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In order to situate Kilwardby and Buridan’s modal theories, it may be helpful to rehearse some of the main details of the development of medieval theories of modality and modal logic. Broadly, there were three main groups whose thinking and writings influenced the development of medieval modal logic. As we have already remarked in a couple of places, the medieval study of modal logic was one that developed and was deeply influenced by the Greek sources that were available to them. The most influential of these, at first, were Boethius’s translations of Aristotle’s De interpretatione and the Categories as well as Porphyrius’sIsogoge.2 Later, as more of Aristotle’s works became available, the rest of theOrganon exerted a considerable influence on medieval theories of modality, as we have already seen in the case of Kilwardby. A second influence came through the theological writings of a number of ‘Church Fathers’ in particular Augustine, who had a considerable impact on the way modality was understood and 1. By gloss we mean, what sort of motivation that is offered to interpret the various parts of the math-

ematical model (e.g. W, R, O, D, candvin the case of Buridan) 2. See [33, p.505]

conceived. Most of these writings influenced the development of modal theories focusing around theological issues such as God’s power and on the sorts of things that it was possible or impossible for God to do. Third, and finally, there was the influence of medieval Arabic logicians/philosophers, who we will not discuss in detail here.

Our survey in this section will cover a few selected interpretations of modality that are relevant to our discussions of Buridan and Kilwardby. We will start by looking at the statistical interpretation of modality, the potential interpretation of modality and (briefly) the temporal interpretation of modality, and discuss the role that essentialism and modal conversion rules played in these 12th and 13thcentury developments. We will then look at theories of modality in the 14th and in particular some of the contours of Duns Scotus’ modal theory.

6.2.1 Modal Logic in the 12th _{& 13}th _centuries

In the 12th and 13th centuries, approaches to modality were closely connected with a number of related Aristotelian metaphysical ideas. Our commentary here follows Knuuttila’s in [33] observing, the standard view of the time can be well summed up in the following passage from Aquinas’ commentary on thePeri Hermeneias:

In necessary matter, all affirmative propositions are determinately true; this holds for propositions in the future tense as well as in the past and present tenses; and negative ones are false. In impossible matter the contrary is the case. In contingent matter, however, universal propositions are false and particular propositions are true. This is the case in future tense propositions as well as those in the past and present tenses. In indefinite ones, both are at once true in the future tense propositions as well as those in the past and present tenses. [33, p.507]

There are a number of things here that are interesting, but our main focus will be to observe that the ideas of necessity, impossibility, and contingency are closely tied to the matter of the proposition. The matter of a proposition can be elucidated in the following way:

If the predicate is per se in the subject, it will be said to be a proposition in necessary or natural matter, for example, ‘Man is an animal’ and ‘Man is risible’. If the predicate is per se repugnant to the subject, as in a way excluding the notion of it, it is said to be a proposition in impossible or remote matter, for example ‘Man is an ass’. If the predicate is related to the subject in a way midway between these two, being neither per se repugnant to the subject nor per se in it, the proposition is said to be in possible or contingent matter. [33, p.508]

It is helpful to observe that by the time we come to Aquinas, we have a very succinct and clear presentation of the ideas of per se predication of terms, and the relationship

between them and the matter of a proposition. Aquinas’s theory is one example of what is referred to as astatistical model for modality. In such a model, the notion of necessity is connected with a notion of containment or of omni-temporal actuality of something within all members of a species, contingency with some members of the species possessing the property and impossibility with no members of the species possessing the property. As we have already seen (see page 13), Kilwardby’s theory of modality falls under this framework, specifically under the second disjunct.

Another view that was considered during the middle ages was the view of possibility as potency or the potential interpretation of modality. As in the previous case, the inspiration for this view came from Aristotle, in this case by way ofMetaphysics V.12 and IX.1. In these passages potency is described as a principle of motion where something is either an activator or a receptor of the relevant influence. Modal notions were then developed based on these ideas, often following Aristotle’s observation that this is one legitimate sense of ‘can’. Something is necessary on this view if its potency is never unrealised, i.e. the thing in question is always actual. Their nature is such that they are always actual.[33, p.513] Similarly, a thing is impossible if its potency is never actualised. The original idea behind this theory was to account for various kinds of changes in objects.

