Peter Abelard had a great influence upon his contemporaries. As he himself reports, many students followed him, and, as is clear from what we know about the history of twelfth-century logic,1 his rivals could not neglect his innovating theories and discussions, feeling it necessary to develop their own theories in response to his. In the next century, however, his direct influence disappeared in logic as well as in theology. The census of Peter Abelard’s works2 shows that very few manuscripts from the thirteenth century preserve his works, and that there are no manuscripts at all for his logical works. He did, however, leave a school – the so-called Nominales, named after his own commitment to nominalism – but it survives for only one or two generations after him. As a result, Abelard is known in the next century only in connection with the name, or rather the notoriety, of the school of the Nominales, together with a few distinctive theories associated with it.
In this chapter, I attempt to provide some indication of Abelard’s overall historical influence. I shall focus, however, on the influence he had on his contemporaries, taking up three areas of twelfth-century logic to which contributed – areas that we would nowadays think of falling within the domain of metaphysics, philosophy of language, and logic, respectively – and then examining his contemporaries’ reactions to them. Along the way, however, I will also have something to say about the school of the Nominales and some of their distinctive theories.
Let us begin with a brief sketch of the development of twelfth-century logic. Our story begins with the arrival of Peter Abelard at Paris around the very beginning of the century. Paris was at this time celebrated for logic, because of the influence and reputation of William of Champeaux. As is well known, Abelard attacked William’s own theory of universals soon after his arrival, and thereby provoked what was to become a long-standing controversy. The theory of universals is but one example of the many topics in the logic of the day to which Abelard made an important contribution.
In the 1130s many masters began to gather at Paris and at Mont Ste. Geneviève and to start logic schools, including Abelard, Adam du Petit-Pont, Robert of Melun, Alberic of Paris, and Gilbert of Poitiers. Although by mid-century, these masters had all died or retired from teaching logic, their schools – the Nominales, Parvipontani, Meludinenses, Albricani, and Porretani, respectively – continued to exist, presumably at Paris, and to carry on their rivalries.3 There remain, moreover, collections of theses which were held by each school.4
By the 1180s, which mark the beginning of the decline of twelfth-century logic, all five of these logical schools had disappeared, with the result that textbooks of logic from this period tend to be patchworks of material taken from this and that school.5 By the very end of the century, twelfth-century logic had died altogether, and a new development of logic had started, with the appearance of the earliest terminist texts (textbooks of logic of a style which becomes very popular from the thirteenth century on). These early terminist texts consist of both a survey of the so-called “old logic” (logica vetus) – cutting off to an elementary level every subtlety of speculation associated with Abelard and the other twelfth-century logicians – as well as a survey of the theory of fallacies based on the newly discovered text of Aristotle, Sophistici Elenchi, which Abelard knew only very superficially. The terminist texts also contain chapters which are proudly entitled “the logic of the moderns” (logica modernorum) in contrast to the aforementioned chapters, which they refer to as “the logic of the ancients” (logica antiquorum). The theories discussed in the logica modernorum systematize theories that were already discussed explicitly or implicitly in the second half of the twelfth century,6 but few of their elements trace back to Peter Abelard.
The best-known topic of twelfth-century logic is, no doubt, the controversy surrounding the nature and ontological status of universals. In order to clarify Abelard’s own contribution to this controversy, I shall begin with a brief historical survey of its development.
To a great extent, what lies behind the universals controversy is a new approach to logic, which is commonly referred to as vocalism,7 and which was first propounded in the late eleventh century by a certain John, who was the master of Roscelin of Compiègne. According to this new approach, logic deals only with “voices” (voces) – i.e., verbal sounds – in contrast to the more traditional approach, represented in Porphyry’s Isagoge, which holds that logic deals not only with verbal sounds, but also with things (res). In all probability, Roscelin was the first to apply the vocalistic approach directly to the three questions which Porphyry raises but declines to answer at the beginning of his Isagoge – namely, the question (i) whether universals subsist or not, (ii) if so, whether they are incorporeal or not, and (iii) if they are incorporeal, whether they are separated from or exist in bodies. Although Roscelin propounded a vocalist theory, the lack of sources makes it uncertain as to how precisely he answered Porphyry’s questions. It was, however, with the vocalist theory in mind that Abelard, who had been Roscelin’s student, attacked William of Champeaux’s view that universals are things. As I have indicated, moreover, Abelard’s attack on William initiated what was to be an on-going controversy about universals.
We should note the difference between Abelard’s formulation of the problem (namely, whether universals are verbal sounds or things), and Porphyry’s original questions (namely, whether universals subsist or not). Abelard’s formulation is quite alien to the ancient controversy over Platonism and Aristotelianism, which is evidently reflected in Porphyry’s questions. As a matter of fact, nobody in the twelfth century thought of any Platonic idea as separate from bodies (choris einai), and all agreed that, at least in our sensible world, there are only individuals. (A Platonic theory, which asserted ideae as exemplary forms in divine mind, did appear in the controversy, but outside the mainstream.8) The answer given to the Abelardian question, therefore, by vocalists (vocales) such as Roscelin and Abelard provided a challenge for the realists (reales), namely, to explain what sort of real things universals are and to conform their own theories about universal things with the foregoing assumption that only individuals exist in the sensible world.
Finally, we should also note that an important prelude to the controversy between Abelard and William is the attack of Anselm of Canterbury on Roscelin in the late eleventh century.9 The occasion for Anselm’s attack was Roscelin’s views about the Christian doctrine of the Trinity, which Anselm thinks rest on a mistaken view of universals. In any case, Anselm has no problem introducing the term “essence” (essentia) to explain how one and the same universal (or essence) can be in many different individuals that exist only in the sensible world. Now it is possible to distinguish two meanings of the term “essence,” corresponding to each of the two meanings of verb “to be” (esse): (i) to be or to exist, when it is used as a free-standing predicate, and (ii) to be something, when it is used as a copula. Now, for Anselm, it is only God who exists in a pure sense, whereas creatures always exist in a qualified or a restricted way as being something. In respect of God, therefore, “essence” can only mean some existing thing, whereas in respect of creatures it can only mean being something. Thus, according to Anselm, a species, such as man, is a universal essence derived from a qualified or restricted way of being, such as Socrates’s being man, Plato’s being man, and so on.10 In this way, he suggests, a universal essence (that is, a species or genus), such as man or animal, can be one and the same thing even while existing in many different individuals.
