A great deal of Peter Abelard’s writing is concerned with what he regarded as logic, but which we would now classify as ontology or philosophical semantics.1 Following Cicero and Boethius, Abelard holds that properly speaking the study of logic has to do with the discovery and evaluation of arguments (LI Isag. 3.10). A necessary preliminary for this is an examination of the issues dealt with by Porphyry in the Isagoge and by Aristotle in the Categories, and De interpretatione (LI Cat. 113.26–114.30). In the present chapter, however, I will ignore most of this material and concentrate on the central issue of logical theory both for Abelard and for us, that is, on the nature of the relation of consequence, or following.2 Even with this limitation there is a great deal of ground to cover. Abelard sets out his theory of entailment and argument in two very extended and dense discussions both of which have suffered considerable textual corruption. The treatment of topics and hypothetical syllogisms in the Dialectica, is apparently the earlier. The other is the surviving fragment of Abelard’s commentary on Boethius’s De topicis differentiis, Glossae super De topicis differentiis, which seems to belong with his other commentaries on the works of the logica vetus published as the Logica “ingredientibus.” The two expositions disagree on some crucial questions, but here I will restrict myself almost entirely to the discussion in the Dialectica.
Abelard was the greatest logician between Aristotle and the Stoics in antiquity and William of Ockham and John Buridan in the fourteenth century. In many ways his achievement is much more remarkable than those of Ockham and Buridan. They had all of Aristotle’s logic and built upon two centuries of intensive work by their medieval predecessors. Abelard had only the Isagoge, Categories, and De interpretatione, accompanied by Boethius’s commentaries on them and his rudimentary paraphrase of Prior Analytics I, 1–7. In addition, Abelard had Boethius’s treatise on the hypothetical syllogism, De syllogismis hypotheticis, his study of division, De divisione, and his two works on topical inference: In Topica Ciceronis, a commentary on Cicero’s Topica, and De topicis differentiis, an introduction to the theory of topical arguments. He also had a brief work by Marius Victorinus on definition, De definitionibus, and some material relevant to his logical concerns in Priscian’s Institutiones grammaticae. Abelard is often critical of the logical theories of his contemporaries but unfortunately, aside from what he himself tells us, relatively little of their work and that of his immediate predecessors has been published.3
Abelard’s logical theory appears to have developed as an attempt, a quite brilliant one, to unify into a single theory various disconnected remarks made by Boethius in his discussions of the topics and hypothetical syllogism. To understand Abelard’s work we must thus first say something about the material which he inherited from Boethius in De topicis differentiis and De syllogismis hypotheticis.
Boethius is the sole Latin representative of the neo-Platonic commentary tradition which flourished in Alexandria and Athens from the third to the sixth centuries CE. These commentators wrote at a time when the original impetus for logical investigation had been lost and many of the issues were no longer well understood, in particular the problem of the relationship between Stoic and Peripatetic accounts of argument. The material which Boethius bequeathed to the philosophers of the eleventh and twelfth centuries, where it is not entirely elementary, is frequently confused and sometimes inconsistent. Its transformation by Abelard into a unified theory of entailment and argument is extraordinarily impressive.
The promise made by Boethius as a reward for the study of the loci, or topics, is irresistible. It will provide access to a rich source of arguments to settle any given question. Boethius proposes in De topicis differentiis to give an account of the role of the loci in the discovery of arguments, to explain what a locus is, how loci differ from one another, and to show which loci “are appropriate for which syllogisms” (De top. diff. 1.1173C). The last of these is especially important for Abelard since he construes it as a claim about the nature of the syllogism which he rejects.
In pursuing his project Boethius first introduces the technical terminology which will provide medieval philosophers with a vocabulary for theorizing about argumentation. He insists that in proving a conclusion a distinction must be made between what, following Cicero, he calls an argumentum, the meaning, or sense, of the words used in proving something (De top. diff. 1.1174C), and an argument (argumentatio), the verbal or written expression of such a proof. The locus for an argument is the “source of the argumentum,” or “that from which there is drawn an argumentum appropriate to the proposed question” (De top. diff. 1.1174D).
Arguments are sequences of propositions consisting of one or more premises and a conclusion.4 If a proposition is not provable because it is known “per se” to be true, Boethius calls it a maximal proposition. His example is “if equals are removed from equals, then equals remain” (De top. diff. 1.1176C).5 A maximal proposition may appear as premise in an argument or as an external principle required to guarantee that the conclusion follows from the premises. In either case, Boethius maintains, though without explanation, the maximal proposition contains the other propositions6 and indeed “contains the whole proof” (De top. diff. 2.1185–86C).
Although Boethius does not say so explicitly, his theory allows arguments with hypothetical as well as categorical conclusions. The proof of a hypothetical proposition does not, according to Boethius, require a syllogism all of whose premises are hypothetical.7 Rather, in De syllogismis hypotheticis 1.2, he accepts the priority of the categorical syllogism and insists that a hypothetical premise is itself provable with a categorical syllogism.8
Abelard finds many problems in Boethius’s account of argument and in particular in his claims about the argumentum. According to Boethius, neither the argumentum nor what it settles doubt about is a proposition. Rather, the argumentum is a ratio, a reason, which removes doubt with respect to a res, that is to say, with respect to a state-of-affairs.9 Argumenta are either necessary or not, and so bearers of modal properties, and in addition either probable or not probable.
The notion of probability appealed to here has its origin in Aristotle’s Topics where it is used to characterize a dialectical syllogism as an argument in one of the canonical moods whose premises are each probable in the sense of being acceptable to everyone or to an appropriate group of experts. For Boethius, however, it is argumenta which are probable and although his examples of what is probable are categorical propositions, the arguments given in De topicis differentiis are not typically categorical syllogisms. In addition Boethius extends the notion of the probable to include whatever is accepted by one’s opponent in an argument or by whoever is judging the exchange.
Like Aristotle in the Topics, Boethius classifies true predicative propositions and the corresponding questions in terms of the relations between the extensions of their subject and predicate terms (De top. diff. 1.1177D).10 There are four possibilities: the predicate may be the genus of the subject, its definition, a property, or an accident. Other kinds of categorical propositions and questions may be reduced to one of these. Unlike Aristotle in the Topics, but perhaps following the suggestions made in Prior Analytics I.27, Boethius also investigates the relationships which may be expressed in a true conditional proposition and asked about with the corresponding conditional question. Conditional propositions, he holds, indicate that one “thing” (res) is accompanied (comitatur) by another and conditional questions ask whether this is so. The “things” are often the same as those with which categorical propositions and questions are concerned but there are some, for example cause and effect, whose association cannot be expressed in a predicative proposition.
Boethius often, though not always, gives conditionals in the form “si est/non est A, est/non est B” which he explains as indicating that if there is something which is/is not A, then there is something which is/is not B.11 Often the same thing is the intended subject of both antecedent and consequent, a member of a species and of its genus, for example, but not always, as with a cause and its effect, or relative opposites such as father and son.12
Boethius catalogues the different kinds of relationships between things which may be indicated with a true conditional in terms of the qualities of the antecedent and consequent. Conditionals whose antecedent and consequent are both affirmative hold, as for example, when the antecedent is a species and the consequent a genus of it (De top. diff. 1.1178D–80A). The equivalence of contrapositives13 guarantees that conditionals whose antecedent and consequent are both negative hold for just the same kinds of things. Simple conditionals whose antecedent and consequent differ in quality can hold, according to Boethius, only of things which are opposed in appropriate ways and, as we will see, they appear in some famous arguments of Abelard’s to support the characteristic thesis of his logic and that of his followers, the Nominales, that all conditionals of mixed quality are false.14 Boethius argues that a conditional with a negative antecedent and an affirmative consequent – “if something is A, then it is not B,” is true whenever A and B simply cannot occur together although both may perhaps fail to be present.15 A conditional with a negative antecedent and an affirmative consequent – “if something is not A, then it is B” is, on the other hand, true only where A and B are immediate contraries – they can neither both be present nor both be absent.16
Boethius proposes to show which loci are appropriate to which syllogisms and in Book II of De topicis differentiis he introduces the syllogism as one kind of argument, the principal kind. His definition differs from that given in the Prior Analytics in a crucial respect. Where Aristotle defines a syllogism as “an expression in which from certain things being posited something other than what is posited follows of necessity from their being so” (Pr. An. 1.1, 24b19) Boethius adds “and conceded” to Aristotle’s “posited” (De top. diff. 2.1183A). The additional requirement will provide Abelard with one clear distinction between syllogisms and conditionals. Enthymemes are arguments obtained from syllogisms by the omission of a premise. They are thus, according to Boethius, imperfect syllogisms.
After all these preliminaries we come finally to the loci. The problem of obtaining an argumentum is solved, according to Boethius, by locating an appropriate maximal proposition. Such a proposition, he tells us, may be found by considering the terms of the question, which he apparently assumes here to be categorical.17 A maximal proposition states that particular kinds of things are related in particular ways. Two examples: (1) the external maximal proposition “that to which the definition of the genus does not apply is not a species of the genus so defined” is invoked to settle the question of whether a tree is an animal by warranting the syllogism which concludes that it is not from the premises “an animal is a sensible animate substance” and “a tree is not an animate sensible substance” (De top. diff. 2.1187A). Abelard, as we will see, objects that in this case the proof does not need the help of a maximal proposition; (2) the external maximal proposition “what holds of the whole holds of the parts” and the premise “human beings are part of the world” support the enthymeme “the world is ruled by providence; therefore human beings are ruled by providence” or the corresponding conditional.18
A maximal proposition is distinguished by what Boethius calls its locus differentia. In the case of the pair just mentioned that is by being from definition and from whole. The locus for an argument, according to Boethius, consists of both the locus differentia and the maximal proposition. To discover an argumentum we consider the question, note, for example, that we are being asked whether the predicate is the genus of the subject, realize, say, that the properties of definition are relevant and then invoke an appropriate maximal proposition in conjunction with the relevant definition.
