. Richard of Campsall

An important figure in the history of syllogistic logic is Richard of Campsall (c. 1280/90–1350/60). Sometime before 1308 he wrote his Questions on the Books of the Prior Analytics(Questiones super librum Priorum Analeticorum), a commentary on the first book of the Prior Analytics that devotes 14 of its 20 questions to modal syllogistic. He seems to think that there is nothing to add to the theory of assertoric syllogistic and his presentation of it is fairly standard, but he has lots of interesting things to say about modal syllogistic.
The main development of modal syllogistic in Campsall's work is his systematic application of the distinction between composite (de dicto) and divided (de re) modal sentences. Campsall seems to have held that the system of modal syllogisms presented in the Prior Analytics was intended for divided modal sentences, and so he tries to prove that what Aristotle said is basically correct when modal sentences are understood in this way. But this turns out to be a very cumbersome task. It is no surprise that he does not quite succeed, as he occasionally admits.
In his reply to one of the questions in his commentary, he makes a brief remark about the difference between composite and divided modal sentences. With regard to universal negative necessity sentences he writes: “[Such a sentence] in the composite sense is singular and signifies that the inherence which it modifies is necessary; in the divided sense it is universal and does not signify that the inherence that is modified is necessary, but solely that whatever is contained under the predicate necessarily is removed from whatever is contained under the subject” (5.38: 110). The universal negative modal sentence is singular when it is taken in the composite sense, that is, when it is read so that the modality is predicated of what a non-modal proposition expresses (dictum) or, as Campsall says, when it is predicated of the inherence. He goes on to explain that a necessity sentence in the composite sense signifies that the corresponding non-modal sentence is necessarily true. ‘That every B is not A is necessary’ is thus not universal but singular. When the universal negative necessity proposition is taken in the divided sense, it is universal. The modality does not qualify the dictum as a whole, but only the mode of removal of whatever is under the predicate term from whatever is under the subject term.
Both the conversion rules and the syllogisms for modal sentences in the composite sense are validated by a small number of consequences, such as:

اقتباس :: (6:1) If the antecedent is necessary, then the consequent is necessary.

The corresponding non-modal sentence is here assumed to be valid. Similar consequences can be formed for possibility and contingency sentences. These exhaust the theory of syllogism for composite modal sentences and Campsall accordingly spends little time elaborating it.
It is natural to assume, as Campsall does, that Aristotle meant his theory of modal syllogisms to cover divided modal sentences, since the reading of composite sentences Campsall proposes entails that they are all singular and that Aristotle’s theory is not a theory for singular sentences. Therefore, he must show how the conversion rules can be made to hold on such a reading of modal sentences.
In his attempt to give Aristotelian modal syllogistic a consistent interpretation, Campsall is forced to adopt a very artificial reading of divided modal sentences. He is clearly influenced by the suppositum approach suggested by Kilwardby, but he thinks that both subject and predicate terms should be taken in this way. Furthermore, he states that the terms in divided modal sentences should be taken as standing for that which is now under them. He believes that with these conditions, the conversion rules and almost all of the moods accepted by Aristotle can be shown to be valid.
Campsall also thinks that on such a reading, the following holds:

اقتباس :: (6:2) C can be one of those that are now under B; therefore, it is one of those that are now under B.

Campsall takes the terms to signify how things actually are now. If the terms in the sentence ‘A can be B’ are taken to stand for the things that are at this very moment under them, then ‘A can be B’ means the same as ‘A is B’. According to Campsall, ‘Socrates can be white’ should read in the divided sense ‘That which is Socrates can now be one of those that are now white’. If that which is Socrates can be one of those that are white now, it is one of them; otherwise, Socrates could not have been that particular white being in the first place. Campsall thinks that Socrates can be this white being (B1) or that white being (B2) or …, that is, (B1, B2, … , Bn), and if Socrates is not actually B1 now, he is B2 now, etc., but Socrates will be one of B1 to Bn now. This is Campsall's reason for stating (6:2).
This is not as crazy as it might first seem. Consider the following schema in quantified modal logic:

اقتباس :: (6:3) ∀x (Bx & ◊(c = x) ⊃ Bc).

