Entailment and Truthmaking: The Consequentia Rerum from Boethius to the Ars Meliduna

Abstract

In Categories 12 (14b11–22), Aristotle famously claims that [1] true sentences and reality stand in a mutually implicative relationship, and that [2] reality causes the truth of sentences but not vice versa. In this paper, I first argue that Boethius’ reading of the above passage led medieval logicians to assess [1] and [2] within the framework of a theory of consequence. Then, I consider two important questions raised by Boethius and later logicians in relation to [1] and [2], and, namely, whether the implication described in [1] really satisfies the requirement for consequence, and whether the truthmaking relation posited in [2] can be boiled down to the relation of consequence. Boethius himself, I argue, answered affirmatively to both questions and thus (much like some contemporary truthmaker theorists) committed himself to an account of truthmaking as entailment. I then consider criticisms to Boethius’ account found in the writings of William of Champeaux, two anonymous authors connected to the school of Jocelyn of Soissons, Peter Abelard, and the author of the Ars Meliduna. Here, I argue that the emergence of a concern for relevance drives the need for abandoning Boethius’ account as well as his conception of truthmaking as entailment.

KEYWORDS:

• Truthmaking
• entailment
• Boethius
• Abelard
• Ars Meliduna

1. Introduction

Reflection on consequence (‘consequentia’) – the relation that holds between the antecedent and the consequent of a true conditional – occupies a central place in twelfth-century logic. Beginning with Garlandus and Abelard, and up to the later treatises by the so-called ‘logico-theological schools’, several accounts of consequence were developed in order to assess the truth of conditional sentences. Alongside this foundational interest, however, the theory of consequence also provided twelfth-century logicians with a tool to unpack specific views contained in authoritative texts. A glaring illustration of this phenomenon, and one which will be explored in some detail in this paper, is found in the exegesis of Aristotle’s Categories 12 (14b11–22) – a locus classicus of the Aristotelian conception of truth.

Here, Aristotle introduces the last of five types of priority, which he calls ‘priority by nature’ (πρότϵρον τῇ φύσϵι). As he goes on to explain, priority by nature is a relation that holds of pairs of items for which each one converts with the other as to the implication (ἀκολούθησις) of being, and one is the cause (αἴτιον) of the being the other, but not conversely. Aristotle famously regards the relation that holds between a thing (πρᾶγμα) and a true sentence (λόγος) about it as an instance of priority by nature. For instance, ‘there being a man’ (τὸ ϵἶναι ἄνθρωπον), Aristotle remarks, converts as to the implication of being with the true sentence about it. But whereas the former is the cause of the being of the latter, the converse does not hold.¹ All this commits Aristotle to two crucial claims about truth and reality, and namely that [1] a true sentence and the portion of reality corresponding to it stand in a relationship of mutual implication; and also that [2] the truth of sentences is caused by reality, but not vice versa – a claim which has led David Armstrong to believe that ‘the notion of the truthmaker may be traced back to Aristotle’.²

As I shall argue in what follows, it was Boethius who first (and more or less justifiably) thought that the theory of consequence provided the correct theoretical framework to unpack Aristotle’s [1] and [2]. Following on the footsteps of Boethius, twelfth-century logicians were prompted to ask Aristotle two crucial questions. The first question is whether the implicative relation described in [1] is at all different from the relation of consequence that is expressed in a true conditional proposition. The second question (one which is very dear also to truthmaker theorists like Armstrong) is whether the reality-to-truth relation described in [2] (what Armstrong calls the ‘truthmaking relation’) can also be boiled down to the relation of consequence.³

The answers which Abelard and his contemporaries provided to the two abovementioned questions represent one of the most fascinating chapters in the history of twelfth-century logic. Before weighing in on this debate, however, let me first examine how Boethius went about connecting the text of Categories 12 to the issue of consequence.

2. Boethius

The medieval reception of Categories 12, 14b11–22 is deeply influenced by the mediation of Boethius. Two elements in particular make Boethius’ understanding of this passage relevant for later logicians. The first element has to do with the way in which Boethius conceives of the implication (henceforth [TR]) that goes from the truth of a sentence to the relevant thing (res) or better state of affairs;⁴ the second element, and arguably the most important one, concerns the theoretical framework in which Boethius places Aristotle’s mutual implication. Let us examine these two in order. Boethius’ characterization of [TR] differs from Aristotle’s in one crucial respect. For whereas Aristotle simply claims that a true sentence about something entails that thing, Boethius maintains that a true sentence about some state of affairs entails that ‘it is necessary’ (necesse est) that the state of affairs obtains. Or, to put it schematically ${\text{T}}^{\prime }{p}^{\prime }\to ◻\text{ that-}p$ where ‘T’ signifies the truth-predicate, ‘p’ is a placeholder for a declarative sentence, and ‘that-p’ stands for the relevant state of affairs. What is even more striking, the phrase ‘necesse est’ occurs not only in Boethius’s commentary on the Categories but also as an addition (!) in his Latin translation.⁵ From a textual point of view, Boethius’ addition of ‘necesse est’ is of course completely unjustified. From an exegetical point of view, however, one can make very good sense of it by pointing to other passages of the Aristotelian corpus that are concerned with the very same issue. More precisely, I contend that Boethius’ insistence on the necessitas consequentis of [TR] is motivated by what Aristotle himself says in De interpretatione 9.

As is well known, De interpretatione 9 contains Aristotle’s discussion about the truth and falsity of future-tensed contingent propositions, as well as several crucial observations about truth and reality in general. An important remark is found in De int. 9, 18a39–b1, where Aristotle puts forward the following inference: If it is true to say of something that is (not) white, then it is necessary for it (not) to be white.⁶ This inference is an instance of what is commonly referred to as the ‘principle of the necessity of the present’, which according to Aristotle applies unrestrictedly to past- present- and future-tensed propositions.⁷ This principle tells us that if a proposition ‘p’ is true at a time t, then it is necessary at t that what is proposed by p was/is/will be the case. Or, to put it schematically ${{\text{T}}_t}^{\prime }{p}^{\prime }\to {◻}_t\text{ that-}p$ where the subscript ‘t’ is a placeholder for a time-index. As we can see, the principle of the necessity of the present and Boethius’ reading of [TR] are formally identical, especially once we take into account that Boethius, too, regards sentences as being true at a time and not eternally.⁸ From this we may conclude that Boethius regards [TR] as an instance of the principle of the necessity of the present – it is this very principle that Boethius has in mind when commenting on Categories 12 (14b11–22), and furthermore it is on the basis of this principle that he inserts the ‘necesse est’ in his translation.