Boethius’s treatment of the loci in De topicis differentiis is extremely sketchy. He gives lists and classifications of differentiae and maximal propositions from both Themistius and Cicero with very brief examples. He apparently intends a topical argument to a categorical conclusion to be a categorical syllogism one of whose premises states that the terms about which the question is asked are related in the appropriate way. Sometimes the examples are categorical syllogisms, sometimes enthymemes, and sometimes conditional propositions. Although Boethius says absolutely nothing to indicate that they are connected in the way that Abelard proposes, the appearance of the last of these as putatively warranted by external maximal propositions, and the claim that conditionals may be proven with categorical arguments, perhaps suggested to him that the theory of the loci might be unified with that of the conditional.
The most striking feature of Boethius’s account of hypothetical syllogisms is that he clearly has no notion of the propositional compounding of propositional contents to form new contents of arbitrary complexity.19 Thus, in his longer commentary on De Interpretatione, he rejects the Stoic practice of preposing the negative particle to a categorical proposition as simply inviting ambiguity over which term is being negated (In De in. maior 10.261–262). Equally importantly he refuses to allow that the copulative conjunction “and” might combine conjuncts into a single proposition and insists that it serves no more than a listing function (In De in. maior 5.109).
Boethius claims in De syllogismis hypotheticis that nothing had been written in Latin on the hypothetical syllogism and only very little in Greek. In his commentary on Cicero’s Topica he does discuss the Stoic indemonstrables at some length but his knowledge of them seems to be entirely derived from Cicero’s very brief remarks. In De syllogismis hypotheticis Boethius proposes to give an account of hypothetical syllogisms on the basis, presumably, of the small amount of work that he knows by Aristotle’s successors, Theoprastus and Eudemus. The result is very curious. Since Boethius has no concept of a propositional operation he considers in turn all the syllogisms in which each form of conditional acceptable to him may occur. In addition to the four forms of simple conditional mentioned above and the corresponding disjunctions, he also allows three compound forms of the conditional which may be obtained by having a categorical as antecedent or consequent to a simple conditional or a simple conditional as both antecedent and consequent.
According to Boethius a conditional proposition, or consequence (consequentia), is a proposition indicating that something holds on a condition which is marked with the connective si or equivalently, he tells us, with cum (De hyp. syll. 1.3.1).20 Simple disjunctions are equivalent to simple conditionals in which the antecedent is the opposite of the first disjunct and the consequent is the second disjunct.21 Although the antecedent and consequent of a conditional may contain one of the modal terms “necessary” or “possible,” they do not usually contain an expression of quantity and so are indefinite.
Having said that si and cum have exactly the same meaning as indications of a condition on which something holds, Boethius goes on to claim that they may be used to mark two distinct relationships between antecedent and consequent. A true conditional formed with cum, he implies,22 is used to indicate that the antecedent and the consequent are associated secundum accidens, that is, whenever the antecedent is true the consequent is true, as in the case of “if fire is hot, the heavens are spherical” (De hyp. syll. 1.3.7). Here the antecedent does not explain the consequent nor the consequent the antecedent. If there is such an explanatory connection between them, the conditional expresses a consequence of nature (consequentia naturae), e.g. “if something is human, it is an animal” and “if the earth lies between the sun and the moon, an eclipse of the moon follows” (De hyp. syll. 1.3.7). Unfortunately Boethius tells us nothing more in De syllogismis hypotheticis about consequences of nature.23
What Boethius goes on to say about conditionals in De syllogismis hypotheticis is presumably intended to apply equally well to both consequences of nature and those which hold secundum accidens: to “oppose” a conditional, he tells us, one must “destroy its substance.” That is to say, since the necessity of a conditional lies in an “immutable consequence,” we must show that when the antecedent is posited, the consequent “does not immediately (statim) follow” (De hyp. syll. 1.9.4). A necessary condition, at least, for the truth of a conditional is thus that it is not possible for the antecedent to be true and the consequent false at the same time.
Although Boethius claims that Aristotle had nothing to say on the hypothetical syllogism, he nevertheless sets at the center of his own account a principle for the logic of conditionals which he takes from Prior Analytics 2.4. Aristotle is concerned there to show that although the truth of the conclusion of a categorical syllogism is compatible with the falsity of its premises, the truth of the conclusion cannot follow from the falsity of the premises. Boethius’s version of the claim is that:
It is not necessary that the same is when the same both is and is not, as when A is, if for this reason it is necessary that B is, if the same A is not, it is not necessary that B is, that is, that it is because A is not.
(De hyp. syll.1.4.2)
Boethius claims to prove the principle by arguing that an impossibility follows from supposing that both “if A is, then B is” and “if A is not, then B is” are true. The proof seems confused and wrongly to conclude, unlike Aristotle, that the impossibility which would follow from both conditionals being true is that B both is and is not. What Aristotle argues, and Abelard sees, is that what follows is “if B is not, then B is” which both regard as impossible. Appeals to this last, so-called connexive principle,24 and others will be the characteristic feature of Abelardian logic and, as we will see, ultimately responsible for its failure.
Unlike any of the ancient authors to whom he had access Abelard clearly understands the nature of propositionality and of propositional combination. If he was the first to achieve this understanding in the Middle Ages, and there is no evidence to the contrary, he must be recognized as one of the greatest of all philosophical logicians. Peter Geach has claimed the central discovery for Frege but it was well known in the twelfth century. What Geach calls the Frege Point provides the foundation for the development of propositional logic with the proposal that a distinction must be made for speech acts between force and propositional content. For example, an assertion of “Socrates is running” has the same propositional content as the question “Is Socrates running?” but a different force. With the first I assert that Socrates is running, with the second I ask whether it is the case that Socrates is running. The propositional content in both is that Socrates is running. The force in the first is assertion, in the second interrogation.
In De interpretatione 4, Aristotle observes that there are various different kinds of speech act and relegates those other than assertion to the rhetorician and the poet. Abelard thus does not have much to say about the differences between them but he does notice that one and the same content, that the King comes, may be given different force – optative force in “Would that the king comes” and, as Abelard construes it, assertoric force in “I hope that the King comes” (LI De in. 3.5.13, G 374.21–26).
Boethius uses the terms “assertion” (enuntiatio) and “proposition” (propositio) interchangeably, and distinguishes a proposition, or assertion, from other kinds of speech act by its being true or false. He thus makes it difficult for later writers to refer separately to the true or false propositional content of each kind of speech act and to the particular speech act of asserting a propositional content. Abelard, too, often fails to make the required distinction but when he is being careful he does make it and notes in particular that although the component propositions of the true conditional “if Socrates is a pearl, then Socrates is a stone” are both false, they are not asserted (proponitur) when the whole is asserted (TI 90).25
An operation “F” is a propositional operation if it takes a propositional content “p” as its argument and yields as its value, “F(p)”, a propositional content available for use in all the various kinds of speech acts. For example a conditional propositional content is formed by combining two propositional contents with the operation “if ( ), then ( ).” A propositional operation is truth-functional if the truth-value of the result of applying the operation is determined solely by the truth-value of its arguments.
The simplest propositional operation, but also the one that is crucial in distinguishing Abelardian from Boethian and Aristotelian logic, is the truth-functional operation of propositional negation which has for its value a propositional content which is true if the argument is false and false if it is true. Neither Aristotle nor Boethius provide such an operation. Both treat an affirmation and the corresponding negation as distinct speech acts, not as assertions of two different propositional contents with the negation obtained from the affirmation by a propositional operation. Abelard does precisely this, calling the operation of propositional negation extinctive, or destructive, negation: “not (S is P),” is true just in case “S is P” is false. For affirmative categorical propositions, he also allows a form of negation that modifies the predicate (Dial. 477.4–26). This “Aristotelian” negation, “S is not P,” Abelard calls separative, or remotive, negation and it is true just in case S exists26 and “S is P” is false.27
The extinctive negation of a proposition is its contradictory and the separative negation a contrary (Dial. 173ff.). Abelard notes that extinctive and separative negation may be combined and that extinctive negation may be iterated to produce, when doubled, a proposition equivalent in the sense of necessarily coinciding in truth-value with the original affirmation. He denies, however, for reasons which will become clear, that an affirmation entails or is entailed by its extinctive double negation (Dial. 178.36–179.33).
Abelard finds interesting support for this distinction of negations in Aristotle’s treatment of universal propositions. In De interpretatione, though not elsewhere, Aristotle gives the contradictory opposite of “Every A is B” as “Not every A is B.” Boethius claims that this means the same as “Some A is not B,” the form which he follows Aristotle in using in his presentation of the theory of the categorical syllogism (e.g. In De in. maior 7.164ff.). Reading Aristotle’s preposed negation as propositional, however, Abelard insists to the contrary that the two propositions have quite distinct meanings. “Not (every A is B)” is the extinctive negation and so the contradictory opposite of “Every A is B.” It is entailed by “No A is B,” itself a contrary of “Every A is B,” and is true if either there are no As or As exist but none of them are Bs. “Some A is not B,” on the other hand, is entailed by “Every A is not B,” a contrary of “Every A is B,” and it is true only if there exist some As. Finally, “No A is B” is the extinctive negation of “Some A is B,”28 and so true either if there are no As or if there are As but none of them are Bs.29 Abelard thus lays out a rectangle of opposition where Aristotle had his famous square (LI De in. 3.07.37–49, G 408–11).
Giving the example of “both Apollo is prophet and Jupiter thunders,” Boethius had claimed that the copulative connective “and” (et) does not form a single proposition from two propositions but serves in effect merely to punctuate a list. The following remark would make a singularly appropriate epitaph for Abelard the logician:
For since he concedes that “if it’s day, then it’s light” is a single proposition in which different propositions are reduced to the sense of one proposition by the preposed conjunction, I do not see why “both Apollo is a prophet and Jupiter thunders,” cannot be said to be a single proposition, just as “when Apollo is a prophet, Jupiter thunders.” Whence each may have a single dividing opposite, so that as we say “not (if it’s day, then it’s light),” we should also say “not (both Apollo is a prophet and Jupiter thunders).”