If (6:3) is an accurate interpretation of (6:2), then it seems true since (◊(t = t′) ⊃ (t = t′)) is true for identity statements in Kripke's S5 if t and t′ are rigid designators.
Given his interpretation of divided modal sentences and consequences like (6:2), Campsall manages to prove the conversion rules. Like Kilwardby, he approximates Aristotle’s original system but in the end does not preserve all of its features. The most interesting features of Campsall's work, however, are not the result of his efforts to prove Aristotle right, but of his apparently successful solutions. His concept of contingency allows for simultaneous alternatives, such that if something exists, it is possible for it not to exist at that very same moment. Campsall thus abandons the fundamental Aristotelian principle of the necessity of the present (see Knuuttila 1993 and the entry on medieval theories of modality for discussion of criticisms of this principle in the late thirteenth century). But Campsall's analysis is complicated by the fact that, as we have seen, he also accepts the principle that what can exist now does exist now, and that what does not exist now is necessarily non-existent now. In other words, he denies the necessity of the present for affirmative sentences and accepts it for negative ones. There is thus an asymmetry between affirmative and negative modal sentences in Campsall's system.
Accepting simultaneous alternatives and denying the necessity of the present are typical of modal semantics and modal logic after Campsall, especially in the work of figures such as William of Ockham and John Buridan. It is historically interesting that Campsall employs these principles in his work, even though they are embedded in a theory whose elements point in another direction, towards Kilwardby. Campsall's problematization of the necessity of the present also indicates that he wants to separate logic from ontology. In many respects, he paves the way for the next generation of logicians. His complicated interpretation also shows that no matter how hard one might try, there is no way to give a consistent interpretation of what Aristotle says in the Prior Analytics (for discussion, see Lagerlund 2000, Thom 2003 and Knuuttila 2008).

7. William of Ockham

Around the time William of Ockham (c. 1287–1347) wrote his Compendium of Logic (Summa logicae), medieval logic began to change. More emphasis was placed on the theory of consequences than the theory of syllogisms. A theory of consequences was developed by Abelard in the course of his discussion of topical inferences and hypothetical syllogisms, and during the thirteenth century the basic idea was further developed in treatments of the topics, but in the fourteenth century works devoted solely to consequences began to appear (Green-Pedersen 1984). The most famous is John Buridan's Treatise on Consequences (Tractatus de consequentiis), though earlier authors such as Walter Burley had also stressed consequences over syllogisms. Burley probably wrote his On the Purity of the Art of Logic (De puritate arte logicae) as a reply to Ockham's famous Summa. For Ockham, however, syllogisms are still the most important formal inferences, and he devotes most of Book III of the Summa to them (see Normore 1999 for a recent study of Ockham's logic).
Like Campsall, Ockham has nothing to add to the theory of assertoric syllogisms, which was by then well understood. Let us, however, have a look at the proof method using the expository syllogism that medieval logicians such as Ockham seem to have preferred over Aristotle’s cumbersome method of ekthesis. Ockham uses this method frequently, though not as frequently as Buridan later did.
The method is used to prove the third figure moods. Expository syllogisms are perfect for this because the middle term is the subject of both premises in that figure. Darapti, for example, runs as follows:

اقتباس :

Every B is A

Every B is C

Some C is A

Proofs by expository syllogism are practically self-evident. To prove Darapti, one has only to take a particular instantiation of the two premises to get:

اقتباس :

b is A

b is C

One C is A

The resulting syllogism is an expository syllogism since it has singular terms as subject terms, and so Darapti is proved. This method is reminiscent of ekthesis since it involves particular instantiation, though it is not the same method.
As we have seen, the theory of modal syllogisms was explored in order to try to save Aristotle’s theory, and this was still the motive of most logicians in the first half of the fourteenth century. But Ockham himself seems no longer to be interested in this project. His aim instead seems purely systematic, and in his desire to extend his basic methods he manages to bring the theory into a whole new light.
The most fundamental distinction in modal syllogistic is of course that between composite and divided modal sentences, but divided modal sentences are also equivocal according to Ockham. Using ideas developed in the theory of supposition, he distinguishes between divided modal sentences with an ampliated subject term and those with a non-ampliated subject term. Let us look at how Ockham draws the more fundamental distinction.
Ockham proceeds by dividing modal sentences into sentences with a dictum (cum dicto) and those without a dictum (sine dicto), dividing modal sentences with a dictum into composite and divided senses. He adds that a modal sentence cum dicto taken in the divided sense is always equivalent to a modal sentence sine dicto. Ockham expresses the dictum in Latin by an accusative and infinitive construction. Thus, in the sentence ‘That every human being is an animal is necessary’ (in Latin ‘Omnem hominem esse animal est necessarium’), ‘That every human being is an animal’ (’Omnem hominem esse animal’) is the dictum of the sentence, which has the mode ‘necessity’ predicated of it. He treats the dictum as the subject term of the modal sentence, and the mode as the predicate term. It is important to distinguish between the dictum and an assertoric sentence. The dictum is what is asserted in an assertoric sentence. Thus, when the dictum is said to be necessary or possible and there is a sentence asserting it, such a sentence is necessarily or possibly true (Summa logicae II, 9).
As noted by Campsall, the fact that both the dictum and the mode are treated as terms has an important consequence for composite modal sentences. ‘That every B is A is necessary’ is, according to the reading suggested, not a universal affirmative but a singular affirmative sentence. Ockham adds that one can also call such sentences universal or particular, depending on whether the original sentence — that is, the sentence referred to by the dictum — is universal or particular.
The syllogistic for composite modal sentences is equally straightforward and reduces to a few valid consequences. He explicitly mentions the following six (Summa logicae III-1):