After these preliminaries, let us turn to the controversy between Abelard and William of Champeaux. According to Abelard’s own report in his Historia calamitatum (HC 65.85–91; Radice 1974, 60), William began his career by asserting that universals are things – or more specifically that essentially the same things (namely, genus and species) are totally present in each individual. The young Abelard attacked this theory, forcing William to reject it in favor of another theory, according to which universals are indifferently the same things, as opposed to being essentially the same. For convenience, let us call the former theory the material essence theory, and the latter the indifference theory. Abelard’s report about his exchanges is well supported by various other sources. Although there is no Porphyry-commentary that defends the material essence theory as such, there are several commentaries that report this theory and Abelard’s arguments against it.11
Briefly stated, the material essence theory maintains that the same essence is present in different individuals of the same species and in different species of the same genus – indeed, the essence just is the species or genus in such cases. Now the essence in each case is referred to as the matter (materia) of the individuals and species, and the matter is said to become an individual or a species when it is combined with certain forms (formae) – certain accidents in the case of individuals (namely, those peculiar to the individual in question), and certain substantial differences in the case of species (namely, those definitive of the species in question). Thus, the essence man becomes an individual, say Socrates, when it is combined with the accidents that are peculiar to Socrates, and the essence animal becomes the species man when it is combined with the substantial differences that are definitive of man, namely, rationality and mortality. Although William is responsible for introducing the terms “matter” and “form,”12 he clearly borrowed the term “essence” from Anselm of Canterbury. We should note, however, that William’s use of “essence” loses the subtle distinction that Anselm had drawn between the existence of God and that of creatures. William’s use of “essence” is ambiguous, or rather it seems simply to mean some existing thing. Abelard’s attack on the material essence theory takes advantage of this very ambiguity or simplification. Indeed, Abelard’s attack is tantamount to saying that if one and the same universal, u, is in different individuals, I1 and I2, as their matter, then u in I1 and u in I2 cannot be the same thing, since I1 and I2 are different things, as William himself admits.13 The indifference theory, which William eventually adopts, attempts to avoid this consequence by asserting, not the sheer sameness of u, but the sameness of u in the sense that there is no intrinsic difference between u in I1and u in I2, although the two u’s are different things in themselves.
Abelard’s attack on the material essence theory was extremely successful, though it did not succeed in persuading people to accept his own theory that universals are mere verbal sounds. Indeed, at this point in the controversy, Abelard himself had not yet developed his own theory fully enough to explain precisely the sense in which universals are voces. His early glosses on Porphyry (IP Isag. and P714), for example, which were written at around the time of his initial attacks on William, do not even touch on such questions at all. In order to resist Abelard’s position, therefore, realists initially thought it was sufficient to appeal to authorities, such as Aristotle, Porphyry, and Boethius, who support the realist position at various places in their writings. Although Abelard gradually elaborated his theory in a semantic direction – eventually speaking of meaningful words (sermones) rather than verbal sounds (voces) – realists stuck to the traditional conviction that universals are things (res), and continued to defend a form of William’s indifference theory.
In his Glossae super Porphyium,15 after attacking William’s material essence theory, Abelard turns to a discussion of two versions of the indifference theory that emerge from the original formulation of it, which we may refer to, respectively, as “the collectio theory” and “the status theory.”16 According to John of Salisbury, Metalogicon II.17, these two theories are propounded by Jocelin of Soissons and Walter of Mortagne, respectively. Jocelin of Soissons taught at the cathedral school of Paris from c. 1110 to c. 1113, before Abelard took the position, and his view is preserved in a work composed by his student, De generibus et speciebus. The collectio theory asserts that a universal, say man, which is indifferently the same but essentially different in individual human beings, is a collection of the essentially different cases of man in all the individuals (namely, man1, man2, man3, etc.). The status theory is presented in two texts: Tractatus de generali et speciali statu rerum universalium and an unpublished Porphyry commentary (P17).17 It asserts that an indivi-dual, say Socrates, is an individual insofar as it is Socrates, or following their terminology secundum status of Socrates; that it is also its species, namely, man, insofar as it is man, or in the status of man; and so on for genera, such as animal, and each of the ten categories. According to the status theory, therefore, there are as many universals as individuals, since man in Socrates is a different thing from man in Plato, although in the status of man they are indifferently the same as each other. Interestingly, the status and collectio theories both retain William’s terminology of “matter” and “form” as well as “indifference.”
In his Logica “nostrorum petitoni sociorum,” Abelard no longer mentions the collectio theory. Instead, after discussing the other two realist theories – namely, material essence theory (LNPS 515.14–518.8) and status theory (LNPS 518.9–521.20) – he adds another realist theory (LNPS 521.21–522.9).18 His discussion of this third theory is too brief to make clear what the theory amounts to. However, in the Tractatus de generali et speciali statu rerum, which also omits the collectio theory, there is some discussion of another realist theory.19 According to the Tractatus, this third realist theory distinguishes three different uses of the term “man.” For it can be used (1) for man simply (simplex or in sua simplicitate); (2) for that which is associated with individual human beings (circa inferiora); or (3) for individual human beings themselves (inferioratus). In the first sense, man neglects the forms that distinguish individual human beings, as when we say “man is a species”; in the second sense, it focuses on individual cases of man in individuals, as when we say “man is an animal”; and in the third sense, man is used to for individual human beings themselves, as when we say “Socrates” or “Plato.”
A very similar theory is developed in the Sententia de universalibus secundum magistrum R. It begins by mentioning the grammatical distinction between proper and appellative (= common) nouns and, appealing to the authority of Priscian, asserting that an appellative noun, say “man,” refers to (nominat) individual human beings, but signifies (significat) a universal nature, namely, rational mortal animal (§2). It then asserts that an appellative noun can function as a proper noun, namely, when it signifies in its simplicity (in sua simplicitate), rather than in respect of individuals (in inferioribus), as is the case when we say “man is a species,” not in its individuals (in inferioribus) (§3). Here we see an evident parallelism of terminology between the Tractatus and the Sententia. Therefore, let us refer to the realist theory in question as Master R.’s, whoever he might have been.20
The Sententia secundum magistrum R. distinguishes man as the potential matter of individuals, when “man” is used as a proper name (§4), and man as the actual matter, when “man” is used as an appellative noun (§5). Then, the Sententia says, man and this man are different by proper nature; this man is the actual matter of Socrates and is actually the same as Socrates, but precedes Socrates by nature since it can exist without Socrateity (the forms which make the potential matter man Socrates) (§6). This explanation of the distinction between man and this man is strongly reminiscent of the third form of realism mentioned by Abelard in his Logica “nostrorum petitoni sociorum.” I conclude, therefore, that the theory Abelard has in mind there is the theory of Master R.
Now, the unpublished commentary, P17, which adheres to the status theory, also mentions this other realist theory as one of its rivals. According to P17, some realists assert that essentially the same animal is the matter of both Socrates and Browny (a name of a fictitious ass), which is just William’s material essence theory. However, according to P17, these realists also assert that a universal in its simple nature (in simplici natura) is opposed (oppositum esse) to its singulars, but a universal affected by individual forms is the same as its singulars,21 the very theory of master R. In the end, therefore, William’s material essence theory is similar to the theory of Master R., but not the same. Master R. never uses the terms “essentially the same” as William does. The fact that the Logica “nostrorum petitoni sociorum,” the Tractatus, and P17 mention the theory of Master R. and that they all omit the collectio theory, strongly suggests that the collectio theory was outdated by the 1120s and from then on Master R. actually propounded a new theory as part of a counterattack of William’s realist party against Abelard.
Philosophers continue to develop various theories of universals in the mid-twelfth century, and here again Abelard’s influence can be detected. For present purposes, a brief survey of the various theories will have to suffice.
