## 1. Aristotle's Theory



In the *Prior Analytics*, Aristotle presents the first system
of logic, the theory of the syllogism (see the entry on
 Aristotle's logic
 and ch. 1 of Lagerlund 2000 for further details). A syllogism is a
deduction consisting of three sentences: two premises and a
conclusion. Syllogistic sentences are categorical sentences involving
a subject and a predicate connected by a copula (verb). These are in
turn divided into four different classes: universal affirmative (a),
particular affirmative (i), universal negative (e) and particular
negative (o), written by Aristotle as follows: 




|  |  |  |
| --- | --- | --- |
| a | - | A belongs to all B (AaB) |
| i | - | A belongs to some B (AiB) |
| e | - | A does not belong to any B (AeB) |
| o | - | A does not belong to some B (AoB) |



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




|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
| I. |
| A - B |
| B - C |
| A - C |

 | 

|  |
| --- |
| II. |
| B - A |
| B - C |
| A - C |

 | 

|  |
| --- |
| III. |
| A - B |
| C - B |
| A - C |

 |



When the four categorical sentences are placed into these three
figures, Aristotle ends up with the following 14 valid moods (in
parentheses are the medieval mnemonic names for the valid moods; see
Spade 2002, pp. 29-33, and Lagerlund 2008, for the significance
of 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 the
Middle Ages. In Aristotelian syllogistic, it has the following
form:




|  |
| --- |
| IV. |
| B - A |
| C - B |
| A - C |



By taking this figure into account we can derive additional valid
moods, all of which are mentioned by Aristotle in the *Prior
Analytics* (see, e.g., *An. Pr.* I.7, 29a19-29). The
fourth 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 categorical
sentences and the four figures, we find that there are 256 possible
combinations of sentences. Of these, 24 have traditionally been
thought to yield valid deductions. To the 19 already mentioned we must
add 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 is
that the syllogisms in the first figure are complete, meaning that
they are immediately evident and do not require proof. This
distinction is important in Aristotle's theory, since it gives the
first figure an axiomatic character, so that the proofs of the
incomplete syllogisms in the other three figures are arrived at
primarily through reduction to the complete syllogisms.



The reductions of the incomplete syllogisms were made by Aristotle
through conversion rules. He states the following conversion rules in
the *Prior 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 (*per
accidens*) conversion and (1:2) and (1:3) simple
(*simpliciter*) conversions. Particular negative sentences do
not convert, according to Aristotle.



Not all incomplete syllogisms were reduced to complete syllogisms;
Aristotle also gave other arguments for them. He used two methods to
prove the incomplete syllogisms: *reductio ad impossibile* and
*ekthesis*. Thus, he proves Baroco by impossibility, from the
assumption 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.



The *ekthesis* proof is more complicated and was not commonly
used by medieval logicians, who preferred proofs through expository
syllogisms, a simplification and refinement of the *ekthesis*.
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, the *ekthesis* method permits
straightforward proofs of the third figure syllogisms. Aristotle
proves Darapti (*An. Pr.* I.6, 28a22-26) and mentions
that Bocardo is provable by *ekthesis* (*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 the *ekthesis* proof is not without its
problems. Even in antiquity, Aristotle was accused of arguing in a
circle, since (1:6) and (1:7) seem to correspond to the third figure
incomplete syllogisms Darapti and Felapton. (1:4)-(1:7) also seem
superfluous, and in fact Alexander of Aphrodisias (fl. c. 200 AD) was
able to show that *ekthesis* is really all Aristotle needed,
since all the valid moods can be proved with it. Aristotle also used
counterexamples to show that a mood is invalid.



In Chapters 3 and 8-22 of Book I of the *Prior
Analytics*, Aristotle extends his theory to include syllogisms
with modally qualified categorical sentences. An Aristotelian modal
syllogism is a syllogism that has at least one premise modalized,
i.e., that in addition to the standard terms also contains the modal
words 'necessarily', 'possibly' or
'contingently'. Aristotle's terminology is not entirely
clear, however. He speaks only of necessity and possibility, though he
works with two notions of possibility. In what seems to be his
preferred sense, used primarily in the *Prior Analytics*,
possibility is defined as that which is not necessary and not
impossible. This sense of possibility was called contingency in the
Middle Ages. But there is another sense of possibility in Aristotle's
*On Interpretation* according to which possibility is
equivalent to what is not impossible. The first concept of
possibility, which I will henceforth call 'contingency',
is used in the modal syllogistic. The second concept is not treated
systematically in the *Prior Analytics*.



If we follow this terminology we get eight modal categorical
sentences, which we can raise to twelve if the notion of possibility
is added. If we then perform the same calculation as before, taking
into account the four figures and also the non-modal propositions, we
get either 6,912 or 16,384 possible moods. It would be a gargantuan
task, of course, to go through them all and see which ones are valid.
Accordingly, Aristotle limits his discussion to those modal syllogisms
whose assertoric counterparts are valid, as did most medieval
logicians.



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



Aristotle uses the same methods to prove the incomplete modal
syllogisms as he uses for the assertoric syllogisms, i.e.,
conversions, *reductio ad impossibile*, and *ekthesis*.
In *An. Pr.* I.3, 25a27-25b26, he accepts the following
conversion rules for necessity, contingency, and possibility
sentences:



(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 or
possibility particular negative sentences, though he does accept two
conversions to the opposite quality for contingency sentences (see
*An. Pr.* I.13, 32a30-32b2):



(1:18)
Contingently AaB \(\equiv\) Contingently AeB,
(1:19)
Contingently AiB \(\equiv\) Contingently AoB.


In the *Prior Analytics* Aristotle gives only vague hints about
how modal sentences are supposed to be interpreted. The problem is
best illustrated by what is often used as a test for all
interpretations of Aristotle, i.e., the problem of the two Barbara
syllogisms. They are discussed at *An. Pr.* I.9:




|  |
| --- |
| It is necessary that AaB |
| BaC |
| It is necessary that AaC |




|  |
| --- |
| AaB |
| It is necessary that BaC |
| It is necessary that AaC |



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



To solve this problem, it has been common in contemporary discussions
to introduce the distinction between *de dicto* and *de
re* modal sentences. I have presented the two syllogisms above
with a *de dicto* reading of the modal sentences, i.e., so that
the modality concerns the way the sentence is or is not true. On this
reading, both Barbara syllogisms seem invalid. But what about the
*de re* reading? The modality in this reading of the sentences
applies to the manner in which the predicate belongs to the subject.
The two syllogisms will then have the following form:




|  |
| --- |
| (A necessarily)aB |
| BaC |
| (A necessarily)aC |




|  |
| --- |
| AaB |
| (B necessarily)aC |
| (A necessarily)aC |



It is equally obvious in these cases that the first syllogism is valid
whereas the latter is not, since the latter involves five different
terms. This suggests that Aristotle's modal syllogistic should be
given a *de re* interpretation (Becker 1933).



