Aristotle’s theory of the syllogism played an important role in theWestern and Near Eastern intellectual traditions for more than twothousand years, but it was during the Middle Ages that it became thedominant model of correct argumentation.

Historically, medieval logic is divided into the old logic (logicavetus), the tradition stretching from Boethius (c. 480–525)until Abelard (1079–1142), and the new logic (logicanova), from the late twelfth century until the Renaissance. Thedivision reflects the availability of ancient logical texts. BeforeAbelard, medieval logicians were only familiar with Aristotle’sCategories andOn Interpretation and Porphyry’sIsagoge orIntroduction to the Categories and notthePrior Analytics, where Aristotle develops the theory ofthe syllogism — though they did know something of his theorythrough secondary sources. Once thePrior Analyticsreappeared in the West in the middle of the twelfth century,commentaries on it began appearing in the late twelfth and earlythirteenth centuries.

Aristotle’s theory of the syllogism for assertoric (non-modal)sentences was a remarkable achievement and virtually complete in thePrior Analytics. To quote Kant, it was “a closed andcompleted body of doctrine.” Medieval logicians could not addmuch to it, though small changes were sometimes made and it wassystematized in different ways. It was not until the mid-fourteenthcentury, when John Buridan reworked logic in general and placed thetheory of the syllogism in the context of the more comprehensive logicof consequence, that people’s understanding of syllogistic logic beganto change.

The theory of the modal syllogism, however, was incomplete in thePrior Analytics, and in the hands of medieval logicians itsaw a remarkable development. The first commentators tried to saveAristotle’s original theory and in the course of doing so producedsome interesting logical theories, though in the end they were unableto make the system work. The next generation of logicians simplyabandoned the idea of saving Aristotle and instead introduced newdistinctions and developed a completely new theory that subsumed thelogic of syllogisms.

The theory of the syllogisms developed independently in the Arabictradition. Although there was some influence on the Latin traditionthrough Averroes, the dominant influence was Avicenna, who madeseveral changes to Aristotle’s theory and eventually became the soleauthority.

• 1. Aristotle’s Theory
• 2. Boethius
• 3. Arabic Logic and Syllogisms
• 4. Peter Abelard
• 5. The Early Commentators on thePrior Analytics
• 6. Richard of Campsall
• 7. William of Ockham
• 8. John Buridan
• 9. Later Medieval Developments of the Theory
• 10. Summary
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Aristotle’s Theory

In thePrior Analytics, Aristotle presents the first systemof logic, the theory of the syllogism (see the entry onAristotle’s logic and ch. 1 of Lagerlund 2000 for further details). A syllogism is adeduction consisting of three sentences: two premises and aconclusion. Syllogistic sentences are categorical sentences involvinga subject and a predicate connected by a copula (verb). These are inturn divided into four different classes: universal affirmative (a),particular affirmative (i), universal negative (e) and particularnegative (o), written by Aristotle as follows:

The subject and predicate in the categorical sentences used in asyllogism are called terms (horoi) by Aristotle. There arethree terms in a syllogism: a major, a minor, and a middle term. Themajor and the minor are called the extremes (akra), i.e., themajor extreme (meizon akron) and the minor extreme(elatton akron), and they form the predicate and the subjectof the conclusion. The middle (meson) term is what joins thetwo premises. These three terms can be combined in different ways toform three figures (skhemata), which Aristotle presents inthePrior Analytics (A is the major, B the middle, and C theminor term):

When the four categorical sentences are placed into these threefigures, Aristotle ends up with the following 14 valid moods (inparentheses are the medieval mnemonic names for the valid moods; seeSpade 2002, pp. 29–33, and Lagerlund 2008, for the significanceof these names):

First figure: AaB, BaC, therefore, AaC (Barbara); AeB, BaC, therefore,AeC (Celarent); AaB, BiC, therefore, AiC (Darii); AeB, BiC, therefore,AoC (Ferio).

Second figure: BaA, BeC, therefore, AeC (Camestres); BeA, BaC,therefore, AeC (Cesare); BeA, BiC, therefore, AoC (Festino); BaA, BoC,therefore, AoC (Baroco).

Third figure: AaB, CaB, therefore, AiC (Darapti); AeB, CaB, therefore,AoC (Felapton); AiB, CaB, therefore, AiC (Disamis); AaB, CiB,therefore, AiC (Datisi); AoB, CaB, therefore, AoC (Bocardo); AeB, CiB,therefore, AoC (Ferison).

A fourth figure was discussed in ancient times as well as during theMiddle Ages. In Aristotelian syllogistic, it has the followingform:

By taking this figure into account we can derive additional validmoods, all of which are mentioned by Aristotle in thePriorAnalytics (see, e.g.,An. Pr. I.7, 29a19–29). Thefourth figure moods are the following:

Fourth figure: BaA, CaB, therefore, AiC (Bramantip); BaA, CeB,therefore, AeC (Camenes); BiA, CaB, therefore, AiC (Dimaris); BeA,CaB, therefore, AoC (Fesapo); BeA, CiB, therefore, AoC (Fresison).

If we perform a simple calculation based on the four categoricalsentences and the four figures, we find that there are 256 possiblecombinations of sentences. Of these, 24 have traditionally beenthought to yield valid deductions. To the 19 already mentioned we mustadd two subalternate moods in the first figure (Barbari and Celaront),two subalternate moods in the second figure (Camestrop and Cesaro),and one subalternate mood in the fourth figure (Camenop).

The difference between the first figure and the other three figures isthat the syllogisms in the first figure are complete, meaning thatthey are immediately evident and do not require proof. Thisdistinction is important in Aristotle’s theory, since it gives thefirst figure an axiomatic character, so that the proofs of theincomplete syllogisms in the other three figures are arrived atprimarily through reduction to the complete syllogisms.

The reductions of the incomplete syllogisms were made by Aristotlethrough conversion rules. He states the following conversion rules inthePrior Analytics (I.2, 25a1–26):

(1:1)

AaB \(\supset\) BiA,

(1:2)

AiB \(\equiv\) BiA,

(1:3)

AeB \(\equiv\) BeA.

During the Middle Ages, (1:1) was called an accidental (peraccidens) conversion and (1:2) and (1:3) simple(simpliciter) conversions. Particular negative sentences donot convert, according to Aristotle.

Not all incomplete syllogisms were reduced to complete syllogisms;Aristotle also gave other arguments for them. He used two methods toprove the incomplete syllogisms:reductio ad impossibile andekthesis. Thus, he proves Baroco by impossibility, from theassumption that the premises are true and the conclusion false(An. Pr. I.5, 27a36-b1):

1. BaA Premise
2. BoC Premise
3. AaC Assumed as the negation of the conclusion
4. BaC From (i) and (iii) by Barbara
5. \(\bot\) From (ii) and (iv)
6. AoC From (iii) and (v)

Medieval logicians used this method as well, following Aristotle.

Theekthesis proof is more complicated and was not commonlyused by medieval logicians, who preferred proofs through expositorysyllogisms, a simplification and refinement of theekthesis.Aristotle’s method can be expressed in terms of the following rules(Patzig 1968 and Smith 1982):

(1:4)

AiB, therefore, AaC, BaC (where C does not occur previously),

(1:5)

AoB, therefore, AeC, BaC (where C does not occur previously),

(1:6)

AaC, BaC, therefore, AiB,

(1:7)

AeC, BaC, therefore, AoB.

Based on these rules, theekthesis method permitsstraightforward proofs of the third figure syllogisms. Aristotleproves Darapti (An. Pr. I.6, 28a22–26) and mentionsthat Bocardo is provable byekthesis (An. Pr. I.6,28b20–21). The proof of Bocardo is as follows:

1. AoB Premise
2. CaB Premise
3. AeD From (i) and (1:5)
4. BaD From (i) and (1:5)
5. CaD From (ii), (iv) and Barbara
6. AoC From (iii), (v) and (1:7)

Yet this account of theekthesis proof is not without itsproblems. Even in antiquity, Aristotle was accused of arguing in acircle, since (1:6) and (1:7) seem to correspond to the third figureincomplete syllogisms Darapti and Felapton. (1:4)–(1:7) also seemsuperfluous, and in fact Alexander of Aphrodisias (fl. c. 200 AD) wasable to show thatekthesis is really all Aristotle needed,since all the valid moods can be proved with it. Aristotle also usedcounterexamples to show that a mood is invalid.

In Chapters 3 and 8–22 of Book I of thePriorAnalytics, Aristotle extends his theory to include syllogismswith modally qualified categorical sentences. An Aristotelian modalsyllogism is a syllogism that has at least one premise modalized,i.e., that in addition to the standard terms also contains the modalwords ‘necessarily’, ‘possibly’ or‘contingently’. Aristotle’s terminology is not entirelyclear, however. He speaks only of necessity and possibility, though heworks with two notions of possibility. In what seems to be hispreferred sense, used primarily in thePrior Analytics,possibility is defined as that which is not necessary and notimpossible. This sense of possibility was called contingency in theMiddle Ages. But there is another sense of possibility in Aristotle’sOn Interpretation according to which possibility isequivalent to what is not impossible. The first concept ofpossibility, which I will henceforth call ‘contingency’,is used in the modal syllogistic. The second concept is not treatedsystematically in thePrior Analytics.