What influences to an even greater extent the medieval reception of Categories 12 (14b11–22), however, is Boethius’ idea that [TR] as well as its converse (henceforth [RT]) are necessary consequences (consequentiae) of the sort he discusses in his logical works. Consider for instance how Boethius presents the two inferences in his shorter commentary on De interpretatione 9 (18a39–b1):

[T1]Aristotle states that the consequence of states of affairs (consequentia rerum) is such that an obtaining states of affairs follows from the truth of a sentence, and that the existence of a state of affairs (about which the sentence speaks) is accompanied by the truth of a proposition.⁹

First, notice that the expression ‘consequentia rerum’ which occurs in [T1] is also used by Boethius in his commentary on Cicero’s Topics as well as in his De hypotheticis syllogismis to indicate the relationship of consequence that holds between two states of affairs and may be expressed by a true conditional proposition such as ‘If it is day, then it is light’.¹⁰ Secondly, notice that the verbs ‘to follow’ (consequor) and ‘to accompany’ (comitor) which occur in [T1] are used consistently by Boethius in his logical works to indicate that the state of affairs signified by the consequent proposition follows necessarily from the state of affairs signified by the antecedent proposition.¹¹ As we shall see, twelfth-century logicians will follow in Boethius’ footsteps and address Aristotle’s mutual implication within the context of a more general theory of consequences. Before moving on to the middle ages, however, let us first determine what sort of true conditional expresses a consequentia rerum according to Boethius.

Boethius maintains that a necessary condition for the truth of a conditional proposition is that it is impossible for the antecedent to be true and the consequent false at the same time.¹² Two kinds of conditional propositions satisfy such a condition: (1) conditionals in which the connection between truth of the antecedent and the truth of the consequent is merely accidental (secundum accidens) and non-explanatory, such as ‘if fires is hot, then the heavens are spherical’, and (2) conditionals in which the connection between the truth of the antecedent and the truth of the consequent is a ‘consequence of nature’ (consequentia naturae), as Boethius calls it. The latter are conditionals in which the truth of the consequent causally explains the truth of the antecedent or vice versa, such as ‘If something is a human being, then something is an animal’ or ‘If there is an interposition of the earth, then an eclipse of the moon follows’.¹³ According to Boethius, conditionals expressing a consequentia rerum are of the latter kind. In particular, both in the case of [TR] and of [RT] the truth of the proposition functions as the explanandum, whereas the obtaining state of affairs as the causal explanans. Sticking to the example provided by Aristotle, Boethius thus goes on to remark that the existence of human being is ‘the cause and principle’ (causa atque principium) of the truth of ‘Human being is’.¹⁴

Boethius thus seems committed to an account of truthmaking as entailment. According to Boethius, that is, the state of affairs a is a truthmaker for the sentence p just in case a’s existence necessarily entails that p.¹⁵ For some time, this view enjoyed a degree of popularity also among contemporary philosophers. Its inadequacy, however, became all too soon apparent. The main problem with a conception of truthmaking as entailment can be stated simply as follows: whereas entailment is commonly regarded as being not relevant, truthmaking always is. And consequently, truthmaking is not entailment. The credit for such a discovery, however, does not belong to present-day philosophers. As we shall see in what follows, the insight that entailment invariably fails to model truthmaking stems from the twelfth-century debate on the consequentia rerum.

3. William of Champeaux

Two commentaries on Aristotle’s De interpretatione preserved in ms. Orléans, Mediathèque municipale 266 and listed as ‘H13’ and ‘H20’ in Marenbon’s catalogue report a debate that occurred between ‘M. W.’ or ‘magister Will.’ on one hand and ‘M. G.’ or ‘M. Gos.’ and his followers on the other.¹⁶ These two masters can be identified, with a reasonable degree of certainty, with William of Champeaux and Jocelyn of Soissons.¹⁷ At the center of the debate is precisely the validity of [TR] and [RT], or, as the author of H20 puts it, of those consequentiae put forward by Aristotle in De interpretatione 9 (18a39–b1) which hold ‘between vocal sounds and things, and between things and vocal sounds’.¹⁸

According to the author of H13, William maintained that whereas [TR] is true without any further qualification (constantia), [RT] requires the qualification that the embedded proposition be uttered. So, for instance, William would not accept that ‘If Socrates is white, then the proposition “Socrates is white” is true’, and yet he would concede that ‘If Socrates is white, then the proposition “Socrates is white”, if uttered, is true’.¹⁹ In this way, William attempted to rule out all those cases in which a state of affairs obtains but the relevant proposition is not true simply because it has not been uttered.

Yukio Iwakuma tried without success to locate the view just discussed in a commentary on De interpretatione that he takes to be William’s.²⁰ I found the same view in two partially overlapping commentaries on the Categories (C7 and C14 in Marenbon’s list) that date from the beginning of the century and appear to be closely related to William’s teaching.²¹ Commenting on Aristotle’s implication of being, the authors of C7 and C14 remark that Aristotle

did not simply say ‘if human being is, then the sentence “human being is” is true’, for it could well be the case that a human being is and the sentence is not true, for clearly it had not been uttered. Rather, he added a qualification. For Aristotle said that the expression saying ‘human being is’ is true as though if he had said: the expression ‘human being is’ is true, if uttered.²²

In sum, William says what any reasonable tokenist should say, and namely that propositions exist only if uttered. Because of this, on his account [RT] holds only as long as the embedded proposition is uttered. However, William says nothing more than this, and overall his account is essentially the same as Boethius’.