However, if this interpretation is accepted, another problem emerges,
namely that the conversion rules are not valid under a *de re*
interpretation, for if the *de re* interpretation means that
the predicate is modified by the mode, the conversion rules will never
be 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 implies
actuality, but 'B' has been transformed to
'necessarily B', which is an invalid move. The same can be
said for all the modal conversion rules under a *de re*
interpretation. If, on the other hand, the *de dicto* reading
is maintained, it is easily seen that they are valid in view of the
validity of the non-modal conversion rules.



However, Aristotle probably did have something like a *de re*
reading of the categorical sentences in mind, as many scholars have
come to realize and as most medievals who read him thought. But if the
conversion rules must be given a *de dicto* interpretation and
the different syllogisms a *de re* interpretation, the whole
system seems to collapse. This problem makes a consistent
reconstruction of Aristotle's modal syllogistic using modern modal
logic very difficult. (See Becker 1933, Lukasiewicz 1957, Rescher
1974, van Rijen 1989, Patterson 1995, Thom 1996, Nortmann 1996 and
Malink 2013 for such attempts, and see Hintikka 1973 and Lagerlund
2000 for critical reflections on these attempts.)


## 2. Boethius



The medieval tradition of logic is generally thought to have begun
with Boethius (c. 475-526), who ambitiously tried to preserve
philosophical learning from the declining culture of late
antiquity. In fact, however, he was only able to save parts of ancient
logic, primarily Aristotelian logic (see Lee 1984 for a discussion of
Aristotle's syllogistic in late ancient thought). He wrote
extensively on the theory of the syllogism, producing a Latin
translation of the *Prior Analytics*, though it was not used
very much before the twelfth century (see Aristotle, *Analytica
Priora*, and the introduction by Minio-Paluello). He also wrote
two textbooks on the categorical syllogism: *On the Categorical
Syllogism* (*De syllogismo categorico*) and
*Introduction to Categorical Syllogisms* (*Introductio ad
syllogismos categoricos*) (for the texts, see Migne 1847 and
Thomsen Thornqvist 2008). In addition, he produced an interesting
book called *On Hypothetical Syllogisms* (*De hypotheticis
syllogismis*), 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 the
syllogism, though he was an important transmitter of the theory to
later logicians and his works offer a clear presentation of the
Aristotelian account. But that presentation differs from Aristotle's
in one important respect. In Boethius, the categorical sentences are
constructed using 'is' ('*est*') and
not 'belongs', as in Aristotle. The four sentences thus
become:




|  |  |  |
| --- | --- | --- |
| a | - | Every B is A |
| i | - | Some B is A |
| e | - | No B is A |
| o | - | Some B is not A |



Put in this way, it is more obvious that they are subject predicate
sentences, and moreover, that the syllogisms are deductions rather
than conditional sentences. As a result, the four figures look
different:




|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
| I. |
| B - A |
| C - B |
| C - A |

 | 

|  |
| --- |
| II. |
| A - B |
| C - B |
| C - A |

 | 

|  |
| --- |
| III. |
| B - A |
| B - C |
| C - A |

 | 

|  |
| --- |
| IV. |
| A - B |
| B - C |
| C - A |

 |



In systematic terms, Boethius' change makes no difference and all
medieval logicians writing after him adopted it, even though it makes
the first figure syllogisms less evident. According to Aristotle, the
first syllogism of the first figure (Barbara) should read: 'A
belongs to all B, B belongs to all C; therefore A belongs to all
C'. This is obviously valid by the transitivity of inclusion.
But if we line up the same syllogism using Boethian formulation we
get: '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 the
premises to get the same transitivity characteristic: 'Every C
is B, Every B is A; therefore Every C is A'. Beyond this small
but significant change, Boethius does not contribute much to the
theory, though he is a little more interested than Aristotle in the
different kinds of conversion. His hypothetical syllogistic is, on the
other hand, rather novel.



Like most things in the history of logic, hypothetical syllogistic
also begins with Aristotle. In the *Prior Analytics*, he says
that every syllogism is either direct or from a hypothesis.
Traditional syllogistic is direct and hence all syllogisms that do not
fall into the patterns of inference defined by the three Aristotelian
figures, but which are nevertheless valid syllogisms, must be
hypothetical. Aristotle's principal example is a syllogism through
impossibility. If we reason from a hypothesis P via a syllogism to a
conclusion Q that is impossible, then we can conclude that not-P is
true and P false (*An. Pr.* 41a23-30).



In the second century C.E., Alexander of Aphrodisias tried to develop
this into a theory of the hypothetical syllogism. What emerged from
his attempt is something quite strange and even confused, though it
has been studied at great length (see esp. Speca 2001 and the list of
further references there). Boethius' *On the Hypothetical
Syllogisms* is the only remaining early work on this topic.



A hypothetical syllogism is a syllogism in which one or more premises
are hypothetical sentences. Boethius draws the distinction between
categorical sentences and hypothetical sentences formally by saying
that a categorical sentence involves a predication whereas a
hypothetical sentence involves a condition, i.e., it says that
something is, *if* something else is. Typically such sentences
are conditional sentences such as 'if P then Q', though
Boethius also treats 'P or Q' as hypothetical, apparently
because he thinks that disjunction can be translated in terms of a
conditional sentence. Another characteristic of hypothetical sentences
is 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 by
modalities such as necessity or possibility, but he never develops
this idea.