Abelard’s followers, the Nominales, continued to develop his most mature theory, according to which “universals are meaningful words (sermones),” though they tended to use terms like termini instead of sermones.22
Gilbert of Poitiers and his followers, the Porretani, assert that a universal is a collection of forms in individuals each of which forms makes its subject (an individual) what it is. Thus, man or white is a collection of such forms in individuals, which make, say, Socrates a man or white. This theory is in a sense a revival of the collectio theory of Jocelin of Soissons. In the case of Gilbert and his followers, however, the theory was developed in a much more subtle theological and ontological context.23
John of Salisbury (Metalogicon II.17) reports that some teachers focus on the status of things, identifying them with genera and species. (Note that this theory is different from the aforementioned status theory of Walter of Mortagne, which does not identify status as universals.) In all probability these are the Meludinenses, followers of Robert of Melun. According to the Ars Meliduna, words (dictiones) signify common and private status, which are able to be participated in by many (namely, in the case of common nouns) or by one (namely, in the case of proper nouns24) – though in reading these works it is important to recognize that “signifying” (significare) is being used in such a way as to contrast with “referring” (appellare). The same work also asserts that universals are things able to be grasped by intellect and to be participated in by many.25
John of Salisbury also reports that there are some who explain universals with a new-fangled term maneria/maneries, whose precise meaning is unclear, even to John (though in other contexts it often means something like type). P21 and/or P2026 hold the maneries theory, sometimes drawing a contrast between maneria rerum (universals) and res maneriei (individuals).27 I would argue that the maneria theory was held by the Parvipontani.28 For the same type of phrasing used to draw this contrast in P21 and/or P20 is also found in the Fallaciae Parvipontani,29 as well as in the Speculum speculationum of Alexander Nequam,30 a former student of the school of the Petit-Pont.31
With the exception of a notio theory mentioned by John of Salisbury in his Metalogicon II.17, of which we know little,32 and of John’s own theory in Metalogicon II.20, these are the only theories of universals we know to have been propounded in the mid-twelfth century. It is worth mentioning that all these theories in the mid-twelfth century discuss universals in relation to predication just as Abelard does. To my knowledge, moreover, only the faint memory of the Parvipontani’s term maneria and the label of Nominales survive into the next century.33
In developing his theory of universals, Abelard articulated a distinctive approach to semantical issues. And although his theory of universals was not accepted by his rivals, his semantical views were influential in many respects. Here I shall take up several topics as examples.
In De Interpretatione, Aristotle defines a noun, a verb, or a sentence as a “sound significant by convention” (vox significativa ad placitum).34 In order to illuminate Aristotle’s meaning, Boethius gives some examples of non-significant sounds in his commentary, namely, phonemes (litterae) and the nonsense word blityri (In De in. maior 5.9, 14). In addition to these Boethian examples, William of Champeaux includes the names of fictitious things like “chimaera,” “goat-stag,” and so on. This is indicative of the fact that, by the late eleventh century, the significative function of words is closely identified with their denotation of things existing in the real world. Although people in William’s day did define “signifying” as “producing understandings in our mind” (generare intellectum), they took these understandings to be only understandings of existing things.
Abelard noticed that William’s interpretation of signification contradicts what Aristotle says in his De interpretatione (1, 16a16–17), namely, that “‘goat-stag’ signifies something.” He points out in his Dialectica that this passage of Aristotle suggests that “goat-stag” is significant. He thus distinguishes signification from denotation (nominatio), saying that although “goat-stag” names (nominat) nothing existing in the world, it is nonetheless significant (Dial. 127.28–32).
Now on this point it may be that Abelard was simply following the vocalist tradition. Another contemporary vocalist, Garlandus of Besançon,35 touches on the idea that “goat-stag” is significant.36 Abelard’s interpretation of signification soon enjoyed widespread acceptance. Fictitious names were never again to appear as examples of non-significant sounds in later Perihermenias commentaries nor indeed in any of the logic textbooks throughout the twelfth century.
Aristotle says that oblique cases of a noun “are not nouns but cases of a noun” (De int. 2, 16a32-b1).37 From a semantic point of view, however, it seems that one should say rather that declensions of nouns according to case, gender, and number make no difference whatsoever with respect to their signification. In all probability, Abelard was the first to propose this anti-Aristotelian view (cf. Dial. 124.36–125.15). Thus, in commenting on the words of Aristotle quoted above, Abelard says that Aristotle thinks cases are different from nouns in a strict sense, but one can at the same time assert that oblique cases of a noun are the same as their nominative insofar as we pay attention to the identification of signification, not to the construction of sentences (LI De in. 33.02.56–69, G 343.40–344.17). Two later Perihermenias commentaries written by realists (H15 and H10) follow Abelard on this point and say that Aristotle in fact intends to be giving a two-fold definition of nouns, a loose one which includes oblique cases as nouns, and a strict one which excludes them.
The Introductiones Montanae maiores, which reports the teachings of Alberic of Paris, a bitter opponent of the nominalist school, also substantially follows Abelard on this issue. It asserts that “Socratis” (“Socrates” in the genitive case), “Socrati” (“Socrates” in the dative case), and so on are names of Socrates, but that Socrates does not have many names, because although “Socratis” is not the same as “Socratem” in ending, nonetheless it is the same as it in signification.38 The Compendium logicae Porretanum, a work by a Porretanus, also asserts (Thesis I.2) that many verbal sounds, namely, different cases of a noun, are the same term, and (Thesis I.3) that different genders of an adjective are the same noun.39
According to the Ars Meliduna, a work by a Melidunensis, several theories about the relationship of oblique cases to nouns in the corresponding nominative were proposed in the third quarter of the twelfth century: (1) that oblique cases differ from nouns in the corresponding nominative, since their endings are different; (2) that oblique cases are neither the same as nor different from such nouns; and (3) that oblique cases are the same as their corresponding nominative, because their signification is the same. The Ars Meliduna divides this third position into two species: (3a) oblique cases are not only the same noun as their corresponding nominative but also the same verbal sound; and (3b) oblique cases are the same noun as their nominative, but different verbal sounds. The Ars Meliduna itself adheres to theory (3a).
The Introductiones Parvipontanae (in MS Berlin, lat. oct. 262)40 follows Aristotle in asserting that only nominatives are nouns, and thus excludes oblique cases from this category, because the imposition of nouns is made in the nominative.41 I suggest that this treatise is a work of a Parvipontanus, since it holds a theory of arguments peculiar to that school.42
Except for the Parvipontani, therefore, all the schools active in the second half of the twelfth century followed Abelard on this issue in one way or another. The next century retains the memory of these views in the form of the claim “‘albus’, ‘alba’, ‘album’ are the same noun,” although during the thirteenth century it is always falsely ascribed only to the nominalist sect in a negative tone.
In his Dialectica, as I mentioned above in §III.1, Abelard uses the term nominare for denotation or reference in the modern sense of the term, rather than for meaning in general (significare). The term nominare originates from the grammar of the earlier period, namely, the Glosulae to the Priscian minor, and was introduced in order to gloss Priscian’s definition of nouns, according to which the property of a noun is to signify substance with quality (Inst. Gr. II.18, 55.6). According to the Glosulae, Priscian’s definition may be paraphrased as follows: a noun names (nominat) only substance (= subject matter) and signifies (significat) a quality belonging to (circa) the substance.43 For example, “man” names an individual human being and signifies, or determines, a quality (namely, humanity) belonging to it. Abelard comes to use the term appellare rather than nominare in his Logica “nostrorum petitoni sociorum.”44 The term appellare is, presumably, derived from Priscian’s nomen appellativum, but Abelard is the first logician, to my knowledge, to use the term.