4. H13 and H20 Against William

After presenting William’s position, the authors of H13 and H20 go on to argue against it. First, an objection is raised against William’s view according to which [TR] does not require any qualification. Second, William’s own reply to the objection is reported. And finally, the authors provide a stronger version of the original objection, thus putting away William’s solution once and for all.

The authors first argue that if William takes conditionals of the type [TR] to be true without qualification, then he must also accept the truth of conditionals that are patently false. Consider for instance the conditional ‘If “Socrates is a human being” is true, then Socrates is a human being’. If this conditional is true, the authors argue, then it is also true that

If ‘Socrates is a human being’ is ‘Brownie is a donkey’, then Socrates is a human being.

The texts are not very clear about why this should be true.²³ Fortunately, a very similar objection against the truth of [TR] is raised by Abelard both in his Dialectica and in his commentary on the Categories, and it seems only reasonable to assume that our authors are following Abelard on this point. In the Dialectica, Abelard reasons that if we define a true proposition as one that says what is the case, then the conditional

[1] If ‘Socrates is a human being’ is true, then Socrates is a human being

is equivalent in meaning to

[2] If ‘Socrates is a human being’ says what is in reality, then Socrates is a human being.

However, as Abelard goes on to show, this equivalence leads to the unflattering consequence that

[C] If ‘Socrates is a human being’ says that Brownie is a donkey, then Socrates is a human being.²⁴

Indeed, if the state of affairs that Socrates is a human being follows from the truth of ‘Socrates is a human being’ simply because ‘Socrates is a human being’ says what is the case, as William would have it, then ‘Socrates is a human being’ could be replaced by any true proposition – that is, by any proposition that says what is the case, such as for instance ‘Brownie is a donkey’ – and the conditional would still be true. The authors of H13 and H20 take this to show that [TR] validates conditionals in which the antecedent is irrelevant to the consequent and is therefore to be dismissed.²⁵

The authors go on to remark that William attempted to overcome the objection by adding to [TR] the qualification that the truth of the embedded proposition must be assigned ‘in virtue of the things that the proposition is about’ (gratia rerum de quibus agit), and not simply because the proposition says something that is in reality.²⁶ William’s criterion would thus be satisfied by the conditional

If ‘Socrates is a human being’ is true in virtue of Socrates and human being, then Socrates is a human being,

but not by the conditional

If ‘Socrates is a human being’ is true in virtue of Brownie and donkey, then Socrates is a human being.

As is clear, William wants to restrict the class of truth-makers for any given sentence in accordance with the meaning of that sentence in the attempt to make the truth of the antecedent relevant to the truth of the consequent. Unfortunately for him, this failed to persuade our authors.

A strengthened version of the objection is put forward by the authors in order to reject William’s ‘gratia rerum’ principle. In H13, the objection is introduced in the first-person plural, which very strongly suggests that it was a product of Jocelyn’s milieu.²⁷ The objection runs as follows: if the conditional

If ‘Socrates is not a human being’ is true, then Socrates is a human being

is judged true on the basis of William’s ‘gratia rerum’ principle, then, the authors argue, the conditional

If ‘Socrates is a human being’, then Socrates is not a human being

must also be judged true on the basis of the same principle.²⁸ Recall that William’s principle establishes that the existence of a given state of affairs follows from the truth of a proposition only under one condition – that the proposition is true because of the things it signifies. But ‘Socrates is a human being’ and ‘Socrates is not a human being’ are both true in virtue of the same things, namely Socrates and human being. Therefore, one could reason that if ‘Socrates is a human being’ is true in virtue of Socrates and human being, then Socrates is not a human being. And more generally, that if ‘p’ is true, not-p follows and vice versa. As it appears, William’s principle fails to account for the fact that negation – which is not a thing! – contributes crucially to determining the truth-conditions of a proposition.

5. Peter Abelard

Abelard (1079–1142) presents his account of the consequentia rerum in two of his logical works, namely his Dialectica and his commentary on the Categories. In both cases, the context for the discussion is provided by what Aristotle says in Categories 12 about priority of nature. Before entering on a detailed analysis of Abelard’s account of the consequentia rerum, however, let me briefly say something about Abelard’s general views on the validity of inference.²⁹

In his Dialectica, Abelard argues against Boethius’ criterion for the truth of a conditional. He correctly notices that if a conditional were true simply because the truth of the antecedent is inseparable from that of the consequent, then one would have to concede that anything follows from an impossibility – a result which he is not willing to accept.³⁰ Abelard instead develops a relevantistic account of consequence. On this account, only those conditionals express a genuine and causally explanatory relation of consequence in which the meaning of the antecedent ‘requires’ (exigit) by itself the meaning of the consequent.³¹ Whether or not this stricter criterion is satisfied is determined by Abelard on the basis of the natures of things (naturae rerum), rules of inference, or both. He explains for instance that the conditional ‘If something is an animal, something is a human being’ is true because the nature of human is ‘contained’, as he says, in the nature of animal.³² Abelard however pushes his relevantistic view so far that many conditionals that would prima facie be judged true are, according to him, false. He argues, for instance, that the conditional ‘If something is a body, something is colored’ is false because the nature of body does not contain the nature of color, and this although bodies are always colored and it cannot be the case that something is a body and is not colored.

Although failing to express a relation of genuine consequence, conditionals such as the one from being a body to being colored capture a weaker yet still necessary relation which Abelard calls either ‘natural’ or ‘necessary accompaniment’ (naturalis/necessaria comitatio). Conditionals expressing a necessary accompaniment only satisfy Boethius’ criterion which states that it is impossible for the antecedent to be true and the consequent false. It is important to point out that, unlike the genuine relation of consequence, the relation of necessary accompaniment is not causally explanatory according to Abelard. And indeed, Abelard maintains that conditionals expressing a necessary accompaniment have the logical form of a necessary conjunction, or ‘□(p∧q)’.³³

The distinction between genuine consequence and necessary accompaniment provides the background for Abelard’s discussion of the consequentia rerum. More precisely, Abelard claims that although conditionals of the type [TR] and [RT] express a necessary accompaniment, they invariably fail to express a genuine consequence.³⁴ As he sees it, conditionals expressing a consequentia rerum do not satisfy the stricter criterion for the truth of a conditional since both in the case of [TR] and of [RT] the antecedent does not require by itself the consequent.³⁵ Abelard takes Aristotle to hold the same view and to propose an accompaniment between the truth of a proposition and the obtaining state of affairs. But before addressing how such an accompaniment should be construed, let us first consider the pars destruens of Abelard’s discussion.