In trying to establish what combination of premises form valid
inferences he proceeds like Aristotle and develops lists or tables in
which he can group the valid patterns. The basic sentences (2:1)-(2:4)
combined with a simple categorical sentence as the second premise boil
down to what we today know as *modus ponens* and *modus
tollens*. This led some modern interpreters to think that Boethius
was developing a sentential logic as the Stoics had done (Durr
1951), but this idea has been rejected by more recent scholars
(Obertello 1969, Martin 1991 and Speca 2001). Whatever Boethius
thought he was doing, he was not trying to develop a sentential logic.
This becomes obvious if one considers a more complex hypothetical
syllogism, such as the following, which he accepts as valid:




|  |
| --- |
| If it is A, then if it is B, then it is C |
| If it is B, then it is not-C |
| It is not-A |



If Boethius' logic is a sentential logic his syllogism would be
translatable into the following:




|  |
| --- |
| A \(\supset\) (B \(\supset\) C) |
| B \(\supset\) \(\sim\)C |
| \(\sim\)A |


 But this is not a valid deduction, since if A is assumed to be
true and B false, the conclusion that follows, not-A, remains false
despite both premises being true. If, on the other hand, Boethius
means his reader not to take them for sentences but rather, following
Aristotle's syllogistic, terms, which carry no truth-value by
themselves, then the inference schema can be given a valid
interpretation, as long as one allows for the way that Boethius
understood negation and compound conditionals (see Martin 2009,
76-78). If a thing is A then it must be that if it is B then it
is C. So, if we also have that if it is B then it is *not*-C,
then it could not be A, so it must be not-A. This example alone thus
already suggests that, despite some formal similarities to sentential
logic, Boethius is actually operating with something more akin to the
term 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 the
logical forms of sentences were distinguished according to their
linguistic form, such that 'if ... then' structures
indicate conditionals and 'or' structures indicate
disjunctions, making these terms rather like operators on sentences.
He seems to be using these ideas to demarcate his hypothetical
sentences, though he is still writing in an Aristotelian fashion and
developing an Aristotelian term logic (see Speca 2001 and Marenbon
2003: 50-56). This mix makes his logic quite confused, and the
confusion was not sorted out until Abelard was able to develop a
proper sentential logic out of Boethius' suggestions. (See
Martin 2009 as well.)


## 3. Arabic Logic and Syllogisms



Arabic logic begins in the middle of the eighth century. As with logic
in the Latin tradition, it has its foundation in Ancient Greek logic,
primarily in Aristotelian logic and syllogistics. The Syriac
Christians adopted a teaching tradition in logic that included
Porphyry's *Isagoge* in addition to Aristotle's
*Categories*, *De interpretatione* and the first seven
chapters of the *Prior Analytics*. This teaching tradition was
adopted and spread through the Arab conquest. During the Abbasid
Caliphate (750-1258), there was a continuous and growing interest in
philosophy and logic. It is this time period that is often referred to
as the 'Golden Age' of Arabic philosophy and logic. 



Gradually, the whole *Organon* was made available in Arabic
translation (the Arabic tradition was unique in treating
Aristotle's
*Rhetoric* and *Poetics* as part of logic, unlike later
traditions in medieval logic. The first more important Arabic logician
was Ishaq al-Kindi (d. 870), who wrote a short overview of the whole
of the *Organon*. After him, more and more substantial works
were produced. Abu Nasr Alfarabi (d. 950) made the first original
contributions, writing a series of commentaries on Aristotle, although
only his commentary on *De interpretatione* has survived.
Avicenna seems to have held his work in very high esteem. By far the
most important logician in the Arabic tradition, however, was Ibn
Sina (d. 1037) or Avicenna, as he was known in the Latin West. 



Avicenna had a different attitude to Aristotle's logic than
logicians before him. He did not think that Aristotle was necessarily
right. Aristotle had a lot of intuitions about logic that do not all
fit together into a coherent whole. They had to be worked out and
Avicenna believed that when that happened, it would become clear that
Aristotle's logic was only a fragment of a much larger system. After
Avicenna, the general character of Arabic logic was no longer
Aristotelian but Avicennan, which is to say that the texts drawn upon
by most logicians were no longer Aristotle's but Avicenna's (with
the notable exception of Averroes, known to the Latin tradition as
'the Commentator', i.e., on Aristotle). One work of Avicenna in
particular became important for subsequent logicians: what is known as
*Al-Isharat wa'l Tanbihat* in Arabic and *Pointers and
Reminders* in English -- or *Remarks and Admonitions* in S.
Inati's translation (Avicenna 1984). 



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



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



Avicenna talks about three kinds of absolute sentences, all of which
are explicated with reference to time. First are absolute sentences
that refer to a definite time, but these play no role in his
discussion. The other two are the general and the special absolute
sentences. General absolute sentences are sentences taken without
limitation with respect to time, which means that they have to take in
all individuals - past, present and future. Furthermore, the copula is
taken to mean that the Bs are As at least sometime, as in 'Every
human is sometimes moving'. A special absolute sentence is a
sentence with limitations with respect to time, its subject term
referring to individuals at a specific moment in time - although it is
not explicated what moment in time that is. The copula is also
understood as a conjunction meaning 'sometimes B and sometimes not
B', as in 'Everything running is sometimes walking and sometimes
not walking'. 



Avicenna is quick to point out that neither general nor special
absolute sentences behave as expected. For example, they do not fit
the traditional square of opposition. 'Every B is A' on the
general reading does not contradict 'Some B is not A'. Thus, he
introduces one other kind of absolute sentence, namely a perpetual
absolute sentence. In a perpetual sentence the copula is simply read
as 'is always'. The contradictory of a general absolute is the
perpetual absolute, and similarly with the special absolute, although
it will contradict a disjunction of two perpetual sentences (see
Street 2002 and Lagerlund 2009). 



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



(3:1)
Everything walking is necessarily moving while walking


The addition of 'while walking' restricts all moving things to
those actually walking, which makes the sentence true. Avicenna
distinguishes descriptional sentences from substantial sentences. The
example 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 for
descriptional sentences. A sentences like (3:2) converts according to
the 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 as *kath' hauto* (*per
se*) by Aristotle in Posterior Analytics I.4 (see Lagerlund 2000,
30-1). Part of what Aristotle said about modal syllogistic is valid
for 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 converted
sentence is false: 



(3:7)
 Some human being is necessarily literate.