Abelard argues elsewhere (LI De in. 3.03.28–34, G 348.37–351.2; cf. also Dial. 139.12–31 and 248.7–249.37) as follows. In order to save the rules of conversion of past- or future-tensed propositions, as well the rules for syllogisms containing them, we must consider that in a proposition such as:
the expression “was young” does not consist of two different words, but functions as if it consisted only one. Otherwise, for example, its simple conversion
should be true if (1) is true. As a matter of fact, he says, the simple conversion of (1) is not (2), but
which is indeed true. The Introductiones Montanae minores attacks this discussion of Abelard’s, saying that “our master,” Alberic of Paris, simply asserts that the rule of conversion does not hold in the case of such propositions.45 This is the beginning of the discussion of what sort of denotation subject and predicate terms have in propositions of various tenses, although the term appellatio is not invoked at this point.
In the second half of the century texts appear that provide many rules to show what sort of denotation terms have in propositions of various tenses or contexts. For example, the Ars Meliduna I, a product of the Melidunenses and dated to the 1170s, provides such rules in a part entitled De appellatione.46 Again the Tractatus Anagnini III47 and another treatise (in MS Vienna, Nationalbibl., VPL 2237, ff. 31v–34v)48 provides the same sort of discussion, though this time under the title of De suppositionibus. Substantially the same rules, though greatly reduced in number, also appear in later terminist texts in chapters entitled De appellatione.
Medieval logicians inherited from Boethius a distinction between two kinds of inferences, namely, topical inferences and syllogisms (the latter of which were further subdivided into categorical and hypothetical). Boethius expresses both kinds of inferences as of the form “. . . , therefore. . .” By the end of the eleventh century, however, philosophers such as William of Champeaux had come to express these inferences as of the form “if . . . , then . . .”49 That is to say, for William, all inferences are simply conditional propositions of various types. Against this background, Peter Abelard scrutinized inferential forms in detail and drew some particular conclusions, which provoked various reactions among his rivals and their schools.
In his Super Topica glossae (LI Top. 296.13–23), Abelard maintains that an argument (argumentum) is a proposition (propositio) introduced to prove a question. For example, in the proof
(P) Socrates is man, therefore Socrates is animal,
the proposition “Socrates is man” is the argument introduced to settle the question of whether or not Socrates is animal. By “a pro-position (or argument)” Abelard means nothing but verbal sounds (voces).
In the course of discussing this view he mentions two other theories of arguments: (a) that an argument is that which is signified or generated in our mind by the proposition, and (b) that an argument is the very thing which has the force of inference, so that, e.g., man is the argument in (P), since it is what enables us to infer animal in the conclusion. The second of these two theories, (b), is William of Champeaux’s.50 Abelard rejects William’s theory in his Dialectica (461.3–462.2), but the Dialectica is an earlier work and it is not yet very clear whether Abelard himself takes arguments to be merely verbal sounds or the things signified by them (cf. Dial. 459.26–461.2). As for theory (a), it is developed in two commentaries on Boethius’ Topics (namely, B8 and B9).51 We do not know yet the authors of these commentaries or who in fact asserted theory (a), but it is safe to suppose that the theory was propounded between Abelard’s writing of the Dialectica (before 1117) and the Super Topica glossae (c. 1120).
Now at this stage of its development, the question “what is an argument?” is closely connected with another, namely, “what is a locus?” where “locus” is defined by Boethius as “the seat of arguments” (De top. diff. 1174D9). And the latter question is itself closely connected with the famous problem of universals. For example, William asserts that a universal thing (res), man, is both the argument and the locus in a proof such as (P). And, it was Abelard who introduced the tri-partite distinction between things (res), understandings (intellectus), and words or other verbal sounds (voces) into the universal controversy. Unfortunately, however, at the present stage of research, we cannot pursue these connections more precisely. We must satisfy ourselves, therefore, with a brief survey of the extant commentaries on Boethius’s Topics in the pioneering work of N. J. Green-Pedersen.52
According to Abelard, those who assert theory (a) argue that mere verbal sounds without understanding are not sufficient to prove anything (LI Top. 296.23–35). Abelard, however, answers this argument in one way as follows. If the understanding of premises alone were sufficient to prove anything, then there would be no use in overtly expressing any conclusion! Abelard also develops many other counter-arguments or his responses in his Super Topica glossae (for other counter-arguments, see LI Top. 296.35–298.22). Indeed, it was in the face of theory (a) that Abelard sharpened his own theory into the more “nominalistic” one, namely, that arguments are premises which are merely verbal, a thesis that was taken up and defended by the Nominales in the mid-twelfth century.
In response to Abelard’s nominalism about arguments, all the rival schools assert that arguments are dicta, or items signified by propositions in one way or another.53 Unfortunately, we do not know the arguments against Abelard’s thesis developed by his rival schools, except in the case of the Meludinenses. Thus, the Ars Meliduna argues that because verbal expressions are conventional, the premise “Socrates is man,” from which the conclusion “therefore Socrates is animal” is supposed to be drawn, cannot be merely verbal. For if it were verbal and therefore conventional, it could also mean the same as “Socrates is stone,”54 in which case the conclusion would not follow. In light of considerations such as these, the Meludinenses assert that arguments are true dicta of premises, so that, e.g., the argument in (P) is the dictum of the premise “Socrates is man,” namely, that Socrates is man (Socratem esse hominem).
Alberic of Paris and his school assert that arguments are dicta of general hypothetical propositions,55 so that, e.g., the argument in (P) is the dictum of the hypothetical proposition:
If something is man, it is animal.
According to the Albricani, this dictum is the argument, not only of (P), but of many other proofs of the same type:
Plato is man, therefore Plato is animal,
and so on. This theory is, in a sense, a revival of William of Champeaux’s theory. For according to both William and Alberic, the force of inference in this type of inferences lies with man, namely, the predicate of the premise.
As for the Porretani, they assert56 that arguments are relations (habitudines) of middle terms to either subjects or predicates, so that, e.g., the argument in (P) is the relation which man as a species has to animal as a genus, namely, the relation which is expressed by the maxim (maxima propositio):
Of whatever a species is predicated,
so is its genus.
We may say that this theory is also a revival, in another sense, of William’s theory. For William, the thing man is the argument, as well as locus, for (P), but not insofar as it considered in itself. Rather man is the argument (and the locus) only insofar as it has a relation to its conclusion.57
Finally, the Parvipontani assert that arguments are the dicta of special hypothetical propositions, so that, e.g., the argument in (P) is the dictum of the hypothetical proposition derived from (P) itself, namely
If Socrates is man, Socrates is animal.
All of these considerations disappear with the arrival of terminist texts, which simply follow, without any further ado, Boethius’s definition of argument: ratio rei dubiae faciens fidem (“reason producing belief in something that was in doubt”).
As I have said, William of Champeaux58 treats topical inferences and syllogisms (both categorical and hypothetical) as various types of conditional propositions. Thus, he treats topical inferences as conditionals consisting of two categoricals (namely, “if {a categorical}, then {another categorical}”), categorical syllogisms as conditionals consisting of a categorical and a conditional (namely, “if {a categorical}, then if {a categorical} then {a categorical}”), and hypothetical syllogisms as conditionals consisting of hypotheticals (namely, “if {a hypothetical}, then {a hypothetical}”). William also introduces new non-Boethian loci (from subject/predicate and from antecedent/consequent) to validate those inferences.
Unlike William, Abelard makes a clear distinction between topical arguments and syllogisms, arguing that syllogisms need no support from loci. In his Dialectica (352.29–353.23), Abelard argues for the distinction as follows. Suppose, as William argues, that a syllogism had to be validated by this locus from predicates:
If something (M) is predicated of another (S) universally,
then if some other (P) is predicated of the predicate (M) universally,
P is predicated, too, of S universally.59
In that case, this locus could be applied to a syllogism consisting of any terms, including:
If every man is a stone,
then if every stone is an ass, every man is an ass.
