EXPERIMENTO N° 1 PROPORCIÓN DE FRUTAS PARA ELABORAR EL NECTAR

In the previous chapter we spoke at length about Buridan’s theory of the expository syllogism and how a slightly extended version of it could be used to provide a basis for the modal syllogism. At this point, two natural questions present themselves. First, ‘How does Kilwardby conceive of the expository syllogism?’ and second, ‘In what ways does Kilwardby’s conception of the expository syllogism differ from Buridan’s?’ It is to these questions that we now turn.

In his writing on Aristotle’s logic, Kilwardby observes that there are three methods by which a syllogism can be perfected.11 According to Aristotle, first figure syllogisms are perfect and require no further justification to make them evident.12 Hence, one way to perfect a particular non first figure syllogism is to show that one can transform that syllogism into a valid first figure one. For example, Ferison, (‘No B is C’, ‘Some B is A’ therefore ‘Not every A is C’), can be reduced to Ferio, by observing that ‘Some B is A’ is equivalent to ‘Some A is B’. Normally such transformations involve either converting one of the premises, switching the major and minor premise, or moving from a universal premise to a particular one.

The second way is to prove the syllogism per impossibile. This form of inference involves assuming the contradictory of the conclusion and showing that a contradiction13 follows from this assumption together with the major and minor premises. The third way of showing an inference is perfect is by use of an expository syllogism. As we have already seen, the expository syllogism usually involves selecting a singular term and using these singular terms to show that a particular inference is valid. According to Kilwardby, the principle of expository syllogism always yields a valid conclusion and one that is evident to the senses.

Kilwardby writes:

And he [presumably, Aristotle] says that it can shown per impossibile, and 11. The notion of perfectibility is connected to our ability to know if a particular syllogism is valid or

not. Aristotle remarks that: “I call that a perfect syllogism which needs nothing other than what has been stated to make plain what necessarily follows; a syllogism is imperfect, if it needs either one or more propositions which are indeed the necessary consequences of the terms set down, but have not been expressly stated as premises.” Prior Analytics 24b23-26.

12. For Barbara and Celarent seePrior Analytics26a3-12, for Darii and Ferio, seePrior Analytics26a16- 30.

13. Here a contradiction amounts to deriving either (AaB and AoB) or (AiB and AeB) for some terms A and B.

also by exposition. To show it by exposition is to descend to some designated individual and to posit a singular beyond the universal, and so to exhibit what was proposed to the senses. So if some designated object is taken under the middle of which each of the extremes is said it is necessary that one extreme is said particularly of the other extreme.14[61, p.149]

What is important to note here is that, in the case of the assertoric syllogism, the terms selected for the expository syllogism are always singular terms. When we move to the modal context, this will not be the case. The main reason for this should be obvious. As we have already seen, for Kilwardby the truth of necessary propositions do not require that there be anything that necessarily or actually falls under the subject. As such, the move from ‘Some B is necessarily A’ to ‘This C is A’ and ‘This same C is B’ would be invalid if C refers to some object that does not currently exist. Kilwardby is well aware of this problem and fixes it in a natural way. According to Kilwardby, the term ‘C’ in the expository syllogism can either refer to a particular singular object (presumably we can do this if we have already shown that such an object exists) or we can descend to a term that is less general than the terms in the premise, but of which it is true to say, ‘This C is necessarily A’ and ‘This same C is necessarily B’.15

Given what we have already seen formally, the idea is that we need to find the following inferences:

Premise Less General Term

‘Every A is per se necessarily B’ entails A≤B

‘Some A is per se necessarily B’ entails ∃C, such thatC≤A and C≤B

‘No A is per se necessarily B’ entails ¬∃C, such thatC≤A andC≤B

‘Not every A isper se necessarily B’ entails A≰B

Given what we have set up, our theory does not have any problem tracking this theory. We can always make the required inferences. In the worst case, we will have to select one of the terms, (either A or B) to stand in as our ‘less general term’.16

We can reconstruct the negative syllogism Thom considers[60, p.152]:

‘Some C is necessarily not A’, ‘Every B is necessarily A’ therefore ‘Some C is necessarily not B’ as follows: First, select some common term D, such that D ≤ C and D ≰ A. 14. Et dicit quod per impossible ostendi potest; similiter etiam per expositionem. Et est ostendere per expositionem descendere ad aliquid individuum signatum et ponere singulare extra suum universale, et sic ad sensum manifestare propositum. Si ergo sub medio accipiatur aliquid signatum de quo dicitur utrumque extremorum necesse est extremum de extremo dici particulariter.

