Aristotle's theory of the syllogism played an important role in the Western and Near Eastern intellectual traditions for more than two thousand years, but it was during the Middle Ages that it became the dominant model of correct argumentation.
Historically, medieval logic is divided into the old logic (logica vetus), the tradition stretching from Boethius (c. 480–525) until Abelard (1079–1142), and the new logic (logica nova), from the late twelfth century until the Renaissance. The division reflects the availability of ancient logical texts. Before Abelard, medieval logicians were only familiar with Aristotle's Categoriesand On Interpretation and Porphyry's Isagoge or Introduction to the Categories and not thePrior Analytics, where Aristotle develops the theory of the syllogism — though they did know something of his theory through secondary sources. Once the Prior Analytics reappeared in the West in the middle of the twelfth century, commentaries on it began appearing in the late twelfth and early thirteenth centuries.
Aristotle's theory of the syllogism for assertoric (non-modal) sentences was a remarkable achievement and virtually complete in the Prior Analytics. To quote Kant, it was “a closed and completed body of doctrine.” Medieval logicians could not add much to it, though small changes were sometimes made and it was systematized in different ways. It was not until the mid-fourteenth century, when John Buridan reworked logic in general and placed the theory of the syllogism in the context of the more comprehensive logic of consequence, that people's understanding of syllogistic logic began to change.
The theory of the modal syllogism, however, was incomplete in the Prior Analytics, and in the hands of medieval logicians it saw a remarkable development. The first commentators tried to save Aristotle's original theory and in the course of doing so produced some interesting logical theories, though in the end they were unable to make the system work. The next generation of logicians simply abandoned the idea of saving Aristotle and instead introduced new distinctions and developed a completely new theory that subsumed the logic of syllogisms.
The theory of the syllogisms developed independently in the Arabic tradition. Although there was some influence on the Latin tradition through Averroes, the dominant influence was Avicenna, who made several changes to Aristotle's theory and eventually became the sole authority.
[size=30]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:
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- اقتباس :
| 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) |
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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):
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- اقتباس :
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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):
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- اقتباس :
- 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).
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A fourth figure was discussed in ancient times as well as during the Middle Ages. In Aristotelian syllogistic, it has the following form:
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- اقتباس :
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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:
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- اقتباس :
- 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).
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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):
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- اقتباس :
- (1:1) AaB⊃BiA,
(1:2) AiB≡BiA,
(1:3) AeB≡BeA.
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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):
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- BaA Premise
- BoC Premise
- AaC Assumed as the negation of the conclusion
- BaC From (i) and (iii) by Barbara
- ⊥ From (ii) and (iv)
- AoC From (iii) and (v)
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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):
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- اقتباس :
- (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.
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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:
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- AoB Premise
- CaB Premise
- AeD From (i) and (1:5)
- BaD From (i) and (1:5)
- CaD From (ii), (iv) and Barbara
- AoC From (iii), (v) and (1:7)
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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:
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- اقتباس :
- (1:8) Necessarily AaB ⊃ Necessarily BiA,
(1:9) Necessarily AiB ≡ Necessarily BiA,
(1:10) Necessarily AeB ≡ Necessarily BeA,
(1:11) Contingently AaB ⊃ Contingently BiA,
(1:12) Contingently AiB ≡ Contingently BiA,
(1:13) Contingently AeB ⊃ Contingently BoA,
(1:14) Contingently AoB ≡ Contingently BoA,
(1:15) Possibly AaB ⊃ Possibly BiA,
(1:16) Possibly AiB ≡ Possibly BiA,
(1:17) Possibly AeB ≡ Possibly BeA.
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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):
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- اقتباس :
- (1:18) Contingently AaB ≡ Contingently AeB,
(1:19) Contingently AiB ≡ Contingently AoB.
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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:
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- اقتباس :
| It is necessary that AaB |
| BaC | | It is necessary that AaC |
|
| AaB |
| It is necessary that BaC | | It is necessary that AaC |
|
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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:
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- اقتباس :
| AaB |
| (B necessarily)aC | | (A necessarily)aC |
|
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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:
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- اقتباس :
- (A necessarily)aB,
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which should convert to:
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- اقتباس :
- (B necessarily)iA.
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‘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 reinterpretation, 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.)[/size]
Medieval Theories of the Syllogism
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