If we follow this terminology we get eight modal categoricalsentences, which we can raise to twelve if the notion of possibilityis added. If we then perform the same calculation as before, takinginto account the four figures and also the non-modal propositions, weget either 6,912 or 16,384 possible moods. It would be a gargantuantask, of course, to go through them all and see which ones are valid.Accordingly, Aristotle limits his discussion to those modal syllogismswhose assertoric counterparts are valid, as did most medievallogicians.

Aristotle treats modal syllogisms with (i) uniform necessity, (ii)uniform contingency, (iii) mixed necessity and assertoric, (iv) mixedcontingency and assertoric, and (v) mixed necessity and contingencypremises. Possibility sentences are not treated as premises of modalsyllogisms. Sometimes, however, mixed syllogisms are only valid inreaching a possibility conclusion.

Aristotle uses the same methods to prove the incomplete modalsyllogisms as he uses for the assertoric syllogisms, i.e.,conversions,reductio ad impossibile, andekthesis.InAn. Pr. I.3, 25a27–25b26, he accepts the followingconversion rules for necessity, contingency, and possibilitysentences:

(1:8)

Necessarily AaB \(\supset\) Necessarily BiA,

(1:9)

Necessarily AiB \(\equiv\) Necessarily BiA,

(1:10)

Necessarily AeB \(\equiv\) Necessarily BeA,

(1:11)

Contingently AaB \(\supset\) Contingently BiA,

(1:12)

Contingently AiB \(\equiv\) Contingently BiA,

(1:13)

Contingently AeB \(\supset\) Contingently BoA,

(1:14)

Contingently AoB \(\equiv\) Contingently BoA,

(1:15)

Possibly AaB \(\supset\) Possibly BiA,

(1:16)

Possibly AiB \(\equiv\) Possibly BiA,

(1:17)

Possibly AeB \(\equiv\) Possibly BeA.

Aristotle accepts no conversion rules for either necessity orpossibility particular negative sentences, though he does accept twoconversions to the opposite quality for contingency sentences (seeAn. Pr. I.13, 32a30–32b2):

(1:18)

Contingently AaB \(\equiv\) Contingently AeB,

(1:19)

Contingently AiB \(\equiv\) Contingently AoB.

In thePrior Analytics Aristotle gives only vague hints abouthow modal sentences are supposed to be interpreted. The problem isbest illustrated by what is often used as a test for allinterpretations of Aristotle, i.e., the problem of the two Barbarasyllogisms. They are discussed atAn. Pr. I.9:

The problem is that Aristotle accepts the former but not the latter.The question then is: Under which interpretation does the former comeout valid but not the latter?

To solve this problem, it has been common in contemporary discussionsto introduce the distinction betweende dicto anddere modal sentences. I have presented the two syllogisms abovewith ade dicto reading of the modal sentences, i.e., so thatthe modality concerns the way the sentence is or is not true. On thisreading, both Barbara syllogisms seem invalid. But what about thede re reading? The modality in this reading of the sentencesapplies to the manner in which the predicate belongs to the subject.The two syllogisms will then have the following form:

It is equally obvious in these cases that the first syllogism is validwhereas the latter is not, since the latter involves five differentterms. This suggests that Aristotle’s modal syllogistic should begiven ade re interpretation (Becker 1933).

However, if this interpretation is accepted, another problem emerges,namely that the conversion rules are not valid under ade reinterpretation, for if thede re interpretation means thatthe predicate is modified by the mode, the conversion rules will neverbe valid. Consider the following example:

(A necessarily)aB,

which should convert to:

(B necessarily)iA.

‘Necessarily A’ has here been transformed to‘A’, which is a valid move since necessity impliesactuality, but ‘B’ has been transformed to‘necessarily B’, which is an invalid move. The same can besaid for all the modal conversion rules under ade reinterpretation. If, on the other hand, thede dicto readingis maintained, it is easily seen that they are valid in view of thevalidity of the non-modal conversion rules.

However, Aristotle probably did have something like ade rereading of the categorical sentences in mind, as many scholars havecome to realize and as most medievals who read him thought. But if theconversion rules must be given ade dicto interpretation andthe different syllogisms ade re interpretation, the wholesystem seems to collapse. This problem makes a consistentreconstruction of Aristotle’s modal syllogistic using modern modallogic very difficult. (See Becker 1933, Lukasiewicz 1957, Rescher1974, van Rijen 1989, Patterson 1995, Thom 1996, Nortmann 1996 andMalink 2013 for such attempts, and see Hintikka 1973 and Lagerlund2000 for critical reflections on these attempts.)

2. Boethius

The medieval tradition of logic is generally thought to have begunwith Boethius (c. 475–526), who ambitiously tried to preservephilosophical learning from the declining culture of lateantiquity. In fact, however, he was only able to save parts of ancientlogic, primarily Aristotelian logic (see Lee 1984 for a discussion ofAristotle’s syllogistic in late ancient thought). He wroteextensively on the theory of the syllogism, producing a Latintranslation of thePrior Analytics, though it was not usedvery much before the twelfth century (see Aristotle,AnalyticaPriora, and the introduction by Minio-Paluello). He also wrotetwo textbooks on the categorical syllogism:On the CategoricalSyllogism (De syllogismo categorico) andIntroduction to Categorical Syllogisms (Introductio adsyllogismos categoricos) (for the texts, see Migne 1847 andThomsen Thörnqvist 2008). In addition, he produced an interestingbook calledOn Hypothetical Syllogisms (De hypotheticissyllogismis), which will be touched on in the discussion below(see Boethius [Obertello 1969] for the text).

Boethius made no substantial contribution to the theory of thesyllogism, though he was an important transmitter of the theory tolater logicians and his works offer a clear presentation of theAristotelian account. But that presentation differs from Aristotle’sin one important respect. In Boethius, the categorical sentences areconstructed using ‘is’ (’est’) andnot ‘belongs’, as in Aristotle. The four sentences thusbecome:

Put in this way, it is more obvious that they are subject predicatesentences, and moreover, that the syllogisms are deductions ratherthan conditional sentences. As a result, the four figures lookdifferent:

In systematic terms, Boethius’ change makes no difference and allmedieval logicians writing after him adopted it, even though it makesthe first figure syllogisms less evident. According to Aristotle, thefirst syllogism of the first figure (Barbara) should read: ‘Abelongs to all B, B belongs to all C; therefore A belongs to allC’. This is obviously valid by the transitivity of inclusion.But if we line up the same syllogism using Boethian formulation weget: ‘Every B is A, Every C is B; therefore Every C is A’.This is not at all as obvious and we have to switch the places of thepremises to get the same transitivity characteristic: ‘Every Cis B, Every B is A; therefore Every C is A’. Beyond this smallbut significant change, Boethius does not contribute much to thetheory, though he is a little more interested than Aristotle in thedifferent kinds of conversion. His hypothetical syllogistic is, on theother hand, rather novel.

Like most things in the history of logic, hypothetical syllogisticalso begins with Aristotle. In thePrior Analytics, he saysthat every syllogism is either direct or from a hypothesis.Traditional syllogistic is direct and hence all syllogisms that do notfall into the patterns of inference defined by the three Aristotelianfigures, but which are nevertheless valid syllogisms, must behypothetical. Aristotle’s principal example is a syllogism throughimpossibility. If we reason from a hypothesis P via a syllogism to aconclusion Q that is impossible, then we can conclude that not-P istrue and P false (An. Pr. 41a23–30).

In the second century C.E., Alexander of Aphrodisias tried to developthis into a theory of the hypothetical syllogism. What emerged fromhis attempt is something quite strange and even confused, though ithas been studied at great length (see esp. Speca 2001 and the list offurther references there). Boethius’On the HypotheticalSyllogisms is the only remaining early work on this topic.

A hypothetical syllogism is a syllogism in which one or more premisesare hypothetical sentences. Boethius draws the distinction betweencategorical sentences and hypothetical sentences formally by sayingthat a categorical sentence involves a predication whereas ahypothetical sentence involves a condition, i.e., it says thatsomething is,if something else is. Typically such sentencesare conditional sentences such as ‘if P then Q’, thoughBoethius also treats ‘P or Q’ as hypothetical, apparentlybecause he thinks that disjunction can be translated in terms of aconditional sentence. Another characteristic of hypothetical sentencesis that they are made up of categorical sentences.

The basic hypothetical sentences he gives are:

(2:1)

If it is A, then it is B

(2:2)

If it is not-A, then it is not-B

(2:3)

If it is A, then it is not-B

(2:4)

If it is not-A, then it is B

He also considers sentences involving three terms:

(2:5)

If, if it is A, then it is B, then it is C

(2:6)

If it is A, then if it is B, then it is C

though a hypothetical sentence can be even more complicated:

(2:7)

If, if it is A, then it is B, then if it is C, then it is D.

Boethius also thinks that hypothetical sentences can be qualified bymodalities such as necessity or possibility, but he never developsthis idea.