Abelard raises several objections against those who believed that [TR] and [RT] express a genuine consequence. As we already know, this same view was set out by Boethius and adopted with negligible modifications by William of Champeaux, one of Abelard’s masters. Abelard starts out by criticizing William’s view that [RT] expresses a genuine consequence provided one adds the qualification that the embedded proposition be uttered. According to Abelard, attaching the qualification the way William does validates conditionals that are clearly false. Consider for instance the true conditional

If two exists, one exists.

Now, William’s method of adding a qualification would allow us to produce the converse of the above conditional in the following way:

If one exists, when the other exists, two exists.

Yet, this flies in the face of Aristotle who argues in the Categories that the inference from two to one does not have a true converse.³⁶ Abelard takes this to show that William’s method of qualifying conditionals never provides a warrant for their truth, and consequently that the conditional reading of [RT] must be rejected if the relevant qualification is added. Abelard also quickly dismisses the conditional reading of [RT] when this is taken without qualification. For consider the conditional

If a human being exists, then the proposition ‘a human being exists’ is true.

This is false according to Abelard, for the existence of a human being does not by itself require the existence of the proposition ‘a human being exists’.³⁷

Abelard’s rejection of [TR] is motivated by analogous concerns about the lack of relevance between the antecedent and the consequent. As has been explained in the previous section, Abelard insists that if a proposition is considered true just in case it says what is in reality, then any antecedent of the form ‘“p” is true’ will entail any obtaining state of affairs, or ‘dictum’, to use Abelard’s jargon:

Whence it seems that one should completely reject the following consequence, too: ‘If “a man exists” is true, then a human being exists’, as well as its converse simply taken, for the antecedent in no way of itself compels the consequent – the fact that this proposition is true – namely, that it says that what is the case – could very well have occurred because of another dictum than because of this one, which is that a human being is.³⁸

How to avoid overgenerating truth-makers? The obvious way would be to restrict what follows from the truth of the embedded proposition to the relevant state of affairs which is signified by that same proposition – precisely what William tried to achieve with his ‘gratia rerum’ principle. Abelard, however, thinks that such a strategy is doomed to failure. More exactly, Abelard assumes that those who take [TR] to express a genuine consequence will adopt the following line of reasoning:

[1] If ‘a human being exists’ is true, i.e. it says what is the case, then what is the case is said by the proposition. [2] If what is the case is said by the proposition, then what the proposition says is the case. [3] If what the proposition says is the case, then that a human being exists is the case.

∴ If that a human being exists is the case, then a human being exists.³⁹ (1, 2, 3, Transitivity)

Conditional [3] is the crucial step of the argument. For if true, it would establish the identity between the meaning of the embedded proposition and the obtaining state of affairs, which in turn would secure a relevant connection between the antecedent and the consequent of [TR]. However, Abelard insists that [3] is false. More exactly, he argues that if [3] were true, then one would have to admit by the same token that

[1*] If Sophroniscus is someone’s father, then someone is Sophroniscus’ son. [2*] If someone is Sophroniscus’ son, then Sophroniscus’ son is someone. [3*] If Sophroniscus’ son is someone, then Socrates is someone. ∴ If Socrates is someone, then Socrates exists.⁴⁰ (1, 2, 3, Transitivity)

But [3*] is clearly false, for the meaning of the antecedent does not contain the meaning of the consequent. Indeed, there is nothing in the meaning of ‘Sophroniscus’ son’ that points to Socrates. For our knowledge that Socrates is indeed the only son of Sophroniscus is entirely a posteriori. And so similarly in the case of [3] the criterion for the truth of a conditional is not satisfied. For there is nothing in the meaning of the expression ‘what the proposition says’ that points to the state of affairs that a human being exists. Once again, this can only be known a posteriori.

Abelard thus concludes that there is no principled way to argue that [TR] and [RT] express a genuine consequence, and moreover that these conditionals only express a relation of mutual accompaniment. Abelard explains that the accompaniment between a true proposition and the relevant state of affairs holds as long as the proposition signifies the relevant state of affairs. And so, sticking to the standard Aristotelian example Abelard claims that

Necessarily, as long as a human being exists, the proposition ‘a human being exists’ is true (if uttered).

Necessarily, as long as the proposition ‘a human being exists’ is true, a human being exists.⁴¹

Although Abelard does not draw attention on this aspect, it is important to remark that the above accompaniments still allow us to overgenerate truth-makers for the embedded proposition by trading on the equivalence of ‘“p” is true’ and ‘“p” says what is in reality’. In this case, however, the price to pay is negligible. More exactly, Abelard only has to concede that every obtaining state of affairs necessarily coexists with every true proposition. And since Abelard maintains that the genuine relation of consequence is both relevant and causally explanatory, he is not bound to concede that every obtaining state of affairs implies or is implied by any true proposition. In this regard, Abelard’s discussion of the consequentia rerum, and in particular his arguments for the over-generation of truth-makers, fascinatingly anticipate some of the criticisms made by contemporary logicians against a conception of truthmaking as entailment.⁴²

Unlike his contemporaries, Abelard pauses over Aristotle’s remark according to which the ‘thing’ – or, as Abelard prefers to call it, the ‘event of a thing’ (eventus rei) – causes the truth of the sentence about it. He points out that the event of a thing (which I take to be an obtaining state of affairs) is not a ‘being’ (essentia), and in fact ‘does not exist at all’ (omnino non est). With these remarks, Abelard at least commits himself to the view that states of affairs do not fall in any of the Aristotelian categories.⁴³ This, however, poses a problem for him. Since the four types of causality countenanced by Aristotle apply only to categorical items, and since states of affairs are not among them, it follows that states of affairs are not Aristotelian causes. Abelard concedes the point but insists that there is a sui-generis form of causality that would hold between non-categorical items.⁴⁴ Had Abelard said more about this issue, we would have his positive account of truthmaking. But alas, he did not.