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



(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 substantial
sentences and Avicenna thinks part of Aristotle's modal syllogistic
can be worked out using descriptional sentences (for a comparison with
similar logics found in the thirteenth-century Latin tradition, see
Lagerlund 2009). But even though Avicenna sketches a syllogistic for
descriptional sentences in *Pointers and Reminders*, he is
mostly concerned with substantial sentences and their logic. 



The third distinctive mark of Avicenna's logic is the distinction
between so-called connective and repetitive syllogisms, which
corresponds roughly to Aristotle's distinction between categorical
and hypothetical syllogisms.



In his history of Arabic logic, Khaled El-Rouayheb divides Arabic
logic 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 disassociated
from Aristotle and more narrowly focused on the predicables,
definitions, propositions, and syllogisms. 



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



Arabic logic began to fragment in the fifteenth and sixteenth
centuries and several centers developed. El-Rouyaheb identifies
distinct Ottoman Turkish, Iranian, Indo-Muslim, North African, and
Christian Arabic traditions. These developed independently of one
other, and, according to El-Rouyaheb, it is the Ottoman Turkish
tradition that is the most important up to the twentieth century. The
basic themes outlined by Avicenna remained dominant in this tradition,
however. 


## 4. Peter Abelard



Peter Abelard (1079-1142) was one of the first original medieval
logicians in the Latin West. His most thorough treatment of the theory
of the syllogism can be found in the *Dialectica*, though he
occasionally discusses it in other works as well, such as the
*Logica ingredientibus* (Minio-Paluello 1958). It is only in
the *Dialectica*, however, that the theory is outlined in full.




Since the logic of the *Dialectica* is based on Boethius'
commentaries and monographs, we find in it a treatise on categorical
sentences and categorical syllogisms (*Tractatus II*), and
another on hypothetical sentences and hypothetical syllogisms
(*Tractatus IV*). But neither of these discussions is very
extensive. Taken together, they are shorter than the discussion of
topical inferences, which indicates that Abelard was most interested
in developing a logic for sentences (Green-Pedersen 1984 and Martin
1987). His presentation of syllogistic is condensed but highly
original. It reveals that he was not able to study the text of
Aristotle's *Prior Analytics* in any detail. He must have seen
it, 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 first
figure with the terms in the conclusion converted) can be reduced to
the first in the standard ways using conversion rules and proofs
through impossibility, but to clarify and simply the theory he also
presents rules showing the validity of the different moods. In the
first figure he gives these rules (I have included the terms A, B, C
to clarify the rules, though they are not in Abelard's text):



(4:1)
If something A is predicated of something else B universally and
a third thing C places the subject B under it universally, then the
same 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 a
third thing C places the subject B under it universally, then the
first predicate A is removed from the second subject C
universally.
(4:3)
If something A is predicated of something else B universally and
some third thing C places the subject B under it particularly, then
that thing C also places the predicate A under it particularly.
(4:4)
If something A is removed from something else B and a third thing
C places the subject B under it particularly, then the first predicate
A 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 third
thing C places that predicate B under it, then the first subject A is
removed from the second subject C universally.
(4:6)
If something B is predicated of some other thing A universally and
that predicate is removed from a third thing C universally, then the
[first] subject is removed from the same [subject] C
universally.


There are three more rules for the third figure:



(4:7)
If two different things A and C are predicated of the same B
universally, then the first A predicated of the second comes together
particularly.
(4.8)
If something B is removed from something A universally and
something third C is predicated of the same subject B universally,
then the first predicate A is removed from the second C
particularly.
(4:9)
If something A is predicated of something B particularly and the
same B with another predicate C supposits universally, then the first
A is predicated of the second C particularly.


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



Abelard's rules 1 and 2 are equivalent to the rules of class inclusion
that later became the subject of much discussion, i.e., the so-called
*dici de omni et nullo* rules. These rules are based on the
transitivity of class inclusion and were the standard way in which
later medieval logicians explained how the first figure moods are
perfect or evident.



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



Abelard's hypothetical syllogistic does not repeat Boethius' mistake
of mixing a term logic like the theory of the syllogism with a
sentential logic. Rather, Abelard's work should be seen as a very
sophisticated development of a sentential logic. I will therefore not
treat it in this overview, since it belongs to the history of
sentential logic rather than syllogistic. It seems that the medievals
also rather quickly stopped associating the word
'syllogism' with this theory.



Abelard is also associated with the history of modal logic. He is
famous as the philosopher who introduced the distinction between
*de dicto* and *de re* modal sentences. The basic
notions of Abelard's modal theory are to be found in the introduction
to Chapters XII and XIII of his longer commentary on Aristotle's
*De interpretatione* (ed. Minio-Paluello 1958). Abelard
concentrates his analysis on the logical structure of modal sentences,
introducing some new distinctions and concepts that were later
commonly used by medieval logicians.



According to Abelard, modal terms are strictly speaking adverbs
expressing how something said of the subject is actualized, e.g.,
'well' or 'quickly' or
'necessarily'. Adverbs that do not modify an actual
inherence, e.g., 'possibly', are called secondary modal
terms due to their position in a sentence. Abelard also noticed that
in *De interpretatione* 12-13, Aristotle operates with
nominal rather than adverbial modes, e.g. 'it is necessary
that' or 'it is possible that'. He seems to have
assumed that Aristotle did this because the nominal modes lead to many
more problems than simple adverbial modes. This is more clearly seen
from the fact that sentences including nominal modes, such as
'*Necesse est Socratem currere*', can be understood
either adverbially, 'Socrates runs necessarily', or, as
suggested by the grammar, 'That Socrates runs is
necessary'. He calls these two alternatives *de re*
necessity sentences and *de sensu* (or *de dicto*)
necessity sentences, respectively. Abelard seems to be the first to
employ this terminology. A *de re* modal sentence expresses the
mode through which the predicate belongs to the subject. The mode is,
therefore, associated with a thing, whereas the mode in the *de
dicto* case (as he also calls it) is said of what is expressed by
a non-modal sentence.



Abelard also referred to this distinction as the distinction between
personal and impersonal readings of a modal sentence, the *de
re* sense corresponding to the personal reading and the *de
dicto* sense to the impersonal reading because when the expression
'*necesse est*' or '*possibile
est*' is used at the beginning of a sentence, it lacks a
personal subject. Abelard states that this distinction is related to
Aristotle's distinction between *per divisionem* and *per
compositionem* in the *Sophistici Elenchi* (4,
166a23-31). What is new is Abelard's contention that modal
discussions should proceed by distinguishing the different possible
readings of modal sentences, moving on to consider their quantity,
quality, and conversion as well as their equipollence and any other
relations holding between them *on these different readings*.
Abelard's program thus became the standard operating procedure in
medieval treatises on logic.