In trying to establish what combination of premises form validinferences he proceeds like Aristotle and develops lists or tables inwhich he can group the valid patterns. The basic sentences (2:1)–(2:4)combined with a simple categorical sentence as the second premise boildown to what we today know asmodus ponens andmodustollens. This led some modern interpreters to think that Boethiuswas developing a sentential logic as the Stoics had done (Dürr1951), but this idea has been rejected by more recent scholars(Obertello 1969, Martin 1991 and Speca 2001). Whatever Boethiusthought he was doing, he was not trying to develop a sentential logic.This becomes obvious if one considers a more complex hypotheticalsyllogism, such as the following, which he accepts as valid:

If Boethius’ logic is a sentential logic his syllogism would betranslatable into the following:

But this is not a valid deduction, since if A is assumed to betrue and B false, the conclusion that follows, not-A, remains falsedespite both premises being true. If, on the other hand, Boethiusmeans his reader not to take them for sentences but rather, followingAristotle’s syllogistic, terms, which carry no truth-value bythemselves, then the inference schema can be given a validinterpretation, as long as one allows for the way that Boethiusunderstood negation and compound conditionals (see Martin 2009,76–78). If a thing is A then it must be that if it is B then itis C. So, if we also have that if it is B then it isnot-C,then it could not be A, so it must be not-A. This example alone thusalready suggests that, despite some formal similarities to sententiallogic, Boethius is actually operating with something more akin to theterm logic that preceded him rather than in some newer, original,pre-modern formulation of sentential logic.

Boethius is conscious of a Stoic logical tradition in which thelogical forms of sentences were distinguished according to theirlinguistic form, such that ‘if … then’ structuresindicate conditionals and ‘or’ structures indicatedisjunctions, making these terms rather like operators on sentences.He seems to be using these ideas to demarcate his hypotheticalsentences, though he is still writing in an Aristotelian fashion anddeveloping an Aristotelian term logic (see Speca 2001 and Marenbon2003: 50–56). This mix makes his logic quite confused, and theconfusion was not sorted out until Abelard was able to develop aproper sentential logic out of Boethius’ suggestions. (SeeMartin 2009 as well.)

3. Arabic Logic and Syllogisms

Arabic logic begins in the middle of the eighth century. As with logicin the Latin tradition, it has its foundation in Ancient Greek logic,primarily in Aristotelian logic and syllogistics. The SyriacChristians adopted a teaching tradition in logic that includedPorphyry’sIsagoge in addition to Aristotle’sCategories,De interpretatione and the first sevenchapters of thePrior Analytics. This teaching tradition wasadopted and spread through the Arab conquest. During the AbbasidCaliphate (750–1258), there was a continuous and growing interest inphilosophy and logic. It is this time period that is often referred toas the ‘Golden Age’ of Arabic philosophy and logic.

Gradually, the wholeOrganon was made available in Arabictranslation (the Arabic tradition was unique in treatingAristotle’sRhetoric andPoetics as part of logic, unlike latertraditions in medieval logic. The first more important Arabic logicianwas Ishâq al-Kindî (d. 870), who wrote a short overview of the wholeof theOrganon. After him, more and more substantial workswere produced. Abû Nasr Alfarabi (d. 950) made the first originalcontributions, writing a series of commentaries on Aristotle, althoughonly his commentary onDe interpretatione has survived.Avicenna seems to have held his work in very high esteem. By far themost important logician in the Arabic tradition, however, was IbnSînâ (d. 1037) or Avicenna, as he was known in the Latin West.

Avicenna had a different attitude to Aristotle’s logic thanlogicians before him. He did not think that Aristotle was necessarilyright. Aristotle had a lot of intuitions about logic that do not allfit together into a coherent whole. They had to be worked out andAvicenna believed that when that happened, it would become clear thatAristotle’s logic was only a fragment of a much larger system. AfterAvicenna, the general character of Arabic logic was no longerAristotelian but Avicennan, which is to say that the texts drawn uponby most logicians were no longer Aristotle’s but Avicenna’s (withthe notable exception of Averroes, known to the Latin tradition as‘the Commentator’, i.e., on Aristotle). One work of Avicenna inparticular became important for subsequent logicians: what is known asAl-Ishârât wa’l Tanbîhat in Arabic andPointers andReminders in English — orRemarks and Admonitions in S.Inati’s translation (Avicenna 1984).

Tony Street (2002, 2004) has identified three things that make logicAvicennan as opposed to Aristotelian: (1) truth-conditions of absolute(or assertoric, i.e., non-modal) sentences are expressed in modalterms; (2) logical properties of so-called descriptional(wasfi) sentences, such as ‘Every B is A while B’, arestudied; and (3) syllogisms are divided into connective andrepetitive. If a logician adopts (1)–(3), he is following Avicenna,according to Street.

In Pointer Two of Path Four, Avicenna introduces distinctions betweendifferent kinds of sentences. The first distinction is betweenabsolute and modal sentences, although absolute sentences turn out tobe modal as well. The basic division is one between absolute sentencesthat are taken to be definite or indefinite with respect to time.

Avicenna talks about three kinds of absolute sentences, all of whichare explicated with reference to time. First are absolute sentencesthat refer to a definite time, but these play no role in hisdiscussion. The other two are the general and the special absolutesentences. General absolute sentences are sentences taken withoutlimitation with respect to time, which means that they have to take inall individuals - past, present and future. Furthermore, the copula istaken to mean that the Bs are As at least sometime, as in ‘Everyhuman is sometimes moving’. A special absolute sentence is asentence with limitations with respect to time, its subject termreferring to individuals at a specific moment in time - although it isnot explicated what moment in time that is. The copula is alsounderstood as a conjunction meaning ‘sometimes B and sometimes notB’, as in ‘Everything running is sometimes walking and sometimesnot walking’.

Avicenna is quick to point out that neither general nor specialabsolute sentences behave as expected. For example, they do not fitthe traditional square of opposition. ‘Every B is A’ on thegeneral reading does not contradict ‘Some B is not A’. Thus, heintroduces one other kind of absolute sentence, namely a perpetualabsolute sentence. In a perpetual sentence the copula is simply readas ‘is always’. The contradictory of a general absolute is theperpetual absolute, and similarly with the special absolute, althoughit will contradict a disjunction of two perpetual sentences (seeStreet 2002 and Lagerlund 2009).

The second distinctive Avicennan thesis is the introduction ofdescriptional (wasfi) sentences. This is again done in thecontext of modal syllogistics, although, such sentences do not need tobe modal at all and Avicenna can be seen to have introduced a logicfor descriptional sentences (Street 2002). The example he gives inPointers and Reminders is:

(3:1)

Everything walking is necessarily moving while walking

The addition of ‘while walking’ restricts all moving things tothose actually walking, which makes the sentence true. Avicennadistinguishes descriptional sentences from substantial sentences. Theexample he gives of a substantial sentence is:

(3:2)

Every human is necessarily an animal

The logic for substantial sentences is different from the logic fordescriptional sentences. A sentences like (3:2) converts according tothe standard Aristotelian conversion rules, so:

(3:3)

Every human being is necessarily an animal

converts into:

(3:4)

Some animal is necessarily a human.

Such sentences are characterized by askath’ hauto (perse) by Aristotle in Posterior Analytics I.4 (see Lagerlund 2000,30–1). Part of what Aristotle said about modal syllogistic is validfor such sentences.

But another group of sentences, such as:

(3:6)

Every literate being is necessarily a human being

are not substantial and hence do not convert, since this convertedsentence is false:

(3:7)

Some human being is necessarily literate.

However, if these are read as descriptional sentences, then they doconvert:

(3:8)

Every literate being is necessarily a human being while literate

converts into

(3:9)

Some human being is necessarily literate while literate.

Descriptional sentences have a syllogistic logic like substantialsentences and Avicenna thinks part of Aristotle’s modal syllogisticcan be worked out using descriptional sentences (for a comparison withsimilar logics found in the thirteenth-century Latin tradition, seeLagerlund 2009). But even though Avicenna sketches a syllogistic fordescriptional sentences inPointers and Reminders, he ismostly concerned with substantial sentences and their logic.

The third distinctive mark of Avicenna’s logic is the distinctionbetween so-called connective and repetitive syllogisms, whichcorresponds roughly to Aristotle’s distinction between categoricaland hypothetical syllogisms.

In his history of Arabic logic, Khaled El-Rouayheb divides Arabiclogic after 1200 into several distinctive periods (El-Rouyaheb 2010).According to him the next period begins with Fakhr al-Din al-Razi.After Razi, the later Arabic logical tradition became disassociatedfrom Aristotle and more narrowly focused on the predicables,definitions, propositions, and syllogisms.

Most thirteenth-century logic can also be described as post-Avicennanin the sense that logicians in this period all took their departurefrom Avicenna rather than from Aristotle. In the fourteenth centuryanother transformation took place and the lengthy summaries found inthe earlier traditions became very rare. Instead of writingcommentaries on the works of Aristotle, Arabic logicians were contentwith writing glosses. Their interest also shifted from formal logic(syllogisms) to semantic concerns.