(LI De in. 3.05.41, G 380.4–11)30
Abelard perfectly understood the nature of propositional operations and their generality. The copulative conjunction and the disjunction “or,” he goes on to tell us, may connect any number of components into a single proposition (LI De in. 3.05.90, G 387.1–21). Tragically for the history of logic, as we will see, he did not realize until too late the consequences of trying to combine plausible principles for the manipulation of the copulative connective with what he took to be equally compelling intuitions about the operation of negation.
Abelard introduces loci at the beginning of Treatise III of the Dialectica by remarking that just as prior to his account of the categorical syllogisms he had to say something about categorical propositions, so, before he deals with hypothetical syllogisms, he must say something about hypothetical propositions. This involves him immediately with loci since it is loci, according to the theory presented in the Dialectica, which are the source of the true conditional and disjunctive propositions employed in such arguments.
Much later in the Dialectica Abelard argues that the classical definitions of a locus as “the source of an argumentum” or as that “from whence an argumentum is drawn to settle a proposed question” are appropriate for the theory of argumentation but not for the theory of the conditional (Dial. 454–455). At the beginning, however, the problem is simply that these definitions are too narrow to connect loci and consequences. The new definition, which is apparently entirely Abelard’s own, is that a locus is the “power of” or, as we would say, provides the warrant for an entailment (vis inferentiae). For example, the conditional “if something is a human being, then it’s an animal” is true because of the relationship, properly called a habitude (habitudo), in which human being stands to animal (Dial. 253.16–23), that is, as a species to its genus. The locus differentia, according to Abelard, is thus the antecedent thing (res) in the proved consequence; in this case human being. If, however, a dialectician is asked to justify his assertion of the conditional with the standard question “whence the locus?” (unde locus?), he should reply “from species.” He is not being asked what the locus is but rather in virtue of what fact about human beings the entailment holds (Dial. 264–266). The next two hundred pages or so of the Dialectica are devoted to exploring the application of this new broad definition of locus.
I translate Abelard’s inferentia as “entailment” because it requires both necessity and relevance.31 The relation of entailment, or consecution (consecutio) is expressed, either in a true consequence32 – that is, in a true conditional or disjunctive proposition – or in an argument whose conditionalization is true,33 and the necessity of a true consequence lies in there being a meaning relation, and so a relevant connection, between antecedent and consequent:
Entailment consists in the necessity of consecution, that is, in that the sense (sententia) of the consequent is required (exigitur) by the sense (sensus) of the antecedent, as is asserted with a hypothetical proposition . . .
(Dial. 253.28–30; cf. LI Top. 238.6–9)
We will see below that Abelard often explicates this meaning relation in terms of the antecedent in some way containing the consequent.
Entailments are either perfect or imperfect. A perfect entailment is one which satisfies the requirement of containment in virtue of what Abelard calls the structure, or form (complexio), of the propositions involved (Dial. 253.31). An imperfect entailment, on the other hand, is necessary not in virtue of its structure but rather because of non-formal truths which hold, as we would say, of every possible or impossible world (de natura rerum). Such truths are expressed in maximal propositions and it is the assignment of the appropriate locus differentia in conjunction with the maximal proposition which provides the required external guarantee of the necessity of an imperfect entailment. Abelard’s main concern here is with such entailments, but to characterize them he has first to give an account of the purely formal character of perfect entailment.
Earlier in the Dialectica Abelard gives Aristotle’s definition of a syllogism from the Prior Analytics (Dial. 232.4–6).34 In glossing the definition, however, it is Boethius’s version to which he refers with its requirement that the premises be conceded as well as posited. This condition is added, he holds, to show that we are concerned with an argumentum and to distinguish syllogisms from conditionals whose structure (complexio) has the same form (forma) (Dial. 232.8–12). The perfection of syllogisms as entailments is indicated in Aristotle’s definition, according to Abelard, by the qualification that the conclusion follows necessarily from the premises themselves in that nothing extrinsic is required to guarantee this.35
All of the various moods of categorical and hypothetical syllogisms are perfect entailments. Aristotle, however, distinguishes the first figure of the categorical syllogism and Boethius the first figures of simple and composite hypothetical syllogisms as having a perfection not possessed by the other figures. Let us call this second kind of perfection evident perfection. According to Abelard, the secondary figures are imperfect in this sense, because they are not immediately accepted but rather have to be proved by reduction to the first figure by means of the external principles of conversion and proof by reductio ad impossibile. So while no external principle is required to warrant the inference of the conclusion from the premises of a categorical or hypothetical syllogism, principles which are more evident must be deployed to show someone who does not see this that the conclusion does indeed follow from the premises in the case of the secondary figures.
It is clear from Abelard’s description that perfection of entailment is a property of form and from his further explanation that his conception of formality is one which we share. The conditionalization of a syllogism is perfect, he tells us, because “whatever terms you substitute, whether they are compatible or incompatible with one another, the consecution can in no way be broken” (Dial. 255.310–334).36 In the context it is clear that uniform substitution is what Abelard has in mind and so the property of “formal truth” which Quine has popularized in the twentieth century and which has generally been said to have had its origins in the work of Bolzano.37
Abelard’s version of the substitutional criterion of formal truth provides him with, as a necessary condition for perfection, the requirement that uniform substitution preserve consecution. He clearly does not regard it as sufficient since he argues that the substitutionally true entailment “if every animal is an animate being, then every animal is an animate being” is not perfect because missing from the antecedent is the proposition “every animate being is an animate being” which is needed to show that “the same thing is contained in itself” (Dial. 255.19–30).38 Abelardian logic thus includes the principle of reflexivity “p ⊨ p” and its conditionalization “⊨ p → p” but classifies the entailment as imperfect. Although he does not address the question, Abelard apparently holds that it is only instances of the canonical moods of categorical and hypothetical syllogisms which are perfect entailments.39
Practically the whole of the rest of Treatise III of the Dialectica is concerned with the fact that uniform substitution plus whatever other condition might be imposed is not a test for entailment but rather for perfect entailment.40 Imperfect entailments are conditionals which fail the substitution test but in which the consequent nevertheless follows necessarily from the antecedent. So, for example, Abelard points out, the conditional which results from dropping one of the conjuncts from the antecedent of a perfect entailment may still be connected necessarily to the consequent in virtue of the nature of things (ex natura rerum). This is so, he claims, in the case of both (C1) “if every human is an animal, then every human is an animate being” and (C2) “if every human is an animal, then no human is a stone” (Dial. 254.31–255.11). His use of (C2) as an example of an entailment is striking since, as I said, one of the defining features of his logic is the theorem that a conditional whose antecedent and consequent differ in quality cannot be true. We should note, however, that he goes on immediately to designate the connection between antecedent and consequent in these two cases as perfection of necessity rather than of construction, and characterizes this not in terms of the antecedent requiring the consequent but rather simply as the inseparable association of the things signified – “the nature of animal . . . does not suffer animal ever to exist without animation” (Dial. 255.19–30). That is to say, it satisfies Boethius’s requirement for the truth of a conditional. As we will see, according to Abelard, this condition is necessary but not sufficient for entailment. We have to wait, however, until we are well into the discussion of the individual loci for this point to be made clearly and before we come to it Abelard has to settle precisely how appeal to a locus may warrant an imperfect entailment.
Abelard observes that all entailments which fail the substitution test pass another in which substitution is restricted to terms signifying things which stand in the same relationship as those signified by the terms substituted for. For example, every conditional is true which is obtained by substituting for “human” and for “animal” in “if something is a human, then it is an animal” terms signifying things standing in the habitude of species to genus. If we restrict substitution in this way, then consecution is preserved in virtue of the nature of things.
What is needed next is an account of the different relations between things which will support true conditionals and an analysis of the nature of this support. Abelard sought to provide both of these by appealing to and radically developing the theory of topical inference which he had inherited from Boethius. His claim is that if we consider entailments which are not perfect we will see that their truth depends on the existence of relationships which can be identified with some of the loci listed by Boethius. Such imperfect entailments, Abelard argues, can be perfected by the assignment of the appropriate locus (Dial. 256.34–257.23).
Boethius as we saw, claims to provide loci appropriate for syllogisms and although in none of his examples is a maximal proposition invoked to guarantee an instance of a canonical mood of a categorical or hypothetical syllogism, it is to such syllogisms that Abelard and his contemporaries took Boethius to be referring (Dial. 258.9). Abelard on the contrary holds that loci are required only where perfection is lacking and that to perfect, or prove, an imperfect entailment one must assign the appropriate habitude and invoke a suitable maximal proposition. The maximal proposition is a rule applicable where the habitude exists which has the differentia corresponding to that habitude. No habitude is required for the application of the rules of syllogistic entailment and so no locus is required for a syllogism. The claim that syllogisms need no support from loci was controversial, however, and became another characteristic thesis of the Nominales (Dial. 256–263).41
Abelard acknowledges, of course, that a given categorical or hypothetical syllogism is an instance of a general rule. The conditional “if every human is an animal and every animal is an animate being, then every human is an animate being” instantiates, for example, the rule for the first mood of the first categorical figure that “if something is predicated of something universally, and some other thing is predicated universally of the predicate, then the second predicate is predicated universally of the first subject” (Dial. 237.6–8). Abelard insists, however, that such rules are not maximal propositions since they have no differentiae restricting the appropriate substitution class to items related by one of the local habitudes.