Before concluding this section, it may be noted in passing that Abelard’s account of the consequentia rerum seems to have gained the support of some later logician. In his commentary on the Categories dating from the end of the century, for instance, the so-called ‘Anonymous d’Orvillensis’ takes up Abelard’s idea that the event of a thing causes the truth of a proposition, and spells out Aristotle’s mutual implication in terms of accompaniment.⁴⁵

6. The Ars Meliduna

Another interesting analysis of the consequentia rerum carried out within the framework of a theory of consequences is found in the anonymous treatise called by De Rijk ‘Ars Meliduna’ (henceforth AM).⁴⁶ Together with a fragment which De Rijk called ‘Secta Meliduna’, AM contains the views of the so-called Melidunenses, the followers of master Robert of Melun (ca. 1100–1167). Although the date of composition of AM is uncertain, it is safe to assume that the treatise belongs to the second half of the twelfth-century.⁴⁷

AM is divided into four sections, the first dealing with terms, the second with the meaning of terms, the third with propositions, and the fourth with the meaning of propositions. In the third section, the author of AM fleshes out his account of consequence by determining what is required for a proposition to follow from one or more propositions. The most striking thesis defended here by the author of AM is that nothing follows from a falsehood, which in turn implies that only those conditionals are judged true in which both the antecedent and the consequent are true.⁴⁸ As the author goes on to explain, a conditional between (simple) affirmative propositions can be warranted either by the ‘relationship among terms’ (habitudo terminorum), the ‘relationship between propositions’ (habitudo propositionum), or the ‘syllogistic arrangement’ (dispositio syllogistica).⁴⁹ According to the author, the relationships among terms that warrant an entailment also define the set of acceptable ‘loci’ – i.e. the non-formal argumentative devices which may legitimately be appealed to when arguing. In the case of AM, however, this set is extremely narrow and only includes the so-called locus from equals, the locus from a part, the locus from the whole, and the locus from opposites. What makes these loci acceptable and not others, the author explains, is the fact that ‘only in these must the antecedent and the consequent be taken to be about the same thing’.⁵⁰ A necessary condition for the truth of a conditional proposition, therefore, is that the antecedent and the consequent deal with the same thing. This ‘identity condition’, as I shall call it, is what ultimately motivates the author’s take on the consequentia rerum.

More exactly, since the author of AM accepts the identity condition, he has to concede that most instances of [TR] and [RT] cannot be expressed in a true conditional proposition. This is because, as the author goes on to explain, in proposing a consequentia rerum one often either infers a de dicto statement from a de re one (in the case of [TR]), or the opposite (in the case of [RT]). What is more, the author of AM takes the identity condition so rigidly that even those conditionals which infer the truth of a dictum from the truth of another dictum are judged false:

Just as we said that every consequence is false whose antecedent and consequent cannot be taken to be about the same thing, so we also reject those consequences in which something different is spoken about in the antecedent than in the consequent. Those consequences are such in which one moves from a thing to a dictum, or conversely. For it is not the case that if Socrates is a human being, then that Socrates is a human being is true. Nor is the case that if that Socrates is a human being is true, then that Socrates is an animal is true – it is not the case that if this is truly said, then that is truly said. Nor is the case that if Socrates is a human being, then the proposition ‘Socrates is an animal’ is true. Nor is the case that if either member of a contradictory pair will be true, then it will be true that either member is true; nor is the case that if it will be true that only one moment exists, then only one moment will exist.⁵¹

This nothwithstanding, the author of AM maintains that some conditionals expressing a consequentia rerum are true. These are conditionals that not only meet the identity condition but also may be warranted by one or more of the abovementioned loci. So, for instance, the conditional ‘If it is true that something exists, then something exists’ is true insofar as it instantiates a double locus a parte: first, the dictum occurring in the antecedent is part of the somethings referred to in the consequent, and secondly the existence of the dictum is part of what simply exists.⁵² Clearly, the number of conditionals that satisfy the identity condition while at the same time instantiating an admissable topical relationship is obviously very limited. Here we come across an important point of difference between the author of AM and Abelard. And that is, whereas for Abelard all attempts of securing a relevant connection between the antecedent and consequent of a consequentia rerum are doomed to failure, for the author of AM the path of relevance does eventually lead us to a class of true conditionals. (This is probably why the author of AM does not consider the possibility of expressing a consequentia rerum in a true conjunction, as Abelard did.) Notice, however, that the said relevance is in no way linked to any intrinsic feature of the consequentia rerum. Rather, relevance is solely due to the existence of a topical relationship. And therefore, there is nothing ‘inferentially special’, so to speak, about the consequentia rerum.

Finally, although the author of AM grants that some instances of [RT] are true, he nonetheless agrees with Abelard that the consequence relation fails to capture the relation of truthmaking. Presumably, this is because none of the loci that support truth of a consequence has anything to do with the truthmaking relation. Accordingly, the fact that a given propositional meaning is true must be explained by appealing to a relation other than entailment, which I shall call the ‘in virtue of’ relation. The author of AM argues in this way that the propositional meaning that Socrates is a human being is true ‘in virtue of Socrates’ (pro Socrates).⁵³ On the authors account, therefore, the truthmaking relation holds between particulars, such as Socrates or his whiteness, and propositional meanings.

7. Concluding Remarks

Three main views have emerged from the analysis carried out in this paper. According to a first view, which was propounded by Boethius and (with minimal modifications) William of Champeaux, [TR] and [RT] express genuine and explanatory consequences. On this view, moreover, the truthmaking relation is itself understood in terms of the consequence relation.

According to a second view, which was propounded most forcefully by Abelard (but gained some traction also in Jocelyn’s school), [TR] and [RT] fail to express genuine and explanatory consequences because the antecedent and the consequent are not relevant – or better, not relevant enough – to each other. Nevertheless, Abelard maintains that [TR] and [RT] do express necessarily true conjunctions of the form ‘□(p∧q)’ (which, incidentally, is enough to give us Tarski’s T-schema). On his view, truthmaking cannot be boiled down to consequence. And yet, since Abelard does not provide us with a fully fledged account of truthmaking, it is not clear in what respects the two should differ.

