conflicts:

In the last line are listed, under the heading test, the differences with regard to the test value one. The comparison of the tables shows that even in the undissolved form, the test result has a tendency to approach much closer to the value one in groups ABD, ABF, than in groups ADF, BDF. Since, on the other hand, the test result for ADF, BDF, remains practically the same before as well as after the dissolution, it appears from our example that the dissolution of conflicts does not radically change the results, but rather clarifies and specifies their meaning by eliminating, at least partially, the irregularities caused by contamination or convergent variation.[30]

In the dissolved table, the difference from one is in groups ABD, ABF so small that it can certainly be disregarded without forcing the results.[31] The test result being .999 on the one side, and .965 on the other, we have in our case a comfortable difference of .034, which at any rate gives us the right of reading the former as equal one, and the latter as not equal one. On this basis, then, we can infer the existence

of the exclusive common ancestor X for the manuscripts D and F, and its absence for A and B. Our four groups therefore appear to have the following partial stemmata:

These four partial stemmata can be overlayed only in one possible way:

The stemma of the actual Lai de l'Ombre group being the same as the stemma of our hypothetical case, we have obtained a solid basis of comparison and can now investigate whether the results of the tests can be verified. Actually, there exist additional ways of crosschecking the result of the quantitative solution. The principle of the ambiguity of three texts suggests, for instance, that from the standpoint of logic any three manuscripts successively derived may be regarded as manuscripts derived independently; we know from our preceding analysis that a higher frequency of the variation A : DF, or B : DF can be produced by the changes introduced by X in successive derivation, as well as by a higher rate of A, or B in independent derivation. For this reason, independent and successive derivation are, logically, ambiguous and our result:

can therefore be also represented in the following way:

These graphs then indicate that, if the groups ADF and BDF are really successively derived; and, if we compute the variables under the wrong hypothesis that they form an independently derived group, we must expect to compute an apparent rate of change for A and B, augmented by the changes introduced in reality by X.

On the other hand, we also know that, if our resulting stemma for ABDF is to be correct, the inferential manuscript X actually lies on the line between "A" and DF and remains undetected in groups ABD and ABF only because the manuscripts D and F are compared, in these groups, with AB without the other member of the branch X. We therefore must again expect that the rates computed for D and F will be apparent rates, augmented by the number of changes introduced by X.

If we then base our schemes on logical analysis only, we may represent the relationship of our four witnesses by the following graphs:

These graphs then indicate clearly that we may treat any group of three manuscripts as independent derivation and proceed to the operations which are expected to produce the values of the variables:

--> However, if the results of our identification tests are to be correct, we must expect that the above operations will yield in our actual case:

• 1. the correct values Pa, Pb, if computed from the groups ABD,

  ---

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  ABF, because these groups actually are independent groups and because X does not lie between "A" and A or B (comp. table II, under f 4, f 3).
• 2. the correct values Pd1, Pf1, if computed from groups ADF, BDF, because these groups actually are successive groups, allowing the computation of the rates of change realized by the copies of X (comp. table III, under f 3, f 2).
• 3. the apparent values Pd1, Pf1, augmented by the changes introduced by X, if computed from groups ABD, ABF, because these groups actually are independent groups in which X lies between "A" and D or F (compare table VI, under f 2).
• 4. the apparent values Pa, Pb, augmented, more or less, by the changes introduced by X, if computed from the groups ADF, BDF; these groups actually are successive groups, but by treating them as independent groups, we ascribe to A and B the changes introduced in reality by X (principle of the ambiguity of three texts). In other words: The reality being: v4 = a o o + a x x + o x x, or:
  v4 = b o o + b x x + o x x, we interpret it incorrectly as if it was: v4 = ao o o + ax o o + x o o, or: v4 = bo o o + bx o o + x o o (compare Table III, under v4 and Table VI, under v2.

The preceding deductions offer us a rather wide possibility of verifying the correctness of the proposed theory, since—if the theory is to be correct—the results of the computations, in the hypothetical case as well as in the actual group, must confirm the conclusions of our speculation.

We are listing in the left column of table X the results of the operations as obtained in the hypothetical group, and in the right column the results of the actual case. The two tables show at the first glance that there is a full agreement between the hypothetical case and the Lai de l'Ombre manuscripts on the one hand, and between both cases and the theory on the other. According to the theoretical expectation as well as to the testimony of the hypothetical example, it is always possible, in our case, to compute the correct rates of change from two different groups of three witnesses, whereas the value computed from the third group appears, as expected, to be augmented by the changes introduced by X. The lines marked number 3 show these apparent rates. We are thus able to decide with certitude that the four scribes copying the archetype of the Lai de l'Ombre realized the following rates of preservation: (1—Pa) = .994; (1—Pb) = .972; (1—Pd1) = .898;

(1—Pf1) = .846 (we again neglect the small inaccuracy of .001 in two of those rates). The result of the experiment thus seems to indicate that our theory correctly describes the reality and that it is well applicable in practice. Yet, the statistics contain still more information which may serve as a basis for an additional verification.