After Abelard, equipollence and other relations between modal
sentences were commonly presented with the help of the square of
opposition, which Abelard mentions though it does not appear as such
in his works. The square can be taken to refer to *de dicto*
modal sentences or to singular *de re* modal sentences.
Although the distinction between *de dicto* and *de re*
modal sentences was common in logical treatises on the properties of
the terms, syncategorematic terms, and the solution of sophisms,
twelfth- and thirteenth-century logicians were mainly interested in
the logical properties of singular *de re* modal sentences.
There is no detailed theory of quantified *de re* modal
sentences from this period, and the first movements in this direction
by Abelard and his followers were rather confused. A satisfactory
theory of *de re* modal sentences did not appear until the
fourteenth century, when the various relations between such sentences
was presented by John Buridan in his octagon of opposition.



Medieval logicians generally assumed that Aristotle dealt with *de
dicto* modal sentences in the *De Interpretatione* and
*de re* modal sentences in the *Prior Analytics*. In
early commentaries on the *Prior Analytics*, there is usually
no mention of Abelard's distinction between them. One reason may be
that the only theory available concentrated on singular *de re*
modal sentences, which are not part of modal syllogistic as developed
by Aristotle.



While the *de dicto/de re* terminology was used, it was not all
that common. Medieval logicians preferred to use what they took to be
Aristotle's terminology, talking about modal sentences in the
composite sense (*in sensu composito*) and divided sense
(*in sensu diviso*). The structure of a composite modal
sentence can be represented as follows:



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



A composite modal sentence corresponds to a *de dicto* modal
sentence. The word 'composite' is used because the mode is
said to qualify the composition of the subject and the predicate. The
structure of a divided modal sentence can be represented as
follows:



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



Here, the mode is thought to qualify the copula and thus to divide the
sentence into two parts (hence the name, 'divided modal
sentence'). This type of modal sentence was characterized as
*de re* because what is modified is how things (*res*)
are related to each other, rather than the truth of what is said by
the sentence (*dictum*) (see Lagerlund 2000: 35-39, and
the entry on
 medieval theories of modality
 for further details).



Like virtually all medievals, Abelard thought that Aristotle's modal
syllogistic was a theory for *de re* modal sentences. He says
very little about it in his logical works, however. In less than five
pages in the *Dialectica* (245-249) he treats modal,
oblique, and temporal syllogistic logic. Earlier in the same work, he
says a little about conversion rules. He argues in both the
*Dialectica* (195-196) and the *Logica*
(15-16) that the conversion rules can be defended even on a
*de re* reading, but the conversions he discusses are not modal
conversions since the mode must be attached to the predicate and
follow the term in the conversion, making the conversion into the
conversion of an assertoric sentence. The conversions of *de
re* modal sentences, as Abelard has defined them, do not hold, as
Paul Thom has convincingly shown. (Thom 2003: 57-58.)



There is no modal syllogistic explicitly outlined in any of Abelard's
logical works, though in the *Dialectica*, he exemplifies some
of the valid mixed moods: M-M in the first figure, MM- in
the second, and M-M in the third (M represents a possibility
sentence and '-' an assertoric). He also shows that
uniform modal syllogisms are not generally valid, so that MMM is not
valid unless the middle term in the major premise is read with the
mode attached to it, as in:




|  |
| --- |
| Everything which is possibly B is possibly A |
| Every C is possibly B |
| Every C is possibly A |



A consequence of this, of course, is that the middle term in the minor
premise is 'possibly B' and hence no longer a modal
sentence. MMM is consequently reduced to M-M.



Anything more systematic than this has to be drawn out from Abelard's
definition of modal sentences and their semantic interpretation. Thom
has done this in his book (Thom 2003), where he claims that there is a
very specific system developed that is not at all similar to
Aristotle's modal system. Abelard was therefore not attempting an
interpretation of Aristotle, but must be seen as developing a new
system based on his reading of *de re* sentences. But this
project must overcome several problems, particularly since Abelard
cannot use the conversion rule.


## 5. The Early Commentators on the *Prior Analytics*



The first known commentary on the *Prior Analytics* in the
Latin West is an anonymous work that has recently been edited (Thomsen
Thornqvist 2015) but is not yet fully studied (see Thomsen
Thornqvist 2010 and 2013 for more details). The author has been
called Anonymous Aurelianensis III by Sten Ebbesen, who studied parts
of the work (Ebbesen 1981). He dates it to c. 1160-80. The
theory of the assertoric syllogism was repeated and summarized in
almost all logic works from this time, but there are no other major
Latin commentaries that we know of until the 1240's when Robert
Kilwardby (d. 1279) wrote his *Literal Commentary on the Books of
the Prior Analytics* (*In libros Priorum Analyticorum
expositio*). 



By the time Kilwardby wrote his commentary, however, a Latin
translation of a commentary by Averroes on the *Prior
Analytics* was becoming known in the West. Averroes wrote three
kinds of commentaries on Aristotle's works, called
'minor', 'middle', and 'major' or
'great' based on their length and detail. In the 1220s and
1230s, William of Luna translated Averroes' middle commentary on
Porphyry's *Isagoge* as well as his middle commentaries
on Aristotle's *Categories*, *On Interpretation*,
and *Prior Analytics*. In addition to these, a major commentary
on the *Posterior Analytics* also became available. In the
middle commentaries, Averroes does not go much beyond Aristotle,
adhering closely to the letter of Aristotle's text and deviating only
on occasion. Nevertheless, his commentaries played an indispensable
role throughout the later Middle Ages in the teaching and study of
these difficult texts.