William and his followers would agree that the locus of this syllogism is stone (since, as we have seen, for William a locus is the thing signified by the middle term). But if stone carries the force of inference by virtue of its being predicated of man, the question arises whether the predication holds in virtue of the way things exist in the world or in virtue of the mere utterance of verbal sounds (secundum rerum cohaerentiam sive secundum vocum enuntiationem). It appears that it must be in virtue of one of them, and yet William could not accept either.
Abelard, thus, clearly distinguishes syllogisms from topical arguments, saying that the former are perfect in the sense that their validity derives from the combination of terms (complexio) itself, whereas the latter need a topical validation and so are imperfect (Dial. 253.28–257.23). Consider, for example, the topical inference:
If Socrates is man, Socrates is animal,
In order for this hypothetical to be true, says Abelard, we need a locus, that is, a relationship between man and animal, or more generally, a relationship between a species and its genus. By contrast, syllogisms of the form
(F1) If every M is P and every S is M, then every S is P,
will always be true solely in virtue of the combinations of terms, since whatever terms we substitute for “S,” “M,” and “P,” the conditional comes out true.
Here we need to be careful to distinguish two points. First, Abelard does deny that syllogisms take the form of the conditionals suggested by William, namely:
(F2) If every M is P,
then if every S is M, every S is P.
Nonetheless, he does not deny that they take the form suggested by (F1). Even so, it remains unclear whether Abelard distinguishes between syllogistic conditionals like (F1) and syllogistic inferences of the form
(F3) Every M is P,
every S is M,
therefore every S is P.
Abelard does eventually argue, in his Super Topica glossae (LI Top. 323.5–328.37), that syllogisms, and in general argumentation of the form
(F4){premise(s) = argumentum}, therefore {a conclusion},
are not conditionals at all. Conditionals, he says, have a single meaning which derives from its constituents being united by a conjunction such as “if.” On the other hand, he says, the conjunction “therefore” does not make a single meaning from antecedent(s) and its consequent, but merely links them as premise(s) and the conclusion respectively. In his long and complicated discussions of this issue, Abelard seems to struggle to show that conditionals of the form (F1) and (F2) are on a different level than inferences of the form of (F3) and (F4).
These arguments of Abelard stimulated many rival masters. First, there is the teaching of Alberic of Paris, at least as it is reported in the Introductiones Montanae maiores. According to the Introductiones, Alberic introduces rules of all types – (F1), (F2), and (F3) – with respect to the first three moods of the first figure of syllogisms. In discussing the second mood of the first figure, the Introductiones60 says that Abelard refuses to accept any rules of form indicated by (F2), on the grounds of counter-arguments such as the following:
If there is no flower,
then if a rose is a flower, there is not a rose.
This counter-argument cannot be found in Abelard’s extant works, and is difficult to understand. According to the Introductiones, Abelard thinks the conditional is false because, while the antecedent is true, the consequent is false – which is, perhaps, just to say: “there is no flower” can be true, say in winter, while the consequent is false, since “a rose is a flower” is always true, whereas “there is not a rose” can be false, say in spring. Thus, the Introductiones says, Abelard denies that any hypotheticals follow from categoricals or vice versa. In the face of such counter-arguments, Alberic responds that such hypotheticals can be true, but only when their terms are such that the subjects cannot exist without the predicates.
The Introductiones also argues elsewhere61 against “the error of somebody” who maintains that syllogisms of type (F3) are the same as conditionals of type (F1). This view is mistaken, the Introductiones says, because it often happens that conditionals of type (F1) are true, whereas the corresponding syllogisms are false in virtue of the falsity of one of their premises. Consider, for example, the following conditional:
If every man is a stone and every stone is imperceptible,
then every man is imperceptible.
This conditional is always true, but the syllogism corresponding to it is false. The Introductiones then assigns loci to syllogisms, pointing out that the loci are not introduced for the sake of confirming the syllogisms, as some (namely, Abelard) falsely maintain, but rather for the sake of showing how to invent them.
As for the Porretani, they concede that in syllogisms of type (F3), the conclusion does follow necessarily from the syllogistic combination of terms (sillogistica dispositio), but assert that necessity involved is due not so much to the combination of terms as to entailment (consecutio).62 The Ars Meliduna briefly asserts that loci should be assigned to syllogisms, because syllogisms have a twofold necessity, one from the combination of terms, the other from loci.63 Both of these two schools ultimately reject Abelard’s final conclusion that syllogisms of type (F3) and (F4) are on a different level from the conditionals of type (F1) and (F2). The Compendium logicae Porretanum discusses syllogisms in part II, of which the subjects are propositions, and the Ars Meliduna discusses them in Part IV, of which the subjects are enuntiabilia, or the items signified by propositions.64 I know of no sources that report the position of the Parvipontani.
As a whole, the concept of “combination of terms” which Abelard proposes is widely accepted, but many sources assert against Abelard that syllogisms have twofold necessity.65 And in early terminist texts, syllogisms are dealt with only in the form (F3), and have nothing more to say about conditionals.
As mentioned above, William of Champeaux introduces certain non-Boethian loci, such as the locus from subject/predicate etc. At the same time, he also reduces the number of loci differentia enumerated by Boethius in his De differentiis topicis to five: namely, from the whole (a toto), from the part (a parte), from the equal (a pari), from opposites (ab oppositis), and from immediates (ab immediatis).66 Although he does not explicitly say why he treats only these five, the explanation is perhaps that he is interested only in those loci which give necessary or at least probable conditionals, as Abelard reports (Dial. 271.38).
In his Dialectica, Abelard discusses these and some other loci, putting each to the test of whether it produces necessary conditionals. The discussion in which Abelard attempts to show that conditionals from locus ab oppositis are not necessary in particular provokes much debate.67
In order to follow Abelard’s discussion, it will be useful to highlight certain rules it presupposes:
Using these rules, Abelard argues as follows.68
(A1)
1 If Socrates is (man and stone), Socrates is man. [by (b)]
2 If Socrates is man, Socrates is not stone. [by (a)]∴ 3 If Socrates is (man and stone), Socrates is not stone. [from 1 and 2 by (e)]
4 If Socrates is not stone, Socrates is not (man and stone). [by (c)]∴ 5 If Socrates is (man and stone), Socrates is not (man and stone). [from 3 and 4 by (e)]
Thus, if a conditional from (a) locus ab oppositis – such as that at 2 – is allowed, then a contradiction can be deduced. Therefore, the locus ab oppositis is not valid.
On the basis of arguments such as this one, Abelard and his followers, the Nominales, refuse to accept the necessity of conditionals like 2, and instead propose the following thesis:
No negatives follow from affirmatives,
and vice versa:
No affirmatives follow from negatives.
Alberic of Paris argues against this view as follows. First of all, he says, rule (b) does not hold; a conditional such as 1 is not a proposition at all, but consists of many propositions (multiplex), one of which is true, namely, “Socrates is man,” and the other of which is false, namely, “Socrates is stone.”69 Secondly, rule (e) should be understood not in such a way that proposition Q plays the role of a middle term, but rather in such a way that the predicate term of Q plays this role. In other words, rule (e) should be understood as follows:
If a predicate causes something else to be predicated of its subject,
which in turn causes a third to be predicated,
then the first thing causes the third to be predicated.