Arabic logic began to fragment in the fifteenth and sixteenthcenturies and several centers developed. El-Rouyaheb identifiesdistinct Ottoman Turkish, Iranian, Indo-Muslim, North African, andChristian Arabic traditions. These developed independently of oneother, and, according to El-Rouyaheb, it is the Ottoman Turkishtradition that is the most important up to the twentieth century. Thebasic themes outlined by Avicenna remained dominant in this tradition,however.

4. Peter Abelard

Peter Abelard (1079–1142) was one of the first original medievallogicians in the Latin West. His most thorough treatment of the theoryof the syllogism can be found in theDialectica, though heoccasionally discusses it in other works as well, such as theLogica ingredientibus (Minio-Paluello 1958). It is only intheDialectica, however, that the theory is outlined in full.

Since the logic of theDialectica is based on Boethius’commentaries and monographs, we find in it a treatise on categoricalsentences and categorical syllogisms (Tractatus II), andanother on hypothetical sentences and hypothetical syllogisms(Tractatus IV). But neither of these discussions is veryextensive. Taken together, they are shorter than the discussion oftopical inferences, which indicates that Abelard was most interestedin developing a logic for sentences (Green-Pedersen 1984 and Martin1987). His presentation of syllogistic is condensed but highlyoriginal. It reveals that he was not able to study the text ofAristotle’sPrior Analytics in any detail. He must have seenit, but he cannot have had access to a copy himself.

Abelard gives the four standard figures and shows how the second,third, and fourth (he treats the fourth figure as part of the firstfigure with the terms in the conclusion converted) can be reduced tothe first in the standard ways using conversion rules and proofsthrough impossibility, but to clarify and simply the theory he alsopresents rules showing the validity of the different moods. In thefirst figure he gives these rules (I have included the terms A, B, Cto clarify the rules, though they are not in Abelard’s text):

(4:1)

If something A is predicated of something else B universally anda third thing C places the subject B under it universally, then thesame thing C also places the predicate A under it with the same mode,namely universally.

(4:2)

If something A is removed from something else B universally and athird thing C places the subject B under it universally, then thefirst predicate A is removed from the second subject Cuniversally.

(4:3)

If something A is predicated of something else B universally andsome third thing C places the subject B under it particularly, thenthat thing C also places the predicate A under it particularly.

(4:4)

If something A is removed from something else B and a third thingC places the subject B under it particularly, then the first predicateA is removed from the second subject C particularly.

To these he adds two more rules for the second figure:

(4:5)

If something B is removed from some other thing A and a thirdthing C places that predicate B under it, then the first subject A isremoved from the second subject C universally.

(4:6)

If something B is predicated of some other thing A universally andthat predicate is removed from a third thing C universally, then the[first] subject is removed from the same [subject] Cuniversally.

There are three more rules for the third figure:

(4:7)

If two different things A and C are predicated of the same Buniversally, then the first A predicated of the second comes togetherparticularly.

(4.8)

If something B is removed from something A universally andsomething third C is predicated of the same subject B universally,then the first predicate A is removed from the second Cparticularly.

(4:9)

If something A is predicated of something B particularly and thesame B with another predicate C supposits universally, then the firstA is predicated of the second C particularly.

If we allow that the conjuncts in the antecedent of these conditionalstatements can switch places, and that a universal implies aparticular, these rules exhaust the 24 valid syllogisms.

Abelard’s rules 1 and 2 are equivalent to the rules of class inclusionthat later became the subject of much discussion, i.e., the so-calleddici de omni et nullo rules. These rules are based on thetransitivity of class inclusion and were the standard way in whichlater medieval logicians explained how the first figure moods areperfect or evident.

It was elegant of Abelard to lay out these rules that entail the validmoods, but then again, the theory of the syllogism is an elegant andsimple system. The simplicity of his nine rules reflects thesimplicity of Aristotelian syllogistic, since on Aristotle’s view onlythe first two syllogisms and the rules of conversion plus the methodof proof by impossibility and a couple of other consequences areneeded to demonstrate all 24 valid moods.

Abelard’s hypothetical syllogistic does not repeat Boethius’ mistakeof mixing a term logic like the theory of the syllogism with asentential logic. Rather, Abelard’s work should be seen as a verysophisticated development of a sentential logic. I will therefore nottreat it in this overview, since it belongs to the history ofsentential logic rather than syllogistic. It seems that the medievalsalso rather quickly stopped associating the word‘syllogism’ with this theory.

Abelard is also associated with the history of modal logic. He isfamous as the philosopher who introduced the distinction betweende dicto andde re modal sentences. The basicnotions of Abelard’s modal theory are to be found in the introductionto Chapters XII and XIII of his longer commentary on Aristotle’sDe interpretatione (ed. Minio-Paluello 1958). Abelardconcentrates his analysis on the logical structure of modal sentences,introducing some new distinctions and concepts that were latercommonly used by medieval logicians.

According to Abelard, modal terms are strictly speaking adverbsexpressing how something said of the subject is actualized, e.g.,‘well’ or ‘quickly’ or‘necessarily’. Adverbs that do not modify an actualinherence, e.g., ‘possibly’, are called secondary modalterms due to their position in a sentence. Abelard also noticed thatinDe interpretatione 12–13, Aristotle operates withnominal rather than adverbial modes, e.g. ‘it is necessarythat’ or ‘it is possible that’. He seems to haveassumed that Aristotle did this because the nominal modes lead to manymore problems than simple adverbial modes. This is more clearly seenfrom the fact that sentences including nominal modes, such as‘Necesse est Socratem currere’, can be understoodeither adverbially, ‘Socrates runs necessarily’, or, assuggested by the grammar, ‘That Socrates runs isnecessary’. He calls these two alternativesde renecessity sentences andde sensu (orde dicto)necessity sentences, respectively. Abelard seems to be the first toemploy this terminology. Ade re modal sentence expresses themode through which the predicate belongs to the subject. The mode is,therefore, associated with a thing, whereas the mode in thededicto case (as he also calls it) is said of what is expressed bya non-modal sentence.

Abelard also referred to this distinction as the distinction betweenpersonal and impersonal readings of a modal sentence, thedere sense corresponding to the personal reading and thededicto sense to the impersonal reading because when the expression‘necesse est’ or ‘possibileest’ is used at the beginning of a sentence, it lacks apersonal subject. Abelard states that this distinction is related toAristotle’s distinction betweenper divisionem andpercompositionem in theSophistici Elenchi (4,166a23–31). What is new is Abelard’s contention that modaldiscussions should proceed by distinguishing the different possiblereadings of modal sentences, moving on to consider their quantity,quality, and conversion as well as their equipollence and any otherrelations holding between themon these different readings.Abelard’s program thus became the standard operating procedure inmedieval treatises on logic.

After Abelard, equipollence and other relations between modalsentences were commonly presented with the help of the square ofopposition, which Abelard mentions though it does not appear as suchin his works. The square can be taken to refer tode dictomodal sentences or to singularde re modal sentences.Although the distinction betweende dicto andde remodal sentences was common in logical treatises on the properties ofthe terms, syncategorematic terms, and the solution of sophisms,twelfth- and thirteenth-century logicians were mainly interested inthe logical properties of singularde re modal sentences.There is no detailed theory of quantifiedde re modalsentences from this period, and the first movements in this directionby Abelard and his followers were rather confused. A satisfactorytheory ofde re modal sentences did not appear until thefourteenth century, when the various relations between such sentenceswas presented by John Buridan in his octagon of opposition.

Medieval logicians generally assumed that Aristotle dealt withdedicto modal sentences in theDe Interpretatione andde re modal sentences in thePrior Analytics. Inearly commentaries on thePrior Analytics, there is usuallyno mention of Abelard’s distinction between them. One reason may bethat the only theory available concentrated on singularde remodal sentences, which are not part of modal syllogistic as developedby Aristotle.

While thede dicto/de re terminology was used, it was not allthat common. Medieval logicians preferred to use what they took to beAristotle’s terminology, talking about modal sentences in thecomposite sense (in sensu composito) and divided sense(in sensu diviso). The structure of a composite modalsentence can be represented as follows:

(quantity/subject/copula, [quality]/predicate)mode

A composite modal sentence corresponds to ade dicto modalsentence. The word ‘composite’ is used because the mode issaid to qualify the composition of the subject and the predicate. Thestructure of a divided modal sentence can be represented asfollows:

quantity/subject/copula, mode, [quality]/predicate

Here, the mode is thought to qualify the copula and thus to divide thesentence into two parts (hence the name, ‘divided modalsentence’). This type of modal sentence was characterized asde re because what is modified is how things (res)are related to each other, rather than the truth of what is said bythe sentence (dictum) (see Lagerlund 2000: 35–39, andthe entry onmedieval theories of modality for further details).