Finally, according to a third view propounded by the anonymous author of AM, [TR] and [RT] express a relation of consequence provided that one may appeal to an appropriate locus. This however goes to show that these inferences are not true qua instances of the consequentia rerum. Furthermore, the author of AM realizes that none of the appropriate topical relationships may account for truthmaking. And therefore, he introduces what I called an ‘in virtue of’ relation that holds between particulars and propositional meanings.

Notes

1 Aristotle 1949, p. 42: ‘τῶν γὰρ ἀντιστρϵφόντων κατὰ τὴν τοῦ ϵἶναι ἀκολούθησιν τὸ αἴτιον ὁπωσοῦν θατέρῳ τοῦ ϵἶναι πρότϵρον ϵἰκότως τῇ φύσϵι λέγοιτ᾿ ἄν. ὅτι δ᾿ ἔστι τινὰ τοιαῦτα, δῆλον· τὸ γὰρ ϵἶναι ἄνθρωπον ἀντιστρέφϵι κατὰ τὴν τοῦ ϵἶναι ἀκολούθησιν πρὸς τὸν ἀληθῆ πϵρὶ αὐτοῦ λόγον. ϵἰ γὰρ ἔστιν ἄνθρωπος, ἀληθὴς ὁ λόγος ᾧ λέγομϵν ὅτι ἔστιν ἄνθρωπος. καὶ ἀντιστρέφϵι γϵ· ϵἰ γὰρ ἀληθὴς ὁ λόγος ᾧ λέγομϵν ὅτι ἔστιν ἄνθρωπος, ἔστιν ἄνθρωπος. ἔστι δὲ ὁ μὲν ἀληθὴς λόγος οὐδαμῶς αἴτιος τοῦ ϵἶναι τὸ πρᾶγμα, τὸ μέντοι πρᾶγμα φαίνϵταί πως αἴτιον τοῦ ϵἶναι ἀληθῆ τὸν λόγον· τῷ γὰρ ϵἶναι τὸ πρᾶγμα ἢ μὴ ἀληθὴς ὁ λόγος ἢ ψϵυδὴς λέγϵται’.

2 Armstrong 2004, p. 4.

3 For a contemporary defense of truthmaking as entailment see Fox 1987 and Bigelow 1988.

4 The term ‘res’ is commonly used in both classical and medieval Latin to denote what we would call states of affairs. Some scholars have credited Aristotle himself with a theory of state of affairs. See, for instance, Simons 1988; Crivelli 2004.

5 Aristoteles Latinus 1961, pp. 38.4–14: ‘Eorum enim quae convertuntur secundum essentiae consequentiam, quod alterius quomodolibet cause est digne prius natura dicitur. Quoniam autem sunt quaedam talia, manifestum est; nam esse hominem convertitur secundum subsistentiae consequentiam ad verum de eo sermonem; nam, si est homo, verus sermo est quo dicimus quoniam est homo, et convertitur (nam, si verus est sermo quo dicimus quoniam est homo, hominem esse necesse est); est autem verus sermo nullo modo causa subsistendi rem, res autem videtur quodammodo causa esse ut sermo verus sit; nam, quoniam est res vel non est, verus sermo vel falsus dicitur’. Boethius 1847a, 286A: ‘Nam si est aliqua res, verum est de ea dicere, quoniam est. Rursus si de ea verum est dicere quoniam est, illam ipsam rem esse necesse est’. Notice that the later translators of the so-called editio composita – the edition which Abelard and his contemporaries refer to – do not make any additions and their translation is overall much more literal than Boethius’. See Aristoteles Latinus 1961, pp. 76.11–20.

6 Aristotle 1949, p. 55: ‘ϵἰ γὰρ ἀληθὲς ϵἰπϵῖν ὅτι λϵυκὸν ἢ οὐ λϵυκὸν ἐστιν, ἀνάγκη ϵἶναι λϵυκὸν ἢ οὐ λϵυκὸν, καὶ ϵἰ ἔστι λϵυκὸν ἢ οὐ λϵυκὸν, ἀληθὲς ἦν φάναι ἢ ἀποφάναι’. Aristoteles Latinus 1965, pp. 14.5–8: ‘Nam si verum est dicere quoniam album vel non album est, necesse est esse album vel non album, et, si est album vel non album, verum est vel affirmare vel negare’.

7 Kirwan 1986; Crivelli 2004, pp. 199–200.

8 Boethius 1847a, pp. 211D–212A.

9 Boethius 1877, pp. 109.25–28: ‘Ait enim hanc esse rerum consequentiam, ut rem subsistentem propositionis veritas consequatur, veritatem propositionis rei, de qua loquitur propositio, essentia comitetur’.

10 Boethius 1847b, p. 1125C; Boethius 1969, p. 328.

11 See, for instance, Boethius 1969, pp. 214, 230.

12 Boethius 1969, pp. 250–252. See Martin 2004, p. 165.

13 Boethius 1969, pp. 219–220.

14 Boethius 1847a, 286B: ‘Ut in hoc ipso sermone de homine, conuertuntur utraque quidem sed homo ut sit sermo uerus, causa est atque principium’.

15 According to standard usage in contemporary philosophy, only sentences or their meanings (the so-called ‘propositions’) entail. In order to capture the cross-categorial relation between sentences and portions of reality, therefore, contemporary philosophers often speak in terms of ‘necessitation’. Boethius, however, does not need to draw this sort of distinction because his notion of entailment or consequence encompasses both the abovementioned relationships. In the context of the present discussion, therefore, I shall be led by charity to take his entailment as our necessitation. On the distinction between entailment and necessiation see Fox 1987, p. 189.

16 Marenbon 1993, pp. 119, 121, where it is claimed that H13 dates ‘probably’ from the ‘second quarter of the twelfth century’ and H20 ‘probably’ from ‘mid-twelfth century’.