We can put this more schematically as follows:
Third and finally, if we accept rule (e), we get into trouble even without assuming (a), as the following argument clearly indicates:71
0′ If Socrates is man, Socrates is animal. [true assumption] 1′ If Socrates is (man and non-animal), Socrates is not animal. [by (b)] 2′ If Socrates is not animal, Socrates is not man. [from 0′ by (d)] ∴ 3′ If Socrates is (man and non-animal), Socrates is not man. [from 1′ and 2′ by (e)] 4′ If Socrates is not man, Socrates is not (man and non-animal). [3′ by (c)] ∴ 5′ If Socrates is (man and non-animal), Socrates is not (man and non-animal). [from 3′ and 4′ by (e)]
Like Alberic of Paris, Gilbert of Poitiers asserts that conditionals are not necessary when there is opposition in their antecedents, as in the case of 1 above.72 His followers, the Porretani, assert73 more generally that it does not follow that (P & Q)→P [or Q], since if one says
If Socrates is man and Socrates is ass, Socrates is man,
‘Socrates is ass’ plays no role in the inference from the antecedent to the consequent (non est causa), and thus commits the fallacy of non causa ut causa. Conditionals conforming to rule (b), such as 1, or
If Socrates is man and ass, Socrates is man
cannot be true at all. Thus, Porretani assert, those who accept (b), namely, Nominales, should admit that a contradiction follows from (b), as can be seen from the following argument:74
(A2)
1″ If Socrates is (Socrates and Plato), Socrates is Plato. [by (b)] 2″ If Socrates is Plato, Socrates is not Socrates. [by (a)] ∴ 3″ If Socrates is (Socrates and Plato), Socrates is not Socrates. [from 1″ and 2″ by (e)] 4″ But if Socrates is (Socrates and Plato), Socrates is Socrates. [by (b)] ∴ 5″ If Socrates is (Socrates and Plato), Socrates is Socrates and Socrates is not Socrates. [from 3″ and 4″]
As for the Meludinenses, they judge that premise 1 is false, asserting the thesis:75
Nothing follows from the false.
Thus, against those who do not accept this thesis, the Ars Meliduna argues that, it is possible to prove (i) that a proposition follows from its contradiction, (ii) that two propositions contradictory to each other follow from the same proposition, and (iii) a proposition (P) follows from another (Q), while both P and Q cannot be true together. In support of (i), the Ars Meliduna gives a number of arguments, including (A1) above. In support of (ii), it gives, among others, an argument very similar to (A2) above. In support of (iii), however, it argues as follows:
If there is everything, there is not nothing. [by (a)] If there is not everything, there is nothing. [by (a)] ∴ If there is or is not everything, there is or is not nothing.
What an awkward argument! Not surprisingly, the terms “everything” and “nothing” become the focus of attention in Sophismata literature from the thirteenth century on.
It is worth noting that in support of (ii), the Ars Meliduna also gives the following argument.
If Socrates speaks truly and he lies, Socrates lies, if he lies, he does not speak truly, ∴ if he speaks truly and he lies, he does not speak truly,
And yet
Even if he speaks truly and he lies, he speaks truly.
Very awkward again! This is one of the earliest known formulation of the Liar’s paradox in the Middle Ages. Note that here the proposition “Socrates speaks truly and he lies” is supposed to be false, a proposition from which nothing follows, according to the Meludinenses.
As for the Parvipontani, they argue that Abelard’s argument at (A1) does not establish anything interesting, because according to their thesis:
Anything follows from the impossible.
Hence, there is nothing surprising about the fact that any conclusion whatsoever follows from propositions such as “Socrates is man and stone,” which are obviously impossible. They attempt to prove their thesis using the following argument, among others:76
If (Socrates is man) and (Socrates is not man), Socrates is man. But if Socrates is man, Socrates is man or stone. ∴ If (Socrates is man) and (Socrates is not man), Socrates is man or stone. But if (Socrates is man) and (Socrates is not man), Socrates is not man. ∴ If (Socrates is man) and (Socrates is not man), Socrates is stone.
Using a similar line of reasoning, one can deduce that Socrates is ass, goat, rose, or anything whatsoever. Thus, the Parvipontani argue, anything follows from an impossible proposition such as “Socrates is man and Socrates is not man.”
Like the Meludinenses, the Parvipontani formulate a version of the Liar’s paradox, though they do so in a much more sophisticated manner:
If Socrates only says that he himself lies, he says something true or false. But if (Socrates only says that he himself lies) and (he says something true), it is true that he himself lies. If it is true that he himself lies, then he says something false; ∴ If (Socrates only says that he himself lies) and (he says something true), then he says something false. Again, if (Socrates only says that he himself lies) and (he says something false), then it is false that he says something false; But if it is false that he says something false, he does not say anything false. ∴ If Socrates only says that he himself lies, he says something false and he does not say anything false.77
According to the Parvipontani, everything follows from this last contradictory consequent.
It is fair to say, therefore, every twelfth-century school of logic develops its own views about inferences in the course of reflecting on Abelard’s argument at (A1). Although these schools discuss the views of one another,78 they do so without ever coming to any agreement among themselves. As it turns out, moreover, among the various theses proposed by these schools, only Parvipontani’s thesis was to survive and to continue to be discussed in the next century on.79
1. Many of the texts cited in this chapter are unedited or untranslated. I quote Latin texts only when the relevant passages have never been published; otherwise I give references to editions and other studies where the Latin text at issue is quoted, allowing readers to find these sources for themselves.
2. Barrow, Burnett, and Luscombe 1984–1985.
3. For the sources mentioning these schools, see Iwakuma and Ebbesen 1992.
4. In the case of the Porretani, we have the Compendium logicae Porretanum; in the case of the Meludinenses, we have the Ars Meliduna and the Secta Meliduna; and in the case of the Albricani, we have a short treatise De sententia magistri nostri Alberici. As for the Parvipontani and the Nominales, we have unfortunately no such texts, though a fragment of MS Avranches 224 f. 3r–v, a partial edition of which can be found in Iwakuma 1993b, might well be part of the Parvipontani’s collection.
5. For example, the Ars Burana (193.10–11) adopts the Parvipontani’s definition of argumentum (cf. §IV.1 below), while at the same time admitting the non-Parvipontanean view that oblique cases are nouns as well as nominatives (cf. §III.2 below).
6. Iwakuma 1987.
7. For vocalism, or early nominalism, see Iwakuma 1992b. Cf. also the discussion in chapter 1, §III.1.
8. For this Platonic stream, see Iwakuma 1996, §7. For the fully developed Platonic theories of Bernard of Chartres and the later Walter of Mortagne, see John of Salisbury, Metalogicon II.17. (As far as I know, there are no logical texts which contain the fully developed Platonic theory.)
9. The realism of Anselm is discussed in more detail in Iwakuma 1996.
10. Hereafter I use “man” where English grammar requires “a man,” since the latter often has misleading connotations for discussions of the Latin, which lacks the indefinite article.
11. LI Isag., Glossae secundum vocales (for its authorship, see Mews 1984), LNPS, another unpublished commentary P17, and the Tractatus de generali et speciali statu. Hereafter I shall follow the standard scholarly convention of referring to these commentaries by the numbers attached to them in Marenbon 2000, for example, P1, P2, etc.
12. This terminology can be traced back to Ps.-Rabanus (P3). I tentatively identify Ps.-Rabanus (P3) with William of Champeaux in Iwakuma 1999.