15. Kilwardby writes: “Et dicendum, ut dicant aliqui, quod non fit hic expositio per singularia vere sed per minus universalia, et illa sumi, dicunt, universaliter sic ‘Necesse est omnem hominem esse animal, necesse est quoddam album non esse animal, ergo necesse est album non esse hominem.’ Et exponi devet ‘album’ non per aliquid signatum sensible, sed per aliquid particulare album cuiusmodi est nix. Et ideo sumi debet universaliter, et fiet sillogismus in secundo secunde sic‘Necesse omnem hominem esse animal, necesse nullam nivem esse animal, ergo necesse est nullam nivem esse hominem.’ Et ita cum nix sit aliquod album, necesse est aliquod album non esse hominem. Consequenter facienda est expositio in quinto tertie, et fiet sillogismus expositorius in secundo tertie. Et ita utrobique fit sillogismus expositorius in eadem figura cum eo qui exponitur, licet non in eodem modo. Sic satis bene dici potest.”

16. This will occur in cases where A is lower bound of a particular sequence in which B occurs and B is immediately above A.

ThenD≰A, together with ‘Every B is necessarily A’ (B ≤A) entails thatD≰B, which together withD≤A entails that ‘Some C is necessarily not B’ as desired.

In fact, what this analysis suggests is that Kilwardby could have, if he so desired, restricted the expository syllogism in the modal case to only allow for the selection of less general terms. If singular terms happened to fall under a particular general term, this would be covered by the inclusion of meaning of the singular terms in the general terms.17 On this reading, it would seem that Kilwardby could have placed a much lower stress on the particular individuals that fall under particular predicates then other medievals (say, Buridan for example) did. What seems to be more important for Kilwardby is the relationships that hold or fail to hold between the particular terms of interest.

As we saw in Buridan, the validity of the expository syllogism was based on two rules, one dealing with the case when one singular term was affirmative and the other negative (the principle of difference) and one when both premises were affirmative (the principle of sameness). What, according to Kilwardby, accounts for the validity of the expository syllogism?

According to Kilwardby, the validity of the expository syllogism is the principle ‘what- ever follows from the consequent also follows from the antecedent.’ He writes:

But it may be asked how the syllogism of this figure is perfected by means of an exposition, since it is necessary that what follows from a universal proposition, follows from a singular proposition because a singular contrac- tion is made in the universal proposition to some of these singulars. And it is said that the necessity of this is seen through the principle: what follows from the consequent follows from the antecedent.18 [61, p.150]

For Kilwardby, the validity of the expository syllogism is grounded in a much simpler way then Buridan’s. His observation is simply that, in each case, the inferences to the singular terms follow from the particular syllogisms, and the inferences from the singular terms/less general terms also meet the request definition of validity.

Unfortunately, while what Kilwardby says is true, it is somewhat uninformative as to why these inferences are valid. Part of this likely stems from Kilwardby’s interest in the syllogism. Unlike Buridan, Kilwardby does not ground his understanding of the syllogism in the expository syllogism. This, combined with Kilwardby’s more textually based approach to the modal syllogism (in contrast with Buridan’s more systematic presentation of a modal syllogism that is weaker then Aristotle’s), suggests that the 17. For this to work, it would be required that singular terms are the ‘lowest’ elements in any sequence of≤. While this assumption was not made in our formal treatment of Kilwardby, it can be accommo- dated. For example, we could select a particular subset,S of terms to be singular terms, and impose requirements on≤andv requiring that each term inS be incomparable with all of the others inS and require that every term not inS is greater than or incomparable with every element inS. 18. Sed queretur de expositione per quam perficit quosdam sillogismos huius figure, qua necessitate se-

quatur ad propositiones universales quod sequitur ad singulares quando fit contractio subiecti in propositionibus universalibus ad hoc aliquod et singulare. Et dicendum quod necessitas patet per hanc maximam: Quod sequitur ad consequens sequitur ad antecedens.

expository syllogism played a much less important role in Kilwardby’s analysis of the syllogism then it did for Buridan.

Likewise, Buridan and Kilwardby’s approach to the expository syllogism in the context of modal propositions is very different. Because of Buridan’s ampliative reading of the subject, Buridan does not run afoul of Kilwardby’s condemnation in 1277 since, in Buridan’s case, the subject is ampliated to cover all of the objects that do or could ever fall under that proposition. However, Buridan’s analysis of the expository syllogism is entirely based on singular terms which is entirely in keeping with the key roles supposition and ampliation play in Buridan’s analysis of the syllogism and his nominalism more generally. In contrast, Kilwardby’s use of the expository syllogism admits both singular terms and less general categorical terms. To a nominalist, the inclusion of terms of various degrees of generality will be unsatisfactory if such terms cannot ultimately be reduced down to the singular terms that either do or do not fall under those other terms. Given Kilwardby’s realist commitments, this is clearly not a problem for his logic.

This reading seems closer to what Aristotle might have had in mind with his use of the expository syllogism, and also allows Kilwardby to preserve the validity of the expository syllogism for modal operations without having to raise any theologically or metaphysically loaded questions. This is not the case for Buridan, as we shall see in the next section.

There is one other tangential question that is worth asking at this point, though Kilwardby does not address the question here: On Kilwardby’s theory, is the expository syllogism naturally valid, like the syllogisms or is it only accidentally valid? From what we have seen formally, we would predict that the inference would be a naturally valid inference, as meaning is preserved when we descend to a common term, and then reascend to the terms above it.