Like virtually all medievals, Abelard thought that Aristotle’s modalsyllogistic was a theory forde re modal sentences. He saysvery little about it in his logical works, however. In less than fivepages in theDialectica (245–249) he treats modal,oblique, and temporal syllogistic logic. Earlier in the same work, hesays a little about conversion rules. He argues in both theDialectica (195–196) and theLogica(15–16) that the conversion rules can be defended even on ade re reading, but the conversions he discusses are not modalconversions since the mode must be attached to the predicate andfollow the term in the conversion, making the conversion into theconversion of an assertoric sentence. The conversions ofdere modal sentences, as Abelard has defined them, do not hold, asPaul Thom has convincingly shown. (Thom 2003: 57–58.)

There is no modal syllogistic explicitly outlined in any of Abelard’slogical works, though in theDialectica, he exemplifies someof the valid mixed moods: M–M in the first figure, MM– inthe second, and M–M in the third (M represents a possibilitysentence and ‘–’ an assertoric). He also shows thatuniform modal syllogisms are not generally valid, so that MMM is notvalid unless the middle term in the major premise is read with themode attached to it, as in:

A consequence of this, of course, is that the middle term in the minorpremise is ‘possibly B’ and hence no longer a modalsentence. MMM is consequently reduced to M–M.

Anything more systematic than this has to be drawn out from Abelard’sdefinition of modal sentences and their semantic interpretation. Thomhas done this in his book (Thom 2003), where he claims that there is avery specific system developed that is not at all similar toAristotle’s modal system. Abelard was therefore not attempting aninterpretation of Aristotle, but must be seen as developing a newsystem based on his reading ofde re sentences. But thisproject must overcome several problems, particularly since Abelardcannot use the conversion rule.

5. The Early Commentators on thePrior Analytics

The first known commentary on thePrior Analytics in theLatin West is an anonymous work that has recently been edited (ThomsenThörnqvist 2015) but is not yet fully studied (see ThomsenThörnqvist 2010 and 2013 for more details). The author has beencalled Anonymous Aurelianensis III by Sten Ebbesen, who studied partsof the work (Ebbesen 1981). He dates it to c. 1160–80. Thetheory of the assertoric syllogism was repeated and summarized inalmost all logic works from this time, but there are no other majorLatin commentaries that we know of until the 1240’s when RobertKilwardby (d. 1279) wrote hisLiteral Commentary on the Books ofthe Prior Analytics (In libros Priorum Analyticorumexpositio).

By the time Kilwardby wrote his commentary, however, a Latintranslation of a commentary by Averroes on thePriorAnalytics was becoming known in the West. Averroes wrote threekinds of commentaries on Aristotle’s works, called‘minor’, ‘middle’, and ‘major’ or‘great’ based on their length and detail. In the 1220s and1230s, William of Luna translated Averroes’ middle commentary onPorphyry’sIsagoge as well as his middle commentarieson Aristotle’sCategories,On Interpretation,andPrior Analytics. In addition to these, a major commentaryon thePosterior Analytics also became available. In themiddle commentaries, Averroes does not go much beyond Aristotle,adhering closely to the letter of Aristotle’s text and deviating onlyon occasion. Nevertheless, his commentaries played an indispensablerole throughout the later Middle Ages in the teaching and study ofthese difficult texts.

One thing Averroes does do in these commentaries, however, is to builda strong connection between logic and a realist metaphysics, which hada clear influence on thirteenth-century logicians in the Latin West(Lagerlund 2000, 2008). In particular, Averroes’s treatment of modalsyllogistics is profoundly metaphysical. In his commentary on thePrior Analytics, he pursues a line of interpretation which ismore developed in theQuaesitum, a short treatise on mixedsyllogisms (see Uckelman and Lagerlund 2016). In theQuaesitum, Averroes focuses on modal syllogistics anddevelops an interpretation based on the metaphysical nature of theterms involved in different syllogisms. It has been claimed that thisshort work is the final result of his inquiries into modalsyllogistics (Elamrani-Jamal 1995, p.74). TheQuaesitum hasbeen studied by scholars in detail insofar as it clearly influencedRobert Kilwardby (c. 1215–79) (Lagerlund 2000, 32–35; Thom 2003,81–91; Lagerlund 2008, 300–302). Although Kilwardby added nothing ofsubstance to the theory of the assertoric syllogism, hisinterpretation of modal syllogistic is quite remarkable. It was alsovery influential in the thirteenth and early fourteenth centuries.Albert the Great, Simon of Faversham, and Radulphus Brito — inother words, all of the major thirteenth-century commentators on thePrior Analytics — followed Kilwardby in theirinterpretations.

Throughout the commentary, Kilwardby assumes that Aristotle’s theoryis correct and makes it his project to find the interpretation thatshows this. He begins by considering a counterexample to theaccidental conversion of necessity sentences:

(5:1)

Every literate being is necessarily a human being.

According to the conversion rules accepted by Kilwardby, (5:1) shouldconvert to:

(5:2)

Some human being is necessarily literate.

But (5:1) is obviously true whereas (5:2) is false.

As we have seen, this is a common issue forde re readings ofthe modal sentences. Kilwardby assumes that Aristotle’s modalsyllogistic is a logic for divided (de re) sentences. Heproceeds to give two separate solutions to this puzzle. The first isbased on a distinction between different readings of (5:1). Kilwardbyexplains that the subject term of a sentence can stand for the subjectof the inherence (thesuppositum), or for the qualificationthrough which the subject is specified (qualitas/forma). Ifthe term ‘white’ stands for itssuppositum, itrefers to a thing that is white or to ‘that which iswhite’, but if it stands for the quality or form, it refers tothe whiteness inhering in that which is white, rather than to thething in which it inheres. Kilwardby says that in (5:1),‘literate being’ stands for itssuppositum, whichexplains why (5:1) is true, whereas in (5:2) the term is takendifferently as standing for the quality or form. According toKilwardby, the meaning of the original subject term is changed when itno longer stands for thesuppositum (literate being), but forthe abstract quality of being literate, and it is this change thatblocks the conversion. (5:2) is true if it is read as:

(5:3)

Something that is a human being is necessarily that which is literate.

Kilwardby, however, preferred another solution to these difficultiesfor the conversion rules of necessity sentences. The second solutionis based on a distinction between sentences that are necessaryperse and those that are necessaryper accidens. He writes(I, fol. 7rb):

When it is said: ‘Every literate being is necessarily a humanbeing’, the subject is not something that can be saidperse of the predicate, but since ‘literate being’ isnot separated from what belongs to a human being in itself, thesentence is conceded as necessary, though when a sentence is necessaryin this way it is necessaryper accidens. Therefore, whenAristotle says that necessity sentences are convertible, he means onlysentences necessaryper se are convertible.

The idea is here that since ‘human being’ is notpredicatedper se of its subject ‘literatebeing’, the sentence (5:1) is not aper se necessitysentence and therefore not convertible. (5:1) is a necessity sentence,though of theper accidens type, since it is necessarily trueonly in the sense that being human and being literate are notseparable. Kilwardby implies that the relation between the subject andpredicate terms must be of a special kind if a sentence is to becalled necessaryper se. In (5:1), ‘literatebeing’ and ‘human being’ do not have the closeper se relation Kilwardby demands of a convertiblesentence.

Kilwardby thinks that sentencesper se should be understoodfollowingAn. Post. I.4–6, where Aristotle discussesfour different notions ofper se (kath’ hauto)predication, though Kilwardby seems only to have the first two in mindwhen discussingper se necessity. Aristotle says that thefirst type ofper se predication (per se primo modo)occurs when the definition of the subject includes the predicate. Thesecond type ofper se predication (per se secundomodo) occurs when the definition of the predicate includes thesubject. The best characterization of the first type is thegenus/species relation, where the definition of a species includes itsgenus. The second type is often characterized by aproprium(property), since aproprium is included in the definition ofa subject, as in ‘a human being is able to laugh’, wherethe term ‘human being’ is included in the definition ofthe predicate ‘able to laugh’. A sentence isperse necessary if it involves either of these two predications,according to Kilwardby. Necessityper accidens belongs to allother necessity sentences, which lack this intrinsic relation betweensubject and predicate.

Kilwardby also stresses that in aper se necessity sentence,the subject must be ‘something belonging in itself to thatpredicate’ (‘per se aliquod ipsiuspredicati’), by which he seems to mean that the subject hasthe predicate as an essential property, i.e., such that it has thepredicate as a necessary property through itself and not throughsomething else. A syllogistic necessity sentence is then understood asa proposition expressing the essential properties of a thing in agenus/species relationship. He seems to assume that in aperse necessity sentence, the subject term is not an accidental termbut an essential or necessary term, and that the subject isessentially (per se) linked to the predicate rather thanmerely through the weaker relation of inseparability. Consequently, ifthe subject term is necessary and the link is necessary, it followsthat the predicate term cannot be merely a contingent (accidental)term. It must be necessary as well. The Aristotelian theory ofnecessity syllogistic is thus limited to a special class of terms, allof which stand for substances. The same terminology is also used toexplain syllogistic for contingency sentences, which suggests thatKilwardby was trying to develop a uniform and highly originalinterpretation of the theory. A number of recent scholars have offeredsimilar interpretations of Aristotle (see van Rijen 1989, Patterson1995, Thom 1996, and Nortmann 1996).