17 Green-Pedersen 1974; Kneepkens 1993, p. 65.

18 Anonymous, H20, f. 260^b: ‘Hic facit Aristoteles consequentiam inter voces et res et inter res et voces’. The author’s decision to insert a digression on the consequentia rerum in his exegisis of De interpretatione 9 (18a39–b1) is influenced by the Boethian reading which has been defended above.

19 Anonymous, H13, f. 247^a: ‘M. W. dicit quod, ubi agimus de vocibus ad res, non est apponenda constantia; sed de rebus ad voces, tunc apponenda constantia. Verbi gratia: si Socrates est albus, vera est propositio si fiat “Socrates est albus”’.

20 Iwakuma 1999, pp. 109–110. The commentary is listed as ‘H11’ in Marenbon’s catalogue.

21 One of the commentaries is contained in ms. Paris, Bibliothèque nationale, Lat. 17813, ff. 19bisr-54v (‘C7’ in Marenbon’s list), and the second in ms. Assisi, Biblioteca del Convento Francescano 573, ff. 15v–48r (‘C14’ in Marenbon’s list). According to Iwakuma, C7 and C14 are later revisions of William’s original commentary on the Categories (‘C8’ in Marenbon’s list). See Marenbon 1993, p. 111. At first, Iwakuma 1999 held that these revisions were made by William himself. Later in Iwakuma 2009, pp. 89–91, however, he argued that the revisions were made by William’s students. The position which we are interested in, however, is not found in C8.

22 Anonymous, C7, f. 54^rb/C14, f. 46^rb (as collated by Iwakuma): ‘Et nota quod non dixit simpliciter “si homo est, vera est haec oratio ‘homo est’”, quia bene posset esse homo, et tantum non esset vera illa oratio, quia nec etiam fieret sed apposuit loco constantiae. Nam <vera> est oratio quae dicit “homo est”, tanquam si diceret: vera est haec oratio “homo est” si fiat’.

23 Anonymous, H20, 261^a: ‘Ex quo tale sequitur inconveniens: si haec propositio “Socrates est homo” est haec alia “Brunellus est asinus”, tunc Socrates est homo. Quod sic probatur. Si, quia “Socrates est homo” est vera, Socrates est homo; et, quia “Socrates est homo” est “Brunellus est asinus”, Socrates est homo’. Compare with Anonymous, H13, 247^a.

24 Abelard 1956, pp. 372.20–21: ‘[S]i, quia “Socrates est homo” dicit illud quod in re est Socrates est homo, et, quia “Socrates est homo” dicit brunellum esse asinum, Socrates est homo’. Compare with Abelard 1921, pp. 291.26–36.

25 Abelard, who certainly inspired our authors, rejects this sort of consequences because of the lack of relevance between the antecedent and the consequent. See Abelard 1921, pp. 291.39–42: ‘[A]ntecedens nullo modo ex se exigit consequens, cum uidelicet id quod haec propositio sit uera, hoc est dicat illud quod in re est, omnino propter aliud dictum contingere posset quam propter hoc quod est hominem esse’.

26 Anonymous, H13, f. 247^a: ‘Hic determinant gratia rerum de quibus agit. Unde non sequitur: si est Brunellus, est asinus, tunc est vera gratia rerum de quibus agit’. Anonymous, H20, f. 261^a: ‘Contra hoc dicebat magister W. sic esse determinandum: si “Socrates est homo” est vera gratia rerum de quibus agit, tunc Socrates est homo’. The ‘gratia rerum’ principle seems to have been a very important tool in William’s arsenal. See Grondeux & Rosier-Catach 2017, pp. 130–132.

27 Anonymous, H13, f. 247^a-b: ‘Nos dicimus hanc determinationem parum valere. Quod potes per hoc videre […]’. And indeed there seems to be no parallel in Abelard’s works.

28 Anonymous, H20, f. 261^a: ‘Sed ex hoc quoque trahitur inconveniens, sic. Si “Socrates non est homo” est “Socrates est homo”, tunc Socrates non est homo. Quod sic probatur. Si, quia vera est “Socrates non est homo”, Socrates non est homo; et, quia “Socrates non est homo” est “Socrates est homo”, Socrates non est homo’.

29 For a more detailed discussion of Abelard’s theory of consequence, see Martin 2004.

30 Abelard 1956, pp. 285.8–12.

31 Abelard 1956, p. 253.29.

32 Abelard 1956, p. 284.7.

33 Abelard 1921, pp. 288.4–6; Abelard 2010, pp. 304.1403–305.1406; pp. 411.502–505. Necessary accompaniments need to be distinguished from accompaniments tout court, which are contingent truths expressed by propositions such as ‘When it rains, it tunders’. See Abelard 1956, pp. 472.16–29. Unfortunately for the reader, Abelard at times uses ‘accompaniment’ (comitatio) to indicate a necessary accompaniment.

34 In order to avoid using the term as much as possible, Abelard glosses most occurrences of ‘consequence’ in Boethius’s translation with the clause ‘namely, an accompaniment of mutual permanence’. See, for instance, Abelard 1921, pp. 290.30–32.

35 Guilfoy 2002, p. 144 says that ‘Abelard is not at all clear as to why and how’ the inference has to be true if taken as an accompaniment. He also adds (2002, p. 146) that Abelard’s argument for showing that the inference fails if taken as a conditional ‘is not quite clear and decisive, or even comprehensible as he seems to think’. Guilfoy, however, does not seem to acknowledge the importance of Abelard’s criterion for the truth of conditional propositions in the context of his discussion about the consequentia rerum. It is such a criterion that explains why the inference fails as a conditional, and therefore why Abelard could only accept the inference as a necessary accompaniment.

36 Abelard 1921, pp. 291.14–20: ‘Sed ad haec dico quod similiter inter unum et duo possent mutuam consecutionem servare vel inter animal et hominem, si videlicet ex una parte constantiam [correxi, Geyer “circumstantiam”] velint adiungere hoc modo: “Si duo sunt, unum est”, “Si unum est, cum alterum sit, duo sunt”. Et rursus: “Si Socrates est homo, est animal” et: “Si est animal, cum sit risibile, est homo”. At vero Aristoteles, qui veritatem propositionis sequi concedit de uno ad duo, consequentiam dicit non conuerti’. Aristoteles Latinus 1961, pp. 75.23–27: ‘Secundo autem quod non conuertitur secundum subsistendi consequentiam, ut unus duobus prior est (duobus enim existentibus, mox consequens est unum esse, uno autem existente, non necessarium est duo esse; idcirco non conuertitur ab uno consequentia ut sit reliquum)’.