One thing Averroes does do in these commentaries, however, is to build
a strong connection between logic and a realist metaphysics, which had
a clear influence on thirteenth-century logicians in the Latin West
(Lagerlund 2000, 2008). In particular, Averroes's treatment of modal
syllogistics is profoundly metaphysical. In his commentary on the
*Prior Analytics*, he pursues a line of interpretation which is
more developed in the *Quaesitum*, a short treatise on mixed
syllogisms (see Uckelman and Lagerlund 2016). In the
*Quaesitum*, Averroes focuses on modal syllogistics and
develops an interpretation based on the metaphysical nature of the
terms involved in different syllogisms. It has been claimed that this
short work is the final result of his inquiries into modal
syllogistics (Elamrani-Jamal 1995, p.74). The *Quaesitum* has
been studied by scholars in detail insofar as it clearly influenced
Robert Kilwardby (c. 1215-79) (Lagerlund 2000, 32-35; Thom 2003,
81-91; Lagerlund 2008, 300-302). Although Kilwardby added nothing of
substance to the theory of the assertoric syllogism, his
interpretation of modal syllogistic is quite remarkable. It was also
very influential in the thirteenth and early fourteenth centuries.
Albert the Great, Simon of Faversham, and Radulphus Brito -- in
other words, all of the major thirteenth-century commentators on the
*Prior Analytics* -- followed Kilwardby in their
interpretations.



Throughout the commentary, Kilwardby assumes that Aristotle's theory
is correct and makes it his project to find the interpretation that
shows this. He begins by considering a counterexample to the
accidental conversion of necessity sentences:



(5:1)
Every literate being is necessarily a human being.


According to the conversion rules accepted by Kilwardby, (5:1) should
convert 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 for *de re* readings of
the modal sentences. Kilwardby assumes that Aristotle's modal
syllogistic is a logic for divided (*de re*) sentences. He
proceeds to give two separate solutions to this puzzle. The first is
based on a distinction between different readings of (5:1). Kilwardby
explains that the subject term of a sentence can stand for the subject
of the inherence (the *suppositum*), or for the qualification
through which the subject is specified (*qualitas/forma*). If
the term 'white' stands for its *suppositum*, it
refers to a thing that is white or to 'that which is
white', but if it stands for the quality or form, it refers to
the whiteness inhering in that which is white, rather than to the
thing in which it inheres. Kilwardby says that in (5:1),
'literate being' stands for its *suppositum*, which
explains why (5:1) is true, whereas in (5:2) the term is taken
differently as standing for the quality or form. According to
Kilwardby, the meaning of the original subject term is changed when it
no longer stands for the *suppositum* (literate being), but for
the abstract quality of being literate, and it is this change that
blocks 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 difficulties
for the conversion rules of necessity sentences. The second solution
is based on a distinction between sentences that are necessary *per
se* and those that are necessary *per accidens*. He writes
(I, fol. 7rb):



> 
> When it is said: 'Every literate being is necessarily a human
> being', the subject is not something that can be said *per
> se* of the predicate, but since 'literate being' is
> not separated from what belongs to a human being in itself, the
> sentence is conceded as necessary, though when a sentence is necessary
> in this way it is necessary *per accidens*. Therefore, when
> Aristotle says that necessity sentences are convertible, he means only
> sentences necessary *per se* are convertible.
> 



The idea is here that since 'human being' is not
predicated *per se* of its subject 'literate
being', the sentence (5:1) is not a *per se* necessity
sentence and therefore not convertible. (5:1) is a necessity sentence,
though of the *per accidens* type, since it is necessarily true
only in the sense that being human and being literate are not
separable. Kilwardby implies that the relation between the subject and
predicate terms must be of a special kind if a sentence is to be
called necessary *per se*. In (5:1), 'literate
being' and 'human being' do not have the close
*per se* relation Kilwardby demands of a convertible
sentence.



Kilwardby thinks that sentences *per se* should be understood
following *An. Post.* I.4-6, where Aristotle discusses
four different notions of *per se* (*kath' hauto*)
predication, though Kilwardby seems only to have the first two in mind
when discussing *per se* necessity. Aristotle says that the
first type of *per se* predication (*per se primo modo*)
occurs when the definition of the subject includes the predicate. The
second type of *per se* predication (*per se secundo
modo*) occurs when the definition of the predicate includes the
subject. The best characterization of the first type is the
genus/species relation, where the definition of a species includes its
genus. The second type is often characterized by a *proprium*
(property), since a *proprium* is included in the definition of
a subject, as in 'a human being is able to laugh', where
the term 'human being' is included in the definition of
the predicate 'able to laugh'. A sentence is *per
se* necessary if it involves either of these two predications,
according to Kilwardby. Necessity *per accidens* belongs to all
other necessity sentences, which lack this intrinsic relation between
subject and predicate.



Kilwardby also stresses that in a *per se* necessity sentence,
the subject must be 'something belonging in itself to that
predicate' ('*per se aliquod ipsius
predicati*'), by which he seems to mean that the subject has
the predicate as an essential property, i.e., such that it has the
predicate as a necessary property through itself and not through
something else. A syllogistic necessity sentence is then understood as
a proposition expressing the essential properties of a thing in a
genus/species relationship. He seems to assume that in a *per
se* necessity sentence, the subject term is not an accidental term
but an essential or necessary term, and that the subject is
essentially (*per se*) linked to the predicate rather than
merely through the weaker relation of inseparability. Consequently, if
the subject term is necessary and the link is necessary, it follows
that the predicate term cannot be merely a contingent (accidental)
term. It must be necessary as well. The Aristotelian theory of
necessity syllogistic is thus limited to a special class of terms, all
of which stand for substances. The same terminology is also used to
explain syllogistic for contingency sentences, which suggests that
Kilwardby was trying to develop a uniform and highly original
interpretation of the theory. A number of recent scholars have offered
similar interpretations of Aristotle (see van Rijen 1989, Patterson
1995, Thom 1996, and Nortmann 1996).



When they interpreted Aristotle's modal syllogistic, most medievals
saw the need to introduce a distinction between different kinds of
assertoric sentences. In the mixed syllogism L-L (L represents a
necessity sentence), the assertoric minor premise cannot be any kind
of assertoric sentence because then the terms could merely be
accidentally connected. Kilwardby therefore introduced a distinction
between absolutely (*simpliciter*) and as-of-now (*ut
nunc*) assertoric sentences. The origins of this distinction can
be found in Aristotle (*An. Pr.* I.15, 34b7-18), but
Kilwardby of course uses his own *per se/per accidens*
terminology to spell out the difference. An absolutely assertoric
sentence involves a *per se* predication whereas an as-of-now
assertoric sentence involves a *per accidens* predication. In
this way, he can guarantee that an essential connection between the
terms in a valid L-L syllogisms is preserved through to the
conclusion. This is not unproblematic (see Lagerlund 2000,
39-42), though the distinction between different assertoric
sentences needed somehow to be made and remained a problem throughout
the later Middle Ages.