Abelard considers the possibility that Boethius might be understood as indicating that the locus for a syllogism is that which warrants the enthymeme obtained by dropping one of the premises from the syllogism. He argues that, in general, loci for categorical syllogisms cannot be obtained in this way since the only loci which may be appealed to where both premise and conclusion are of the same quality and there is no local habitude between their terms are the non-Boethian loci from the predicate and from the subject. If the enthymeme contains negative propositions, on the other hand, the only locus which could be invoked is that from opposites – “if one of a pair of opposites belongs to something, the other does not.” According to Abelard, however, the maximal propositions associated with each of these loci are false and so cannot provide the necessity required by the syllogism (Dial. 262.1–28). Likewise, he argues, an attempt to appeal in a similar way to the loci from antecedent and consequent to warrant hypothetical syllogisms fails because the associated maximal propositions are entirely lacking in probative force (Dial. 262.1–28). Someone who is in doubt about whether “q” follows from “p” will be equally doubtful about the premise asserting that “p” is antecedent to “q” and the maximal proposition “if the antecedent is posited, then so is the consequent” will be of no help in convincing him of the truth of the entailment.
Abelard suggests that rather than giving loci for syllogisms Boethius was proposing a way of providing evidence for non-evident perfect entailments by appealing to the principle that if an enthymeme is necessary then so is the corresponding syllogism. That is to say, Abelard accepts monotonicity, or weakening: “p ⊨ r / p, q ⊨ r” and its conditionalization: “p → r / ⊨ p&q → r.” Since he also accepts reflexivity he is thus committed to simplification: “p, q ⊨ p,” “p, q ⊨ q,” and conditional simplification “⊨ p&q → p,” “⊨ p&q → q”42 and he explicitly accepts these principles here.43
The problem for the logician is to show that the consequent of a putative entailment does in fact follow from the antecedent. In the case of “if something is a human being, then it is an animal,” for example, the habitude is assigned by noting that human being is a species of animal. The assignment alone, however, is not enough to prove the truth of the conditional. What is required in addition are principles which connect appropriate habitudes to true conditionals. These principles, according to Abelard, are the maximal propositions of topical theory.
In accordance with Abelard’s new definition of locus as the warrant for an entailment, Boethius’s division of locus into locus differentia and maximal proposition thus provides the two elements required in the proof of a true non-complexional conditional:
The locus differentia is that thing (res) in the habitude of which to the other consists the firmness of the consecution, so when we assert “if something is a human being, then it is an animal” human being which is posited in the antecedent in order that animal, which follows, is entailed, is brought forward in virtue of its being a species (of animal). The maximal proposition is a proposition containing the sense of many consequences which shows, according to the power of the same habitude, the common mode of proof which their differentiae have in them. As in the case of all these consequences: “if something is a human being, then it is an animal,” “if something is a rose, then it is a flower,” “if something is redness, then it is a color,” etc. . . . a maximal proposition such as the following is invoked: “of whatever the species is predicated, the genus is also predicated.”
(Dial. 263.7–18)
According to the theory that Abelard presents in the Dialectica, true conditionals may thus be proved in arguments with the following structure:
Maximal propositionAssignment of locus differentiaConclusion (a consecution)
An Abelardian proof of consecution and so of the truth of a conditional thus seems at first sight to be obtained from a Boethian topical categorical syllogism by conditionalizing and introducing the maximal proposition into the argument. Abelard will insist, however, that this is precisely what one cannot in general do. There are many valid enthymemes, he holds, which cannot be conditionalized to form true conditionals and the role played by the maximal proposition in the proof of a conditional is quite different from the role played by the maximal proposition in providing the argumentum to prove a categorical conclusion.
According to Abelard maximal propositions employed in the proof of conditionals must “contain the sense of many conditionals” and by means of the same differentia, or habitude, show that the antecedent follows from the consequent in each of the contained conditionals. So, for example, although the maximal proposition “of whatever the species is predicated the genus is predicated” is verbally categorical it is equivalent to the conditional: “if a species is predicated of something, then every genus of that species is predicated the same thing” (Dial. 263.18–20; cf. also 267.33, 317.26). Abelard argues that it is the thing (res) which appears in the proved consequences which is the locus differentia and it is this thing which is contained in the maximal proposition. This leads him to interpret the general terms appearing in maximal propositions not as the names of second order properties – “species” for example standing for the second order property of being a species – but rather as names of the res appearing in the proved consequence. “Species” here thus stands for human being as well as for all other species.
To serve in the proof of a consequence a maximal proposition must be true and so, since according to the Dialectica it is a conditional,44 it must itself satisfy the conditions to be met for the truth of such a proposition. The first condition noted by Abelard is a version of Boethius’s requirement that the truth of the antecedent is inseparable from that of the consequent. I will call this condition N, for necessity:
The sense of a hypothetical proposition lies in consecution, that is in that one thing follows . . . on another. (N) The truth of consecution lies in necessity, in that what is said (id quod dicitur) in the antecedent cannot be without that which is proposed in the consequent.
(Dial. 271.26–30)
This requirement of necessary association was not uncontroversial. I noted above that, according to Boethius, argumenta in dialectical arguments must be probable but that his examples of topical arguments are not always, or even typically, categorical syllogisms. Without Aristotle’s Topics to help them, twelfth-century logicians thus had to decide just where the necessity and probability of the argumentum are located. Abelard argues that they must be properties of the connection between the argumentum and the conclusion rather than of the argumentum considered in itself. Since a dialectical argumentum requires probability but not necessity it follows that in a local argument the loci differentiae and maximal propositions are required to guarantee only a probable connection between premise and conclusion.
Perhaps prompted by Boethius’s practice in De topicis differentiis, some of Abelard’s contemporaries apparently did not distinguish between the use of loci to warrant arguments and their use to warrant conditionals and allowed, in effect, that one may obtain a true conditional by the conditionalization of any locally warranted enthymeme. They thus held that the connection between the antecedent and consequent in a true conditional is probable but did not require it to be necessary (Dial. 271.38–272.1);45 and so, according to Abelard, identified truth with opinion since what is probable is what appears to be so to the appropriate audience.
Abelard easily finds authorities to support condition N against this probabilisitic account of the conditional and points out that it manifestly fails to be satisfied by many conditionals which the probabilisitic reading should concede (Dial. 271.35–272.25). For example, since it is more likely that a peasant will be thrashed than a soldier, the locus from the less likely warrants for the probabilist the easily falsifiable conditional “if a soldier is thrashed, then a peasant is thrashed” (Dial. 275.1–16). More compelling, and much more interesting, is the objection that the transitivity of entailment (i.e., “p → q, q → r ⊨ p → r”) allows one to infer conditionals which even the probabilist must reject from conditionals which he accepts. In particular, Abelard argues, the locus from immediate contraries has the greatest probability but nevertheless the conditional “if something is not well, then it is sick” is false since taken in conjunction with conditionals agreed to be true it entails the embarrassing impossibility (inconvenientia) “if something does not exist, then it exists.”
Abelard does not give the argument, but it is reported by other writers (cf. Introductiones Montane minores, 67) and can be reconstructed from what he says elsewhere in the Dialectica. I will call it the Embarrassing Argument Against Immediates (EI):
if something does not exist, then it is not well;
if something is not well, then it is sick;
if something is sick, then it exists; so, by transitivity,
if something does not exist, then it is sick, and
if something does not exist, then it exists.
(EI1) is warranted by appeal to the locus from part to whole for the predicate. It applies here since “does not exist” includes all and only things which do not exist while the extension of the infinite noun “not-well,” according to Abelard, includes both all non-existent things and all existing things which are not well. The former is thus contained in the extension of the latter. (EI2) is warranted by the locus from immediates with the maximal proposition “if one of a pair of immediates is removed from something, the other is predicated of the same.” (EI3) holds by the locus from part to whole for the predicate. Abelard insists that both (EI4) and (EI5) are impossible and entirely lacking in probability. What is true rather is (EI4*) if something does not exist, then it is not sick.
Abelard argues that the use of the locus from immediates leads to embarrassing problems here because it is employed as if it were completely general in its application whereas in fact immediates are exhaustive and exclusive only with respect to their proper subjects. In the case of being well and being sick this subject is a member of the genus animal. A qualification (constantia) indicating that we are concerned only with existing animals thus has to be included in some way in the conditional and the putative maximal proposition. Qualifications of this kind are marked, according to Abelard, with the Latin word cum. We have already seen that Boethius uses the word in his account of the conditional and Abelard cites the example of “when (cum) fire is hot, the heavens are spherical” to illustrate its temporal use. It may also, he says, be used to mark a condition, as in “if (cum) something is human, then it is an animal,” or a cause, for example “hang him, for (cum) he is a thief.”
There are three ways in which the qualification might be incorporated into the conditional. In the case of cum as a conditional connective we have the following:46 (i) “if something is an animal, then (if it is not well, it is sick)”; (ii) “if something is not well, then (if it is an animal, it is sick)”; and (iii) “if (if something is an animal, then if it is not well), then it is sick.” (i) is false, Abelard argues, because the antecedent is true, say, in the case of Socrates, but we have just proved with (EI) that the consequent is false. (ii) is false because the term “sick” applies only to animals but not to all animals, and the less general cannot follow from the more general; Socrates is an animal, for example, but he is not sick. (iii) is false, finally, because no affirmative categorical proposition follows from a conditional since the truth of the conditional entails nothing about the existence of the subject required for the truth of the categorical affirmation. Similar arguments hold against the interpretation of cum as a causal connective and against its use as a temporal connective – except in the case in which it is added to the conditional to qualify the antecedent “if (something is not well when it is an animal), then it is sick,” that is to say, “if at the same time something is both an animal and not well, then it is sick.”