37 Abelard 1921, pp. 291.26–28: ‘[H]anc consequentiam simpliciter factam reprobamus: si homo est, vera est haec propositio homo est, eo videlicet quod saepe contingit rem esse non facta propositione’.

38 Abelard 1921, pp. 291.36–42: ‘Unde omnino reprobanda esse videtur haec quoque consequentia: si vera est “homo est”, tunc homo est, sicut eius conversa simpliciter facta, quippe antecedens nullo modo ex se exigit consequens, cum videlicet id quod haec propositio sit vera, hoc est dicat illud quod in re est, omnino propter aliud dictum contingere posset quam propter hoc quod est hominem esse’.

39 Abelard 1921, pp. 292.1–5: ‘Si “homo est” est vera, hoc est dicit illud quod in re est, tunc illud quod in re est dicitur ab ea, et ita illud quod ab ea dicitur est in re, et ita hominem esse est in re. A pari. Unde et homo est. Itaque per medium ostensum est, quod si homo est, est vera “homo est”’.

40 Assuming of course that Socrates is Sophroniscus’ only son. Abelard 1921, pp. 292.1–5: ‘Sed profecto secundum huiusmodi argumentationem posset ostendi solo Socrate existente filio Sophronici, quod si Sophronicus est pater alicuius, Socrates est, hoc modo: Si Sophronicus est pater alicuius, aliquis est filius Sophronici et ita filius Sophronici est aliquis; unde Socrates est aliquis. A pari. Et ita est. Itaque per medium ostensa est consequentia haec: si Sophronicus est pater alicuius, Socrates est, quae nullo modo procedit’.

41 If Abelard’s view that [TR] and [RT] fail to express genuine consequences were motivated by the view that meaning is conventional, as Guilfoy argues (2002), then Abelard would have rejected the above accompaniments as well. But he does not. Therefore, it follows by modus ponens that the motivation for Abelard’s view is has nothing to do with the conventionality of meaning (but it has a lot to do with the lack of relevance, as I argued).

42 It has been proved, for instance, that under a classical conception of entailment every truthmaker makes true every truth, be it necessary or contingent. See Restall 1996.

43 As a matter of fact, some scholars have taken the above remarks by Abelard to indicate that states of affairs are not part of the ontology. Others instead have taken them to mean simply that states of affairs do not fall in any of the Aristotelian categories (and that they ultimately do in fact belong to the ontology). Since the issue has no direct bearing on our discussion, I will only credit Abelard with the view that states of affairs do not fall in any Aristotelian category – which is true regardless of whether one thinks that they exist or not.

44 Abelard 1921, pp. 293.13–30.

45 Ebbesen 1999, p. 408: ‘Et ostendit quae: esse enim hominem i.e. talis eventus convertitur secundum essendi consequentiam ad veram de se orationem i.e. talem eventum comitatur veritas propositionis qua hoc dicitur: Et hoc probat: nam si homo est, vera est oratio i.e. propositio qua nos dicimus hoc sc. Quia homo est; et ita eventus et veritas propositionis sese comitantur, et tamen unum est causa alterius, quia eventus rei causa est veritatis propositionis. Et ostendit quod oratio non est causa eventus sed econverso […], quia veritas propositionis variatur secundum eventum rei’. According to Ebbesen 1999, ‘Introduction’ the author belongs to the sect of nominales, which squares well with the fact that he follows Abelard on this issue.

46 The treatise is preserved in Oxford, Bodleian Library Digby 174, ff. 211^ra–241^rb. De Rijk’s edition, from which I cite, is partial.

47 De Rijk et al. 1982, p. 165; Ebbesen 2013, p. 70.

48 AM 1967, p. 386. See Martin 2022.

49 AM 1967, p. 347: ‘Harum aliae sunt verae ex habitudine terminorum sive eorum quae terminis significantur, ut “si est homo, est animal”; aliae ex habitudine propositionum, ut “si omnis homo currit, quidam homo currit”; locus a subalternis. Aliae ex dispositione syllogistica, ut “si omnis homo est animal et omne risibile est homo, omne risibile est animal”’.

50 AM 1967, p. 347: ‘Non tamen secundum quemlibet locorum vera sumitur consequentia, sed tantum locus a pari, locus a parte, locus a toto, et locus ab oppositis de quo prius diximus, necessitatem tenent, quia in his solis circa idem est sumere antecedens et consequens. Alii in sola probabilitate consistunt, et improprie loci appellantur’.

51 AM 1967, p. 347: ‘Sicut diximus omnem consequentiam falsam esse cuius antecedens et consequens non possunt sumi circa idem, ita et illas repudiamus in quibus de alio fit sermo in antecedenti quam in consequenti. Quales sunt illae in quibus de re fit progressus ad dictum vel e converso. Nec enim si Socrates est homo, Socratem esse hominem est verum; nec si Socratem esse hominem est verum, Socratem esse animal est verum; velut nec si vere dicitur hoc, et illud; nec si Socrates est homo, vera est haec propositio “Socrates est animal”; nec si utrumque contradictorie oppositorum erit verum, verum erit utrumque esse verum; nec si verum erit unum solum momentum esse, unum solum momentum erit’.

52 AM 1967, p. 347: ‘Tamen si aliquid esse est verum, aliquid est, quia hic duplex operatur habitudo. Est enim locus a parte tam inter subiecta quam inter praedicata. Et similiter si unum vel aliud esse est verum, unum vel aliud est’.

53 AM 1967, pp. 365–366: ‘Si quis tamen altius inspiciat, intelliget convenientius et magis exquisite dictum pro aliquo aliquid esse verum quam de aliquo, ut pro homine hominem esse animal’.