In the end, Kilwardby did not arrive at just the moods accepted by
Aristotle. 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 some
other moods he does not succeed in validating and others still he
grants but which are not accepted by Aristotle. But perhaps Kilwardby
gets as close as one can possibly get to making Aristotle's system
consistent. (See Knuuttila 1996, Lagerlund 2000, and Thom 2003 and
2007.)


## 6. 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:



(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:



(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:



(6:3)
\(\forall x (Bx \amp \Diamond(c = x) \supset Bc)\).


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



Given his interpretation of divided modal sentences and 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:




|  |
| --- |
| 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:




|  |
| --- |
| 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):



(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:



(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:



> 
> 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:



(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:



(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.



(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 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.)


## 8. John Buridan



John Buridan (c. 1300-1361) was the foremost logician of the
later Middle Ages and in his hands the theory of the syllogism was
reworked and developed well beyond anything seen before in the history
of logic. His two most important logical works are the *Treatise on
Consequence* and the *Summulae de Dialectica*. The
presentation here is primarily based on the *Treatise* (for
further discussion, see Lagerlund 2000, Chapter 5, and Zupko 2003,
Chapters 5-6). 



In the *Treatise*, Buridan bases his discussion of the
syllogism on a philosophical semantics that views syllogistic
inference as a special case of the much more comprehensive theory of
consequences. Like his immediate predecessors, he was for the most
part uninterested in assertoric syllogisms and moves on quickly to
temporal, oblique, variation, and modal syllogisms, though this does
not prevent him from making some original contributions to the theory
of the assertoric syllogism.



According to Buridan, a syllogism is a formal consequence, and so
syllogistic becomes a branch of the theory of formal consequence. As
consequences, syllogisms are distinguished by their conjunctive
antecedent and single-sentence consequent, and furthermore by their
three terms -- though this last condition is not necessary since
Buridan also treats of syllogisms with more than three terms.



Buridan treats of the three famous figures and notes that a conclusion
can be either direct or indirect. In an indirect conclusion, the minor
term is predicated of the major instead of the other way around. Since
the premises are part of a conjunction and together form the
antecedent of a consequent, they can easily switch places, which means
that Buridan can define the fourth figure as a first figure with
transposed premises and an indirect conclusion. Hence, he does not
need to discuss it independently of the first figure.



For Buridan, a formal consequence holds by the principle of uniform
substitution. It is valid for any uniform substitution of its
categorematic terms. A syllogism is a special kind of formal
consequence since it requires for its validity that terms be conjoined
across sentences. How the principle of uniform substitution is
supposed to work here is a bit tricky and forces him to bring into
play his general semantics as well as the notion of distribution. To
spell out the relation of their terms and hence the validity of the
first figure syllogisms, he reformulates the traditional *dici de
omni et nullo* rules (see King 1985: 71):



(8:1)
Any two terms, which are called the same as a third by reason of
the 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 term
of which the other is called not the same by reason of the same thing
for which that third term supposits, are correctly called not the same
as each other.


One could say a great deal about these rules, but the term that does
most of the heavy lifting is 'supposits'. Supposition is a
theory of reference and it is the coreferentiality of terms in the
different sentences in a syllogism that is the decisive factor in
determining whether the principle of uniform substitution is satisfied
or not. It is at this point that he introduces the theory of
distribution.



The rules governing the distribution of terms in a sentence are given
as part of his account of common personal supposition. A term is
distributed in a sentence if it is taken to refer to everything it
signifies, such as if the term is in the scope of a universal
quantifier. To indicate when a term is distributed he gives five
rules, according to which universals distribute subjects, negatives
distribute predicates, and no other terms are distributed. If we stay
within the square of opposition (Buridan's theory of distribution, and
hence his syllogistic, has wide application, extending far beyond the
traditional A, E, I, and O sentences), this implies that universal
affirmative sentences have their subject terms distributed, universal
negatives have both terms distributed, particular affirmatives have
neither term distributed, and particular negatives have only their
predicate terms distributed. (For an influential criticism of this
theory of distribution see Geach 1962, and King 1985 for a reply.)



With his theory of distribution in place, Buridan turns to the
syllogisms, and we see now that in order for a combination of premises
to be acceptable, the middle terms must be distributed --
otherwise we will not have a formally acceptable consequence. Buridan
approaches the problem in combinatorial fashion. Given the four
sentences and two possible positions for each we get 16 possible
combinations. Some of these can be ruled out immediately based on the
rules for distribution. A combination with only negative premises will
not work at all; hence EE, EO, OE, and OO must be rejected. II has
both middle terms undistributed and can thus be rejected. In the first
figure, IA, OA, and OI have an undistributed middle. The other eight
are accepted. In the second figure, we see that AA, AI, and IA must be
rejected because of an undistributed middle. The other eight are
accepted. In the third figure, IO and OI have an undistributed middle
but the remaining nine combinations are accepted.



At first glance, there are some surprises in Buridan's presentation of
assertoric syllogistic. In the second figure he accepts indirect
conclusions for IEO (Tifesno) and OAO (Robaco), and in the third
figure 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 the
predicate term precedes the negation, as in 'Some B A is
not' (*Quaedam B A est non*). Such sentences make no
sense in English, but Buridan treats them as equivalent to sentences
where the predicate term quantified, as in 'Some B is not some
A'. He writes the sentences in this way because otherwise they
would violate his rules for distribution and scope. Syllogisms
concluding to an "uncommon idiom for negatives" add
another three valid forms in the first figure and two in the second.
If we tally this up and include all of the indirect conclusions, we
get 33 valid moods, as opposed to 19 in Aristotelian syllogistic. If
we then add the supplementary subalternate conclusions, we get 24
valid moods in traditional syllogistic but 38 in Buridan's
systematization.