Abelard, as I said, gives Boethius’s consequentia secundum accidens as his example of the temporal use of cum. He discusses temporal propositions at length in the Dialectica as part of his treatment of Boethius’s curious claims about compound conditionals, explaining that they are true if their components coincide in truth value at all times. If this is how he wishes us to understand the temporal use of cum as a qualification, however, there is an interesting problem with Abelard’s treatment of the locus from immediates. Construed as asserting an omnitemporal coincidence in truth-value, the antecedent “something is not well when it is an animal” is true only if being an animal coincides with not being well, that is, only if all animals are always sick. But this is false so there cannot be a sound argument from immediates.
What Abelard needs is a temporal connective true at a time just in case both components are true at that time. The proposition asserting omnitemporal coincidence in truth-value is simply the generalization of this connective over all times.47 In fact he seems nowhere to make this distinction but the related one between “as-of-now” and “simple” consequences will play an important role in later accounts of the conditional.
Abelard goes on to argue that one qualification to the antecedent is not enough to avoid embarrassment and that the appropriate form of the conditional is rather “if something is not well when it is an animal and every animal which is not well is sick, then it is sick,” an instance of the rule “if some one of a pair of immediates is removed from something and it remains one of the kind of things to which the immediates apply, then the other immediate is predicated of it.” This rule is not, however, a maximal proposition, because it includes the specification of the habitude which it warrants and so holds for all uniform substitutions (Dial. 404.15–26).
Since Abelard agrees with Boethius that the only candidates for true conditionals with a simple negation as antecedent and a simple affirmation as consequent are those whose predicate terms are related as immediates,48 his argument, he believes, shows that no such conditional is true.
With probabilism defeated Abelard concludes that necessity is required for consecution and proceeds to an examination of the relationship between conditional and categorical assertions. A conditional such as “if something is human, then it is an animal” expresses what Abelard calls a law of nature (lex naturae) which holds eternally and independently of the existence of humans and animals (Dial. 280.13). Categorical affirmations and separative negations, on the other hand, whether they signify mere inherence (de inesse) or its necessity, are true only if the extensions of their subject terms are not empty.
Earlier in the Dialectica Abelard famously distinguishes the de re from the de sensu account of the semantics of modal propositions. Read de re “every A is necessarily B” asserts of each existing A that it itself is necessarily B. This is false if A is any kind of created being, since God alone exists necessarily (Dial. 200.33–201.17). Read de sensu, on the other hand, it is the claim that the proposition “every A is a B,” or what it asserts, is necessarily, and so eternally, true, which is again false if A is any kind of creature. Furthermore, Abelard argues, the extinctive negation “no A is B” does not entail the conditional “if something is an A, then it is not a B.” The entailment obviously fails where “no A is B” may be true but is not necessarily so; but Abelard goes on to insist that it fails even where A and B are opposed, as, for example, man and ass, and “no A is B” is a necessary truth, since no conditional of the form “if something is an A, then it is not a B” is true.
For Abelard the truth of a conditional requires more than the inseparability of the truth of the antecedent from that of the consequent. Something more, that is, than the satisfaction of condition N. Although it is not possible for “every human is an animal” to be true and “if something is a human, then it is an animal” to be false, the latter is not entailed by the former. Furthermore, in general, Abelard claims, because consecution does not entail existence, the only categoricals which follow from conditionals are separative negations. Thus, although the conditional, “if something is human, it is not a stone,” is false, it entails “no human is a stone.” On the other hand, the only categorical which entails a conditional is one which states the habitude which warrants that conditional. For example, “animal is the genus of human” provides the warrant for the consecution “if something is a human then it is an animal” and so entails it (Dial. 283.12–19). To establish just when such an entailment holds requires, however, a “more subtle investigation”:
There seem to be two kinds of necessity of consecution. A broader kind, which is found where the antecedent cannot hold without the consequent. Another narrower kind, where not only can the antecedent not be true without the consequent but also of itself requires (exigit) the consequent. This latter necessity is the proper sense of consecution and the guarantee of immutable truth. As, for example, when it is said “if something is human, then it is an animal,” human is properly antecedent to animal since it of itself requires animal. Because animal is contained in the substance of human, animal is always predicated with human.
(Dial. 283.37–284.6)
One authority cited by Abelard for the distinction between necessities is Boethius’s observation in In Topica Ciceronis that “antecedents are such that when they are posited something else immediately (statim) follows of necessity” (In Cic. Top. 1123D–1124A). The reference to “necessity” has already provided evidence for condition N and Abelard now claims that the qualification “immediately” indicates that for consecution the sense (sensus) of the consequent must be contained in that of the antecedent. Curiously he does not mention the appearance of the same qualification in Boethius’s test for falsity of conditionals. We saw above that in his initial general characterization of entailment Abelard stipulated that the antecedent requires (exigit) the consequent and he repeats that here with the explanation that being animal is contained in being human. Elsewhere he says, among other things, that the antecedent contains the consequent, that the consequent is understood in the antecedent, and, in the case of perfect entailments, that the construction, or form, of the antecedent requires the consequent. Let us call this stricter requirement for the truth of a conditional condition R, for relevance.
Abelard characterizes his distinction as between two kinds of necessity, but in our terms what he is proposing is that something more than necessity is required for the truth of a conditional. Condition N already guarantees that there is no possible situation in which the antecedent is true and the consequent false. Condition R requires in addition that there be some genuine connection between them and the various glosses cover familiar suggestions about how to ensure this. The antecedent is required to be relevant to the consequent in that its truth is genuinely sufficient for that of the consequent and this is guaranteed by the consequent being in some way contained in the antecedent.49
While the satisfaction of condition R is required for entailment and the truth of a conditional, all that is needed in argument is that the truth of the premises guarantee the truth of the conclusion. It must not be possible for the premises all to be true and the conclusion false. The validity of an argument may thus be defined as the satisfaction of condition N alone:
Every argumentum is said to be necessary which is so connected with its conclusion, either by the nature of things or the property of terms or the construction itself, that things are not able to come about as the argumentum says without their being as the conclusion proposes. On the other hand, only that is necessarily antecedent which includes in its sense the sense of the consequent. Whence although all necessary antecedents may be necessary argumenta, the converse does not hold. For even if “Socrates is human” necessarily argues that he is not a stone, the former is not necessarily antecedent to the latter.
(LI Top. 309.13–22)
Thus, one cannot conditionalize a valid argument to obtain a true conditional and so the Deduction Theorem does not hold for Abelard’s logic, a feature which shocked his student John of Salisbury.50
The explanation of Abelard’s reference to two kinds of necessity is to be found in his theory of the relation of substances to their properties and accidents. In the Isagoge Porphyry introduces a distinction between separable accidents such as being seated for humans, and inseparable accidents such as being black for crows,51 and gives as the general definition of an accident that it is something which may be present or absent without the corruption of its subject. While separable accidents are features which are present at one time but not at another, inseparable accidents cannot actually be separated from their subjects. A property, for example the ability to laugh, or to learn geometry is, according to Boethius, an inseparable accident of all and only the members of a particular species, in this case humans.
To reconcile the existence of inseparable accidents with the general definition of an accident, Porphyry proposes that although such accidents cannot actually be removed from their subjects they may be removed in thought in the sense that, for example, we can conceive of, or understand, a crow which is not black. It is no part of the nature, and so no part of the definition of a crow that it is black even though a crow which is not black cannot exist. Abelard follows Porphyry in referring to such conceivability as an “ability” to be separated but conceivability certainly does not for him imply the possibility of being actual.52 Rather, the connection between being a crow and being black and the connection between being human and being able to laugh is that of condition N necessity.
Abelard notices in his discussion of inseparable accidents that although the conditional “if Socrates is a stone, then Socrates is a pearl” satisfies condition N, it is false since being a stone does not require being a particular kind of stone. In the Dialectica this feature of condition N provides another argument for condition R. If we required only inseparability for entailment, the conditional “if Socrates is a stone, then Socrates is an ass” would be true, since “what entirely cannot be, cannot be without the consequent” (Dial. 285.8–12). Abelard never announces as a principle that anything follows from an impossibility but he clearly sees that it holds if the satisfaction of condition N is all that is required for the truth of a conditional.53 He thus suggests that the requirement of inseparability, “this cannot be without that” cannot capture consecution because it is categorical rather than hypothetical (Dial. 285.5).
Abelard explicates condition R in terms of containment and in particular of the containment of understanding in understanding or of sense in sense. In De intellectibus, however, he notes that when we speak of the sense of a word we are using “sense” to mean “understanding” (TI 3; cf. also Dial. 582.16–20). To understand the appeal to sense as understanding in the formulation of condition R we must refer to his theory of meaning.
The basic units of meaning for Abelard are proper and general names. Finite forms of verbs are general names, which in addition to naming, serve to indicate time and propositional combination. Names acquire their meanings in acts of baptism, or imposition, in which an “impositor” introduces names with injunctions of the form “let this be named P,” indicating the individual to be so named, or of the form “let items of the same kind as this, or these, be named N” (e.g. LI De in. 3.10.84, G 460.27–35), indicating a representative, or representatives, of the kind.54 When a name acquires its reference either to a single individual or to all the individuals of a kind, it also acquires its sense. For with imposition a causal association is established between the utterance of a name and the occurrence of acts of understanding (intellectus) in the minds of the impositor and his audience.
Abelard maintains that general names signify understandings – that is, acts of intellectual attention (attentiones), which are actualizations of the power of rational discernment possessed by humans. In the Logica “ingredientibus” he maintains that these acts of understanding are directed at suitable images constructed by the mind (LI De in. 3.01.78, G 322.12–14);55 but in the De intellectibus the role, if any, played by image in understanding is much less clear. In both works, however, Abelard argues that an utterance of the word, homo, in the presence of a Latin speaker who understands it causes him to think about what it is that makes a human to be a human by directing his attention at only these features. As Abelard puts it: someone who understands homo understands mortal, rational, animal.56 That is, on hearing the term homo he directs his intellectual attention at human nature.