A third approach to understanding possibility and necessity is to employ various no- tions of time and tense. The view was motivated by Aristotle’s comments in the Peri Hermeneias 9.19 A23-24. In this much studied passage, Aristotle draws a distinction between what is necessary without qualification and what is actual. Interpreting this passage and the distinctions that Aristotle was using gave rise to a number of differ- ent interpretations.3 However, what eventually developed was a view of modality that sought to unpack modal notions by thinking of them in temporal terms. One very sim- ple view is to read ‘possibly φ’ as saying that either φis true, was true or will be true. Likewise, ‘necessarilyφ’ is read as saying thatφis true, was always true and will always be true. This is thetemporal interpretation of modality. We note this view, only to set it aside, as it does not play an important role in either Kilwardby or Buridan’s modal logic.

6.2.2 Modal Logic in the 14th _century

As we move to look at the time when Buridan wrote, one observes that in the intellectual landscape of 14th century, a number of important changes had occurred over the past 200 years. Most important among them for our purposes was that:

In contrast to the twelfth and thirteenth centuries, many Scholastics of the fourteenth and fifteenth viewed the world as radically contingent, depending upon a divine will able to will other than it does will. Belief in God’s ab- solute power to do anything that does not involve a contradiction meant a concomitant belief that only some of the logically possible possibilities could be actual at any one time. . . Moreover, since this world unfolds its events over 3. See [33][p.516] and references given there.

time, it seemed possible that alternative futures still lay open to divine and human choice– that what would come to pass was not ultimately determined or perhaps even determinate.[18][p.2]

It is these two commitments, combined with the idea of God’s omniscience, that led to a number of interesting and striking problems for medieval theologians. The devel- opments are equally important for the analysis of possibility. The impact of Aquinas’ statistical views on modality seems to have been underwhelming. For example, when we look at Dominican discussions of possibility and necessity in the 1320’s and 1330’s we find that

And yet absent from their works are the Thomist analysis of necessity and contingency according to Aristotelian causal theory, the Thomist emphasis on providence and its allied compatibilism, the Principle of Plenitude, and other hallmarks of Aquinas system. [18][p.123]

What we find in place of this view is the final view in the history of modal logic that we are interested in, the view that something is possible just in case it is within (or compatible with) the absolute power of God to bring it about. According to Knuuttila, this view has its origin in the bible[30, p.199] and was present in Christian writers as early as Tertullian. However, it is John Duns Scotus who is usually credited with the development of this view and of seeing the implications this had for logic and meta- physics more generally. In his writings Scotus rejects Aristotle’s thesis that the present is necessary, and develops a conception of synchronic alternatives.[31, p.129] Scotus’s view can be described as follows:

The main lines of Scotus’ theory of modality are easy to understand without any speculative interpretations. This depends on the fact that his starting point is a criticism of the statistical interpretation of modality. An extensive discussion about the theory of modality is to be found in distinction 39 of the first book of Scotus’ commentary on the Sentences. . . According to Scotus a causatum can be contingent only if the first cause functions in a contingent way. The contingency of phenomena in a causal chain depends on whether the whole universe (which is ultimately reducible to the first cause) could be different. Scotus take this to be conceptually possible. [30, p.225]

What is important here is the idea that the actual world does not need to be the way it is. Scotus’ ideas cut against the statistical interpretation of modality and

Scotus often treated modal notions in a way which shows similarities to what has been done in the contemporary possible worlds semantics. [31, p.130]

As Knuuttila goes on to observe, there are a number of interpretations of Scotus that make his views appear to be very modern in spirit.