According to our analysis, for instance, f 1 is expected to have in the four different groups the following expectancies:

If then our analysis and the computations of table X are to be correct, we must be able to compute four times the correct value (1-Px) by dividing the numerical value f 1 of each group by the product of the three complementary rates of preservation constituting the expectancy of this particular variation. In group ADF, for example, one divides .680 by .755 which, according to table X, is the product of (1 — Pa) (1 — Pd1) (1 — Pf1); one proceeds analogically in all remaining groups. A simple computation thus again confirms our theoretical expectation because we obtain four times the result (1 — Px) = .900; that means that the corresponding rate Px is equal to .100. To carry out the same operations for the hypothetical group, we use tables XI and X and obtain four times the value (1 — Px) = .600 which we chose, as we know, to represent the numerical value of this particular variable.

The consistency of our results as well as the repeated confirmation of our theoretical expectations seems to indicate first, that our analysis of the observable phenomena is basically correct, second, that our mathematical formulae correctly describe the facts of the reality and, finally, that the calculus actually permits us to detect the existence of the inferential manuscript X and even to estimate its rate of change with great accuracy.

Our next concern is to verify the decisive question whether or not the proposed method is able to locate the position of the inferential manuscript by a formal procedure.

According to the results of the identification tests, the inferential manuscript X is, in our group ABDF, located on the side of DF:

If this result is actually correct, we represented the reality adequately by drawing the two following schemes:

However, we completely misrepresented reality by drawing:

Since the original source of our information is the archetype, and since from this source the information comes to us in a direction determined by the genealogical and "historical" relationship of the lost and extant manuscripts of the group, the correct representation of our four groups, expressed by a three branch scheme, is the following:

The arrows indicate the direction in which the information is delivered to us and point out the fact that the reality will remain unchanged whichever way we may bend our diagrams. This implies the obvious fact, to be sure, that the reality will remain unaffected by our misinterpretation of the data and that we therefore can expect to obtain the correct values of the variables only if we are computing them under the correct hypothesis. In other words: The correctness of the hypothesis under which we are working will be verified by proving the identity of a given variable.

Let us therefore consider now the groups under the hypothesis that they, all four, form an independently derived group. If then our identification tests correctly establish the genealogy of the four groups, this hypothesis will be correct in two instances and wrong in the other two instances; the unchangeable fact of the reality being the circumstance that the critical variations v2 = AB : D, and v2 = AB : F, will be composed of variants different from v4 = DF : A, and v4 = DF : B. Let us recall these facts and list the theoretical expectancies of the four critical variations (comp. table III, for groups ADF, BDF, and table VI, for groups ABD, ABF):

The relationship between f1 and f2, established by our analysis and defined in table VI, indicates that, if the identification tests correctly established the position of the inferential manuscript X, we must expect to obtain the correct values (1—Px) (1—Pd1) and (1—Px) (1—Pf1), if we compute them from the groups ABD and ABF, respectively. Because in

We obtain an analogical result in group ABF.

On the other hand, the relationship between f 1 and f 4, defined in Table III, indicates that we must not expect to compute the correct values (1—Px) (1—Pa) and (1—Px)(1—Pb), if we apply the analogical operation to groups ADF and BDF, respectively. Because in

One obtains an analogical result in group BDF.

We then can easily verify the correctness of these theoretical expectations by dividing the product of the four above operations by the value (1—Px), identified previously by four different operations (compare Table XI), and by comparing the result with the values: (1 — Pd1), (1 — Pf1), (1 — Pa), (1 — Pb), respectively, identified previously by two different operations each (compare Table X). In the following table, we list the results of these operations and comparisons:

The full agreement between the result of the test operation in groups ABD, ABF and the correct values (1 — Pd1), (1 — Pf1) proves that we have worked under the correct hypothesis; the incorrect or even nonsensical result of the test operation in groups ADF, BDF, on the other hand, shows that we have worked under a wrong hypothesis. In other words, we have verified that the inferential manuscript really lies on the line between "A" and DF, and that it certainly does not lie on the line between "A" and AB.

I have included the last demonstration in my report, although I am fully aware of the fact that it does not add any really new information to the one obtained by the application of the identification test. It does, however, spell out the mechanics of the test and thereby clearly describe the principle on which the test is based. The intention was to show that the method of inferring the existence and position of the inferential manuscript by means of probability calculus is based on rigid logic and not merely on a numerical coincidence. We share Housman's contention that in textual criticism, as indeed in any branch of knowledge, everything which conflicts with either common sense or reason should be rejected as mere hocus pocus.[32]

If then our reasoning contains no basic fallacy, we hope to have demonstrated, that the quantitative method offers us the possibility of overcoming the ambiguity of three texts by a formal procedure.

Notes