The understanding of “human” is simple in that it is not constructed from temporally distinct acts of understanding but nevertheless includes, or contains, as its parts the understandings of animal, rational, and mortal. To understand the expression “mortal, rational, animal” on the other hand three successive acts of intellectual attention are required.57
For Abelard, then, one cannot understand “human” without understanding mortal, rational, and animal. It is impossible intellectually to separate any of the latter features from a human being and in this sense being human requires being mortal, rational, and animal. Just the same account holds for proper names. However, the only features of Socrates which cannot be separated from him with the understanding are all and only those required for him to be human. The sense of the proper name “Socrates” is thus just that of the definite description “this human” (LI Isag. 49.12–26)58 and so the conditional “if something is Socrates, then it is human” is true. Humans are essentially mortal, rational, animals and Socrates is essentially human. The satisfaction of condition R is a consequence of Abelard’s semantics for names and his essentialism.
The signification of a conditional proposition as a whole is produced, according to Abelard, by a series of acts of understanding. Acts of conjoining in the case of affirmation, and acts of disjoining in the case of separative negation, produce understandings of the antecedent and consequent from the understandings of their subject and predicate terms. A further intellectual act associated with the conditional sign “si” produces the understanding signified by the complete proposition.59 This is, of course, only a part of theory of the meaning of propositions and the rest has to be supplied with accounts of truth-conditions and of force (LI De in. 3.04.12ff., G 365.13ff.).
The conditional “if Socrates is a pearl, then Socrates is a stone” – a conditional which satisfies condition R because a pearl is a kind of stone – uttered assertively will be true just in case if things are as the antecedent says they are, then they are required to be as the consequent says they are. The necessary connection which must hold for the truth of a conditional is, Abelard argues, not a relationship between the words spoken in uttering the antecedent and consequent, nor one between the occurrences of the understandings that they signify, but rather a relationship between the states-of-affairs which, if uttered assertively, they indicate to be so – a relationship, that is, between what Abelard sometimes calls the dicta of the corresponding propositions.60 This connection holds, he insists, independently of the existence of the conventions of imposition and independently of the existence of individuals of the kinds in questions. Even the creator is bound by entailment.
Abelard maintains that to understand a general kind term is to understand the nature of that kind (cf. esp. TI 94). He acknowledges, however, both that the differentiae of a species may not be directly accessible to sense and that the original impositor may have no notion of the nature of the thing for which he introduces a name. What is important are his intentions:
Each name of every existing thing insofar as it can generates an understanding rather than an opinion, because their inventor intended to impose them in accordance with some natures or properties of things, even if he did not properly know how to think out the nature or property.
This suggests that original imposition fixes the reference of a general term as all and only the individuals of a kind, but that initially at least the associated mental actions may amount to no more than confused conceptions of the surface features of the kind. Abelard calls such confused conception “imagination” and notes that it marks the beginning of understanding (LI De in. 3.01.44, G 317.8–21). He barely hints at it, but his theory of understanding thus seems to allow progression from a pre-scientific acquaintance with its immediately sensible properties to a full understanding of a natural kind. Although he does not say so, Abelard could it seems maintain that the signification of a general term is fixed as an act of understanding but changes as the understanding comes more and more to discern the nature of the kind.62
The work required to establish the truth or falsity of conditionals is, as Abelard points out, just the work required to establish the meaning of general kind terms and this he claims is a cooperative enterprise involving both the dialectician and the natural scientist (Dial. 286–287). Abelard appeals to the method of division to obtain definitions (Dial. 591.5–8), but he tells us nothing about the procedures employed by natural scientists in investigating the “properties of things” (Dial. 286.30–287.5). The problem is, as he points out, that the dialectician has to distinguish between conditionals such as “if something is human, then it is able to laugh,” “if something is human, then it is not an ass,” and “if something is human, then it rational.”63 The first two are false and the third is true but in each case the consequent follows from the antecedent with condition N necessity and so no physical investigation can distinguish between them.64 In practice Abelard proceeds by producing arguments to show that a given maximal proposition does not yield conditionals satisfying condition R, so let us turn now to some of these.
Redefining the classical terms once again, Abelard divides loci into intrinsic which according to him have to do with inherence, extrinsic which have to do with various kinds of difference, and mediate which involve both relationships. Among the intrinsic loci there is a distinction between those having to do with substances considered in themselves and those having to do with features which “follow substance.”65
The loci from substance concern the various forms of definition (Dial. 331.7–11) and there are eight candidate maximal propositions connecting definition and defined term which might yield true conditionals. There are four with both antecedent and consequent affirmative, with the definition either in the antecedent or consequent and as either subject or predicate. And there are four more which are the contrapositives of these.
Abelard argues that for definition in the strictest sense, by genus and differentiae, only four of the rules guarantee that condition R is satisfied. These are the rules “of whatever the defined term holds the definition holds,” and “whatever holds of the defined term holds of the definition” and their contrapositives. He proves that the remaining maximal propositions do not satisfy condition R both directly and by showing that embarrassments would follow if they were accepted. In doing so he reveals a little more about his theory of definition and its role in his philosophical semantics.
The rule that “of whatever the definition is predicated the defined term is predicated” has as an instance “if Socrates is a mortal rational animal, then Socrates is human.” This conditional is false, according to Abelard, since rationality and mortality are not alone sufficient to constitute a human being from the genus animal. Rather, every substantial form of a human is required, since “there is no superfluity in nature,” and there are, according to Abelard, many such forms. Bipedality,66 for example, and the ability to walk, and many others for which we have no names (Dial. 332.11–13)!
The indirect argument to embarrassment appeals to the same fact about the nature of humans but relies on the logic of the syncategorematic term “only” (tantum) which will be of much interest to later logicians but of which Abelard, again, provides the earliest published treatment. His argument is absolutely typical of hundreds that fill twelfth-century logic texts. If the conditional warranted by the locus from definition as predicate were true, we could argue that if something is a human if it is an animal informed with rationality and mortality, then it is human if it is an animal informed only with rationality and mortality. But then it would follow, by the locus from definition, that if it is an animal informed only with rationality and mortality, it is informed with bipedality – which is false. The last conditional holds because “all the substantial forms of a human being are understood in the name ‘human’” (Dial. 332.26–27).
Abelard agrees that “human being” and “mortal rational animal” are cointensive, that is to say equivalent with condition N necessity. His embarrassing argument with “only” shows that substitution of condition N equivalents into contexts which it governs is not truth-preserving. Condition R equivalence, on the other hand, preserves truth for substitutions into opaque contexts. In the modern jargon, its logic is hyperintensional since it distinguishes between concepts whose extensions coincide in all possible worlds:
The defined word and the definition name (notant) just the same thing; “human” and “mortal rational animal” are imposed on the same thing and said of the same thing; but they do not indicate the same thing under the same guise. For in “human” all the differentiae are bound to be understood; in the definition only two are apposed.
(Dial. 334.25–39)
Abelard is famous for the confrontations which punctuated his life, such as that with William of Champeaux over universals and that with Bernard of Clairvaux over theology. Less well known but, I think, much more important for the long-term development of philosophy is his confrontation with Alberic of Paris in the 1130s over the principles of inference. Their dispute about the locus from opposites was crucial for the history of logic, but Abelard and Alberic also disagreed interestingly over the locus from a universal whole.
Abelard calls such wholes “general wholes” in the Dialectica and maintains that the maximal propositions that “from whatever the genus is removed the species is removed” and that “whatever does not hold of the genus does not hold of the species” satisfy condition R. During Abelard’s last stay in Paris the second of these became the object of controversy in the more general form “whatever is removed from a universal whole universally, is removed from each part.” The rule seems to support the conditional “if no body is made by Socrates, then no knife is made by Socrates.” In his discussion of the locus from efficient cause in the Dialectica, however, Abelard insists on the difference between creation, which is the work of God alone, and the activities of human beings, which involve combining into accidental unities what has already been created; humans make knives but only God can make a body (Dial. 417.23–28).
Abelard’s response to the apparently embarrassing conditional is reported in two treatises written by a follower, or followers, of Alberic.67 We are told that he maintained that the conditional is false and argued, again in a way that anticipated later developments, that the meaning of the subject term affects that of the predicate and that conditional must be expounded as “if no body is made to be a body by Socrates, then no knife is made to be a knife by Socrates” – which is false but not an instance of the maximal proposition which Abelard accepts.68
To Alberic’s response that Socrates today combined bone and iron to make a knife and so a body which did not exist yesterday Abelard is reported to have replied that:
he [Abelard] and the Queen of France were one body and likewise he and the Appenine mountains because he did not wish to concede that this body did not exist yesterday. Indeed he said that the horn which is on the head of a cow and the iron from which there will be a knife, are already one body before they are conjoined, and likewise himself and the Queen of France.
(Introductiones Montanae maiores 69ra)69
Unfortunately no such flippant answer was available to the most important of Alberic’s objections.
Abelard argues that eight principles govern the propositional relationship of antecedence and consequence expressed in a true conditional (Dial. 288.23–34). The first two are familiar, modus ponens, “p → q, p ⊨ q” and modus tollens, “p → q,¬ q ⊨ ¬ p”. The reductio proof of modus tollens shows at the same time that if a conditional is true, then its contrapositive is true. We can prove that a conditional satisfies condition R if from the hypothesis that its antecedent is true and its consequent false we can derive a formal contradiction using only inferences which satisfy condition R (Dial. 289.3–23). That is, if we can derive both “q” and “not q” for some “q”. The derivation of an impossibility which is not a formal contradiction will not do since this would show only that the antecedent in question is inseparable from the consequent with condition N necessity.