13. Tweedale 1976, 95–111, gives a more precise analysis to the counter-argument based on the texts of Abelard’s LI and LNPS. The other texts mentioned in n. 12 above gives essentially the same counter-arguments against the material essence theory.
14. Iwakuma 1999 argues that P7 is a work of Abelard.
15. LI Isag. 10.17–11.9, Spade 1994, 23–27 (introduction of material essence theory); and 11.10–13.17, Spade 1994, 28–40 (counter-arguments). Geyer 1919 reads, 10.31–32, “quam – trahunt”; but MS reads “quae – transit.”
16. LI Isag.13.18–14.6, Spade 1994, 41–44 (indifference theory); 14.7–17, Spade 1994, 45–56 (collectio theory); and 14.18–31, Spade 1994, 47 (status theory). Strictly speaking, it is incorrect to refer to this theory as the status theory, since the key-term status is not used in this text. The text does, however, report the characteristic thesis of the status theory, namely, that there are as many universals (species and genus) as there are individuals.
17. MS Paris BN lat. 3237, ff. 125ra–130rb and ff. 123ra–124va. The discussion of universals are in ff. 125va–126ra = ff. 123rb–124rb.
18. This part of the discussion has been neglected, to my knowledge, by all commentators except Tweedale. See the brief discussion in Tweedale 1976, 128–129.
19. Dijs 1990, §§2–3 (material essence theory), §§4–7 (another theory), and counter-arguments for both §§8–25. This treatise does not mention vocalism either.
20. For the various attempts that have been made so far to identify master R., see Dijs 1990, 91, n. 21. I tentatively propose that we identify him with Radulph, who was master at Laon by 1115 and successor of his brother Anselm’s school at his death c. 1117 (cf. Lesne 1940, 308). He would seem to be the best candidate to try to revive the once defeated theory of his former co-disciple, William of Champeaux.
21. P17, Paris BN lat. 3237, f. 125va = f. 123rb: “Quorundam enim eorum est sententia eandem rem universalem totam indivisam in diversis et oppositis individuis esse, ut vere dici possit idem animal in essentia est materia Socratis et Brunelli. Ponunt etiam et genus et quodlibet universale in simplici natura acceptum rei singulari oppositum esse, inferioribus vero formis vestitum idem esse cum singulari.”
22. See the texts by the Nominales ed. in Iwakuma 1995, 68–88.
23. For Gilbert’s theory of universals, as developed in his theological works, see Nielsen 1982, 68. As for the Porretani’s theory of universals, see Martin 1983, xxxvii–xcii.
24. See de Rijk 1967, I.295.
26. De Rijk 1966, 24 shows that P21 (a treatise on universals in MS Wien Östereichishe Staatsbibl., VPL 2486, ff. 1r–4r, ed. by Grabmann 1947) is an extract from unedited P20 (in the same MS, ff. 45r–60v).
27. For example, see Grabmann 1947, 69, checked against the mss (the words in [] are omitted in P20): “Et notandum quod . . . cum dicimus ‘homo est [animal]’, genus praedicatur [de specie, quia dicitur] homo esse (P21] est P20) de illa maneria rerum (om. Grabmann) quae est animal, quia ostenditur quod res huius maneriei est illius.”
28. de Rijk 1966, 29–30, and following him Marenbon 1993, 107, suggest that P21 and/or P20 is a product of the school of Alberic of Paris. But I doubt this attribution for various reasons, which I cannot discuss here. P25, a Porphyry-commentary undoubtedly by an Albricanus, simply gives a resume of Boethius’s arguments concerning Porphyry’s questions, and does not develop any theory of its own about universals.
29. de Rijk 1967, II.52.20–26.
30. Thomson 1988, II, vii 6. Cf. also I, xxxi 4 and II, vi 3.
31. The maneria theory is also held by William of Conches. See his Glossae super Platonem, 149 and Glossae super Boethium, 326.290–292. Adam of the Petit-Pont, the founder of the school of Parvipontani, never uses the term maneria. Did Parvipontani learn the maneria theory from William, not from Adam?
32. As for the notio theory, see William of Conches, Glossae super Boethium, lxx–lccxiv. I owe this information to the editor, L. Nauta.
33. The term manerialiter appears in William of Sherwood’s Introductiones in logicam, 268.89. As for the Nominales, see Iwakuma and Ebbeson 1992.
34. For this topic, see the detailed discussion in Iwakuma forthcoming, §5. The relevant texts are edited in Iwakuma 1993a. Cf. also the notes there (52) to Vienna I 1.4 and Escorial I 2.2.
35. See Iwakuma 1992a, 47–54, where it is argued that Garlandus of Besançon, author of Dialectica, was active in the beginning of the twelfth century, and that he is not the Garlandus Compostista, author of the Compotus (hence “compostista”), active in the eleventh century (as de Rijk 1959, xlix, suggests).
36. Garlandus of Besançon, Dialectica (ed. in de Rijk 1959) 65.16 and 68.30. Garlandus elsewhere explicitly asserts that “chimaera” was significant before but is not now. Cf. de Rijk 1959, 70.33. Does this mean that Garlandus thought that chimaera existed really before?
38. For the full context, see Iwakuma and Ebbesen 1992, 98.
39. Compendium logicae Porretanum, 2–3. Note that adjectives were considered as a group of nouns in this period.
40. This MS was thought to have been lost in the Second World War. Constant Mews, however, informs me that it is fortunately preserved in Biblioteka Jagiellionska, Krakov.
41. F. 1va: “‘Recta’ additur (in descriptione nominis) ut excludantur obliqui, qui non sunt nomina sed casus nominum. Solus enim nominativus nomen est eo quod per eum fiat impositio nominis.” The Ars Emmerana also says that nouns are only nominatives, although the “recta” do not need to be added in the definition of nouns (150.12–21, cf. also 151.17–21). I suggest that the Ars Emmerana is a work of a Parvipontanus (cf. also the next note).
42. F. 4va: “Dicimus ergo argumentum esse dictum conditionalis hypotheticae quae transformatur ab argumentatione.” The Ars Emmerana also adopts this definition of argumentum (164.26–165.4). For this definition of arguments, see §IV.1 below.
43. For the text of the Glosulae, see de Rijk 1967, I.228, n. 1.
44. See LNPS 527.24 for “appellare vel nominare”; and 527.35 and 528.15 for “per appellationem.”
45. The Introductiones Montanae minores 36.16–32. The Introductiones Montanae maiores (MS Paris BN lat. 15141, ff. 59vb–60ra) holds still another theory.
46. See de Rijk 1967, I.300–305. As for the date, see Hunt 1980, 112, n. 7.
47. Cf. de Rijk 1967, II.260–282.
48. This treatise is, on the whole, word-for-word the same as the Tractatus Anagnini III, though at certain points it is more detailed. The MS contains in ff. 27r–34v several texts which I consider to be products of the schools that Alberic of Paris left in Italy.
50. See Green-Pedersen 1974, 16ff., Fragment 1.
51. For relevant sources, see Iwakuma 1995, 57ff. The designations “B8” and “B9” refer to the list in Green-Pedersen 1984.