When they interpreted Aristotle’s modal syllogistic, most medievalssaw the need to introduce a distinction between different kinds ofassertoric sentences. In the mixed syllogism L–L (L represents anecessity sentence), the assertoric minor premise cannot be any kindof assertoric sentence because then the terms could merely beaccidentally connected. Kilwardby therefore introduced a distinctionbetween absolutely (simpliciter) and as-of-now (utnunc) assertoric sentences. The origins of this distinction canbe found in Aristotle (An. Pr. I.15, 34b7–18), butKilwardby of course uses his ownper se/per accidensterminology to spell out the difference. An absolutely assertoricsentence involves aper se predication whereas an as-of-nowassertoric sentence involves aper accidens predication. Inthis way, he can guarantee that an essential connection between theterms in a valid L–L syllogisms is preserved through to theconclusion. This is not unproblematic (see Lagerlund 2000,39–42), though the distinction between different assertoricsentences needed somehow to be made and remained a problem throughoutthe later Middle Ages.

In the end, Kilwardby did not arrive at just the moods accepted byAristotle. For example, he accepts –LL for the first figure,which is not accepted by Aristotle, and does not manage to get–CC and LCC for Disamis in the third figure. There are also someother moods he does not succeed in validating and others still hegrants but which are not accepted by Aristotle. But perhaps Kilwardbygets as close as one can possibly get to making Aristotle’s systemconsistent. (See Knuuttila 1996, Lagerlund 2000, and Thom 2003 and2007.)

6. Richard of Campsall

An important figure in the history of syllogistic logic is Richard ofCampsall (c. 1280/90–1350/60). Sometime before 1308 he wrote hisQuestions on the Books of the Prior Analytics (Questionessuper librum Priorum Analeticorum), a commentary on the firstbook of thePrior Analytics that devotes 14 of its 20questions to modal syllogistic. He seems to think that there isnothing to add to the theory of assertoric syllogistic and hispresentation of it is fairly standard, but he has lots of interestingthings to say about modal syllogistic.

The main development of modal syllogistic in Campsall’s work is hissystematic application of the distinction between composite (dedicto) and divided (de re) modal sentences. Campsallseems to have held that the system of modal syllogisms presented inthePrior Analytics was intended for divided modal sentences,and so he tries to prove that what Aristotle said is basically correctwhen modal sentences are understood in this way. But this turns out tobe a very cumbersome task. It is no surprise that he does not quitesucceed, as he occasionally admits.

In his reply to one of the questions in his commentary, he makes abrief remark about the difference between composite and divided modalsentences. With regard to universal negative necessity sentences hewrites: “[Such a sentence] in the composite sense is singularand signifies that the inherence which it modifies is necessary; inthe divided sense it is universal and does not signify that theinherence that is modified is necessary, but solely that whatever iscontained under the predicate necessarily is removed from whatever iscontained under the subject” (5.38: 110). The universal negativemodal 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 anon-modal proposition expresses (dictum) or, as Campsallsays, when it is predicated of the inherence. He goes on to explainthat a necessity sentence in the composite sense signifies that thecorresponding non-modal sentence is necessarily true. ‘Thatevery B is not A is necessary’ is thus not universal butsingular. When the universal negative necessity proposition is takenin the divided sense, it is universal. The modality does not qualifythedictum as a whole, but only the mode of removal ofwhatever is under the predicate term from whatever is under thesubject term.

Both the conversion rules and the syllogisms for modal sentences inthe composite sense are validated by a small number of consequences,such as:

(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 contingencysentences. These exhaust the theory of syllogism for composite modalsentences and Campsall accordingly spends little time elaboratingit.

It is natural to assume, as Campsall does, that Aristotle meant histheory of modal syllogisms to cover divided modal sentences, since thereading of composite sentences Campsall proposes entails that they areall singular and that Aristotle’s theory is not a theory for singularsentences. Therefore, he must show how the conversion rules can bemade to hold on such a reading of modal sentences.

In his attempt to give Aristotelian modal syllogistic a consistentinterpretation, Campsall is forced to adopt a very artificial readingof divided modal sentences. He is clearly influenced by thesuppositum approach suggested by Kilwardby, but he thinksthat both subject and predicate terms should be taken in this way.Furthermore, he states that the terms in divided modal sentencesshould be taken as standing for that which is now under them. Hebelieves that with these conditions, the conversion rules and almostall of the moods accepted by Aristotle can be shown to be valid.

Campsall also thinks that on such a reading, the following holds:

(6:2)

C can be one of those that are now under B; therefore, it is oneof those that are now under B.

Campsall takes the terms to signify how things actually are now. Ifthe terms in the sentence ‘A can be B’ are taken to standfor the things that are at this very moment under them, then ‘Acan be B’ means the same as ‘A is B’. According toCampsall, ‘Socrates can be white’ should read in thedivided sense ‘That which is Socrates can now be one of thosethat are now white’. If that which is Socrates can be one ofthose that are white now, it is one of them; otherwise, Socrates couldnot have been that particular white being in the first place. Campsallthinks that Socrates can be this white being (B₁) or thatwhite being (B₂) or …, that is, (B₁,B₂, …, B_n), and if Socrates is notactually B₁ now, he is B₂ now, etc., butSocrates will be one of B₁ to B_n now. This isCampsall’s reason for stating (6:2).

This is not as crazy as it might first seem. Consider the followingschema in quantified modal logic:

(6:3)

\(\forall x (Bx \amp \Diamond(c = x) \supset Bc)\).

If (6:3) is an accurate interpretation of (6:2), then it seems truesince \((\Diamond(t = t') \supset(t = t'))\) is true for identity statements in Kripke’s S5 if\(t\) and \(t'\) are rigid designators.

Given his interpretation of divided modal sentences and consequenceslike (6:2), Campsall manages to prove the conversion rules. LikeKilwardby, he approximates Aristotle’s original system but in the enddoes not preserve all of its features. The most interesting featuresof Campsall’s work, however, are not the result of his efforts toprove Aristotle right, but of his apparently successful solutions. Hisconcept of contingency allows for simultaneous alternatives, such thatif something exists, it is possible for it not to exist at that verysame moment. Campsall thus abandons the fundamental Aristotelianprinciple of the necessity of the present (see Knuuttila 1993 and theentry onmedieval theories of modality for discussion of criticisms of this principle in the late thirteenthcentury). But Campsall’s analysis is complicated by the fact that, aswe have seen, he also accepts the principle that what can exist nowdoes exist now, and that what does not exist now is necessarilynon-existent now. In other words, he denies the necessity of thepresent for affirmative sentences and accepts it for negative ones.There is thus an asymmetry between affirmative and negative modalsentences in Campsall’s system.

Accepting simultaneous alternatives and denying the necessity of thepresent are typical of modal semantics and modal logic after Campsall,especially in the work of figures such as William of Ockham and JohnBuridan. It is historically interesting that Campsall employs theseprinciples in his work, even though they are embedded in a theorywhose elements point in another direction, towards Kilwardby.Campsall’s problematization of the necessity of the present alsoindicates that he wants to separate logic from ontology. In manyrespects, he paves the way for the next generation of logicians. Hiscomplicated interpretation also shows that no matter how hard onemight try, there is no way to give a consistent interpretation of whatAristotle says in thePrior Analytics (for discussion, seeLagerlund 2000, Thom 2003 and Knuuttila 2008).

7. William of Ockham

Around the time William of Ockham (c. 1287–1347) wrote hisCompendium of Logic (Summa logicae), medieval logicbegan to change. More emphasis was placed on the theory ofconsequences than the theory of syllogisms. A theory of consequenceswas developed by Abelard in the course of his discussion of topicalinferences and hypothetical syllogisms, and during the thirteenthcentury the basic idea was further developed in treatments of thetopics, but in the fourteenth century works devoted solely toconsequences began to appear (Green-Pedersen 1984). The most famous isJohn Buridan’sTreatise on Consequences (Tractatus deconsequentiis), though earlier authors such as Walter Burley hadalso stressed consequences over syllogisms. Burley probably wrote hisOn the Purity of the Art of Logic (De puritate artelogicae) as a reply to Ockham’s famousSumma. ForOckham, however, syllogisms are still the most important formalinferences, and he devotes most of Book III of theSumma tothem (see Normore 1999 for a recent study of Ockham’s logic).

Like Campsall, Ockham has nothing to add to the theory of assertoricsyllogisms, which was by then well understood. Let us, however, have alook at the proof method using the expository syllogism that medievallogicians such as Ockham seem to have preferred over Aristotle’scumbersome method ofekthesis. Ockham uses this methodfrequently, though not as frequently as Buridan later did.

The method is used to prove the third figure moods. Expositorysyllogisms are perfect for this because the middle term is the subjectof both premises in that figure. Darapti, for example, runs asfollows:

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

The resulting syllogism is an expository syllogism since it hassingular terms as subject terms, and so Darapti is proved. This methodis reminiscent ofekthesis since it involves particularinstantiation, though it is not the same method.

As we have seen, the theory of modal syllogisms was explored in orderto try to save Aristotle’s theory, and this was still the motive ofmost logicians in the first half of the fourteenth century. But Ockhamhimself seems no longer to be interested in this project. His aiminstead seems purely systematic, and in his desire to extend his basicmethods he manages to bring the theory into a whole new light.