The remaining six principles exclude all other inferences from a true conditional and the negation or affirmation of either its antecedent or its consequent to the affirmation or negation of the other. They thus exclude the inference from a conditional of anything apart from its contrapositive. Two of these principles stand at the center of Abelard’s logic and provide, as it were, the rules of proof corresponding to the semantics of containment. The first of them is Abelard’s version of Aristotle’s Principle, noted above, that the same cannot follow both from something and its opposite. Abelard’s version is propositional and properly represented as: “not {(p → q) & (¬p → q)}” (Ar1). What we may call Abelard’s first principle may be represented as “not {(p → q) & (p → ¬q)}” (Ab1). Their proofs are by reductio:70
| (Ar1) | (Ab1) | ||
|---|---|---|---|
| (1) p → q | Hypothesis | (1*) p → q | Hypothesis |
| (2) ¬p → q | Hypothesis | (2*) p → ¬q | Hypothesis |
| (3) ¬q → ¬p | 1, Modus Tollens | (3*) ¬q → ¬p | 1, Modus Tollens |
| (4) ¬q → q | 3, 2, Transitivity | (4*) p → ¬p | 3, 2, Transitivity |
(4) and (4*) cannot possibly be true since:
no one doubts them to be embarrassing, or inconsistent, because the truth of one of a pair of dividing propositions not only does not require the truth of the other but rather entirely expels and extinguishes it.
(Dial. 290.25–27)
That is, “not (¬p → p)” (Ar2) and “not (p → ¬p)” (Ab2).
The principles (Ar1), (Ar2), (Ab1), and (Ab2) to which Abelard commits himself here have become known as connexive principles. The associated intuition regarding negation is sometimes called the deletion, or cancellation, theory.71 According to it, the negation of a proposition cancels its content. The conjunction of a proposition with its negation has no content at all and so nothing follows from it if following requires containment.
Abelard appeals to the connexive principle (Ab2) in his argument to show that the locus from opposites does not warrant true conditionals. The maximal proposition at issue is that “of whatever one opposite is asserted the other is removed” (Dial. 394.4–19). Applied to the disparate opposites human and stone this yields “if Socrates is a human, then Socrates is not a stone.” Abelard argues against it with what I call the Embarrassing Argument from Opposites (EO):72
If Socrates is a human and a stone, then Socrates is a stone.
If Socrates is a human and a stone, then Socrates is a human.
If Socrates is a human, then Socrates is not a stone.
If Socrates is not a stone, then Socrates is not (a human and a stone); so
If Socrates is a human and a stone, then Socrates is not (a human and a stone).
(EO1) and (EO2) are applications of the principle of conditional simplification which, as we saw above, Abelard accepts and indeed seems to be a paradigm for an inference warranted by the containment of the consequent in the antecedent. (EO3) is the conditional whose truth is at issue. (EO4) follows from (EO1) by contrapositon and (EO5) follows from (EO2), (EO3), and (EO4) by transitivity.
The argument shows, Abelard holds, that no conditional of the form “p → ¬q” can be true and, with (EI), that no conditional of mixed quality can be true. We can see now why Abelard cannot accept the double negation equivalence “p ↔ ¬¬ p”
| (1) p → ¬¬p | Hypothesis |
| (2) (p & ¬p) → ¬p | Simplification |
| (3) (p & ¬p) → p | Simplification |
| (4) (p & ¬p) → ¬¬p | 3, 1, Transitivity |
| (5) ¬¬p → ¬(p & ¬p) | 2, Contraposition |
| (6) (p & ¬p) → ¬(p & ¬p) | 4, 5, Transitivity |
and likewise for the other conditional.
Since, according to Abelard, there are no negative substantial forms, the definition of a natural kind cannot contain a negative term and so Abelardian connexive logic allows one to make inferences from impossible hypotheses such as that Socrates is Browny, an ass, or that a man is an ass, without risk of arriving at a formal contradiction. In the decades after Abelard’s death this kind of inference was formalized in the procedure known as the obligatio of impossible positio and put to use in theology.73
Unfortunately Abelard’s various intuitions about the propositional connectives cannot be reconciled. The deletion account of negation is not compatible with simplification; hence modern connexive logics thus give up at least the conditional version.74 More generally connexive logic is non-monotonic but, as we saw above, Abelard accepts monotonicity.
It was Alberic of Paris who first noticed the problem and produced against Abelard the most embarrassing of all twelfth-century arguments. Abelard accepts as a paradigm of true conditionals “if something is human, then it is an animal.” But as reported, for example, in Introductiones Montanae minores (65–66), Alberic argued by simplification that:
But this is in contravention of (Ab2).
One source tells us that this argument was too much for Abelard and that he simply accepted the conclusion.75 Though he was an old man nearing the end of his eventful life, this seems quite out of character and another source indicates that he did see what might have been a way out and suggested that the principle of simplification needs to be qualified.76 Unfortunately he seems never to have developed this suggestion and, although his followers, the Nominales, remained faithful to the principles of Abelardian connexivism, Alberic’s argument provoked a crisis in the history of logic that was finally resolved with the general acceptance of condition N as providing both the truth conditions for conditional propositions and definition of validity.77 A place remained for condition R, however, since in answering certain philosophical and theological questions, hypotheses acknowledged to be impossible had to be made. In reasoning about such a hypothesis the principle that anything follows from an impossibility, which was recognized explicitly as characteristic of the logic of condition N in the 1150s,78 had to be suspended. In such cases only inferences satisfying condition R were permitted.79 Philosophers continued for the next one hundred and fifty years to work with logical tools developed by Abelard though certainly without realizing that they were doing so. It was only at the beginning of the fourteenth century that logicians appeared whose work could compare with Abelard’s and who were able to rethink the theory of entailment.
1. Abelard observes that he refers to the science in question indifferently as logic (logica) or dialectic (dialectica) (LNPS 506ff.). Logic is the science of arguing which, together with physics and ethics, is one of the three parts of philosophy. Logic is also an instrument of philosophical research and in particular logic is employed in answering questions about logic. Cf. LI Isag. 1.14–25.
2. Limitations of space also prevent me covering other topics in logic to which Abelard makes an outstanding contribution, in particular his theory of modality and the modal syllogism and his treatment of the logic of future contingents. For a discussion of the first, see Martin 2001.
3. The most significant of which is the Dialectica of Garlandus, a radically nominalist treatment of logic roughly contemporary with Abelard’s own Dialectica. Cf. de Rijk 1959.
4. Propositio in Boethius and Abelard means a propositional token.
5. In his LI Top. 243.7–16, Abelard argues that this proposition is not in fact maximal since it does not contain a locus differentia.
6. Cf. De top. diff. 2.1185–1186A: “Universal and maximal propositions are thus called loci because they contain the other propositions and through them a consequent and confirmed conclusion.”
7. Hypothetical syllogisms of, e.g., the form “if A is, B is, if B is, C is; therefore if A is, C is.” Boethius does not have a special name for this type of syllogism but he curiously appeals to it in De hyp. syll. 2.9, to demonstrate the “imperfect” syllogisms of the form “A is, if A is, B is, if B is C is; therefore C is.”
8. Cf. De hyp. syll. 1.2.4–5: “[The hypothetical premises] take the warrant for their proper consequences (vis propriae consequentiae) from categorical . . . syllogisms. For if there is doubt about whether the first [conditional] premise of a hypothetical syllogism is true, it will be demonstrated with a predicative conclusion.” Boethius gives no hint of a proof procedure but if he knew of one it was presumably some form of conditionalization – “p, q ⊨ r / p ⊨ q→r.”
9. That’s not to say that Boethius had a clear understanding of the distinction between thing, state-of-affairs, and fact, for all of which he uses res. Far from it. See the remarks on propositionality, §§I.2 and II.1 below.
10. In his LI Top., 228.20–24, Abelard notes that we may falsely predicate terms of lesser extension of those of greater extension and that he refers to true predications as “regular.”
11. Cf. De hyp. syll. 1.2.2. In De hyp. syll., but not in his other works, Boethius uses the schema “si est A, est B” where “A” and “B” are term variables.
12. The translation has to be varied accordingly.
13. The equivalence of “if A is, B is” to “if B is not, A is not.” Boethius doesn’t use the term “contraposition” in this sense but rather for the formation of “every non-B is non-A” from “every A is B.”
14. Cf. Iwakuma and Ebbesen 1992.
15. E.g. “If something’s a man, it is not a horse.” Such conditionals only hold for predications of the same subject.
16. E.g. “If it is not day, it is night,” “if it is not dark, it is light.”
17. In his commentary on Cicero’s Topica, Boethius claims that the discovery of an argumentum is simply the discovery of the middle term in a syllogism.
18. Here, as elsewhere, Boethius apparently intends a claim about a syllogism or enthymeme but uses a conditional construction. Cf. De top. diff. 2.1188C.
19. Cf. Martin 1991, 304.
20. According to Priscian, Inst. 16.2, a conditional signifies a consequentia rerum. Boethius, however, uses “consequence” both for the conditional and for what it signifies.
21. Boethius has nothing to say about compound disjunctions.
22. I say “implies” here because although Boethius claims that this distinction is a distinction between the use of cum and si, his examples of each kind of conditional are all formed with cum.
23. Though he does put them to use elsewhere. Cf. Martin 1999. In In Cic. Top. 1165A, he makes apparently the same distinction in terms of consequences which are substantial and those which are accidental.
24. The name is due to Stors McCall (1966) on the basis of Sextus Empiricus’s remark that “those who introduce the notion of connexion say that a conditional is sound in which the contradictory of its consequent is incompatible with its antecedent.” (Translated in Kneale and Kneale 1984, 129.)
25. When he is being very careful Abelard contrasts assertion with other speech acts as “proposing assertively what is true or false.” LI De in 3.05.17, G375.29–35. Cf. Dial. 151–33.05.
26. According to Abelard, provided the term “B” standardly applies to existents, the truth of both “A is B” and “A is not B” requires the existence of A, if “A” is singular and of some As if it is quantified as “every S” or “some S.” Propositions such as “Homer is a poet” and “a chimaera is something which can be thought about” are non-standard. A consequence of Abelard’s account of negation is that while for any individual A, “A does not exist” is necessarily false, “A exists” is only contingently true if A is a creature and divides truth and falsity with “not (A exists).”