Punch’s interpretation of Scotus’s modal theory is possibilistic and that of Mastrius is conceptualistic and mind dependent. I have argued that Scotus’s theory of purely logical possibilities is not possibilistic in any standard sense and that that it is not mind dependent at this level either.[31, p.142] 4

What is important for our purposes here is to observe that, broadly speaking, there are important philosophical parallels that exist between modern theories of modality and the theory of modality advocated by Scotus. In part, this is important in our attempts to situate Buridan’s theory, and it is also important when we turn to think about the relationship between the formal systems we have developed for these two figures and the faithfulness of our representations of them.

This leaves us with the question of situating Buridan within these theories. As we have already remarked in a few places, Buridan does not attempt to offer an interpretation or a gloss on what it means for a proposition to be necessary or possible in theTreatise on Consequences. As such, this means that any attempt to situate Buridan’s analysis must be somewhat speculative in nature. From what we know of the life of Scotus, he was lecturing on theSentences at the University of Paris from 1302, subsequently being expelled from France in June 1303, returning to Paris in April 1304. He remained in Paris, likely around October of 1307. From this, it is seems likely that Scotus’ views, both theological and philosophical (including his views on modality) would have been in the air while Buridan was a student in Paris.5 The difficulties become further complicated when we observe that Buridan’s philosophical positions are, generally speaking, not consistent with Scotus. However, there seems to be at least one place where Buridan does make implicit use of Scotus’ views on modality.

Buridan seems to show some awareness of Scotus’ views on modality in hisQuaestiones super octo Physicorum libros Aristotelis. According to Knuuttila,

In question 22 of the first book of the questions on Aristotle’s Physics, Buri- dan analyzes the terms “potency” and “possibility” as follows. One can take these terms to refer (1) to a proposition which expresses something that is possible, (2) to something which can be actualized by the interplay of an active potency and a passive potency which exist in nature, or (3) to some- thing which can be realized by the supreme active potency which is God’s omnipotence. He says that things which are possible in sense (2) are also possible in sense (3) and the same holds of (1) as far as propositions are treated. . . Even though divine omnipotence is the ultimate executive power, the possibilities which can be realised are possible by themselves. As for 4. By ‘possibilistic’ Knuuttila writes: “In the possibilist theories, possible worlds are treated as having

some kind of reality, either all of them as having equal being as Lewis thinks, or the actual world of ours as having the superior sort of being.”[31, p.142]

5. Even this is not entirely clear. Scotus would have likely been a master in the English nation and as such, it is unclear how quickly or completely Scotus’ views on modality would have been disseminated to the other nations in the university. However, it is not unreasonable to suppose that by the time Buridan was an Art’s master (some 15-20 years later) these views would have been known.

the unrealized possible beings (possibilia), Buridan states that they have no existence and are not founded on anything.[32, p.71]

As Knuuttila goes on to point out, the view of possibility that Buridan sketches in the Questions on the Physics is one that only makes sense in the framework of the modal metaphysics developed by Scotus. What this suggests is that by the time Buridan wrote his Quaestiones he was aware of Scotus’ position and was willing to make use of it in understanding the distinction between something that is physically possible/impossible and something that is possible simpliciter. This offers some further evidence for seeing Buridan’s modal theory in the Summulae and perhaps theTreatise on Consequences as being aware of Scotus’ modal theory, albeit developed in a different direction.

One final point, if this sort of theologically informed modal view is what Buridan is working with, then it does make some sense as to why he might not have wanted to offer a gloss or an interpretation of necessity and possibility. As is well-known, the Arts faculty at the University of Pairs had, by the time of Buridan, gotten itself into theological troubles with the church a number of times, perhaps most famously in the condemnation of 1277, but other times as well. By Buridan’s time Arts masters were normally required to promise to not speak on matters outside of their discipline (i.e. on matters of theology) and at least one point in the Treatise on Consequences Buridan makes reference to this.6 Because of this, if Buridan wanted to avoid courting theological controversy, it would be prudent of him to avoid mentioning any more theologically relevant ideas concerning modality than he needs to in order to establish his conclusions about modal consequences.