اقتباس :

(7:1)	If the premises of a valid argument are necessary, so is the conclusion
(7:2)	If the premises of a valid argument are possible and compossible, then the conclusion is possible
(7:3)	If the premises of a valid argument are contingent and compossible, then the conclusion is contingent
(7:4)	A necessity sentence, whether in the composite or divided sense, always entails the corresponding assertoric sentence
(7:5)	An assertoric sentence entails the corresponding possibility sentence

(7:4) and (7:5) are used by Ockham to get:

اقتباس :: (7:6) A necessity sentence entails the corresponding possibility sentence

It is consequences such as these that give Ockham a syllogistic for composite modal sentences. He simply takes the standard assertoric syllogistic and applies these rules to them. Ockham's syllogistic for divided modal sentences, however, is much less straightforward.
In Book III-1, Ockham expresses the equivocal nature of divided modal sentences as follows:

اقتباس :: But if the possibility proposition is taken in the divided sense or if one takes a proposition equivalent to it — such as the propositions, ‘Every human being can be white’, ‘A white being can be black’ and the like — then this proposition must be distinguished by virtue of the third mode of equivocation, in the way that a subject can supposit for those that are or for those that can be, that is, in the way that a subject can supposit for that about which a thing is verified with a word about the present or for that about which a thing is verified with a word about the possible; or else it denotes what it can supposit for, which I say to exclude quibbling. If this is said about ‘Every white being can be a human being’, then one sense is, ‘Everything that is white can be a human being’, and in this sense it is true as long as nothing is white except a human being. Another sense is, ‘Everything that can be white can be a human being’, and this is false, provided either only a human being is white or something other than a human being [is white].

Here he clearly states that a possibility sentence has two readings, namely:

اقتباس :: (7:7) (Quantity) what is B can be (quality) A
(7:8) (Quantity) what can be B can be (quality) A

(7:8) has generated some scholarly debate as to what Ockham really meant by this reading of divided possibility sentences. Does he mean by (7:8) that the subject term is ampliated to stand for possible beings as well as for actual beings? Can a strict nominalist such as Ockham really accept quantification over possible beings? (For the details, see Karger 1980, Freddoso 1980, McGrade 1985, Knuuttila 1993: 139–49, and Lagerlund 2000: 107–112.)
Ockham also thinks that contingency sentences are equivocal in the same sense as possibility sentences, but not necessity sentences. The only reading he accepts for these sentences is:

اقتباس :: (7:10) (Quantity) what is B is necessarily (quality) A.

(7:10) implies that only actually existing things have necessary properties. It is unclear why he thinks this (see Lagerlund 2000: 112–114), but it gives his syllogistic an unattractive feature that has awkward consequences. For example, no conversion rules for divided necessity sentences are valid and there are also no valid moods in the second figure.
Ockham also discusses syllogisms with mixed composite and divided modal premises. He mentions some very interesting consequences in the course of this discussion.

اقتباس :: (7:11) That every B is A is possible ⊃ Some B is possibly A
(7:12) That some B is A is possible ⊃ Some B is possibly A
(7:13) That this is A is possible ≡ This is possibly A
(7:14) That this is A is contingent ≡ This is contingently A
(7:15) That this is A is necessary ≡ This is necessarily A

In (7:11) and (7:12), the subject terms of divided modal sentences must be ampliated for the consequences to hold. It is interesting to note that for categorical sentences with singular subject terms, the distinction between composite and divided senses collapses. His example is ‘That Socrates is white is possible’ implies ‘Socrates is possibly white’. With the help of these consequences he can prove some additional moods to be valid in the different figures. (See Lagerlund 2000: 124–29, and for a systematic reconstruction see Klima 2008.)

. Richard of Campsall

7. William of Ockham

. Richard of Campsall :: تعاليق

. Richard of Campsall