52. Green-Pedersen 1984, III-C, 163–221.
53. See Iwakuma 1995, 53–58.
54. Concerning this and other counter-arguments by the Meludinenses against the Nominales’ thesis, see Iwakuma 1995, 55.
55. In Iwakuma 1995, 56ff., I attributed this theory to the Albricani only tentatively. Since then, however, I have discovered further evidence for my view in MS Wien Nationalbibl., VPL 2237, f. 31r, “De sententia magistri nostri Alberici”, which says explicitly, among positiones nostrae, “Septima est quod argumentum est dictum hypotheticae generaliter propositiae, ut dictum huius hypotheticae ‘si aliquid est homo, ipsum est animal.’ Dictum huius naturalis est argumentum ad istas omnes argumentationes ‘Socrates est homo, ergo est animal’ ‘Plato est homo, ergo est animal,’ et sic de ceteris. Dictum vero illius est hoc: aliquid esse animal si ipsum est homo.”
56. See Compendium logicae Porretanum, Thesis III.40, 56ff.
57. See Green-Pedersen 1974, 17 (Fragment 1) and 18 (Fragment 3).
59. All S are M, all M are P, therefore all S are P.
60. Paris BN lat. 15141, f. 83ra: “Sed m(agister) P(etrus) hypotheticam huiusmodi quae constat ex categorica et hypothetica naturali in nullo modo concedit propterea quod in quibusdam terminis eam falsam esse contingit, ut si nullus flos est, tunc si rosa est flos, rosa non est . . . Sed quod ista fal/sa sit, apparet ex hoc quod ex vero nunquam sequitur falsum, cum prima sit vera et quae sequitur sit falsa. Et hic communibus M.P. hypotheticam aliquam ex categorica sequi vel e converso negabat. Ad quod m(agister) Al(bericus), quamvis in quibusdam terminis qui leviter huiusmodi propositiones falsae numerantur, tamen non omnes falsae sunt iudicandae . . . Dicendum est igitur quod huiusmodi hypotheticae verae sunt, non tamen in quibuslibet terminis propositae, sed in his tantum in quibus subiectum nunquam potest esse sine praedicato vel nunquam sine eo contingit, . . .”
61. Paris BN lat. 15141, f. 84rb. I cite this passage in Iwakuma 1995, 58–59. The passage omitted there runs: “Sed primo errorem quorundam putantium hypotheticam, quae constat ex categoricis duabus iunctis per ‘et’ consequenti, eam(?) non esse aliam a syllogismo secundum cuius formam et regulam constituta est, removere liceat. Dicunt namque syllogismum istum ‘omnis homo est animal, sed omne risibile est homo, ergo omne risibile est animal’, esse istam ‘Si omnis homo est animal et omne risibile est homo, [ergo] omne risibile est animal’, putantes scilicet quod ponitur in assumptione esse positum pro conclusione. Sed in utroque errant. Propositio namque hypothetica composita ex duabus categoricis iunctis per ‘et’ consequenti (] et tertia ms), in multis vera inveniretur, in quibus ille syllogismus, qui sic secundum regulam constitutus est, falsus est, ut ‘si omnis homo est lapis <et omnis lapis est> insensatus, omnis homo est insensatus’, ista hypothetica vera est, et syllogismus |84va| falsus est, quia habet falsam propositionem. Et quod syllogismus dici debeat falsus propter hoc quod habet falsam partem, dicit Aristoteles plane in Elenchis. Iterum si verum huiusmodi syllogismum concedant, oportet eos concedere ratione simili quod vero syllogismo probatur quod ipsi sunt asini, quod est inconveniens. Unde manifestum quod nullo modo syllogismus est illa hypothetica quae est constituta secundum(?) se. Ista vero hypothetica semper vera est in omnibus terminis, et nunquam fallit si secundum regulam alicuius fiat syllogismi. Illa vero quae constat ex categorica et hypothetica naturalibus non deberet, ut superius demonstravi, proponi [in quibus] in quibuscumque terminis syllogismus proponitur, sed in illis in quibus praedicatum nullo modo relinquat subiectum. Revertendum est igitur ad id quod demonstrare proposuimus, scilicet quod loci singulos ‘syllogismos’ contineat.”
62. Compendium logicae Porretanum, 28, II-36.
63. See Ars Meliduna, II.17 (de Rijk 1967, i.347).
64. For the discussion of syllogisms in the Ars Meliduna, see de Rijk 1967, II.378–383.
65. For a survey of this issue, see Green-Pedersen 1984, III-C-2, esp. 198–201.
66. For a survey of the number of loci discussed in the twelfth century, see Green-Pedersen 1984, III-C-3, 203–210. Green-Pedersen leaves open the question of this tradition’s origin, but it appears to me to have begun with William. As for William’s loci, see Iwakuma 1993b, note on 53.
67. There are already a number of studies treating this discussion of Abelard’s and the reactions of his rivals. See the list in Iwakuma 1995, 48, n. 5. Among these Martin 1987b is the most perspicuous, and therefore my discussion mainly follows his.
68. Abelard’s discussion of this issue in the Dialectica (395.6–397.13) is corrupted, as well as too brief to be certain of his views. I follow, therefore, the discussion reported in the Introductiones Montanae minores, 63.18–64.27 with emendations in Martin 1987a, 391, n. 29.
69. Introductiones Montanae maiores, MS Paris BN lat. 15141, f. 63rb: “Fit autem illa obiectio. ‘Socrates est homo et est asinus.’ Haec propositio est hypothetica, ergo est simplex vel composita. Et nolumus in huius laborare solutione, cum hanc et consimiles propositiones minime esse concedamus. Nunquam enim qui dicit hanc orationem ‘Socrates est homo et asinus’, unum significat, † Et ideo multiplex est iudicanda et una vera et altera falsa. M.P. vero huiusmodi propositiones iunctas per ‘et’ (] et per ms) hypo(theticas) iudicat etiam usque ad denarium numerum iudicat unam hypothetariam(!).”
70. Introductiones Montanae minores, 64.28–65.12.
71. Introductiones Montanae minores, 65.23–66.4, where “expositione” on line 25 should be read “ex positione.” Cf. Martin 1987a, 394–395.
72. Compendium logicae Porretanum, 23: “Dicit enim magister (whom I interpret as Gilbert): non est necessitas consequentiae ubi est conflictus positionis et nature.”
73. Compendium logicae Porretanum, Thesis II.26 (22–23).
74. Compendium logiae Porretanum (23.77–81), where I emend “hoc est ‘Plato non est Socrates; . . .” to “‘si Socrates est Plato, non est Socrates; . . .”
75. For the Meludinenses’ thesis (as well as the Parvipontani’s thesis discussed below), see Iwakuma 1993a and relevant sources edited there.
76. I follow the arguments given by Alexander Nequam, De naturis rerum, cited in de Rijk 1967, II.290ff.
77. I follow the version of Alexander Nequam, De naturis rerum, cited in de Rijk 1967, II.290, with some modification.
78. The Ars Meliduna mentions all the theses of the schools. As for the discussion between the Meludinenses and the Parvipontani, there are a number of sources to draw on (see Iwakuma 1993a). That Alberic of Paris knew well the theses of other schools is proved by the source, De sententia magistri nostri Alberici (MS Wien Nationalbibl. 2237, f. 31r) which mentions as positio secunda, ex falso aliquid sequitur (against the Meludinenses’ thesis), as tertia, ex impossibili aliquid sequitur (against the Parvipontani’s thesis), and as quarta, negativa sequitur ex affirmativa (against the Nominales’ thesis).
79. See Spruyt 1993, D’Ors 1993, and Ashworth 1974, 133–136.