Lesson 13 · The syllogism II: the general rules, the four figures, Figure 1
Sources: Book pp. 41–45; S5 slides 9–10, 14.
1. Five rules for every iqtirānī syllogism
Break any of these and nothing follows.
1. The middle term must be repeated exactly, with the same meaning. "There is a mouse in the wall; every mouse has two ears" does not give "the wall has two ears". The predicate of the first premise is "has a mouse in it", not "mouse". (To get that conclusion you would need "everything with a mouse in it has two ears", which is false.) Likewise "Every water is a liquid; some liquids catch fire" fails: the liquid that is water and the liquid that burns (oil) are not the same "liquid".
2. No conclusion from two negative premises. "No human is a horse; no horse is rational" gives nothing (and "no human is rational" is false). Being kept apart from the same third thing tells you nothing about how two things relate to each other. S5 slide 7: A ≠ B and B ≠ C doesn't imply A ≠ C.
3. No conclusion from two particular premises. "Some humans are animals; some animals are horses" does not give "some humans are horses". The two "somes" may be different parts of the middle.
4. The conclusion follows the weaker premise. If either premise is negative, the conclusion is negative; if either is particular, the conclusion is particular. A conclusion grows out of its premises, so it can never assert more than they do.
5. No conclusion from a negative minor premise with a particular major premise. "No crow is human; some humans are white" gives neither "some crows are white" nor "some crows are not white".
2. The four figures
A figure (shakl; S5: "type") is fixed by where the middle term (M) sits. Write S for the minor term and P for the major term.
| Figure | Minor premise | Major premise | Where M sits (S5 slide 9) |
|---|---|---|---|
| 1 | S – M | M – P | predicate of the minor, subject of the major |
| 2 | S – M | P – M | predicate of both |
| 3 | M – S | M – P | subject of both |
| 4 | M – S | P – M | subject of the minor, predicate of the major (Figure 1 reversed) |
Book slip (p. 43): the book's definition of Figure 1 is reversed. Its own examples ("Every A is B; every B is C") and S5 slide 9 have it right: in Figure 1 the middle is the predicate of the minor premise and the subject of the major.
How easy each is (S5 slide 14): Figure 1 is the most natural; its conclusion is obvious and needs no proof. Figure 2 comes next (the minor term keeps its place), then Figure 3. Figure 4 is the hardest, and many logicians dropped it altogether.
Moods. Each premise can be A, E, I or O, so every figure has 4 × 4 = 16 possible pairings (ḍurūb, "moods"). The ones that yield a conclusion are productive; the rest are barren (ʿaqīm).
3. Figure 1
Conditions
- The minor premise must be affirmative. It is what places the minor term under the middle. "No rock is a plant; every plant grows" gives nothing about rocks and growing.
- The major premise must be universal. It must say something about all of the middle, so that it surely reaches the part the minor term occupies. "Every water is a liquid; some liquids catch fire" gives nothing.
The idea behind both: whatever is said of all of the middle is said of whatever falls under the middle.
The four productive moods (minor premise first, as in Arabic)
| Mood | Form | Example |
|---|---|---|
| 1. A + A → A | Every S is M; every M is P; ∴ every S is P | Every khamr is an intoxicant; every intoxicant is forbidden; ∴ every khamr is forbidden. |
| 2. A + E → E | Every S is M; no M is P; ∴ no S is P | Every masjid is a waqf; no waqf may be sold; ∴ no masjid may be sold. (S5 slide 10) |
| 3. I + A → I | Some S is M; every M is P; ∴ some S is P | Some beggars are poor; every poor person deserves charity; ∴ some beggars deserve charity. |
| 4. I + E → O | Some S is M; no M is P; ∴ not every S is P | Some beggars have what they need; no one who has what he needs deserves charity; ∴ not every beggar deserves charity. |
Figure 1 is the only figure that yields all four kinds of conclusion (A, E, I and O). That is why the logicians call it the perfect figure.
A singular (shakhṣiyya) minor premise works like a universal one: "Zayd is a man; every man is mortal; ∴ Zayd is mortal" is mood 1.
Checking any syllogism, step by step
- Find the conclusion; its subject is S, its predicate is P.
- Find the middle term M (in both premises, not in the conclusion). Make sure it means the same thing both times.
- Find which premise has S (minor) and which has P (major).
- Read off the figure from where M sits.
- Mark each premise A, E, I or O, and check the general rules and the figure's conditions.
ʿAqīda link: al-Māʾida 5:75. "The Messiah, son of Maryam, was only a messenger; messengers had passed before him. His mother was a woman of truth. They both used to eat food." The commentators read the last sentence as a proof in two Figure 1 steps:
They both ate food. Whoever eats food is in need (of food and of what it passes through). ∴ They were in need. (mood 1)
They were in need. Nothing in need is a god. ∴ They were not gods. (mood 2)
Notice how much the Qurʾān leaves unsaid; Lesson 16 calls this an enthymeme.
Quiz 13
- State the five general rules.
- What is wrong with: "The Qurʾān is in the muṣḥaf; the muṣḥaf is made of paper; ∴ the Qurʾān is made of paper"? Which rule does it break?
- Name the figure:
- a) "Every prophet is truthful; every truthful person is trusted; ∴ every prophet is trusted."
- b) "Every believer is sincere; no hypocrite is sincere; ∴ no believer is a hypocrite."
- c) "Every thief is a fraud; every thief has his hand cut; ∴ some frauds have their hands cut."
- Why must the minor premise of Figure 1 be affirmative, and the major universal?
- Give the mood and the conclusion:
- a) "Some of the Companions were poets; every poet knows the rules of metre."
- b) "Every created thing is in need; nothing in need is a god."
- c) "Some narrators are weak; no weak narrator's lone report is accepted as ṣaḥīḥ."
- Why is Figure 1 called "perfect"?
- Set out the al-Māʾida 5:75 argument as two syllogisms and name the mood of each.
- Review: convert "No idol creates" and "Every messenger is a prophet".
Show answers
- The middle term is repeated with the same meaning; no conclusion from two negatives; none from two particulars; the conclusion follows the weaker premise; none from a negative minor with a particular major.
- Rule 1: the middle term is not repeated. The predicate of the first premise is "in the muṣḥaf", but the subject of the second is "the muṣḥaf": the same trap as the mouse in the wall. To reach the conclusion you would need "whatever is in the muṣḥaf is made of paper", and that is exactly the point in dispute. (How the Qurʾān relates to its written copies is a famous question of ʿaqīda; the logical point here is only the missing middle term.)
- a) Figure 1 · b) Figure 2 (the middle, "sincere", is the predicate of both) · c) Figure 3 (the middle, "thief", is the subject of both)
- The minor premise must place the minor term under the middle, which only an affirmative can do; the major must say something of all of the middle, so that it is sure to reach the part where the minor term sits.
- a) Mood 3 (I + A → I): "Some of the Companions knew the rules of metre." · b) Mood 2 (A + E → E): "No created thing is a god." · c) Mood 4 (I + E → O): "Not every narrator's lone report is accepted as ṣaḥīḥ."
- Its conclusions are self-evident, and it is the only figure that yields all four kinds: A, E, I and O.
- "They both ate food; whoever eats food is in need; ∴ they were in need" (mood 1). "They were in need; nothing in need is a god; ∴ they were not gods" (mood 2).
- "No creator is an idol" (E → E); "Some prophets are messengers" (A → I).