The most fundamental distinction in modal syllogistic is of coursethat between composite and divided modal sentences, but divided modalsentences are also equivocal according to Ockham. Using ideasdeveloped in the theory of supposition, he distinguishes betweendivided modal sentences with an ampliated subject term and those witha non-ampliated subject term. Let us look at how Ockham draws the morefundamental distinction.

Ockham proceeds by dividing modal sentences into sentences with adictum (cum dicto) and those without adictum (sine dicto), dividing modal sentences with adictum into composite and divided senses. He adds that amodal sentencecum dicto taken in the divided sense is alwaysequivalent to a modal sentencesine dicto. Ockham expressesthedictum in Latin by an accusative and infinitiveconstruction. Thus, in the sentence ‘That every human being isan animal is necessary’ (in Latin ‘Omnem hominem esseanimal est necessarium’), ‘That every human being isan animal’ (’Omnem hominem esse animal’) isthedictum of the sentence, which has the mode‘necessity’ predicated of it. He treats thedictum as the subject term of the modal sentence, and themode as the predicate term. It is important to distinguish between thedictum and an assertoric sentence. Thedictum iswhat is asserted in an assertoric sentence. Thus, when thedictum is said to be necessary or possible and there is asentence asserting it, such a sentence is necessarily or possibly true(Summa logicae II, 9).

As noted by Campsall, the fact that both thedictum and themode are treated as terms has an important consequence for compositemodal sentences. ‘That every B is A is necessary’ is,according to the reading suggested, not a universal affirmative but asingular affirmative sentence. Ockham adds that one can also call suchsentences universal or particular, depending on whether the originalsentence — that is, the sentence referred to by thedictum — is universal or particular.

The syllogistic for composite modal sentences is equallystraightforward and reduces to a few valid consequences. He explicitlymentions the following six (Summa logicae III-1):

(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 andcompossible, then the conclusion is possible

(7:3)

If the premises of a valid argument are contingent andcompossible, 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 possibilitysentence

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

(7:6)

A necessity sentence entails the corresponding possibilitysentence

It is consequences such as these that give Ockham a syllogistic forcomposite modal sentences. He simply takes the standard assertoricsyllogistic and applies these rules to them. Ockham’s syllogistic fordivided modal sentences, however, is much less straightforward.

In Book III-1, Ockham expresses the equivocal nature of divided modalsentences as follows:

But if the possibility proposition is taken in the divided sense or ifone takes a proposition equivalent to it — such as thepropositions, ‘Every human being can be white’, ‘Awhite being can be black’ and the like — then thisproposition must be distinguished by virtue of the third mode ofequivocation, in the way that a subject can supposit for those thatare or for those that can be, that is, in the way that a subject cansupposit for that about which a thing is verified with a word aboutthe present or for that about which a thing is verified with a wordabout the possible; or else it denotes what it can supposit for, whichI say to exclude quibbling. If this is said about ‘Every whitebeing can be a human being’, then one sense is,‘Everything that is white can be a human being’, and inthis sense it is true as long as nothing is white except a humanbeing. Another sense is, ‘Everything that can be white can be ahuman being’, and this is false, provided either only a humanbeing is white or something other than a human being [is white].

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

(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 reallymeant by this reading of divided possibility sentences. Does he meanby (7:8) that the subject term is ampliated to stand for possiblebeings as well as for actual beings? Can a strict nominalist such asOckham really accept quantification over possible beings? (For thedetails, 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 thesame sense as possibility sentences, but not necessity sentences. Theonly reading he accepts for these sentences is:

(7:10)

(Quantity) what is B is necessarily (quality) A.

(7:10) implies that only actually existing things have necessaryproperties. It is unclear why he thinks this (see Lagerlund 2000:112–114), but it gives his syllogistic an unattractive featurethat has awkward consequences. For example, no conversion rules fordivided necessity sentences are valid and there are also no validmoods in the second figure.

Ockham also discusses syllogisms with mixed composite and dividedmodal premises. He mentions some very interesting consequences in thecourse of this discussion.

(7:11)

That every B is A is possible \(\supset\) Some B is possibly A

(7:12)

That some B is A is possible \(\supset\) Some B is possibly A

(7:13)

That this is A is possible \(\equiv\) This is possibly A

(7:14)

That this is A is contingent \(\equiv\) This is contingently A

(7:15)

That this is A is necessary \(\equiv\) This is necessarily A

In (7:11) and (7:12), the subject terms of divided modal sentencesmust be ampliated for the consequences to hold. It is interesting tonote that for categorical sentences with singular subject terms, thedistinction between composite and divided senses collapses. Hisexample is ‘That Socrates is white is possible’ implies‘Socrates is possibly white’. With the help of theseconsequences he can prove some additional moods to be valid in thedifferent figures. (See Lagerlund 2000: 124–29, and for asystematic reconstruction see Klima 2008.)

8. John Buridan

John Buridan (c. 1300–1361) was the foremost logician of thelater Middle Ages and in his hands the theory of the syllogism wasreworked and developed well beyond anything seen before in the historyof logic. His two most important logical works are theTreatise onConsequence and theSummulae de Dialectica. Thepresentation here is primarily based on theTreatise (forfurther discussion, see Lagerlund 2000, Chapter 5, and Zupko 2003,Chapters 5–6).

In theTreatise, Buridan bases his discussion of thesyllogism on a philosophical semantics that views syllogisticinference as a special case of the much more comprehensive theory ofconsequences. Like his immediate predecessors, he was for the mostpart uninterested in assertoric syllogisms and moves on quickly totemporal, oblique, variation, and modal syllogisms, though this doesnot prevent him from making some original contributions to the theoryof the assertoric syllogism.

According to Buridan, a syllogism is a formal consequence, and sosyllogistic becomes a branch of the theory of formal consequence. Asconsequences, syllogisms are distinguished by their conjunctiveantecedent and single-sentence consequent, and furthermore by theirthree terms — though this last condition is not necessary sinceBuridan also treats of syllogisms with more than three terms.

Buridan treats of the three famous figures and notes that a conclusioncan be either direct or indirect. In an indirect conclusion, the minorterm is predicated of the major instead of the other way around. Sincethe premises are part of a conjunction and together form theantecedent of a consequent, they can easily switch places, which meansthat Buridan can define the fourth figure as a first figure withtransposed premises and an indirect conclusion. Hence, he does notneed to discuss it independently of the first figure.

For Buridan, a formal consequence holds by the principle of uniformsubstitution. It is valid for any uniform substitution of itscategorematic terms. A syllogism is a special kind of formalconsequence since it requires for its validity that terms be conjoinedacross sentences. How the principle of uniform substitution issupposed to work here is a bit tricky and forces him to bring intoplay his general semantics as well as the notion of distribution. Tospell out the relation of their terms and hence the validity of thefirst figure syllogisms, he reformulates the traditionaldici deomni et nullo rules (see King 1985: 71):

(8:1)

Any two terms, which are called the same as a third by reason ofthe same thing for which that third term supposits, not collectively,are correctly called the same as each other.

(8:2)

Any two terms, of which one is called the same as some third termof which the other is called not the same by reason of the same thingfor which that third term supposits, are correctly called not the sameas each other.

One could say a great deal about these rules, but the term that doesmost of the heavy lifting is ‘supposits’. Supposition is atheory of reference and it is the coreferentiality of terms in thedifferent sentences in a syllogism that is the decisive factor indetermining whether the principle of uniform substitution is satisfiedor not. It is at this point that he introduces the theory ofdistribution.

The rules governing the distribution of terms in a sentence are givenas part of his account of common personal supposition. A term isdistributed in a sentence if it is taken to refer to everything itsignifies, such as if the term is in the scope of a universalquantifier. To indicate when a term is distributed he gives fiverules, according to which universals distribute subjects, negativesdistribute predicates, and no other terms are distributed. If we staywithin the square of opposition (Buridan’s theory of distribution, andhence his syllogistic, has wide application, extending far beyond thetraditional A, E, I, and O sentences), this implies that universalaffirmative sentences have their subject terms distributed, universalnegatives have both terms distributed, particular affirmatives haveneither term distributed, and particular negatives have only theirpredicate terms distributed. (For an influential criticism of thistheory of distribution see Geach 1962, and King 1985 for a reply.)

With his theory of distribution in place, Buridan turns to thesyllogisms, and we see now that in order for a combination of premisesto be acceptable, the middle terms must be distributed —otherwise we will not have a formally acceptable consequence. Buridanapproaches the problem in combinatorial fashion. Given the foursentences and two possible positions for each we get 16 possiblecombinations. Some of these can be ruled out immediately based on therules for distribution. A combination with only negative premises willnot work at all; hence EE, EO, OE, and OO must be rejected. II hasboth middle terms undistributed and can thus be rejected. In the firstfigure, IA, OA, and OI have an undistributed middle. The other eightare accepted. In the second figure, we see that AA, AI, and IA must berejected because of an undistributed middle. The other eight areaccepted. In the third figure, IO and OI have an undistributed middlebut the remaining nine combinations are accepted.