Buridan is quite right to accept these additional 14 moods. They are
valid. But his result is not as dramatic as it seems since the middle
terms are either in the subject or predicate position. In the second
and third figure an indirect conclusion becomes equivalent to
transposing 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 the
names of these syllogisms, which suggest that Buridan has only
reshuffled the letters of the names of the standard Aristotelian
syllogisms. After having done all this over just a few pages --
as mentioned above, Buridan is rather uninterested in assertoric
syllogistic -- he turns to the temporal, oblique, and modal
syllogistic. Of these, it is modal syllogistic to which he devotes the
most time.



A temporal syllogism consists of sentences whose copulas involve
temporal ampliation. In such sentences, the supposition of the subject
term is extended to include past and future things as well as present
things. The syllogistic for sentences involving oblique terms is
important for Buridan's general theory of consequence, since this is
where we find rules governing the behavior of oblique terms in
distributive contexts. His investigation is extraordinarily detailed
and extremely rigorous, qualities all the more impressive when we
consider that he did not have the representational tools of modern
symbolic logic.



The syllogistic for composite modal sentences is straightforward and
Buridan uses only a couple of pages in the *Treatise* to sketch
its basic structure. The theory of the syllogism for divided modal
sentences is given a much more thorough treatment. For Buridan, a
modal copula always ampliates its subject term to stand not only for
present, past, and future things but also possible things, unless the
supposition of the subject term is explicitly restricted to what is
actual. On this basis, he can give an exhaustive account of the
logical relations between quantified divided modal sentences, which he
presents in the octagon of opposition. Slightly simplified, and
assuming that the complete octagon can be formed by some trivial
equivalences holding between the modalities, it can be depicted as in
Figure 1:



![Figure 1](octagon.jpg)
Figure 1




Together with the octagon he also uses some consequences to prove the
valid 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 is
contingently not A


All these are valid assuming their subject terms are ampliated. He
also 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) B
is 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, Buridan
has presented a virtually exhaustive syntactical account of modal
logic, and, together with his semantics of supposition and
distribution, constructed a powerful logic unmatched by anything
presented in the history of logic before him.



Buridan uses four methods to prove the valid syllogistic moods. All
first figure moods are proved by the rules of class inclusion, that
is, the *dici de omni* and *dici de nullo* rules. The
second and third figure moods are proved using three different
methods: either by conversion -- that is, by (8:3), (8:4), (8:5)
or (8:6) -- by *reductio ad impossibile*, or by expository
syllogism. Proof by impossibility is used on a few occasions, but
Buridan's approach here differs in no way from Aristotle's. This
fourth way is frequently used to prove the valid third figure moods.
Since the number of possible combinations of premises and conclusions
in modal syllogistic is quite extensive, he limits himself to
discussing those moods whose assertoric counterparts are valid, but
even so he manages to discuss a large number of valid and invalid
syllogisms (see Hughes 1989, Lagerlund 2000, Thom 2003, Klima 2008,
and Dutilh Novaes 2008).



There has been considerable scholarly discussion of Buridan's modal
syllogistic. It has been asked in particular whether it corresponds to
any modern system of modal logic. The most popular answer is S5 (King
1985). Some have argued that Buridan must have been thinking in terms
of some kind of possible worlds model (Hughes 1989 and Knuuttila
1993). Such comparisons of course reflect the extent to which these
scholars have been impressed by Buridan's modal logic, which was
without equal until the late twentieth century.(For an example of just
how 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 and
for the next two hundred years, little was said about it. Buridan's
younger associates at Paris, Albert of Saxony and Marsilius of Inghen,
were both competent logicians, but neither made any substantive
additions to the theory developed by their master. Paul of Venice was
a well-known early fifteenth-century logician, but he had little to
say about the theory of the syllogism. In the late fifteenth and
sixteenth centuries, several very good logicians wrote books on logic,
perhaps the most skillful being Jodocus Trutfetter, a follower of
Ockham who is better known as a teacher of Martin Luther. But
Trutfetter'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 beyond
Buridan to include discussions of epistemic and doxastic modalities.
His treatment of syllogistic is perhaps the most extensive in the
medieval tradition. 



As noted above, the syllogistic logic of Ockham and Buridan was not
primarily aimed at saving Aristotle. But historical interest in
Aristotle returned in the latter part of the fifteenth century, and
some scholars, mainly from the Thomistic and Albertist traditions,
wanted to know what Aristotle had said about syllogistic. There was
also the nominalist commentator George of Brussels, who tried to offer
a historically accurate interpretation of Aristotle together with a
systematic account along the lines of Buridan. It is interesting to
note that the modal syllogistic these philosophers ascribe to
Aristotle is identical to that provided under Kilwardby's
interpretation. (For further discussion of modal logic in the later
Middle Ages, see Coombs 1990, Roncaglia 1996, and Lagerlund 2000,
Chapter 8.)


## 10. Summary



The theory of the syllogism was the most important logical theory
during the Middle Ages and for a long time it was practically
synonymous with logic as a discipline. Buridan altered this picture by
making syllogistic part of a much larger and more complex logic of
consequence. 



At first, medieval commentators on Aristotle's *Prior
Analytics* sought to save what they took to be Aristotle's
original system. Kilwardby thought this could be done by interpreting
modal sentences in light of Aristotle's metaphysics of essence
together with his account of essential prediction in the *Posterior
Analytics*. This was a very influential interpretation, but it was
ultimately abandoned because it did not succeed in saving
Aristotle.



In the early fourteenth century, Campsall tried to save Aristotle by
developing a more radical interpretation restricting the supposition
of subject and predicate terms in modal sentences. This enabled him to
prove the conversion rules and many of the syllogistic moods accepted
by Aristotle, but not even this interpretation could make sense of the
*Prior Analytics*.



By the second quarter of the fourteenth century, modal logic had begun
to change and new distinctions were used to develop the theory of the
modal syllogism, such as the distinction between *de dicto* and
*de re* modal sentences. Ockham was the first simply to abandon
Aristotle's theory in favor of a newer and more systematic account. He
was not quite successful, however, and it was left to Buridan to
subsume modal syllogistic as part of his larger project of
systematizing the whole of logic.



The history of syllogistic does not end with the Middle Ages, of
course, but it is fair to say that the theory did not really change in
the six centuries since Buridan. What did change, and for the worse,
was people's knowledge of the original sources and hence also of the
richness and sophistication of medieval logic, a state of ignorance
that made the doctrine easy for logicians of the early twentieth
century to ridicule.