27. Abelard does not, it seems, explicitly make the distinction between separative and destructive negations for singular propositions. It is attributed to him, however, in the Glossa doctrinae sermonum, MS Paris BN lat. fol. 15015 ff.180ra01–199ra46, 187va20–188va38. Where Aristotle appears to claim that a separative negation is true if its subject fails to exist, Abelard must interpret him as referring to destructive negation. Cf. LI De in. 3.06.29, G 396.4–17.
28. Following Boethius, In De in. maior 7.146, Abelard reads “nullus A” (no A) as “non ullus A” and this as “not one A.”
29. William Kneale (Kneale and Kneale 1984, 210–211) puzzles over Abelard’s claims about “No A is B” in the Dialectica but does not notice that the text has him identifying it with both “Every A is not B” for the truth of which it is necessary that some As exist and with “Not some A is B,” which is true if there are no As, as well as insisting that it is not equivalent to the latter! In LI De in. 3.07 (G 401.8–414.27), this confusion is absent. Abelard notes that some people identify “No A is B” with the separative negation “Every A is not B” but he rejects this. “No A is B” is equivalent, he insists, to the extinctive negation “Not (some A is B).”
30. Abelard’s claim was controversial. The author of Glossa doctrinae sermonum, writing no earlier than the late 1130s, after fairly stating Abelard’s position, rejects the claim that copulative conjunction forms a single proposition: “Whence we do not agree with this theory but judge that Boethius is to be followed.” (MS Paris BN lat. ff. 15015 ff. 196va.)
31. Cf. Anderson and Belnap 1975, vol. I, §1.4, 23 and generally Routley, Meyer, Plumwood, and Brady 1983, vol. i.
32. Abelard sometimes also uses consecutio and inferentia to mean a true conditional.
33. That is, the conditional proposition whose antecedent is the conjoined premises and whose consequent is the conclusion.
34. Abelard apparently quotes from the Prior Analytics on two other occasions to give Aristotle’s definition of perfection – discussed below – and to note that he speaks of “inherence” where Abelard and his contemporaries refer to “predication” (cf. Dial. 239.20–27).
35. Cf. Dial. 232.6–8, referring to Prior Analytic 1.1, 24b21: “I mean by ‘from their being so’ to fall out through them; and by ‘to fall out through them’, that nothing extrinsic to the terms is required for this to come about necessarily.”
36. Abelard’s example has “if every human is a stone and every stone is a wood, then every human is a wood” obtained from “if every human is an animal and every animal is animate, then every human is animate.”
37. Proust 1989, ch. 3.
38. The examples given here, as often in the Dialectica, seem to have become corrupted but the rejection of repetition is explicit and repeated.
39. Abelard allows as canonical, however, figures mentioned by neither Aristotle nor Boethius. For example: “Some A is B and the same B is C; therefore some B is C” has a complexio syllogismi (Dial. 320.27–321.11). Abelard also has syllogisms which mix non-modal and modal categoricals, syllogisms with tensed propositions, and syllogisms which mixed general and singular categoricals.
40. Later twelfth-century writers follow Abelard, LNPS 508.9–15, in distinguishing between complexional and local arguments.
41. Cf. Iwakuma and Ebbesen 1992.
42. Since (1) p ⊨ p (reflexivity), (2) p, q ⊨ p (1, monotonicity), etc.
43. See Dial. 260.28–33: “Two propositions taken together are antecedent to each one of them.”
44. One of the differences between the Dialectica and the Gloss on the Topics is that in the latter Abelard asserts that maximal propositions are categorical. Cf. LI Top. 239.1–8.
45. Probably referring to William of Champeaux, Abelard notes that his own Master held this view.
46. Cf. Dial. 403.12–18; Abelard did not have the apparatus of parentheses to indicate scope.
47. In the qualified conditional connecting immediates the generalization over time includes the whole conditional in its scope.
48. That is a conditional whose antecedent and consequent contain no propositional connectives – separative negation is not a propositional connective.
49. Cf. Routley, Meyer, Plumwood, and Brady 1983.
50. John of Salisbury, Metalogicon iii.6: “I am amazed that the Peripatetic of Pallet so narrowly laid down the law for hypotheticals that he judged only those to be accepted the consequent of which is included in the antecedent . . . indeed while he freely accepted argumenta, he rejected hypotheticals unless forced by the most manifest necessity.” See Martin 1987b.
51. Apparently Abelard would have held that albino birds aren’t crows. Cf. LI Cat. 128, where Abelard says “we know all crows are black.”
52. Cf. LI Isag. 91.11–16: “Granted that in a certain way a human being cannot exist without the ability to laugh, because, that is, it cannot come about that there is a human being who is not able to laugh. In another way it can come about, as Porphyry holds that a human being may be without a property, that is, understood negatively as follows: the property is not required by being human.”
53. We first hear of the principle and of a proof by William of Soissons in the Metalogicon of John of Salisbury, written in 1159. See Martin 1986.
54. Adam was the original impositor. Cf. Hex. 127, on Genesis 2:19, where Abelard describes Adam as “first inspecting the natures of things, which he would afterwards provide words to designate.”
55. Abelard holds that when the named thing is present intellectual attention is directed at it rather than at an image, but it is not clear to me that he intends to claim this for general names and indefinite descriptions as well as for proper names and definite descriptions. Cf. chapter 6.
56. Cf. TI 39: “Someone who hears the name ‘animal’ attends simultaneously to these three: body and animation and sensibility, as conjoined in the substance of animal.”
57. Cf. TI 33–35, esp. 34: “For the name ‘human’ simultaneously determines the matter, animal, rationality, and mortality, and all are understood at once in the name and not by succession. And perhaps there are several simultaneous actions in one understanding of a simple significant word, following upon the soul’s conceiving several things, so that there is one action for each thing it deliberates upon.”
58. Cf. also Dial. 332.32–333.15: “it follows from the imposition of ‘Socrates’ that he is a mortal, rational, animal.”
59. Cf. LI De in. 3.01.118, G 330.18–26: “in ‘if Socrates is a pearl, then Socrates is a stone’, the force of the conjunction brings an act of attention to bear on the whole consequence with a certain part of the understanding, which necessarily conjoins this with that. This attention is a third act which with the actions of the two [component] propositions composes the action of a single understanding.”
60. Abelard seems never to say that one dictum requires or does not require another but he does speak this way about status. Cf. LNPS 561.20–28.
61. Cf. also Dial. 595.25–31: “Granted, moreover, that the impositor did not distinctly understand all the differentiae of human, he intended, nevertheless, that the word [‘human’] to be taken for all of them as he confusedly conceived them. Or if he imposed the name [‘human’] only for certain differentiae, the sense (sententia) should consist only of them, and in accordance with them the definition of the sense should be assigned.”
62. But cf. Dial. 583.32–34: “many, because they know the signification of the substance of the name ‘human’ but do not adequately perceive from the name the [substantial] qualities, require the definition only in order to have a demonstration of the qualities.”
63. Cf. Dial. 285.16–286.33: “When it is proposed ‘if something is body, it is corporeal,’ and ‘if something is a body, it is colored,’ although the same substance of the body which is corporeal is colored, and whatever is colored is corporeal and vice versa, so that there is in reality no difference (distantia) between a substance which is informed with color or constituted with corporeity, the first assertion is, nevertheless, true and the second false.”
64. Cf. Dial. 284.24–49: “It is clear that the consequent is not contained in the antecedent of the consequences ‘if something is human, then it is not a stone’ and ‘if there is paternity, then there is filiation’, and that the senses of the consequents do not hold on account of the senses of the antecedents, but rather that by our discernment of the nature and cognition of the property of the nature, we are certain with the antecedent of the consequent. That is, because we know the nature of human and stone to be so disparate that they cannot exist together in the same thing at the same time.”
65. Abelard tries for a principled distinction here, but classical loci do not fit particularly well into it. As the loci most useful to dialecticians, Abelard gives the following: (1) Intrinsic, (a) from substance: from definition, from description, from interpretation; (b) from what follows substance: from genus, from integral whole, from parts, from equals, from predicate or subject, from antecedent or consequent. (2) Extrinsic, from opposites: from relative things, from simultaneous things, from what is prior, from contraries, from privation and habit, from affirmation and negation. (3) Mediate: relatives (also extrinsic), integral whole and part (also intrinsic), exceeding and exceeded. Cf. Dial. 413.1–35.
66. On the curious problems presented by bipedality as a differentiae, see my discussion of Abelard on amputees in Martin 2001.
67. Cf. the Introductiones Montanae minores and the unpublished Introductiones Montanae maiores, Paris BN lat. 15.141, ff. 47r–104r.
68. Introductiones Montanae minores, 49–52; Introductiones Montanae maiores, 69rb–69va.
69. The Introductiones Montanae minores has the King of France rather than the Queen!
70. Cf. Dial. 290–292, which makes explicit all the steps in the argument. The text of the Dialectica is rather corrupt at this point.
71. Cf. Routley, Meyer, Plumwood, and Brady 1983, 2.4.
72. Cf. esp. Dial. 395.6–35. I have reconstructed the full argument with the help of Introductiones Montanae minores, 63–64.
73. Cf. Martin 1992a and Martin 1998.
74. Cf. Routley, Meyer, Plumwood, and Brady 1983, 2.4.
75. In the commentary on De hyp. syll. in MS Berlin lat. f. 624, quoted in de Rijk 1966, 57.
76. Introductiones Montanae minores, 66. “They said that ‘if Socrates is human and Socrates is not an animal, then Socrates is not an animal’ does not hold because a negation is not so powerful (vehemens) when joined with an affirmation as it is when it is alone, and something follows from a negation alone which does not follow from it when it is conjoined with an affirmation.”
78. Cf. Martin 1986.
79. Cf. Martin 2000.