At first glance, there are some surprises in Buridan’s presentation ofassertoric syllogistic. In the second figure he accepts indirectconclusions for IEO (Tifesno) and OAO (Robaco), and in the thirdfigure indirect conclusions for AOO (Carbodo), AEO (Lapfeton), and IEO(Rifeson). He also accepts syllogisms concluding to what he calls an“uncommon idiom for negatives,” that is, when thepredicate term precedes the negation, as in ‘Some B A isnot’ (Quaedam B A est non). Such sentences make nosense in English, but Buridan treats them as equivalent to sentenceswhere the predicate term quantified, as in ‘Some B is not someA’. He writes the sentences in this way because otherwise theywould violate his rules for distribution and scope. Syllogismsconcluding to an “uncommon idiom for negatives” addanother three valid forms in the first figure and two in the second.If we tally this up and include all of the indirect conclusions, weget 33 valid moods, as opposed to 19 in Aristotelian syllogistic. Ifwe then add the supplementary subalternate conclusions, we get 24valid moods in traditional syllogistic but 38 in Buridan’ssystematization.

Buridan is quite right to accept these additional 14 moods. They arevalid. But his result is not as dramatic as it seems since the middleterms are either in the subject or predicate position. In the secondand third figure an indirect conclusion becomes equivalent totransposing the premises. Hence Buridan’s Tifesno, Robaco, Carbodo,Lapfeton, and Rifeson reduce to Festino, Baroco, Bocardo, Felapton,and Ferison, respectively. This is also obvious if we look at thenames of these syllogisms, which suggest that Buridan has onlyreshuffled the letters of the names of the standard Aristoteliansyllogisms. After having done all this over just a few pages —as mentioned above, Buridan is rather uninterested in assertoricsyllogistic — he turns to the temporal, oblique, and modalsyllogistic. Of these, it is modal syllogistic to which he devotes themost time.

A temporal syllogism consists of sentences whose copulas involvetemporal ampliation. In such sentences, the supposition of the subjectterm is extended to include past and future things as well as presentthings. The syllogistic for sentences involving oblique terms isimportant for Buridan’s general theory of consequence, since this iswhere we find rules governing the behavior of oblique terms indistributive contexts. His investigation is extraordinarily detailedand extremely rigorous, qualities all the more impressive when weconsider that he did not have the representational tools of modernsymbolic logic.

The syllogistic for composite modal sentences is straightforward andBuridan uses only a couple of pages in theTreatise to sketchits basic structure. The theory of the syllogism for divided modalsentences is given a much more thorough treatment. For Buridan, amodal copula always ampliates its subject term to stand not only forpresent, past, and future things but also possible things, unless thesupposition of the subject term is explicitly restricted to what isactual. On this basis, he can give an exhaustive account of thelogical relations between quantified divided modal sentences, which hepresents in the octagon of opposition. Slightly simplified, andassuming that the complete octagon can be formed by some trivialequivalences holding between the modalities, it can be depicted as inFigure 1:

Figure 1

Together with the octagon he also uses some consequences to prove thevalid syllogisms. He first states the valid conversion rules:

(8:3)

Every B is possibly A \(\supset\) Some A is possibly B

(8:4)

Some B is possibly A \(\equiv\) Some A is possibly B

(8:5)

Every B is necessarily not A \(\equiv\) Every A is necessarily not B

(8:6)

(Quantity) B is contingently A \(\equiv\) (Quantity) B iscontingently not A

All these are valid assuming their subject terms are ampliated. Healso employs the following consequences:

(8:7)

Every B is necessarily not A \(\supset\) Every B is not A

(8:8)

(Quantity) which is B is necessarily A \(\supset\) (Quantity) B is A

(8:9)

(Quantity) B is A \(\supset\) Some B is possibly A

(8:10)

(Quantity) B is necessarily (quality) A \(\supset\) (Quantity) Bis possibly (quality) A

(8:11)

(Quantity) B is contingently (quality) A \(\supset\) (Quantity) B is possibly (quality) A

By stating these consequences and the octagon of opposition, Buridanhas presented a virtually exhaustive syntactical account of modallogic, and, together with his semantics of supposition anddistribution, constructed a powerful logic unmatched by anythingpresented in the history of logic before him.

Buridan uses four methods to prove the valid syllogistic moods. Allfirst figure moods are proved by the rules of class inclusion, thatis, thedici de omni anddici de nullo rules. Thesecond and third figure moods are proved using three differentmethods: either by conversion — that is, by (8:3), (8:4), (8:5)or (8:6) — byreductio ad impossibile, or by expositorysyllogism. Proof by impossibility is used on a few occasions, butBuridan’s approach here differs in no way from Aristotle’s. Thisfourth way is frequently used to prove the valid third figure moods.Since the number of possible combinations of premises and conclusionsin modal syllogistic is quite extensive, he limits himself todiscussing those moods whose assertoric counterparts are valid, buteven so he manages to discuss a large number of valid and invalidsyllogisms (see Hughes 1989, Lagerlund 2000, Thom 2003, Klima 2008,and Dutilh Novaes 2008).

There has been considerable scholarly discussion of Buridan’s modalsyllogistic. It has been asked in particular whether it corresponds toany modern system of modal logic. The most popular answer is S5 (King1985). Some have argued that Buridan must have been thinking in termsof some kind of possible worlds model (Hughes 1989 and Knuuttila1993). Such comparisons of course reflect the extent to which thesescholars have been impressed by Buridan’s modal logic, which waswithout equal until the late twentieth century.(For an example of justhow powerful his general logic was see Parsons 2014.)

9. Later Medieval Developments of the Theory

Syllogistic logic reached the height of its development in Buridan andfor the next two hundred years, little was said about it. Buridan’syounger associates at Paris, Albert of Saxony and Marsilius of Inghen,were both competent logicians, but neither made any substantiveadditions to the theory developed by their master. Paul of Venice wasa well-known early fifteenth-century logician, but he had little tosay about the theory of the syllogism. In the late fifteenth andsixteenth centuries, several very good logicians wrote books on logic,perhaps the most skillful being Jodocus Trutfetter, a follower ofOckham who is better known as a teacher of Martin Luther. ButTrutfetter’s logic is wholly based on Buridan. In his massive work,the modestly titled,Little Compendium of the Whole of Logic(Summulae totius logicae), he extends modal logic beyondBuridan to include discussions of epistemic and doxastic modalities.His treatment of syllogistic is perhaps the most extensive in themedieval tradition.

As noted above, the syllogistic logic of Ockham and Buridan was notprimarily aimed at saving Aristotle. But historical interest inAristotle returned in the latter part of the fifteenth century, andsome scholars, mainly from the Thomistic and Albertist traditions,wanted to know what Aristotle had said about syllogistic. There wasalso the nominalist commentator George of Brussels, who tried to offera historically accurate interpretation of Aristotle together with asystematic account along the lines of Buridan. It is interesting tonote that the modal syllogistic these philosophers ascribe toAristotle is identical to that provided under Kilwardby’sinterpretation. (For further discussion of modal logic in the laterMiddle Ages, see Coombs 1990, Roncaglia 1996, and Lagerlund 2000,Chapter 8.)

10. Summary

The theory of the syllogism was the most important logical theoryduring the Middle Ages and for a long time it was practicallysynonymous with logic as a discipline. Buridan altered this picture bymaking syllogistic part of a much larger and more complex logic ofconsequence.

At first, medieval commentators on Aristotle’sPriorAnalytics sought to save what they took to be Aristotle’soriginal system. Kilwardby thought this could be done by interpretingmodal sentences in light of Aristotle’s metaphysics of essencetogether with his account of essential prediction in thePosteriorAnalytics. This was a very influential interpretation, but it wasultimately abandoned because it did not succeed in savingAristotle.

In the early fourteenth century, Campsall tried to save Aristotle bydeveloping a more radical interpretation restricting the suppositionof subject and predicate terms in modal sentences. This enabled him toprove the conversion rules and many of the syllogistic moods acceptedby Aristotle, but not even this interpretation could make sense of thePrior Analytics.

By the second quarter of the fourteenth century, modal logic had begunto change and new distinctions were used to develop the theory of themodal syllogism, such as the distinction betweende dicto andde re modal sentences. Ockham was the first simply to abandonAristotle’s theory in favor of a newer and more systematic account. Hewas not quite successful, however, and it was left to Buridan tosubsume modal syllogistic as part of his larger project ofsystematizing the whole of logic.

The history of syllogistic does not end with the Middle Ages, ofcourse, but it is fair to say that the theory did not really change inthe six centuries since Buridan. What did change, and for the worse,was people’s knowledge of the original sources and hence also of therichness and sophistication of medieval logic, a state of ignorancethat made the doctrine easy for logicians of the early twentiethcentury to ridicule.

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• Parsons, T., 2014,List of Errata inArticulating Medieval Logic, Oxford: Oxford University Press.
• Uckelman, S., 2017, “Syllogism Mnemonics”, blog post at Medieval Logic and Semantics (medievallogic.wordpress.com).

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